On the theory of electron-positron pair production in crystals

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On the theory of electron-positron pair production in crystals

V.V. Tikhomirov

To cite this version:

V.V. Tikhomirov. On the theory of electron-positron pair production in crystals. Journal de Physique,

1987, 48 (6), pp.1009-1016. �10.1051/jphys:019870048060100900�. �jpa-00210508�

On the theory ^of electron-positron pair production ⁱⁿ crystals

V. V. Tikhomirov

Department ^of Physics, V. I. Lenin Byelorussian ^State University, ^Minsk 220080, ^U.S.S.R.

(Requ le 30 octobre 1986, accepté ^{le 16} janvier 1986)

Résumé.

On présente ^une théorie décrivant les processus de rayonnement ^{cohérents et} incohérents quand

des quanta 03B3, des électrons ou des positrons ^se déplacent ^{dans un} cristal. On montre que, dans le cadre d’une

approximation logarithmique, ^les probabilités locales des processus incohérents ^sont proportionnelles ^aux

sections efficaces des mêmes processus relatifs à des noyaux placés ^{dans un} champ ^{uniforme. Quand le taux de} production ^des paires cohérentes augmente, ^la probabilité totale de production ^de paires incohérentes commence à décroître proportionnellement ^à l’énergie ^des quanta 03B3 suivant une puissance 2014 ^2/3.

Abstract.

The theory describing both coherent and incoherent radiational processes when energetic 03B3-quanta, ^{electrons or} positrons ^move through crystals in directions nearly parallel ^{to the} crystal ^{axes or} planes

is presented. ^{It is shown} that, within the logarithmic approximation, ^{the local} probabilities of incoherent radiation processes ^are proportional ^to the cross-sections of the same processes pertaining ^{to a} separate nucleus placed ^{in a} uniform field. When the coherent pair production ^rate increases, ^{the total} probability ^of

incoherent pair production ^{starts to decrease} proportionally ^{to the} 03B3-quantum energy ^to the 2014 2/3 power.

Classification

Physics ^Abstracts

12.20

41.70

61.80

1. Introduction.

A new pair-production mechanism appears ^when

sufficiently energetic y-quanta ^move nearly along

the crystallographic ^{axes or} planes [1-4]. ^{It is known} [3, 5, 6] ^{that, in this case, the} high-energy particle

interaction with the crystal ^{axes or} planes ^{is de-}

scribed with a good accuracy by ^the averaged potential ^{of the axes or} planes (i. e. by ^{the atomic} potential averaged along ^{the axes or} planes and per lattice vibration). ^{Near the} perfect crystal alignment

the averaged crystal ^{field is} practically ^uniform

within the narrow pair ^formation region. ^{As a}

consequence, in order ^to calculate the pair produc-

tion (PP) probability ^{in a} crystal it is sufficient to average the corresponding probability ^{in a uniform}

electric field over the averaged field distribution.

The uniform field approximation has allowed us to

predict ^{a PP rate} enhancement observed in [7]. ^The

account of the averaged ^{axis field} inhomogeneity ^has provided ^to investigate ^the orientational dependence

of PP probability [8-10]. However, ^the averaged potential ^describes only ^coherent particle interaction without changing ^the crystal ^state. Up ^{to now the}

incoherent PP processes in crystals ^{were considered}

in the Bom approximation only [11-13]. ^{It was found}

that such processes ^are accompanied by sufficiently

large ^{momentum transfers to the} separate ^nuclei and, consequently, they ^are quite ^{similar to the}

processes taking place ^{in an} amorphous ^medium.

With sufficiently high yquantum energies, ^incohe-

rent PP processes become more various. For

example, ^the produced ^e± moving ^{inside an atomic} string undergo strong multiple scattering ^{within the} pair ^formation region, ^the length ^{of which} grows with the yquantum energy. The nuclear density

inside the atomic string ^{is some} hundred times larger

than the _average density ^{in a} crystal. ^{Under such}

conditions the Landau-Pomeranchuk effect [14, 15]

can manifest itself at the T-quantum energy exceed-

ing ^{10 GeV. The} consideration of the PP process ⁱⁿ this case is not a simple problem [16].

