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The ionisation loss of relativistic charged particles in thin gas samples and its use for particle identification I. Theoretical predictions
NUCLEAR
INSTRUMENTS
AND
METHODS
I33
0976) 3 1 5 - 3 2 3 ;
©
NORTH-HOLLAND
PUBLISHING
CO.
T H E I O N I S A T I O N L O S S OF R E L A T I V I S T I C C H A R G E D P A R T I C L E S I N T H I N GAS S A M P L E S AND ITS USE FOR P A R T I C L E I D E N T I F I C A T I O N
I. Theoretical predictions J. H. COBB*, W. W. M. A L L I S O N and J. N. B U N C H
Nuclear Physics Laboratory Oxford, England Received 21 N o v e m b e r 1975 A brief review s h o w s a significant discrepancy between available data and theoretical predictions o n the ionisation loss o f charged particles in thin gas-filled proportional counters. T h e discrepancy relates both to the increase o f the m o s t probable loss at relativistic velocities (" relativistic rise") and to the s p e c t r u m o f such losses at a given velocity (the " L a n d a u distribution"). T h e origin o f this relativistic rise is discussed in simple terms a n d related to the p h e n o m e n a o f transition radiation and C h e r e n k o v radiation. W e s h o w that the failure o f the predictions is due to the small n u m b e r o f i o n i s i n g collisions in a gas. This problem is overcome by using a M o n t e Carlo m e t h o d rather t h a n a c o n t i n u o u s integral over the spectrum o f single collision processes. A specific model o f the atomic f o r m factors is used with a modified Born a p p r o x i m a t i o n to yield the differential cross-sections needed for the calculation. T h e new predictions give improved agreement with experiment and are used to investigate the p r o b l e m o f identifying particles o f k n o w n m o m e n t a in the relativistic region. We show that by m e a s u r i n g the ionisation loss o f each particle several h u n d r e d times over 5 m or more, kaon, pion a n d proton separation with good confidence level m a y be achieved. M a n y gases are considered and a c o m p a r i s o n is made. T h e results are also c o m p a r e d with the velocity resolution achievable by m e a s u r i n g primary ionisation.
1. Introduction The mean energy loss particles in thin absorbers velocity, fl, of the particle fact it is more convenient to
p/mc =
fly
=
f(1 _f12)--~.
of high velocity charged is a function only of the and not of its massl). In consider its dependence on (1)
In terms of this there are three relatively well defined regions: a) For p/me<4, the familiar non relativistic region in which the energy loss goes like f - 2 for all media. b) Forp/me>4, there is a slow logarithmic increase (the "relativistic rise") which extends up to a value of p/mc depending on the nature of the absorber and its density. c) A flat region beyond the region of the rise where no further increase takes place (the "Fermi plateau "). In this paper we explore what is known about the logarithmic increase and the plateau region as observed in thin gas counters and how it may be applied to the problem of particle identification at high energy. In section 2 we review existing data and current models, in section 3 we discuss the origin of the "relativistic rise" and other relativistic phenomena in terms of simple models. In section 4 we set up a * N o w at C E R N .
specific model for the calculation of energy loss spectra and in section 5 we apply it to a variety of gases. These results are used to estimate the particle identification efficiency of ionisation sampling chambers 2) with different gas fillings.
2. Status of theory and experiment The experimental and theoretical status of the relativistic rise region has been reviewed by Crispin
Various PC data on relativistic/ / i rise in argon a t 1 a t m . ,
1.6-
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1.5 ¸ t/
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1.o-
10
I00
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Fig. I. Calculations and m e a s u r e m e n t s o f the relativistic rise o f energy loss in a r g o n filled proportional chambers. Curve I is based on the m e t h o d o f Sternheimer (1971) ref. 6. Curve II is based on the m e t h o d described in this paper.
316
J . H . COBB et al.
and Fowler1). In the case of measurements which can be simply related to calculations there is good agreement for dense materials for which the relativistic increase rarely exceeds 20 %. Some difficulty surrounds the interpretation of emulsion and bubble chamber data since they are not related in a simple way to the energy loss. This is not serious. In the case of ionisation loss in gases on the other hand the interpretation of the measurements is simpler but the discrepancy with calculation appears large. Fig. 1 shows the results of a variety of measurements of the relative ionisation loss of relativistic particles in thin proportional chambers filled with argon at 1 atm3-S). The dashed curve 1 has been calculated according to the model of Sternheimer 6) for the most probable ionisation loss in a chamber 1.5 cm thick. The measurements favour a relativistic rise of about 50% whereas the model predicts about 7 0 % 5 ' 7 ' 8 ) . Fig. 2 shows that the discrepancy goes further. The histogram is the observed ionisation loss spectrum of 3 GeV/c ~- in 1.5 cm (2.7 mg/cm 2) of argon 3' 9). The distribution is remarkable for its width and long tail. Curve I is the prediction of the original Landau theory 1°) which is used explicitly in the derivation of the most probable energy loss [curve 1 in fig. 16)]. It is an extremely bad fit to the experimental distribution. The model of Blunck and Leisegang ~2) does not fit the data either (curve II). The reason is clear. The number of ionising collisions is small ( ~ 30/cm) and the most probable energy loss cannot involve any contribution from inner electrons with large binding energies ~1). For instance, in argon the most probable loss is about 2 keV/cm, whereas the K shell binding energy is ! .~~1
2000, 1800-
Pulse Height Distribution 1.5crns argon
i •
1600.
3GcV fr-Expcr[mcnt { histogram) t i t 3 3GcV/fMont¢ Carto(cor~inuou$ |in¢ )
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1200 1000 8OO
600 400 200 0
J 1
2
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i 3
4
5
6
7
8
9
10 11 12 13 14 15 KeV
Fig. 2. A histogram o f a typical pulse height distribution from a thin proportional chamber. The smooth curve is based on the model described in this paper and ref. 9. It does not include any instrumental or resolution effects.
already 3 keV. This shows that we should try first to abandon the use of a " m e a n ionisation potential" and work with a specific spectrum of atomic oscillator strengths and second to replace the continuous integral over possible collisions by a corresponding random summation of discrete energy losses by a Monte Carlo method, lspiryan et al. ~3) have already shown that the technique gives improved agreement with the Landau distribution.
