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Pair production in a slowly varying electromagnetic field and the pair production process
Volume 113A, number 6
PHYSICS LETTERS
30 December 1985
P A I R P R O D U C T I O N IN A S L O W L Y V A R Y I N G E L E C T R O M A G N E T I C F I E L D AND THE PAIR PRODUCTION PROCESS
V.G. B A R Y S H E V S K I I and V.V. T I K H O M I R O V Department of Physics, V.L Lenin Byelorussian State University, Minsk 220080, U¢~R
Received 8 January 1985; revised manuscript received 9 October 1985; accepted for publication 24 October 1985
The numerical analysis of pair production (PP) by y-quanta intersecting the crystal axis with a small angle is given on the basis of the formulae describing the PP in a slightly inhomogeneous electric field.
The new C-pair production (PP) process predicted in refs. [ 1 - 3 ] received extensive interest [ 4 - 9 ] . It should be recalled that the PP process predicted in refs. [ 1 - 5 ] is closely related with the mechanism o f PP in a strong electric field [ 1 0 - 1 3 ]. Recently characteristic features of the process (the significant increase of PP rate above the Bethe-Heitler (BH) rate, a very narrow range of angles over which this process extends and the significant difference of the e +- energy distribution from the BH predictions) have been observed experimentally [14]. The quantum formulae of refs. [ 2 - 5 ] turn into the simple formulae of the theory of the PP in a constant electromagnetic field [ 1 0 - 1 3 ] when ~,-quanta are directed along crystallographic axes or planes. At the same time it is very difficult to examine carefully the consequence o f misaligning a 7-quantum beam by some tilt angle on the basis of the quantum formulae [6]. In this paper we have obtained the expression for the PP probability in a slightly inhomogeneous electric field. It enables us to analyze the PP by a non-divergent 7-quantum beam intersecting the crystal axis or plane at some non-zero angle. It will be shown below that a significant change (fall) of the PP rate may occur only when the tilt angles ~b are one order of magnitude larger than those predicted in ref. [6] and said to be observed in ref. [14]. Thus, an enhanced PP rate may be observed even if the 7-quantum beam is not perfectly directed along the axes (planes). We also show that the symmetry o f the e ± energy distribution is preserved with good accuracy when the 3,-quantum beam is misaligned. In the case of PP in the averaged two-dimensional string potential, a considerable simplification of the quantum electrodynamical formulae becomes possible. It is based on the opportunity o f neglecting the quantum character of ultrarelativistic particle motion in the string potential. But it is necessary to account for another quantum effect corresponding to the recoil effect in the case of bremsstrahlung. Following the operator method [ 12,13,15,16] this problem is solved by taking into account the non-commutativity o f operators of the e -+ dynamic variables only with the 3,-quantum field (exp - i k . r ) . If one represents the expression for the PP probability [ 12,13,16] in the form of an integral over the volume where the PP occurs, the integrand may naturally be interpreted as the PP probability at the point r. The latter may be represented in the form: dW(r) - e2e2 f d20+ i de+ 4rr2~ ~
dr - { - ( e + [ e _ ) [ v 2 . t ~ 1 - ( n . v 2 ) ( n . v l )
] + (6o2/2e2)002.Vl - 1 + 1["/2)}
--oo
X exp (i(e+/e_)[6or - k . ( r 2 - r l ) ] ) .