Here we present ^a theory successfully describing

the incoherent PP processes in the ^case of a crystal

axis orientation along ^the y-quantum ^{momentum. In} section 2, ^{the e±} scattering process in the pair

formation region ^is considered. The logarithmic approximation ^{allows us to} distinguish ^two types ^of incoherent PP processes. ^The first one is connected with the e± incoherent multiple scattering ^{in the} pair

formation region. The second one arises due to the

sufficiently large angle scattering ^{of e±} by separate

nuclei. It is shown in sections 2 and 3 that the

probabilities ^{of both} types of incoherent PP proces-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048060100900

1010

ses as well as their sum are proportional ^{to the PP}

cross-section on the separate ^nucleus placed ^{in a}

uniform field. In section 4 the obtained expression

for the PP probability ^is put ⁱⁿ accordance with that in the Born approximation by choosing ^{a certain} parameter. ^{In the same section we} evaluate the differential and integral probabilities of the PP in Ge

(11°). Section 5 is a brief conclusion.

2. Pair production ^{in a} uniform electric field in the presence of multiple scattering.

To understand the basic features of the incoherent PP processes it is sufficient to study ^{the case of} perfect alignment ^{between the} crystal ^{axis and the y.}

quantum ^{momentum. The nature of} the PP process is completely determined by ^{the e±} scattering ^within

the pair ^formation region. ^{It should be recalled that}

in the above mentioned case, the coherent e±

scattering is described by ^{the uniform field} approxi-

mation. The pair formation process in the transversal uniform electric field 6 (see ^Refs. [17-21]) ^{is charac-}

terized by ^{a coherent} length (the length ^{of a} pair

formation region)

and by ^a characteristic angle

where

w, E+ and ^{E_ - w -} 8 + ^are y-quantum, ^{e+ and}

e energies, correspondingly, m ^{and e are the}

e- mass and the absolute value of the e- charge. ^We

use the unit system h

^{c = 1. Since under the} y.

quantum energies w > ^{10 GeV} the coherent length (1) greatly exceeds the axis interatomic distance, e± experience ^numerous both coherent and incohe- rent collisions with the nuclei while the e+ e- -pair ^is forming. ^The small-angle ^{coherent e±} scattering ^is

described by the uniform field (the averaged ^{field of} axes). ^{Below, while} integrating the cross-section of incoherent Coulomb scattering ^over angles ^{we shall}

use the logarithmic approximation. Consequently,

one believes that the incoherent scattering processes take place ^{under the e± -nuclei momentum transfers} åP.l > I /u, ^{or under the e:t} scattering ^{at the} angles

v :> ð _min =1 / u £+ , ^{where u is} the one-dimensional mean-square displacement ^{of the atom in the} string [22], ^while integrating ^over incoherent scattering

angles ifi > v _min the main contribution to the integral

is given by ^the angles ifi > ’tgmin. ^Since the collision parameters corresponding ^{to such} scattering angles

are b == 1 /,0 c, -r, u, ^the change ^{in the} scattering

nuclear density ^{at the} distance I Ap I ~ ^{b u} away from the e+ trajectory ^{can be} neglected. ^Further-

more, the process of incoherent scattering ^at angles

ia > v- min ^{may be} considered as the scattering ^{on the}

free nuclei not being connected with a crystal. ^{As a}

consequence, ^to calculate the root-mean-square (r.m.s.) ^e+ multiple scattering angle ^{in the} pair

formation region ^{0 z} I coh’ ^one may ^use the well- known formulae for uniform amorphous ^media (see,

for example, ^Refs. [12, 19]) ^{which is written in the}

form

where

and, ^more precisely,

is the local nuclear density averaged along ^the crystal axis, p = (x, y) ^{is the} projection ^{of the} positron

vector onto the plane perpendicular ^{to a} crystallogra- phic ^{direction taken as the z-axis, d is the axis}

interatomic distance, ^{and Z} is the atomic number.

