Experimental Methods and Colliders

25. Accelerator physics of colliders

261

25. ACCELERATOR PHYSICS OF COLLIDERS Revised August 2005 by K. Desler and D. A. Edwards (DESY).

25.1. Luminosity The event rate R in a collider is proportional to the interaction cross section σint and the factor of proportionality is called the luminosity: R = L σint .

(25.1)

If two bunches containing n1 and n2 particles collide with frequency f , the luminosity is n1 n2 (25.2) L =f 4πσx σy where σx and σy characterize the Gaussian transverse beam profiles in the horizontal (bend) and vertical directions and to simplify the expression it is assumed that the bunches are identical in transverse profile, that the profiles are independent of position along the bunch, and the particle distributions are not altered during collision. Whatever the distribution at the source, by the time the beam reaches high energy, the normal form is a good approximation thanks to the central limit theorem of probability and the diminished importance of space charge effects. The beam size can be expressed in terms of two quantities, one termed the transverse emittance, , and the other, the amplitude function, β. The transverse emittance is a beam quality concept reflecting the process of bunch preparation, extending all the way back to the source for hadrons and, in the case of electrons, mostly dependent on synchrotron radiation. The amplitude function is a beam optics quantity and is determined by the accelerator magnet configuration. When expressed in terms of σ and β the transverse emittance becomes  = πσ 2 /β . (25.3) Of particular significance is the value of the amplitude function at the interaction point, β ∗ . Clearly one wants β ∗ to be as small as possible; how small depends on the capability of the hardware to make a near-focus at the interaction point. Eq. (25.2) can now be recast in terms of emittances and amplitude functions as n1 n 2 L =f  . (25.4) 4 x βx∗ y βy∗ Thus, to achieve high luminosity, all one has to do is make high population bunches of low emittance to collide at high frequency at locations where the beam optics provides as low values of the amplitude functions as possible.

25.2. Beam dynamics Today’s operating HEP colliders are all synchrotrons, and the organization of this section reflects that circumstance. A major concern of beam dynamics is stability: conservation of adequate beam properties over a sufficiently long time scale. Several time scales are involved, and the approximations used in writing the equations of motion reflect the time scale under consideration. For example, when, in Sec. 25.2.1 below, we write the equations for transverse stability no terms associated with phase stability or synchrotron radiation appear; the time scale associated with the last two processes is much longer than that demanded by the need for transverse stability. 25.2.1. Betatron oscillations : Present-day high-energy accelerators employ alternating gradient focussing provided by quadrupole magnetic fields [1], [2]. The equations of motion of a particle undergoing oscillations with respect to the design trajectory are

with

x + Kx (s) x = 0 , y  + Ky (s) y = 0 ,

(25.5)

x ≡ dx/ds , y  ≡ dy/ds

(25.6)

Kx ≡ B  /(Bρ) + ρ−2 , Ky ≡ −B  /(Bρ)

(25.7)



B ≡ ∂By /∂x .

(25.8)

The independent variable s is path length along the design trajectory. This motion is called a betatron oscillation because it was initially studied in the context of that type of accelerator. The functions Kx and Ky reflect the transverse focussing—primarily due to quadrupole fields except for the radius of curvature, ρ, term in Kx for a synchrotron—so each equation of motion resembles that for a harmonic oscillator but with spring constants that are a function of position. No terms relating to synchrotron oscillations appear, because their time scale is much longer and in this approximation play no role. These equations have the form of Hill’s equation and so the solution in one plane may be written as  x(s) = A β(s) cos(ψ(s) + δ), (25.9) where A and δ are constants of integration and the phase advances according to dψ/ds = 1/β. The dimension of A is the square root of length, reflecting the fact that the oscillation amplitude is modulated by the square root of the amplitude function. In addition to describing the envelope of the oscillation, β also plays the role of an ‘instantaneous’ − λ. The wavelength of a betatron oscillation may be some tens of meters, and so typically values of the amplitude function are of the order of meters rather than on the order of the beam size. The beam optics arrangement generally has some periodicity and the amplitude function is chosen to reflect that periodicity. As noted above, a small value of the amplitude function is desired at the interaction point, and so the focussing optics is tailored in its neighborhood to provide a suitable β ∗ . The number of betatron oscillations per turn in a synchrotron is called the tune and is given by ν=

1 2π



ds . β

(25.10)

Expressing the integration constant A in the solution above in terms of x, x yields the Courant-Snyder invariant A2 =γ(s) x(s)2 + 2α(s) x(s) x (s) + β(s) x (s)2 where α ≡ − β  /2, γ ≡

1 + α2 . β

(25.11)

(The Courant-Snyder parameters α, β and γ employ three Greek letters which have other meanings and the significance at hand must often be recognized from context.) Because β is a function of position in the focussing structure, this ellipse changes orientation and aspect ratio from location to location but the area πA2 remains the same. As noted above the transverse emittance is a measure of the area in x, x (or y, y  ) phase space occupied by an ensemble of particles. The definition used in Eq. (25.3) is the area that encloses 39% of a Gaussian beam. For electron synchrotrons the equilibrium emittance results from the balance between synchrotron radiation damping and excitation from quantum fluctuations in the radiation rate. The equilibrium is reached in a time small compared with the storage time. For present-day hadron synchrotrons, synchrotron radiation does not play a similar role in determining the transverse emittance. Rather the emittance during storage reflects the source properties and the abuse suffered by the particles throughout acceleration and storage. Nevertheless it is useful to argue as follows: Though x and x can serve as canonically conjugate variables at constant energy this definition of the emittance would not be an adiabatic invariant when the energy changes during the acceleration cycle. However, γ(v/c)x , where here γ is the Lorentz factor, is proportional to the transverse momentum and so qualifies as a variable conjugate to x. So often one sees a normalized emittance defined according to N = γ

v , c

(25.12)

which is an approximate adiabatic invariant, e.g. during acceleration.

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25. Accelerator physics of colliders

25.2.2. Phase stability : The particles in a circular collider also undergo synchrotron oscillations. This is usually referred to as motion in the longitudinal degree-of-freedom because particles arrive at a particular position along the accelerator earlier or later than an ideal reference particle. This circumstance results in a finite bunch length, which is related to an energy spread. For dynamical variables in longitudinal phase space, let us take ΔE and Δt, where these are the energy and time differences from that of the ideal particle. A positive Δt means a particle is behind the ideal particle. The equation of motion is the same as that for a physical pendulum and therefore is nonlinear. But for small oscillations, it reduces to a simple harmonic oscillator:

In the electron case the longitudinal emittance is determined by the synchrotron radiation process just as in the transverse degrees of freedom. For the hadron case the history of acceleration plays a role and because energy and time are conjugate coordinates, the longitudinal emittance is a quasi-invariant. For HEP bunch length is a significant quantity because if the bunch length becomes larger than β ∗ the luminosity is adversely affected. This is because β grows parabolically as one proceeds away from the IP and so the beam size increases thus lowering the contribution to the luminosity from such locations. Further discussion and modified expressions for the luminosity may be found in Furman and Zisman [3].

d2 Δt = −(2πνs )2 Δt dn2

25.2.3. Synchrotron radiation [4] : A relativistic particle undergoing centripetal acceleration radiates at a rate given by the Larmor formula multiplied by the 4th power of the Lorentz factor:

(25.13)

where the independent variable n is the turn number and νs is the number of synchrotron oscillations per turn, analogous to the betatron oscillation tune defined earlier. Implicit in this equation is the approximation that n is a continuos variable. This approximation is valid provided νs  1, which is usually well satisfied in practice. In the high-energy limit, where v/c ≈ 1,  νs =

 hη eV cos φs 1/2 . 2πE

(25.14)

There are four as yet undefined quantities in this expression: the harmonic number h, the slip factor η, the maximum energy eV gain per turn from the acceleration system, and the synchronous phase φs . The frequency of the RF system is normally a relatively high multiple, h, of the orbit frequency. The slip factor relates the fractional change in the orbit period τ to changes in energy according to Δτ ΔE =η . τ E

(25.15)

At sufficiently high energy, the slip factor just reflects the relationship between path length and energy, since the speed is a constant; η is positive for all the synchrotrons in the “Tables of Collider Parameters” (Sec. 26). The synchronous phase is a measure of how far up on the RF wave the average particle must ride in order to maintain constant energy to counteract of synchrotron radiation. That is, sin φs is the ratio of the energy loss per turn to the maximum energy per turn that can be provided by the acceleration system. For hadron colliders built to date, sin φs is effectively zero. This is not the case for electron storage rings; for example, the electron ring of HERA runs at a synchronous phase of 45◦ .  and Now if one has a synchrotron oscillation with amplitudes Δt  ΔE,  sin(2πνs n) , Δt = Δt

 cos(2πνs n) ΔE = ΔE

(25.16)

then the amplitudes are related according to  = 2πνs E Δt  . ΔE ητ

(25.17)

The longitudinal emittance  may be defined as the phase space  and ΔE.  In general, area bounded by particles with amplitudes Δt the longitudinal emittance for a given amplitude is found by numerical integration. For sin φs = 0, an analytical expression is as follows:   =

2π 3 EeV h τ 2η

1/2

 2 (Δt)

(25.18)

Again, a Gaussian is a reasonable representation of the longitudinal profile of a well-behaved beam bunch; if σΔt is the standard deviation of the time distribution, then the bunch length can be characterized by  = c σΔt . (25.19)

P =

1 e 2 a2 4 γ . 6π0 c3

(25.20)

Here, a = v 2 /ρ is the centripetal acceleration of a particle with speed v undergoing deflection with radius of curvature ρ. In a synchrotron that has a constant radius of curvature within bending magnets, the energy lost due to synchrotron radiation per turn is the above multiplied by the time spent in bending magnets, 2πρ/v. Expressed in familiar units, this result may be written W = 8.85 × 10−5 E 4 /ρ MeV per turn

(25.21)

for electrons at sufficiently high energy that v ≈ c. The energy E is in GeV and ρ is in kilometers. The radiation has a broad energy spectrum which falls off rapidly above the critical energy, Ec = (3c/2ρ)γ 3 . Typically, Ec is in the hard x-ray region. The characteristic time for synchrotron radiation processes is the time during which the energy must be replenished by the acceleration system. If f0 is the orbit frequency, then the characteristic time is given by E τ0 = . (25.22) f0 W Oscillations in each of the three degrees of freedom either damp or antidamp depending on the design of the accelerator. For a simple separated-function alternating gradient synchrotron, all three modes damp. The damping time constants are related by Robinson’s Theorem, which, expressed in terms of τ0 , is 1 1 1 1 + + =2 . τx τy τs τ0

(25.23)

Even though all three modes may damp, the emittances do not tend toward zero. Statistical fluctuations in the radiation rate excite synchrotron oscillations and radial betatron oscillations. Thus there is an equilibrium emittance at which the damping and excitation are in balance. The vertical emittance is non-zero due to horizontal-vertical coupling. Polarization can develop from an initially unpolarized beam as a result of synchrotron radiation. A small fraction ≈ Ec /E of the radiated power flips the electron spin. Because the lower energy state is that in which the particle magnetic moment points in the same direction as the magnetic bend field, the transition rate toward this alignment is larger than the rate toward the reverse orientation. An equilibrium polarization of 92% is predicted, and despite a variety of depolarizing processes, polarization above 80% has been observed at a number of facilities. The radiation rate for protons is of course down by a factor of the fourth power of the mass ratio, and is given by W = 7.8 × 10−3 E 4 /ρ keV per turn

(25.24)

where E is now in TeV and ρ in km. For the LHC, synchrotron radiation presents a significant load to the cryogenic system, and impacts magnet design due to gas desorption and secondary electron emission from the wall of the cold beam tube. The critical energy for the LHC is 44 eV.

25. Accelerator physics of colliders

25.2.4. Beam-beam tune shift [5] : In a bunch-bunch collision the particles of one bunch see the other bunch as a nonlinear lens. Therefore the focussing properties of the ring are changed in a way that depends on the transverse oscillation amplitude. Hence there is a spread in the frequency of betatron oscillations. There is an extensive literature on the subject of how large this tune spread can be. In practice, the limiting value is hard to predict. It is consistently larger for electrons because of the beneficial effects of damping from synchrotron radiation. In order that contributions to the total tune spread arise only at the detector locations, the beams in a multibunch collider are kept apart elsewhere in the collider by a variety of techniques. For equal energy particles of opposite charge circulating in the same vacuum chamber, electrostatic separators may be used assisted by a crossing angle if appropriate. For particles of equal energy and of the same charge, a crossing angle is needed not only for tune spread reasons but to steer the particles into two separate beam pipes. In HERA, because of the large ratio of proton to electron energy, separation can be achieved by bending magnets. 25.2.5. Luminosity lifetime : In electron synchrotrons the luminosity degrades during the store primarily due to particles leaving the phase stable region in longitudinal phase space as a result of quantum fluctuations in the radiation rate and bremsstrahlung. For hadron colliders the luminosity deteriorates due to emittance dilution resulting from a variety of processes. In practice, stores are intentionally terminated when the luminosity drops to the point where a refill will improve the integrated luminosity, while synchrotron radiation facilitates a “topping-up” process in electron-positron rings to provide continuous luminosity. 25.2.6. Luminosity: Achieved and Desired : Many years ago, the CERN Intersecting Storage Rings set an enviable luminosity record for proton-proton collisions at 1.3 × 1032 cm−2 s−1 . Not until the Summer of 2005 was this level matched in a proton-antiproton collider as the Tevatron approached an integrated luminosity of 1 fb−1 in its Run II. Thus far, electrons have proved more amenable; KEKB has reached 1.5 × 1034 cm−2 s−1 , with the implication of integrated luminosity in the 100 fb−1 yr−1 range. The luminosity specification for a yet-to-be-built system is a complicated process, involving considerations of organization, funding,

263

politics and so forth outside the scope of this paper. In the electron world, the 1/s dependence of cross sections such as “Higgs-strahlung” plays an important role. In the complex hadron environment, the higher luminosity potential of the LHC helped offset the otherwise totally adverse consequences of the SSC project cancellation. As a historical observation, it is interesting that although improvement directions of present and past facilities could not spelled out in advance, such opportunities have proved to be of critical value to the advance of HEP.

25.3. Prospects While this update is in preparation, it is interesting to recall that exactly 20 years ago, the Tevatron was entering its commissioning phase as a proton-antiproton collider to provide a c.m.s. energy at the 2 TeV level. Now, the next major step in discovery reach is approaching. The LHC is scheduled to begin operation in 2007, with its proton beams colliding at 14 TeV c.m.s., and at a luminosity two orders of magnitude above that reached in the Tevatron. LHC progress may be followed at the website provided by CERN [6]. A concerted international effort is underway aimed toward the construction of the electron-positron counterpart of the LHC – a linear collider to operate at the 0.5–1.0 TeV c.m.s. level. A recent decision of an International Technical Review Panel decided in favor of the superconducting RF approach for the main linacs [7]. A current goal is to develop the design report by the end of 2006. References: 1. E. D. Courant and H. S. Snyder, Ann. Phys. 3, 1 (1958). This is the classic article on the alternating gradient synchrotron. 2. A. W. Chao and M. Tigner (eds.), Handbook of Accelerator Physics and Engineering, World Science Publishing Co. (Singapore, 2nd printing, 2002.), Sec. 2.1. 3. M. A. Furman and M. S. Zisman, Handbook of Accelerator Physics and Engineering, op cit, Sec. 4.1. 4. H. Wiedemann, Handbook of Accelerator Physics and Engineering, op cit, Sec. 3.1. 5. K. Hirata, P. Chen, J. M. Jowett, Handbook of Accelerator Physics and Engineering, op cit, Secs. 2.6.1, 2.6.2, 2.6.3, respectively. 6. http://lhc-new-homepage.web.cern.ch/lhc-new-homepage/. 7. http://www.interactions.org/linearcollider/gde/.

264

26. High-energy collider parameters

HIGH-ENERGY COLLIDER PARAMETERS: e+ e− Colliders (I) Updated in early 2008 with numbers received from representatives of the colliders (contact J. Beringer, LBNL). For existing (future) colliders the latest achieved (design) values are given. Quantities are, where appropriate, r.m.s.; H and V indicate horizontal and vertical directions; s.c. stands for superconducting. Parameters for the defunct SPEAR, DORIS, PETRA, PEP, SLC, TRISTAN, and VEPP-2M colliders may be found in our 1996 edition (Phys. Rev. D54, 1 July 1996, Part I). VEPP-2000 (Novosibirsk)

VEPP-4M (Novosibirsk)

BEPC (China)

BEPC-II (China)

DAΦNE (Frascati)

Physics start date

2008

1994

1989

2008

1999

Physics end date

—

—

2005

—

2008

Maximum beam energy (GeV)

1.0

6

2.2

1.89 (2.3 max)

0.700

Luminosity (1030 cm−2 s−1 )

100

20

12.6 at 1.843 GeV/beam 5 at 1.55 GeV/beam

1000

150 (500 achievable)

Time between collisions (μs)

0.04

0.6

0.8

0.008

0.0027 (2.5 to 3.2)×104

Full crossing angle (μ rad)

0

0

0

2.2 × 104

Energy spread (units 10−3 )

0.64

1

0.58 at 2.2 GeV

0.52

0.40

4

5

≈5

1.3

low current: 1 high current: 3

125 (round)

H : 1000 V : 30

H : 890 V : 37

H : 380 V : 5.7

H : 800 V : 4.8

Free space at interaction point (m)

±1

±2

±2.15

±0.63

±0.40

Luminosity lifetime (hr)

continuous

2

7–12

1.5

0.7

Turn-around time (min)

continuous

18

32

26

0.8 (topping up)

Injection energy (GeV)

0.2–1.0

1.8

1.55

1.89

on energy

Transverse emittance (10−9 π rad-m)

H : 250 V : 250

H : 200 V : 20

H : 660 V : 28

H : 144 V : 2.2

H : 300 V: 1

H : 0.06 − 0.11 V : 0.06 − 0.10

H : 0.75 V : 0.05

H : 1.2 V : 0.05

H : 1.0 V : 0.015

H : 0.25 V : 0.009

H : 750 V : 750

500

350

400

250

RF frequency (MHz)

172

180

199.53

499.8

356

Particles per bunch (units 1010 )

16

15

20 at 2 GeV 11 at 1.55 GeV

4.8

e− : 3.3 e+ : 2.4

Bunches per ring per species

1

2

1

93

120 (incl. 10 bunch gap)

150

80

40 at 2 GeV 22 at 1.55 GeV

910

e− : 1800 e+ : 1300

0.024

0.366

0.2404

0.23753

0.098

2

1

2

1

2

Magnetic length of dipole (m)

1.2

2

1.6

Outer ring: 1.6 Inner ring: 1.41

1

Length of standard cell (m)

12

7.2

6.6

Outer ring: 6.6 Inner ring: 6.2

12

Phase advance per cell (deg)

H : 738 V : 378

65

≈ 60

60–90 no standard cell

360

Dipoles in ring

8

78

40 + 4 weak

84 + 8 weak

8

Quadrupoles in ring

20

150

68

134+2 s.c.

48

Peak magnetic field (T)

2.4

0.6

0.903 at 2.8 GeV

Outer ring: 0.677 Inner ring: 0.766

1.7

Bunch length (cm) Beam radius (10−6 m)

β ∗ , amplitude function at interaction point (m) Beam-beam tune shift per crossing (units 10−4 )

Average beam current per species (mA) Circumference or length (km) Interaction regions

265

26. High-energy collider parameters

HIGH-ENERGY COLLIDER PARAMETERS: e+ e− Colliders (II) Updated in early 2008 with numbers received from representatives of the colliders (contact J. Beringer, LBNL). For existing (future) colliders the latest achieved (design) values are given. Quantities are, where appropriate, r.m.s.; H and V indicate horizontal and vertical directions; s.c. stands for superconducting. CESR (Cornell)

CESR-C (Cornell)

KEKB (KEK)

PEP-II (SLAC)

LEP (CERN)

ILC (TBD)

Physics start date

1979

2002

1999

1999

1989

TBD

Physics end date

2002

2008

—

2008

2000

—

6

6

e− × e+ : 8 × 3.5

e− : 7–12 (9.0 nominal) e+ : 2.5–4 (3.1 nominal) (nominal Ecm = 10.5 GeV)

100 - 104.6

(upgradeable to 500)

Luminosity (1030 cm−2 s−1 )

1280 at 5.3 GeV/beam

76 at 2.08 GeV/beam

17120

12069 (design: 3000)

24 at Z 0 100 at > 90 GeV

2 × 104

Time between collisions (μs)

0.014 to 0.22

0.014 to 0.22

0.00590 or 0.00786

0.0042

22

0.3‡

0

0

14000

Maximum beam energy (GeV)

250

Full crossing angle (μ rad)

±2000

±3300

±11000†

Energy spread (units 10−3 )

0.6 at 5.3 GeV/beam

0.82 at 2.08 GeV/beam

0.7

e− /e+ : 0.61/0.77

0.7→1.5

1

Bunch length (cm)

1.8

1.2

0.65

e− /e+ : 1.1/1.0

1.0

0.03

Beam radius (μm)

H : 460 V: 4

H : 340 V : 6.5

H : 110 V : 1.9

H : 157 V : 4.7

H : 200 → 300 V : 2.5 → 8

H : 0.639 V : 0.0057

Free space at interaction point (m)

±2.2 (±0.6 to REC quads)

±2.2 (±0.3 to PM quads)

+0.75/−0.58 (+300/−500) mrad cone

±0.2, ±300 mrad cone

±3.5

±3.5

Luminosity lifetime (hr)

2–3

2–3

continuous

continuous

20 at Z 0 10 at > 90 GeV

n/a

Turn-around time (min)

5 (topping up)

1.5 (topping up)

continuous

continuous

50

n/a

Injection energy (GeV)

1.8–6

1.5–6

e− /e+ : 8/3.5

2.5–12

22

n/a

48 (H), 1.5 (V ) e+ : 24 (H), 1.5 (V )

H : 20–45 V : 0.25 → 1

H : 0.02 V : 8 × 10−5

Transverse emittance (π rad-nm)

H : 210 V: 1

H : 120 V : 3.5

e− :

24 (H), 0.61 (V ) e+ : 18 (H), 0.56 (V )

e− :

(at 250 GeV) β ∗ , amplitude function at interaction point (m) Beam-beam tune shift per crossing (units 10−4 ) RF frequency (MHz) Particles per bunch (units 1010 )

e− :

e− :

H : 1.0 V : 0.018

H : 0.94 V : 0.012

0.56 (H), 0.0059 (V ) e+ : 0.59 (H), 0.0065 (V )

0.50 (H), 0.012 (V ) e+ : 0.50 (H), 0.012 (V )

H : 1.5 V : 0.05

H : 0.02 V : 0.0004

H : 250 V : 620

e− : 420 (H), 280 (V ) e+ : 410 (H), 270 (V )

e− : 750 (H), 560 (V ) e+ : 1150 (H), 1010 (V )

e− : 703 (H), 498 (V ) e+ : 510 (H), 727 (V )

830

n/a

500

500

508.887 e− /e+ :

1300 2

4.7

9 trains of 5 bunches

8 trains of 3 bunches

1389

1732

4 trains of 1 or 2

2625

Average beam current per species (mA)

340

72

e− /e+ : 1330/1650

e− /e+ : 1960/3026

4 at Z 0 4→6 at > 90 GeV

9 (in pulse)

Beam polarization (%)

—

—

—

—

55 at 45 GeV 5 at 61 GeV

e− : > 80% e+ : > 60%

0.768

0.768

3.016

2.2

26.66

31

1

1

1

1

4

1

11.66/pair

n/a

79

n/a

102/90

n/a

Circumference or length (km) Interaction regions Magnetic length of dipole (m)

1.6–6.6

1.6–6.6

e− /e+

: 5.86/0.915

e− /e+

: 75.7/76.1

e− /e+ :

5.2/8.0

352.2 45 in collision 60 in single beam

1.15

Bunches per ring per species

6.1/7.5

476 e− /e+ :

5.4/0.45

Length of standard cell (m)

16

16

Phase advance per cell (deg)

45–90 (no standard cell)

45–90 (no standard cell)

450

86

84

e− /e+ : 116/112

e− /e+ : 192/192

3280+24 inj. + 64 weak

n/a

101 + 4 s.c.

101 + 4 s.c.

e− /e+ : 452/452

e− /e+ : 290/326

520+288 + 8 s.c.

n/a

0.3 / 0.8 at 8 GeV

0.3 / 0.8 at 8 GeV, 2.1 wigglers at

e− /e+ : 0.25/0.72

e− /e+ : 0.18/0.75

0.135

n/a

Dipoles in ring Quadrupoles in ring Peak magnetic field (T)

1.9 GeV † KEKB ‡ Time

is operating with crab crossing since February 2007.

between bunch trains: 200ms.

15.2 e− /e+ :

60/90

266

26. High-energy collider parameters

HIGH-ENERGY COLLIDER PARAMETERS: ep, pp, pp, and Heavy Ion Colliders Updated in early 2008 with numbers received from representatives of the colliders (contact J. Beringer, LBNL). For existing (future) colliders the latest achieved (design) values are given. Quantities are, where appropriate, r.m.s.; H and V indicate horizontal and vertical directions; s.c. stands for superconducting; pk and ave denote peak and average values. HERA (DESY)

TEVATRON∗ (Fermilab)

Physics start date

1992

1987

Physics end date

2007

—

Particles collided

ep

pp

Maximum beam energy (TeV)

e: 0.030 p: 0.92

Luminosity (1030 cm−2 s−1 ) Time between collisions (ns)

RHIC (Brookhaven) 2001

2000

2004

LHC (CERN) 2002

2008

2009

— Au Au

Cu Cu

d Au

pp

Pb Pb

0.980

0.1 60% pol

0.1 TeV/n

0.1 TeV/n

0.1 TeV/n

7.0

2.76 TeV/n

75

286

35 (pk) 20 (ave)

0.0030 (pk) 0.0012 (ave)

0.020 (pk) 0.0008 (ave)

0.23 (pk) 0.11 (ave)

1.0 × 104

1.0 × 10−3 (5.4 × 10−5 )†

96

396

107

107

321

107

24.95

99.8 (1347)†

0

0

≈ 300

≤ 100 (0)†

e: 0.91 p: 0.2

0.14

0.45

0.75

0.75

0.65

0.113

0.11

e: 0.83 p: 8.5

p: 50 p¯: 45

100

30

30

25

7.55

7.94

e: 280(H), 50(V ) p: 265(H), 50(V )

p: 28 p¯: 16

165 (β ∗ =1 m)

145 (β ∗ =1m)

145 (β ∗ =0.9m)

155 (β ∗ =2m)

16.6

15.9 (22.5)†

Free space at interaction point (m)

±2

±6.5

38

38

Initial luminosity decay time, −L/(dL/dt) (hr)

10

6 (average)

3.9

14.9

10.9 - 3.6‡ (22 - 7.5)†‡

Turn-around time (min)

e: 75, p: 135

150

155

150

145

145

0.011 TeV/n

0.012 TeV/n

0.450

0.1774 TeV/n

Full crossing angle (μ rad) Energy spread (units

10−3 )

Bunch length (cm) Beam radius (10−6 m)

pp (pol.)

—

0

16 1.5

1.8

1.8

60

e: 0.012 p: 0.040

0.15

0.023

0.011 TeV/n

e: 20(H), 3.5(V ) p: 5(H), 5(V )

p: 3 p¯: 1

28

26

23

28

0.5

0.5

β ∗ , ampl. function at interaction point (m)

e: 0.6(H), 0.26(V ) p: 2.45(H), 0.18(V )

0.28

> 1.0

> 0.8

> 0.9

> 0.85

0.55

0.5 (1.0)†

Beam-beam tune shift per crossing (units 10−4 )

e: 190(H), 450(V ) p: 12(H), 9(V )

p: 120 p¯: 120

56

15

30

d: 21 Au: 17

34

—

e: 499.7 p: 208.2/52.05

53

accel: 28 store: 28

accel: 28 store: 197

accel: 28 store: 197

accel: 28 store: 197

400.8

400.8

e: 3 p: 7

p: 26 p¯: 9

13.5

0.11

0.45

d: 10 Au: 0.1

11.5

0.007

Bunches per ring per species

e: 189 p: 180

36

111

103

37

95

2808

592 (62)†

Average beam current per species (mA)

e: 40 p: 90

p: 70 p¯: 24

187

112

60

d: 119 Au: 94

584

6.12 (0.641)†

Circumference (km)

6.336

6.28

Interaction regions

2 colliding beams 1 fixed target (e beam)

2 high L

6 total, 2 high L

Magnetic length of dipole (m)

e: 9.185 p: 8.82

6.12

9.45

14.3

Length of standard cell (m)

e: 23.5 p: 47

59.5

29.7

106.90

Phase advance per cell (deg)

e: 60 p: 90

67.8

Dipoles in ring

e: 396 p: 416

774

192 per ring + 12 common

1232 main dipoles

Quadrupoles in ring

e: 580 p: 280

216

246 per ring

482 2-in-1 24 1-in-1

e: C-shaped p: s.c., collared, cold iron

s.c. cos θ warm iron

s.c. cos θ cold iron

s.c. 2 in 1 cold iron

e: 0.274 p: 5

4.4

3.5

8.3

Injection energy (TeV) Transverse emittance (10−9 π rad-m)

RF frequency (MHz) Particles per bunch (units 1010 )

Magnet type Peak magnetic field (T)

3.834

26.659 2 high L +1

d: 84 Au: 93

84

1 dedicated +2

90

TEVATRON parameters: p source accum. rate: 25×1010 hr−1 ; max. no. of p stored: 3.1×1012 (Accumulator), 4.6×1012 (Recycler). in parentheses refer to settings for ”Early” PbPb running. ‡ For 1 - 3 experiments. ∗ Additional † Numbers

27. Passage of particles through matter

267

27. PASSAGE OF PARTICLES THROUGH MATTER 27.1. Notation

Revised April 2008 by H. Bichsel (University of Washington), D.E. Groom (LBNL), and S.R. Klein (LBNL).

Table 27.1: Summary of variables used in this section. The kinematic variables β and γ have their usual meanings.

27. PASSAGE OF PARTICLES THROUGH MATTER . . . . . . . . . . . . . . . . . . . . . 27.1. Notation

267

. . . . . . . . . . . . . . . . . . .

27.2. Electronic energy loss by heavy particles . . . . . . . . . . . . . . 27.2.1. Energy loss at low energies . . . 27.2.2. Density effect . . . . . . . . . 27.2.3. Energetic knock-on electrons (δ rays) . . . . . . . . . . . . . . . 27.2.4. Restricted energy loss rates for relativistic ionizing particles . . . . 27.2.5. Fluctuations in energy loss . . . 27.2.6. Energy loss in mixtures and compounds . . . . . . . . . . . . 27.2.7. Ionization yields . . . . . . . .

. . . . . . . . . . . . . . . . . .

268 269 269

. . . . . .

270

. . . . . . . . . . . .

270 270

. . . . . . . . . . . .

271 271

27.3. Multiple scattering through small angles . . . . . . . . . . . . . . . . . . . . .

271

27.4. Photon and electron interactions matter . . . . . . . . . . . . . 27.4.1. Radiation length . . . . . 27.4.2. Energy loss by electrons . . 27.4.3. Critical energy . . . . . . 27.4.4. Energy loss by photons . . 27.4.5. Bremsstrahlung and pair production at very high energies

in . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . .

275 276

27.6. Muon energy loss at high energy . . . . . . . .

277

27.7. Cherenkov and transition radiation . . . . . . .

278

Stopping power [MeV cm2/g]

Definition

Units or Value

α

Fine structure constant (e2 /4π0 c) M Incident particle mass E Incident part. energy γM c2 T Kinetic energy me c2 Electron mass × c2 re Classical electron radius e2 /4π0 me c2 NA Avogadro’s number ze Charge of incident particle Z Atomic number of absorber A Atomic mass of absorber K/A 4πNA re2 me c2 /A

1/137.035 999 11(46) MeV/c2 MeV MeV 0.510 998 918(44) MeV 2.817 940 325(28) fm 6.022 1415(10) × 1023 mol−1

g mol−1 0.307 075 MeV g−1 cm2 for A = 1 g mol−1 I Mean excitation energy eV (Nota bene! ) δ(βγ) Density effect correction to ionization energy loss(a) ωp Plasma energy 28.816 ρZ/A eV  ( 4πNe re3 me c2 /α) Nc Electron density (units of re )−3 wj Weight fraction of the jth element in a compound or mixture nj ∝ number of jth kind of atoms in a compound or mixture (716.408 g cm−2 )−1 for A = 1 g mol−1 — 4αre2 NA /A X0 Radiation length g cm−2 Ec Critical energy for electrons MeV GeV Eμc Critical energy  for muons Es Scale energy 4π/α me c2 21.2052 MeV RM Moli`ere radius g cm−2

272 272 273 274 274

. . . . . . . . . . .

27.5. Electromagnetic cascades

Symbol

267

(a)

For ρ in g cm−3 .