(1)
Here e+ and e_ = 6o - e+ are e + and e - energies, correspondingly, m and e are the e - mass and the charge value, 7 = e+/m, k and 60 are the momentum and the energy o f the 7-quantum. We use units such that ff = c = 1. r(r) 335
Volume 113A, n u m b e r 6
PHYSICS LETTERS
30 December 1985
and u (r) = dr(r)/dr are the e + radius-vector and velocity at the moment r. The radius-vector r(r) obeys the usual relativistic equation of motion with the initial conditions r(0) = r and
o(O) = (Vex, roy, o(1 - 102)),
o ~ 1 - 1/2y 2 ,
(2)
where the z-axis is chosen to be parallel to the vector n z = k/co. The vector O = (0 x, Oy, 0), 0 @ 1 characterizes the e ÷ and y momenta misaligning at the point r of PP. Summation over the final states of the produced e + in eq. (1) can easily be carried out by integration over O. In eq. ( 1 ) r 1,2 = r(gr/2) and u 1,2 = I~(-Tr/2). The e -+ pair is formed in the region with length lcoh = %e-/m2co ~ c°/m2, called the coherence length [17,18]. The angles 0+ made by the e e velocities with the y-quantum momentum inside the region of pair formation are of order m/e. ~ m/co. The characteristic transverse size of the region of the pair formation A ~ Icoh0+_ ~ Re = m -1 is small in comparison with the mean thermal displacement amplitude u, which determines the characteristic space scale of variation of the transversal electric field strength associated with the crystal string potential. Therefore we may neglect the e -+ deviation from the y-quantum path under the calculation of the electric field strength acting on the e -+ during the time of pair formation. It should be pointed out that the correctness of this approximation is not connected with the value of the tilt angle ~k. It concides with the approximation used when one goes from the quantum formulae of refs. [ 2 - 5 ] to the uniform field limit when y-quanta are directed along crystallographic axes or planes. We shall use this approximation when the incident y-quantum intersects the crystal axis at same finite angle ~k ~ 0 and the electric field strength is not constant along the y-quantum path. We shall start from the fact that the qualitative picture of the pair formation in the uniform field does not change when the relative variation of the electric field strength A g / g within the pair formation region is the small parameter. We have the same situation in the case of ~ @ Vmax/m [2,3,19] or more exactly when if2 ,~ (Vmax/m)2 (see below). Expanding the expression (1) in terms of the small parameter, we shall represent it as a sum of the PP probability in the uniform field and a small correction to it. We shall consider PP in the field of a string, as a more general case (see fig. 1). Let the e ÷, created with the e at the point A, move almost along the z-axis which is parallel to the y-quantum path, and undergo a transversal displacement z ~ ~ lcohff "~ P, where p is the distance from the point A to the axis OO'. It should be pointed out that the latter inequality is broken in the region of small p ~ u and g, where the PP is exponentially damped [2, 3,8]. The small value of the parameter z~/p enables us to expand the expressions &x(Z) = g(B) sin a' and &y(Z) = &(B) cos a' for the x and y electric field strength projections in terms of this parameter. Then the equations 5¢'(r) = e~x(r)/e +, 5~'(r) = e~y(r)/e+ and the equality v z ~ o - (02 + o2)/2 (r is the time interval of e + motion from the point A, in the ultrarelativistic limit z ~ r) allow us to obtain the time dependence of the e + velocity components and coordinates entering (1). Expanding the expression (1) in terms of r~b/p and integrating it over 0+ we obtain
D
~o'
,iI
,
I
i.
I!,
fre) y
u
8
336
Fig. 1. The e + trajectory AB under the calculation o f the electric field acting on it during the f o r m a t i o n o f e +--pair, created at the point A. The "/-quantum intersects the crystal axis OO' at the angle ~. The e -+ transversal displacement A'B = qJz is small in comparison with the distance o between the point A and the axis O O ' . a and ~' are the angles which the electric field ~ m a k e s with the plane Z A Y (which is parallel to OO' and AB) at the points A and B, respectively.
Volume 113A, number 6
PHYSICS LETTERS
dW(~b, o0 _ die(~O = O)
de÷
aa:
e2m2
diepo dieo~o dW~o~o + d--ag--,+ d-aTU + d--g--+ + -de+
x/~ ¢o2
30 December 1985
Ai(y) dy + - ~ ( 2
-
2
+ X/-~-w2 { 1 gp cos2~ {10Ai'/x + (8/x 2 + x) Ai - (w2/e÷e_) [6Ai'/x + (4/x 2 + x) Ai] }
+ ~I gpg~o sin2ct {3(x -- 2/x 2) Ai + (3co2/e+e_) [(1/x 2 - x) Ai - Ai'/x] } i 2 sinEtx { 10Ai'/x + (14/x 2 - 2x) Ai - (6o2/e+e_) [3Ai'/x + (7/x 2 - 2x) Ai] } + ~g~o
+ I gpp c°s2ct {3(x - 2/x 2) Ai + (3w2/e+e) [(1/x 2 - x) Ai - Ai'/x]}} ,
(3)
where Ai = Ai(x) =
cos(xt + ~t 3) dt,
Ai' = dAi(x)/dt,
x = (m3co/ege+e_j -/3
are the Airy function of the parameter x and its derivative,
go =go (P) = mS'(P)~/eg2(O)' 8'(0) = dS(p)/dp,
g~o =g~o(P) = ml,O/eS(o)O,
8"(0) = d28(p)/do 2 .