Formula (4) takes into account the e± scattering by separate ^{nuclei at} angles Vmin ^V VmaX. ^The

minimum scattering angle 0 i. - 1 / eu will be rede- termined in section 4. When the angle ^{of the}

e± scattering by ^a separate nucleus exceeds the

r.m.s. angle (4) ^the scattering ^{in the} pair ^formation region ^{can be} considered as a single ^{rather than as a} multiple ^{one. Therefore we} put

and consider such scattering processes separately ⁱⁿ

section 3. Note that in future use of formula (4), ^we

shall neglect ^{the small local e±} scattering asymmetry existing inside both atomic layer [22] ^and string.

Thus, ^{one can conclude that in order to} study ^{the PP}

process ⁱⁿ the case of perfect alignment, ^{it would be}

sufficient to consider the pair formation process in ^a uniform transversal field in the _presence of incohe- rent multiple scattering ^described by ^formulae (4)

and (5).

It was shown by ^Landau and Pomeranchuk [14]

that e± multiple scattering substantially ^{modified the}

nature of the radiational _processes (PP ^or yquan- tum emission) taking place ^{in an} amorphous ^medium

when

where 1,.h ^and ’Vchar ^are the coherent length ^{and the}

characteristic angle ^{of the} radiational _process, corres-

pondingly. ^{In the case} of PP in an amorphous

medium

and, therefore, ’&, (1 h ) OC CO - "2 (we ^assume

E; - w ). ^{Such an} energy dependence ^{on both sides}

of inequality (8) ^{results in the} amplification ^{of the} multiple scattering ^{influence on the} pair ^or y-quan- tum formation with increasing particle energy. ^{As a} consequence, inequality (8) ^is necessarily ^satisfied

for high energies. ^For example, ^{in Pb this} happens when w =103 GeV [15]. ^A further energy growth

leads to the suppression ^of radiational _processes which is known as the Landau-Pomeranchuk effect.

Note that the total PP probability ^decreases propor-

tionally ^{to úJ -1/2} [14, 15].

Let us now consider the process of pair ^formation developing inside the atomic string. ^{The nuclear} density ^{inside the} string (6) typically ^{exceeds the} averaged ^atomic density ^{in a} crystal ^{some hundred}

times. In such a dense medium one can expect ^the manifestation of the Landau-Pomeranchuk effect for yquantum energies w ^{$: 10} GeV. However the _pre-

sence of a uniform electric field (coherent scattering) substantially ^modifies the character of the multiple scattering ^{influence on the} radiational _processes.

Really, ^from equalities (1) ^and (2) ^one easily ^obtains that it & oc ^{úJ - 2/3} and ils (le) ^{oc w -516 in the case of}

intense PP (or x 1, ^see Eq. (3)). ^{Such an energy} dependence ^of both sides of inequality (8) ^indicates

a decrease of the multiple scattering ^{influence on the}

process of ^a pair (a -y-quantum) ^{formation with the} particle energy growth. ^{As a} consequence, the latter

inevitably ^{leads to the violation of} inequality (8) ^for sufficiently high energies. ^{For limited} particle ^ener- gies, inequality (8) ^{is satisfied} within the narrow

cylindrical region around the line _p

0 where the electric field strength ^{is not too} large. ^{The Gauss}

theorem allows us to obtain the averaged ^axis

electric field strength ^as

On the basis of equalities (1), (2) ^and (10), inequality (8) may be rewritten in the form

where Ae

h/mc

^{3.862 x 10-11 cm is the electron} Compton wavelength. ^The theory describing ^the

radiational processes in region (11) ^is quite complex [16]. ^{It can be} substantially simplified ^{in the case}

when the inequality ^{inverted to} (8) ^{is fulfilled.}

Fortunately ^{the fraction of} string ^nuclei occupying

the region p p min is ^a value of p min2/ u 2 ^which

decreases both with increasing -y-quantum ^energy

and with decreasing ^atomic number. As a conse-

quence ^some fundamental features of the incoherent PP process ^can be established without satisfying inequality (8) ^{inside the narrow} region (11). ^{In other} words, ^{we shall} suppose that ^the characteristic angle (2) ^{exceeds the r.m.s. e±} scattering angle (4) ^inside

the pair ^formation region everywhere ^{in a} crystal.

