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Limiting velocity resolution of Cherenkov counters of total internal reflection
63
Nuclear Instruments and Methods m Physics Research A300 (1991) 63-66 North-Holland
Limiting velocity resolution of Cherenkov counters of total internal reflection V.P. Zrelov Joint Institute for Nuclear Research, Dubna, USSR
Received 1 August 1990
A simple method for the practically complete elmunation of the effect produced by VChR cone smearing due to dispersion on the resolution of Cherenkov counters operating under conditions close to the angle of total internal reflection (TIR) is proposed . This is achieved by tilting the counter radiator output side (parallel to the particle velocity vector) by a small angle a . A formula for the calculation of a and conditions for its applications are given. It is pointed out that the main difficulty m making use of the advantages of the obtained achromathzation method is VChR diffraction. In view of its high sensitivity to the angular divergence of particles to be detected, this TIR counter is proposed to be used m experiments on particle scattering in the TeV energy region.
1 . Introduction Since it first appeared, the Cherenkov counter based on the phenomenon of total internal reflection (TIR) [1] (or the Fitch-Motli counter) has been used many times (see ref. [2]) without significant upgrading. Since the Vavilov-Cherenkov radiation (VChR) leaves these counters through the radiator side, which is perpendicular to the direction of particle motion, the radiation cone is greatly blurred because of dispersion, which results in a low velocity resolution Aß _ 10-2 . For a Cherenkov counter [31 with TIR it was supposed that the VChR leaving the radiator through the focussing side could be detected in the form of a disc and that cone blurring could be neutralized by a doublet of circular prisms . However, the difficulties m making this neutralizer have probably prevented manufacturing of the counter. Ref. [4] describes the use of a TIR Cherenkov counter with an original divergence neutralizer for the VChR (due to dispersion) released through a side perpendicular to the direction of particle motion . The intrinsic velocity resolution of this counter was Ap --- 10-° . Ref. [5] considered a version of the TIR Cherenkov counter where the VChR, as in ref. [6], was proposed to be released through a radiator side parallel to the direction of particle motion . In this case the velocity resolution of the counter is determined by the expression O,ß/,ß = tg o 0~5, where 0 = arccos(1/8n 2 (Á)) is the angle between the VChR direction to a medium of reflective index nZ around the radiator and the output side of the radiator . Elsevier Science Publishers B.V . (North-Holland)
Though the considerable dispersion of the radiator itself does not affect the Oß resolution of this counter, it is limited by the divergence An z ,8 n2 sin
Below it will be shown how one can practically completely eliminate this effect as well . 2. Condition for VChR achromatization in a TIR counter As pointed out in the introduction, replacing a TIR Cherenkov counter with the output side perpendicular to the direction of particle motion by one with a parallel output side (rotating this side by 90 ° anticlockwise), one eliminates the radiator dispersion effect on the Oß resolution of the counter . An additional slight rotation of the output side in the same direction by an angle a, as shown in fig. 1, results in the complete elimination of the dispersion effect in a wide range of wavelengths and particle velocities (at the same angle a) .
n,
Fig. 1 On deriving the achromatization condition.