goa =goo (o) = m28" (P)~2/ e283(P) ' (4)
It should be recalled that P is the distance between the point A of PP and the crystal axis 0 0 : tx is the angle made by the vector of electric field strength 8 and the plane A'AB parallel to the axis 0 0 ' and the ")'-quantum momentum (z-axis), see fig. 1. The parameters (4) characterize the relative variation of the field components in the space (time) region Izl ~/cola ([rl ~ lcoh) of the pair formation. The parameters go' goo and g¢ are connected with the variation of the value and of the direction of the electric field 8, respectively. The terms dWpo/de + ~ g2 ..... dle 2/de+ ~ gpo are the corrections to the PP probability die(~ = 0)/de÷ in the uniform field or by the 7-quano tum directed along the axis. They appear due to the heterogeneity of the string field when the 7-quanta intersect the crystal axis at some angle ~ 4= 0. The analysis of the relative value of these corrections makes it possible to understand the qualitative features and the variation of PP with tilt angle. We shall not consider the PP in the averaged field of the atomic planes here, but it should be pointed out that the expression (3) may be applied to this problem if one substitutes a = 0 and p = oo or g~ = 0 into (3). The expansion (3) does not work for p ~ 0 because of &(p) -+ 0. Therefore we shall analyze the PP in the region of p > Pmin > 0. Under the experimental conditions of ref. [ 14] ((110) axis of a Ge crystal at 100 K) it is reasonable to take Omin = 0.03 fit and to analyse the case of ~b --- 0.2 mrad and 60 = 100 GeV in detail. The quadratic dependence of the correction terms in (3) from the value of the tilt angle ~ makes it possible to recalculate the following results to the case of any other value of ~b. The radial distribution of the PP probability 2~r to
die(O, ~)]dp =dnp
f f 0
[die(O, 4, a)/de] de da
0
and the corrections diepo(p, ~)/dp = . ...... to the probability dW(p, ~ = 0)/do are shown in fig. 2. Here n is the atomic density in the crystal and d is the axis interatomic distance. It should be noted that the PP occurs in the region p < 0.4 A where the averaged crystal potential possesses axial symmetry. Fig. 3 presents the energy distribution of e -+ produced in the region of P > 0.03 A. It is not difficult to see that the relative corrections are of the order of some percent in a wide region of % and p. It should be noted that this fact is in full agreement with the statement of refs. [2,3] that the mechanism of the PP in the uniform electromagnetic field manifests itself 337
Volume 113A, n u m b e r 6
PHYSICS LETTERS
30 December 1985
~:~0.2:azc:d ;5
< l/O.":eTOOkJ
20
-'5
\
:o,c:w:,id:
-:0 Fig. 2. Radial distributions of the corrections dWpp(p, ¢)/dp ~ g2 .... for ~V = 0.2 mrad and of the PP probability dW(p, ¢,)/dp for tp = 0 and ¢ = 0.2 mrad, normalized to the BH PP probability.
dW(.p~.40s,:)Idz (W~) ~'-02 mzsd
-2 338
< :':O>GeIOOk'
Fig. 3. Energy distribution of the e + produced at the region o f p > 0.03 A for ~0 = 0 and ~ = 0.2 mrad and the energy dependences of corrections dWpp(p > 0.03 A, z, ~V)/dz .... for ~k = 0.2 mrad, z = e+/to.
Volume 113A, number 6
PHYSICS LETTERS
30 December 1985
o.aos#).w,. (v~,.) < 7I0> 6'ei00/¢ 12 I0 8 6'
~ 2
10mw , 2~:'W2q
70xW77 ....