Under such a condition the incoherent multiple

e± scattering ^{can be} considered as a perturbation against the background ^{of a} uniform field (coherent scattering). ^{To evaluate the} probability ^{of PP in a} perturbed ^uniform field it is suitable to start from the semi-classical expression [19, 20]

where

r(t) ^and v(t) ^{are the} e+ radius-vector and the

velocity ^{at the moment t} (e+ ^{is chosen for} accuracy), rl, 2 = r (tl, 2 ), Vl, 2V (tl, 2 ), v, (t ) ^{is the} projection

of e+ velocity ^{onto the} plane perpendicular ^{to the} y.

quantum ^{momentum, and a} = 1/137. ^{Note that} expression (12) ^{describes the} probability ^{of e+}

e- -pair production ^{when its} positron ^moves along ^a

fixed trajectory. ^{In the case of a} slightly perturbed

uniform electric field as well as in the case of a

perfect uniform electric field, ^{the internal} integral ⁱⁿ (12) ^{is formed in a} small time interval close to the

moment t1 and characterizes the local PP probability

at the point r(tl). The external integral ^over

t1 plays the role of the integral ^{over different} points

of PP. As a consequence, ^to evaluate the local PP

probability ^{at some} point ^r

r (t1 ) ^{we can omit the} integral over t1 ⁱⁿ (12) ^and integrate the obtained

expression ^{over e+} trajectories crossing ^the point ^r.

To perform ^this operation ^we represent ^{the e+}

velocity ^{in the} following ^form by using the small-

angle approximation

where

is the e+ acceleration in the uniform transversal electric field, ⁿ

k/w ^{is the} yquantum ^momentum direction, ^{0 is a solid} angle ^{between the vector n and}

the e+ velocity ^{at the} point ^{of its} production ^{r at the}

moment T = t2 - tl

^{0. Since in} logarithmic ap-

proximation ^the typical e+ -nuclei collision par-

ameters greatly exceed the transversal e+ displace-

1012

ment inside the pair ^formation region, ^{the e+}

velocity perturbation ^{8 v does not} depend ^{on the}

e+ emission angle 0. Expanding ^the integrand ⁱⁿ

terms of the velocity perturbation ^and integrating

expression (12), ^{where the first} integral ^{is omitted,}

over 0, ^{we obtain the} following expression ^{for the}

local PP probability ^{in the} perturbed ^{uniform trans-}

versal electric field

where

The modification of the integration path ^{near the} point ^T

^{0 is} connected with the change ^{of the}

order of integration ^{over 0 and T} [10]. Equation (14)

has allowed us to evaluate the PP probability ^{in the} slowly varying ^{electric field in reference} [9] ^and

allows us to evaluate ^{the PP} probability ^{in a uniform}

field in the presence of incoherent multiple ^scat- tering. ^{It is not difficult to conclude that} the terms, which do not contain the velocity perturbation, give

the PP probability ^{in the} uniform electric field (see

Refs. [17-21])

where

is the Airy function. To calculate the PP probability

in the presence of e± multiple scattering, the differ-

ence dW /dE+ - dWe/dE, ^{needs to be} averaged

over the stochastic velocity variations. From the well-known equalities 8v.L (T)

^0, ( dv)1) -

(6vf (T )) = ^{Us I T I} (see Eqs. (4), (5)) ^and dVl.L ==

d vl (0 ) ^{= 0 and} by using the standard method [12]

we obtain

Using ^these equations ^{one can} easily average the

probability (14) ^and represent ^{it as a} linear combi- nation of the upsilon ^function

and its derivatives. All derivatives, besides the first one, ^can be excluded from such a combination with the help ^of equation [17]

As a result, ^one obtains the local PP probability ⁱⁿ

the transversal uniform electric field in the _presence of e± multiple scattering

where

Equation (22) ^is applicable when the characteristic

angle ifig exceeds ^{the r.m.s. e±} multiple scattering

angle vS (le ) inside the pair ^formation region. ^Since

the multiple scattering process ^{in the pair formation}

region only includes the acts of e’ single scattering

at the angles 15 vmax== vs (1 e) ^the incoherent PP processes, corresponding ^{to the e±} single scattering

at the angles v > v max’ ^{need to be considered} separately.