3. The origin of the relativistic rise Before pursuing quantitative questions further we pause to discuss the origin of the relativistic rise and the Fermi plateau. This will suggest qualitatively what rise we can expect in particular gases. In section 5 we confirm these expectations with detailed calculations. The electromagnetic field of a relativistic charged particle can be represented as a Fourier integral over plane wave components with wave number q, frequency co. The energy loss process involves the scattering of one of these virtual photons by the medium resulting in an energy loss heJ and momentum transfer hq. The interaction through virtual longitudinal photons (E parallel to q) represents the instantaneous Coulomb interaction and is effectively constant at high velocities15). The increased range of the transverse photons on the other hand is responsible for the "relativistic rise" of the energy loss. These virtual photons have a very broad spectrum simply because they relate to the transform of the field of a point charge. This broad spectrum leads in turn to significant ionisation cross-sections over a wide range of energy transfer even for atomic electrons with a unique binding energy. In considering a finite thickness of absorber we sum over all energy loss processes including the large energy loss collisions which have a low probability. The latter are responsible for the long tail observed in the energy loss spectrum shown in fig. 2. Why does the interaction through transverse photons increase and then saturate? It is well known that in vacuum the electric field of a relativistic particle expands in the transverse dimension as ? increases as a kinematic consequence of special relativity. In a medium on the other hand this increase does not continue indefinitely. The effect of the medium may be described by a screening length, 2, which is the Compton wavelength of the renormalised photon in a medium of dielectric constant ~(~o). In simple terms, since electromagnetic waves do not travel with velocity c in a medium, the corresponding photons have mass,
317
I O N I S A T I O N LOSS I
II, given by
same direction, ~,, with the same energy, ha).
~ = -5- (I - v Z / c Z ) ~ = c
I -
(w,,/'~)
(2)
where t, is the group velocity at frequency co. Exchange of such quanta will be characterised by a range equal to the Compton wavelength*,
Apx -
hco
he)
fie
c
_
C
/2C
1--
d
(wx/~)
.
- \/e cos
.
At high frequencies when the binding energy of the electrons may be neglected this reduces to
Of course, when I/fl = x/e cos ~k, the range is infinite (for real e). This describes Cherenkov radiation. Making further approximations to emphasise the dependence on velocity, we have
(4) Rx ~~ __2c (~,- 2 + ~2 + ~ 2 ) - 1 ,
where coo is the plasma frequency. We may compare this screening length with the maximum impact parameter for which the collision is sufficiently " s u d d e n " to transfer energy h~o to an atom. Jackson1) shows that
b .... = 7v/oJ. It follows that we may expect the relativistic rise due to the increase of b ~ to saturate when bm~ = 2 or when fly "-
w -
-
wp
=
ionisation energy plasma energy
,
(5)
in the high frequency case. Before discussing this result we derive it in a different way 9) which brings out relationships with the phenomena of Cherenkov and transition radiation and is based on the uncertainty principle. A virtual photon associated with a particle of velocity tic along the xaxis must itself have a phase velocity tic along x. Otherwise it would not be seen to an observer travelling in the particle frame as a component of a static Coulomb field. It therefore has a momentum projection along x given by hw p~ =
-
(9)
(3)
W
2 = cI~%,
(8)
where \,/e is the refractive index of the medium. This gives a range • = --
li
),_
x/~ c o s ~ ,
-
(0)
The range of such a virtual photon is given by Heisenberg's uncertainty principle:
Rx = h/Apx,
(7)
where Ap~ is the difference between its momentum and that of a free photon in the medium travelling in the * This is only true in the forward direction. In particular the argument fails completely in the case o f emission at the Cherenkov angle.
(10)
(2)
where we have taken ~, small, y = (I _f12)-~ large and e = 1 _ ( 2 . This formula is the same as the expression for the "formation zone" discussed by Garibyan 14) for transition radiation. It shows that the range of virtual photons increases with y until ? is of order ~- 1, where for high frequencies ~-~,~ho~/plasma energy. So again we expect the relativistic rise to give way to the Fermi plateau region when ?~
ionisation energy plasma energy
(11)
This suggests that the largest relativistic rise will be seen in gases because of their low plasma energies, especially those with high Z, for which the ionisation potentials increase faster than the plasma energy. Further, of those with similar numbers of electrons, the rare gases with the tightest binding will show the largest rise. In addition different electron shells will saturate at different values of ? and calculations should take account of this. In particular if measurements are made which effectively exclude the contribution from inner electrons - such as the most probable ionisation loss in thin gas counters - the relativistic rise will be less than conventional calculations might suggest. 4. The model
Following Fano iS) we start by considering the differential cross section per atomic electron for the inelastic collision of an isolated atom with the incident particle according to the Born approximation:
d2a dQdE
2he4 [[F(E'q)[2 + [~"G(E'q)[2 I my z [ Q2 (Q--U~_~)~],
(12)
318
J . H . COBB et al.
where v = fic is the incident particle velocity, Q = h2qE/2m and m is the electron mass. F and G are inelastic form factors discussed by Fano 15). We have d r o p p e d some terms in Q/mc 2 which are only i m p o r t a n t for large energy losses and are consequently of no interest when considering thin absorbers. The first term represents the interaction of charges and as such is essentially constant in the relativistic region. It describes ionisation due to the absorption of longitudinal virtual photons. These photons have a range ~ 1/q as in the static c o u l o m b field hence the d e n o m i n a t o r Q2. The second term is the currentcurrent term involving the exchange of transverse virtual photons. These two terms add incoherently because they involve transitions to final states of opposite parity. When q is small, dipole transitions will predominate and we can follow F a n o ~5) by approximating: IF(E,q)l 2 dE = (Q/E) f ( E ) d E ,
(13)
and lilt" G(E,q)i 2 dE = ~2 f ( E ) dE E / 2 m c 2 ,
(14)
w h e r e f ( E ) d E is the optical dipole oscillator strength. The definition of small q in this respect is
q < (electron orbital size)-1, which is essentially: Q < E , , where E i is the electron binding energy. The longitudinal term is evaluated in two different regions: 1) Low Q, i.e. Q < El with dipole approximation and the choice f ( E ) = 6 ( E - E3, so [F(E,q)I 2 dE = (Q/E,) b ( E - E , ) d E .
(15)
do" L
dE
2 7re'* 1 - -my 2 E E '
for E > E,,
(18)
the latter being of course the Rutherford crosssection. There are two important points here. The first is that our crude approximations satisfy the sum rule*, are consistent with the longitudinal part of the BetheBloch formula for the mean energy l o s s 9) and incorporate some not too implausible low energy behaviour. The second is that, by using the sum rule to integrate over energy, traditional calculations avoid having to make some of our crude assumptions. We are unable to do this since an energy integration is inappropriate when the discrete energy loss collisions are few. This integration must be deferred to the end of our calculation and be carried out as a Monte Carlo process using the spectrum of energy loss probabilities. The transverse term only contributes significantly in the low Q region 15) and m a y therefore be evaluated in the dipole approximation. If ~, is the angle between the virtual photon direction and the particle momentum, we have fi2 = fi2 sinZt~ and Q = E2/Emv 2 C O S 2 I ~ /
.