64
V P. Zrelou / Limiting uelocity resolution of Cherenkov counters
Using the Snellius refraction law we have n, cos(B-a)=n 2 cos 0,
(1)
where n, and 17 2 are the absolute refractive indices, B(X) = arccos(1/ßn,(X)), and ß = v/c is the velocity of a particle emitting the VChR in a medium with n, . Using the condition for the VChR achromatization in a medium with 17 2 in the form ~'(N,) =P(/\2) as well as eq . (1), one can obtain a formula for the angle a : n2(~l)-n2(~2) (2) n2(at) tg 0(~2) n2(À2) tg a(X,) Formula (2) should be used with care, because at the given refractive index n, of the radiator a restriction must be imposed on dispersion 17 2 of the gas surrounding the radiator . In its turn, this restriction is related to the one to be imposed on angle a. When angle a is large, the VChR will not enter the medium with 17 2 because of TIR. To avoid this, a should satisfy the following inequality : tg a
_
a
(3)
where X 2 is the longest of the achromatized wavelengths. Using (2) and (3) and taking into account that angle a is small and n2(ß) -- 1, one obtains the restriction on gas dispersion 17 2 in the form : On e < larccos &b l - arccos rei 1 ß17t (/X2) l nt (~2)
X(tg 0(~`t) - tg 0 (À2)) . (4) As seen from (4), it is better to take n, with large dispersion . The use of formula (2) without condition (3) yields wrong results. For example, if one uses melted quartz (Si0 2 ) as the radiator and air as the medium with 17 2 , one will not get the desired result because the combination of n, and 17 2 does not satisfy condition (4). Calculation shows that the most suitable gas for the radiator of melted quartz (Si02 ) is helium ; according to ref. [7], its refractive index depends on the wavelength (at t = 0 ° C and P = 760 mm Hg) as 2.24 z 10 5 + 5.94 X 10 10 ntie - 1 = 6 .927 X 10 -5 (1 + X + 1 .72X10 16 ) , À6
where a is in Á. Table 1 demonstrates the efficiency of this simple achromatization method showing angles 0, at which the VChR leaves the quartz radiator, for different X, and values of n He , n air and relative values of ns,o, . As seen from the table, divergence 0(P does not exceed ±2" in a wide range of X from 202.55 mm to 656.9 nm . Note that 0$ = 40" if a = 0', i.e. a slope of
Table 1 Angles -p at which the VChR leaves the quartz radiator, for different a, values of nH ., n,_ and relative values of ns,o, [nm] 202.55 250.39 303 412 404656 546.072 656.3
niie
n_
1 .0000346 1 .0000340 1 .0000336 1 .0000333 1.0000331 1.0000330
1 .0003222 1 .0003013 1 .0002911 10002825 10002779 1 .0002762
(t=15 ° C, P=760 mm Hg) `
(t=15 ° C. P=760 mm Hg)
oo) s
1 .54727 1 .50745 1 .48594 146968 146013 1.45640
0'21'14 5" 0'21'17 5" 0'21'17 2" 0 ° 21'16.6" 0 ° 21'16.0" 0 ° 21'14.5"
`' Figures are from the reference book Tables of Physical
Quantities (Moscow, Atomizdat, 1976) pp . 634 and 636. These values of (P(X) are obtained for the angle a= .3]X10-5 rad) calculated using formula (2) 0.0007516o (1 for the values of n s , o ,, n He , n, , given m table 1 ; XIh)= 202.55 nm, X'2' = 546.072 nm, ,ß =1 ` Values obtained with eq . (5), with a correction for temperature (At =15 ° C) the side at angle a allows at least a tenfold decrease in 0~ . Noteworthy is that for the same angle a = 0.0007516' the degree of achromatization does not change at ß = 0.999990 as well (q) = 0 °14'42" ± 2"). For the LiF-He pair at 1 atm the VChR divergence in helium at a = 0 .001708 ° (2 .98 X 10 -5 rad) in the same range AX is somewhat larger : 0$ _ ±5" (at a = 0'09'51"). Ignoring the ultraviolet part of the VChR spectrum one can select a radiator-gas pair where 17 2 i~ air at normal pressure . For example, calculation by formula (2) for a TF-5 radiator (n D = 1.7550) with X, = 404.656 rim, ~'2 = 486.13 nm and ß = 1 yields 0 = 1 ° 01'16 " ± 3" (at a = 0.0047 ° ) in the range of A from 365 to 656.3 nm (if a=0 ° , 0(p=76") . 3. Velocity resolution of the counter From eq . (1) one can obtain ,ß
-
n 2ß sin (P 0$ n, sin a /3 sin 0 - (cos a + sin atg0)
At a - 0 one can write down with a high accuracy (for the above example with S'02-He the denominator in eq . (6) is 1.0000145) -- n2ß sin (P A~ = tg q> 0$, ßß because at a = 0, cos -~ = 1 /ßn 2. According to eq. (7), the resolution will be O,ß//3 = 6.0 X 10 -8 at 4, = 0 ° 21'15" and AO = ± 2", which is
65