~
10~W££
0
m
18o
W~
2q.O
ca(~ev)
Fig. 4. The PP probabilities in tne region o f p < 0.03 f o r V~ = 0 and in the zegion o f p > 0.03 A f o r VJ = 0.2 mrad and ~ = 0 and the corrections to the last Wpp(p > 0.03 A ) .... for ~ = 0.2 mrad,
in the crystal when the 7-quantum tilt angles qJ < "~, "~ = Vmax/m (Vmax is the maximum value of the e + potential energy in the field of crystal planes or axes~. Indeed the expansion (3) fails for ~ >~ ~ because the value of the corrections are nine times as large for ~ = ~ = 0.6 mrad as for ~ = 0.2 mrad. The correction becomes comparable with the value of the main term in (3). The authors of refs. [6,14] have concluded that the PP rate falls for > 0c ~ (V m a x / ~ ) 1/2 ,~ ~. They have taken into account only the pairs with channeling e - , but the e - produced by the 7-quanta intersecting the crystal axes at the angle 0 c < ~ < ~b are not channeled. It should be noted that the channeling effect cannot influence the pair formation process because the channeling period Tch >~/cob" Therefore, in our opinion, the experimental results of ref. [ 14] showing the fall of PP rate beyond the critical angle 0 c ~ 0.05 mrad need additional examination. The expression(3) and fig. 3 show the high degree of symmetry of the e +- energy distributions both for ~ = 0 and for 0 < $ < ~b, which contradicts ref. [6]. Therefore the precise measurement of the e e energy distribution is of great interest too. Fig. 4 displays the ~o dependence of the PP probability in the region of p > 0.03 A of 2~ Pmin 0
~o 0
for ~J = 0.2 mrad and ~b = 0. Our results indicate an increase of the PP rate with ~k for co < 200 GeV. At the same time it seems very natural that the PP rate should fall beyond some sufficiently larg~ tilt angle. This fall can be described only by further corrections to (3) which became considerablewhen ~ ~ 0- Therefore the rate of PP by 7-quanta with the energy co < 200 GeV reaches a maximum at 0 < ~0 < ~ (the authors of ref. [6] have predicted a maximum at ~ ~ 0c). It should be recalled that the PP rate predicted on the basis of the formulae of refs. [4-6] exceeds the experimental one by a factor of about 3 (see the text of ref. [14]). Calculations based on the formulae of refs. [2,3] have led us to the same result (see fig. 4). It should be noted that the PP by the 3,-quanta intersecting the crystal axis at the angles $ > ~ has been analysed in ref. [19].
339
Volume 113A, number 6
PHYSICS LETTERS
30 December 1985
References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [ 17]
V.G. Baryshevskii, Proc. XlVth Winter School of LIYaF (Leningrad, 1978) p. 158. V.G. Baryshevskii and V.V. Tikhomirov, Sov. J. Nucl. Phys. 36 (1982) 408. V.G. Baryshevskii and V.V. Tikhomirov, Phys. Lett. 90A (1982) 153. J.C. Kimball, N. Cue, L.M. Roth and B.B. Marsh, Phys. Rev. Lett. 50 (1983) 950. J.C. Kimball and N. Cue, Nucl. Instrum. Methods B2 (1984) 25. N. Cue and J.C. Kimball, Nucl. Instrum. Methods B2 (1984) 29. V.N. Baier, V.M. Katkov and V.M. Strakhovenko, Phys. Lett. 104A (1984) 231. A.M. Frotov and V.L. Mikhaljov, IHEP 84-24 (Serpukhov, 1984). A.H. SCrensen, E. Uggerh~j, J. Bak and S.P. M~ller, CERN-EP/84-149. N.P. Klepikov, Zh. Eksp. Teor. Fiz. 26 (1954) 19. A.I. Nikishov and V.I. Ritus, Tr. Fiz. Inst. Akad. Nauk SSSR 111 (1978). V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Quantum electrodynamics (Pergamon, London, 1982). V.N. Baier, V.M. Katkov and V.S. Fadin, Radiation from relativistic electrons (Atomizdat, Moscow, 1973), in Russian. A. Belkacem et al., Phys. Rev. Lett. 53 (1984) 2371. J. Schwinger, Proc. Nat. Acad. Sci. 40 (1954) 132. V.N. Baler and V.N. Katkov, Phys. Lett. 25A (1967) 492. M.L. Ter-Mikaelian, Interscience tracts in physics and astronomy, Vol. 29. High energy electromagnetic processes in condensed media (Wiley, New York, 1972). [18] G. Diambrini Palazzi, Rev. Mod. Phys. 40 (1968) 611. [19] V.G. Baryshevskii and V.V. Tikhomirov, Nucl. Instrum. Methods 234A (1985) 430.