Substituting we get 15) d Etr
27re 4 sin E ~b f ( E ) d(cos E ¢)
dE
Emv 2
(1/fi2 _ cos E ~k)E
This formula applies to an electron of an isolated a t o m rather than one in a medium - i.e. in the approximation e = 1. To see how it is to be modified we refer back to eq. (9). The d e n o m i n a t o r there contains a V/e factor and a similar d e n o m i n a t o r appears in eq. (19) representing the effect of the finite range or p r o p a g a t o r of the virtual photon. Instead of eq. (19) we write: d20 " _ 2roe 4 sin 2 t~ f ( E ) d(cos 2 ~)
This choice satisfies the sum rule ~EIF[ Ed E = Q. 2) High Q, i.e. Q > E, with impulse or free electron approximation IF(E,q)l 2 dE = 6 ( E - Q )
dE,
(16)
which also satisfies the sum rule. Taking these two together and integrating over Q we get
(19)
dE
Emv 2
(20)
(1/fi E- 5 cos E t/J)2
which is related to the formula given by Fano t 5). Further evaluation is simplified by the observation 16) that f (E) -
(21)
moo Jm(e((o)), 2 7~2 e 2 h N
where N is the electron density. dO'L --
2rte4 6 ( E - E i ) log 2mv---~2,
dE
my E
for E ~< Ei;
E,
dEa
E,
eE~o
sine (J d(cos2 @)
dE - ETrhvE~--Nf l m ( e ( w ) ) [l/fiE _ e(co) COS2 ~b[2' (22)
(17)
*
Eq. (13}.
IONISATION
319
LOSS I
TABLE l Atomic data used in calculations. Helium
Plasma energy Binding energies (eV) and electrons per molecule
Neon
Argon
Krypton
Xenon
0.272
0.609
0.816
1.156
1.414
24.5 (2)
870 (2) 54.4 (8)
3196 (2) 294 (8) 39.5 (8)
14280 (2) 1754 (8) 152 (18) 39.4 (8)
34612 5073 831 169 25.8
(2) (8) (18) (18) (8)
Methane
0.609 313 55.8 17.7 13.6
(2) (2) (2) (4)
Ammonia
Nitrogen
Carbon dioxide
0.609
0.721
0.902
412 (4) 47.6 (2) 31.3 (6)
313 (2) 55.8 (2) 17.7(2) 575 (4) 54.4 (4) 39.4 (8)
412 47.6 31.3 13.6
(2) (2) (3) (3)
Argon/20% Co2, Plasma energy = 0.834 eV He/50% Ne, Plasma energy = 0.472 eV Atomic data from Sternheimer (1952), ref. 6. Chemical effects neglected.
_-
dE
+
nh2 v2 N
(23)
arctan
by integrating cos2~b from 0 to I and taking e = el For e we use
~.
f,
,
+je2.
(24)
where we take Yl = E*. 71 is the "width" of the effective ionisation level which takes account of the finite kinetic energy of the electron in the continuum on the one hand and energy loss through excitation levels on the other. This is crude but it is a significant improvement over the use of a mean ionisation potential. 5. R e s u l t s o f c a l c u l a t i o n s
We have calculated 17) the energy loss spectra in 1.5 cm of gas at NTP at values of p/mc from 2 to 2000 in powers of 2 and also 5 × l04. For each gas or gas mixture a knowledge of the atomic levels Ei and the plasma frequency were sufficient to determine the different parts of the cross-section according to the model described in the previous section. The results are only weakly dependent on the values of E~ used. These are shown in table I and are taken from Sternheimer (1962)6). A simple Monte Carlo calculation then gave the differential energy loss distribution in 1.5 cm of gas. The full curve in fig. 2 shows the result for 3 GeV/c *
The results o f calculations are essentially unaffected by other choices e.g. ~i = 0.2El,
pions in argon9). The agreement with the data is significantly better than for the Landau and Blunck and Leisegang models. By plotting the peak of the Monte Carlo energy loss distribution as a function of p/mc, a prediction for the relativistic rise is obtained which is also in better agreement with available data. This is shown as curve II in fig. 1. Of course different experiments use different gas mixtures, sample thicknesses and methods of estimating the peak or most probable ionisation loss. We defer to paper I118) a comparison of the model with new data where these effects are treated in detail. However, neither in the case of the old data nor in the case of the new is there any indication of a real discrepancy with the model. Next we use the model to discuss the use of the relativistic rise in different gases for identifying particles of known momentum. As discussed elsewhere2), the combined effect of the slow rise giving little difference between masses and the broad energy loss spectrum giving poor resolution is to require more than 100 energy-loss measurements on each track if individual pions, kaons and protons are to be distinguished. In a second Monte Carlo program we have simulated such a multiple sampling device and studied the mass resolution that may be derived from the measurements on a single track. The larger and less probable measurements (the Landau tail) contain little information and only serve to degrade the variance of the smaller more probable measurements. Traditionally the peak of the distribution, the most probable energy-loss, is used but this is notoriously hard to estimate with small statistics. It has been shown 5) that the resolution is close to optimum if a fixed fraction of the measurements (20-50% of the largest) is discarded and the
320
J.H. COBB et al.
~
ARGON I 2O% CO2
~5
HAMBER I 1.5crn SAMPLE NO CROSS TALK)
rn SAMPLES
5
I
4
d-3
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22 bJ 1
lb
o
zb
3b
Lo
do
so
SAMPLES REJECTED BY CUT
o
(a)
ARGON/ 20% CO2 7metre CHAMBER I I-5cm SAMPLE THEORY
S
4'
3' 2"
1
-20'
-1's
-lb
~
6
~
(b)
3E
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~
DEVICE LENGTH ( m )
%
-g
g
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l's
zb
(c) mean of the remainder is used as the estimator. This is the method we have used. Fig. 3a shows how the resolution of 330 x 1.5 cm 2 samples of argon varies as a function of this cut. Both this resolution and the corresponding relativistic rise are rather insensitive to the size of this c u t - in fact according to the calculations the rise is less than 5 % different for a 50 % cut than for a 10% cut. In all of the following results we have used a 40 % cut. in fig. 3b we show how the resolution behaves as a
Fig. 3. The dependence o f ionisation resolution on various parameters o f a multiple sampling technique as found in Monte Carlo studies: (a) Fraction o f samples rejected by cut to remove the Landau tail (see text). (b) Device length and sample size. (c) Nearest neighbour intersample cross-talk.
function of number of samples and device length. Typically a 5 m device with 1.5 cm samples is comparable to a 7 m device with 3 cm samples. There is no easy cheap way to get the required resolution (5-6 %). A device of several metres depth and several hundred samples is needed. In the real world there are systematic effects also. Fig. 3c shows the effect of a small amount of nearest neighbour inter-sample crosstalk as simulated by the Monte Carlo program. The effect on the resolution is small. These problems are discussed at greater length in paper 1II8). In the following analysis we are concerned primarily with the efficacy of different gases and therefore restrict our calculations to a 5 m device yielding 3 3 0 x l . 5 cm 2 samples and ignore possible systematic effects. Fig. 4a shows the response* expected from such a device filled with argon and exposed to a mixture of pions, kaons and protons at 25 GeV/c in the ratio 10:1:1. In this case we can identify particles with good confidence. At 60GeV/c on the other hand the separation is marginal as shown in fig. 4b. An optimum cut between kaons and pions would result in a loss of about 10% of kaons into the pion peak and the same number of pions misidentified as kaons. In order to * The " r e s p o n s e " is the mean o f the 60% smallest signals discussed above.