V P. Zrelov / Limning velocity resolution of Cherenkov counters
enough to distinguish -rr and K mesons down to the momentum p _< 1.4 TeV/c. O&K = Om!,K/2P2) . At ß= 0.999990 (E R = 31 GeV) and ¢~ = 0°14'42" ± 2" the resolution will be Oß = tg 0 00 = 4.1 X 10 -8 . Thus, according to the achromatization conditions, a Sí02 radiator with a = 1 .31 X 10 -5 rad allows in principle a A/3 resolution which is enough to separate iT and K mesons from approximately 30 GeV/c to 1 .4 TeV/c. For a TF-5-air counter with p = 1 ° 01'16" ± 3" the resolution will be Oß/,ß = 3 X 10 -7 , i.e . a high one as well . However, other effects to be discussed below (see sections 6 and 7) do not allow this resolution to be achieved . 4. Threshold condition While m a TIR counter with a = 0 the threshold condition is ß > 1/n2, for a * 0 it is slightly different : cos a _ 1 thresh n2 - n i sin 0 sin a - n 2 - n i sin 0 sin a ' i .e. ßthresh increases by 0/3 = n l sin 0 sin a . In our example (S'02 -He) ßthre,h( X = 656 .3 nm) = 0.9999781 (at a = 0, /30 = 0.9999670), and the threshold shift is Nthresh -ßo =1 .1X10 -5 . 5. VChR intensity in an achromatic TIR counter Since the VChR emitted in the radiator (n i ) has a 100% polarization (the electric vector of radiation is in the plane of crossing the interphase boundary), the part of the VChR that enters the medium with n 2 can be calculated either by the precise Fresnel formula 1-R ij =Itransm-1_
_ tg2(4, 4+9), op) tg
(9)
where (p = 90 ° - (B - a) and ¢ = arcsin ni/n2 COO a), or by the approximate Landau-Lifshitz formula [81 which is valid for incidence angles close to the angle of total internal reflection 0'=arcsin n 2 /n i : 1-Ru-Itransm=4
20Bn1
(ní2-1)-i/4,
(10)
where in our case AB = 0 * + 0 - (90' + a), n,' = n,1"2. The calculation shows that at 06-- 1 .5 X 10 -5 rad formulae (9) and (10) coincide within 2.5% (formula (10) yields a larger value) . In the case of a radiator with n i and thickness 1 the intensity of the VChR (by the number of photons) from a particle with ß = 1 released into the medium with n2 will be N = Itransmk sin 2 91 = ItransmNO,
(11 )
Fig. 2. VChR diffraction on the output side of an achromatic TIR Cherenkov counter where k=2ma Ä2-Xt , X,Ä2
a = 1/137, No is the number of photons emitted in the radiator, and Itanen is determined by eq. (10) . In the wavelength range from X i = 202.55 nm to_ X2 = 656.3 nm (0 for X = 404.66 nm), 1 = 1 cm, N = 42 .6 photons/cm, which is about 5% * of the VChR photons emitted to a Si02 radiator (No = 856.6 photons) . Let us see what length L the differential Cherenkov counter must have when working with helium at the same angle 0 = 0 ° 21'16" at which the VChR leaves the SiO, radiator . Using eqs. (10) and (11) we obtain (at ß = 1) L=
171
2OB(ni 2- 1) 3/4,
(12)
where r1=n2-1 ; n, =ni/n2 ; 'B=0 * +0-(90o+ a) . For lS,o z = 5 cm and il = 3.33 X 10 -5 : LHe = 10 M6. Fundamental limitation of the velocity resolution of the achromatic TIR counter When the VChR slides out of the radiator (small angles (p) the photons are concentrated in a narrow cone of " thickness" - 1 sin $o, which leads to its blurring because of diffraction . Following fig. 2 with allowance for difference in beam paths in media ni and n 2 , we obtain for the radiator of length 1: ntl
sin B ctg(0 - a ) -
n21
sin 0 cos ~ sin(B - a)
(13)
On condition that nj cos(B - a) = n 2 cos $o, a - 0, m = 1 and n 2 - 1 one can obtain an expression for the VChR divergence in the form (A~5 = q) - (pv) : X _ X A,P _ = 1 sin $o = 1q),
(14)
" If the pressure P of the gas n 2 (helium) increases, the part of the VChR leaving the radiator will increase - F worsening the degree of achromatization. For example, at P = 4 aim Itransm =10.4% (N=88 photons/cm) but Oß/ß = ±4 x 10-7 .
66
VP. Zrelou / Limiting velocity resolution of Cherenkov counters
According to eq . (7) :
á = tg
$0
0-~ .- $o A ~5 ,
so, using eq . (14), we obtain 0p
_A
(15) 8 ~ "m l . Since the VChR photon spectrum - 1/A2 , formula (15) must involve a spectrum-averaged quantity 1
AlA2 À2 In À 2 > À,) . X2 - XI X( i
For the wavelength range Ai = 202.55 nm and A 2 = 656.3 nm : )~ = 344 nm. Since the function describing the angular distribution of a diffraction peak of the form sine(p/$2 has a half-width at half maximum equal to 0.44x, formula (15) takes the form Opl/S
/hm
- 0.44
ï.