μ+ on Cu μ−

10

LindhardScharff

100

Bethe-Bloch

Radiative

AndersonZiegler

Eμc Radiative losses

Radiative effects Minimum ionization reach 1%

Nuclear losses

Without δ 1 0.001

0.01

0.1

1

10

0.1

1

10

100

1

[MeV/c]

βγ

100

1000

10 4

10

100

1

[GeV/c] Muon momentum

10 5

10 6

10

100

[TeV/c]

Fig. 27.1: Stopping power (= −dE/dx) for positive muons in copper as a function of βγ = p/M c over nine orders of magnitude in momentum (12 orders of magnitude in kinetic energy). Solid curves indicate the total stopping power. Data below the break at βγ ≈ 0.1 are taken from ICRU 49 [2], and data at higher energies are from Ref. 1. Vertical bands indicate boundaries between different approximations discussed in the text. The short dotted lines labeled “μ− ” illustrate the “Barkas effect,” the dependence of stopping power on projectile charge at very low energies [3].

268

27. Passage of particles through matter

27.2. Electronic energy loss by heavy particles

10

[1–22, 24–30, 82]

〈– dE/dx〉min (MeV g –1cm2)

2.5 H2 gas: 4.10 H2 liquid: 3.97

6 5

H2 liquid

4 He gas

3 2

1 0.1

Sn Pb

1.0

10 100 βγ = p/Mc

0.1

0.1

0.1

1.0

Solids Gases 1.0

H

He

1

2

Li Be B C NO Ne 5

10

Z

Fe 20

Sn 50

100

Figure 27.2: Stopping power at minimum ionization for the chemical elements. The straight line is fitted for Z > 6. A simple functional dependence on Z is not to be expected, since −dE/dx also depends on other variables.

Al

C

1000

10 000

1.0 10 100 Muon momentum (GeV/c)

1000

10 100 1000 Proton momentum (GeV/c)

1000

10 000

Figure 27.3: Mean energy loss rate in liquid (bubble chamber) hydrogen, gaseous helium, carbon, aluminum, iron, tin, and lead. Radiative effects, relevant for muons and pions, are not included. These become significant for muons in iron for βγ > ∼ 1000, and at lower momenta for muons in higher-Z absorbers. See Fig. 27.21. Eq. (27.1) may be integrated to find the total (or partial) “continuous slowing-down approximation” (CSDA) range R for a particle which loses energy only through ionization and atomic excitation. Since dE/dx depends only on β, R/M is a function of E/M or pc/M . In practice, range is a useful concept only for low-energy hadrons (R < ∼ λI , where λI is the nuclear interaction length), and for muons below a few hundred GeV (above which radiative effects dominate). R/M as a function of βγ = p/M c is shown for a variety of materials in Fig. 27.4. The mass scaling of dE/dx and range is valid for the electronic losses described by the Bethe-Bloch equation, but not for radiative losses, relevant only for muons and pions. For a particle with mass M and momentum M βγc, Tmax is given by Tmax =

1.5

Fe

1.0 10 100 Pion momentum (GeV/c)

2.35 – 0.28 ln(Z)

2.0

0.5

8

− dE/dx (MeV g−1cm2)

Moderately relativistic charged particles other than electrons lose energy in matter primarily by ionization and atomic excitation. The mean rate of energy loss (or stopping power) is given by the Bethe-Bloch equation,   dE Z 1 1 2me c2 β 2 γ 2 Tmax δ(βγ) 2 − = Kz 2 ln . − β − dx A β2 2 I2 2 (27.1) Here Tmax is the maximum kinetic energy which can be imparted to a free electron in a single collision, and the other variables are defined in Table 27.1. With K as defined in Table 27.1 and A in g mol−1 , the units are MeV g−1 cm2 . In this form, the Bethe-Bloch equation describes the energy loss of pions in a material such as copper to about 1% accuracy for energies between about 6 MeV and 6 GeV (momenta between about 40 MeV/c and 6 GeV/c). At lower energies various corrections discussed in Sec. 27.2.1 must be made. At higher energies, radiative effects begin to be important. These limits of validity depend on both the effective atomic number of the absorber and the mass of the slowing particle. The function as computed for muons on copper is shown by the solid curve in Fig. 27.1, and for pions on other materials in Fig. 27.3. A minor dependence on M at the highest energies is introduced through Tmax , but for all practical purposes in high-energy physics dE/dx in a given material is a function only of β. Except in hydrogen, particles of the same velocity have similar rates of energy loss in different materials; there is a slow decrease in the rate of energy loss with increasing Z. The qualitative difference in stopping power behavior at high energies between a gas (He) and the other materials shown in Fig. 27.3 is due to the density-effect correction, δ(βγ), discussed below. The stopping power functions are characterized by broad minima whose position drops from βγ = 3.5 to 3.0 as Z goes from 7 to 100. The values of minimum ionization as a function of atomic number are shown in Fig. 27.2. In practical cases, most relativistic particles (e.g., cosmic-ray muons) have mean energy loss rates close to the minimum, and are said to be minimum ionizing particles, or mip’s. As discussed below, the most probable energy loss in a detector is considerably below the mean given by the BetheBloch equation.

2me c2 β 2 γ 2 . 1 + 2γme /M + (me /M )2

(27.2)

In older references [4,5] the “low-energy” approximation Tmax = 2me c2 β 2 γ 2 , valid for 2γme /M  1, is often implicit. For a pion in copper, the error thus introduced into dE/dx is greater than 6% at 100 GeV. At energies of order 100 GeV, the maximum 4-momentum transfer to the electron can exceed 1 GeV/c, where hadronic structure effects significantly modify the cross sections. This problem has been investigated by J.D. Jackson [6], who concluded that for hadrons (but not for large nuclei) corrections to dE/dx are negligible below energies where radiative effects dominate. While the cross section for rare hard collisions is modified, the average stopping power, dominated by many softer collisions, is almost unchanged. “The determination of the mean excitation energy is the principal non-trivial task in the evaluation of the Bethe stopping-power formula” [7]. Recommended values have varied

27. Passage of particles through matter

22

50000 20000

C

20

Fe

10000

Pb

18

2000

Iadj/Z (eV)

5000

R/M (g cm−2 GeV−1)

269

H2 liquid He gas

1000 500 200

ICRU 37 (1984) (interpolated values are not marked with points)

16 14

Barkas & Berger 1964

12

100 50

Bichsel 1992

10

20

8

10 5 2 1 0.1

2

0.02

1.0

5

0.05

0.1

2

5

0.2

0.5

βγ = p/Mc

10.0

1.0

2

5

2.0

5.0

100.0

10.0

Muon momentum (GeV/c) 0.02

0.05

0.1

0.2

0.5

1.0

2.0

5.0

10.0

0

10

20

30

40

50 Z

60

70

80

90

100

Figure 27.5: Mean excitation energies (divided by Z) as adopted by the ICRU [8]. Those based on experimental measurements are shown by symbols with error flags; the interpolated values are simply joined. The grey point is for liquid H2 ; the black point at 19.2 eV is for H2 gas. The open circles show more recent determinations by Bichsel [10]. The dotted curve is from the approximate formula of Barkas [11] used in early editions of this Review.

Pion momentum (GeV/c) 0.1

0.2

0.5

1.0

2.0

5.0

10.0 20.0

50.0

Proton momentum (GeV/c)

Figure 27.4: Range of heavy charged particles in liquid (bubble chamber) hydrogen, helium gas, carbon, iron, and lead. For example: For a K + whose momentum is 700 MeV/c, βγ = 1.42. For lead we read R/M ≈ 396, and so the range is 195 g cm−2 . substantially with time. Estimates based on experimental stopping-power measurements for protons, deuterons, and alpha particles and on oscillator-strength distributions and dielectricresponse functions were given in ICRU 49 [2]. See also ICRU 37 [8]. These values, shown in Fig. 27.5, have since been widely used. Machine-readable versions can also be found [9]. These values are widely used. 27.2.1. Energy loss at low energies : Shell corrections C/Z must be included in the square brackets of of Eq. (27.1) [2,8,10,11] to correct for atomic binding having been neglected in calculating some of the contributions to Eq. (27.1). The Barkas form [11] was used in generating Fig. 27.1. For copper it contributes about 1% at βγ = 0.3 (kinetic energy 6 MeV for a pion), and the correction decreases very rapidly with energy. Eq. (27.1) is based on a first-order Born approximation. Higher-order corrections, again important only at lower energy, are normally included by adding the “Bloch correction” z 2 L2 (β) inside the square brackets (Eq.(2.5) in [2]) . An additional “Barkas correction” zL1 (β) makes the stopping power for a negative particle somewhat smaller than for a positive particle with the same mass and velocity. In a 1956 paper, Barkas et al. noted that negative pions had a longer range than positive pions [3]. The effect has been measured for a number of negative/positive particle pairs, most recently for antiprotons at the CERN LEAR facility [12]. A detailed discussion of low-energy corrections to the Bethe formula is given in ICRU Report 49 [2]. When the corrections are properly included, the accuracy of the Bethe-Bloch treatment is accurate to about 1% down to β ≈ 0.05, or about 1 MeV for protons.

For 0.01 < β < 0.05, there is no satisfactory theory. For protons, one usually relies on the phenomenological fitting formulae developed by Andersen and Ziegler [2,13]. For particles moving more slowly than ≈ 0.01c (more or less the velocity of the outer atomic electrons), Lindhard has been quite successful in describing electronic stopping power, which is proportional to β [14]. Finally, we note that at low energies, e.g., for protons of less than several hundred eV, non-ionizing nuclear recoil energy loss dominates the total energy loss [2,14,15]. As shown in ICRU 49 [2] (using data taken from Ref. 13), the nuclear plus electronic proton stopping power in copper is 113 MeV cm2 g−1 at T = 10 keV, rises to a maximum of 210 MeV cm2 g−1 at 100–150 keV, then falls to 120 MeV cm2 g−1 at 1 MeV. Above 0.5–1.0 MeV the corrected Bethe-Bloch theory is adequate. 27.2.2. Density effect : As the particle energy increases, its electric field flattens and extends, so that the distant-collision contribution to Eq. (27.1) increases as ln βγ. However, real media become polarized, limiting the field extension and effectively truncating this part of the logarithmic rise [4–5,16–18]. At very high energies, δ/2 → ln(ωp /I) + ln βγ − 1/2 , (27.3)

where δ(βγ)/2 is the density effect correction introduced in Eq. (27.1) and ωp is the plasma energy defined in Table 27.1. A comparison with Eq. (27.1) shows that |dE/dx| then grows as ln βγ rather than ln β 2 γ 2 , and that the mean excitation energy I is replaced by the plasma energy ωp . The ionization stopping power as calculated with and without the density effect correction is shown in Fig. 27.1. Since the plasma frequency scales as the square root of the electron density, the correction is much larger for a liquid or solid than for a gas, as is illustrated by the examples in Fig. 27.3.

27. Passage of particles through matter

The density effect correction is usually computed using Sternheimer’s parameterization [16]: ⎧ ⎪ 2(ln 10)x − C if x ≥ x1 ; ⎪ ⎨ 2(ln 10)x − C + a(x1 − x)k if x0 ≤ x < x1 ; δ(βγ) = ⎪ 0 if x < x0 (nonconductors); ⎪ ⎩ if x < x0 (conductors) δ0 102(x−x0 ) (27.4) Here x = log10 η = log10 (p/M c). C (the negative of the C used in Ref. 16) is obtained by equating the high-energy case of Eq. (27.4) with the limit given in Eq. (27.3). The other parameters are adjusted to give a best fit to the results of detailed calculations for momenta below M c exp(x1 ). Parameters for elements and nearly 200 compounds and mixtures of interest are published in a variety of places, notably in Ref. 18. A recipe for finding the coefficients for nontabulated materials is given by Sternheimer and Peierls [20], and is summarized in Ref. 1. The remaining relativistic rise comes from the β 2 γ 2 growth of Tmax , which in turn is due to (rare) large energy transfers to a few electrons. When these events are excluded, the energy deposit in an absorbing layer approaches a constant value, the Fermi plateau (see Sec. 27.2.4 below). At extreme energies (e.g., > 332 GeV for muons in iron, and at a considerably higher energy for protons in iron), radiative effects are more important than ionization losses. These are especially relevant for high-energy muons, as discussed in Sec. 27.6. The distribution of secondary electrons with kinetic energies T I is [4] d2 N 1 Z 1 F (T ) = Kz 2 (27.5) dT dx 2 A β2 T 2 27.2.3.

Energetic knock-on electrons (δ rays) :

for I  T ≤ Tmax , where Tmax is given by Eq. (27.2). Here β is the velocity of the primary particle. The factor F is spindependent, but is about unity for T  Tmax . For spin-0 particles F (T ) = (1 − β 2 T /Tmax ); forms for spins 1/2 and 1 are also given by Rossi [4]. For incident electrons, the indistinguishability of projectile and target means that the range of T extends only to half the kinetic energy of the incident particle. Additional formulae are given in Ref. 19. Equation (27.5) is inaccurate for T close to I. δ rays of even modest energy are rare. For β ≈ 1 particle, for example, on average only one collision with Te > 1 keV will occur along a path length of 90 cm of Ar gas [25]. A δ ray with kinetic energy Te and corresponding momentum pe is produced at an angle θ given by cos θ = (Te /pe )(pmax /Tmax ) , (27.6) where pmax is the momentum of an electron with the maximum possible energy transfer Tmax . 27.2.4. Restricted energy loss rates for relativistic ionizing particles : Further insight can be obtained by examining the

mean energy deposit by an ionizing particle when energy transfers are restricted to T ≤ Tcut ≤ Tmax . The restricted energy loss rate is  dE 2me c2 β 2 γ 2 Tcut 2Z 1 1 − = Kz ln 2 dx T
 δ Tcut β2 − 1+ . (27.7) − 2 Tmax 2 This form approaches the normal Bethe-Bloch function (Eq. (27.1)) as Tcut → Tmax . It can be verified that the difference between Eq. (27.1) and Eq. (27.7) is equal to  Tmax 2 2 Tcut T (d N/dT dx)dT , where d N/dT dx is given by Eq. (27.5). Since Tcut replaces Tmax in the argument of the logarithmic term of Eq. (27.1), the βγ term producing the relativistic rise in

the close-collision part of dE/dx is replaced by a constant, and |dE/dx|T
where ξ = (K/2)Z/A(x/β 2 ) MeV for a detector with a thickness x in g cm−2 , and j = 0.200 [24]. † While dE/dx is independent of thickness, Δp /x scales as a ln x + b. The density correction δ(βγ) was not included in Landau’s or Vavilov’s work, but it was later included by Bichsel [24]. The high-energy behavior of δ(βγ) (Eq. (27.3)), is such that   2mc2 ξ + j . (27.9) Δp −→ ξ ln (ωp )2 βγ > ∼100 Thus the Landau-Vavilov most probable energy loss, like the restricted energy loss, reaches a Fermi plateau. The Bethe-Bloch dE/dx and Landau-Vavilov-Bichsel Δp /x in silicon are shown as a function of muon energy in Fig. 27.6. The case x/ρ = 1600 μm was chosen since it has about the same stopping power as does 3 mm of plastic scintillator. Folding in experimental resolution displaces the peak of the distribution, usually toward a higher value. 3.0

MeV g−1 cm2 (Electonic loses only)

270

Silicon 2.5 Bethe-Bloch 2.0

Restricted energy loss for: Tcut = 10 dE/dx|min Tcut = 2 dE/dx|min

1.5

Landau/Vavilov/Bichsel Δp /x for: x/ρ = 1600 μm 320 μm 80 μm

1.0

0.5 0.1

1.0

10.0

100.0

1000.0

Muon kinetic energy (GeV)

Figure 27.6: Bethe-Bloch dE/dx, two examples of restricted energy loss, and the Landau most probable energy per unit thickness in silicon. The change of Δp /x with thickness x illustrates its a ln x + b dependence. Minimum ionization (dE/dx|min ) is 1.664 MeV g−1 cm2 . Radiative losses are excluded. The incident particles are muons. The mean of the energy-loss given by the Bethe-Bloch equation, Eq. (27.1), is ill-defined experimentally and is not useful for describing energy loss by single particles. (It finds its application in dosimetry, where only bulk deposit is of relevance.) It rises as ln βγ because Tmax increases as β 2 γ 2 . The * G< ∼ 0.05–0.1, where G is given by Rossi [Ref. 4, Eq. 2.7.10]. It is Vavilov’s κ [23]. † Rossi [4], Talman [26], and others give somewhat different values for j. The most probable loss is not sensitive to its value.

27. Passage of particles through matter

large single-collision energy transfers that increasingly extend the long tail are rare, making the mean of an experimental distribution consisting of a few hundred events subject to large fluctuations and sensitive to cuts as well as to background. The most probable energy loss should be used. For very thick absorbers the distribution is less skewed but never approaches a Gaussian. In the case of Si illustrated in Fig. 27.6, the most probable energy loss per unit thickness for x ≈ 35 g cm−2 is very close to the restricted energy loss with Tcut = 2 dE/dx|min . The Landau distribution fails to describe energy loss in thin absorbers such as gas TPC cells [25] and Si detectors [24], as shown clearly in Fig. 1 of Ref. 25 for an argon-filled TPC cell. Also see Talman [26]. While Δp /x may be calculated adequately with Eq. (27.8), the distributions are significantly wider than the Landau width w = 4ξ [Ref. 24, Fig. 15]. Examples for thin silicon detectors are shown in Fig. 27.7. Δ/x (MeV g−1 cm2) 0.50

1.00

1.50

1.0

2.00

640 μm (149 mg/cm2) 320 μm (74.7 mg/cm2) 160 μm (37.4 mg/cm2) 80 μm (18.7 mg/cm2)

0.6

w 0.4

Mean energy loss rate

Δp/x 0.2

0.0 100

200

300

400

500

Δ/x (eV/μm)

27.3. Multiple scattering through small angles

1.00 0.95

(Δp/x) / dE/dx min

0.90 0.85 2

/cm ) x = 640 μm (149 mg 2) 320 μm (74.7 mg/cm2 ) 160 μm (37.4 mg/cm 2) /cm mg .7 (18 80 μm

0.75 0.70 0.65 0.60 0.55 0.50 0.3

1

3

10

30

βγ (= p/m)

where dE/dx|j is the mean rate of energy loss (in MeV g cm−2 ) in the jth element. Eq. (27.1) can be inserted into Eq. (27.10) to  find expressions  for Z/A,  I , and δ; for example, Z/A = wj Zj /Aj = nj Zj / nj Aj . However, I  as defined this way is an underestimate, because in a compound electrons are more tightly bound than in the free elements, and δ as calculated this way has little relevance, because it is the electron density that matters. If possible, one uses the tables given in Refs. 18 and 27, which include effective excitation energies and interpolation coefficients for calculating the density effect correction for the chemical elements and nearly 200 mixtures and compounds. If a compound or mixture is not found, then one uses the recipe for δ given in Ref. 20 (repeated in Ref. 1), and calculates I according to the discussion in Ref. 7. (Note the “13%” rule!)

600

Figure 27.7: Straggling functions in silicon for 500 MeV pions, normalized to unity at the most probable value δp /x. The width w is the full width at half maximum.

0.80

27.2.6. Energy loss in mixtures and compounds : A mixture or compound can be thought of as made up of thin layers of pure elements in the right proportion (Bragg additivity). In this case,  dE dE , (27.10) = wj dx dx j

27.2.7. Ionization yields : Physicists frequently relate total energy loss to the number of ion pairs produced near the particle’s track. This relation becomes complicated for relativistic particles due to the wandering of energetic knock-on electrons whose ranges exceed the dimensions of the fiducial volume. For a qualitative appraisal of the nonlocality of energy deposition in various media by such modestly energetic knock-on electrons, see Ref. 28. The mean local energy dissipation per local ion pair produced, W , while essentially constant for relativistic particles, increases at slow particle speeds [29]. For gases, W can be surprisingly sensitive to trace amounts of various contaminants [29]. Furthermore, ionization yields in practical cases may be greatly influenced by such factors as subsequent recombination [30].

500 MeV pion in silicon

0.8

f (Δ/x)

2.50

271

100

300

1000

Figure 27.8: Most probable energy loss in silicon, scaled to the mean loss of a minimum ionizing particle, 388 eV/μm (1.66 MeV g−1 cm2 ).

A charged particle traversing a medium is deflected by many small-angle scatters. Most of this deflection is due to Coulomb scattering from nuclei, and hence the effect is called multiple Coulomb scattering. (However, for hadronic projectiles, the strong interactions also contribute to multiple scattering.) The Coulomb scattering distribution is well represented by the theory of Moli`ere [31]. It is roughly Gaussian for small deflection angles, but at larger angles (greater than a few θ0 , defined below) it behaves like Rutherford scattering, with larger tails than does a Gaussian distribution. If we define 1 rms θ0 = θ rms (27.11) plane = √ θspace . 2 then it is sufficient for many applications to use a Gaussian approximation for the central 98% of the projected angular distribution, with a width given by [32,33]   13.6 MeV  θ0 = (27.12) z x/X0 1 + 0.038 ln(x/X0 ) . βcp Here p, βc, and z are the momentum, velocity, and charge number of the incident particle, and x/X0 is the thickness of the scattering medium in radiation lengths (defined below). This value of θ0 is from a fit to Moli`ere distribution [31] for singly charged particles with β = 1 for all Z, and is accurate to 11% or better for 10−3 < x/X0 < 100. Eq. (27.12) describes scattering from a single material, while the usual problem involves the multiple scattering of a particle traversing many different layers and mixtures. Since it is from a fit to a Moli`ere distribution, it is incorrect to add the individual θ0 contributions in quadrature; the result is systematically too

27. Passage of particles through matter

small. It is much more accurate to apply Eq. (27.12) once, after finding x and X0 for the combined scatterer. Lynch and Dahl have extended this phenomenological approach, fitting Gaussian distributions to a variable fraction of the Moli`ere distribution for arbitrary scatterers [33], and achieve accuracies of 2% or better.

x x /2

Ψplane

splane

yplane θplane

Figure 27.9: Quantities used to describe multiple Coulomb scattering. The particle is incident in the plane of the figure. The nonprojected (space) and projected (plane) angular distributions are given approximately by [31] ⎫ ⎧ 2 θspace ⎪ ⎪ 1 ⎪ ⎪ ⎪ (27.13) exp ⎪ ⎭ dΩ , ⎩− 2π θ02 2θ02 ⎧ 2 ⎫ ⎪ θplane ⎪ 1 ⎪ ⎪ ⎪ √ exp ⎪ − (27.14) ⎩ ⎭ dθplane , 2θ02 2π θ0

27.4. Photon and electron interactions in matter 27.4.1. Radiation length : High-energy electrons predominantly lose energy in matter by bremsstrahlung, and high-energy photons by e+ e− pair production. The characteristic amount of matter traversed for these related interactions is called the radiation length X0 , usually measured in g cm−2 . It is both (a) the mean distance over which a high-energy electron loses all but 1/e of its energy by bremsstrahlung, and (b) 79 of the mean free path for pair production by a high-energy photon [35]. It is also the appropriate scale length for describing high-energy electromagnetic cascades. X0 has been calculated and tabulated by Y.S. Tsai [36]:  1 NA  2 = 4αre2 Z [Lrad − f (Z)] + Z Lrad . (27.20) X0 A

For A = 1 g mol−1 , 4αre2 NA /A = (716.408 g cm−2 )−1 . Lrad and Lrad are given in Table 27.2. The function f (Z) is an infinite sum, but for elements up to uranium can be represented to 4-place accuracy by f (Z) = a2 [(1 + a2 )−1 + 0.20206 −0.0369 a2 + 0.0083 a4 − 0.002 a6 ] , where a = αZ [37]. Table 27.2: Tsai’s Lrad and Lrad , for use in calculating the radiation length in an element using Eq. (27.20).

where θ is the deflection angle. In this approximation, 2 2 2 θspace ≈ (θplane,x +θplane,y ), where the x and y axes are orthogonal to the direction of motion, and dΩ ≈ dθplane,x dθplane,y . Deflections into θplane,x and θplane,y are independent and identically distributed. Figure 27.9 shows these and other quantities sometimes used to describe multiple Coulomb scattering. They are 1 rms 1 ψ rms = √ θ0 , (27.15) plane = √ θ plane 3 3 1 1 rms y rms (27.16) plane = √ x θ plane = √ x θ0 , 3 3 1 1 rms s plane = √ x θ rms (27.17) plane = √ x θ0 . 4 3 4 3

Note that the second term for y plane equals x θplane /2 and represents the displacement that would have occurred had the deflection θplane all occurred at the single point x/2. For heavy ions the multiple Coulomb scattering has been measured and compared with various theoretical distributions [34].

Element

Z

H He Li Be Others

1 2 3 4 >4

1.0

Lrad

Lrad

5.31 6.144 4.79 5.621 4.74 5.805 4.71 5.924 ln(184.15 Z −1/3 ) ln(1194 Z −2/3 )

Positrons

− 1 dE (X0−1) E dx

All the quantitative estimates in this section apply only in the limit of small θ rms plane and in the absence of large-angle scatters. The random variables s, ψ, y, and θ in a given plane are distributed in a correlated fashion (see Sec. 31.1 of this Review for the definition of the correlation coefficient). Obviously, y ≈ xψ. √ In addition, y and θ have the correlation coefficient ρyθ = 3/2 ≈ 0.87. For Monte Carlo generation of a joint (y plane , θplane ) distribution, or for other calculations, it may be most convenient to work with independent Gaussian random variables (z1 , z2 ) with mean zero and variance one, and then set √ √ yplane =z1 x θ0 (1 − ρ2yθ )1/2 / 3 + z2 ρyθ x θ0 / 3 √ =z1 x θ0 / 12 + z2 x θ0 /2 ; (27.18) θplane =z2 θ0 . (27.19)

(27.21)

0.20

Lead (Z = 82)

Electrons

0.15 Bremsstrahlung

0.10

Ionization

0.5

Møller (e −) Bhabha (e +)

0 1

0.05

Positron annihilation

10

100

1000

E (MeV)

Figure 27.10: Fractional energy loss per radiation length in lead as a function of electron or positron energy. Electron (positron) scattering is considered as ionization when the energy loss per collision is below 0.255 MeV, and as Møller (Bhabha) scattering when it is above. Adapted from Fig. 3.2 from Messel and Crawford, Electron-Photon Shower Distribution Function Tables for Lead, Copper, and Air Absorbers, Pergamon Press, 1970. Messel and Crawford use X0 (Pb) = 5.82 g/cm2 , but we have modified the figures to reflect the value given in the Table of Atomic and Nuclear Properties of Materials (X0 (Pb) = 6.37 g/cm2 ).

(cm2 g−1)

272

27. Passage of particles through matter

Results using this formula agree with Tsai’s values to better than 2.5% for all elements except helium, where the result is about 5% low.

400 200 100

Ec (MeV)

Although it is easy to use Eq. (27.20) to calculate X0 , the functional dependence on Z is somewhat hidden. Dahl provides a compact fit to the data [38]: 716.4 g cm−2 A √ X0 = . (27.22) Z(Z + 1) ln(287/ Z)

50

Solids Gases

20 (X0 NA/A) ydσLPM/dy

Bremsstrahlung

10

100 GeV 1 TeV

5

0.8

10 TeV

1 PeV 10 PeV 0

0

0.25

0.5

H

He

1

2

Li Be B C NO Ne

5

10 Z

Fe

20

Sn

50

100

Figure 27.13: Electron critical energy for the chemical elements, using Rossi’s definition [4]. The fits shown are for solids and liquids (solid line) and gases (dashed line). The rms deviation is 2.2% for the solids and 4.0% for the gases. (Computed with code supplied by A. Fass´o.)

100 TeV

0.4

710 MeV ________ Z + 0.92

610 MeV ________ Z + 1.24

10 GeV 1.2

273

0.75

1

y = k/E

Figure 27.11: The normalized bremsstrahlung cross section k dσLP M /dk in lead versus the fractional photon energy y = k/E. The vertical axis has units of photons per radiation length.

Complete discussions and tables can be found in Refs. 7, 8, and 27. At very high energies and except at the high-energy tip of the bremsstrahlung spectrum, the cross section can be approximated in the “complete screening case” as [36] dσ/dk = (1/k)4αre2 {( 43 − 43 y + y 2 )[Z 2 (Lrad − f (Z)) + Z Lrad ] + 19 (1 − y)(Z 2 + Z)} , (27.24) where y = k/E is the fraction of the electron’s energy transfered to the radiated photon. At small y (the “infrared limit”) the term on the second line ranges from 1.7% (low Z) to 2.5% (high Z) of the total. If it is ignored and the first line simplified with the definition of X0 given in Eq. (27.20), we have 4 4  A dσ − y + y2 . = (27.25) dk X0 NA k 3 3

200

70 Rossi: Ionization per X0 = electron energy

50 40 30

l ta To

Br em Ex s≈ ac tb E re m ss tr ah

dE/dx × X0 (MeV)

100

lu ng

Copper X0 = 12.86 g cm−2 Ec = 19.63 MeV

Ionization

20

Brems = ionization 10

2

5

10 20 50 Electron energy (MeV)

100

200

Figure 27.12: Two definitions of the critical energy Ec . The radiation length in a mixture or compound may be approximated by  1/X0 = wj /Xj , (27.23) where wj and Xj are the fraction by weight and the radiation length for the jth element. 27.4.2. Energy loss by electrons : At low energies electrons and positrons primarily lose energy by ionization, although other processes (Møller scattering, Bhabha scattering, e+ annihilation) contribute, as shown in Fig. 27.10. While ionization loss rates rise logarithmically with energy, bremsstrahlung losses rise nearly linearly (fractional loss is nearly independent of energy), and dominates above a few tens of MeV in most materials Ionization loss by electrons and positrons differs from loss by heavy particles because of the kinematics, spin, and the identity of the incident electron with the electrons which it ionizes.

This cross section (times k) is shown by the top curve in Fig. 27.11. This formula is accurate except in near y = 1, where screening may become incomplete, and near y = 0, where the infrared divergence is removed by the interference of bremsstrahlung amplitudes from nearby scattering centers (the LPM effect) [39,40] and dielectric suppression [41,42]. These and other suppression effects in bulk media are discussed in Sec. 27.4.5. With decreasing energy (E < ∼ 10 GeV) the high-y cross section drops and the curves become rounded as y → 1. Curves of this familar shape can be seen in Rossi [4] (Figs. 2.11.2,3); see also the review by Koch & Motz [43]. Except at these extremes, and still in the complete-screening approximation, the number of photons with energies between kmin and kmax emitted by an electron travelling a distance d  X0 is   

2 2 d 4 − kmin 4(kmax − kmin ) kmax kmax Nγ = − . ln + X0 3 kmin 3E 2E 2 (27.26) We obtain 610 MeV 710 MeV Ec = (solids and liquids) , = (gases) . Z + 1.24 Z + 0.92

274

27. Passage of particles through matter

(a) Carbon ( Z = 6) - experimental σtot

1 Mb

Cross section (barns/atom)

27.4.3. Critical energy : An electron loses energy by bremsstrahlung at a rate nearly proportional to its energy, while the ionization loss rate varies only logarithmically with the electron energy. The critical energy Ec is sometimes defined as the energy at which the two loss rates are equal [44]. Berger and Seltzer [44] also give the approximation Ec = (800 MeV)/(Z + 1.2). This formula has been widely quoted, and has been given in older editions of this Review [45]. Among alternate definitions is that of Rossi [4], who defines the critical energy as the energy at which the ionization loss per radiation length is equal to the electron energy. Equivalently, it is the same as the first definition with the approximation |dE/dx|brems ≈ E/X0 . This form has been found to describe transverse electromagnetic shower development more accurately (see below). These definitions are illustrated in the case of copper in Fig. 27.12. The accuracy of approximate forms for Ec has been limited by the failure to distinguish between gases and solid or liquids, where there is a substantial difference in ionization at the relevant energy because of the density effect. We distinguish these two cases in Fig. 27.13. Fits were also made with functions of the form a/(Z + b)α , but α was found to be essentially unity. Since Ec also depends on A, I, and other factors, such forms are at best approximate.

σp.e. 1 kb

σRayleigh

1b

κ nuc σCompton

κe

10 mb

(b) Lead ( Z = 82) - experimental σtot

σp.e.

1.00

σRayleigh

(X0 NA/A) dσLPM/dx

Cross section (barns/atom)

1 Mb

1 kb

σg.d.r. 1b

κ nuc κe

σCompton

0.75

1 TeV 10 TeV

0.50 100 TeV

1 EeV

0.25 1 PeV 10 PeV

100 PeV 0

10 mb 10 eV

Pair production

0

0.25

0.5

0.75

1

x = E/k 1 keV

1 MeV

1 GeV

100 GeV

Photon Energy

Figure 27.14: Photon total cross sections as a function of energy in carbon and lead, showing the contributions of different processes: σp.e. = Atomic photoelectric effect (electron ejection, photon absorption) σRayleigh = Rayleigh (coherent) scattering–atom neither ionized nor excited σCompton = Incoherent scattering (Compton scattering off an electron) κnuc = Pair production, nuclear field κe = Pair production, electron field σg.d.r. = Photonuclear interactions, most notably the Giant Dipole Resonance [46]. In these interactions, the target nucleus is broken up. Data from [47]; parameters for σg.d.r. from [48]. Curves for these and other elements, compounds, and mixtures may be obtained from http://physics.nist.gov/PhysRefData. The photon total cross section is approximately flat for at least two decades beyond the energy range shown. Original figures courtesy J.H. Hubbell (NIST).