321
I O N I S A T I O N LOSS I
/ 1.2
1.3
~o
1.4
1:5
1.3
1!4
~/lo
1'.s
1.6
Fig. 4. Ideal response of a 5 m detector to a mixture of:'t, K and protons in flux ratio 10: I : 1 ; (a) shows the separation at 25 GeV/c, (b) shows the separation at 60 GeV/c. This is very close to the case described as " 9 0 % s e p a r a t i o n " (see text).
make a definitive comparison between different gases we have calculated at what momentum the kaon/pion separation is 90 % pure in the above sense. For each gas and value ofp/mc in turn we generated an energy loss distribution appropriate to 1.5 cm of gas at NTP (20 000 points). In the second Monte Carlo program we simulated 2000 traversals of the detector described above. Some of the results are given in table 2. We estimate that the statistical inaccuracies on these results are less than 1%. For reference we have also calculated the value of the most probable ionisation loss. The full results will be available elsewhere' 7). The rare gases are treated in the first section of table 2. In spite of a big increase in the energy loss with atomic number the resolution is essentially constant at around 5%. The relativistic rise on the other hand increases from 4 0 % to 70%. This is due to the binding energy which is increasing faster than the plasma frequency (see section 3). The kaonlpion separation limit which is 55-60 GeV/c for argon and krypton is 95 GeV/e for pure xenon. (The lower limit for Kin separation is 1.8 GeV/c for all gases). The second group in table 2 compares the first chemical period gases methane, ammonia, neon and nitrogen*. The first three of these all have the same plasma frequency. They show that as the binding energy increases, so does the relativistic rise, while the
* We ignore all chemical effects,
mean energy loss gets smaller. There is also an improved resolution when the binding energy is smaller due to the increased statistics of small energy-loss collisions. However, these have a small contribution to the rise and therefore the K/~ separation limit is still worse for methane and ammonia than neon. The third section of table 2 shows how gas mixtures behave. The argon-carbon dioxide system is the example. It shows that the addition of small quantities of a low-Z quenching gas has a small effect. The ionisation of gas mixtures is essentially additive in the Born approximation except for the modification due to the dielectric constant. In the last section of table 2 we show figures for the total ionisation cross-section in the form of the number of ionising collisions per metre of gas ("primary ionisation"). Davidenko ' 9) has proposed that particles could be identified by measuring their primary ionisation in a streamer chamber operating in the avalanche mode. The resolution of such a technique would be limited under ideal conditions by Poisson statistics. In table 2 we assume such a limit and calculate the K/~ separation limit discussed above for a helium/neon mixture and an argon/CO 2 mixture. The relativistic rise is smaller due to the equal weighting of large and small energy-loss collisions. In dE/dx measurements on the other hand the large energy loss collisions which show the larger relativistic rise are weighted more than the small energy loss collisions. On the other hand the ideal resolution is much better than in dE/dx measure-
322
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increase o f i o n i s a t i o n loss a n d its s a t u r a t i o n at the h i g h e s t velocities. T h e s e c a l c u l a t i o n s give a g r e e m e n t with existing d a t a a n d s h o w q u a n t i t a t i v e l y the efficacy o f different gases for m e a s u r i n g velocities o r identifying c h a r g e d particles o f k n o w n m o m e n t u m . A r g o n is b o t h effective, practical a n d cheap. X e n o n is the o n l y gas e x a m i n e d which is m o r e effective.
20%
-1 3 -1.2 1.1
~0
16o0 Pl~
10 10~oo
Fig. 5. Plots of the calculated ionisation loss for some interesting cases (table 2 and ref. 17).
m e n t s a l t h o u g h there are m a j o r technical difficulties in a c h i e v i n g this. Finally, we n o t e t h a t t h e r e is no d i s c e r n i b l e difference b e t w e e n the relativistic rise o f the m o s t p r o b a b l e a n d the t r u n c a t e d m e a n i o n i s a t i o n . T h i s is r e l a t e d to the fact t h a t t h e s h a p e o f the e n e r g y loss d i s t r i b u t i o n d o e s n o t a p p e a r to c h a n g e m u c h as a f u n c t i o n o f p/mc. Fig. 5 s h o w s a few o f the c a l c u l a t e d relativistic rise curves.
6. Conclusion The discrepancy between theoretical predictions and e x p e r i m e n t a l results f o r the e n e r g y - l o s s in thin gas s a m p l e s seems to be resolved. S i m p l e q u a l i t a t i v e pictures a n d d e t a i l e d q u a n t i t a t i v e c a l c u l a t i o n s t o g e t h e r give insight i n t o the m e c h a n i s m o f the relativistic
References 1) j. D. Jackson, Classical electrodynamics (J. Wiley, New York) Ch. 13; A. Crispin and G. N. Fowler, Rev. Mod. Phys. 42 (1970) 290. 2) W. W. M. Allison et al., Nucl. Instr. and Meth. 119 (1974) 499. 5) F. Harris et al., Nucl. Instr. and Meth. 107 (1973) 413. a) p. V. Ramana Murthy, Nucl. Instr. and Meth. 63 (1968) 77. 5) D. Jeanne et al., Nucl. Instr. and Meth. 111 (1973) 287; M. Aderholz et al., Nucl. Instr. and Meth. 118 (1974) 419. 6) R. M. Sternheimer and R. F. Peierls, Phys. Rev. B3 (1971) 3681 ; R. M. Sternheimer, Phys. Rev. 88 0952) 851. 7) Z. Dimcovski et al., Nucl. Instr. and Meth. 94 (1971) 151. 8) A. V. Alakoz et al., Nucl. Instr. and Meth. 124 (1975) 41. 9) j. H. Cobb, D. Phil. Thesis (Univ. of Oxford, 1975). Available from Rutherford as HEP/T/55. 10) L. Landau, J. Phys. (USSR) 8 (1944) 201. la) G. Knop et al., Z. Physik 165 (1961) 533. 12) O. Blunck and S. Leisegang, Z. Physik 128 (1950) 500. 13) K. A. lspiryan et al., Nucl. Instr. and Meth. 117 (1974) 125. 14) G. M. Garibyan, Yerevan Report: Theoretical foundations of transition radiation (1970); See also L. Durand, Phys. Rev. O l l (1975) 89. 15) U. Fano, Ann. Rev. Nucl. Sci, 13 (1963) 1. 16) L. Landau and E. Lifshitz, Electrodynarnics of continuous media, (Pergamon Press, 1960) Ch. 62, eq. (13). 17) j. N. Bunch, D. Phil. Thesis (Univ. of Oxford) in preparation. 18) W. W. M. Allison et al., following paper (this issue), p. 325. 19) V. A. Davidenko, Nucl. Instr. and Meth. 67 (1969) 325.
INSTRUMENTS
AND
METHODS
I33
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NORTH-HOLLAND
PUBLISHING
CO.