(16)
If l = 5 cm and )~ = 344 nm then { A,ß/,ß) 1,m = 3 X 10 -6 *. This resolution allows one to separate X and K mesons to p = 193 GeV/c, and K mesons and protons to p = 325 GeV/c. 7. Limitations of divergence of detected particles Angular deviations of particles, e, from the initial direction practically change the incidence angle of the VChR with the output side (or angle 0) by ±~ . So it follows from eq . (1) that p.P
ni sin(0-a)
Vnz-ni cos
2 (B - a)
(17)
For the S'02-He pair with A = 303 .41 nm (a = 0.0007516 ° ) 0 /0~ = 175 .8 . If ( AB/,8 ) i,m = 3 x 10 -6 _ and ~5=0 0 21'16" (6 .19x 10 -3 rad), the permissible 0$ = 4 .85 X 10 -4 rad, which, according to eq . (17), imposes a limit 0~ < 2.8 X 10 -6 rad. These requirements to the divergence of the beam of detected particles are very severe and can hardly be met. So a Cherenkov counter with such a high sensitivity to the direction of the particles can be used in experiments of particle scattering at energies of 10-20 TeV when angles of the diffractive scattering are - 10 ~Lrad [9] and multiple scattering is approximately an order smaller. In this counter detection of circular VChR images * For the TF-5-air pair working m the range from Ai = 365 nm to Az = 656.3 rim, A = 482.3 nm and ( op/p ) I,m = 4.2 x 10 -6 , which allows one to separate a and K mesons to p-163 GeV/c, and K mesons and protons to p = 275 GeV/c.
Fig. 3. Schematic view of an achromatic TIR Cherenkov counter 1 - flat parallel radiator with output side slope a, 2 tilted spherical mirror with focal distance f; 3 - set of multianode photomultipliers with 100 anodes each . must be done by position-sensitive detectors, as schematically shown in fig. 3. One can use, for example, multianode photomultipliers with microchannel plates of the FEU-2MKP-100 type (100 anodes) as VChR detectors installed to the focal plane of a spherical mirror whose axis coincides with the radiator axis. The photocathodes of these photomultipliers are 2.3 cm in diameter and are placed in a ring with a radius R = 5 cm . The tolerable ring shift region is Or = ± 1 cm . If the radiator is made of TF-5 glass, n2 is air, p = 1, ~ = 1 ° 01'16" (1 .78 X 10 -2 rad), and a = 0.0047 ° , then R can be obtained with a spherical mirror with f = R/tg v, where v = 0 + a. In our example f -- 280 cm . In this case the coordinate determination accuracy will be OR = fw, where Ov --- 0q) is determined by the diffractive divergence . If 00 = 2.3 X 10 -4 rad (determined from ( Op/p ) I,m = 4.2 X 10-6), R = ± 0.65 mm, which can probably be achieved with these photomultipliers that have 100 anodes of area 1 .5 X 1 .5 mm2 with a gap of 1 .6 mm between them . References [1] V. Fitch and R. Motly, Phys Rev. 101 (1956) 496. [2] V.P. Zrelov, Vavilov-Cherenkov Radiation and its Application to High Energy Physics (Atomizdat, Moscow, 1968) pp . 183-208. [3] G. Von Dardel, Proc . Int . Conf. on Instrumentation m High-Energy Physics, Berkeley, 1960, p. 166. [4] V.I . Solyanik, Preprint IHEP 81-63, Serpukhov (1981) . [5] V.P . Zrelov, Prib . Tekhn. Eksp. 3 (1965) 100. [6] G.W . Hutchinson, Progr Nucl Phys. 8 (1960) 226. [7] A. Dalgarno and A.E. Kingston, Proc. Roy. Soc. 259 (1960) 24 . [8] L.D . Landau and E.M . Lifshitz, Electrodynamics of Continuous Media (GITTL, Moscow, 1957) p 352. [9] Yu.D . Prokoshkm, Proc . 2nd ICFA Workshop on Possibility and Limitations of Accelerators and Detectors, Les Diablerets, Switzerland, 1979, CERN, RD/450-1500 (1980) p. 347.