Figure 27.15: The normalized pair production cross section dσLP M /dy, versus fractional electron energy x = E/k. 27.4.4. Energy loss by photons : Contributions to the photon cross section in a light element (carbon) and a heavy element (lead) are shown in Fig. 27.14. At low energies it is seen that the photoelectric effect dominates, although Compton scattering, Rayleigh scattering, and photonuclear absorption also contribute. The photoelectric cross section is characterized by discontinuities (absorption edges) as thresholds for photoionization of various atomic levels are reached. Photon attenuation lengths for a variety of elements are shown in Fig. 27.16, and data for 30 eV< k <100 GeV for all elements is available from the web pages given in the caption. Here k is the photon energy. The increasing domination of pair production as the energy increases is shown in Fig. 27.17. Using approximations similar to those used to obtain Eq. (27.25), Tsai’s formula for the differential cross section [36] reduces to  dσ A  = 1 − 43 x(1 − x) (27.27) dx X0 NA in the complete-screening limit valid at high energies. Here x = E/k is the fractional energy transfer to the pair-produced electron (or positron), and k is the incident photon energy. The cross section is very closely related to that for bremsstrahlung, since the Feynman diagrams are variants of one another. The

27. Passage of particles through matter

275

100

Absorption length λ (g/cm 2 )

10

Sn

1

Si

Fe

Pb

0.1

C

H 0.01 0.001 10 10 10

–4

–5 –6

10 eV

100 eV

1 keV

10 keV

100 keV 1 MeV 10 MeV 100 MeV 1 GeV 10 GeV 100 GeV Photon energy Fig. 27.16: The photon mass attenuation length (or mean free path) λ = 1/(μ/ρ) for various elemental absorbers as a function of photon energy. The mass attenuation coefficient is μ/ρ, where ρ is the density. The intensity I remaining after traversal of thickness t (in mass/unit area)  is given by I = I0 exp(−t/λ). The accuracy is a few percent. For a chemical compound or mixture, 1/λeff ≈ elements wZ /λZ , where wZ is the proportion by weight of the element with atomic number Z. The processes responsible for attenuation are given in Fig. 27.10. Since coherent processes are included, not all these processes result in energy deposition. The data for 30 eV < E < 1 keV are obtained from http://www-cxro.lbl.gov/optical constants (courtesy of Eric M. Gullikson, LBNL). The data for 1 keV < E < 100 GeV are from http://physics.nist.gov/PhysRefData, through the courtesy of John H. Hubbell (NIST). cross section is of necessity symmetric between x and 1 − x, as can be seen by the solid curve in Fig. 27.15. See the review by Motz, Olsen, & Koch for a more detailed treatment [49]. Eq. (27.27) may be integrated to find the high-energy limit for the total e+ e− pair-production cross section: σ = 79 (A/X0 NA ) . (27.28)

1.0 0.9 NaI

0.8 0.7

C

Ar Pb

0.6

H2O

P 0.5

Equation Eq. (27.28) is accurate to within a few percent down to energies as low as 1 GeV, particularly for high-Z materials.

H

Fe

27.4.5. Bremsstrahlung and pair production at very high energies : At ultrahigh energies, Eqns. 27.24–27.28 will fail

0.4 0.3 0.2 0.1 0.0

1

2

5

10

20

50

100 200

500 1000

Photon energy (MeV)

Figure 27.17: Probability P that a photon interaction will result in conversion to an e+ e− pair. Except for a few-percent contribution from photonuclear absorption around 10 or 20 MeV, essentially all other interactions in this energy range result in Compton scattering off an atomic electron. For a photon attenuation length λ (Fig. 27.16), the probability that a given photon will produce an electron pair (without first Compton scattering) in thickness t of absorber is P [1 − exp(−t/λ)].

because of quantum mechanical interference between amplitudes from different scattering centers. Since the longitudinal momentum transfer to a given center is small (∝ k/E(E − k), in the case of bremsstrahlung), the interaction is spread over a comparatively long distance called the formation length (∝ E(E − k)/k) via the uncertainty principle. In alternate language, the formation length is the distance over which the highly relativistic electron and the photon “split apart.” The interference is usually destructive. Calculations of the “Landau-Pomeranchuk-Migdal” (LPM) effect may be made semi-classically based on the average multiple scattering, or more rigorously using a quantum transport approach [39,40]. In amorphous media, bremsstrahlung is suppressed if the photon energy k is less than E 2 /(E + ELP M ) [40], where* X0 (me c2 )2 αX0 ELP M = = (7.7 TeV/cm) × . (27.29) 4π cρ ρ * This definition differs from that of Ref. 50 by a factor of two. ELP M scales as the 4th power of the mass of the incident particle, so that ELP M = (1.4 × 1010 TeV/cm) × X0 /ρ for a muon.

276

27. Passage of particles through matter

Since physical distances are involved, X0 /ρ, in cm, appears. The energy-weighted bremsstrahlung spectrum for lead, k dσLP M /dk, is shown in Fig. 27.11. With appropriate scaling by X0 /ρ, other materials behave similarly. For photons, pair production is reduced for E(k − E) > k ELP M . The pair-production cross sections for different photon energies are shown in Fig. 27.15. If k  E, several additional mechanisms can also produce suppression. When the formation length is long, even weak factors can perturb the interaction. For example, the emitted photon can coherently forward scatter off of the electrons in the media. Because of this, for k < ωp E/me ∼ 10−4 , bremsstrahlung is suppressed by a factor (kme /ωp E)2 [42]. Magnetic fields can also suppress bremsstrahlung. In crystalline media, the situation is more complicated, with coherent enhancement or suppression possible. The cross section depends on the electron and photon energies and the angles between the particle direction and the crystalline axes [51].

27.5. Electromagnetic cascades When a high-energy electron or photon is incident on a thick absorber, it initiates an electromagnetic cascade as pair production and bremsstrahlung generate more electrons and photons with lower energy. The longitudinal development is governed by the high-energy part of the cascade, and therefore scales as the radiation length in the material. Electron energies eventually fall below the critical energy, and then dissipate their energy by ionization and excitation rather than by the generation of more shower particles. In describing shower behavior, it is therefore convenient to introduce the scale variables t = x/X0 , y = E/Ec , (27.30) so that distance is measured in units of radiation length and energy in units of critical energy. 0.125

dE (bt)a−1 e−bt = E0 b dt Γ(a)

(27.31)

The maximum tmax occurs at (a − 1)/b. We have made fits to shower profiles in elements ranging from carbon to uranium, at energies from 1 GeV to 100 GeV. The energy deposition profiles are well described by Eq. (27.31) with tmax = (a − 1)/b = 1.0 × (ln y + Cj ) , j = e, γ , (27.32) where Ce = −0.5 for electron-induced cascades and Cγ = +0.5 for photon-induced cascades. To use Eq. (27.31), one finds (a − 1)/b from Eq. (27.32) and Eq. (27.30), then finds a either by assuming b ≈ 0.5 or by finding a more accurate value from Fig. 27.19. The results are very similar for the electron number profiles, but there is some dependence on the atomic number of the medium. A similar form for the electron number maximum was obtained by Rossi in the context of his “Approximation B,” [4] (see Fabjan’s review in Ref. 53), but with Ce = −1.0 and Cγ = −0.5; we regard this as superseded by the EGS4 result.

30 GeV electron incident on iron

80

0.075

60 Energy

0.050

40 Photons × 1/6.8

0.025

20

Electrons

Number crossing plane

100

0.100

(1/E0) dE/dt

electron number falls off more quickly than energy deposition. This is because, with increasing depth, a larger fraction of the cascade energy is carried by photons. Exactly what a calorimeter measures depends on the device, but it is not likely to be exactly any of the profiles shown. In gas counters it may be very close to the electron number, but in glass Cherenkov detectors and other devices with “thick” sensitive regions it is closer to the energy deposition (total track length). In such detectors the signal is proportional to the “detectable” track length Td , which is in general less than the total track length T . Practical devices are sensitive to electrons with energy above some detection threshold Ed , and Td = T F (Ed /Ec ). An analytic form for F (Ed /Ec ) obtained by Rossi [4] is given by Fabjan [53]; see also Amaldi [54]. The mean longitudinal profile of the energy deposition in an electromagnetic cascade is reasonably well described by a gamma distribution [55]:

0.8

Carbon

0.7

0.6

Aluminum

b

Iron 0.5 Uranium

0.000

0

5 10 15 t = depth in radiation lengths

20

0

Figure 27.18: An EGS4 simulation of a 30 GeV electroninduced cascade in iron. The histogram shows fractional energy deposition per radiation length, and the curve is a gamma-function fit to the distribution. Circles indicate the number of electrons with total energy greater than 1.5 MeV crossing planes at X0 /2 intervals (scale on right) and the squares the number of photons with E ≥ 1.5 MeV crossing the planes (scaled down to have same area as the electron distribution). Longitudinal profiles from an EGS4 [52] simulation of a 30 GeV electron-induced cascade in iron are shown in Fig. 27.18. The number of particles crossing a plane (very close to Rossi’s Π function [4]) is sensitive to the cutoff energy, here chosen as a total energy of 1.5 MeV for both electrons and photons. The

0.4

0.3

10

100

1000

10 000

y = E/Ec

Figure 27.19: Fitted values of the scale factor b for energy deposition profiles obtained with EGS4 for a variety of elements for incident electrons with 1 ≤ E0 ≤ 100 GeV. Values obtained for incident photons are essentially the same. The “shower length” Xs = X0 /b is less conveniently parameterized, since b depends upon both Z and incident energy, as shown in Fig. 27.19. As a corollary of this Z dependence, the number of electrons crossing a plane near shower maximum is underestimated using Rossi’s approximation for carbon and seriously overestimated for uranium. Essentially the same b

27. Passage of particles through matter

At sufficiently high energies, radiative processes become more important than ionization for all charged particles. For muons and pions in materials such as iron, this “critical energy” occurs at several hundred GeV. (There is no simple scaling with particle mass, but for protons the “critical energy” is much, much higher.) Radiative effects dominate the energy loss of energetic muons found in cosmic rays or produced at the newest accelerators. These processes are characterized by small cross sections, hard spectra, large energy fluctuations, and the associated generation of electromagnetic and (in the case of photonuclear interactions) hadronic showers [59–67]. As a consequence, at these energies the treatment of energy loss as a uniform and continuous process is for many purposes inadequate. It is convenient to write the average rate of muon energy loss as [68] −dE/dx = a(E) + b(E) E . (27.36) Here a(E) is the ionization energy loss given by Eq. (27.1), and b(E) is the sum of e+ e− pair production, bremsstrahlung, and photonuclear contributions. To the approximation that these slowly-varying functions are constant, the mean range x0 of a muon with initial energy E0 is given by x0 ≈ (1/b) ln(1 + E0 /Eμc ) , (27.37)

106 b(E) (g−1 cm2)

Iron

7

btotal

6 5 4

bpair

3 bbremsstrahlung

2 1

bnuclear

0 1

10

103 102 Muon energy (GeV)

104

105

Figure 27.20: Contributions to the fractional energy loss by muons in iron due to e+ e− pair production, bremsstrahlung, and photonuclear interactions, as obtained from Groom et al. [1] except for post-Born corrections to the cross section for direct pair production from atomic electrons.

1000

ta

l

/home/sierra1/deg/dedx/rpp_mu_E_loss.pro

U

to

Thu Apr 4 13:55:40 2002

100

10

l

ta

Fe

to

H (gas) total

nu Fe

1

cl

Fe ion di a Fe tive pa tot al Fe ir br em s

27.6. Muon energy loss at high energy

8

ra

where R is a phenomenological function of x/X0 and ln E. At high enough energies, the LPM effect (Sec. 27.4.5) reduces the cross sections for bremsstrahlung and pair production, and hence can cause significant elongation of electromagnetic cascades [40].

9

Fe

where Es ≈ 21 MeV (Table 27.1), and the Rossi definition of Ec is used. In a material containing a weight fraction wj of the element with critical energy Ecj and radiation length Xj , the Moli`ere radius is given by 1 1  wj Ecj = . (27.34) RM Es Xj Measurements of the lateral distribution in electromagnetic cascades are shown in Refs. 57 and 58. On the average, only 10% of the energy lies outside the cylinder with radius RM . About 99% is contained inside of 3.5RM , but at this radius and beyond composition effects become important and the scaling with RM fails. The distributions are characterized by a narrow core, and broaden as the shower develops. They are often represented as the sum of two Gaussians, and Grindhammer [56] describes them with the function 2r R2 , (27.35) f (r) = 2 (r + R2 )2

where Eμc = a/b. Figure 27.20 shows contributions to b(E) for iron. Since a(E) ≈ 0.002 GeV g−1 cm2 , b(E)E dominates the energy loss above several hundred GeV, where b(E) is nearly constant. The rates of energy loss for muons in hydrogen, uranium, and iron are shown in Fig. 27.21 [1].

dE/dx (MeV g −1 cm 2 )

values are obtained for incident electrons and photons. For many purposes it is sufficient to take b ≈ 0.5. The gamma function distribution is very flat near the origin, while the EGS4 cascade (or a real cascade) increases more rapidly. As a result Eq. (27.31) fails badly for about the first two radiation lengths; it was necessary to exclude this region in making fits. Because fluctuations are important, Eq. (27.31) should be used only in applications where average behavior is adequate. Grindhammer et al. have developed fast simulation algorithms in which the variance and correlation of a and b are obtained by fitting Eq. (27.31) to individually simulated cascades, then generating profiles for cascades using a and b chosen from the correlated distributions [56]. The transverse development of electromagnetic showers in different materials scales fairly accurately with the Moli`ere radius RM , given by [57,58] RM = X0 Es /Ec , (27.33)

277

0.1 1

10

102

103

104

105

Muon energy (GeV)

Figure 27.21: The average energy loss of a muon in hydrogen, iron, and uranium as a function of muon energy. Contributions to dE/dx in iron from ionization and the processes shown in Fig. 27.20 are also shown. The “muon critical energy” Eμc can be defined more exactly as the energy at which radiative and ionization losses are equal, and can be found by solving Eμc = a(Eμc )/b(Eμc ). This definition corresponds to the solid-line intersection in Fig. 27.12, and is different from the Rossi definition we used for electrons. It serves the same function: below Eμc ionization losses dominate, and above Eμc radiative effects dominate. The dependence of Eμc on atomic number Z is shown in Fig. 27.22. The radiative cross sections are expressed as functions of the fractional energy loss ν. The bremsstrahlung cross section goes roughly as 1/ν over most of the range, while for the pair production case the distribution goes as ν −3 to ν −2 [69]. “Hard” losses are therefore more probable in bremsstrahlung,

278

27. Passage of particles through matter

27.7. Cherenkov and transition radiation [72,73,82]

4000

A charged particle radiates if its velocity is greater than the local phase velocity of light (Cherenkov radiation) or if it crosses suddenly from one medium to another with different optical properties (transition radiation). Neither process is important for energy loss, but both are used in high-energy physics detectors.

7980 GeV ___________ (Z + 2.03) 0.879

2000

Eμc (GeV)

1000

5700 GeV ___________ (Z + 1.47) 0.838

700

Cherenkov Radiation. The angle θc of Cherenkov radiation, relative to the particle’s direction, for a particle with velocity βc in a medium with index of refraction n is

400

Gases Solids

200

100

H

He

1

2

Li Be B C N O Ne 5

Fe

10

or

Sn

20

50

100

Z

Figure 27.22: Muon critical energy for the chemical elements, defined as the energy at which radiative and ionization energy loss rates are equal [1]. The equality comes at a higher energy for gases than for solids or liquids with the same atomic number because of a smaller density effect reduction of the ionization losses. The fits shown in the figure exclude hydrogen. Alkali metals fall 3–4% above the fitted function, while most other solids are within 2% of the function. Among the gases the worst fit is for radon (2.7% high).

cos θc = (1/nβ)  tan θc = β 2 n2 − 1  2(1 − 1/nβ) ≈

for small θc , e.g. in gases. (27.38)

The threshold velocity βt is 1/n, and γt = 1/(1 − βt2 )1/2 . Therefore, βt γt = 1/(2δ + δ2 )1/2 , where δ = n − 1. Values of δ for various commonly used gases are given as a function of pressure and wavelength in Ref. 74. For values at atmospheric pressure, see Table 6.1. Data for other commonly used materials are given in Ref. 75.

γc Ch

0.10

dN/dp [1/(GeV/c)]

0.08

Median

1 TeV muons on 3 m Fe

987 GeV/c

0.06

Mean 977 GeV/c

0.04

FWHM 9 GeV/c 0.02

0.00 950

960

970

980

990

1000

Final momentum p [GeV/c]

Figure 27.23: The momentum distribution of 1 TeV/c muons after traversing 3 m of iron as calculated with the MARS15 Monte Carlo code [67] by S.I. Striganov [1].

vg

er

v=

and in fact energy losses due to pair production may very nearly be treated as continuous. The simulated [67] momentum distribution of an incident 1 TeV/c muon beam after it crosses 3 m of iron is shown in Fig. 27.23. The most probable loss is 8 GeV, or 3.4 MeV g−1 cm2 . The full width at half maximum is 9 GeV/c, or 0.9%. The radiative tail is almost entirely due to bremsstrahlung, although most of the events in which more than 10% of the incident energy lost experienced relatively hard photonuclear interactions. The latter can exceed detector resolution [70], necessitating the reconstruction of lost energy. Tables [1] list the stopping power as 9.82 MeV g−1 cm2 for a 1 TeV muon, so that the mean loss should be 23 GeV (≈ 23 GeV/c), for a final momentum of 977 GeV/c, far below the peak. This agrees with the indicated mean calculated from the simulation. Electromagnetic and hadronic cascades in detector materials can obscure muon tracks in detector planes and reduce tracking efficiency [71].

θc

en

ko

v

η

wa

ve

fro

nt

Particle velocity v = βc

Figure 27.24: Cherenkov light emission and wavefront angles. In a dispersive medium, θc + η = 900 . Practical Cherenkov radiator materials are dispersive. Let ω be the photon’s frequency, and let k = 2π/λ be its wavenumber. The photons propage at the group velocity vg = dω/dk = c/[n(ω) + ω(dn/dω)]. In a non-dispersive medium, this simplies to vg = c/n. In his classical paper, Tamm [76] showed that for dispersive media the radiation is concentrated in a thin conical shell whose vertex is at the moving charge, and whose opening half-angle η is given by   d cot η = (ω tan θc ) dω ω0   dn 2 cot θc = tan θc + β ω n(ω) , (27.39) dω ω0 where ω0 is the central value of the small frequency range under consideration. (See Fig. 27.24.) This cone has a opening halfangle η, and, unless the medium is non-dispersive (dn/dω = 0), θc + η = 900 . The Cherenkov wavefront ‘sideslips’ along with the particle [77]. This effect may have timing implications for ring imaging Cherenkov counters [78], but it is probably unimportant for most applications. The number of photons produced per unit path length of a particle with charge ze and per unit energy interval of the photons is

 d2 N α2 z 2 1 αz 2 1 − = sin2 θc = dEdx c re me c2 β 2 n2 (E) ≈ 370 sin2 θc (E) eV−1 cm−1

(z = 1) , (27.40)

27. Passage of particles through matter

or, equivalently, d2 N dxdλ

=

2παz 2 λ2

1−

1 β 2 n2 (λ)

 .

(27.41)

The index of refraction is a function of photon energy E = ω, as is the sensitivity of the transducer used to detect the light. For practical use, Eq. (27.40) must be multiplied by the the transducer response function and integrated over the region for which β n(ω) > 1. Further details are given in the discussion of Cherenkov detectors in the Particle Detectors section (Sec. 28 of this Review). When two particles are close together (within < ∼ 1 wavelength), the electromagnetic fields from the particles may add coherently, affecting the Cherenkov radiation. The radiation from an e+ e− pair at close separation is suppressed compared to two independent leptons [79]. Coherent radio Cherenkov radiation from electromagnetic showers (containing a net excess of e− over e+ ) is significant [80], and has been used to study cosmic ray air showers [81] and to search for νe induced showers. Transition radiation. The energy radiated when a particle with charge ze crosses the boundary between vacuum and a medium with plasma frequency ωp is (27.42) I = αz 2 γ ωp /3 , where   ωp = 4πNe re3 me c2 /α = 4πNe a3∞ 2 × 13.6 eV .

(27.43)

Here Ne is the electron density in the medium, re is the classical electron radius, and  a∞ is the Bohr radius. For styrene and similar materials, 4πNe a3∞ ≈ 0.8, so that ωp ≈ 20 eV. The typical emission angle is 1/γ. The radiation spectrum is logarithmically divergent at low energies and decreases rapidly for ω/γ ωp > 1. About half the energy is emitted in the range 0.1 ≤ ω/γ ωp ≤ 1. For a particle with γ = 103 , the radiated photons are in the soft x-ray range 2 to 20 keV. The γ dependence of the emitted energy thus comes from the hardening of the spectrum rather than from an increased quantum yield. For a typical radiated photon energy of γ ωp /4, the quantum yield is 1 αz 2 γ ωp  γ ωp (27.44) ≈ 23 αz 2 ≈ 0.5% × z 2 . Nγ ≈ 2 3 4 More precisely, the number of photons with energy ω > ω0 is given by [82]   2 γ ω p αz 2 π2 Nγ (ω > ω0 ) = −1 + ln , (27.45) π ω 0 12 within corrections of order (ω0 /γ ωp )2 . The number of photons above a fixed energy ω0  γ ωp thus grows as (ln γ)2 , but the number above a fixed fraction of γ ωp (as in the example above) is constant. For example, for ω > γ ωp /10, Nγ = 2.519 αz 2 /π = 0.59% × z 2 . The yield can be increased by using a stack of plastic foils with gaps between. However, interference can be important, and the soft x rays are readily absorbed in the foils. The first problem can be overcome by choosing thicknesses and spacings large compared to the “formation length” D = γc/ωp , which in practical situations is tens of μm. Other practical problems are discussed in Sec. 28. References: 1. D.E. Groom, N.V. Mokhov, and S.I. Striganov, “Muon stopping-power and range tables: 10 MeV–100 TeV” Atomic Data and Nuclear Data Tables 78, 183–356 (2001). Since submission of this paper it has become likely that post-Born corrections to the direct pair production cross section should be made. Code used to make Figs. 27.20,

2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

15. 16. 17. 18.

19.

20. 21. 22. 23. 24. 25.

26. 27.

28. 29.

279

27.21, and 27.12 included these corrections [D.Yu. Ivanov et al., Phys. Lett. B442, 453 (1998)]. The effect is negligible for except at high Z. (It is less than 1% for iron); More extensive printable and machine-readable tables are given at http://pdg.lbl.gov/AtomicNuclearProperties/. “Stopping Powers and Ranges for Protons and Alpha Particles,” ICRU Report No. 49 (1993); Tables and graphs of these data are available at http://physics.nist.gov/PhysRefData/. W.H. Barkas, W. Birnbaum, and F.M. Smith, Phys. Rev. 101, 778 (1956). B. Rossi, High Energy Particles, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1952. U. Fano, Ann. Rev. Nucl. Sci. 13, 1 (1963). J. D. Jackson, Phys. Rev. D59, 017301 (1999). S.M. Seltzer and M.J. Berger, Int. J. of Applied Rad. 33, 1189 (1982). “Stopping Powers for Electrons and Positrons,” ICRU Report No. 37 (1984). http://physics.nist.gov/PhysRefData/XrayMassCoef/tab1.html. H. Bichsel, Phys. Rev. A46, 5761 (1992). W.H. Barkas and M.J. Berger, Tables of Energy Losses and Ranges of Heavy Charged Particles, NASA-SP-3013 (1964). M. Agnello et al., Phys. Rev. Lett. 74, 371 (1995). H.H. Andersen and J.F. Ziegler, Hydrogen: Stopping Powers and Ranges in All Elements. Vol. 3 of The Stopping and Ranges of Ions in Matter (Pergamon Press 1977). J. Lindhard, Kgl. Danske Videnskab. Selskab, Mat.Fys. Medd. 28, No. 8 (1954); J. Lindhard, M. Scharff, and H.E. Schiøtt, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 33, No. 14 (1963). J.F. Ziegler, J.F. Biersac, and U. Littmark, The Stopping and Range of Ions in Solids, Pergamon Press 1985. R.M. Sternheimer, Phys. Rev. 88, 851 (1952). A. Crispin and G.N. Fowler, Rev. Mod. Phys. 42, 290 (1970). R.M. Sternheimer, S.M. Seltzer, and M.J. Berger, “The Density Effect for the Ionization Loss of Charged Particles in Various Substances,” Atomic Data and Nuclear Data Tables 30, 261 (1984). Minor errors are corrected in Ref. 1. Chemical composition for the tabulated materials is given in Ref. 7. For unit-charge projectiles, see E.A. Uehling, Ann. Rev. Nucl. Sci. 4, 315 (1954). For highly charged projectiles, see J.A. Doggett and L.V. Spencer, Phys. Rev. 103, 1597 (1956). A Lorentz transformation is needed to convert these center-of-mass data to knock-on energy spectra. R.M. Sternheimer and R.F. Peierls, Phys. Rev. B3, 3681 (1971). N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions, Oxford Press, London, 1965. L.D. Landau, J. Exp. Phys. (USSR) 8, 201 (1944). P.V. Vavilov, Sov. Phys. JETP 5, 749 (1957). H. Bichsel, Rev. Mod. Phys. 60, 663 (1988). H. Bichsel, “A method to improve tracking and particle identification in TPCs and silicon detectors,” Nucl. Inst. & Meth. A (in press, 2006). R. Talman, Nucl. Instrum. Methods 159, 189 (1979). S.M. Seltzer and M.J. Berger, Int. J. of Applied Rad. 35, 665 (1984). This paper corrects and extends the results of Ref. 7. L.V. Spencer “Energy Dissipation by Fast Electrons,” Nat’l Bureau of Standards Monograph No. 1 (1959). “Average Energy Required to Produce an Ion Pair,” ICRU Report No. 31 (1979).

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30. 31.

32. 33. 34. 35. 36. 37. 38. 39.

40. 41.

42. 43. 44.

45. 46. 47. 48. 49. 50. 51. 52.

53. 54. 55. 56.

27. Passage of particles through matter

N. Hadley et al., “List of Poisoning Times for Materials,” Lawrence Berkeley Lab Report TPC-LBL-79-8 (1981). H.A. Bethe, Phys. Rev. 89, 1256 (1953). A thorough review of multiple scattering is given by W.T. Scott, Rev. Mod. Phys. 35, 231 (1963). However, the data of Shen et al., (Phys. Rev. D20, 1584 (1979)) show that Bethe’s simpler method of including atomic electron effects agrees better with experiment that does Scott’s treatment. For a thorough discussion of simple formulae for single scatters and methods of compounding these into multiple-scattering formulae, see W.T. Scott, Rev. Mod. Phys. 35, 231 (1963). Detailed summaries of formulae for computing single scatters are given in J.W. Motz, H. Olsen, and H.W. Koch, Rev. Mod. Phys. 36, 881 (1964). V.L. Highland, Nucl. Instrum. Methods 129, 497 (1975), and Nucl. Instrum. Methods 161, 171 (1979). G.R. Lynch and O.I Dahl, Nucl. Instrum. Methods B58, 6 (1991). M. Wong et al., Med. Phys. 17, 163 (1990). E. Segr´e, Nuclei and Particles, New York, Benjamin (1964) p. 65 ff. Y.S. Tsai, Rev. Mod. Phys. 46, 815 (1974). H. Davies, H.A. Bethe, and L.C. Maximon, Phys. Rev. 93, 788 (1954). O.I. Dahl, private communication. L.D. Landau and I.J. Pomeranchuk, Dokl. Akad. Nauk. SSSR 92, 535 (1953); 92, 735 (1953). These papers are available in English in L. Landau, The Collected Papers of L.D. Landau, Pergamon Press, 1965; A.B. Migdal, Phys. Rev. 103, 1811 (1956). S. Klein, Rev. Mod. Phys. 71, 1501 (1999). M. L. Ter-Mikaelian, SSSR 94, 1033 (1954); M. L. Ter-Mikaelian, High Energy Electromagnetic Processes in Condensed Media (John Wiley & Sons, New York, 1972). P. Anthony et al., Phys. Rev. Lett. 76, 3550 (1996). H. W. Koch and J. W. Motz, Rev. Mod. Phys. 31, 920 (1959). M.J. Berger and S.M. Seltzer, “Tables of Energy Losses and Ranges of Electrons and Positrons,” National Aeronautics and Space Administration Report NASASP-3012 (Washington DC 1964). K. Hikasa et al., Review of Particle Properties, Phys. Rev. D46 (1992) S1. B. L. Berman and S. C. Fultz, Rev. Mod. Phys. 47, 713 (1975). J.S. Hubbell, H. Gimm, and I Øverbø, J. Phys. Chem. Ref. Data 9, 1023 (1980). A. Veyssiere et al., Nucl. Phys. A159, 561 (1970). J. W. Motz, H. A. Olsen, and H. W. Koch, Rev. Mod. Phys. 41, 581 (1969). P. Anthony et al., Phys. Rev. Lett. 75, 1949 (1995). U.I. Uggerhoj, Rev. Mod. Phys. 77, 1131 (2005). W.R. Nelson, H. Hirayama, and D.W.O. Rogers, “The EGS4 Code System,” SLAC-265, Stanford Linear Accelerator Center (Dec. 1985). Experimental Techniques in High Energy Physics, ed. by T. Ferbel (Addision-Wesley, Menlo Park CA 1987). U. Amaldi, Phys. Scripta 23, 409 (1981). E. Longo and I. Sestili, Nucl. Instrum. Methods 128, 283 (1975). G. Grindhammer et al., in Proceedings of the Workshop on Calorimetry for the Supercollider, Tuscaloosa, AL, March 13–17, 1989, edited by R. Donaldson and M.G.D. Gilchriese (World Scientific, Teaneck, NJ, 1989), p. 151.

57. 58. 59.

60. 61. 62. 63.

64. 65. 66. 67.

68. 69. 70. 71.

72. 73.

74.

75. 76. 77. 78. 79. 80. 81. 82.

W.R. Nelson et al., Phys. Rev. 149, 201 (1966). G. Bathow et al., Nucl. Phys. B20, 592 (1970). H.A. Bethe and W. Heitler, Proc. Roy. Soc. A146, 83 (1934); H.A. Bethe, Proc. Cambridge Phil. Soc. 30, 542 (1934). A.A. Petrukhin and V.V. Shestakov, Can. J. Phys. 46, S377 (1968). V.M. Galitskii and S.R. Kel’ner, Sov. Phys. JETP 25, 948 (1967). S.R. Kel’ner and Yu.D. Kotov, Sov. J. Nucl. Phys. 7, 237 (1968). R.P. Kokoulin and A.A. Petrukhin, in Proceedings of the International Conference on Cosmic Rays, Hobart, Australia, August 16–25, 1971, Vol. 4, p. 2436. A.I. Nikishov, Sov. J. Nucl. Phys. 27, 677 (1978). Y.M. Andreev et al., Phys. Atom. Nucl. 57, 2066 (1994). L.B. Bezrukov and E.V. Bugaev, Sov. J. Nucl. Phys. 33, 635 (1981). N.V. Mokhov, “The MARS Code System User’s Guide,” Fermilab-FN-628 (1995); N. V. Mokhov et al., Radiation Protection and Dosimetry, vol. 116, part 2, pp. 99 (2005); Fermilab-Conf-04/053 (2004); N. V. Mokhov et al., in Proc. of Intl. Conf. on Nuclear Data for Science and Tech, (Santa Fe, NM, 2004), AIP C onf. Proc. 769, part 2, p. 1618; Fermilab-Conf-04/269-AD (2004); http://www-ap.fnal.gov/MARS/. P.H. Barrett et al., Rev. Mod. Phys. 24, 133 (1952). A. Van Ginneken, Nucl. Instrum. Methods A251, 21 (1986). U. Becker et al., Nucl. Instrum. Methods A253, 15 (1986). J.J. Eastman and S.C. Loken, in Proceedings of the Workshop on Experiments, Detectors, and Experimental Areas for the Supercollider, Berkeley, CA, July 7–17, 1987, edited by R. Donaldson and M.G.D. Gilchriese (World Scientific, Singapore, 1988), p. 542. Methods of Experimental Physics, L.C.L. Yuan and C.S. Wu, editors, Academic Press, 1961, Vol. 5A, p. 163. W.W.M. Allison and P.R.S. Wright, “The Physics of Charged Particle Identification: dE/dx, Cherenkov Radiation, and Transition Radiation,” p. 371 in Experimental Techniques in High Energy Physics, T. Ferbel, editor, (Addison-Wesley 1987). E.R. Hayes, R.A. Schluter, and A. Tamosaitis, “Index and Dispersion of Some Cherenkov Counter Gases,” ANL-6916 (1964). T. Ypsilantis, “Particle Identification at Hadron Colliders”, CERN-EP/89-150 (1989), or ECFA 89-124, 2 661 (1989). I. Tamm, J. Phys. U.S.S.R., 1, 439 (1939). H. Motz and L. I. Schiff, Am. J. Phys. 21, 258 (1953). B. N. Ratcliff, Nucl. Instrum & Meth. A502, 211 (2003). S. K. Mandal, S. R. Klein, and J. D. Jackson, Phys. Rev. D72, 093003 (2005). E. Zas, F. Halzen and T. Stanev, Phys. Rev. D45, 362 (1991). H. Falcke et al, Nature 435, 313 (2005). J.D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, New York, 1998).