T H E I O N I S A T I O N L O S S OF R E L A T I V I S T I C C H A R G E D P A R T I C L E S I N T H I N GAS S A M P L E S AND ITS USE FOR P A R T I C L E I D E N T I F I C A T I O N
I. Theoretical predictions J. H. COBB*, W. W. M. A L L I S O N and J. N. B U N C H
Nuclear Physics Laboratory Oxford, England Received 21 N o v e m b e r 1975 A brief review s h o w s a significant discrepancy between available data and theoretical predictions o n the ionisation loss o f charged particles in thin gas-filled proportional counters. T h e discrepancy relates both to the increase o f the m o s t probable loss at relativistic velocities (" relativistic rise") and to the s p e c t r u m o f such losses at a given velocity (the " L a n d a u distribution"). T h e origin o f this relativistic rise is discussed in simple terms a n d related to the p h e n o m e n a o f transition radiation and C h e r e n k o v radiation. W e s h o w that the failure o f the predictions is due to the small n u m b e r o f i o n i s i n g collisions in a gas. This problem is overcome by using a M o n t e Carlo m e t h o d rather t h a n a c o n t i n u o u s integral over the spectrum o f single collision processes. A specific model o f the atomic f o r m factors is used with a modified Born a p p r o x i m a t i o n to yield the differential cross-sections needed for the calculation. T h e new predictions give improved agreement with experiment and are used to investigate the p r o b l e m o f identifying particles o f k n o w n m o m e n t a in the relativistic region. We show that by m e a s u r i n g the ionisation loss o f each particle several h u n d r e d times over 5 m or more, kaon, pion a n d proton separation with good confidence level m a y be achieved. M a n y gases are considered and a c o m p a r i s o n is made. T h e results are also c o m p a r e d with the velocity resolution achievable by m e a s u r i n g primary ionisation.
1. Introduction The mean energy loss particles in thin absorbers velocity, fl, of the particle fact it is more convenient to
p/mc =
fly
=
f(1 _f12)--~.
of high velocity charged is a function only of the and not of its massl). In consider its dependence on (1)
In terms of this there are three relatively well defined regions: a) For p/me<4, the familiar non relativistic region in which the energy loss goes like f - 2 for all media. b) Forp/me>4, there is a slow logarithmic increase (the "relativistic rise") which extends up to a value of p/mc depending on the nature of the absorber and its density. c) A flat region beyond the region of the rise where no further increase takes place (the "Fermi plateau "). In this paper we explore what is known about the logarithmic increase and the plateau region as observed in thin gas counters and how it may be applied to the problem of particle identification at high energy. In section 2 we review existing data and current models, in section 3 we discuss the origin of the "relativistic rise" and other relativistic phenomena in terms of simple models. In section 4 we set up a * N o w at C E R N .
specific model for the calculation of energy loss spectra and in section 5 we apply it to a variety of gases. These results are used to estimate the particle identification efficiency of ionisation sampling chambers 2) with different gas fillings.
2. Status of theory and experiment The experimental and theoretical status of the relativistic rise region has been reviewed by Crispin
Various PC data on relativistic/ / i rise in argon a t 1 a t m . ,
1.6-
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®
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1.5 ¸ t/
///
O 0
*
0
1.4.
1.3/ 1.2/ 1.1-
/
//
0 Ramana Murthy rcf4 ® Harris ref 3 ÷ Jeanne, Aderhc4z rcf 5
j/ Plmc
1.o-
10
I00
I0()0
I0000
Fig. I. Calculations and m e a s u r e m e n t s o f the relativistic rise o f energy loss in a r g o n filled proportional chambers. Curve I is based on the m e t h o d o f Sternheimer (1971) ref. 6. Curve II is based on the m e t h o d described in this paper.
316
J . H . COBB et al.
and Fowler1). In the case of measurements which can be simply related to calculations there is good agreement for dense materials for which the relativistic increase rarely exceeds 20 %. Some difficulty surrounds the interpretation of emulsion and bubble chamber data since they are not related in a simple way to the energy loss. This is not serious. In the case of ionisation loss in gases on the other hand the interpretation of the measurements is simpler but the discrepancy with calculation appears large. Fig. 1 shows the results of a variety of measurements of the relative ionisation loss of relativistic particles in thin proportional chambers filled with argon at 1 atm3-S). The dashed curve 1 has been calculated according to the model of Sternheimer 6) for the most probable ionisation loss in a chamber 1.5 cm thick. The measurements favour a relativistic rise of about 50% whereas the model predicts about 7 0 % 5 ' 7 ' 8 ) . Fig. 2 shows that the discrepancy goes further. The histogram is the observed ionisation loss spectrum of 3 GeV/c ~- in 1.5 cm (2.7 mg/cm 2) of argon 3' 9). The distribution is remarkable for its width and long tail. Curve I is the prediction of the original Landau theory 1°) which is used explicitly in the derivation of the most probable energy loss [curve 1 in fig. 16)]. It is an extremely bad fit to the experimental distribution. The model of Blunck and Leisegang ~2) does not fit the data either (curve II). The reason is clear. The number of ionising collisions is small ( ~ 30/cm) and the most probable energy loss cannot involve any contribution from inner electrons with large binding energies ~1). For instance, in argon the most probable loss is about 2 keV/cm, whereas the K shell binding energy is ! .~~1
2000, 1800-
Pulse Height Distribution 1.5crns argon
i •
1600.
3GcV fr-Expcr[mcnt { histogram) t i t 3 3GcV/fMont¢ Carto(cor~inuou$ |in¢ )
1400,
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1200 1000 8OO
600 400 200 0
J 1
2
\~
i 3
4
5
6
7
8
9
10 11 12 13 14 15 KeV
Fig. 2. A histogram o f a typical pulse height distribution from a thin proportional chamber. The smooth curve is based on the model described in this paper and ref. 9. It does not include any instrumental or resolution effects.
already 3 keV. This shows that we should try first to abandon the use of a " m e a n ionisation potential" and work with a specific spectrum of atomic oscillator strengths and second to replace the continuous integral over possible collisions by a corresponding random summation of discrete energy losses by a Monte Carlo method, lspiryan et al. ~3) have already shown that the technique gives improved agreement with the Landau distribution.
3. The origin of the relativistic rise Before pursuing quantitative questions further we pause to discuss the origin of the relativistic rise and the Fermi plateau. This will suggest qualitatively what rise we can expect in particular gases. In section 5 we confirm these expectations with detailed calculations. The electromagnetic field of a relativistic charged particle can be represented as a Fourier integral over plane wave components with wave number q, frequency co. The energy loss process involves the scattering of one of these virtual photons by the medium resulting in an energy loss heJ and momentum transfer hq. The interaction through virtual longitudinal photons (E parallel to q) represents the instantaneous Coulomb interaction and is effectively constant at high velocities15). The increased range of the transverse photons on the other hand is responsible for the "relativistic rise" of the energy loss. These virtual photons have a very broad spectrum simply because they relate to the transform of the field of a point charge. This broad spectrum leads in turn to significant ionisation cross-sections over a wide range of energy transfer even for atomic electrons with a unique binding energy. In considering a finite thickness of absorber we sum over all energy loss processes including the large energy loss collisions which have a low probability. The latter are responsible for the long tail observed in the energy loss spectrum shown in fig. 2. Why does the interaction through transverse photons increase and then saturate? It is well known that in vacuum the electric field of a relativistic particle expands in the transverse dimension as ? increases as a kinematic consequence of special relativity. In a medium on the other hand this increase does not continue indefinitely. The effect of the medium may be described by a screening length, 2, which is the Compton wavelength of the renormalised photon in a medium of dielectric constant ~(~o). In simple terms, since electromagnetic waves do not travel with velocity c in a medium, the corresponding photons have mass,
317
I O N I S A T I O N LOSS I
II, given by
same direction, ~,, with the same energy, ha).