340
PHYSICS LETTERS
30 December 1985
P A I R P R O D U C T I O N IN A S L O W L Y V A R Y I N G E L E C T R O M A G N E T I C F I E L D AND THE PAIR PRODUCTION PROCESS
V.G. B A R Y S H E V S K I I and V.V. T I K H O M I R O V Department of Physics, V.L Lenin Byelorussian State University, Minsk 220080, U¢~R
Received 8 January 1985; revised manuscript received 9 October 1985; accepted for publication 24 October 1985
The numerical analysis of pair production (PP) by y-quanta intersecting the crystal axis with a small angle is given on the basis of the formulae describing the PP in a slightly inhomogeneous electric field.
The new C-pair production (PP) process predicted in refs. [ 1 - 3 ] received extensive interest [ 4 - 9 ] . It should be recalled that the PP process predicted in refs. [ 1 - 5 ] is closely related with the mechanism o f PP in a strong electric field [ 1 0 - 1 3 ]. Recently characteristic features of the process (the significant increase of PP rate above the Bethe-Heitler (BH) rate, a very narrow range of angles over which this process extends and the significant difference of the e +- energy distribution from the BH predictions) have been observed experimentally [14]. The quantum formulae of refs. [ 2 - 5 ] turn into the simple formulae of the theory of the PP in a constant electromagnetic field [ 1 0 - 1 3 ] when ~,-quanta are directed along crystallographic axes or planes. At the same time it is very difficult to examine carefully the consequence o f misaligning a 7-quantum beam by some tilt angle on the basis of the quantum formulae [6]. In this paper we have obtained the expression for the PP probability in a slightly inhomogeneous electric field. It enables us to analyze the PP by a non-divergent 7-quantum beam intersecting the crystal axis or plane at some non-zero angle. It will be shown below that a significant change (fall) of the PP rate may occur only when the tilt angles ~b are one order of magnitude larger than those predicted in ref. [6] and said to be observed in ref. [14]. Thus, an enhanced PP rate may be observed even if the 7-quantum beam is not perfectly directed along the axes (planes). We also show that the symmetry o f the e ± energy distribution is preserved with good accuracy when the 3,-quantum beam is misaligned. In the case of PP in the averaged two-dimensional string potential, a considerable simplification of the quantum electrodynamical formulae becomes possible. It is based on the opportunity o f neglecting the quantum character of ultrarelativistic particle motion in the string potential. But it is necessary to account for another quantum effect corresponding to the recoil effect in the case of bremsstrahlung. Following the operator method [ 12,13,15,16] this problem is solved by taking into account the non-commutativity o f operators of the e -+ dynamic variables only with the 3,-quantum field (exp - i k . r ) . If one represents the expression for the PP probability [ 12,13,16] in the form of an integral over the volume where the PP occurs, the integrand may naturally be interpreted as the PP probability at the point r. The latter may be represented in the form: dW(r) - e2e2 f d20+ i de+ 4rr2~ ~
dr - { - ( e + [ e _ ) [ v 2 . t ~ 1 - ( n . v 2 ) ( n . v l )
] + (6o2/2e2)002.Vl - 1 + 1["/2)}
--oo
X exp (i(e+/e_)[6or - k . ( r 2 - r l ) ] ) .