Nuclear Instruments and Methods m Physics Research A300 (1991) 63-66 North-Holland
Limiting velocity resolution of Cherenkov counters of total internal reflection V.P. Zrelov Joint Institute for Nuclear Research, Dubna, USSR
Received 1 August 1990
A simple method for the practically complete elmunation of the effect produced by VChR cone smearing due to dispersion on the resolution of Cherenkov counters operating under conditions close to the angle of total internal reflection (TIR) is proposed . This is achieved by tilting the counter radiator output side (parallel to the particle velocity vector) by a small angle a . A formula for the calculation of a and conditions for its applications are given. It is pointed out that the main difficulty m making use of the advantages of the obtained achromathzation method is VChR diffraction. In view of its high sensitivity to the angular divergence of particles to be detected, this TIR counter is proposed to be used m experiments on particle scattering in the TeV energy region.
1 . Introduction Since it first appeared, the Cherenkov counter based on the phenomenon of total internal reflection (TIR) [1] (or the Fitch-Motli counter) has been used many times (see ref. [2]) without significant upgrading. Since the Vavilov-Cherenkov radiation (VChR) leaves these counters through the radiator side, which is perpendicular to the direction of particle motion, the radiation cone is greatly blurred because of dispersion, which results in a low velocity resolution Aß _ 10-2 . For a Cherenkov counter [31 with TIR it was supposed that the VChR leaving the radiator through the focussing side could be detected in the form of a disc and that cone blurring could be neutralized by a doublet of circular prisms . However, the difficulties m making this neutralizer have probably prevented manufacturing of the counter. Ref. [4] describes the use of a TIR Cherenkov counter with an original divergence neutralizer for the VChR (due to dispersion) released through a side perpendicular to the direction of particle motion . The intrinsic velocity resolution of this counter was Ap --- 10-° . Ref. [5] considered a version of the TIR Cherenkov counter where the VChR, as in ref. [6], was proposed to be released through a radiator side parallel to the direction of particle motion . In this case the velocity resolution of the counter is determined by the expression O,ß/,ß = tg o 0~5, where 0 = arccos(1/8n 2 (Á)) is the angle between the VChR direction to a medium of reflective index nZ around the radiator and the output side of the radiator . Elsevier Science Publishers B.V . (North-Holland)
Though the considerable dispersion of the radiator itself does not affect the Oß resolution of this counter, it is limited by the divergence An z ,8 n2 sin
Below it will be shown how one can practically completely eliminate this effect as well . 2. Condition for VChR achromatization in a TIR counter As pointed out in the introduction, replacing a TIR Cherenkov counter with the output side perpendicular to the direction of particle motion by one with a parallel output side (rotating this side by 90 ° anticlockwise), one eliminates the radiator dispersion effect on the Oß resolution of the counter . An additional slight rotation of the output side in the same direction by an angle a, as shown in fig. 1, results in the complete elimination of the dispersion effect in a wide range of wavelengths and particle velocities (at the same angle a) .
n,
Fig. 1 On deriving the achromatization condition.