28. Particle detectors

281

28. PARTICLE DETECTORS 28.1. Summary of detector spatial resolution, temporal resolution, and deadtime

Revised 2008 (see the various sections for authors). 28. PARTICLE DETECTORS . . . . . . . . . . . . .

281

28.1. Summary of detector spatial resolution, temporal resolution, and deadtime . . . . . . . . . . . . . . . . . . . .

281

28.2. Photon detectors . . . . . . . 28.2.1. Vacuum photodetectors . . 28.2.1.1. Photomultiplier tubes . 28.2.1.2. Microchannel plates . . 28.2.1.3. Hybrid photon detectors 28.2.2. Gaseous photon detectors . 28.2.3. Solid-state photon detectors

. . . . . . .

281 282 282 283 283 283 284

. . . . . . . . . . . . . . . . . . . . .

285 285 286

. . . . . . .

286

. . . . . . .

28.3. Organic scintillators . . . . . . . 28.3.1. Scintillation mechanism . . . 28.3.2. Caveats and cautions . . . . 28.3.3. Scintillating and wavelengthshifting fibers . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

28.4. Inorganic scintillators: . . . . . . . . . . . . .

286

28.5. Cherenkov detectors . . . . . . . . . . . . . . 28.5.1. Threshold counters . . . . . . . . . . . . 28.5.2. Imaging counters . . . . . . . . . . . . .

288 289 289

28.6. Cherenkov tracking calorimeters . . . . . . . .

290

28.7. Gaseous detectors . . . . . . . . . . 28.7.1. Energy loss and charge transport in gases . . . . . . . . . . . . . . 28.7.2. Multi-Wire Proportional Chambers . . . . . . . . . . . . . 28.7.3. Micro-pattern Gas Detectors . . . 28.7.4. Time-projection chambers . . . . 28.7.5. Transition radiation detectors (TRD’s) . . . . . . . . . . . . . . 28.7.6. Resistive-plate chambers . . . . .

. . . . .

292

. . . . .

292

. . . . . . . . . . . . . . .

294 295 297

. . . . . . . . . .

298 299

. . . . . . . .

300

. . . . . . . . . . . . .

301

In this section we give various parameters for common detector components. The quoted numbers are usually based on typical devices, and should be regarded only as rough approximations for new designs. More detailed discussions of detectors and their underlying physics can be found in books by Ferbel [1], Grupen [2], Kleinknecht [3], Knoll [4], Green [5], and Leroy & Rancoita [6]. In Table 28.1 are given typical resolutions and deadtimes of common detectors. Table 28.1: Typical resolutions and deadtimes of common detectors. Revised September 2003 by R. Kadel (LBNL). Detector Type

Bubble chamber 10–150 μm Streamer chamber 300 μm Proportional chamber 50–300 μmb,c,d Drift chamber 50–300 μm Scintillator — Emulsion 1 μm Liquid Argon Drift [7] ∼175–450 μm Gas Micro Strip [8] 30–40 μm  10 μm Resistive Plate chamber [9] Silicon strip pitch/(3 to 7)g Silicon pixel 2 μmi a

28.8. Silicon semiconductor detectors 28.9. Low-noise electronics

28.10. Calorimeters . . . . . . . . . . . 28.10.1. Electromagnetic calorimeters . 28.10.2. Hadronic calorimeters . . . . 28.10.3. Free electron drift velocities in liquid ionization sensors . . . . . .

d

303 304 304

. . . . . .

306

. . . . . . . . . . . .

307 307

. . . . . . . . . . . .

308 308

28.12. Measurement of particle momenta in a uniform magnetic field . . . . . . . . . . .

308

References

309

28.11. Superconducting magnets for collider detectors . . . . . . . . . . 28.11.1. Solenoid Magnets . . . . . . 28.11.2. Properties of collider detector magnets . . . . . . . . . . . . . 28.11.3. Toroidal magnets . . . . . . .

b c

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

Accuracy (rms)

e f g

h i

Resolution Dead Time Time 1 ms 50 msa 2 μs 100 ms 2 ns 200 ns 100 ns 2 nse 100 ps/nf 10 ns — — ∼ 200 ns ∼ 2 μs < 10 ns — 1–2 ns — h h h h

Multiple pulsing time. 300 μm is for 1 mm pitch. Delay line cathode readout can give ±150 μm parallel to anode wire. √ wirespacing/ 12. For two chambers. n = index of refraction. The highest resolution (“7”) is obtained for small-pitch detectors (  25 μm) with pulse-height-weighted center finding. Limited by the readout electronics [10]. (Time resolution of ≤ 25 ns is planned for the ATLAS SCT.) Analog readout of 34 μm pitch, monolithic pixel detectors.

28.2. Photon detectors Updated August 2007 by D. Chakraborty (Northern Illinois U) and T. Sumiyoshi (Tokyo Metro U). Most detectors in high-energy, nuclear, and astrophysics rely on the detection of photons in or near the visible range, 100 nm  λ  1000 nm, or E ≈ a few eV. This range covers scintillation and Cherenkov radiation as well as the light detected in many astronomical observations. Generally, photodetection involves generating a detectable electrical signal proportional to the (usually very small) number of incident photons. The process involves three distinct steps: 1. Generation of a primary photoelectron or electron-hole (e-h) pair by an incident photon by the photoelectric or photoconductive effect, 2. Amplification of the p.e. signal to detectable levels by one or more multiplicative bombardment steps and/or an avalanche process (usually), and, 3. Collection of the secondary electrons to form the electrical signal. The important characteristics of a photodetector include the following in statistical averages: 1. Quantum efficiency (QE or Q ): the number of primary photoelectrons generated per incident photon (0 ≤ Q ≤ 1;

282

2. 3. 4. 5.

28. Particle detectors

in silicon more than one e-h pair per incident photon can be generated for λ < ∼ 165 nm), Collection efficiency (CE or C ): the overall acceptance factor other than the generation of photoelectrons (0 ≤ C ≤ 1), Gain (G): the number of electrons collected for each photoelectron generated, Dark current or dark noise: the electrical signal when there is no photon, Energy resolution: electronic noise (ENC or Ne ) and statistical fluctuations in the amplification process compound the Poisson distribution of nγ photons from a given source:  σ(E) = E

fN + n γ Q C

Ne Gnγ Q C

2 ,

(28.1)

where fN , or the excess noise factor (ENF), is the contribution to the energy distribution variance due to amplification statistics [11], 6. Dynamic range: the maximum signal available from the detector (this is usually expressed in units of the response to noise-equivalent power, or NEP, which is the optical input power that produces a signal-to-noise ratio of 1), 7. Time dependence of the response: this includes the transit time, which is the time between the arrival of the photon and the electrical pulse, and the transit time spread, which contributes to the pulse rise time and width, and 8. Rate capability: inversely proportional to the time needed, after the arrival of one photon, to get ready to receive the next.

Other important considerations also are highly applicationspecific. These include the photon flux and wavelength range, the total area to be covered and the efficiency required, the volume available to accommodate the detectors, characteristics of the environment such as chemical composition, temperature, magnetic field, ambient background, as well ambient radiation of different types and, mode of operation (continuous or triggered), bias (high-voltage) requirements, power consumption, calibration needs, aging, cost, and so on. Several technologies employing different phenomena for the three steps described above, and many variants within each, offer a wide range of solutions to choose from. The salient features of the main technologies and the common variants are described below. Some key characteristics are summarized in Table 28.2. 28.2.1. Vacuum photodetectors : Vacuum photodetectors can be broadly subdivided into three types: photomultiplier tubes, microchannel plates, and hybrid photodetectors. 28.2.1.1. Photomultiplier tubes: A versatile class of photon detectors, vacuum photomultiplier tubes (PMT) has been employed by a vast majority of all particle physics experiments to date [11]. Both “transmission-” and “reflection-type” PMT’s are widely used. In the former, the photocathode material is deposited on the inside of a transparent window through which the photons enter, while in the latter, the photocathode material rests on a separate surface that the incident photons strike. The cathode material has a low work function, chosen for the wavelength band of interest. When a photon hits the cathode and liberates an electron (the photoelectric effect), the latter

Table 28.2: Representative characteristics of some photodetectors commonly used in particle physics. The time resolution of the devices listed here vary in the 10–2000 ps range. Type

λ (nm)

Q C

Gain

Risetime (ns)

Area (mm2 )

PMT∗ 115–1100 0.15–0.25 103 –107 0.7–10 102 –105 ∗ 3 7 100–650 0.01–0.10 10 –10 0.15–0.3 102 –104 MCP 0.1–0.3 103 –104 7 102 –105 HPD∗ 115–850 ∗ 3 6 GPM 115–500 0.15–0.3 10 –10 O(0.1) O(10) O(1) 10–103 APD 300–1700 ∼0.7 10–108 PPD 400–550 0.15–0.3 105 –106 ∼1 1–10 VLPC 500–600 ∼0.9 ∼5 × 104 ∼ 10 1

1-p.e noise (Hz) 10–104 0.1–200 10–103 10–103 1–103 O(106 ) O(104 )

HV (V)

Price (USD)

500–3000 100–5000 500–3500 10–6000 ∼2 × 104 ∼600 300–2000 O(10) 400–1400 O(100) 30–60 O(10) ∼7 ∼1

∗These devices often come in multi-anode configurations. In such cases, area, noise, and price are to be considered on a “per readout-channel” basis.

The QE is a strong function of the photon wavelength (λ), and is usually quoted at maximum, together with a range of λ where the QE is comparable to its maximum. Spatial uniformity and linearity with respect to the number of photons are highly desirable in a photodetector’s response. Optimization of these factors involves many trade-offs and vary widely between applications. For example, while a large gain is desirable, attempts to increase the gain for a given device also increases the ENF and after-pulsing (“echos” of the main pulse). In solid-state devices, a higher QE often requires a compromise in the timing properties. In other types, coverage of large areas by focusing increases the transit time spread.

is accelerated and guided by electric fields to impinge on a secondary-emission electrode, or dynode, which then emits a few (∼ 5) secondary electrons. The multiplication process is repeated typically 10 times in series to generate a sufficient number of electrons, which are collected at the anode for delivery to the external circuit. The total gain of a PMT depends on the applied high voltage V as G = AV kn , where k ≈ 0.7–0.8 (depending on the dynode material), n is the number of dynodes in the chain, and A a constant (which also depends on n). Typically, G is in the range of 105 –106 . Pulse risetimes are usually in the few nanosecond range. With e.g. two-level discrimination the effective time resolution can be much better.

28. Particle detectors

A large variety of PMT’s, including many just recently developed, covers a wide span of wavelength ranges from infrared (IR) to extreme ultraviolet (XUV) [12]. They are categorized by the window materials, photocathode materials, dynode structures, anode configurations, etc. Common window materials are borosilicate glass for IR to near-UV, fused quartz and sapphire (Al2 O3 ) for UV, and MgF2 or LiF for XUV. The choice of photocathode materials include a variety of mostly Cs- and/or Sb-based compounds such as CsI, CsTe, bi-alkali (SbRbCs, SbKCs), multi-alkali (SbNa2 KCs), GaAs(Cs), GaAsP, etc. Sensitive wavelengths and peak quantum efficiencies for these materials are summarized in Table 28.3. Typical dynode structures used in PMT’s are circular cage, line focusing, box and grid, venetian blind, and fine mesh. In some cases, limited spatial resolution can be obtained by using a mosaic of multiple anodes. PMT’s are vulnerable to magnetic fields—sometimes even the geomagnetic field causes large orientation-dependent gain changes. A high-permeability metal shield is often necessary. However, proximity-focused PMT’s, e.g. the fine-mesh types, can be used even in a high magnetic field (≥ 1 T) if the electron drift direction is parallel to the field. 28.2.1.2. Microchannel plates: A typical Microchannel plate (MCP) photodetector consists of one or more ∼2 mm thick glass plates with densely packed O(10 μm)-diameter cylindrical holes, or “channels”, sitting between the transmission-type photocathode and anode planes, separated by O(1 mm) gaps. Instead of discrete dynodes, the inner surface of each cylindrical tube serves as a continuous dynode for the entire cascade of multiplicative bombardments initiated by a photoelectron. Gain fluctuations can be minimized by operating in a saturation mode, whence each channel is only capable of a binary output, but the sum of all channel outputs remains proportional to the number of photons received so long as the photon flux is low enough to ensure that the probability of a single channel receiving more than one photon during a single time gate is negligible. MCP’s are thin, offer good spatial resolution, have excellent time resolution (∼20 ps), and can tolerate random magnetic fields up to 0.1 T and axial fields up to ∼ 1 T. However, they suffer from relatively long recovery time per channel and short lifetime. MCP’s are widely employed as image-intensifiers, although not so much in HEP or astrophysics. 28.2.1.3. Hybrid photon detectors: Hybrid photon detectors (HPD) combine the sensitivity of a vacuum PMT with the excellent spatial and energy resolutions of a Si sensor [13]. A single photoelectron ejected from the photocathode is accelerated through a potential difference of ∼20 kV before it impinges on the silicon sensor/anode. The gain nearly equals the maximum number of e-h pairs that could be created from the entire kinetic energy of the accelerated electron: G ≈ eV /w, where e is the electronic charge, V is the applied potential difference, and w ≈ 3.7 eV is the mean energy required to create an e-h pair in Si at room temperature. Since the gain is achieved in a single step, one might expect to have the excellent resolution of a simple Poisson statistic with large mean, but in fact it is even better, thanks to the Fano effect discussed in Sec. 28.8. Low-noise electronics must be used to read out HPD’s if one intends to take advantage of the low fluctuations in gain, e.g. when counting small numbers of photons. HPD’s can have the same Q C and window geometries as PMT’s and can be segmented down to ∼50 μm. However, they require rather high

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biases and will not function in a magnetic field. The exception is proximity-focused devices (⇒ no (de)magnification) in an axial field. With time resolutions of ∼10 ps and superior rate capabiility, proximity-focused HPD’s can be an alternative to MCP’s. Current applications of HPD’s include the CMS hadronic calorimeter and the RICH detector in LHCb. Largesize HPD’s with sophisticated focusing may be suitable for future water Cherenkov experiments. Hybrid APD’s (HAPD’s) add an avalanche multiplication step following the electron bombardment to boost the gain by a factor of ∼50. This affords a higher gain and/or lower electrical bias, but also degrades the signal definition. Table 28.3: Properties of photocathode and window materials commonly used in vacuum photodetectors [12]. Photocathode material CsI CsTe Bi-alkali Multi-alkali GaAs(Cs)∗ GaAsP(Cs) ∗ Reflection

λ (nm)

Window material

115–200 MgF2 115–240 MgF2 300–650 Borosilicate 160-650 Quartz 300–850 Borosilicate 160-850 Quartz 160–930 Quartz 300-750 Borosilicate

Peak Q (λ/nm) 0.15 0.18 0.27 0.27 0.20 0.23 0.23 0.42

(135) (210) (390) (390) (360) (280) (280) (560)

type photocathode is used.

28.2.2. Gaseous photon detectors : In gaseous photomultipliers (GPM) a photoelectron in a suitable gas mixture initiates an avalanche in a high-field region, producing a large number of secondary impact-ionization electrons. In principle the charge multiplication and collection processes are identical to those employed in gaseous tracking detectors such as multiwire proportional chambers, micromesh gaseous detectors (Micromegas), or gas electron multipliers (GEM). These are discussed in Sec. 28.7.3. The devices can be divided into two types depending on the photocathode material. One type uses solid photocathode materials much in the same way as PMT’s. Since it is resistant to gas mixtures typically used in tracking chambers, CsI is a common choice. In the other type, photoionization occurs on suitable molecules vaporized and mixed in the drift volume. Most gases have photoionization work functions in excess of 10 eV, which would limit their sensitivity to wavelengths far too short. However, vapors of TMAE (tetrakis dimethyl-amine ethylene) or TEA (tri-ethyl-amine), which have smaller work functions (5.3 eV for TMAE and 7.5 eV for TEA), are suited for XUV photon detection [14]. Since devices like GEM’s offer sub-mm spatial resolution, GPM’s are often used as positionsensitive photon detectors. They can be made into flat panels to cover large areas (O(1 m2 )), can operate in high magnetic fields, and are relatively inexpensive. Many of the ring imaging Cherenkov (RICH) detectors to date have used GPM’s for the detection of Cherenkov light [15]. Special care must be taken to suppress the photon-feedback process in GPM’s. It is also important to maintain high purity of the gas as minute traces of O2 can significantly degrade the detection efficiency.

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28. Particle detectors

28.2.3. Solid-state photon detectors : In a phase of rapid development, solid-state photodetectors are competing with vacuum- or gas-based devices for many existing applications and making way for a multitude of new ones. Compared to traditional vacuum- and gaseous photodetectors, solid-state devices are more compact, lightweight, rugged, tolerant to magnetic fields, and often cheaper. They also allow fine pixelization, are easy to integrate into large systems, and can operate at low electric potentials, while matching or exceeding most performance criteria. They are particularly well suited for detection of γand X-rays. Except for applications where coverage of very large areas or dynamic range is required, solid-state detectors are proving to be the better choice. Some hybrid devices attempt to combine the best features of different technologies while applications of nanotechnology are opening up exciting new possibilities. Silicon photodiodes (PD) are widely used in high-energy physics as particle detectors and in a great number of applications (including solar cells!) as light detectors. The structure is discussed in some detail in Sec. 28.8. In its simplest form, the PD is a reverse-biased p-n junction. Photons with energies above the indirect bandgap energy (wavelengths shorter than about 1050 nm, depending on the temperature) can create e-h pairs (the photoconductive effect), which are collected on the p and n sides, respectively. Often, as in the PD’s used for crystal scintillator readout in CLEO, L3, Belle,BaBar, and GLAST, intrinsic silicon is doped to create a p-i-n structure. The reverse bias increases the thickness of the depleted region; in the case of these particular detectors, to full depletion at a depth of about 100 μm. Increasing the depletion depth decreases the capacitance (and hence electronic noise) and extends the red response. Quantum efficiency can exceed 90%, but falls toward the red because of the increasing absorption length of light in silicon. The absorption length reaches 100 μm at 985 nm. However, since G = 1, amplification is necessary. Optimal low-noise amplifiers are slow, but, even so, noise limits the minimum detectable signal in room-temperature devices to several hundred photons. Very large arrays containing O(107 ) of O(10 μm2 )-sized photodioides pixelizing a plane are widely used to photograph all sorts of things from everyday subjects at visible wavelengths to crystal structures with X-rays and astronomical objects from infrared to UV. To limit the number of readout channels, these are made into charge-coupled devices (CCD), where pixel-topixel signal transfer takes place over thousands of synchronous cycles with sequential output through shift registers [16]. Thus, high spatial resolution is achieved at the expense of speed and timing precision. Custom-made CCD’s have virtually replaced photographic plates and other imagers for astronomy and in spacecraft. Typical QE’s exceed 90% over much of the visible spectrum, and “thick” CCD’s have useful QE up to λ = 1 μm. Active Pixel Sensor (APS) arrays with a preamplifier on each pixel and CMOS processing afford higher speeds, but are challenged at longer wavelengths. Much R&D is underway to overcome the limitations of both CCD and CMOS imagers. In avalanche photodiodes (APD), an exponential cascade of impact ionizations initiated by the initial photogenerated e-h pair under a large reverse-bias voltage leads to an avalanche breakdown [17]. As a result, detectable electrical response can be obtained from low-intensity optical signals down to single

photons. Excellent junction uniformity is critical, and a guard ring is generally used as a protection against edge breakdown. Well-designed APD’s, such as those used in CMS’ crystal-based electromagnetic calorimeter, have achieved Q C ≈ 0.7 with sub-ns response time. The sensitive wavelength window and gain depend on the semiconductor used. The gain is typically 10–200 in linear and up to 108 in Geiger mode of operation. Stability and close monitoring of the operating temperature are important for linear-mode operation, and substantial cooling is often necessary. Position-sensitive APD’s use time information at multiple anodes to calculate the hit position. One of the most promising recent developments in the field is that of devices consisting of large arrays (O(103 )) of tiny APD’s packed over a small area (O(1 mm2 )) and operated in a limited Geiger mode [18]. Among different names used for this class of photodetectors, “PPD” (for “Pixelized Photon Detector”) is most widely accepted (formerly “SiPM”). Although each cell only offers a binary output, linearity with respect to the number of photons is achieved by summing the cell outputs in the same way as with a MCP in saturation mode (see above). PPD’s are being adopted as the preferred solution for various purposes including medical imaging, e.g. positron emission tomography (PET). These compact, rugged, and economical devices allow auto-calibration through decent separation of photoelectron peaks and offer gains of O(106 ) at a moderate bias voltage (∼50 V). However, the single-photoelectron noise of a PPD, being the logical “or” of O(103 ) Geiger APD’s, is rather large: O(1 MHz/mm2 ) at room temperature. PPD’s are particularly wellsuited for applications where triggered pulses of several photons are expected over a small area, e.g. fiber-guided scintillation light. Intense R&D is expeced to lower the noise level and improve radiation hardness, resulting in coverage of larger areas and wider applications. Attempts are being made to combine the fabrication of the sensors and the front-end electronics (ASIC) in the same process with the goal of making PPD’s and other finely pixelized solid-state photodetectors extremely easy to use. Of late, much R&D has been directed to p-i-n diode arrays based on thin polycrystalline diamond films formed by chemical vapor deposition (CVD) on a hot substrate (∼1000 K) from a hydrocarbon-containing gas mixture under low pressure (∼100 mbar). These devices have maximum sensitivity in the extreme- to moderate-UV region [19]. Many desirable characteristics, including high tolerance to radiation and temperature fluctuations, low dark noise, blindness to most of the solar radiation spectrum, and relatively low cost make them ideal for space-based UV/XUV astronomy, measurement of synchrotron radiation, and luminosity monitoring at (future) lepton collider(s). Visible-light photon counters (VLPC) utilize the formation of an impurity band only 50 meV below the conduction band in As-doped Si to generate strong (G ≈ 5 × 104 ) yet sharp response to single photons with Q ≈ 0.9 [20]. The smallness of the band gap considerably reduces the gain dispersion. Only a very small bias (∼7 V) is needed, but high sensitivity to infrared photons requires cooling below 10 K. The dark noise increases sharply and exponentially with both temperature and bias. The Run 2 DØ detector uses 86000 VLPC’s to read the optical signal from its scintillating-fiber tracker and scintillator-strip preshower detectors.

28. Particle detectors

28.3. Organic scintillators

Ionization excitation of base plastic

Revised September 2007 by K.F. Johnson (FSU). Organic scintillators are broadly classed into three types, crystalline, liquid, and plastic, all of which utilize the ionization produced by charged particles (see Sec. 27.2) of this Review) to generate optical photons, usually in the blue to green wavelength regions [21]. Plastic scintillators are by far the most widely used. Crystal organic scintillators are practically unused in high-energy physics. Densities range from 1.03 to 1.20 g cm−3 . Typical photon yields are about 1 photon per 100 eV of energy deposit [22]. A one-cm-thick scintillator traversed by a minimum-ionizing particle will therefore yield ≈ 2 × 104 photons. The resulting photoelectron signal will depend on the collection and transport efficiency of the optical package and the quantum efficiency of the photodetector. Plastic scintillators do not respond linearly to the ionization density. Very dense ionization columns emit less light than expected on the basis of dE/dx for minimum-ionizing particles. A widely used semi-empirical model by Birks posits that recombination and quenching effects between the excited molecules reduce the light yield [23]. These effects are more pronounced the greater the density of the excited molecules. Birks’ formula is dL dE/dx = L0 , dx 1 + kB dE/dx

(28.2)

where L is the luminescence, L0 is the luminescence at low specific ionization density, and kB is Birks’ constant, which must be determined for each scintillator by measurement. Decay times are in the ns range; rise times are much faster. The combination of high light yield and fast response time allows the possibility of sub-ns timing resolution [24]. The fraction of light emitted during the decay “tail” can depend on the exciting particle. This allows pulse shape discrimination as a technique to carry out particle identification. Because of the hydrogen content (carbon to hydrogen ratio ≈ 1) plastic scintillator is sensitive to proton recoils from neutrons. Ease of fabrication into desired shapes and low cost has made plastic scintillators a common detector component. Recently, plastic scintillators in the form of scintillating fibers have found widespread use in tracking and calorimetry [25]. 28.3.1.

Scintillation mechanism :

Scintillation: A charged particle traversing matter leaves behind it a wake of excited molecules. Certain types of molecules, however, will release a small fraction (≈ 3%) of this energy as optical photons. This process, scintillation, is especially marked in those organic substances which contain aromatic rings, such as polystyrene (PS) and polyvinyltoluene (PVT). Liquids which scintillate include toluene and xylene. Fluorescence: In fluorescence, the initial excitation takes place via the absorption of a photon, and de-excitation by emission of a longer wavelength photon. Fluors are used as “waveshifters” to shift scintillation light to a more convenient wavelength. Occurring in complex molecules, the absorption and emission are spread out over a wide band of photon energies, and have some overlap, that is, there is some fraction of the emitted light which can be re-absorbed [26]. This “self-absorption” is undesirable for detector applications because it causes a shortened attenuation length. The wavelength difference between the major absorption and emission peaks is called the Stokes’ shift. It is usually the case that the greater the Stokes’ shift, the smaller the self absorption—thus, a large Stokes’ shift is a desirable property for a fluor.

10−8 m

base plastic

Forster energy transfer

emit UV, ~340 nm

10−4 m

primary fluor (~1% wt/wt )

γ absorb UV photon

1m

285

γ

emit blue, ~400 nm absorb blue photon

secondary fluor (~0.05% wt/wt )

photodetector

Figure 28.1: Cartoon of scintillation “ladder” depicting the operating mechanism of plastic scintillator. Approximate fluor concentrations and energy transfer distances for the separate sub-processes are shown.

Scintillators: The plastic scintillators used in high-energy physics are binary or ternary solutions of selected fluors in a plastic base containing aromatic rings. (See the appendix in Ref. 27 for a comprehensive list of components.) Virtually all plastic scintillators contain as a base either PVT or PS. PVT-based scintillator can be up to 50% brighter. Ionization in the plastic base produces UV photons with short attenuation length (several mm). Longer attenuation lengths are obtained by dissolving a “primary” fluor in high concentration (1% by weight) into the base, which is selected to efficiently re-radiate absorbed energy at wavelengths where the base is more transparent. The primary fluor has a second important function. The decay time of the scintillator base material can be quite long—in pure polystyrene it is 16 ns, for example. The addition of the primary fluor in high concentration can shorten the decay time by an order of magnitude and increase the total light yield. At the concentrations used (1% and greater), the average distance between a fluor molecule and an excited base unit is around 100 ˚ A, much less than a wavelength of light. At these distances the predominant mode of energy transfer from base to fluor is not the radiation of a photon, but a resonant dipole-dipole interaction, first described by Foerster, which strongly couples the base and fluor [28]. The strong coupling sharply increases the speed and the light yield of the plastic scintillators. Unfortunately, a fluor which fulfills other requirements is usually not completely adequate with respect to emission wavelength or attenuation length, so it is necessary to add yet another waveshifter (the “secondary” fluor), at fractional percent levels, and occasionally a third (not shown in Fig. 28.1). External wavelength shifters: Light emitted from a plastic scintillator may be absorbed in a (nonscintillating) base doped with a wave-shifting fluor. Such wavelength shifters are widely used to aid light collection in complex geometries. The wavelength shifter must be insensitive to ionizing radiation and Cherenkov light. A typical wavelength shifter uses an acrylic base because of its good optical qualities, a single fluor to shift the light emerging from the plastic scintillator to the blue-green, and contains ultra-violet absorbing additives to deaden response to Cherenkov light.

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28. Particle detectors

28.3.2. Caveats and cautions : Plastic scintillators are reliable, robust, and convenient. However, they possess quirks to which the experimenter must be alert.

Aging and Handling: Plastic scintillators are subject to aging which diminishes the light yield. Exposure to solvent vapors, high temperatures, mechanical flexing, irradiation, or rough handling will aggravate the process. A particularly fragile region is the surface which can “craze”—develop microcracks—which rapidly destroy the capability of plastic scintillators to transmit light by total internal reflection. Crazing is particularly likely where oils, solvents, or fingerprints have contacted the surface. Attenuation length: The Stokes’ shift is not the only factor determining attenuation length. Others are the concentration of fluors (the higher the concentration of a fluor, the greater will be its self-absorption); the optical clarity and uniformity of the bulk material; the quality of the surface; and absorption by additives, such as stabilizers, which may be present. Afterglow: Plastic scintillators have a long-lived luminescence which does not follow a simple exponential decay. Intensities at the 10−4 level of the initial fluorescence can persist for hundreds of ns [21,29]. Atmospheric quenching: Plastic scintillators will decrease their light yield with increasing partial pressure of oxygen. This can be a 10% effect in an artificial atmosphere [30]. It is not excluded that other gases may have similar quenching effects. Magnetic field: The light yield of plastic scintillators may be changed by a magnetic field. The effect is very nonlinear and apparently not all types of plastic scintillators are so affected. Increases of ≈ 3% at 0.45 T have been reported [31]. Data are sketchy and mechanisms are not understood. Radiation damage: Irradiation of plastic scintillators creates color centers which absorb light more strongly in the UV and blue than at longer wavelengths. This poorly understood effect appears as a reduction both of light yield and attenuation length. Radiation damage depends not only on the integrated dose, but on the dose rate, atmosphere, and temperature, before, during and after irradiation, as well as the materials properties of the base such as glass transition temperature, polymer chain length, etc. Annealing also occurs, accelerated by the diffusion of atmospheric oxygen and elevated temperatures. The phenomena are complex, unpredictable, and not well understood [32]. Since color centers are less intrusive at longer wavelengths, the most reliable method of mitigating radiation damage is to shift emissions at every step to the longest practical wavelengths, e.g., utilize fluors with large Stokes’ shifts (aka the “Better red than dead” strategy). 28.3.3.

Scintillating and wavelength-shifting fibers :

The clad optical fiber is an incarnation of scintillator and wavelength shifter (WLS) which is particularly useful [33]. Since the initial demonstration of the scintillating fiber (SCIFI) calorimeter [34], SCIFI techniques have become mainstream [35]. SCIFI calorimeters are fast, dense, radiation hard, and can have leadglass-like resolution. SCIFI trackers can handle high rates and are radiation tolerant, but the low photon yield at the end of a long fiber (see below) forces the use of sensitive photodetectors. WLS scintillator readout of a calorimeter allows a very high level of hermeticity since the solid angle blocked by the fiber on its way to the photodetector is very small. The sensitive region of scintillating fibers can be controlled by splicing them onto clear (non-scintillating/non-WLS) fibers. A typical configuration would be fibers with a core of polystyrene-based scintillator or WLS (index of refraction

n = 1.59), surrounded by a cladding of PMMA (n = 1.49) a few microns thick, or, for added light capture, with another cladding of fluorinated PMMA with n = 1.42, for an overall diameter of 0.5 to 1 mm. The fiber is drawn from a boule and great care is taken during production to ensure that the intersurface between the core and the cladding has the highest possible uniformity and quality, so that the signal transmission via total internal reflection has a low loss. The fraction of generated light which is transported down the optical pipe is denoted the capture fraction and is about 6% for the single-clad fiber and 10% for the double-clad fiber. The number of photons from the fiber available at the photodetector is always smaller than desired, and increasing the light yield has proven difficult. A minimum-ionizing particle traversing a high-quality 1 mm diameter fiber perpendicular to its axis will produce fewer than 2000 photons, of which about 200 are captured. Attenuation may eliminate 95% of these photons in a large collider tracker. A scintillating or WLS fiber is often characterized by its “attenuation length,” over which the signal is attenuated to 1/e of its original value. Many factors determine the attenuation length, including the importance of re-absorption of emitted photons by the polymer base or dissolved fluors, the level of crystallinity of the base polymer, and the quality of the total internal reflection boundary. Attenuation lengths of several meters are obtained by high quality fibers. However, it should be understood that the attenuation length is not necessarily a measure of fiber quality. Among other things, it is not constant with distance from the excitation source and it is wavelength dependent. So-called “cladding light” causes some of the distance dependence [36], but not all. The wavelength dependence is usually related to the higher re-absorption of shorter wavelength photons—once absorbed, re-emitted isotropically and lost with 90% probability—and to the lower absorption of longer wavelengths by polystyrene. Experimenters should be aware that measurements of attenuation length by a phototube with a bialkali photocathode, whose quantum efficiency drops below 10% at 480 nm, should not be na¨ıvely compared to measurements utilizing a silicon photodiode, whose quantum efficiency is still rising at 600 nm.

28.4. Inorganic scintillators: Revised September 2007 by R.-Y. Zhu (California Institute of Technology) and C.L. Woody (BNL). Inorganic crystals form a class of scintillating materials with much higher densities than organic plastic scintillators (typically ∼ 4–8 g/cm3 ) with a variety of different properties for use as scintillation detectors. Due to their high density and high effective atomic number, they can be used in applications where high stopping power or a high conversion efficiency for electrons or photons is required. These include total absorption electromagnetic calorimeters (see Sec. 28.10.1), which consist of a totally active absorber (as opposed to a sampling calorimeter), as well as serving as gamma ray detectors over a wide range of energies. Many of these crystals also have very high light output, and can therefore provide excellent energy resolution down to very low energies (∼ few hundred keV). Some crystals are intrinsic scintillators in which the luminescence is produced by a part of the crystal lattice itself. However, other crystals require the addition of a dopant, typically fluorescent ions such as thallium (Tl) or cerium (Ce) which is responsible for producing the scintillation light. However, in both cases, the scintillation mechanism is the same. Energy is deposited in the crystal by ionization, either directly by charged particles, or by the conversion of photons into electrons or positrons which subsequently produce ionization.