(1)
Here e+ and e_ = 6o - e+ are e + and e - energies, correspondingly, m and e are the e - mass and the charge value, 7 = e+/m, k and 60 are the momentum and the energy o f the 7-quantum. We use units such that ff = c = 1. r(r) 335
Volume 113A, n u m b e r 6
PHYSICS LETTERS
30 December 1985
and u (r) = dr(r)/dr are the e + radius-vector and velocity at the moment r. The radius-vector r(r) obeys the usual relativistic equation of motion with the initial conditions r(0) = r and
o(O) = (Vex, roy, o(1 - 102)),
o ~ 1 - 1/2y 2 ,
(2)
where the z-axis is chosen to be parallel to the vector n z = k/co. The vector O = (0 x, Oy, 0), 0 @ 1 characterizes the e ÷ and y momenta misaligning at the point r of PP. Summation over the final states of the produced e + in eq. (1) can easily be carried out by integration over O. In eq. ( 1 ) r 1,2 = r(gr/2) and u 1,2 = I~(-Tr/2). The e -+ pair is formed in the region with length lcoh = %e-/m2co ~ c°/m2, called the coherence length [17,18]. The angles 0+ made by the e e velocities with the y-quantum momentum inside the region of pair formation are of order m/e. ~ m/co. The characteristic transverse size of the region of the pair formation A ~ Icoh0+_ ~ Re = m -1 is small in comparison with the mean thermal displacement amplitude u, which determines the characteristic space scale of variation of the transversal electric field strength associated with the crystal string potential. Therefore we may neglect the e -+ deviation from the y-quantum path under the calculation of the electric field strength acting on the e -+ during the time of pair formation. It should be pointed out that the correctness of this approximation is not connected with the value of the tilt angle ~k. It concides with the approximation used when one goes from the quantum formulae of refs. [ 2 - 5 ] to the uniform field limit when y-quanta are directed along crystallographic axes or planes. We shall use this approximation when the incident y-quantum intersects the crystal axis at same finite angle ~k ~ 0 and the electric field strength is not constant along the y-quantum path. We shall start from the fact that the qualitative picture of the pair formation in the uniform field does not change when the relative variation of the electric field strength A g / g within the pair formation region is the small parameter. We have the same situation in the case of ~ @ Vmax/m [2,3,19] or more exactly when if2 ,~ (Vmax/m)2 (see below). Expanding the expression (1) in terms of the small parameter, we shall represent it as a sum of the PP probability in the uniform field and a small correction to it. We shall consider PP in the field of a string, as a more general case (see fig. 1). Let the e ÷, created with the e at the point A, move almost along the z-axis which is parallel to the y-quantum path, and undergo a transversal displacement z ~ ~ lcohff "~ P, where p is the distance from the point A to the axis OO'. It should be pointed out that the latter inequality is broken in the region of small p ~ u and g, where the PP is exponentially damped [2, 3,8]. The small value of the parameter z~/p enables us to expand the expressions &x(Z) = g(B) sin a' and &y(Z) = &(B) cos a' for the x and y electric field strength projections in terms of this parameter. Then the equations 5¢'(r) = e~x(r)/e +, 5~'(r) = e~y(r)/e+ and the equality v z ~ o - (02 + o2)/2 (r is the time interval of e + motion from the point A, in the ultrarelativistic limit z ~ r) allow us to obtain the time dependence of the e + velocity components and coordinates entering (1). Expanding the expression (1) in terms of r~b/p and integrating it over 0+ we obtain
D
~o'
,iI
,
I
i.
I!,
fre) y
u
8
336
Fig. 1. The e + trajectory AB under the calculation o f the electric field acting on it during the f o r m a t i o n o f e +--pair, created at the point A. The "/-quantum intersects the crystal axis OO' at the angle ~. The e -+ transversal displacement A'B = qJz is small in comparison with the distance o between the point A and the axis O O ' . a and ~' are the angles which the electric field ~ m a k e s with the plane Z A Y (which is parallel to OO' and AB) at the points A and B, respectively.
Volume 113A, number 6
PHYSICS LETTERS
dW(~b, o0 _ die(~O = O)
de÷
aa:
e2m2
diepo dieo~o dW~o~o + d--ag--,+ d-aTU + d--g--+ + -de+
x/~ ¢o2
30 December 1985
Ai(y) dy + - ~ ( 2
-
2
+ X/-~-w2 { 1 gp cos2~ {10Ai'/x + (8/x 2 + x) Ai - (w2/e÷e_) [6Ai'/x + (4/x 2 + x) Ai] }
+ ~I gpg~o sin2ct {3(x -- 2/x 2) Ai + (3co2/e+e_) [(1/x 2 - x) Ai - Ai'/x] } i 2 sinEtx { 10Ai'/x + (14/x 2 - 2x) Ai - (6o2/e+e_) [3Ai'/x + (7/x 2 - 2x) Ai] } + ~g~o
+ I gpp c°s2ct {3(x - 2/x 2) Ai + (3w2/e+e) [(1/x 2 - x) Ai - Ai'/x]}} ,
(3)
where Ai = Ai(x) =
cos(xt + ~t 3) dt,
Ai' = dAi(x)/dt,
x = (m3co/ege+e_j -/3
are the Airy function of the parameter x and its derivative,
go =go (P) = mS'(P)~/eg2(O)' 8'(0) = dS(p)/dp,
g~o =g~o(P) = ml,O/eS(o)O,
8"(0) = d28(p)/do 2 .