~ = -5- (I - v Z / c Z ) ~ = c
I -
(w,,/'~)
(2)
where t, is the group velocity at frequency co. Exchange of such quanta will be characterised by a range equal to the Compton wavelength*,
Apx -
hco
he)
fie
c
_
C
/2C
1--
d
(wx/~)
.
- \/e cos
.
At high frequencies when the binding energy of the electrons may be neglected this reduces to
Of course, when I/fl = x/e cos ~k, the range is infinite (for real e). This describes Cherenkov radiation. Making further approximations to emphasise the dependence on velocity, we have
(4) Rx ~~ __2c (~,- 2 + ~2 + ~ 2 ) - 1 ,
where coo is the plasma frequency. We may compare this screening length with the maximum impact parameter for which the collision is sufficiently " s u d d e n " to transfer energy h~o to an atom. Jackson1) shows that
b .... = 7v/oJ. It follows that we may expect the relativistic rise due to the increase of b ~ to saturate when bm~ = 2 or when fly "-
w -
-
wp
=
ionisation energy plasma energy
,
(5)
in the high frequency case. Before discussing this result we derive it in a different way 9) which brings out relationships with the phenomena of Cherenkov and transition radiation and is based on the uncertainty principle. A virtual photon associated with a particle of velocity tic along the xaxis must itself have a phase velocity tic along x. Otherwise it would not be seen to an observer travelling in the particle frame as a component of a static Coulomb field. It therefore has a momentum projection along x given by hw p~ =
-
(9)
(3)
W
2 = cI~%,
(8)
where \,/e is the refractive index of the medium. This gives a range • = --
li
),_
x/~ c o s ~ ,
-
(0)
The range of such a virtual photon is given by Heisenberg's uncertainty principle:
Rx = h/Apx,
(7)
where Ap~ is the difference between its momentum and that of a free photon in the medium travelling in the * This is only true in the forward direction. In particular the argument fails completely in the case o f emission at the Cherenkov angle.
(10)
(2)
where we have taken ~, small, y = (I _f12)-~ large and e = 1 _ ( 2 . This formula is the same as the expression for the "formation zone" discussed by Garibyan 14) for transition radiation. It shows that the range of virtual photons increases with y until ? is of order ~- 1, where for high frequencies ~-~,~ho~/plasma energy. So again we expect the relativistic rise to give way to the Fermi plateau region when ?~
ionisation energy plasma energy
(11)
This suggests that the largest relativistic rise will be seen in gases because of their low plasma energies, especially those with high Z, for which the ionisation potentials increase faster than the plasma energy. Further, of those with similar numbers of electrons, the rare gases with the tightest binding will show the largest rise. In addition different electron shells will saturate at different values of ? and calculations should take account of this. In particular if measurements are made which effectively exclude the contribution from inner electrons - such as the most probable ionisation loss in thin gas counters - the relativistic rise will be less than conventional calculations might suggest. 4. The model
Following Fano iS) we start by considering the differential cross section per atomic electron for the inelastic collision of an isolated atom with the incident particle according to the Born approximation:
d2a dQdE
2he4 [[F(E'q)[2 + [~"G(E'q)[2 I my z [ Q2 (Q--U~_~)~],
(12)
318
J . H . COBB et al.
where v = fic is the incident particle velocity, Q = h2qE/2m and m is the electron mass. F and G are inelastic form factors discussed by Fano 15). We have d r o p p e d some terms in Q/mc 2 which are only i m p o r t a n t for large energy losses and are consequently of no interest when considering thin absorbers. The first term represents the interaction of charges and as such is essentially constant in the relativistic region. It describes ionisation due to the absorption of longitudinal virtual photons. These photons have a range ~ 1/q as in the static c o u l o m b field hence the d e n o m i n a t o r Q2. The second term is the currentcurrent term involving the exchange of transverse virtual photons. These two terms add incoherently because they involve transitions to final states of opposite parity. When q is small, dipole transitions will predominate and we can follow F a n o ~5) by approximating: IF(E,q)l 2 dE = (Q/E) f ( E ) d E ,
(13)
and lilt" G(E,q)i 2 dE = ~2 f ( E ) dE E / 2 m c 2 ,
(14)
w h e r e f ( E ) d E is the optical dipole oscillator strength. The definition of small q in this respect is
q < (electron orbital size)-1, which is essentially: Q < E , , where E i is the electron binding energy. The longitudinal term is evaluated in two different regions: 1) Low Q, i.e. Q < El with dipole approximation and the choice f ( E ) = 6 ( E - E3, so [F(E,q)I 2 dE = (Q/E,) b ( E - E , ) d E .
(15)
do" L
dE
2 7re'* 1 - -my 2 E E '
for E > E,,
(18)
the latter being of course the Rutherford crosssection. There are two important points here. The first is that our crude approximations satisfy the sum rule*, are consistent with the longitudinal part of the BetheBloch formula for the mean energy l o s s 9) and incorporate some not too implausible low energy behaviour. The second is that, by using the sum rule to integrate over energy, traditional calculations avoid having to make some of our crude assumptions. We are unable to do this since an energy integration is inappropriate when the discrete energy loss collisions are few. This integration must be deferred to the end of our calculation and be carried out as a Monte Carlo process using the spectrum of energy loss probabilities. The transverse term only contributes significantly in the low Q region 15) and m a y therefore be evaluated in the dipole approximation. If ~, is the angle between the virtual photon direction and the particle momentum, we have fi2 = fi2 sinZt~ and Q = E2/Emv 2 C O S 2 I ~ /
.