3. Pair production processes ^on separate ^nuclei.

The logarithmic approximation ^{allows us to divide}

incoherent PP processes into ^two parts. ^{The first}

one, considered in the previous ^section, corresponds

to the e± multiple scattering ^{inside the} pair ^formation region. The second one can be connected with the e± scattering by separate ^nuclei. Really, ^{in the} logarithmic approximation ^the probability ^{of the} repeated ^ert scattering ^{at the} angles v> vmax

vs(le) ^{inside the formation} region ^{of a} pair ^is negligible. ^{As a} result, the PP processes, ^accom-

panied by ^{the isolated e±} scattering ^at angles

ifi > vmax , ^can be considered independently. ^Let vmax ifig (the ^{results, obtained below,} keep ^their

sense for i) max - ve too). ^{In this case under the} integration ^{over the} angles of the isolated scattering

the main contribution to the integral ^is given by ^the angles & > vmax ^{At the same time} multiple ^scat- tering ^{deflects the e± at the} angles - vs(le)

vmax ve and, therefore, weakly changes ^the

e± trajectories. ^{As a} result, multiple scattering ^can

lead only ^to small corrections. Therefore in the

logarithmic approximation the PP process, ^accom-

panied by ^the e± isolated scattering ^{at the} angles

v > v max inside the crystal, ^{can be} considered as a

PP process taking place ^{on a} free nucleus placed ⁱⁿ

the uniform field. Such a process ^was studied in references [23, 24] ^{where the} corresponding ^cross-

section was derived. The results of references [23, 24] ^{can be} easily reproduced ^on the basis of the

quasiclassical ^method [19, 20]. ^This procedure ^is greatly simplified by using ^the logarithmic approxi-

mation which allows us to consider the incoherent e± scattering by ^{a nucleus as a} perturbation ^{and to}

start from expression (14) ^{with T} = t2 - tl,

and with the substitution

The pair ^formation length (1) considerably ^exceeds

the interatomic distance d and especially ^{the atomic} screening ^radius aF. As ^a result, ^the positron velocity

perturbation against ^the background of its evolution in the uniform electric field which is caused by ^an

isolated nucleus at the moment t

0 can be _rep- resented in the form

where 0 is a solid angle ^{of e+} scattering, 0 ^{-L k,}

ifi « 1. After substitution of the equation ^of (26) ^into

the integrals ^of (24) ^{it is useful to} divide the infinite

integrals ^of (25) over t1 and t2 ^{into two} parts : ^over the positive ^{and the} negative ^semiaxes. Using ^the

new variables T ⁼ tl + t2 ^{and u}

X (t2 - t1)/a ^one

can as certain that the pair ^of integrals ^with

tl . t2 > 0 is equal ^{to zero. The second} pair ^of integrals with tl . t2 0 includes the terms pro-

portional ^to null, ^first and second powers of the

angle ^{D. The terms, not} containing ^the angle D, describe the PP in a pure uniform field. The latter is

already taken into account by ^the probability ^of (22).

Under the integration ^{over D} (see below) ^{the terms} proportional ^{to b} equal ^{zero. After the} integration

over T the remaining ^terms take the form

A subsequent simplification ^{of the} expression (27) ^is

realized by introducing ^the upsilon-function (20) ^and by eliminating ^its derivatives with the help ^of equation (21). ^{In order to evaluate} the contribution of the isolated e± scattering process ^to the local PP

probability expression (27) ^{needs to be} multiplied by

the local nuclear density n (p ) ^and by ^the relativistic Rutherford cross-section

and then be integrated ^{over the} scattering angles it max ^v ve ^{As a result, we obtain}

where

The upper limit of the integration ^{over U is deter-}

mined by ^the condition of applicability ^{of the} expansion (14). ^{Note that the} contribution of scat-

tering angles ifi >. ve ^is negligible ^{in the} logarithmic

approximation. After summation of the contribu-

tions of (22) ^and (29) ^{we obtain for the local PP}

1014

probability ^{at the distance p} far from the axis (the plane)

where

Note that the logarithmic approximation ^{has allowed}

us to combine the contributions of (22) ^and (29) quite naturally.