28. Particle detectors

This energy is transferred to the luminescent centers which then radiate scintillation photons. The efficiency η for the conversion of energy deposit in the crystal to scintillation light can be expressed by the relation [37] η =β·S·Q .

(28.3)

where β is the efficiency of the energy conversion process, S is the efficiency of energy transfer to the luminescent center, and Q is the quantum efficiency of the luminescent center. The value of η ranges between 0.1 and ∼ 1 depending on the crystal, and is the main factor in determining the intrinsic light output of the scintillator. In addition, the scintillation decay time is primarily determined by the energy transfer and emission process. The decay time of the scintillator is mainly dominated by the decay time of the luminescent center. For example, in the case of thallium doped sodium iodide (NaI(Tl)), the value of η is ∼ 0.5, which results in a light output ∼ 40,000 photons per MeV of energy deposit. This high light output is largely due to the high quantum efficiency of the thallium ion (Q ∼ 1), but the decay time is rather slow (τ ∼ 250 ns). Table 28.4 lists the basic properties of some commonly used inorganic crystal scintillators. NaI(Tl) is one of the most common and widely used scintillators, with an emission that is well matched to a bialkali photomultiplier tube, but it is highly hygroscopic and difficult to work with, and has a rather low density. CsI(Tl) has high light yield, an emission that is well matched to solid state photodiodes, and is mechanically robust (high plasticity and resistance to cracking). However, it needs careful surface treatment and is slightly hygroscopic. Compared with CsI(Tl), pure CsI has identical mechanical properties, but faster emission at shorter wavelengths and light output approximately an order of magnitude lower. BaF2 has a fast component with a sub-nanosecond decay time, and is the fastest known scintillator. However, it also has a slow component with a much longer decay time (∼ 630 ns). Bismuth gemanate (Bi4 Ge3 O12 or BGO) has a high density, and consequently a short radiation length X0 and Moli`ere radius RM . BGO’s emission is well-matched to the spectral sensitivity of photodiodes, and it is easy to handle and not hygroscopic. Lead tungstate (PbWO4 or PWO) has a very high density, with a very short X0 and RM , but its intrinsic light yield is rather low. Cerium doped lutetium oxyorthosilicate (Lu2 SiO5 :Ce, or LSO:Ce) [38], cerium doped lutetium-yttrium oxyorthosilicate (Lu2(1−x) Y2x SiO5 , LYSO:Ce) [39] and cerium doped gadolinium orthosilicate (Gd2 SiO5 :Ce, or GSO:Ce) [40] are dense crystal scintillators which have a high light yield and a fast decay time. Only properties of LSO:Ce and GSO:Ce are listed in Table 28.4 since the properties of LYSO:Ce are similar to that of LSO:Ce except a little lower density than LSO:Ce depending on the yttrium fraction in LYSO:Ce [41]. Beside the crystals listed in Table 28.4, a number of new crystals are being developed that may have potential applications in high energy or nuclear physics. Of particular interest is the family of yttrium and lutetium perovskites, which include YAP (YAlO3 :Ce) and LuAP (LuAlO3 :Ce) and their mixed compositions. These have been shown to be linear over a large energy range [42], and have the potential for providing extremely good intrinsic energy resolution. In addition, other fluoride crystals such as CeF3 have been shown to provide excellent energy resolution in calorimeter applications. Table 28.4 gives the light output of other crystals relative to NaI(Tl) and their dependence to the temperature variations measured for crystal samples of 1.5 X0 cube with a Tyvek paper wrapping and a full end face coupled to a photodetector [43]. The quantum efficiencies of the photodetector is taken out to facilitate a direct comparison of crystal’s light output. However,

287

the useful signal produced by a scintillator is usually quoted in terms of the number of photoelectrons per MeV produced by a given photodetector. The relationship between the number of photons/MeV produced and photoelectrons/MeV detected involves the factors for the light collection efficiency L and the quantum efficiency QE of the photodetector: Np.e. /MeV = L · QE · Nγ /MeV

(28.4)

L includes the transmission of scintillation light within the crystal (i.e., the bulk attenuation length of the material), reflections and scattering from the surfaces, and the size and shape of the crystal. These factors can vary considerably depending on the sample, but can be in the range of ∼10–60%. The internal light transmission depends on the intrinsic properties of the material, e.g. the density and type of the scattering centers and defects that can produce internal absorption within the crystal, and can be highly affected by factors such as radiation damage, as discussed below. The quantum efficiency depends on the type of photodetector used to detect the scintillation light, which is typically ∼15–20% for photomultiplier tubes and ∼70% for silicon photodiodes for visible wavelengths. The quantum efficiency of the detector is usually highly wavelength dependent and should be matched to the particular crystal of interest to give the highest quantum yield at the wavelength corresponding to the peak of the scintillation emission. Fig. 28.2 shows the quantum efficiencies of two photodetectors, a Hamamatsu R2059 PMT with bialkali cathode and quartz window and a Hamamatsu S8664 avalanche photodiode (APD) as a function of wavelength. Also shown in the figure are emission spectra of three crystal scintillators, BGO, LSO:Ce/LYSO:Ce and CsI(Tl), and the numerical values of the emission weighted quantum efficiency. The area under each emission spectrum is proportional to crystal’s light yield, as shown in Table 28.4, where the quantum efficiencies of the photodetector has been taken out. Results with different photodetectors can be significantly different. For example, the response of CsI(Tl) relative to NaI(Tl) with a standard photomultiplier tube with a bialkali photocathode, e.g. Hamamatsu R2059, would be 45 rather than 165 because of the photomultiplier’s low quantum efficiency at longer wavelengths. For scintillators which emit in the UV, a detector with a quartz window should be used. One important issue related to the application of a crystal scintillator is its radiation hardness. Stability of its light output, or the ability to track and monitor the variation of its light output in a radiation environment, is required for high resolution and precision calibration [44]. All known crystal scintillators suffer from radiation damage. A common damage phenomenon is the appearance of radiation induced absorption caused by the formation of color centers originated from the impurities or point defects in the crystal. This radiation induced absorption reduces the light attenuation length in the crystal, and hence its light output. For crystals with high defect density, a severe reduction of light attenuation length may cause a distortion of the light response uniformity, leading to a degradation of the energy resolution. Additional radiation damage effects may include a reduced intrinsic scintillation light yield (damage to the luminescent centers) and an increased phosphorescence (afterglow). For crystals to be used in the construction a high precision calorimeter in a radiation environment, its scintillation mechanism must not be damaged and its light attenuation length in the expected radiation environment must be long enough so that its light response uniformity, and thus its energy resolution, does not change [45]. Most of the crystals listed in Table 28.4 have been used in high energy or nuclear physics experiments when the ultimate

288

28. Particle detectors

28.5. Cherenkov detectors

7

Revised September 2007 by B.N. Ratcliff (SLAC). Although devices using Cherenkov radiation are often thought of as particle identification (PID) detectors, in practice, they are widely used over a much broader range of applications; including (1) fast particle counters; (2) hadronic particle identification; and (3) tracking detectors performing complete event reconstruction. A few examples of specific applications from each category include; (1) the polarization detector of the SLD [46]; (2) the hadronic PID detectors at the B factory detectors (DIRC in BaBar [9] and the aerogel threshold Cherenkov in Belle [47]) ; and (3) large water Cherenkov counters such as Super-Kamiokande [49]. Cherenkov counters contain two main elements; (1) a radiator through which the charged particle passes, and (2) a photodetector. As Cherenkov radiation is a weak source of photons, light collection and detection must be as efficient as possible. The presence of the refractive index n and the path length of the particle in the radiator in the Cherenkov relations allows tuning these quantities for a particular experimental application. Cherenkov detectors utilize one or more of the properties of Cherenkov radiation discussed in the Passages of Particles through Matter section (Sec. 27 of this Review): the prompt emission of a light pulse; the existence of a velocity threshold for radiation; and the dependence of the Cherenkov cone half-angle θc and the number of emitted photons on the velocity of the particle. The number of photoelectrons (Np.e. ) detected in a given device is

Figure 28.2: The quantum efficiencies of two photodetectors, a Hamamatsu R2059 PMT with bi-alkali cathode and a Hamamatsu S8664 avalanche photodiode (APD), are shown as a function of wavelength. Also shown in the figure are emission spectra of three crystal scintillators, BGO, LSO and CsI(Tl), and the numerical values of the emission weighted quantum efficiency. The area under each emission spectrum is proportional to crystal’s light yield. energy resolution for electrons and photons is desired. Examples are the Crystal Ball NaI(Tl) calorimeter at SPEAR, the L3 BGO calorimeter at LEP, the CLEO CsI(Tl) calorimeter at CESR, the KTeV CsI calorimeter at the Tevatron, the BaBar and BELLE CsI(Tl) calorimeters at PEP-II and KEK. Because of its high density and low cost, PWO calorimeters are widely used by CMS and ALICE at LHC, by CLAS and PrimEx at CEBAF, and are the leading option for PANDA at GSI. Recently, investigations have been made aiming at using LSO:Ce or LYSO:Ce crystals for future high energy or nuclear physics experiments [41].

Np.e. = L

α2 z 2 re me c2

(E) sin2 θc (E)dE ,

where L is the path length in the radiator, (E) is the efficiency for collecting the Cherenkov light and transducing it in photoelectrons, and α2 /(re me c2 ) = 370 cm−1 eV−1 .

Table 28.4: Properties of several inorganic crystal scintillators. Most of the notation is defined in Sec. 6 of this Review. Parameter: ρ Units:

MP 3 ◦

g/cm

C

X0∗

∗ RM

cm

cm MeV/cm cm

dE/dx

λ∗I

τdecay λmax ns

n

nm

Relative Hygro- d(LY)/dT output† scopic? %/◦ C‡

NaI(Tl)

3.67

651 2.59

4.13

4.8

42.9

230

410

1.85

100

yes

−0.2

BGO BaF2

7.13 1050 1.12 4.89 1280 2.03

2.23 3.10

9.0 6.6

22.8 30.7

621 1.86

3.57

5.6

39.3

1.79

21 36s 3.4f 165

no no

4.51

480 300s 220f 560

2.15 1.50

CsI(Tl)

300 630s 0.9f 1300

slight

−0.9 −1.3s ∼0f 0.3

CsI(pure) 4.51

621 1.86

3.57

5.6

39.3

35s

420s

1.95

3.6s

slight

−1.3

310f 425s 420f 420

no

−2.7

no

−0.2

430

no

−0.1

2.00

10.2

20.7

LSO(Ce) 7.40 2050 1.14

2.07

9.6

20.9

6f 30s 10f 40

GSO(Ce) 6.71 1950 1.38

2.23

8.9

22.2

600s

PbWO4

8.3 1123 0.89

56f ∗  †

(28.5)

1.82

1.1f 0.083s 0.29f 83

1.85

3s

2.20

30f

Numerical values calculated using formulae in this review. Refractive index at the wavelength of the emission maximum. Relative light output measured for samples of 1.5 X0 cube with a Tyvek paper wrapping and a full end face coupled to a photodetector. The quantum efficiencies of the photodetector is taken out. ‡ Variation of light yield with temperature evaluated at the room temperature. f = fast component, s = slow component

28. Particle detectors

The quantities  and θc are functions of the photon energy E. However, since the typical energy dependent variation of the index of refraction is modest, a quantity called the Cherenkov detector quality factor N0 can be defined as N0 =

α2 z 2 re me c2

 dE ,

(28.6)

so that Np.e. ≈ LN0 sin2 θc  .

(28.7)

We take z = 1, the usual case in high-energy physics, in the following discussion. This definition of the quality factor N0 is not universal, nor, indeed, very useful for situations where the geometrical photon collection efficiency (coll ) varies substantially for different tracks. In this case, separate factors for photon collection and detection (det ), so that  = coll det , are sometimes included on the right hand side of the equation. A typical value of N0 for a photomultiplier (PMT) detection system working in the visible and near UV, and collecting most of the Cherenkov light, is about 100 cm−1 . Practical counters, utilizing a variety of different photodetectors, have values ranging between about 30 and 180 cm−1 . Radiators can be chosen from a variety of transparent materials (Sec. 27 of this Review and Table 6.1). In addition to refractive index, the choice requires consideration of factors such as material density, radiation length, transmission bandwidth, absorption length, chromatic dispersion, optical workability (for solids), availability, and cost. Long radiator lengths are required to obtain sufficient numbers of photons when the momenta of the particle species to be separated are high. Recently, the gap in refractive index that has traditionally existed between gases and liquid or solid materials has been partially closed with transparent silica aerogels with indices that range between about 1.007 and 1.13. Cherenkov counters may be classified as either imaging or threshold types, depending on whether they do or do not make use of Cherenkov angle (θc ) information. Imaging counters may be used to track particles as well as identify them. 28.5.1. Threshold counters : Threshold Cherenkov detectors [50],

in their simplest form, make a yes/no decision based on whether the particle is above or below the Cherenkov threshold velocity βt = 1/n. A straightforward enhancement of such detectors uses the number of observed photoelectrons (or a calibrated pulse height) to discriminate between species or to set probabilities for each particle species [51]. This strategy can increase the momentum range of particle separation by a modest amount (to a momentum some 20% above the threshold momentum of the heavier particle in a typical case). Careful designs give coll    90%. For a photomultiplier with a typical bialkali cathode, det dE ≈ 0.27, so that Np.e. /L ≈ 90 cm−1 sin2 θc 

(i.e., N0 = 90 cm−1 ) .

(28.8)

Suppose, for example, that n is chosen so that the threshold for species a is pt ; that is, at this momentum species a has velocity βa = 1/n. A second, lighter, species b with the same momentum has velocity βb , so cos θc = βa /βb , and Np.e. /L ≈ 90 cm−1

m2a − m2b . p2t + m2a

(28.9)

For K/π separation at p = pt = 1(5) GeV/c, Np.e. /L ≈ 16(0.8) cm−1 for π’s and (by design) 0 for K’s. For limited path lengths Np.e. can be small, and a minimum number is required to trigger external electronics. The overall

289

efficiency of the device is controlled by Poisson fluctuations, which can be especially critical for separation of species where one particle type is dominant. The effective number of photoelectrons is often less than the average number calculated above due to additional equivalent noise from the photodetector. It is common to design for at least 10 photoelectrons for the high velocity particle in order to obtain a robust counter. As rejection of the particle that is below threshold depends on not seeing a signal, electronic and other background noise can be important. Physics sources of light production for the below threshold particle, such as decay of the above threshold particle or the production of delta rays in the radiator, often limit the separation attainable, and need to be carefully considered. Well designed, modern multi-channel counters, such as the ACC at Belle [47], can attain good particle separation performance over a substantial momentum range for essentially the full solid angle of the spectrometer. 28.5.2. Imaging counters : The most powerful use of the information available from the Cherenkov process comes from measuring the ring-correlated angles of emission of the individual Cherenkov photons. Since low-energy photon detectors can measure only the position (and, perhaps, a precise detection time) of the individual Cherenkov photons (not the angles directly), the photons must be “imaged” onto a detector so that their angles can be derived [52]. In most cases the optics map the Cherenkov cone onto (a portion of) a distorted circle at the photodetector. Though this imaging process is directly analogous to the familiar imaging techniques used in telescopes and other optical instruments, there is a somewhat bewildering variety of methods used in a wide variety of counter types with different names. Some of the imaging methods used include (1) focusing by a lens; (2) proximity focusing (i.e., focusing by limiting the emission region of the radiation); and (3) focusing through an aperture (a pinhole). In addition, the prompt Cherenkov emission coupled with the speed of modern photon detectors allows the use of time imaging, a method which is used much less frequently in conventional imaging technology. Finally, full tracking (and event reconstruction) can be performed in large water counters by combining the individual space position and time of each photon together with the constraint that Cherenkov photons are emitted from each track at a constant polar angle (Sec. 28.6 of this Review). In a simple model of an imaging PID counter, the fractional error on the particle velocity (δβ ) is given by

δβ = where

σβ = tan θc σ(θc ) , β

σ(θi ) σ(θc ) =  ⊕C , Np.e.

(28.10)

(28.11)

where σ(θi ) is the average single photoelectron resolution, as defined by the optics, detector resolution and the intrinsic chromaticity spread of the radiator index of refraction averaged over the photon detection bandwidth. C combines a number of other contributions to resolution including, (1) correlated terms such as tracking, alignment, and multiple scattering, (2) hit ambiguities, (3) background hits from random sources, and (4) hits coming from other tracks. In many practical cases, the resolution is limited by these effects. For a β ≈ 1 particle of momentum (p) well above threshold entering a radiator with index of refraction (n), the number of σ separation (Nσ ) between particles of mass m1 and m2 is approximately |m21 − m22 | √ Nσ ≈ . (28.12) 2p2 σ(θc ) n2 − 1

290

28. Particle detectors

In practical counters, the angular resolution term σ(θc ) varies between about 0.1 and 5 mrad depending on the size, radiator, and photodetector type of the particular counter. The range of momenta over which a particular counter can separate particle species extends from the point at which the number of photons emitted becomes sufficient for the counter to operate efficiently as a threshold device (∼20% above the threshold for the lighter species) to the value in the imaging region given by the equation above. For example, for σ(θc ) = 2mrad, a fused silica radiator(n = 1.474), or a flourocarbon gas radiator (C5 F12 , n = 1.0017), would separate π/K’s from the threshold region starting around 0.15(3) GeV/c through the imaging region up to about 4.2(18) GeV/c at better than 3σ. Many different imaging counters have been built during the last several decades [53]. Among the earliest examples of this class of counters are the very limited acceptance Differential Cherenkov detectors, designed for particle selection in high momentum beam lines. These devices use optical focusing and/or geometrical masking to select particles having velocities in a specified region. With careful design, a velocity resolution of σβ /β ≈ 10−4 –10−5 can be obtained [50]. Practical multi-track Ring-Imaging Cherenkov detectors (generically called RICH counters) are a more recent development. They have been built in small-aperture and 4π geometries both as PID counters and as stand-alone detectors with complete tracking and event reconstruction as discussed more fully below. PID RICH counters are sometimes further classified by ‘generations’ that differ based on performance, design, and photodetection techniques. A typical example of a first generation RICH used at the Z factory e+ e− colliders [54,55] has both liquid (C6 F14 , n = 1.276) and gas (C5 F12 , n = 1.0017) radiators, the former being proximity imaged using the small radiator thickness while the latter use mirrors. The phototransducers are a TPC/wire-chamber combination having charge division or pads. They are made sensitive to photons by doping the TPC gas (usually, ethane/methane) with ∼ 0.05% TMAE (tetrakis(dimethylamino)ethylene). Great attention to detail is required, (1) to avoid absorbing the UV photons to which TMAE is sensitive, (2) to avoid absorbing the single photoelectrons as they drift in the long TPC, and (3) to keep the chemically active TMAE vapor from interacting with materials in the system. In spite of their unforgiving operational characteristics, these counters attained good e/π/K/p separation over wide momentum ranges during several years of operation. In particular, their π/K separation range extends over momenta from about 0.25 to 20 GeV/c. Later generation counters [53] generally must operate at much higher particle rates than the first generation detectors, and utilize different photon detection bandwidths, with higher readout channel counts, and faster, more forgiving photon detection technology than the TMAE doped TPC’s just described. Radiator choices have broadened to include materials such as lithium flouride, fused silica, and aerogel. Vacuum based photodetection systems (e.g., single or multi anode photomultiplier tubes (PMT), multi channel plate PMTs (MCPPMT), or hybrid photodiodes (HPD)) have become increasingly common. They handle very high rates, may be used with a wide choice of radiators, and may be sufficiently fast to allow time imaging or the use of time of flight information. Other fast detection systems that use solid cesium iodide (CSI) photocathodes or triethylamine (TEA) doping in proportional chambers are useful with certain radiator types and geometries. A DIRC (Detector of Internally Reflected Cherenkov light) is a third generation subtype of a RICH first used in the BaBar detector [48]. It “inverts” the usual principle for use of light

from the radiator of a RICH by collecting and imaging the total internally reflected light, rather than the transmitted light. A DIRC utilizes the optical material of the radiator in two ways, simultaneously; first as a Cherenkov radiator, and second, as a light pipe for the Cherenkov light trapped in the radiator by total internal reflection. The DIRC makes use of the fact that the magnitudes of angles are preserved during reflection from a flat surface. This fact, coupled with the high reflection coefficients of the total internal reflection process (> 0.9995 for higly polished SiO2 ), and the long attenuation length for photons in high purity fused silica, allows the photons of the ring image to be transported to a detector outside the path of the particle where they may be imaged in up to three dimensions (two in space and one in time). The BaBar DIRC uses 144 fused silica radiator bars (1.7 × 3.5 × 490 cm) with the light being focused onto 11 000 conventional PMT’s located about 120 cm from the end of the bars by the “pinhole” of the bar end. DIRC performance can be understood using the formula for (Nσ ) discussed above. Typically, Np.e. is rather large (between 15 and 60) and the Cherenkov polar angle is measured to about 2.5 mrad. The momentum range with good π/K separation extends up to about 4 GeV/c, matching the B decay momentum spectrum observed in BaBar.

28.6. Cherenkov tracking calorimeters Written August 2003 by D. Casper (UC Irvine). In addition to the specialized applications described in the previous section, Cherenkov radiation is also exploited in large, ring-imaging detectors with masses measured in kilotons or greater. Such devices are not subdetector components, but complete experiments with triggering, tracking, vertexing, particle identification and calorimetric capabilities, where the large mass of the transparent dielectric medium serves as an active target for neutrino interactions (or their secondary muons) and rare processes like nucleon decay. For volumes of this scale, absorption and scattering of Cherenkov light are non-negligible, and a wavelength-dependent factor e−d/L(λ) (where d is the distance from emission to the sensor and L(λ) is the attenuation length of the medium) must be included in the integral of Eq. (28.5) for the photoelectron yield. The choice of medium is therefore constrained by the refractive index and transparency in the region of photodetector sensitivity; highly-purified water is an inexpensive and effective choice; sea-water, mineral oil, polar ice, and D2 O are also used. Photo-multiplier tubes (PMTs) on either a volume or surface lattice measure the time of arrival and intensity of Cherenkov radiation. Hemispherical PMTs are favored for the widest angular acceptance, and sometimes mounted with reflectors or wavelength-shifting plates to increase the effective photosensitive area. Gains and calibration curves are measured with pulsed laser signals transmitted to each PMT individually via optical fiber or applied to the detector as a whole through one or more diffusing balls. Volume instrumentation [56] is only cost-effective at low densities, with a spacing comparable to the attenuation (absorption and scattering) length of Cherenkov light in the medium (15–40 m for Antarctic ice and ∼45 m in the deep ocean). PMTs are deployed in vertical strings as modular units which include pressure housings, front-end electronics and calibration hardware. The effective photocathode coverage of such arrays is less than 1% but still adequate (using timing information and the Cherenkov angular constraint) to reconstruct the direction of TeV muons to 1◦ or better. The size of such “neutrino telescopes” is limited only by cost once the technical challenges of deployment, power, signal extraction and calibration in an inaccessible and inhospitable environment

28. Particle detectors

are addressed; arrays up to (1 km)3 in size are under study or development. Surface instrumentation [57] allows the target volume to be viewed with higher photocathode density by a number of PMTs which scales like (volume)2/3 . To improve hermeticity and shielding, and to ensure that an outward-going particle’s Cherenkov cone illuminates sufficient PMTs for reconstruction, a software-defined fiducial volume begins some distance (∼ 2 m) inside the photosensor surface. Events originating within the fiducial volume are classified as fully-contained if no particles exit the inner detector, or partially-contained otherwise. An outer (veto) detector, optically separated from the inner volume and instrumented at reduced density, greatly assists in making this determination and also simplifies the selection of contained events. The maximum size of a pure surface array is limited by the attenuation length (∼ 100 m has been achieved for large volumes using reverse-osmosis water purification), pressure tolerance of the PMTs (< 80 meters of water, without pressure housings) and structural integrity of the enclosing cavity, if underground. In practice, these limitations can be overcome by a segmented design involving multiple modules of the nominal maximum size; megaton-scale devices are under study. Cherenkov detectors are excellent electromagnetic calorimeters, and the number of Cherenkov photons produced by an e/γ is nearly proportional to its kinetic energy. For massive particles, the number of photons produced is also related to the energy, but not linearly. For any type of particle, the visible energy Evis is defined as the energy of an electron which would produce the same number of Cherenkov photons. The number of photoelectrons collected depends on a detector-specific scale factor, with event-by-event corrections for geometry and attenuation. For typical PMTs, in water Np.e. ≈ 15 ξ Evis (MeV), where ξ is the effective fractional photosensor coverage; for other materials, the photoelectron yield scales with the ratio of sin2 θc over density. At  solar neutrino energies, the visible energy resolution (∼ 30%/ ξ Evis (MeV)) is about 20% worse than photoelectron counting statistics would imply. For higher energies, multi-photoelectron hits are likely and the charge collected by each PMT (rather the number of PMTs firing) must be used;  this degrades the energy resolution to approximately 2%/ ξ Evis (GeV). In addition, the absolute energy scale must be determined with sources of known energy. Using an electron LINAC and/or nuclear sources, 0.5–1.5% has been achieved at solar neutrino energies; for higher energies, cosmic-ray muons, Michel electrons and π 0 from neutrino interactions allow ∼ 3% absolute energy calibration. A trigger can be formed by the coincidence of PMTs within a window comparable to the detector’s light crossing time; the coincidence level thus corresponds to a visible energy threshold. Physics analysis is usually not limited by the hardware trigger, but rather the ability to reconstruct events. The interaction vertex can be estimated using timing and refined by applying the Cherenkov angle constraint to identified ring edges. Multi-ring events are more strongly constrained, and their vertex resolution is 33–50% better than single rings. Vertex resolution depends on the photosensor density and detector size, with smaller detectors performing somewhat better than large ones (∼ 25 cm is typical for existing devices). Angular resolution is limited by multiple scattering at solar neutrino energies (25–30◦ ) and improves to a few degrees around Evis = 1 GeV. A non-showering (μ, π ± , p) track produces a sharp ring with small contributions from delta rays and other radiated secondaries, while the more diffuse pattern of a showering (e, γ) particle is actually the superposition of many individual rings from charged shower products. Using maximum likelihood techniques and the Cherenkov angle constraint, these two

291

topologies can be distinguished with an efficiency which depends on the photosensor density and detector size [58]. This particle identification capability has been confirmed by using cosmicrays and Michel electrons, as well as charged-particle [59] and neutrino [60] beams. Large detectors perform somewhat better than smaller ones with identical photocathode coverage; a misidentification probability of ∼ 0.4%/ξ in the sub-GeV range is consistent with the performance of several experiments for 4% < ξ < 40%. Detection of a delayed coincidence from muon decay offers another, more indirect, means of particle identification; with suitable electronics, efficiency approaches 100% for μ+ decays but is limited by nuclear absorption (22% probability in water) for μ− . Reconstruction of multiple Cherenkov rings presents a challenging pattern recognition problem, which must be attacked by some combination of heuristics, maximum likelihood fitting, Hough transforms and/or neural networks. The problem itself is somewhat ill-defined since, as noted, even a single showering primary produces many closely-overlapping rings. For π 0 → γγ two-ring identification, performance falls off rapidly with increasing π 0 momentum, and selection criteria must be optimized with respect to the analysis-dependent cost-function for e ↔ π 0 mis-identification. Two representative cases for ξ = 39% will be illustrated. In an atmospheric neutrino experiment, where π 0 are relatively rare compared to e± , one can isolate a > 90% pure 500 MeV/c π 0 sample with an efficiency of ∼ 40%. In a νe appearance experiment at Eν ≤ 1 GeV, where e± are rare compared to π 0 , a 99% pure 500 MeV/c electron sample can be identified with an efficiency of ∼ 70%. For constant ξ, a larger detector (with, perforce, a greater number of pixels to sample the light distribution) performs somewhat better at multi-ring separation than a smaller one. For a more detailed discussion of event reconstruction techniques, see Ref. 49. Table 28.5: Properties of Cherenkov tracking calorimeters. LSND was a hybrid scintillation/Cherenkov detector; the estimated ratio of isotropic to Cherenkov photoelectrons was about 5:1. MiniBooNE’s light yield also includes a small scintillation component. Detector

Fiducial mass (kton)

PMTs (diameter, cm)

ξ

p.e./ Dates MeV

IMB-1 IMB-3 KAM I KAM II LSND SK-1 SK-2 K2K SNO MiniBooNE

3.3 H2 O 3.3 H2 O 0.88/0.78 H2 O 1.04 H2 O 0.084 oil+scint. 22.5 H2 O 22.5 H2 O 0.025 H2 O 1.0 D2 O 0.445 oil

2048 (12.5) 2048 (20 +plate) 1000/948 (50) 948 (50) 1220 (20) 11146 (50) 5182 (50) 680 (50) 9456 (20+cone) 1280 (20)

1% 4.5% 20% 20% 25% 39% 18% 39% 55% 10%

0.25 1.1 3.4 3.4 33 6 3 6 9 3–4

1982–85 1987–90 1983–85 1986–90 1993–98 1997–01 2002– 1999– 1999– 2002–

292

28. Particle detectors

28.7. Gaseous detectors 28.7.1. Energy loss and charge transport in gases : Written April 2008 by F. Sauli (CERN) and M. Titov (CEA Saclay). Gas-filled counters detect and localize the ionization produced by charged particles, generally after charge multiplication. The peculiar statistics of ionization processes, with asymmetries in the ionization trails, affect the coordinate determination deduced from the measurement of drift time or of the center of gravity of the collected charge. For thin gas layers, the width of the energy loss distribution can be larger than the average, requiring multiple sample or truncated mean analysis to achieve particle identification (see Sec. 28.7.4). The energy loss of charged particles and photons in various materials as a function of energy is discussed in Sec. 27. Table 28.6 provides values of relevant parameters in commonly used gases at NTP for unit charge, minimum-ionizing particles [61–67]. Numbers often differ depending on source, and values in the table should be taken only as approximate. For different conditions and mixtures, and neglecting internal energy transfer processes, one can use gas-density-dependent composition rules.

Table 28.6: Properties of rare and molecular gases at normal temperature and pressure (NTP: 20◦ C, one atm). EX , EI : first excitation and ionization energy; WI : average energy per ion pair; dE/dx|min , NP , NT : differential energy loss, primary and total number of electron-ion pairs per cm, for unit charge, minimum ionizing particles. Gas Ne Ar Xe CH4 C2 H6 iC4 H10 CO2 CF4

Density, mg cm−3

Ex eV

EI eV

WI dE/dx|min eV keV cm−1

0.839 1.66 5.495 0.667 1.26 2.49 1.84 3.78

16.7 11.6 8.4 8.8 8.2 6.5 7.0 10.0

21.6 15.7 12.1 12.6 11.5 10.6 13.8 16.0

30 25 22 30 26 26 34 54

1.45 2.53 6.87 1.61 2.91 5.67 3.35 6.38

NP NT cm−1 cm−1 13 25 41 37 48 90 35 63

50 106 312 54 112 220 100 120

When an ionizing particle passes through the gas it creates electron-ion pairs, but often the ejected electrons have sufficient energy to further ionize the medium. On average, the total number of electron-ion pairs (NT ) is between two and three times larger than the number of primaries (NP ) (see Table 28.6). The probability of releasing an electron of energy E or larger follows an approximate 1/E 2 dependence (Rutherford law), shown in Fig. 28.3 for argon at NTP (dotted line, left scale). More detailed estimates taking into account the electronic structure of the medium are shown in the figure, for three values of the particle velocity factor βγ [62]. The dot-dashed line provides, on the right scale, the practical range of electrons of energy E. As an example, there is a 1% probability of creating of an electron of 1 keV or more in 10 mm of argon, substantially increasing the ionization losses. The practical range of 1 keV electrons in argon (dot-dashed line, right scale) is 70 μm; this contributes to the error in the coordinate determination. The number of electron-ion pairs per primary interaction, or cluster size, has an exponentially decreasing probability; for argon, there is about 1% probability for primary clusters to contain ten or more electron-ion pairs [63]. Once released in the gas, and under the influence of an applied electric field, electrons and ions drift in opposite directions and diffuse towards the electrodes. The value of drift velocity

Figure 28.3: Probability of production of an electron of energy equal or larger than E (left scale), and range of electrons in argon at NTP (dot-dashed curve, right scale) [62]. and diffusion depends very strongly on the nature of the gas, namely on the detailed structure of the elastic and inelastic electron-molecule cross-sections. In noble gases, the inelastic cross section is zero below excitation and ionization thresholds. Fast gas mixtures are achieved by adding polyatomic gases (usually CH4 or CO2 ), having large inelastic cross sections at moderate energies, which results in “cooling” electrons into an energy range of the Ramsauer-Townsend minimum (located at ∼0.5 eV) of the elastic scattering cross-section of argon. The reduction in both the total electron scattering cross-section and the electron energy results in a large increase of electron drift velocity (for a compilation of electron-molecule cross sections see Ref. 64). Another principal role of the polyatomic gas is to absorb the ultraviolet (UV) photons emitted by the excited inert gas atoms. The quenching of UV photons occurs through the photo-decomposition of polyatomic molecules. Extensive collections of experimental data [65] and theoretical calculations based on transport theory [66] permit estimates of drift and diffusion properties in pure gases and their mixtures. In a simple approximation, gas kinetic theory provides the following relation between drift velocity v and mean electron-molecule collision time τ (Townsend’s expression): v = eEτ /m. Examples for commonly used gases at NTP are given in Fig. 28.4 and Fig. 28.5, computed with the program MAGBOLTZ [67] as a function of electric field. For different conditions, the horizontal axis has to be scaled with the gas density, proportional to 1/P , where P is the pressure. Standard deviations for longitudinal (σL ) and transverse diffusion (σT ) are given for one cm of drift, and scale with the square root of distance. Since the collection time is inversely related to the drift velocity, diffusion is smaller in fast-counting gases, as can be seen, for example, in CF4 . In presence of an external magnetic field, the Lorentz force acting on electrons between collisions deflects the drifting swarm and modifies the drift properties. The electron trajectories, velocities and diffusion parameters can be computed with the program mentioned above. A simple theory (the friction force model) provides an expression for the vector drift velocity v as a function of electric and magnetic field vectors E and B, of the Larmor frequency ω = eB/m and the mean collision time τ between electrons and molecules: 

e ωτ τ ω2τ 2 v= (E · B)B (28.13) E + (E × B) + m 1 + ω2τ 2 B B2 To a good approximation, and for moderate fields, one can assume that the energy of the electrons is not affected by

28. Particle detectors

293

B, and use for τ the values deduced from the drift velocity at B = 0 (Townsend expression). For E perpendicular to B, the drift angle to the √electric field vector is tan θB = ωτ and |v | = (E/B)(ωτ / 1 + ω 2 τ 2 ). For parallel electric and magnetic fields, drift velocity and longitudinal diffusion are not affected, while the transverse diffusion can be strongly reduced: √ σT (B) = σT (B = 0)/ 1 + ω 2 τ 2 . For example, the large values of ωτ ∼ 20 at 5 T have been measured for Ar/CF4 /iC4 H10 (95:3:2). The dotted line in Fig. 28.5 represents σT for the classic P10 mixture Ar/CH4 (90:10) at 4 T. This reduction is exploited in time projection chambers (Sec. 28.7.4) to improve localization accuracy.