Substituting we get 15) d Etr
27re 4 sin E ~b f ( E ) d(cos E ¢)
dE
Emv 2
(1/fi2 _ cos E ~k)E
This formula applies to an electron of an isolated a t o m rather than one in a medium - i.e. in the approximation e = 1. To see how it is to be modified we refer back to eq. (9). The d e n o m i n a t o r there contains a V/e factor and a similar d e n o m i n a t o r appears in eq. (19) representing the effect of the finite range or p r o p a g a t o r of the virtual photon. Instead of eq. (19) we write: d20 " _ 2roe 4 sin 2 t~ f ( E ) d(cos 2 ~)
This choice satisfies the sum rule ~EIF[ Ed E = Q. 2) High Q, i.e. Q > E, with impulse or free electron approximation IF(E,q)l 2 dE = 6 ( E - Q )
dE,
(16)
which also satisfies the sum rule. Taking these two together and integrating over Q we get
(19)
dE
Emv 2
(20)
(1/fi E- 5 cos E t/J)2
which is related to the formula given by Fano t 5). Further evaluation is simplified by the observation 16) that f (E) -
(21)
moo Jm(e((o)), 2 7~2 e 2 h N
where N is the electron density. dO'L --
2rte4 6 ( E - E i ) log 2mv---~2,
dE
my E
for E ~< Ei;
E,
dEa
E,
eE~o
sine (J d(cos2 @)
dE - ETrhvE~--Nf l m ( e ( w ) ) [l/fiE _ e(co) COS2 ~b[2' (22)
(17)
*
Eq. (13}.
IONISATION
319
LOSS I
TABLE l Atomic data used in calculations. Helium
Plasma energy Binding energies (eV) and electrons per molecule
Neon
Argon
Krypton
Xenon
0.272
0.609
0.816
1.156
1.414
24.5 (2)
870 (2) 54.4 (8)
3196 (2) 294 (8) 39.5 (8)
14280 (2) 1754 (8) 152 (18) 39.4 (8)
34612 5073 831 169 25.8
(2) (8) (18) (18) (8)
Methane
0.609 313 55.8 17.7 13.6
(2) (2) (2) (4)
Ammonia
Nitrogen
Carbon dioxide
0.609
0.721
0.902
412 (4) 47.6 (2) 31.3 (6)
313 (2) 55.8 (2) 17.7(2) 575 (4) 54.4 (4) 39.4 (8)
412 47.6 31.3 13.6
(2) (2) (3) (3)
Argon/20% Co2, Plasma energy = 0.834 eV He/50% Ne, Plasma energy = 0.472 eV Atomic data from Sternheimer (1952), ref. 6. Chemical effects neglected.
_-
dE
+
nh2 v2 N
(23)
arctan
by integrating cos2~b from 0 to I and taking e = el For e we use
~.
f,
,
+je2.
(24)
where we take Yl = E*. 71 is the "width" of the effective ionisation level which takes account of the finite kinetic energy of the electron in the continuum on the one hand and energy loss through excitation levels on the other. This is crude but it is a significant improvement over the use of a mean ionisation potential. 5. R e s u l t s o f c a l c u l a t i o n s
We have calculated 17) the energy loss spectra in 1.5 cm of gas at NTP at values of p/mc from 2 to 2000 in powers of 2 and also 5 × l04. For each gas or gas mixture a knowledge of the atomic levels Ei and the plasma frequency were sufficient to determine the different parts of the cross-section according to the model described in the previous section. The results are only weakly dependent on the values of E~ used. These are shown in table I and are taken from Sternheimer (1962)6). A simple Monte Carlo calculation then gave the differential energy loss distribution in 1.5 cm of gas. The full curve in fig. 2 shows the result for 3 GeV/c *
The results o f calculations are essentially unaffected by other choices e.g. ~i = 0.2El,
pions in argon9). The agreement with the data is significantly better than for the Landau and Blunck and Leisegang models. By plotting the peak of the Monte Carlo energy loss distribution as a function of p/mc, a prediction for the relativistic rise is obtained which is also in better agreement with available data. This is shown as curve II in fig. 1. Of course different experiments use different gas mixtures, sample thicknesses and methods of estimating the peak or most probable ionisation loss. We defer to paper I118) a comparison of the model with new data where these effects are treated in detail. However, neither in the case of the old data nor in the case of the new is there any indication of a real discrepancy with the model. Next we use the model to discuss the use of the relativistic rise in different gases for identifying particles of known momentum. As discussed elsewhere2), the combined effect of the slow rise giving little difference between masses and the broad energy loss spectrum giving poor resolution is to require more than 100 energy-loss measurements on each track if individual pions, kaons and protons are to be distinguished. In a second Monte Carlo program we have simulated such a multiple sampling device and studied the mass resolution that may be derived from the measurements on a single track. The larger and less probable measurements (the Landau tail) contain little information and only serve to degrade the variance of the smaller more probable measurements. Traditionally the peak of the distribution, the most probable energy-loss, is used but this is notoriously hard to estimate with small statistics. It has been shown 5) that the resolution is close to optimum if a fixed fraction of the measurements (20-50% of the largest) is discarded and the
320
J.H. COBB et al.
~
ARGON I 2O% CO2
~5
HAMBER I 1.5crn SAMPLE NO CROSS TALK)
rn SAMPLES
5
I
4
d-3
<3-
22 bJ 1
lb
o
zb
3b
Lo
do
so
SAMPLES REJECTED BY CUT
o
(a)
ARGON/ 20% CO2 7metre CHAMBER I I-5cm SAMPLE THEORY
S
4'
3' 2"
1
-20'
-1's
-lb
~
6
~
(b)
3E
IL.
~
DEVICE LENGTH ( m )
%
-g
g
INTERSAMPLE
CROSSTALK %
Ib
l's
zb
(c) mean of the remainder is used as the estimator. This is the method we have used. Fig. 3a shows how the resolution of 330 x 1.5 cm 2 samples of argon varies as a function of this cut. Both this resolution and the corresponding relativistic rise are rather insensitive to the size of this c u t - in fact according to the calculations the rise is less than 5 % different for a 50 % cut than for a 10% cut. In all of the following results we have used a 40 % cut. in fig. 3b we show how the resolution behaves as a
Fig. 3. The dependence o f ionisation resolution on various parameters o f a multiple sampling technique as found in Monte Carlo studies: (a) Fraction o f samples rejected by cut to remove the Landau tail (see text). (b) Device length and sample size. (c) Nearest neighbour intersample cross-talk.
function of number of samples and device length. Typically a 5 m device with 1.5 cm samples is comparable to a 7 m device with 3 cm samples. There is no easy cheap way to get the required resolution (5-6 %). A device of several metres depth and several hundred samples is needed. In the real world there are systematic effects also. Fig. 3c shows the effect of a small amount of nearest neighbour inter-sample crosstalk as simulated by the Monte Carlo program. The effect on the resolution is small. These problems are discussed at greater length in paper 1II8). In the following analysis we are concerned primarily with the efficacy of different gases and therefore restrict our calculations to a 5 m device yielding 3 3 0 x l . 5 cm 2 samples and ignore possible systematic effects. Fig. 4a shows the response* expected from such a device filled with argon and exposed to a mixture of pions, kaons and protons at 25 GeV/c in the ratio 10:1:1. In this case we can identify particles with good confidence. At 60GeV/c on the other hand the separation is marginal as shown in fig. 4b. An optimum cut between kaons and pions would result in a loss of about 10% of kaons into the pion peak and the same number of pions misidentified as kaons. In order to * The " r e s p o n s e " is the mean o f the 60% smallest signals discussed above.
