Semiclassical Theory of Crystal-Assisted Pair Production:
Beyond the Uniform Field Approximation
H.Nitta, Y.Nagata, and S.Onuki
Department of Physics, Tokyo Gakugei University
M.Kh.Khokonov
Department of Physics, Kabardino-Balkarian State University
Within the semiclassical theory of QED, a simple expression of pair production
in strong field is obtained that approaches to the constant field approximation
at the strong field limit while it gradually behaves like the Bethe-Heitler typewhen the field becomes weaker. By using the impact approximation for thecalculation of trajectory, a simple expression is obtained. Though simple, itagrees well with the experimental result of crystal-assisted pair production.

1Introduction

It is widely known that electron-positron pair production is well described by the BetheHeitler formula as far as the process can be regarded as the incident photon collides
with an isolated atom. However, when photons enter a strong electromagnetic field, theperturbative approach such as the Bethe-Heitlers becomes inappropriate for describing
the pair production process. For such strong field QED effect, Baier and Katokov[1]proposed a semiclassical expression for radiationdNαc=(Jc + Js ),
dη
πλ0 γ

JcJs
Z ∞
dτγ2π2=
1 + (δβ⊥ ) sin∆
− ,2τ2
Z ∞ 200γ
dτγ−γ(δβ⊥ )2 sin∆,=02γγ
2
τ
0(1)

(2)
(3)

where ω ∗ = (γ/γ 0 )ω, η = ~ω/E, δβ⊥ = β⊥ (t+ ) − β⊥ (t− ), t± = t0 ± τ /2, E 0 = E − ~ω,γ(γ 0 ) = E/(m0 c2 )(E 0 /(m0 c2 )), E being the initial energy of the particle and m0 its rest
mass. The phase determined by the trajectory of a radiating particle is given byZωτωω t+ 2 0 02∆(τ ) =−(δρ) +β (τ )dτ ,(4)2γ 2 2c2 τ2 t− ⊥where δρ = ρ(t+ ) − ρ(t− ), ρ(t) being the transverse coordinate. Though there are somediscussion on the derivation of the Baier-Katokov formula [2, 3], this formula explains
very well the radiation as well as pair production in strong fields.

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2“Th-trajectory” and pair production

The pair production probability is obtained by using the crossing symmetry for Eq.(5)[3].

ǫGGG

ǫ

Figure 1: Crossing symmetry. The left-hand figure represents the channeling radiation
while the right-hand one represents the pair production.
By changing the variables as→ −E+ (produced positron) E (initial energy)
0
E (final energy)→ E−(produced electron)ω (emitted photon) → −ω
(absorbed photon)and multiplying the ratio of the density of final states,
E+2 dE+,(~ω)2 d(~ω)

we obtain the pair production probability for the th-trajectory approximation as well as
the synchrotron approximation, though one may feel that the idea of particle trajectoryis rather “spooky”[3]. We obtain,
αcm0 c2dN+=J,(5)dη+πλ0~ω
where dN+ /dη+ represents the number of produced positrons, η+ = E+ /~ω, E+ being
the energy of the produced positrons, α the fine-structure constant, and λ0 the Comptonwavelength.
The factor J for the CFA is given byZ ∞11∗Jcf = √1 − η+ +K 2 (ξ ) −K 1 (λ)dλ ,(6)33
1 − η+3ξ∗
where ξ ∗ = 2/[3η+ (1 − η+ )χ] and χ = ~ωλ0 F (t0 )/(m0 c2 )2 and F (t0 ) is the force by thefield acting on the positron at time t0 . Eq.(6) is obtained by assuming that the trajectoryof produced positrons are circular.
The CFA is applied to evaluate the pair production process when high-energy photons
enters a crystal along the major crystal axis. It is reported that when photons are2

directed exactly parallel to the crystal axis, CFA agrees very well with the experimental
result [5]. However, naturally, CFA does not explain the angular dependence of thepair production rate. For the purpose to calculate the angular dependence, the authorsof Ref.[5] calculated the Baier-Katkov formula with some “numerical experiments” [6]and a good agreement of their numerical approach with the experimental result has
been demonstrated. However, the process of the “numerical experiments” has not been
described in their paper. Therefore, nobody can reproduce the calculations.
Recently, two of the present authors obtained a radiation probability [4] by using amodel trajectory called the “th-trajectory”
 tβ ⊥ (t) = b0 + b tanh,(7)Twhereb0 =β − β ⊥1β ⊥1 + β ⊥2, b = ⊥2,2( 2β ⊥1 ≡ β ⊥ (t → −∞)β ⊥2 ≡ β ⊥ (t → +∞).