Figure 28.5: Electron longitudinal diffusion (σL ) (dashed lines) and transverse diffusion (σT ) (full lines) for 1 cm of drift. The dotted line shows σT for the P10 mixture at 4T [67].

Figure 28.4: Computed electron drift velocity as a function of electric field in several gases at NTP [67]. In mixtures containing electronegative molecules such as oxygen, water or CF4 , electrons can be captured to form negative ions. Capture cross-sections are strongly energy-dependent, and therefore the capture probability is a function of applied field and may differ to a great extent for the same amount of additions other mixtures. As an example, at moderate fields (up to 1 kV/cm) the addition of 0.1% of oxygen to an Ar/CO2 mixture results in an electron capture probability about twenty times larger than the same addition to Ar/CH4 . As they experience increasing fields, electrons get enough energy to undergo ionizing collisions with molecules. Above a gas-dependent threshold, the mean free path for ionization λi , decreases exponentially with the field; its inverse, α = 1/λi , is the first Townsend coefficient. As most of the avalanche growth occurs very close to the anodes, simple electrostatic consideration show that the largest fraction of the detected signal is due to the motion of ions receding from the wires; the electron component, although very fast, is rarely seen. This determines the characteristic shape of the detected signals in proportional mode, with a fast rise followed by a gradually slower increase. The so-called ion tail that limits the time resolution of the counter is usually removed by differentiation of the signal. In uniform fields, N0 initial electrons multiply over a length x forming an electron avalanche of size N = N0 eαx ; N/N0 is the gain of the counter. Fig. 28.6 shows examples of Townsend coefficients for several gas mixtures, computed with MAGBOLTZ. Ions released by ionization or produced in the avalanches drift and diffuse under the influence of the electric field; their drift velocity in the fields encountered in gaseous counters (up to few kV/cm) is typically three orders of magnitude lower than for electrons. The ion mobility μ, the ratio of drift velocity to electric field, is constant for a given ion up to very high fields. For a different temperature and pressure, the mobility can be computed from the expression μ(P, T ) = μ(P0 , T0 )(T /P )(P0 /T0 ).

Figure 28.6: Computed first Townsend coefficient α as a function of electric field in several gases at NTP [67]. Values of mobility at NTP (μ0 ) for ions in their own and other gases are given in Table 28.7 [68]. For mixtures, due to a very effective charge transfer mechanism, only ions with the lowest ionization potential survive after a short path in the gas. The diffusion of ions follows a simple square root dependence on the drift time, with a coefficient that depends on temperature but not on the ion mass. Accumulation of ions in the gas counters may induce gain reduction and field distortions. Table 28.7: Mobility of ions in gases at NTP [68]. Gas

Ion

Mobility (cm2 V−1 s−1 )

He Ne Ar Ar Ar CH4 CO2

He+ Ne+ Ar+ CH+ 4 CO+ 2 CH+ 4 CO+ 2

10.4 4.7 1.54 1.87 1.72 2.26 1.09

28. Particle detectors

E(r) =

CV0 1 ; 2π0 r

ED =

CV0 ; 20 s

C=

2π0 π(/s) − ln(2πa/s)

0.20 0.15

(a) Multiwire proportional chamber

0.10

y-axis [cm]

28.7.2. Multi-Wire Proportional Chambers : Written April 2008 by Fabio Sauli (CERN) and Maxim Titov (CEA Saclay). Single-wire counters that detect the ionization produced in a gas by charged particle energy deposit, followed by charge multiplication and collection around a thin wire, have been used for decades. Good energy resolution is obtained in the streamer mode, and very large saturated pulses can be detected in the Geiger mode. For an overview see e.g. Ref. 4. Multiwire proportional chambers (MWPC’s) [69,70], introduced in the late sixties, detect and localize energy deposit by charged particles over large areas. A mesh of parallel anode wires at a suitable potential, inserted between two cathodes, acts (almost) as an independent set of proportional counters (see Fig. 28.7). Electrons released in the gas volume drift towards the anodes and avalanche in the increasing field. Analytical expressions for the electric field can be found in many textbooks; the fields close to the wires E(r) and in the drift region ED are given by the approximations:

0.05 0.00 −0.05 −0.10 −0.15

0.15

(b) Drift chamber

0.10

y-axis [cm]

294

0.05 0.00 −0.05 −0.10

where r is the distance from the center of the anode, s the wire spacing,  and V0 the distance and difference of potential between anode and cathodes. C the capacitance per unit length of the wires and a the anode wire radius. Because of electrostatic forces, anode wires are in equilibrium only for a perfect geometry. Small deviations result in forces displacing the wires alternatively below and above the symmetry plane, often with catastrophic results. These displacement forces are countered by the mechanical tension of the wire, up to a maximum stable length LM [71]: LM =

−0.15 −0.20 −0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

x-axis [cm]

Figure 28.7: Electric field lines and equipotentials in (a) a multiwire proportional chamber and (b) a drift chamber.

s  4π0 TM CV0

The maximum tension TM depends on the wire diameter and modulus of elasticity. Table 28.8 gives approximate values for tungsten, and the corresponding maximum stable wire length under reasonable assumptions for the operating voltage V0 [72]. Internal supports and spacers have been used in the construction of longer detectors. Table 28.8: Maximum tension TM and stable length LM for tungsten wires with spacing s. Wire diameter (μm) TM (newton) s (mm) LM (cm) 10 20

0.16 0.65

1 2

25 85

Detection of charge over a predefined threshold on the wires provides the event coordinates with the accuracy of the wire spacing; longitudinal localization can be obtained by measuring the ratio of collected charge at the two ends of resistive wires. Making use of the charge profile induced on segmented cathodes, the so-called center-of gravity (COG) method permits localization of tracks to sub-mm accuracy. Due to the statistics of energy loss and asymmetric ionization clusters, the position accuracy is 50 μm rms for tracks perpendicular to the wire plane, but degrades to ∼250 μm at 30◦ to the normal [73]. The intrinsic bi-dimensional characteristic of the COG readout has found numerous applications in medical imaging. Drift chambers, developed in the early ’70’s, can estimate the position of a track by exploiting the arrival time of electrons at the anodes if the time of the interaction is known. They

Figure 28.8: Electric field lines and equipotentials in a multiwire drift module. Each anode wire is surrounded by six cathode wires, and each cathode wire is surrounded by three anode wires.

Field-shaping electrodes Equipotential planes

α

Figure 28.9: Schematics drawing of a JET Chamber sector. can achieve rms localization accuracies of 50 μm or better. The

28. Particle detectors

distance between anode wires, several cm, allows covering large areas at reduced cost. Many designs have been introduced, all aimed at improving performance. In the original design, a thicker wire at proper voltage between anodes (field wire) reduces the field at the middle point between anodes, improving charge collection (Fig. 28.8) [74]. Symmetrically decreasing potentials applied on cathode wires make more uniform and reinforce the drift fields, resulting in a linear space-to-drift-time relation [75]. In some drift chambers design, and with the help of suitable voltages applied to field-shaping electrodes, the electric field structure is adjusted to compensate the distortions. Sampling the drift time on rows of anodes within the same gas volume, together with longitudinal localization through charge division on wires led to the concept of multiple arrays such as the multi-drift module [76]( Fig. 28.8) and the JET chamber [77]( Fig. 28.9). The time projection chamber [78] combines a measurement of drift time and charge induction on cathodes to obtain excellent tracking for high multiplicity topologies occurring at moderate rates (see Sec. 28.7.4). In all cases, a good knowledge of electron drift velocity and diffusion properties is required. This has to be combined with the knowledge of the electric fields in the structures, computed with commercial or custom-developed software [67,79]. For an overview of detectors exploiting the drift time for coordinate measurement see Refs. 2 and 71. Although very powerful in terms of performance, multiwire structures have reliability problems when used in harsh or hard-to access environments, as a single broken wire can affect the whole detector. Introduced in the eighties, Straw and drift tube systems make use of large arrays of wire counters encased in individual envelopes, each acting independently [80]. Suitable techniques for low-cost mass production have been developed for the needs of large experiments, as the Transition Radiation Tracker and the Monitored Drift Tubes array for CERN’s LHC experiments (for a review see Ref. 81). The production of positive ions in the avalanches and their slow drift before neutralization result in a rate-dependent accumulation of positive charge in the detector, with consequent field distortion, gain reduction and degradation of spatial resolution. As shown in Fig. 28.10 [82], independently from the avalanche size, the proportional gain drops above a charge production rate around 109 electrons per second and mm of wire; for a proportional gain of 104 and 100 electrons released per track, this corresponds to a particle flux of 103 s−1 mm−1 (1 kHz/mm2 for 1 mm wire spacing). Despite various improvements, positionsensitive detectors based on wire structures are limited by basic diffusion processes and space charge effects in the gas to localization accuracies of 50–100 μm (see for example Ref. 83).

Figure 28.10: Charge rate dependence of normalized gain in a MWPC [82].

295

Multiwire and drift chambers have been operated with a variety of gas fillings and operating modes, depending on experimental requirements. The so-called “Magic Gas” (a mixture of argon, isobutane and freon) [70] permits very high and saturated gains (∼ 106 ), overcoming the electronics limitations of the time, at the cost of reduced survival due to aging processes. With present-day electronics, proportional gains around 104 are sufficient for detection of minimum ionizing particles, and noble gases with moderate amounts of polyatomic gases, such as methane or carbon dioxide, are used. At high radiation fluxes, a fast degradation of detectors due to the formation of polymers deposits (aging) is often observed. The process has been extensively investigated, often with conflicting results; a number of culprits has been identified (organic pollutants, silicone oils); addition of small amounts of water in many (but not all) cases has been demonstrated to extends the lifetime of the detectors. Addition of fluorinated gases (e.g., CF4 ) or oxygen results in an etching action that can overcome polymer formation, or even eliminate deposits, but the issue of long-term survival of gas detectors in these gases is controversial [84]. Under optimum operating conditions, a total collected charge of a few coulombs per cm of wire can be reached before degradation, corresponding, for one mm spacing and at a gain of 104 , to a total particle flux of ∼ 1014 MIP’s/cm2 . A new generation of wireless gaseous detectors, micro-pattern gas detectors, has been developed in the recent years and promises to overcome many of the limitations of MWPC devices—namely the multi-particle resolution and the rate capability. Their operating principles and performances are described in Sec. 28.7.3. 28.7.3.

Micro-pattern Gas Detectors : Written October 2007

by M. Titov (CEA Saclay) Modern photolithographic technology has enabled a series of inventions of novel Micro-Pattern Gas Detector (MPGD) concepts: Micro-Strip Gas Chamber (MSGC) [85], GEM [86], Micromegas [87] and many others [88], revolutionizing cell size limits for many gas detector applications. The MSGC, a concept invented in 1988 by A. Oed, was the first of the micro-structure gas detectors. Consisting of a set of tiny metal strips laid on a thin insulating substrate, and alternatively connected as anodes and cathodes, the MSGC turned out to be easily damaged by discharges induced by heavily ionizing particles and destroying the fragile electrode structure [89]. The more powerful GEM and Micromegas concepts fulfill the needs of high-luminosity colliders with increased reliability in harsh radiation environments. By using fine pitch size compared to classical wire chambers, these detectors offer intrinsic high rate capability (fast signals with risetimes of a few ns and full widths of 20–100 ns), excellent spatial resolution (∼ 30 μm), double track resolution (∼ 500 μm), and single photo-electron time resolution in the ns range. The GEM detector was introduced by Fabio Sauli. It consists of a thin-foil copper-Kapton-copper sandwich chemically perforated to obtain a high density of holes. The hole diameter is typically between 25 μm and 150 μm, while the pitch varies between 50 μm and 200 μm. Application of a potential difference between the two sides of the GEM generates the electric fields indicated in Fig. 28.11. Each hole acts as an independent proportional counter; electrons released by the ionization in the gas drift into the hole and multiply in the high electric field (50–70 kV/cm). Most of avalanche electrons are transferred into the gap below the GEM. Distributing the avalanche multiplication among several cascading electrodes allows the multi-GEM detectors to operate at overall gas gain above 104 in the presence of highly ionizing particles, while eliminating

296

28. Particle detectors

50 μm

140 μm

Figure 28.11: Schematic view and typical dimensions of the hole structure in the GEM amplification cell. Electric field lines (solid) and equipotentials (dashed) are shown. On application of a potential difference between the two metal layers electrons released by ionization in the gas volume above the GEM are guided into the holes, where charge multiplication occurs in the high field. the risk of discharges (< 10−12 per hadron). This is the major advantage of the GEM technology [90]. A unique property of the GEM detector is the complete decoupling of the amplification stage (GEM) and the readout electrode (PCB), which operates at unity gain and serves only as a charge collector.

100 μm

HV1

Drift gap

40 kV/cm

e− Micromesh Anode plane

Amplification gap

HV2

E2 Particle

Figure 28.12: Schematic drawing and typical dimensions of the Micromegas detector. Charges produced in the drift gap are drifting to the small amplification region, limited by the mesh and the anode, where they are amplified. Ioannis Giomataris introduced the micro-mesh gaseous structure (Micromegas), which is a parallel-plate avalanche counter (Fig. 28.12). It consists of a few mm drift region (electric field ∼1 kV/cm) and a narrow multiplication gap (25-150 μm, 50–70 kV/cm), located between a thin metal grid (micromesh) and the readout electrode (strips/pads of conductor printed on an insulator board). The electric field is homogeneous both in the drift and amplification gaps. Due to the narrow multiplication region in Micromegas, locally small variations of the amplification gap are compensated by an inverse variation of the amplification coefficient and therefore do not induce gain fluctuations. The small amplification gap is a key element in Micromegas operation, giving rise to its excellent spatial resolution: 12 μm accuracy (limited by the the micromesh pitch) for MIPs [91], and very good energy resolution (∼12% FWHM with 6 keV x rays). Over the past decade GEM and Micromegas detectors have become increasingly important. COMPASS is a first highluminosity experiment at CERN which pioneered the use of large-area (∼ 40 × 40 cm2 ) GEM and Micromegas detectors for high-rate particle tracking, reaching 25 kHz/mm2 in the near-beam area. Both technologies have achieved a tracking

efficiency of close to 100% at gas gains of about 104 , a spatial resolution of 70–100 μm and a time resolution of ∼ 10 ns. GEM’s have entered the LHC program; they will be used for triggering in the LHCb Muon System and in the TOTEM Telescopes. A time projection chamber (TPC) using GEM or Micromegas as a gas amplification device is also one of the main options for high-precision tracking at the International Linear Collider (ILC). The performance and robustness of MPGD’s have encouraged their applications in high-energy and neutrino physics, astrophysics, UV and visible photon detection, nuclear physics and neutron detection domain, radiation therapy and electronic portal imaging devices. A big step in the direction of the industrial manufacturing of large-size MPGD’s is the development of the “bulk” Micromegas technology [92]. The basic idea is to build the whole detector in a single process: the anode plane with copper strips, a photo-imageable polyimide film and the woven mesh are laminated together at a high temperature forming a single object. Employing the “bulk” technology, 72 large Micromegas (34 × 36 cm2 ) planes with an active area of 9 m2 are being built for the T2K TPC detector. Sensitive and low-noise electronics will enlarge the range of the MPGD applications. Recently, GEM and Micromegas were read out by high-granularity (50 μm pitch) CMOS pixel chips assembled directly below the GEM or Micromegas amplification structure and serving as an integrated charge collecting anode [93,94,95]. With this arrangement avalanche electrons are collected on the top metal layer of the CMOS ASIC; every input pixel is then directly connected to the amplification, digitization and sparsification circuits integrated in the underlying active layers of the CMOS technology. GEM coupled to a VLSI pixel array could serve as a highly efficient x-ray polarimeter, which is able to reconstruct simultaneously initial direction and dynamics of photoelectron energy loss for the low energy (<10 keV) x rays. A fine-pitch GEM matching the pitch of pixel ASIC (50 μm) allows to achieve a superior single-electron avalanche reconstruction accuracy of 4 μm, which makes it suitable candidate for fast gas photo-multipliers. For minimum ionizing particle tracks a spatial resolution down to 20 μm was achieved with Medipix2 and Timepix CMOS chips coupled to GEM devices. Similar performance is expected for Micromegas. An attractive solution for the construction of MPGD’s with pixel anode readout is the integration of the Micromegas amplification and CMOS chip by means of the “wafer post-processing” technology. The sub-μm precision of the grid dimensions and avalanche gap size results in a uniform gas gain; the grid hole size, pitch and pattern can be easily adapted to match the geometry of any pixel readout chip. Recent developments in radiation hardness research with stateof-the-art MPGD’s are reviewed in Ref. 96. Better properties of MPGD’s, which are rather insensitive to aging compared to wire chambers, can be explained by the separation of multiplication (GEM or parallel plate Micromegas amplification) and anode readout structures, and the lower electric field strength (∼ 50 kV/cm) in the multiplication region, compared to the anode wire surface field (∼ 250 kV/cm).

28. Particle detectors

28.7.4. Time-projection chambers : Written September 2007 by D. Karlen (U. of Victoria and TRIUMF, Canada)

The Time Projection Chamber (TPC) concept, invented by David Nygren in the late 1970’s [78], is the basis for charged particle tracking in a large number of particle and nuclear physics experiments. A uniform electric field drifts tracks of ions produced by charged particles traversing a medium, either gas or liquid, towards a surface segmented into 2D readout pads. The signal amplitudes and arrival times are recorded to provide full 3D measurements of the particle trajectories. The intrinsic 3D segmentation gives the TPC a distinct advantage over other large volume tracking detector designs which record information only in a 2D projection with less overall segmentation, particularly for pattern recognition in events with large numbers of particles. Gaseous TPC’s are often designed to operate within a strong magnetic field (typically parallel to the drift field) so that particle momenta can be estimated from the track curvature. For this application, precise spatial measurements in the plane transverse to the magnetic field are most important. Since the amount of ionization along the length of the track depends on the velocity of the particle, ionization and momentum measurements can be combined to identify the types of particles observed in the TPC. The estimator for the energy deposit by a particle is usually formed as the truncated mean of the energy deposits, using the 50%–70% of the samples with the smallest signals. Variance due to energetic δ-ray production is thus reduced. Gas amplification of 103 –104 at the readout endplate is usually required in order to provide signals with sufficient amplitude for conventional electronics to sense the drifted ionization. Until recently, the gas amplification system used in TPC’s have exclusively been planes of anode wires operated in proportional mode placed close to the readout pads. Performance has been recently improved by replacing these wire planes with micropattern gas detectors, namely GEM [86] and Micromegas [87] devices. Advances in electronics miniaturization have been important in this development, allowing pad areas to be reduced to the 10 mm2 scale or less, well matched to the narrow extent of signals produced with micro-pattern gas detectors. Presently, the ultimate in fine segmentation TPC readout are silicon sensors, with 0.05 mm × 0.05 mm pixels, in combination with GEM or Micromegas [97]. With such fine granularity it is possible to count the number of ionization clusters along the length of a track which, in principle, can improve the particle identification capability. Examples of two modern large volume gaseous TPC’s are shown in Fig. 28.13 and Fig. 28.14. The particle identification performance is illustrated in Fig. 28.15, for the original TPC in the PEP-4/9 experiment [98]. The greatest challenges for a large TPC arise from the long drift distance, typically 100 times further than in a comparable wire chamber design. In particular, the long drift distance can make the device sensitive to small distortions in the electric field. Distortions can arise from a number of sources, such as imperfections in the TPC construction, deformations of the readout surface, or the presence of ions in the active medium. For a gaseous TPC operated in a magnetic field, the electron drift velocity v is defined by Eq. (28.13). With a strong magnetic field parallel to the electric field and a gas with a large value of ωτ (also favored to reduce transverse diffusion as discussed below), the transverse displacements of the drifting electrons due to electric field distortions are reduced. In this mode of operation, it is essential to precisely map the magnetic field as the electron drift lines closely follow the magnetic field lines. Corrections for electric and/or magnetic field non-uniformities can be determined from control samples of electrons produced by ionizing the gas with UV laser beams, from photoelectrons

297

Outer containment volume Central electrode Inner containment volume

End-plate

Figure 28.13: The ALICE TPC shown in a cutaway view. The drift volume is 5 m long with a 5 m diameter. Gas amplification is provided by planes of anode wires.

Outer wall E, B directions

Inner wall and field cage

Beam direction Front end cards Central cathode

Central cathode HV

Figure 28.14: One of the 3 TPC modules for the near detector of the T2K experiment. The drift volume is 2 m×2 m×0.8 m. Micromegas devices are used for gas amplification and readout. produced on the cathode, or from tracks emanating from calibration reactions. The long drift distance means that there is a delay, typically 10–100 μs in a large gaseous TPC, for signals to arrive at the endplate. For experiments with shorter intervals between events, this can produce ambiguities in the starting time for the drift of ionization. This can be resolved by matching the TPC data with that from an auxiliary detector providing additional spatial or timing information. In a gaseous TPC, the motion of positive ions is much slower than the electrons, and so the positive ions produced by many events may exist in the active volume. Of greatest concern is the ions produced in the gas amplification stage. Large gaseous TPC’s built until now with wire planes have included a gating grid that prevent the positive ions from escaping into the drift volume in the interval between event triggers. Micro-pattern gas detectors release much less positive ions than wire planes operating at the same gain, which may allow operation of a TPC without a gating grid. Given the long drift distance in a large TPC, the active medium must remain very pure, as small amounts of contamination can absorb the ionization signal. For example, in a typical large gaseous TPC, O2 must be kept below a few parts in 105 , otherwise a large fraction of the drifting electrons will become

298

28. Particle detectors

non-uniform ionization. Micro-pattern gas detectors exhibit a much smaller E×B effect, since their feature size is much smaller than that of a wire grid.

Energy deposit per unit length (keV/cm)

32

28

μ

π

K

p

D

24

20

e 16

12

8

0.1

1

10

Momentum (GeV/c)

Figure 28.15: The PEP4/9-TPC energy deposit measurements (185 samples, 8.5 atm Ar-CH4 80:20). The ionization rate at the Fermi plateau (at high β) is 1.4 times that for the minimum at lower β. This ratio increases to 1.6 at atmospheric pressure. attached. Special attention must be made in the choice of construction materials in order to avoid the release of other electronegative contaminants. Diffusion degrades the position information of ionization that drifts a long distance. For a gaseous TPC, the effect can be alleviated by the choice of a gas with low intrinsic diffusion or by operating in a strong magnetic field parallel to the drift field with a gas which exhibits a significant reduction in transverse diffusion with magnetic field. For typical operation without magnetic field, the transverse extent of the electrons, σDx , is a few mm after drifting 1 m due to diffusion. With a strong magnetic field, σDx can be reduced by as much as a factor of 10, 1 σDx (B)/σDx (0) = √ 1 + ω2τ 2

(28.14)

where ωτ is defined above. The diffusion limited position resolution from the information collected by a single row of pads is σDx σx = √ (28.15) n where n is the effective number of electrons collected by the pad row, giving an ultimate single row resolution of order 100 μm. Diffusion is significantly reduced in a negative-ion TPC [99], which uses a special gas mixture that attaches electrons immediately as they are produced. The drifting negative ions exhibit much less diffusion than electrons. The slow drift velocity and small ωτ of negative ions must be compatible with the experimental environment. The spatial resolution achieved by a TPC is determined by a number of factors in addition to diffusion. Non-uniform ionization along the length of the track is a particularly important factor, and is responsible for the so-called “track angle” and “E×B ” effects. If the boundaries between pads in a row are not parallel to the track, the ionization fluctuations will increase the variance in the position estimate from that row. For this reason, experiments with a preferred track direction should have pad boundaries aligned with that direction. Traditional TPC’s with wire plane amplification suffer from the effects of non-parallel electric and magnetic fields near the wires that rotate ionization segments, thereby degrading the resolution because of the

28.7.5. Transition radiation detectors (TRD’s) : Written August 2007 by P. Nevski (BNL), A. Romaniouk (Moscow Eng. & Phys. Inst.) Transition radiation (TR) x rays are produced when a highly 3 relativistic particle (γ > ∼ 10 ) crosses a refractive index interface, as discussed in Sec. 27.7. The x rays, ranging from a few keV to a few dozen keV, are emitted at a characteristic angle 1/γ from the particle trajectory. Since the TR yield is about 1% per boundary crossing, radiation from multiple surface crossings is used in practical detectors. In the simplest concept, a detector module might consist of low-Z foils followed by a high-Z active layer made of proportional counters filled with a Xe-rich gas mixture. The atomic number considerations follow from the dominant photoelectric absorption cross section per atom going roughly as Z n /Ex3 , where n varies between 4 and 5 over the region of interest, and the x-ray energy is Ex .* To minimize self-absorption, materials such as polypropylene, Mylar, carbon, and (rarely) lithium are used as radiators. The TR signal in the active regions is in most cases superimposed upon the particle’s ionization losses. These drop a little faster than Z/A with increasing Z, providing another reason for active layers with high Z. The TR intensity for a single boundary crossing always increases with γ, but for multiple boundary crossings interference √ leads to saturation near a Lorentz factor γ sat = 0.6 ω1 1 2 /c [100,101], where ω1 is the radiator plasma frequency, 1 is its thickness, and 2 the spacing. In most of the detectors used in particle physics the radiator parameters are chosen to provide γ sat ≈ 2000. Those detectors normally work as threshold devices, ensuring the best electron/pion separation in the momentum range 1 GeV/c < ∼p< ∼ 150 GeV/c. One can distinguish two design concepts—“thick” and “thin” detectors: 1. The radiator, optimized for a minimum total radiation length at maximum TR yield and total TR absorption, consists of few hundred foils (for instance 300 20 μm thick polypropylene foils). A dominant fraction of the soft TR photons is absorbed in the radiator itself. To increase the average TR photon energy further, part of the radiator far from the active layers is often made of thicker foils. The detector thickness, about 2 cm for Xe-filled gas chambers, is optimized to absorb the shaped x-ray spectrum. A classical detector is composed of several similar modules which respond nearly independently. Such detectors were used in the NA34 [102], NOMAD [103], and are being used in the ALICE [104] experiment. 2. In another TRD concept a fine granular radiator/detector structure exploits the soft part of the TR spectrum more efficiently. This can be achieved, for instance, by distributing small-diameter straw-tube detectors uniformly or in thin layers throughout the radiator material (foils or fibers). Even with a relatively thin radiator stack, radiation below 5 keV is mostly lost in the radiators themselves. However for photon energies above this value the absorption becomes smaller and the radiation can be registered by several consecutive detector layers, thus creating a strong TR build-up effect. Examples of the detectors using this approach can be found in both accelerator (ATLAS [105]) and space (PAMELA [106], AMS [107]) experiments. For example, in the ATLAS TR tracker charged particles cross about 35 effective straw tube

* Photon absorption coefficients for the elements (via a NIST link), and dE/dx|min and plasma energies for many materials are given in pdg.lbl.gov/AtomicNuclearProperties.

28. Particle detectors

layers embedded in the radiator material. The effective thickness of the Xe gas per straw is about 2.3 mm and the average number of foils per straw is about 40 with an effective foil thickness of about 20 μm. Both TR photon absorption and the TR build-up significantly affect the detector performance. Although the values mentioned above are typical for most of the plastic radiators used with Xe-based detectors, they vary significantly depending on detector parameters: radiator material, thickness and spacing, the construction of the sensitive chambers, their position, etc. Thus careful simulations are usually needed to build a detector optimized for a particular application. The discrimination between electrons and pions can be based on the charge deposition measured in each detection module, on the number of clusters—energy depositions observed above an optimal threshold (usually in the 5 to 7 keV region), or on more sophisticated methods analyzing the pulse shape as a function of time. The total energy measurement technique is more suitable for thick gas volumes, which absorb most of the TR radiation and where the ionization loss fluctuations are small. The cluster-counting method works better for detectors with thin gas layers, where the fluctuations of the ionization losses are big. Cluster-counting replaces the Landau-Vavilov distribution of background ionization energy losses with the Poisson statistics of δ-electrons, responsible for the distribution tails. The latter distribution is narrower that the Landau-Vavilov distribution.

KEK UA2 H.Butt et al. D0 M.Holder et al. H.Weidkamp H.Gr ssler et al. ATLAS NOMAD AMS

0.01 ALICE PAMELA NA34 (HELIOS) C.Fabjan et al. R 806 A. Bungener et al. ZEUS

28.7.6.

Resistive-plate chambers :

Revised September 2007

by H.R. Band (U. Wisconsin). The resistive-plate chamber (RPC) was developed by Santonico and Cardarelli in the early 1980’s [111] as a lowcost alternative to large scintillator planes.* Most commonly, an RPC is constructed from two parallel high-resistivity (109 – 1013 Ω-cm) glass or phenolic (Bakelite)/melamine laminate plates with a few-mm gap between them which is filled with atmospheric-pressure gas. The gas is chosen to absorb UV photons in order to limit transverse growth of discharges. The backs of the plates are coated with a lower-resistivity paint or ink (∼105 Ω/), and a high potential (7–12 kV) is maintained between them. The passage of a charged particle initiates an electric discharge, whose size and duration are limited since the current reduces the local potential to below that needed to maintain the discharge. The sensitivity of the detector outside of this region is unaffected. The signal readout is via capacitive coupling to metallic strips on both sides of the detector which are separated from the high voltage coatings by thin insulating sheets. The x and y position of the discharge can be measured if the strips on opposite sides of the gap are orthogonal. When operated in streamer mode, the induced signals on the strips can be quite large (∼300 mV), making sensitive electronics unnecessary. An example of an RPC structure is shown in Fig. 28.17. Aluminum x pickup strips

:

0.001

detection. The excellent coordinate resolution of the Si detectors allows spatial separation of the TR photons from particle ionization tracks with relatively modest distances between radiator and detector. Recent TRDs for particle astrophysics are designed to directly measure the Lorentz factor of high-energy nuclei by using the quadratic dependence of the TR yield on nuclear charge [109,110]. The radiator configuration (1 , 2 ) is tuned to 5 extend the TR yield rise up to γ < ∼ 10 using more energetic part of the TR spectrum (up to 100 keV). Exotic radiator materials such as aluminum and unusual TR detection methods (Compton scattering) are used such cases [109].