64
V P. Zrelou / Limiting uelocity resolution of Cherenkov counters
Using the Snellius refraction law we have n, cos(B-a)=n 2 cos 0,
(1)
where n, and 17 2 are the absolute refractive indices, B(X) = arccos(1/ßn,(X)), and ß = v/c is the velocity of a particle emitting the VChR in a medium with n, . Using the condition for the VChR achromatization in a medium with 17 2 in the form ~'(N,) =P(/\2) as well as eq . (1), one can obtain a formula for the angle a : n2(~l)-n2(~2) (2) n2(at) tg 0(~2) n2(À2) tg a(X,) Formula (2) should be used with care, because at the given refractive index n, of the radiator a restriction must be imposed on dispersion 17 2 of the gas surrounding the radiator . In its turn, this restriction is related to the one to be imposed on angle a. When angle a is large, the VChR will not enter the medium with 17 2 because of TIR. To avoid this, a should satisfy the following inequality : tg a
_
a
(3)
where X 2 is the longest of the achromatized wavelengths. Using (2) and (3) and taking into account that angle a is small and n2(ß) -- 1, one obtains the restriction on gas dispersion 17 2 in the form : On e < larccos &b l - arccos rei 1 ß17t (/X2) l nt (~2)
X(tg 0(~`t) - tg 0 (À2)) . (4) As seen from (4), it is better to take n, with large dispersion . The use of formula (2) without condition (3) yields wrong results. For example, if one uses melted quartz (Si0 2 ) as the radiator and air as the medium with 17 2 , one will not get the desired result because the combination of n, and 17 2 does not satisfy condition (4). Calculation shows that the most suitable gas for the radiator of melted quartz (Si02 ) is helium ; according to ref. [7], its refractive index depends on the wavelength (at t = 0 ° C and P = 760 mm Hg) as 2.24 z 10 5 + 5.94 X 10 10 ntie - 1 = 6 .927 X 10 -5 (1 + X + 1 .72X10 16 ) , À6
where a is in Á. Table 1 demonstrates the efficiency of this simple achromatization method showing angles 0, at which the VChR leaves the quartz radiator, for different X, and values of n He , n air and relative values of ns,o, . As seen from the table, divergence 0(P does not exceed ±2" in a wide range of X from 202.55 mm to 656.9 nm . Note that 0$ = 40" if a = 0', i.e. a slope of
Table 1 Angles -p at which the VChR leaves the quartz radiator, for different a, values of nH ., n,_ and relative values of ns,o, [nm] 202.55 250.39 303 412 404656 546.072 656.3
niie
n_
1 .0000346 1 .0000340 1 .0000336 1 .0000333 1.0000331 1.0000330
1 .0003222 1 .0003013 1 .0002911 10002825 10002779 1 .0002762
(t=15 ° C, P=760 mm Hg) `
(t=15 ° C. P=760 mm Hg)
oo) s
1 .54727 1 .50745 1 .48594 146968 146013 1.45640
0'21'14 5" 0'21'17 5" 0'21'17 2" 0 ° 21'16.6" 0 ° 21'16.0" 0 ° 21'14.5"
`' Figures are from the reference book Tables of Physical
Quantities (Moscow, Atomizdat, 1976) pp . 634 and 636. These values of (P(X) are obtained for the angle a= .3]X10-5 rad) calculated using formula (2) 0.0007516o (1 for the values of n s , o ,, n He , n, , given m table 1 ; XIh)= 202.55 nm, X'2' = 546.072 nm, ,ß =1 ` Values obtained with eq . (5), with a correction for temperature (At =15 ° C) the side at angle a allows at least a tenfold decrease in 0~ . Noteworthy is that for the same angle a = 0.0007516' the degree of achromatization does not change at ß = 0.999990 as well (q) = 0 °14'42" ± 2"). For the LiF-He pair at 1 atm the VChR divergence in helium at a = 0 .001708 ° (2 .98 X 10 -5 rad) in the same range AX is somewhat larger : 0$ _ ±5" (at a = 0'09'51"). Ignoring the ultraviolet part of the VChR spectrum one can select a radiator-gas pair where 17 2 i~ air at normal pressure . For example, calculation by formula (2) for a TF-5 radiator (n D = 1.7550) with X, = 404.656 rim, ~'2 = 486.13 nm and ß = 1 yields 0 = 1 ° 01'16 " ± 3" (at a = 0.0047 ° ) in the range of A from 365 to 656.3 nm (if a=0 ° , 0(p=76") . 3. Velocity resolution of the counter From eq . (1) one can obtain ,ß
-
n 2ß sin (P 0$ n, sin a /3 sin 0 - (cos a + sin atg0)
At a - 0 one can write down with a high accuracy (for the above example with S'02-He the denominator in eq . (6) is 1.0000145) -- n2ß sin (P A~ = tg q> 0$, ßß because at a = 0, cos -~ = 1 /ßn 2. According to eq. (7), the resolution will be O,ß//3 = 6.0 X 10 -8 at 4, = 0 ° 21'15" and AO = ± 2", which is
65