321
I O N I S A T I O N LOSS I
/ 1.2
1.3
~o
1.4
1:5
1.3
1!4
~/lo
1'.s
1.6
Fig. 4. Ideal response of a 5 m detector to a mixture of:'t, K and protons in flux ratio 10: I : 1 ; (a) shows the separation at 25 GeV/c, (b) shows the separation at 60 GeV/c. This is very close to the case described as " 9 0 % s e p a r a t i o n " (see text).
make a definitive comparison between different gases we have calculated at what momentum the kaon/pion separation is 90 % pure in the above sense. For each gas and value ofp/mc in turn we generated an energy loss distribution appropriate to 1.5 cm of gas at NTP (20 000 points). In the second Monte Carlo program we simulated 2000 traversals of the detector described above. Some of the results are given in table 2. We estimate that the statistical inaccuracies on these results are less than 1%. For reference we have also calculated the value of the most probable ionisation loss. The full results will be available elsewhere' 7). The rare gases are treated in the first section of table 2. In spite of a big increase in the energy loss with atomic number the resolution is essentially constant at around 5%. The relativistic rise on the other hand increases from 4 0 % to 70%. This is due to the binding energy which is increasing faster than the plasma frequency (see section 3). The kaonlpion separation limit which is 55-60 GeV/c for argon and krypton is 95 GeV/e for pure xenon. (The lower limit for Kin separation is 1.8 GeV/c for all gases). The second group in table 2 compares the first chemical period gases methane, ammonia, neon and nitrogen*. The first three of these all have the same plasma frequency. They show that as the binding energy increases, so does the relativistic rise, while the
* We ignore all chemical effects,
mean energy loss gets smaller. There is also an improved resolution when the binding energy is smaller due to the increased statistics of small energy-loss collisions. However, these have a small contribution to the rise and therefore the K/~ separation limit is still worse for methane and ammonia than neon. The third section of table 2 shows how gas mixtures behave. The argon-carbon dioxide system is the example. It shows that the addition of small quantities of a low-Z quenching gas has a small effect. The ionisation of gas mixtures is essentially additive in the Born approximation except for the modification due to the dielectric constant. In the last section of table 2 we show figures for the total ionisation cross-section in the form of the number of ionising collisions per metre of gas ("primary ionisation"). Davidenko ' 9) has proposed that particles could be identified by measuring their primary ionisation in a streamer chamber operating in the avalanche mode. The resolution of such a technique would be limited under ideal conditions by Poisson statistics. In table 2 we assume such a limit and calculate the K/~ separation limit discussed above for a helium/neon mixture and an argon/CO 2 mixture. The relativistic rise is smaller due to the equal weighting of large and small energy-loss collisions. In dE/dx measurements on the other hand the large energy loss collisions which show the larger relativistic rise are weighted more than the small energy loss collisions. On the other hand the ideal resolution is much better than in dE/dx measure-
322
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I O N I S A T I O N LOSS l
17 Iniooisationtoss Ill. 16 1.5 14 13 ~ ~ r 12 11
17
Calcolal¢0 increases
lb
d E ~ZO~ ~-~. CC
-16 -1~ 11~
Argon CO 2
323
increase o f i o n i s a t i o n loss a n d its s a t u r a t i o n at the h i g h e s t velocities. T h e s e c a l c u l a t i o n s give a g r e e m e n t with existing d a t a a n d s h o w q u a n t i t a t i v e l y the efficacy o f different gases for m e a s u r i n g velocities o r identifying c h a r g e d particles o f k n o w n m o m e n t u m . A r g o n is b o t h effective, practical a n d cheap. X e n o n is the o n l y gas e x a m i n e d which is m o r e effective.
20%
-1 3 -1.2 1.1
~0
16o0 Pl~
10 10~oo
Fig. 5. Plots of the calculated ionisation loss for some interesting cases (table 2 and ref. 17).
m e n t s a l t h o u g h there are m a j o r technical difficulties in a c h i e v i n g this. Finally, we n o t e t h a t t h e r e is no d i s c e r n i b l e difference b e t w e e n the relativistic rise o f the m o s t p r o b a b l e a n d the t r u n c a t e d m e a n i o n i s a t i o n . T h i s is r e l a t e d to the fact t h a t t h e s h a p e o f the e n e r g y loss d i s t r i b u t i o n d o e s n o t a p p e a r to c h a n g e m u c h as a f u n c t i o n o f p/mc. Fig. 5 s h o w s a few o f the c a l c u l a t e d relativistic rise curves.
6. Conclusion The discrepancy between theoretical predictions and e x p e r i m e n t a l results f o r the e n e r g y - l o s s in thin gas s a m p l e s seems to be resolved. S i m p l e q u a l i t a t i v e pictures a n d d e t a i l e d q u a n t i t a t i v e c a l c u l a t i o n s t o g e t h e r give insight i n t o the m e c h a n i s m o f the relativistic
References 1) j. D. Jackson, Classical electrodynamics (J. Wiley, New York) Ch. 13; A. Crispin and G. N. Fowler, Rev. Mod. Phys. 42 (1970) 290. 2) W. W. M. Allison et al., Nucl. Instr. and Meth. 119 (1974) 499. 5) F. Harris et al., Nucl. Instr. and Meth. 107 (1973) 413. a) p. V. Ramana Murthy, Nucl. Instr. and Meth. 63 (1968) 77. 5) D. Jeanne et al., Nucl. Instr. and Meth. 111 (1973) 287; M. Aderholz et al., Nucl. Instr. and Meth. 118 (1974) 419. 6) R. M. Sternheimer and R. F. Peierls, Phys. Rev. B3 (1971) 3681 ; R. M. Sternheimer, Phys. Rev. 88 0952) 851. 7) Z. Dimcovski et al., Nucl. Instr. and Meth. 94 (1971) 151. 8) A. V. Alakoz et al., Nucl. Instr. and Meth. 124 (1975) 41. 9) j. H. Cobb, D. Phil. Thesis (Univ. of Oxford, 1975). Available from Rutherford as HEP/T/55. 10) L. Landau, J. Phys. (USSR) 8 (1944) 201. la) G. Knop et al., Z. Physik 165 (1961) 533. 12) O. Blunck and S. Leisegang, Z. Physik 128 (1950) 500. 13) K. A. lspiryan et al., Nucl. Instr. and Meth. 117 (1974) 125. 14) G. M. Garibyan, Yerevan Report: Theoretical foundations of transition radiation (1970); See also L. Durand, Phys. Rev. O l l (1975) 89. 15) U. Fano, Ann. Rev. Nucl. Sci, 13 (1963) 1. 16) L. Landau and E. Lifshitz, Electrodynarnics of continuous media, (Pergamon Press, 1960) Ch. 62, eq. (13). 17) j. N. Bunch, D. Phil. Thesis (Univ. of Oxford) in preparation. 18) W. W. M. Allison et al., following paper (this issue), p. 325. 19) V. A. Davidenko, Nucl. Instr. and Meth. 67 (1969) 325.