HV

Pion efficiency

0.1

299

Foam

Insulator

Bakelite

2 mm 2 mm 2 mm Graphite Insulator y pickup strips

Graphite 10

20

50 100 Total detector length (cm)

200

Figure 28.16: Pion efficiency measured (or predicted) for different TRDs as a function of the detector length for a fixed electron efficiency of 90%. The plot is taken from [102] with efficiencies of more recent detectors [105–106] added (ATLAS to PAMELA). The major factor in the performance of any TRD is its overall length. This is illustrated in Fig. 28.16, which shows, for a variety of detectors, the pion efficiency at a fixed electron efficiency of 90% as a function of the overall detector length. The experimental data, covering a range of particle energies from a few GeV to 40 GeV, are rescaled to an energy of 10 GeV when possible. Phenomenologically, the rejection power against pions increases as 5 · 10L/38 , where the range of validity is L ≈ 20–100 cm. Many recent TRDs combine particle identification with charged-track measurement in the same detector [104,105]. This provides a powerful tool for electron identification even at very high particle densities. Another example of this combination is described in Ref. 108. In this work Si-microstrip detectors operating in a magnetic filed are used both for particle and TR

Gas Bakelite

1 mm

Foam 1 cm

Spacers Aluminum

Figure 28.17: Schematic cross section of a typical RPC, in this case the single-gap streamer-mode BaBar RPC. RPC’s have inherent rate limitations since the time needed to re-establish the field after a discharge is proportional to the chamber capacitance and plate resistance. The average charge per streamer is 100–1000 pC. Typically, the efficiency of streamer-mode glass RPC’s begins to fall above ∼0.4 Hz/cm2 . Because of Bakelite’s lower bulk resistivity, Bakelite RPC’s can be efficient at 10–100 Hz/cm2 . The need for higher rate capability led to the development of avalanche-mode RPC’s, in which the gas and high voltage have been tuned to limit the * It was based on earlier work on a spark counter with one high-resistivity plate [112].

300

28. Particle detectors

growth of the electric discharge, preventing streamer formation. Typical avalanche-mode RPC’s have a signal charge of about 10 pC and can be efficient at 1 kHz/cm2 . The avalanche discharge produces a much smaller induced signal on the pickup strips (∼1 mV) than streamers, and thus requires a more sophisticated and careful electronic design. Many variations of the initial RPC design have been built for operation in either mode. Efficiencies of > ∼ 92% for single gaps can be improved by the use of two or more gas gaps with shared pickup strips. Non-flammable and more environmentally friendly gas mixtures have been developed. In streamer mode, various mixtures of argon with isobutane and tetrafluoroethane have been used. For avalanche mode operation, a gas mixture of tetrafluoroethane (C2 H2 F4 ) with 2–5% isobutane and 0.4–10% sulfur hexafluoride (SF6 ) is typical. An example of large-scale RPC use is provided by the muon system being built for the ATLAS detector, where three layers of pairs of RPC’s are used to trigger the drift tube arrays between the pairs. The total area is about 10,000 m2 . These RPC’s provide a spatial resolution of 1 cm and a time resolution of 1 ns at an efficiency ≥ 99%. Developments of multiple-gap RPC’s [113] lead to RPC designs with much better timing resolution (∼ 50 ps) for use in time-of-flight particle identification systems. A pioneering design used by the HARP experiment [114] has two sets of 2 thin gas gaps (0.3 mm) separated by thin(0.7 mm) glass plates. The outer plates are connected to high voltage and ground while the inner plate is electrically isolated and floats to a stable equilibrium potential. The observed RPC intrinsic time resolution of 127 ps may have been limited by amplifier noise. Fonte provides useful review [115] of other RPC designs. Operational experience with RPC’s has been mixed. Several experiments (e.g., L3 and HARP) have reported reliable performance. However, the severe problems experienced with the BaBar RPC’s have raised concerns about the long-term reliability of Bakelite RPC’s. Glass RPC’s have had fewer problems, as seen by the history of the BELLE chambers. A rapid growth in the noise rate and leakage current in some of the BELLE glass RPC’s was observed during commissioning. It was found that water vapor in the input gas was reacting with fluorine (produced by the disassociation of the tetrafluoroethane in the streamers) to produce hydrofluoric acid. The acid etched the glass surfaces, leading to increased noise rates and lower efficiencies. The use of copper gas piping to insure the dryness of the input gas stopped the problem. The BELLE RPC’s have now operated reliably for more than 5 years. Several different failure modes diagnosed in the first-generation BaBar Bakelite RPC’s caused the average efficiency of the barrel RPC’s to fall from > ∼ 90% to 35% in five years. The linseed oil which is used in Bakelite RPC’s to coat the inner surface [116] had not been completely cured. Under warm conditions (32◦ C) and high voltage, oil collected on the spacers between the gaps or formed oil-drop bridges between the gaps. This led to large leakage currents (50–100 μA in some chambers) which persisted even when the temperature was regulated at 20◦ C. In addition, the graphite layer used to distribute the high voltage over the Bakelite became highly resistive (100 kΩ/ → 10 MΩ/), resulting in lowered efficiency in some regions and the complete death of whole chambers. The BaBar problems and the proposed use of Bakelite RPC’s in the LHC detectors prompted detailed studies of RPC aging and have led to improved construction techniques and a better understanding of RPC operational limits. The graphite layer has been improved and should be stable with integrated currents 2 of < ∼ 600 mC/cm . Molded gas inlets and improved cleanliness during construction have reduced the noise rate of new chambers.

Unlike glass RPC’s, Bakelite RPC’s have been found to require humid input gases to prevent drying of the Bakelite (increasing the bulk resistivity) which would decrease the rate capability. Second-generation BaBar RPC’s incorporating many of the above improvements have performed reliably for over two years [117]. With many of these problems solved, new-generation RPC’s are now being or soon will be used in about a dozen cosmic-ray and HEP detectors. Their comparatively low cost, ease of construction, good time resolution, high efficiency, and moderate spatial resolution make them attractive in many situations, particularly those requiring fast timing and/or large-area coverage.

28.8. Silicon semiconductor detectors Updated September 2007 by H. Spieler (LBNL). Semiconductor detectors are widely used in modern highenergy physics experiments. They are the key ingredient of high-resolution vertex and tracking detectors and are also used as photodetectors in scintillation calorimeters. The most commonly used material is silicon, but germanium, gallium-arsenide, CdTe, CdZnTe, and diamond are also useful in some applications. Integrated circuit technology allows the formation of highdensity micron-scale electrodes on large (10–15 cm diameter) wafers, providing excellent position resolution. Furthermore, the density of silicon and its small ionization energy result in adequate signals with active layers only 100–300 μm thick, so the signals are also fast (typically tens of ns). Semiconductor detectors depend crucially on low-noise electronics (see Sec. 28.9), so the detection sensitivity is determined by signal charge and capacitance. For a comprehensive discussion of semiconductor detectors and electronics see Ref. 118. Silicon detectors are p-n junction diodes operated at reverse bias. This forms a sensitive region depleted of mobile charge and sets up an electric field that sweeps charge liberated by radiation to the electrodes. Detectors typically use an asymmetric structure, e.g. a highly doped p electrode and a lightly doped n region, so that the depletion region extends predominantly into the lightly doped volume. The thickness of the depleted region is   W = 2 (V + Vbi )/N e = 2ρμ(V + Vbi ) , (28.16) whereV = external bias voltage Vbi = “built-in” voltage (≈ 0.5 V for resistivities typically used in detectors) N = doping concentration e = electronic charge  = dielectric constant = 11.9 0 ≈ 1 pF/cm ρ = resistivity (typically 1–10 kΩ cm) μ = charge carrier mobility = 1350 cm2 V−1 s−1 for electrons = 450 cm2 V−1 s−1 for holes or  √ W = 0.5 [μm/ Ω-cm · V] × ρ(V + Vbi ) for n-type material, and  √ W = 0.3 [μm/ Ω-cm · V] × ρ(V + Vbi ) for p-type material. The conductive p and n regions together with the depleted volume form a capacitor with the capacitance per unit area C = /W ≈ 1 [pF/cm] /W .

(28.17)

In strip and pixel detectors the capacitance is dominated by the fringing capacitance. For example, the strip-to-strip fringing capacitance is ∼ 1–1.5 pF cm−1 of strip length at a strip pitch of 25–50 μm.

28. Particle detectors

Measurements on silicon photodiodes [119] show that for photon energies below 4 eV one electron-hole (e-h) pair is formed per incident photon. The mean energy Ei required to produce an e-h pair peaks at 4.4 eV for a photon energy around 6 eV. It assumes a constant value, 3.67 eV at room temperature, above ∼ 1.5 keV. It is larger than the bandgap energy because phonon excitation is required for momentum conservation. For minimum-ionizing particles, the most probable charge deposition in a 300 μm thick silicon detector is about 3.5 fC (22000 electrons). Since both electronic and lattice excitations are involved, the variance in the number of charge carriers N = E/Ei produced by an absorbed energy E is√reduced by the Fano factor F (about 0.1 in Si). Thus, σN = F N and the energy resolution σE /E = F Ei /E. However, the measured signal fluctuations are usually dominated by electronic noise or energy loss fluctuations in the detector. Charge collection time decreases with increasing bias voltage, and can be reduced further by operating the detector with “overbias,” i.e. a bias voltage exceeding the value required to fully deplete the device. The collection time is limited by velocity saturation at high fields (approaching 107 cm/s at E > 104 V/cm); at an average field of 104 V/cm the collection time is about 15 ps/μm for electrons and 30 ps/μm for holes. In typical fully-depleted detectors 300 μm thick, electrons are collected within about 10 ns, and holes within about 25 ns. Position resolution is limited by transverse diffusion during charge collection (typically 5 μm for 300 μm thickness) and by knock-on electrons. Resolutions of 2–4 μm (rms) have been obtained in beam tests. In magnetic fields, the Lorentz drift deflects the electron and hole trajectories and the detector must be tilted to reduce spatial spreading (see “Hall effect” in semiconductor textbooks). Electrodes can be in the form of cm-scale pads, strips, or μm-scale pixels. Various readout structures have been developed for pixels, e.g. CCD’s, DEPFET’s, monolithic pixel devices that integrate sensor and electronics (MAPS), and hybrid pixel devices that utilize separate sensors and readout IC’s connected by two-dimensional arrays of solder bumps. For an overview and further discussion see Ref. 118. Radiation damage occurs through two basic mechanisms: 1. Bulk damage due to displacement of atoms from their lattice sites. This leads to increased leakage current, carrier trapping, and build-up of space charge that changes the required operating voltage. Displacement damage depends on the nonionizing energy loss and the energy imparted to the recoil atoms, which can initiate a chain of subsequent displacements, i.e., damage clusters. Hence, it is critical to consider both particle type and energy. 2. Surface damage due to charge build-up in surface layers, which leads to increased surface leakage currents. In strip detectors the inter-strip isolation is affected. The effects of charge build-up are strongly dependent on the device structure and on fabrication details. Since the damage is proportional to the absorbed energy (when ionization dominates), the dose can be specified in rad (or Gray) independent of particle type. The increase in reverse bias current due to bulk damage is ΔIr = αΦ per unit volume, where Φ is the particle fluence and α the damage coefficient (α ≈ 3×10−17 A/cm for minimum ionizing protons and pions after long-term annealing; α ≈ 2×10−17 A/cm for 1 MeV neutrons). The reverse bias current depends strongly on temperature IR (T2 ) = IR (T1 )

T2 T1

2



E exp − 2k

T1 − T2 T1 T2

 (28.18)

301

where E = 1.2 eV, so rather modest cooling can reduce the current substantially (∼ 6-fold current reduction in cooling from room temperature to 0◦ C). Displacement damage forms acceptor-like states. These trap electrons, building up a negative space charge, which in turn requires an increase in the applied voltage to sweep signal charge through the detector thickness. This has the same effect as a change in resistivity, i.e., the required voltage drops initially with fluence, until the positive and negative space charge balance and very little voltage is required to collect all signal charge. At larger fluences the negative space charge dominates, and the required operating voltage increases (V ∝ N ). The safe limit on operating voltage ultimately limits the detector lifetime. Strip detectors specifically designed for high voltages have been extensively operated at bias voltages >500V. Since the effect of radiation damage depends on the electronic activity of defects, various techniques have been applied to neutralize the damage sites. For example, additional doping with oxygen increases the allowable charged hadron fluence roughly three-fold [120]. Detectors with columnar electrodes normal to the surface can also extend operational lifetime Ref. 121. The increase in leakage current with fluence, on the other hand, appears to be unaffected by resistivity and whether the material is n or p-type. At fluences beyond 1015 cm−2 decreased carrier lifetime becomes critical [122,123]. Strip and pixel detectors have remained functional at fluences beyond 1015 cm−2 for minimum ionizing protons. At this damage level, charge loss due to recombination and trapping also becomes significant and the high signal-to-noise ratio obtainable with low-capacitance pixel structures extends detector lifetime. The occupancy of the defect charge states is strongly temperature dependent; competing processes can increase or decrease the required operating voltage. It is critical to choose the operating temperature judiciously (−10 to 0◦ C in typical collider detectors) and limit warm-up periods during maintenance. For a more detailed summary see Ref. 124 and and the web-sites of the ROSE and RD50 collaborations at RD48.web.cern.ch/rd48 and RD50.web.cern.ch/rd50. Currently, the lifetime of detector systems is still limited by the detectors; in the electronics use of standard “deep submicron” CMOS fabrication processes with appropriately designed circuitry has increased the radiation resistance to fluences > 1015 cm−2 of minimum ionizing protons or pions. For a comprehensive discussion of radiation effects see Ref. 125.

28.9. Low-noise electronics Revised August 2003 by H. Spieler (LBNL). Many detectors rely critically on low-noise electronics, either to improve energy resolution or to allow a low detection threshold. A typical detector front-end is shown in Fig. 28.18. DETECTOR BIAS

Cb PREAMPLIFIER BIAS RESISTOR

PULSE SHAPER

Rb

DETECTOR Cd

Cc

Rs

OUTPUT

Figure 28.18: Typical detector front-end circuit. The detector is represented by a capacitance Cd , a relevant model for most detectors. Bias voltage is applied through resistor Rb and the signal is coupled to the preamplifier through

302

28. Particle detectors

a blocking capacitor Cc . The series resistance Rs represents the sum of all resistances present in the input signal path, e.g. the electrode resistance, any input protection networks, and parasitic resistances in the input transistor. The preamplifier provides gain and feeds a pulse shaper, which tailors the overall frequency response to optimize signal-to-noise ratio while limiting the duration of the signal pulse to accommodate the signal pulse rate. Even if not explicitly stated, all amplifiers provide some form of pulse shaping due to their limited frequency response. The equivalent circuit for the noise analysis (Fig. 28.19) includes both current and voltage noise sources. The leakage current of a semiconductor detector, for example, fluctuates due to electron emission statistics. This “shot noise” ind is represented by a current noise generator in parallel with the detector. Resistors exhibit noise due to thermal velocity fluctuations of the charge carriers. This noise source can be modeled either as a voltage or current generator. Generally, resistors shunting the input act as noise current sources and resistors in series with the input act as noise voltage sources (which is why some in the detector community refer to current and voltage noise as “parallel” and “series” noise). Since the bias resistor effectively shunts the input, as the capacitor Cb passes current fluctuations to ground, it acts as a current generator inb and its noise current has the same effect as the shot noise current from the detector. Any other shunt resistances can be incorporated in the same way. Conversely, the series resistor Rs acts as a voltage generator. The electronic noise of the amplifier is described fully by a combination of voltage and current sources at its input, shown as ena and ina . BIAS RESISTOR

DETECTOR

SERIES RESISTOR

Rs

AMPLIFIER + PULSE SHAPER

ens ena

Cd

Rb

inb

ina

ind

Figure 28.19: Equivalent circuit for noise analysis. Shot noise and thermal noise have a “white” frequency distribution, i.e. the spectral power densities dPn /df ∝ di2n /df ∝ de2n /df are constant with the magnitudes i2nd = 2eId , 4kT i2nb = , Rb e2ns = 4kT Rs ,

(28.19)

where e is the electronic charge, Id the detector bias current, k the Boltzmann constant and T the temperature. √Typical amplifier √ noise parameters ena and ina are of order nV/ Hz and pA/ Hz. Trapping and detrapping processes in resistors, dielectrics and semiconductors can introduce additional fluctuations whose noise power frequently exhibits a 1/f spectrum. The spectral density of the 1/f noise voltage is e2nf =

Af , f

(28.20)

where the noise coefficient Af is device specific and of order 10−10 –10−12 V2 . A fraction of the noise current flows through the detector capacitance, resulting in a frequency-dependent noise voltage

in /(ωCd ), which is added to the noise voltage in the input circuit. Since the individual noise contributions are random and uncorrelated, they add in quadrature. The total noise at the output of the pulse shaper is obtained by integrating over the full bandwidth of the system. Superimposed on repetitive detector signal pulses of constant magnitude, purely random noise produces a Gaussian signal distribution. Since radiation detectors typically convert the deposited energy into charge, the system’s noise level is conveniently expressed as an equivalent noise charge Qn , which is equal to the detector signal that yields a signal-to-noise ratio of one. The equivalent noise charge is commonly expressed in Coulombs, the corresponding number of electrons, or the equivalent deposited energy (eV). For a capacitive sensor Q2n = i2n Fi TS + e2n Fv

C2 + Fvf Af C 2 , TS

(28.21)

where C is the sum of all capacitances shunting the input, Fi , Fv , and Fvf depend on the shape of the pulse determined by the shaper and Ts is a characteristic time, for example, the peaking time of a semi-gaussian pulse or the sampling interval in a correlated double sampler. The form factors Fi , Fv are easily calculated 1 Fi = 2TS

∞

2

[W (t)] dt , −∞

TS Fv = 2

∞ −∞



dW (t) dt

2 dt ,

(28.22) where for time-invariant pulse-shaping W (t) is simply the system’s impulse response (the output signal seen on an oscilloscope) with the peak output signal normalized to unity. For more details see Refs. 126 and 127. A pulse shaper formed by a single differentiator and integrator with equal time constants has Fi = Fv = 0.9 and Fvf = 4, independent of the shaping time constant. The overall noise bandwidth, however, depends on the time constant, i.e. the characteristic time Ts . The contribution from noise currents increases with shaping time, i.e., pulse duration, whereas the voltage noise decreases with increasing shaping time. Noise with a 1/f spectrum depends only on the ratio of upper to lower cutoff frequencies (integrator to differentiator time constants), so for a given shaper topology the 1/f contribution to Qn is independent of Ts . Furthermore, the contribution of noise voltage sources to Qn increases with detector capacitance. Pulse shapers can be designed to reduce the effect of current noise, e.g., mitigate radiation damage. Increasing pulse symmetry tends to decrease Fi and increase Fv (e.g., to 0.45 and 1.0 for a shaper with one CR differentiator and four cascaded integrators). For the circuit shown in Fig. 28.19, ! " Q2n = 2eId + 4kT /Rb + i2na Fi TS + (4kT Rs + e2na )Fv Cd2 /TS + Fvf Af Cd2 .

(28.23)

As the characteristic time TS is changed, the total noise goes through a minimum, where the current and voltage contributions are equal. Fig. 28.20 shows a typical example. At short shaping times the voltage noise dominates, whereas at long shaping times the current noise takes over. The noise minimum is flattened by the presence of 1/f noise. Increasing the detector capacitance will increase the voltage noise and shift the noise minimum to longer shaping times. For quick estimates, one can use the following equation, which assumes an FET amplifier (negligible ina ) and a simple CR–RC

28. Particle detectors

where σn is the rms noise and the derivative of the signal dS/dt is evaluated at the trigger level ST . To increase dS/dt without incurring excessive noise, the amplifier bandwidth should match the rise-time of the detector signal. The 10 to 90% rise time of an amplifier with bandwidth fU is 0.35/fU . For example, an oscilloscope with 350 MHz bandwidth has a 1 ns rise time. When amplifiers are cascaded, which is invariably necessary, the individual rise times add in quadrature. $ tr ≈ t2r1 + t2r2 + ... + t2rn

10000

Equivalent noise charge (e)

5000 Increasing V noise 2000

Total

1000 Total 500

1/f noise

200

t ren

ise

no

Vo

lta

ur

C 100 0.01

0.1

ge

no

ise

1

Shaping time (μs)

303

10

100

Figure 28.20: Equivalent noise charge vs shaping time. Changing the voltage or current noise contribution shifts the noise minimum. Increased voltage noise is shown as an example.

Increasing signal-to-noise ratio also improves time resolution, so minimizing the total capacitance at the input is also important. At high signal-to-noise ratios, the time jitter can be much smaller than the rise time. The timing distribution may shift with signal level (“walk”), but this can be corrected by various means, either in hardware or software [10]. For a more detailed introduction to detector signal processing and electronics see Ref. 118.

28.10. Calorimeters shaper with time constants τ (equal to the peaking time): 







1 kΩ τ Id τ + 6 × 105 nA · ns ns Rb   2 ns 2C e . + 3.6 × 104 n τ (pF)2 (nV)2 /Hz (28.24) Noise is improved by reducing the detector capacitance and leakage current, judiciously selecting all resistances in the input circuit, and choosing the optimum shaping time constant. The noise parameters of the amplifier depend primarily on the input device. In field effect transistors, the noise current contribution is very small, so reducing the detector leakage current and increasing the bias resistance will allow long shaping times with correspondingly lower noise. In bipolar transistors, the base current sets a lower bound on the noise current, so these devices are best at short shaping times. In special cases where the noise of a transistor scales with geometry, i.e., decreasing noise voltage with increasing input capacitance, the lowest noise is obtained when the input capacitance of the transistor is equal to the detector capacitance, albeit at the expense of power dissipation. Capacitive matching is useful with field-effect transistors, but not bipolar transistors. In bipolar transistors, the minimum obtainable noise is independent of shaping time, but only at the optimum collector current IC , which does depend on shaping time. (Qn /e)2 = 12

where βDC βDC = 100,

C  kT  Fi Fv at Ic = C βDC e βDC

#

Fv 1 , Fi TS (28.25) is the DC current gain. For a CR–RC shaper and

Q2n,min = 4kT √

 Qn,min /e ≈ 250 C/pF .

A calorimeter is designed to measure the energy deposited in a contained electromagnetic (EM) or hadronic shower. The characteristic interaction distance for an electromagnetic interaction is the radiation length X0 , which ranges from 13.8 g cm−2 in iron to 6.0 g cm−2 in uranium.* Similarly, the characteristic nuclear interaction length λI varies from 132.1 g cm−2 (Fe) to 209 g cm−2 (U). In either case, the calorimeter must be many interaction lengths deep, where “many” is determined by physical size, cost, and other factors. EM calorimeters tend to be 15–30 X0 deep, while hadronic calorimeters are usually compromised at 5–8 λI . Moreover, in a real experiment there is likely to be an EM calorimeter in front of the hadronic section, and perhaps a more poorly sampling catcher in the back, so the hadronic cascade is contained in a succession of different structures. In all cases there is a premium on high density, to contain the shower as compactly as possible, and, especially in the EM case, high atomic number. There are homogeneous and sampling calorimeters. In a homogeneous calorimeter the entire volume is sensitive, i.e., contributes signal. Homogeneous calorimeters (usually electromagnetic) may be built with inorganic heavy (highZ) scintillating crystals such as BGO, CsI, NaI, and PWO, non-scintillating Cherenkov radiators such as lead glass and lead fluoride, or ionizing noble liquids. Properties of commonly used inorganic crystal scintillators can be found in Table 28.4. A sampling calorimeter consists of an active medium which generates signal and a passive medium which functions as an absorber. The active medium may be a scintillator, an ionizing noble liquid, a gas chamber, a semiconductor, or a Cherenkov radiator. The passive medium is usually a material of high density, such as lead, iron, copper, or depleted uranium.

(28.26)

Practical noise levels range from ∼ 1e for CCD’s at long shaping times to ∼ 104 e in high-capacitance liquid argon calorimeters. Silicon strip detectors typically operate at ∼ 103 e electrons, whereas pixel detectors with fast readout provide noise of several hundred electrons. In timing measurements, the slope-to-noise ratio must be optimized, rather than the signal-to-noise ratio alone, so the rise time tr of the pulse is important. The “jitter” σt of the timing distribution is σn tr , (28.27) σt = ≈ (dS/dt)ST S/N

* λI ≈ 35 g cm−2 A1/3 ; for actual values see pdg.lbl.gov/AtomicNuclearProperties .

304

28. Particle detectors

28.10.1. Electromagnetic calorimeters : Written August 2003 by R.-Y. Zhu (California Inst. of Technology).

The development of electromagnetic showers is discussed in the section on “Passage of Particles Through Matter” (Sec. 27 of this Review). Formulae are given which approximately describe average showers, but since the physics of electromagnetic showers is well understood, detailed and reliable Monte Carlo simulation is possible. EGS4 [128] and GEANT [129] have emerged as the standards. The energy resolution σE /E of a calorimeter can be √ parametrized as a/ E ⊕ b ⊕ c/E, where ⊕ represents addition in quadrature and E is in GeV. The stochastic term a represents statistics-related fluctuations such as intrinsic shower fluctuations, photoelectron statistics, dead material at the front of the calorimeter, and sampling fluctuations. For a fixed number of radiation lengths, the stochastic term a for a sampling calorimeter is expected to be proportional to t/f , where t is plate thickness and f is sampling fraction [130,131]. While a is at a few percent level for a homogeneous calorimeter, it is typically 10% for sampling calorimeters. The main contributions to the systematic, or constant, term b are detector non-uniformity and calibration uncertainty. In the case of the hadronic cascades discussed below, non-compensation also contributes to the constant term. One additional contribution to the constant term for calorimeters built for modern high-energy physics experiments, operated in a high-beam intensity environment, is radiation damage of the active medium. This can be minimized by developing radiation-hard active media [45] and by frequent in situ calibration and monitoring [44,131]. With effort, the constant term b can be reduced to below one percent. The term c is due to electronic noise summed over readout channels within a few Moli`ere radii. The best energy resolution for electromagnetic shower measurement is obtained in total absorption homogeneous calorimeters, e.g. calorimeters built with heavy crystal scintillators. These are used when ultimate performance is pursued. The position resolution depends on the effective Moli`ere radius and the transverse granularity of the√calorimeter. Like the energy resolution, it can be factored as a/ E ⊕ b, where a is a few to 20 mm and b can be as small as a fraction of mm for a dense calorimeter with fine granularity. Electromagnetic calorimeters may also provide direction measurement for electrons and photons. This is important for photon-related physics when there are uncertainties in event origin, since photons do not leave information in the particle tracking √ system. Typical photon angular resolution is about 45 mrad/ E, which can be provided by implementing longitudinal segmentation [132] for a sampling calorimeter or by adding a preshower detector [133] for a homogeneous calorimeter without longitudinal segmentation. Novel technologies have been developed for electromagnetic calorimetry. New heavy crystal scintillators, such as PWO, LSO:Ce, and GSO:Ce (see Sec. 28.4), have attracted much attention for homogeneous calorimetry. In some cases, such as PWO, it has received broad applications in high-energy and nuclear physics experiments. The “spaghetti” structure has been developed for sampling calorimetry with scintillating fibers as the sensitive medium. The “accordion” structure has been developed for sampling calorimetry with ionizing noble liquid as the sensitive medium. Table 28.9 provides a brief description of typical electromagnetic calorimeters built recently for high-energy physics experiments. Also listed in this table are calorimeter depths in radiation lengths (X0 ) and the achieved energy resolution. Whenever possible, the performance of calorimeters in situ is quoted, which is usually in good agreement with prototype test beam results as well as EGS or

GEANT simulations, provided that all systematic effects are properly included. Detailed references on detector design and performance can be found in Appendix C of reference [131] and Proceedings of the International Conference series on Calorimetry in Particle Physics. Table 28.9: Resolution of typical electromagnetic calorimeters. E is in GeV. Technology (Exp.)

Depth

NaI(Tl) (Crystal Ball) Bi4 Ge3 O12 (BGO) (L3) CsI (KTeV) CsI(Tl) (BaBar) CsI(Tl) (BELLE) PbWO4 (PWO) (CMS) Lead glass (OPAL) Liquid Kr (NA48)

20X0 2.7%/E1/4 √ 22X0 2%/ E ⊕ 0.7% √ 2%/ E ⊕ 0.45% 27X0 16–18X0 2.3%/E 1/4 ⊕ 1.4% 16X0 1.7% for Eγ > 3.5 GeV √ 25X0 3%/ E ⊕ 0.5% ⊕ 0.2/E √ 20.5X0 5%/ E √ 27X0 3.2/% E ⊕ 0.42% ⊕ 0.09/E √ 20–30X0 18%/ E √ 18X0 13.5%/ E √ 15X0 5.7%/ E ⊕ 0.6%

Scintillator/depleted U (ZEUS) Scintillator/Pb (CDF) Scintillator fiber/Pb spaghetti (KLOE) Liquid Ar/Pb (NA31) Liquid Ar/Pb (SLD) Liquid Ar/Pb (H1) Liquid Ar/depl. U (DØ) Liquid Ar/Pb accordion (ATLAS)

27X0 21X0 20–30X0 20.5X0 25X0

Energy resolution

√ 7.5%/ E ⊕ 0.5% ⊕ 0.1/E √ 8%/ E √ 12%/ E ⊕ 1% √ 16%/ E ⊕ 0.3% ⊕ 0.3/E √ 10%/ E ⊕ 0.4% ⊕ 0.3/E

Date 1983 1993 1996 1999 1998 1997 1990 1998 1988 1988 1995 1988 1993 1998 1993 1996

28.10.2. Hadronic calorimeters : [1–6,131] Written April 2008 by D. E. Groom (LBNL). Most large hadron calorimeters are sampling calorimeters which are parts of complicated 4π detectors at colliding beam facilities. Typically, the basic structure is plates of absorber (Fe, Pb, Cu, or occasionally U or W) alternating with plastic scintillators (plates, tiles, bars), liquid argon (LAr), or gaseous detectors. The ionization is measured directly, as in LAr calorimeters, or via scintillation light observed by photodetectors (usually PMT’s). Waveshifting fibers are often used to solve difficult problems of geometry and light collection uniformity. Silicon sensors are being studied for ILC detectors; in this case e-h pairs are collected. There are as many variants of these schemes as there are calorimeters, including variations in geometry of the absorber and sensors, e.g., scintillating fibers threading an absorber [134], and the “accordion” LAr detector, with zig-zag absorber plates to minimize channeling effects. Another departure from the traditional sandwich structure is the LAr-tube design shown in Fig. 28.21(a). A relatively new variant is the use of Cerenkov light in hadron calorimetry. Such a calorimeter is sensitive to e± ’s in the EM showers plus a few relativistic pions. An example is the radiation-hard forward calorimeter in CMS, with iron absorber and quartz fiber readout by PMT’s. Ideally, the calorimeter is segmented in φ and θ (or η = − ln tan(θ/2)). Fine segmentation, while desirable, is limited by cost, readout complexity, practical geometry, and the transverse size of the cascades. An example, a wedge of the ATLAS central barrel calorimeter, is shown in Fig. 28.21(b). In an inelastic hadronic collision a significant fraction fem of the energy is removed from further hadronic interaction by the production of secondary π 0 ’s and η’s, whose decay photons

28. Particle detectors

(b)

(a) W (Cu) absorber

≈ 1 − (1 − h/e)(E/E0 )m−1 . PMT waveshifter fiber

LAr filled tubes

Hadrons

scintillator tile φ r

z Hadrons

Figure 28.21: (a) ATLAS forward hadronic calorimeter structure (FCal2, 3). Tubes containing LAr are embedded in a mainly tungsten matrix. (b) ATLAS central calorimeter wedge; iron with plastic scintillator tile with wavelength-shifting fiber readout.

generate high-energy electromagnetic (EM) cascades. Charged secondaries (π ± , p, . . . ) deposit energy via ionization and excitation, but also interact with nuclei, producing spallation protons and neutrons, evaporation neutrons, and recoiling nuclei in highly excited states. The charged collision products produce detectable ionization, as do the showering γ-rays from the prompt de-excitation of highly excited nuclei. The recoiling nuclei generate little or no detectable signal. The neutrons lose kinetic energy in elastic collisions over hundreds of ns, gradually thermalize and are captured, with the production of more γ-rays—usually outside the acceptance gate of the electronics. Between endothermic spallation losses, nuclear recoils, and late neutron capture, a significant fraction of the hadronic energy (20%–35%, depending on the absorber and energy of the incident particle) is invisible. In contrast to EM showers, hadronic cascade processes are characterized by relatively few high-energy particles being produced. The lost energy and the π 0 → γγ fraction fem are highly variable from event to event. Until there is event-by-event knowledge of both the invisible energy loss and EM deposit (to be discussed below), the energy resolution of a hadron calorimeter will remain significantly worse than that of an EM calorimeter. It has been shown by a simple induction argument and verified by experiment that the decrease in the average value of the hadronic energy fraction (fh  = 1 − fem ) as the projectile energy E increases is fairly well described by the power law [135,136] fh  ≈ (E/E0 )m−1

(for E > E0 ) ,

(28.28)

up to at least a few hundred GeV. The exponent m depends logarithmically on the mean multiplicity and the mean fractional loss to π 0 production in a single interaction. It is in the range 0.80–0.87, but must be obtained experimentally for each calorimeter configuration. E0 is roughly the energy for the onset of inelastic collisions. It is 1 GeV or a little less for incident pions. In a hadron-nucleus collision a large fraction of the incident energy is carried by a “leading particle” with the same quark content as the incident hadron. If the projectile is a charged pion, the leading particle is usually a pion, which can be neutral and hence contributes to the EM sector. This is not true for incident protons. The result is an increased mean hadronic fraction for incident protons: in Eq. (28.29b) E0 ≈ 2.6 GeV [135,137]. The EM energy deposit is usually detected more efficiently than the hadronic energy deposit. If the detection efficiency for the EM sector is e and that for the hadronic sector is h, then the ratio of the mean response to a pion to that for an electron is π/e = fem  + fh h/e = 1 − (1 − h/e)fh 

(28.29a)

305

(28.29b)

If h = e the hadronic response is not a linear function of energy. Only the product (1 − h/e)E01−m can be obtained by measuring π/e as a function of energy. Since 1 − m is small and E0 ≈ 1 GeV for the usual pion-induced cascades, this fact is usually ignored and h/e is reported. The discussion above assumes an idealized calorimeter, with the same structure throughout and without leakage. “Real” calorimeters usually have an EM detector in front and a coarse “catcher” in the back. Complete containment is generally impractical. By definition, 0 ≤ fem ≤ 1. Its variance changes only slowly with energy, but perforce fem  → 1 as the projectile energy increases. An empirical power law σfem = (E/E1 )1− (where  < 1) describes the energy dependence adequately and has the right asymptotic properties. For h/e = 1, fluctuations in fem significantly contribute to the resolution, in particular contributing a larger fraction of the variance at high energies. Since the fem distribution has a tail on the high side, the calorimeter response is non-Gaussian with a high-energy tail if h/e < 1. Noncompensation (h/e = 1) thus seriously degrades resolution as well as producing a nonlinear response. It is clearly desirable to compensate the response, i.e., to design the calorimeter such that h/e = 1. This is possible only in a sampling calorimeter, where several variables can be chosen or tuned: 1. Decrease the EM sensitivity. Because the EM cross sections increase with Z,* and the absorber usually has higher Z than does the sensor, the EM energy deposit rate, relative to minimum ionization, is greater than this ratio in the sensor. Lower-Z inactive cladding, such as the steel cladding on ZEUS U plates, preferentially absorbs low-energy γ’s in EM showers and thus also lowers the electronic response. G10 signal boards in the DØ calorimeters have the same effect. 2. Increase the hadronic sensitivity. The abundant neutrons have a large n-p scattering cross section, with the production of low-energy scattered protons in hydrogenous sampling materials such as butane-filled proportional counters or plastic scintillator. (When scattering off a nucleus with mass number A, a neutron can at most lose 4/(1 + A)2 of its kinetic energy.) The down side in the scintillator case is that the signal from a highly-ionizing proton stub can be reduced buy as much as 90% by recombination and quenching (Birk’s Law, Eq. (28.2)). Fabjan and Willis proposed that the additional signal generated in the aftermath of fission in 238 U absorber plates should compensate nuclear fluctuations [138]. The production of fission fragments due to fast n capture was later observed [139]. However, while a very large amount of energy is released, it is mostly carried by low-velocity fission fragments which produce very little observable signal. The approach seemed promising for awhile. But, for example, the compensation observed with the ZEUS 238 U/scintillator calorimeter was the result of the two mechanisms discussed above. Motivated very much by the work of Brau, Gabriel, Br¨ uckmann, and Wigmans [140], several groups built calorimeters which were very nearly compensating. The degree of compensation was sensitive to the acceptance gate width, and so could be somewhat tuned. These included (a) HELIOS with 2.5 238 mm thick scintillator plates sandwiched between √2 mm thick U plates (one of several structures); σ/E = 0.34/ E was obtained, 238 U (b) ZEUS, 2.6 cm thick √ scintillator plates between 3.3 mm plates; σ/E = 0.35/ E, (c) a ZEUS prototype with 10 mm * The asymptotic pair-production cross section scales roughly as Z 0.75 , and |dE/dx| slowly decreases with increasing Z.