4. Pair production ^{in the field of} crystal ^axes.

The local PP probability (31) ^{allows us to calculate}

the PP probability ^{in the} crystal ^for perfect alignment

between the y-quantum ^{momentum and the} crystal

axis or plane. ^In simple crystals ^{with all} strings equivalent ^{it is} enough ^to average the local prob- ability (31) ^over the field distribution and over the nuclear density ^{of a} string, ^{i.e. to} carry ^out the

integration

where no is the average atomic density ^{in a} crystal.

The main contribution to the integral (33) gives ^the region ^{in the} vicinity ^{of a} string, so one can extend the integration ^to infinity. ^{Under the} logarithmic approximation ^{the ratio} ð f,1 ð miD ^{contained in ex-} pression (31) ^is defined with the accuracy of the factor compared ^with unity. ^This uncertainty ^allows

us to assume the angle ð miD to ^be independent ^from

the distance _p measured from the string. Choosing

the value of this angle ^{we are based on the fact that}

under the condition Zalcoh/d .c ^{1, where} lcoh

2 E+ ^E- /m 2ow, the PP process in crystals may be described by ^{the Bom} approximation [25]. ^{In accord-}

ance with references [11-13] ^{for the case of} perfect alignment ^{the PP} probability, calculated in the Born

approximation, ^is equal ^to the so-called « amor-

phous » probability. ^{In the} logarithmic approxi-

mation this probability ^{can be written} in the form

where

is the Bethe-Heitler probability, and aF is the atomic

screening ^{radius. Under the} conditions of applicabili- ty ^of equation (34) ^the parameter x ^{= x} (p ) ^consider- ably ^exceeds unity ^at any distance far from the axis.

It is quite ^{natural to} require ^{that the} probability (33)

with the local probability (31) ^{should be} equal ^{to the}

« amorphous ^» probability under such a condition.

Using ^the expansion Y’ (x ) =1/x ⁺ 2/x4 ^{+ ... and}

the equality ve

m/ e+ - 1 / y ^{which are valid in}

the limit of x > 1 one may get ^{convinced that the}

right hand side of equality (31) depends ^{on the}

transversal coordinates _p

(x, y ) only through ^the

factor a (p ) proportional ^{to the} local nuclear density n (p ). ^{Since d} f ^{n (p ) d2p = 1, under the condition}

of applicability ^{of the Born} approximation, ^the probability (33) ^becomes

Putting expression (33) equal ^{to the} right ^{hand side}

of equality (34) ^{we obtain the} equation determining ,&mi.. ^{Now we can} write the final expression ^{for the}

local PP probability:

where

0 (x’ ) ^{= 1 for x’ . 6, 8} (x’ )

^{0 for x 0, we recall}

that expressions (31) ^and (38) ^{are not} applicable ⁱⁿ

the narrow region (11). However, ^{the small value of} the ratio p minl2 ^{U2 allows us to use these} expressions

practically everywhere ⁱⁿ crystals ^of sufficiently light

elements. The averaged ^{field of the} (110) ^{axis of}

the Ge crystal ^{cooled to 100 K was} calculated on the basis of the Moliere atomic potential [6] ⁱⁿ neglecting

the contribution of the fields of neighbouring ^axes.

All calculated probabilities ^are normalized here to

the integral Bethe-Heitler probability WBH

0.331 cm - ¹ obtained by integrating ^the differential

probability (36) ^{over the} positron energy. In figure ¹

solid lines represent ^the differential PP probabilities

calculated on the basis of formulae (33) ^and (38) ^for

the different y-quantum energies. ^For comparison,

the ^« naive ^» differential probabilities ^{of the PP in}

the crystal dWam/ds+ ⁺ dWh/dE+ ^are represented by ^{dashed lines, where}

is the differential probability ^{of the coherent PP in}

the field of axes. Finally, ^{the dotted line in} figure ¹

represents ^the «amorphous» probability (34).

Though ^the logarithmic approximation accuracy is

not too high, putting probability (33) into accordance with probability (34) by choosing ^{the value of} 1L’tmi. ^one may believe in a physical origin ^{of the}

deviation of the solid lines from the corresponding

dashed lines in figure ^1.