V P. Zrelov / Limning velocity resolution of Cherenkov counters
enough to distinguish -rr and K mesons down to the momentum p _< 1.4 TeV/c. O&K = Om!,K/2P2) . At ß= 0.999990 (E R = 31 GeV) and ¢~ = 0°14'42" ± 2" the resolution will be Oß = tg 0 00 = 4.1 X 10 -8 . Thus, according to the achromatization conditions, a Sí02 radiator with a = 1 .31 X 10 -5 rad allows in principle a A/3 resolution which is enough to separate iT and K mesons from approximately 30 GeV/c to 1 .4 TeV/c. For a TF-5-air counter with p = 1 ° 01'16" ± 3" the resolution will be Oß/,ß = 3 X 10 -7 , i.e . a high one as well . However, other effects to be discussed below (see sections 6 and 7) do not allow this resolution to be achieved . 4. Threshold condition While m a TIR counter with a = 0 the threshold condition is ß > 1/n2, for a * 0 it is slightly different : cos a _ 1 thresh n2 - n i sin 0 sin a - n 2 - n i sin 0 sin a ' i .e. ßthresh increases by 0/3 = n l sin 0 sin a . In our example (S'02 -He) ßthre,h( X = 656 .3 nm) = 0.9999781 (at a = 0, /30 = 0.9999670), and the threshold shift is Nthresh -ßo =1 .1X10 -5 . 5. VChR intensity in an achromatic TIR counter Since the VChR emitted in the radiator (n i ) has a 100% polarization (the electric vector of radiation is in the plane of crossing the interphase boundary), the part of the VChR that enters the medium with n 2 can be calculated either by the precise Fresnel formula 1-R ij =Itransm-1_
_ tg2(4, 4+9), op) tg
(9)
where (p = 90 ° - (B - a) and ¢ = arcsin ni/n2 COO a), or by the approximate Landau-Lifshitz formula [81 which is valid for incidence angles close to the angle of total internal reflection 0'=arcsin n 2 /n i : 1-Ru-Itransm=4
20Bn1
(ní2-1)-i/4,
(10)
where in our case AB = 0 * + 0 - (90' + a), n,' = n,1"2. The calculation shows that at 06-- 1 .5 X 10 -5 rad formulae (9) and (10) coincide within 2.5% (formula (10) yields a larger value) . In the case of a radiator with n i and thickness 1 the intensity of the VChR (by the number of photons) from a particle with ß = 1 released into the medium with n2 will be N = Itransmk sin 2 91 = ItransmNO,
(11 )
Fig. 2. VChR diffraction on the output side of an achromatic TIR Cherenkov counter where k=2ma Ä2-Xt , X,Ä2
a = 1/137, No is the number of photons emitted in the radiator, and Itanen is determined by eq. (10) . In the wavelength range from X i = 202.55 nm to_ X2 = 656.3 nm (0 for X = 404.66 nm), 1 = 1 cm, N = 42 .6 photons/cm, which is about 5% * of the VChR photons emitted to a Si02 radiator (No = 856.6 photons) . Let us see what length L the differential Cherenkov counter must have when working with helium at the same angle 0 = 0 ° 21'16" at which the VChR leaves the SiO, radiator . Using eqs. (10) and (11) we obtain (at ß = 1) L=
171
2OB(ni 2- 1) 3/4,
(12)
where r1=n2-1 ; n, =ni/n2 ; 'B=0 * +0-(90o+ a) . For lS,o z = 5 cm and il = 3.33 X 10 -5 : LHe = 10 M6. Fundamental limitation of the velocity resolution of the achromatic TIR counter When the VChR slides out of the radiator (small angles (p) the photons are concentrated in a narrow cone of " thickness" - 1 sin $o, which leads to its blurring because of diffraction . Following fig. 2 with allowance for difference in beam paths in media ni and n 2 , we obtain for the radiator of length 1: ntl
sin B ctg(0 - a ) -
n21
sin 0 cos ~ sin(B - a)
(13)
On condition that nj cos(B - a) = n 2 cos $o, a - 0, m = 1 and n 2 - 1 one can obtain an expression for the VChR divergence in the form (A~5 = q) - (pv) : X _ X A,P _ = 1 sin $o = 1q),
(14)
" If the pressure P of the gas n 2 (helium) increases, the part of the VChR leaving the radiator will increase - F worsening the degree of achromatization. For example, at P = 4 aim Itransm =10.4% (N=88 photons/cm) but Oß/ß = ±4 x 10-7 .
66
VP. Zrelou / Limiting velocity resolution of Cherenkov counters
According to eq . (7) :
á = tg
$0
0-~ .- $o A ~5 ,
so, using eq . (14), we obtain 0p
_A
(15) 8 ~ "m l . Since the VChR photon spectrum - 1/A2 , formula (15) must involve a spectrum-averaged quantity 1
AlA2 À2 In À 2 > À,) . X2 - XI X( i
For the wavelength range Ai = 202.55 nm and A 2 = 656.3 nm : )~ = 344 nm. Since the function describing the angular distribution of a diffraction peak of the form sine(p/$2 has a half-width at half maximum equal to 0.44x, formula (15) takes the form Opl/S
/hm
- 0.44
ï.