The th-trajectory approaches to the free motion (i.e. straight paths) at t → ±∞ while
the trajectory is substantially bent at |t| . T depending on the strength of the field.
Since Eq.(7) is integrated analytically, we obtain an analytic expression for J: Z ∞hidz 2 1 − η+η+2˜ − µ tanh z) + π ,Jth =ν+tanh z − 1 sin ξ(z(8)zη+1 − η+20
where ν = γ+ b and ξ˜ = ν(1 + ν 2 )/[η+ (1 − η+ )χ].

3Numerical results

A typical ν dependence of Eq.(8) is shown in Fig. 2. When ν becomes larger, the resultof Jth approaches to Jcf . From Fig. 2 it is clear that the spectra become CFA-like as theangle decreases while Bethe-Heitler-like as the angle increases.
Based on Eq.(8), we have calculated the crystal-assisted pair production (CAPP)
probability as a function of the photon incident angle in comparison with the experimental data by Belckacem et al.[5]. For simplicity, first we calculate the scattering angle
by the impact approximation where the momentum change during the collision is proportional to the force multiplied by the “interaction time”. The interaction time has
been estimated by aρ/v⊥ , where ρ is the impact parameter of the radiation point (forCAPP, we should have called it the “pair production point” ) and a is the parameter(a ∼ 1). In Fig. 3 we have shown the results of the impact approximation. Taking
account of the simplicity of the impact approximation, the agreement is satisfactory. Jthwith the impact approximation will be useful for planning experiments.
In Fig. 4, a more involved calculation has been made by using thermally-averaged
one-string Molière potential [7] at T = 100K. Though in this case we have no free parameter, the agreement is well. The peaks at higher photon energies look like somewhat
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PAIR CREATION RATE J

0.16

0.14

← 0.5

0.12

← 0.7

← 0.4

← 0.3

←10.1

← 0.2
0.08←50.06

← 0.1

0.04

0.02

00

0.1

0.2

0.3

0.4
0.5
0.6
POSITRON ENERGY

0.7

0.8

0.91

Figure 2: The ν-dependence of pair production probability calculated by using Eq.(8) asa function of the energy of produced positrons E+ /~ω. The dotted line represents theconstant field approximation (CFA) of Eq.(6). (χ = 1)

Figure 3: Comparison of the impact approximation results with the experimental databy Belkacem et al. [5]. The symbols mean the experimental results corresponding theenergy of photons in the range of: ◦: 150-120GeV, :120-90GeV, 5:90-60GeV, 4:5040GeV,√ ♦:40-20GeV. The dotted lines, solid lines, and broken lines correspond to a = 2,a = 2 2, and a = 4, respectively.

4

Figure 4: Comparison with experimental data by the rigorous scattering calculation withthe use of the Molière one-string potential. No fitting parameter is included.
steeper than the experimental data, which may be due to neglecting of the many-stringeffect.
More detailed discussion will be published elsewhere.

References[1] V. N. Baier, and V. M. Katkov, Sov. Phys. JETP,26,854 (1968).
[2] J. Lindhard, Phys. Rev. A43, 6032 (1991).
[3] J. C. Kimball and N. Cue, Radiat. Eff. Defect in Solids 122-123, 481 (1991).
[4] M. Kh. Khokonov and H. Nitta, Phys. Rev. Lett. 89, 094801 (2002).
[5] A. Belkacem et al., Phys. Rev. Lett. 58, 1196 (1987).
[6] A. Belkacem, N. Cue, and J. C. Kimball, Phys.Lett. 111A, 86 (1985).
[7] Y.H.Ohtsuki, Charged Beam Interaction with Solids (Taylor and Frances, London,1983).

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