goa =goo (o) = m28" (P)~2/ e283(P) ' (4)
It should be recalled that P is the distance between the point A of PP and the crystal axis 0 0 : tx is the angle made by the vector of electric field strength 8 and the plane A'AB parallel to the axis 0 0 ' and the ")'-quantum momentum (z-axis), see fig. 1. The parameters (4) characterize the relative variation of the field components in the space (time) region Izl ~/cola ([rl ~ lcoh) of the pair formation. The parameters go' goo and g¢ are connected with the variation of the value and of the direction of the electric field 8, respectively. The terms dWpo/de + ~ g2 ..... dle 2/de+ ~ gpo are the corrections to the PP probability die(~ = 0)/de÷ in the uniform field or by the 7-quano tum directed along the axis. They appear due to the heterogeneity of the string field when the 7-quanta intersect the crystal axis at some angle ~ 4= 0. The analysis of the relative value of these corrections makes it possible to understand the qualitative features and the variation of PP with tilt angle. We shall not consider the PP in the averaged field of the atomic planes here, but it should be pointed out that the expression (3) may be applied to this problem if one substitutes a = 0 and p = oo or g~ = 0 into (3). The expansion (3) does not work for p ~ 0 because of &(p) -+ 0. Therefore we shall analyze the PP in the region of p > Pmin > 0. Under the experimental conditions of ref. [ 14] ((110) axis of a Ge crystal at 100 K) it is reasonable to take Omin = 0.03 fit and to analyse the case of ~b --- 0.2 mrad and 60 = 100 GeV in detail. The quadratic dependence of the correction terms in (3) from the value of the tilt angle ~ makes it possible to recalculate the following results to the case of any other value of ~b. The radial distribution of the PP probability 2~r to
die(O, ~)]dp =dnp
f f 0
[die(O, 4, a)/de] de da
0
and the corrections diepo(p, ~)/dp = . ...... to the probability dW(p, ~ = 0)/do are shown in fig. 2. Here n is the atomic density in the crystal and d is the axis interatomic distance. It should be noted that the PP occurs in the region p < 0.4 A where the averaged crystal potential possesses axial symmetry. Fig. 3 presents the energy distribution of e -+ produced in the region of P > 0.03 A. It is not difficult to see that the relative corrections are of the order of some percent in a wide region of % and p. It should be noted that this fact is in full agreement with the statement of refs. [2,3] that the mechanism of the PP in the uniform electromagnetic field manifests itself 337
Volume 113A, n u m b e r 6
PHYSICS LETTERS
30 December 1985
~:~0.2:azc:d ;5
< l/O.":eTOOkJ
20
-'5
\
:o,c:w:,id:
-:0 Fig. 2. Radial distributions of the corrections dWpp(p, ¢)/dp ~ g2 .... for ~V = 0.2 mrad and of the PP probability dW(p, ¢,)/dp for tp = 0 and ¢ = 0.2 mrad, normalized to the BH PP probability.
dW(.p~.40s,:)Idz (W~) ~'-02 mzsd
-2 338
< :':O>GeIOOk'
Fig. 3. Energy distribution of the e + produced at the region o f p > 0.03 A for ~0 = 0 and ~ = 0.2 mrad and the energy dependences of corrections dWpp(p > 0.03 A, z, ~V)/dz .... for ~k = 0.2 mrad, z = e+/to.
Volume 113A, number 6
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30 December 1985
o.aos#).w,. (v~,.) < 7I0> 6'ei00/¢ 12 I0 8 6'
~ 2
10mw , 2~:'W2q
70xW77 ....