(16)
If l = 5 cm and )~ = 344 nm then { A,ß/,ß) 1,m = 3 X 10 -6 *. This resolution allows one to separate X and K mesons to p = 193 GeV/c, and K mesons and protons to p = 325 GeV/c. 7. Limitations of divergence of detected particles Angular deviations of particles, e, from the initial direction practically change the incidence angle of the VChR with the output side (or angle 0) by ±~ . So it follows from eq . (1) that p.P
ni sin(0-a)
Vnz-ni cos
2 (B - a)
(17)
For the S'02-He pair with A = 303 .41 nm (a = 0.0007516 ° ) 0 /0~ = 175 .8 . If ( AB/,8 ) i,m = 3 x 10 -6 _ and ~5=0 0 21'16" (6 .19x 10 -3 rad), the permissible 0$ = 4 .85 X 10 -4 rad, which, according to eq . (17), imposes a limit 0~ < 2.8 X 10 -6 rad. These requirements to the divergence of the beam of detected particles are very severe and can hardly be met. So a Cherenkov counter with such a high sensitivity to the direction of the particles can be used in experiments of particle scattering at energies of 10-20 TeV when angles of the diffractive scattering are - 10 ~Lrad [9] and multiple scattering is approximately an order smaller. In this counter detection of circular VChR images * For the TF-5-air pair working m the range from Ai = 365 nm to Az = 656.3 rim, A = 482.3 nm and ( op/p ) I,m = 4.2 x 10 -6 , which allows one to separate a and K mesons to p-163 GeV/c, and K mesons and protons to p = 275 GeV/c.
Fig. 3. Schematic view of an achromatic TIR Cherenkov counter 1 - flat parallel radiator with output side slope a, 2 tilted spherical mirror with focal distance f; 3 - set of multianode photomultipliers with 100 anodes each . must be done by position-sensitive detectors, as schematically shown in fig. 3. One can use, for example, multianode photomultipliers with microchannel plates of the FEU-2MKP-100 type (100 anodes) as VChR detectors installed to the focal plane of a spherical mirror whose axis coincides with the radiator axis. The photocathodes of these photomultipliers are 2.3 cm in diameter and are placed in a ring with a radius R = 5 cm . The tolerable ring shift region is Or = ± 1 cm . If the radiator is made of TF-5 glass, n2 is air, p = 1, ~ = 1 ° 01'16" (1 .78 X 10 -2 rad), and a = 0.0047 ° , then R can be obtained with a spherical mirror with f = R/tg v, where v = 0 + a. In our example f -- 280 cm . In this case the coordinate determination accuracy will be OR = fw, where Ov --- 0q) is determined by the diffractive divergence . If 00 = 2.3 X 10 -4 rad (determined from ( Op/p ) I,m = 4.2 X 10-6), R = ± 0.65 mm, which can probably be achieved with these photomultipliers that have 100 anodes of area 1 .5 X 1 .5 mm2 with a gap of 1 .6 mm between them . References [1] V. Fitch and R. Motly, Phys Rev. 101 (1956) 496. [2] V.P. Zrelov, Vavilov-Cherenkov Radiation and its Application to High Energy Physics (Atomizdat, Moscow, 1968) pp . 183-208. [3] G. Von Dardel, Proc . Int . Conf. on Instrumentation m High-Energy Physics, Berkeley, 1960, p. 166. [4] V.I . Solyanik, Preprint IHEP 81-63, Serpukhov (1981) . [5] V.P . Zrelov, Prib . Tekhn. Eksp. 3 (1965) 100. [6] G.W . Hutchinson, Progr Nucl Phys. 8 (1960) 226. [7] A. Dalgarno and A.E. Kingston, Proc. Roy. Soc. 259 (1960) 24 . [8] L.D . Landau and E.M . Lifshitz, Electrodynamics of Continuous Media (GITTL, Moscow, 1957) p 352. [9] Yu.D . Prokoshkm, Proc . 2nd ICFA Workshop on Possibility and Limitations of Accelerators and Detectors, Les Diablerets, Switzerland, 1979, CERN, RD/450-1500 (1980) p. 347.