28. Particle detectors

Q = E[fem + h/e|Q (1 − fem )] S = E[fem + h/e|S (1 − fem )]

(28.30)

These equations are linear in 1/E and fem , and are easily solved for estimators of the corrected energy and fem for each event. Both are subject to resolution effects, but effects due to fluctuations in fem are eliminated. The solution for the corrected energy is given by [136]: E=

1 − h/e|Q RS − Q , where R = R−1 1 − h/e|S

(28.31)

R is the energy-independent slope of the event locus on a plot of Q vs S. It can be found either from the fitted slope or by measuring π/e as a function of E. The DREAM collaboration expects to build a “triple-readout calorimeter” in which a neutron signal, proportional to the the missing energy, is measured as well on an event-by-event basis. It is hoped that such a hadronic calorimeter can approach the resolution of an electromagnetic calorimeter. The fractional resolution can be represented by  σ h E 1− a1 (E) √ ⊕ 1 − = E e E1 E

(28.32)

The coefficient a1 is expected to have mild energy dependence for a number of reasons. For example, the sampling variance is (π/e)E rather than E. (E/E1 )1− is the parameterization of σfem discussed above. At a time when data were of lower quality, a plot of (σ/E)2 vs 1/E was apparently well-described by a straight line (constant a1 ) with a finite intercept—the square of the right term in Eq. (28.32), then called “the constant term.” Modern data show the slight downturn [134]. The average longitudinal distribution rises to a smooth peak about one nuclear interaction length (λI ) into the calorimeter. It then falls somewhat faster than exponentially. Proton-induced cascades are somewhat shorter and broader than pion-induced cascades. In either case, the falloff distance increases with energy. In Fig. 28.22 experimental results for 90% and 95% containment are shown, as are calculations using Bock’s parameterization [143]. A slightly modified form has been used to fit recent measurements, e.g. to fit profiles measured in the ATLAS central barrel wedges [144]. The transverse energy deposit is characterized by a central core dominated by EM cascades, where the neutral meson parents are themselves produced at a variety of angles. There is a wide “skirt” produced by wide-angle hadronic interactions. The energy deposited in an annulus dA is adequately described by an exponential core and a Gaussian halo: dE/dA = (B1 /r) exp(−r/a1 ) + (B2 /r) exp(−r 2 /a2 ) [145].

200

12 11

99%

150

9 95%

8 7

100

6 / Bock param. / CDHS data / CCFR data

5 4

50 5

10

Depth in Iron (λI)

10 Depth in Iron (cm)

√ Pb plates and 2.5 mm scintillator sheets; σ/E = 0.44/ E, and (d) DØ, where the sandwich cell consists of a 4–6 mm thick 238 U plate, 2.3 mm LAr, a G-10 signal board, and another 2.3 mm LAr gap. A more versatile approach to compensation is provided by a dual-readout calorimeter, in which the signal is sensed by two readout systems with highly contrasting h/e. Although the concept is more than two decades old [141], it has only recently been implemented by the DREAM collaboration [142]. The test beam calorimeter consisted of copper tubes, each filled with scintillator and quartz fibers. If the two signals Q and S (quartz and scintillator) are both normalized to electrons, then for each event Eq. (28.29) takes the form:

50 100 500 1000 Single Hadron Energy (GeV)

3

Figure 28.22: Required calorimeter thickness for 95% and 99% hadronic cascade containment in iron, on the basis of data from two large neutrino detectors and Bock’s parameterization [143]. 28.10.3. Free electron drift velocities in liquid ionization sensors : Velocities as a function of electric field strength

are given in Refs. 146–147 and are plotted in Fig. 28.23. Recent precise measurements of the free electron drift velocity in LAr have been published by W. Walkowiak [148]. The new measurements were motivated by the design of the ATLAS electromagnetic calorimeter and by inconsistencies in the previous literature. Velocities are temperatuer depedent and are systematically higher than those shown in Fig. 28.23. 10 TMS

9 8 Drift Velocity (μm/ns)

306

7

LAr+CH4 (0.5%)

6 5 4 LAr

3

2,2,4,4 TMP

2 1 0

0

2

4 6 8 10 12 Field Strength (kV cm−1)

14

16

Figure 28.23: Electron drift velocity as a function of field strength for commonly used liquids.

28. Particle detectors

28.11. Superconducting magnets for collider detectors

If the coil thin, (which is the case if it is to superconducting coil), then (28.38) E ≈ (B 2 /2μ0 )πR2 L .

Revised September 2005 by A. Yamamoto (KEK); revised October 2001 by R.D. Kephart (FNAL) 28.11.1.

307

Solenoid Magnets : In all cases SI unit are assumed,

so that the magnetic field, B, is in Tesla, the stored energy, E, is in joules, the dimensions are in meters, and μ0 = 4π × 10−7 .

For a detector in which the calorimetry is outside the aperture of the solenoid, the coil must be thin in terms of radiation and absorption lengths. This usually means that the coil is superconducting and that the vacuum vessel encasing it is of

Table 28.10: Progress of superconducting magnets for particle physics detectors. Experiment

Laboratory

TOPAZ* CDF VENUS* AMY* CLEO-II ALEPH* DELPHI* ZEUS* H1* BaBar D0 BELLE BES-III† ATLAS-CS† ATLAS-BT† ATLAS-ET† CMS†

KEK Tsukuba/Fermi KEK KEK Cornell Saclay/CERN RAL/CERN INFN/DESY RAL/DESY INFN/SLAC Fermi KEK IHEP ATLAS/CERN ATLAS/CERN ATLAS/CERN CMS/CERN

∗ † ‡

B [T]

1.2 1.45 5.4 20 0.70 1.5 1.5 5.07 30 0.84 0.75 1.75 5.64 12 0.52 3 1.29 3 40 ‡ 1.5 1.55 3.8 25 2.5 1.5 2.75 7.0 130 2.0 1.2 2.8 7.4 109 1.7 1.8 1.5 2.85 11 0.9 1.2 2.8 5.75 120 1.8 1.5 1.5 3.46 27 ‡ 2.0 0.6 2.73 5.6 0.9 1.5 1.8 4 42 ‡ 1.0 1.475 3.5 9.5 ‡ 2.0 1.25 5.3 38 0.66 1 4.7–9.75 26 1080 (Toroid) 1 0.825–5.35 5 2 × 250 (Toroid) 4 6 12.5 2600 ‡

4.3 5.4 2.8 3.7 5.5 4.2 5.5 4.8 3.6 3.7 5.3 2.6 7.0

12

No longer in service Detector under construction EM calorimeter is inside solenoid, so small X/X0 is not a goal

The magnetic field (B) in an ideal solenoid with a flux return iron yoke, in which the magnetic field is < 2 T, is given by B = μ0 n I

(28.33)

where n is the number of turns/meter and I is the current. In an air-core solenoid, the central field is given by L B(0, 0) = μ0 n I √ , L2 + 4R2

(28.34)

where L is the coil length and R is the coil radius. In most cases, momentum analysis is made by measuring the circular trajectory of the passing particles according to p = mvγ = q rB, where p is the momentum, m the mass, q the charge, r the bending radius. The sagitta, s, of the trajectory is given by s = q B 2 /8p , (28.35) where  is the path length in the magnetic field. In a practical momentum measurement in colliding beam detectors, it is more effective to increase the magnetic volume than the field strength, since dp/p ∝ p/B 2 , (28.36) where  corresponds to the solenoid coil radius R. The energy stored in the magnetic field of any magnet is calculated by integrating B 2 over all space: E=

Radius Length Energy X/X0 E/M [m] [m] [MJ] [kJ/kg]

1 2μ0

B 2 dV

(28.37)

minimum real thickness and fabricated of a material with long radiation length. There are two major contributors to the thickness of a thin solenoid: 1) The conductor consisting of the current-carrying superconducting material (usually NbTi/Cu) and the quench protecting stabilizer (usually aluminum) are wound on the inside of a structural support cylinder (usually aluminum also). The coil thickness scales as B 2 R, so the thickness in radiation lengths (X0 ) is tcoil /X0 = (R/σh X0 )(B 2 /2μ0 ) ,

(28.39)

where tcoil is the physical thickness of the coil, X0 the average radiation length of the coil/stabilizer material, and σh is the hoop stress in the coil [151]. B 2 /2μ0 is the magnetic pressure. In large detector solenoids, the aluminum stabilizer and support cylinders dominate the thickness; the superconductor (NbTI/Cu) contributes a smaller fraction. The coil package including the cryostat typically contributes about 2/3 of the total thickness in radiation lengths. 2) Another contribution to the material comes from the outer cylindrical shell of the vacuum vessel. Since this shell is susceptible to buckling collapse, its thickness is determined by the diameter, length and the modulus of the material of which it is fabricated. The outer vacuum shell represents about 1/3 of the total thickness in radiation length.

308

28. Particle detectors

Properties of collider detector magnets :

The physical dimensions, central field stored energy and thickness in radiation lengths normal to the beam line of the superconducting solenoids associated with the major collider are given in Table 28.10 [150]. Fig. 28.24 shows thickness in radiation lengths as a function of B 2 R in various collider detector solenoids. 3

15

CMS

10 E/M [kJ/kg]

28.11.2.

SSC-SDC Prototype ATLAS-CS

5

TOPAZ D0 CLEO-II

CLEO-II

Thickness in radiation lengths

CDF

ZEUS

2.5

BELLE BABAR VENUS

BES-III ALEPH

2

0

H1

1 DELPHI

1.5

ZEUS TOPAZ

VENUS 0.5 BESS, WASA

0

CELLO

1

2

PEP4- CDF TPC

3

4 5 B 2 R [T2 m]

ATLAS-CS

6

28.11.3.

7

8

Figure 28.24: Magnet wall thickness in radiation length as a function of B 2 R for various detector solenoids. Gray entries are for magnets not listed in Table 28.10. Open circles are for magnets not designed to be “thin.” The SSC-SDC prototype provided important R&D for LHC magnets. The ratio of stored energy to cold mass (E/M ) is a useful performance measure. It can also be expressed as the ratio of the stress, σh , to twice the equivalent density, ρ, in the coil [151]:  2 (B /2μ0 )dV σh E = (28.40) ≈ M ρ Vcoil 2ρ The E/M ratio in the coil is approximately equivalent to H,* the enthalpy of the coil, and it determines the average coil temperature rise after energy absorption in a quench: E/M = H(T2 ) − H(T1 ) ≈ H(T2 )

10

100 Stored Energy [MJ]

1000

104

Figure 28.25: Ratio of stored energy to cold mass for thin detector solenoids. Open circles indicate magnets under construction.

SSC-SDC prototype

D0

1

0

ALEPH H1 DELPHI

(28.41)

where T2 is the average coil temperature after the full energy absorption in a quench, and T1 is the initial temperature. E/M ratios of 5, 10, and 20 kJ/kg correspond to ∼65, ∼80, and ∼100 K, respectively. The E/M ratios of various detector magnets are shown in Fig. 28.25 as a function of total stored energy. One would like the cold mass to be as small as possible to minimize the thickness, but temperature rise during a quench must also be minimized. An E/M ratio as large as 12 kJ/kg is designed into the CMS solenoid, with the possibility that about half of the stored energy can go to an external dump resistor. Thus the coil temperature can be kept below 80 K if the energy extraction system work well. The limit is set by the maximum temperature that the coil design can tolerate during a quench. This maximum local temperature should be <130 K (50 K + 80 K), so that thermal expansion effects in the coil are manageable.

* The enthalpy, or heat content, is called H in the thermodynamics literature. It is not to be confused with the magnetic field intensity B/μ.

Toroidal magnets :

Toroidal coils uniquely provide a closed magnetic field without the necessity of an iron flux-return yoke. Because no field exists at the collision point and along the beam line, there is, in principle, no effect on the beam. On the other hand, the field profile generally has 1/r dependence. The particle momentum may be determined by measurements of the deflection angle combined with the sagitta. The deflection (bending) power BL is R0 Bi R i Bi Ri dR BL ≈ = ln(R0 /Ri ) , (28.42) R sin θ sin θ Ri where Ri is the inner coil radius, R0 is the outer coil radius, and θ is the angle between the particle trajectory and the beam line axis . The momentum resolution given by the deflection may be expressed as p p sin θ Δp ∝ ≈ . p BL Bi Ri ln(R0 /Ri )

(28.43)

The momentum resolution is better in the forward/backward (smaller θ) direction. The geometry has been found to be optimal when R0 /Ri ≈ 3–4. In practical designs, the coil is divided into 6–12 lumped coils in order to have reasonable acceptance and accessibility. This causes the coil design to be much more complex. The mechanical structure needs to sustain the decentering force between adjacent coils, and the peak field in the coil is 3–5 times higher than the useful magnetic field for the momentum analysis [149].

28.12. Measurement of particle momenta in a uniform magnetic field [152,153] The trajectory of a particle with momentum p (in GeV/c) and → − charge ze in a constant magnetic field B is a helix, with radius of curvature R and pitch angle λ. The radius of curvature and → − momentum component perpendicular to B are related by p cos λ = 0.3 z B R , (28.44) where B is in tesla and R is in meters. The distribution of measurements of the curvature k ≡ 1/R is approximately Gaussian. The curvature error for a large number of uniformly spaced measurements on the trajectory of a charged particle in a uniform magnetic field can be approximated by (δk)2 = (δkres )2 + (δkms )2 , (28.45) where δk = curvature error δkres = curvature error due to finite measurement resolution δkms = curvature error due to multiple scattering.

28. Particle detectors

If many (≥ 10) uniformly spaced position measurements are made along a trajectory in a uniform medium, # 720  δkres =  2 , (28.46) L N +4 whereN = number of points measured along track L = the projected length of the track onto the bending plane  = measurement error for each point, perpendicular to the trajectory. If a vertex constraint is applied at the origin of the track, the coefficient under the radical becomes 320. For arbitrary spacing of coordinates si measured along the projected trajectory and with variable measurement errors i the curvature error δkres is calculated from: 4 Vss , (δkres ) = w Vss Vs2 s2 − (Vss2 )2 2

(0.016)(GeV/c)z Lpβ cos2 λ

#

L , X0

14. 15.

16. 17.

(28.47)

where V are covariances as Vsm sm sn  − sm sn  sn =  m defined 2 −2 m −1 (si /i ) and w = i . with s  = w The contribution due to multiple Coulomb scattering is approximately δkms ≈

13.

18.

(28.48)

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29. Radioactivity and radiation protection

29. RADIOACTIVITY AND RADIATION PROTECTION Revised Sept. 2007 by S. Roesler (CERN) and J.C. Liu (SLAC).

29.1. Definitions The International Commission on Radiation Units and Measurements (ICRU) recommends the use of SI units. Therefore we list SI units first, followed by cgs (or other common) units in parentheses, where they differ. • Activity (unit: Becquerel): 1 Bq = 1 disintegration per second (= 27 pCi). • Absorbed dose (unit: Gray): The absorbed dose is the energy imparted by ionizing radiation in a volume element of a specified material divided by the mass of this volume element. 1 Gy = 1 J/kg (= 104 erg/g = 100 rad) = 6.24 × 1012 MeV/kg deposited energy. • Kerma (unit: Gray): Kerma is the sum of the initial kinetic energies of all charged particles liberated by indirectly ionizing particles in a volume element of the specified material divided by the mass of this volume element. • Exposure (unit: C/kg of air [= 3880 Roentgen† ]): The exposure is a measure of photon fluence at a certain point in space integrated over time, in terms of ion charge of either sign produced by secondary electrons in a small volume of air about the point. Implicit in the definition is the assumption that the small test volume is embedded in a sufficiently large uniformly irradiated volume that the number of secondary electrons entering the volume equals the number leaving (so-called charged particle equilibrium). • Equivalent dose (unit: Sievert [= 100 rem (roentgen equivalent in man)]): The equivalent dose HT in an organ or tissue T is equal to the sum of the absorbed doses DT,R in the organ or tissue caused by different radiation types R weighted with so-called radiation weighting factors wR :  HT = wR × DT,R . (29.1) R

It expresses long-term risks (primarily cancer and leukemia) from low-level chronic exposure. The values for wR recommended recently by ICRP [1] are given in the following table:

wR

Photons Electrons and muons Neutrons, En < 1 MeV 1 MeV ≤ En ≤ 50 MeV En > 50 MeV Protons and charged pions Alpha particles, fission fragments, heavy ions



29.3. Prompt neutrons at accelerators

1 1 2.5 + 18.2 × exp[−(ln En )2 /6] 5.0 + 17.0 × exp[−(ln(2En ))2 /6] 2.5 + 3.25 × exp[−(ln(0.04En ))2 /6] 2

29.3.1. Electron beams : At electron accelerators, neutrons are generated via photonuclear reactions from bremsstrahlung photons. Neutron yields from semi-infinite targets per unit electron beam power are plotted in Fig. 29.1 as a function of electron beam energy [4]. In the photon energy range 10–30 MeV, neutron production results from the giant photonuclear resonance mechanism. Neutrons are produced roughly isotropically (within a factor of 2) and with a Maxwellian energy distribution described as:

20

En dN = 2 e−En /T , dEn T

• Effective dose (unit: Sievert): The sum  of the equivalent doses, weighted by the tissue weighting factors wT ( T wT = 1) of several organs and tissues T of the body that are considered to be most sensitive [2], is called “effective dose” E: E=

• Natural annual background, all sources: Most world areas, whole-body equivalent dose rate ≈ (0.4–4) mSv (40–400 mrem). Can range up to 50 mSv (5 rem) in certain areas. U.S. average ≈ 3.6 mSv, including ≈ 2 mSv (≈ 200 mrem) from inhaled natural radioactivity, mostly radon and radon daughters. (Average is for a typical house and varies by more than an order of magnitude. It can be more than two orders of magnitude higher in poorly ventilated mines. 0.1–0.2 mSv in open areas.) • Cosmic ray background (sea level, mostly muons): ∼ 1 min−1 cm−2 sr−1 . For more accurate estimates and details, see the Cosmic Rays section (Sec. 24 of this Review). • Fluence (per cm2 ) to deposit one Gy, assuming uniform irradiation: ≈ (charged particles) 6.24×109/(dE/dx), where dE/dx (MeV g−1 cm2 ), the energy loss per unit length, may be obtained from Figs. 27.3 and 27.4 in Sec. 27 of this Review, and pdg.lbl.gov/AtomicNuclearProperties. ≈ 3.5 × 109 cm−2 minimum-ionizing singly-charged particles in carbon. ≈ (photons) 6.24×109/[Ef /], for photons of energy E (MeV), attenuation length  (g cm−2 ), and fraction f  1 expressing the fraction of the photon’s energy deposited in a small volume of thickness   but large enough to contain the secondary electrons. ≈ 2 × 1011 photons cm−2 for 1 MeV photons on carbon (f ≈ 1/2). • Recommended limits of effective dose to radiation workers (whole-body dose):∗ EU/Switzerland: 20 mSv yr−1 U.S.: 50 mSv yr−1 (5 rem yr−1 )† • Lethal dose: The whole-body dose from penetrating ionizing radiation resulting in 50% mortality in 30 days (assuming no medical treatment) is 2.5–4.5 Gy (250–450 rad), as measured internally on body longitudinal center line. Surface dose varies due to variable body attenuation and may be a strong function of energy. • Cancer induction by low LET radiation: The cancer induction probability is about 5% per Sv on average for the entire population [2].

Neutrons dominate the particle environment outside thick shielding (e.g., > 1 m of concrete) for high energy (> a few hundred MeV) electron and hadron accelerators.

Table 29.1: Radiation weighting factors, wR . Radiation type

29.2. Radiation levels [3]

wT × HT .

(29.2)

T

† This unit is somewhat historical, but appears on some measuring instruments. One R is the amount of radiation required to liberate positive and negative charges of one electrostatic unit of charge in 1 cm3 of air at standard temperature and pressure (STP)

(29.3)

where T is the nuclear temperature characteristic of the target nucleus, generally in the range of T = 0.5–1.0 MeV. For higher energy photons, the quasi-deuteron and photo-pion production mechanisms become important. 29.3.2. Proton beams : At proton accelerators, neutron yields emitted per incident proton by different target materials are roughly independent [5] of proton energy between 20 MeV and 1 GeV, and are given by the ratio C:Al:Cu-Fe:Sn:Ta-Pb = 0.3 : 0.6 : 1.0 : 1.5 : 1.7. Above 1 GeV, the neutron yield [6] is proportional to E m , where 0.80 ≤ m ≤ 0.85. A typical neutron spectrum [7] outside a proton accelerator concrete shield is shown in Fig. 29.2. The shape of these spectra are generally characterized as having a low-energy evaporation peak around 1 − 2 MeV, and a high-energy spallation shoulder at around 70 − 80 MeV.

313

29. Radioactivity and radiation protection

150

× 1012 Attenuation length (g cm−2 )

U

W Pb Ta Au Ba

2

Ag

Cu

1 Fe Cu Ag Au Pb Ba

Ni Al Ta

C

40

60 40 100 Electron Energy E0 (MeV)

Figure 29.1: Neutron yields from semi-infinite targets, per kW of electron beam power, as a function of electron beam energy, disregarding target self-shielding.

−3

E × dφ/dE [cm−2 per interaction]

10

concrete iron

−4

10

−5

10

75 50

Concrete ρ = 2.4 g cm− 3

25

1

2

5

29.4. Dose conversion factors 104 Protons π+

103 Muons 100

Photons

Neutrons

10

10−10

10−8

10−6

10−4

0.01

1

100

104

Energy (GeV) Figure 29.4: Fluence to effective dose conversion factors for anterior-posterior irradiation and various particles [9].

−7

10

10−2 10−1 100 101 102 103 104 105 106 107 108 109 Neutron energy [eV]

Figure 29.2: Calculated neutron spectrum from 205 GeV/c hadrons (2/3 protons and 1/3 π + ) on a thick copper target [7]. Spectra are evaluated at 90◦ to beam and through 80 cm of normal density concrete or 40 cm of iron. The neutron-attenuation length, is shown in Fig. 29.3 for concrete and mono-energetic broad-beam conditions. Letaw’s [8] formula for the energy-dependence of the inelastic proton cross-section for E < 2 GeV is:

(29.4)

and for E > 2 GeV: σasympt = 45A0.7 [1 + 0.016 sin(5.3 − 2.63 ln A)] ,

100

1

−6

10

  σ(E) = σasympt 1 − 0.62e−E/200 sin(10.9E −0.28 ) ,

High energy limit

10 20 50 100 200 500 1000 Neutron Energy (MeV) Figure 29.3: The variation of the attenuation length for mono-energetic neutrons in concrete as a function of neutron energy [5].

0 U W

125

0

Fe Ni Al C

Effective Dose Conversion Factor (pSv cm2)

Y (neutrons s−1 kW−1)

3

(29.5)

where σ is in mb, E is the proton energy in MeV and A is the mass number.

Conversion coefficients from fluence to effective dose are given for anterior-posterior irradiation and various particles in Fig. 29.4 [9]. These factors can be used for converting particle fluence to dose for personnel protection purposes. For example, the effective dose from an anterior-posterior irradiation in a field of 1-MeV neutrons with a fluence of 1 neutron / cm2 is about 290 pSv.

29.5. Accelerator-induced activity The dose rate at 1 m due to spallation-induced activity by high energy hadrons in a 1 g medium atomic weight target can be estimated [10] from the following expression: D = D0 Φ ln[(T + t)/t] ,

(29.6)

where T is the irradiation time, t is the decay time since irradiation, Φ is the flux of irradiating hadrons (hadrons cm−2 s−1 ), and D0 has a value of 5.2 × 10−17 [(Sv hr−1 )/(hadron cm−2 s−1 )]. This relation is essentially independent of hadron energy above 200 MeV. Dose due to accelerator-produced induced activity can also be estimated with the use of “ω factors” [5]. These factors give the dose rate per unit star density (inelastic reaction for E > 50 MeV) after a 30-day irradiation and 1-day decay. The ω factor for steel or iron is  3 × 10−12 (Sv cm3 /star). This does not include possible contributions from thermal-neutron activation. Induced activity

314

29. Radioactivity and radiation protection

in concrete can vary widely depending on concrete composition, particularly with the concentration of trace quantities such as sodium. Additional information can be found in Barbier [11].

29.6. Photon sources The dose rate in air from a gamma point source of C Curies emitting one photon of energy 0.07 < E < 4 MeV per disintegration at a distance of 30 cm is about 6 C E (rem/hr), or 60 C E (mSv/hr), ±20%. In general, the dependence of the dose rate from a point source on the distance r follows a 1/r2 behaviour D(r) =

D(r0 ) . (r/r0 )2

(29.7)

The dose rate in air from a semi-infinite uniform photon source of specific activity C (μCi/g) and gamma energy E (MeV) is about 1.07CE (rem/hr), or 10.7CE (mSv/hr). Footnotes: ∗ †

The ICRP recommendation [2] is 20 mSv yr−1 averaged over 5 years, with the dose in any one year ≤ 50 mSv. Many laboratories in the U.S. and elsewhere set lower limits.

References: 1. Recommendation of the International Commission on Radiological Protection, (2007), (in press). 2. ICRP Publication 60, 1990 Recommendation of the International Commission on Radiological Protection, Pergamon Press (1991). 3. See E. Pochin, Nuclear Radiation: Risks and Benefits, Clarendon Press, Oxford, 1983. 4. W.P. Swanson, Radiological Safety Aspects of the operation of Electron Linear Accelerators, IAEA Technical Reports Series No. 188 (1979). 5. R.H. Thomas and G.R. Stevenson, Radiological Safety Aspects of the Operation of Proton Accelerators, IAEA Technical Report Series No. 283 (1988). 6. T.A. Gabriel et al., Nucl. Instrum. Methods A338, 336 (1994). 7. C. Birattari et al., Nucl. Instrum. Methods A338, 534 (1994). 8. J.R. Letaw, R. Silberberg, and C.H. Tsao, Astrophysical Journal Supplement Series, 51, 271 (1983); For improvements to this formula see Shen Qing-bang, “Systematics of intermediate energy proton nonelastic and neutron total cross section,” International Nuclear Data Committee INDC(CPR)-020 (July 1991). 9. M. Pelliccioni, “Overview of fluence-to-effective dose and fluenceto-ambient dose equivalent conversion coefficients for high energy radiation calculated using the FLUKA code,” Radiation Protection Dosimetry 88, 279 (2000). 10. A.H. Sullivan A Guide To Radiation and Radioactivity Levels Near High Energy Particle Accelerators, Nuclear Technology Publishing, Ashford, Kent, England (1992). 11. M. Barbier, Induced Activity, North-Holland, Amsterdam (1969).

30. Commonly used radioactive sources

315

30. COMMONLY USED RADIOACTIVE SOURCES Table 30.1. Revised November 1993 by E. Browne (LBNL). Particle Nuclide 22 Na 11 54 Mn 25

Photon

Type of Energy Emission Half-life decay (MeV) prob. 2.603 y β + , EC 0.545 90%

Energy Emission (MeV) prob. 0.511 Annih. 1.275 100%

0.855 y

EC

0.835 100% Cr K x rays 26%

55 Fe 26

2.73 y

EC

Mn K x rays: 0.00590 24.4% 0.00649 2.86%

57 Co 27

0.744 y

EC

0.014 9% 0.122 86% 0.136 11% Fe K x rays 58%

60 Co 27

5.271 y

β−

0.316

100%

1.173 1.333

100% 100%

68 Ge 32

0.742 y EC Ga K x rays 44% ----------------------------------------------------→ 68 β + , EC 1.899 90% 0.511 Annih. 31 Ga 1.077 3% 28.5 y β − 0.546 100% ----------------------------------------------------→ 90 β− 2.283 100% 39 Y

90 Sr 38

1.020 y β − 0.039 100% ----------------------------------------------------β− 3.541 79% 0.512 21% → 106 45 Rh 0.622 10%

106 Ru 44

109 Cd 48

1.267 y

EC

0.063 0.084 0.087

e− e− e−

41% 45% 9%

0.088 3.6% Ag K x rays 100%

113 Sn 50

0.315 y

EC

0.364 e− 0.388 e−

29% 6%

0.392 65% In K x rays 97%

137 Cs 55

30.2 y

β−

0.514 1.176

94% 6%

0.662

133 Ba 56

10.54 y

EC

0.045 e− 0.075 e−

50% 6%

0.081 34% 0.356 62% Cs K x rays 121%

207 Bi 83

31.8 y

EC

0.481 0.975 1.047

e− e− e−

2% 7% 2%

0.569 98% 1.063 75% 1.770 7% Pb K x rays 78%

228 Th 90

1.912 y

6α: 3β − :

241 Am/Be 95

432.2 y

6 × 10−5 neutrons (4–8 MeV) and 4 × 10−5 γ’s (4.43 MeV) per Am decay

244 Cm 96

18.11 y

252 Cf 98

2.645 y α (97%) 6.076 15% 6.118 82% Fission (3.1%) ≈ 20 γ’s/fission; 80% < 1 MeV ≈ 4 neutrons/fission; En  = 2.14 MeV

85%

0.239 44% 0.583 31% 2.614 36% → 220 → 216 → 212 → 212 → 212 (→224 88 Ra 86 Rn 84 Po 82 Pb 83 Bi 84 Po) 241 Am 432.7 y α 5.443 13% 0.060 36% 95 5.486 85% Np L x rays 38%

α

5.341 to 8.785 0.334 to 2.246

5.763 5.805

24% 76%

Pu L x rays ∼ 9%

“Emission probability” is the probability per decay of a given emission; because of cascades these may total more than 100%. Only principal emissions are listed. EC means electron capture, and e− means monoenergetic internal conversion (Auger) electron. The intensity of 0.511 MeV e+ e− annihilation photons depends upon the number of stopped positrons. Endpoint β ± energies are listed. In some cases when energies are closely spaced, the γ-ray values are approximate weighted averages. Radiation from short-lived daughter isotopes is included where relevant. Half-lives, energies, and intensities are from E. Browne and R.B. Firestone, Table of Radioactive Isotopes (John Wiley & Sons, New York, 1986), recent Nuclear Data Sheets, and X-ray and Gamma-ray Standards for Detector Calibration, IAEA-TECDOC-619 (1991). Neutron data are from Neutron Sources for Basic Physics and Applications (Pergamon Press, 1983).