~
10~W££
0
m
18o
W~
2q.O
ca(~ev)
Fig. 4. The PP probabilities in tne region o f p < 0.03 f o r V~ = 0 and in the zegion o f p > 0.03 A f o r VJ = 0.2 mrad and ~ = 0 and the corrections to the last Wpp(p > 0.03 A ) .... for ~ = 0.2 mrad,
in the crystal when the 7-quantum tilt angles qJ < "~, "~ = Vmax/m (Vmax is the maximum value of the e + potential energy in the field of crystal planes or axes~. Indeed the expansion (3) fails for ~ >~ ~ because the value of the corrections are nine times as large for ~ = ~ = 0.6 mrad as for ~ = 0.2 mrad. The correction becomes comparable with the value of the main term in (3). The authors of refs. [6,14] have concluded that the PP rate falls for > 0c ~ (V m a x / ~ ) 1/2 ,~ ~. They have taken into account only the pairs with channeling e - , but the e - produced by the 7-quanta intersecting the crystal axes at the angle 0 c < ~ < ~b are not channeled. It should be noted that the channeling effect cannot influence the pair formation process because the channeling period Tch >~/cob" Therefore, in our opinion, the experimental results of ref. [ 14] showing the fall of PP rate beyond the critical angle 0 c ~ 0.05 mrad need additional examination. The expression(3) and fig. 3 show the high degree of symmetry of the e +- energy distributions both for ~ = 0 and for 0 < $ < ~b, which contradicts ref. [6]. Therefore the precise measurement of the e e energy distribution is of great interest too. Fig. 4 displays the ~o dependence of the PP probability in the region of p > 0.03 A of 2~ Pmin 0
~o 0
for ~J = 0.2 mrad and ~b = 0. Our results indicate an increase of the PP rate with ~k for co < 200 GeV. At the same time it seems very natural that the PP rate should fall beyond some sufficiently larg~ tilt angle. This fall can be described only by further corrections to (3) which became considerablewhen ~ ~ 0- Therefore the rate of PP by 7-quanta with the energy co < 200 GeV reaches a maximum at 0 < ~0 < ~ (the authors of ref. [6] have predicted a maximum at ~ ~ 0c). It should be recalled that the PP rate predicted on the basis of the formulae of refs. [4-6] exceeds the experimental one by a factor of about 3 (see the text of ref. [14]). Calculations based on the formulae of refs. [2,3] have led us to the same result (see fig. 4). It should be noted that the PP by the 3,-quanta intersecting the crystal axis at the angles $ > ~ has been analysed in ref. [19].
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References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [ 17]
V.G. Baryshevskii, Proc. XlVth Winter School of LIYaF (Leningrad, 1978) p. 158. V.G. Baryshevskii and V.V. Tikhomirov, Sov. J. Nucl. Phys. 36 (1982) 408. V.G. Baryshevskii and V.V. Tikhomirov, Phys. Lett. 90A (1982) 153. J.C. Kimball, N. Cue, L.M. Roth and B.B. Marsh, Phys. Rev. Lett. 50 (1983) 950. J.C. Kimball and N. Cue, Nucl. Instrum. Methods B2 (1984) 25. N. Cue and J.C. Kimball, Nucl. Instrum. Methods B2 (1984) 29. V.N. Baier, V.M. Katkov and V.M. Strakhovenko, Phys. Lett. 104A (1984) 231. A.M. Frotov and V.L. Mikhaljov, IHEP 84-24 (Serpukhov, 1984). A.H. SCrensen, E. Uggerh~j, J. Bak and S.P. M~ller, CERN-EP/84-149. N.P. Klepikov, Zh. Eksp. Teor. Fiz. 26 (1954) 19. A.I. Nikishov and V.I. Ritus, Tr. Fiz. Inst. Akad. Nauk SSSR 111 (1978). V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Quantum electrodynamics (Pergamon, London, 1982). V.N. Baier, V.M. Katkov and V.S. Fadin, Radiation from relativistic electrons (Atomizdat, Moscow, 1973), in Russian. A. Belkacem et al., Phys. Rev. Lett. 53 (1984) 2371. J. Schwinger, Proc. Nat. Acad. Sci. 40 (1954) 132. V.N. Baler and V.N. Katkov, Phys. Lett. 25A (1967) 492. M.L. Ter-Mikaelian, Interscience tracts in physics and astronomy, Vol. 29. High energy electromagnetic processes in condensed media (Wiley, New York, 1972). [18] G. Diambrini Palazzi, Rev. Mod. Phys. 40 (1968) 611. [19] V.G. Baryshevskii and V.V. Tikhomirov, Nucl. Instrum. Methods 234A (1985) 430.
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