The most reliable prediction that formula (38)

allows us to make concerns the high-energy depen-

dence of the total incoherent PP probability

W - W,,oh, ^where probabilities ^{W and} Wcoh ^{may be}

obtained by integrating the differential probabilities (33) ^and (40), correspondingly. ^For sufficiently high energies ^{one has x 1 and} Y’ (x ) = r(I/3)/2. ^{32/3 = 0.644,} Y, (x) == r (2/3)/2 - ^{31/3 =}

0.469. As a result, ^{the local} differential incoherent PP probability

Fig. ^1.

Differential PP probabilities in the field of Ge

(11°) ^{axis at} 100 K for various y-quantum energies.

dW/dE+ - ^{solid lines,} dWam/de:t ⁺ dWh/d --, ^-dashed

lines and dWam/dE, - dotted line. _s± is the energy of ^one of the produced particles.

is proportional ^to the combination

The integral ^local incoherent probability ^{is, obvi-} ously, proportional ^{to le) - 213 and starts to fall when}

K

e&oj /M3 _ 1. ^{Since the area of the} region (11)

decreases with the y-quantum energy such ^an energy

dependence ^{of the local} probability is, absolutely,

correct ^{for the} total incoherent PP probability

W - Wcoh ^{in the} crystal ^with any value of Z. Recall that under high energies the total coherent PP

probability Wcoh ^is proportional ^{to to - 15.} Besides,

the local PP probability starts to fall only ^when

K =11,6. The sufficiently rapid decrease of the incoherent PP probability ^{with the} y-quantum energy is illustrated in figure ^{2 where} the energy dependence

of the coherent PP probability ^{is also} presented.

Fig. ^2.

The coherent

W coh and incoherent

W - Wcoh contributions to the total PP probability ^{W in a}

field of (11°) axis of Ge being ^{cooled to 100 K is shown}

as a function of incident -y-quantum energy. Wam ⁺ Wh ^is

the « naive ^» PP probability.

Both the differential and the integral probabilities

calculated on the basis of formulae (33) ^and (38)

differ from the « naive » probabilities by the relative value of about 5 percent. Besides, they ^{start to} grow for y-quantum energies smaller than the ^« naive ^»

probabilities ^{do. In our} opinion ^{one has a reason to}

believe that this numerical result is not an artifact of

the logarithmic approximation.

1016

5. Conclusion.

When electrons, positrons ^and y-quanta ^{with ener-}

gies E:t’ (J) è:: ^{10 GeV move} along crystal ^{axes or} planes ^the strong crystal ^{fields make it} possible ^to investigate effects of quantum electrodynamics ^{in a} strong ^uniform electric field in experiments using secondary ^{e± and} y-beams produced ⁱⁿ large proton accelerators. However, ^the crystal ^field is, obviously,

not a purely uniform electric field. The expressions

obtained above side by side with the expressions

obtained in references [9, 10] ^{can be used to describe}

quantum electrodynamic ^effects taking ^{into account}

the main special properties ^of crystal ^fields.

Note that effects of quantum electrodynamics ^{in a} strong uniform electric field manifest themselves in

atmospheres ^of pulsars also in the presence ^of incoherent Coulomb e± scattering.

The pair production ^and yquantum ^emission

processes ^near the perfect alignment are accom-

panied by polarization ^effects [1, 2, 26] ^{which may}

be widely ^{used to} produce quite ^intensive highly polarized ^{e± and} yquantum secondary ^{beams in} large proton accelerators such as Super ^Proton Synchrotron (SPS), ^{Tevatron and, in} particular, Accelerating-Storage Complex (UNK, Serpukhov)

and Superconducting Super ^Collider (SSC). ^The developed approach ^to the incoherent pair produc-

tion and y-quantum emission processes may substan-

tially modify ^the predicted values of e± and _y- quantum polarizations ^{in the} energy regions ^where

the coherent contributions to the pair production

and y-quantum ^emission probabilities ^{do not exceed}

the Bethe-Heitler probabilities.

The author is grateful ^{to Professor V. G.}

Baryshevskii for his interest in this work.

,

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