On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target
The ionization loss of an ultrarelativistic e⁺e⁻ pair in a system of two separated targets { thick upstream, in which the pair is created, and thin downstream { is considered. The main attention is drawn to investigation of the effect of enhancement of the pair loss in thin target comparing to the s...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1360912018-06-16T03:03:10Z On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target Trofymenko, S.V. Электродинамика The ionization loss of an ultrarelativistic e⁺e⁻ pair in a system of two separated targets { thick upstream, in which the pair is created, and thin downstream { is considered. The main attention is drawn to investigation of the effect of enhancement of the pair loss in thin target comparing to the sum of independent electron and positron losses (the one opposite to the Chudakov effect of pair ionization loss suppression). The problem is studied without the use of approximation of parallel electron and positron velocities which is usually applied for theoretical consideration of pair ionization loss. The dependence of the magnitude of the effect on the pair energy and its divergence angle is studied. The possibility of existence of such effect in the asymptotic case of large separations between the targets is shown. A simplified analytical expression describing the pair ionization loss dependence on distance between the targets under the conditions of manifestation of the discussed effect is obtained. Розглянуто іонізаційні втрати ультрарелятивістської e⁺e⁻ - пари в системі з двох мішеней - товстої, в якій пара народжується, і тонкої, - що розташовані одна за одною на певному проміжку. Основну увагу приділено дослідженню ефекта збільшення втрат пари в тонкій мішені порівняно з сумою незалежних втрат електрона і позитрона (який є протилежним до ефекту Чудакова зменшення іонізаційних втрат пари). Задачу розв'язано без використання наближення паралельних швидкостей електрона і позитрона, яке зазвичай застосовується при теоретичному розгляді іонізаційних втрат пар. Вивчено залежність величини даного ефекту від енергії і кута розльоту пари. Показана можливість існування такого ефекту в асимптотичному випадку при великих відстанях між мішенями. Отримано спрощений аналітичний вираз, що описує залежність іонізаційних втрат пари від відстані між мішенями в умовах прояву ефекту, що розглядається. Рассмотрены ионизационные потери ультрарелятивистской e⁺e⁻ - пары в системе из двух мишеней - толстой, в которой пара рождается, и тонкой - расположенных одна за другой на некотором промежутке. Основное внимание уделено исследованию эффекта увеличения потерь пары в тонкой мишени по сравнению с суммой независимых потерь электрона и позитрона (который является обратным эффекту Чудакова подавления ионизационных потерь пары). Задача решена без использования приближения параллельных скоростей электрона и позитрона, которое обычно применяется при теоретическом рассмотрении ионизационных потерь пар. Изучена зависимость величины данного эффекта от энергии и угла разлета пары. Показана возможность существования такого эффекта в асимптотическом случае при больших расстояниях между мишенями. Получено упрощенное аналитическое выражение, описывающее зависимость ионизационных потерь от расстояния между мишенями в условиях проявления рассматриваемого эффекта. 2017 Article On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target / S.V. Trofymenko // Вопросы атомной науки и техники. — 2017. — № 3. — С. 60-66. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 41.20.-q, 41.60.-m http://dspace.nbuv.gov.ua/handle/123456789/136091 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Электродинамика Электродинамика |
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Электродинамика Электродинамика Trofymenko, S.V. On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target Вопросы атомной науки и техники |
| description |
The ionization loss of an ultrarelativistic e⁺e⁻ pair in a system of two separated targets { thick upstream, in which the pair is created, and thin downstream { is considered. The main attention is drawn to investigation of the effect of enhancement of the pair loss in thin target comparing to the sum of independent electron and positron losses (the one opposite to the Chudakov effect of pair ionization loss suppression). The problem is studied without the use of approximation of parallel electron and positron velocities which is usually applied for theoretical consideration of pair ionization loss. The dependence of the magnitude of the effect on the pair energy and its divergence angle is studied. The possibility of existence of such effect in the asymptotic case of large separations between the targets is shown. A simplified analytical expression describing the pair ionization loss dependence on distance between the targets under the conditions of manifestation of the discussed effect is obtained. |
| format |
Article |
| author |
Trofymenko, S.V. |
| author_facet |
Trofymenko, S.V. |
| author_sort |
Trofymenko, S.V. |
| title |
On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target |
| title_short |
On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target |
| title_full |
On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target |
| title_fullStr |
On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target |
| title_full_unstemmed |
On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target |
| title_sort |
on anti-chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2017 |
| topic_facet |
Электродинамика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/136091 |
| citation_txt |
On anti-Chudakov effect in ultrarelativistic electron-positron pair ionization loss in thin target / S.V. Trofymenko // Вопросы атомной науки и техники. — 2017. — № 3. — С. 60-66. — Бібліогр.: 13 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-09T21:23:34Z |
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2025-07-09T21:23:34Z |
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1837206041773735936 |
| fulltext |
ELECTRODYNAMICS
ON ANTI-CHUDAKOV EFFECT IN ULTRARELATIVISTIC
ELECTRON-POSITRON PAIR IONIZATION LOSS IN THIN
TARGET
S.V.Trofymenko∗
1Akhiezer Institute for Theoretical Physics of NSC KIPT, 61108, Kharkiv, Ukraine;
2Karazin Kharkiv National University, 61022, Kharkiv, Ukraine
(Received April 18, 2017)
The ionization loss of an ultrarelativistic e+e− pair in a system of two separated targets – thick upstream, in which
the pair is created, and thin downstream – is considered. The main attention is drawn to investigation of the effect
of enhancement of the pair loss in thin target comparing to the sum of independent electron and positron losses (the
one opposite to the Chudakov effect of pair ionization loss suppression). The problem is studied without the use of
approximation of parallel electron and positron velocities which is usually applied for theoretical consideration of pair
ionization loss. The dependence of the magnitude of the effect on the pair energy and its divergence angle is studied.
The possibility of existence of such effect in the asymptotic case of large separations between the targets is shown. A
simplified analytical expression describing the pair ionization loss dependence on distance between the targets under
the conditions of manifestation of the discussed effect is obtained.
PACS: 41.20.-q, 41.60.-m
1. INTRODUCTION
When a charged particle moves through substance
it experiences inelastic collisions with atomic elec-
trons, which result in the loss of its kinetic energy.
The average amount of the particle energy decrease
per unit path due to such collisions is known as ion-
ization loss of the particle. Let us note that it does
not include the radiative energy loss due to parti-
cle multiple scattering on atomic nuclei but just the
loss on ionization of the atoms and excitation of their
electron subsystem. Ionization loss of a single particle
moving in homogeneous and boundless substance is
defined by Bethe-Bloch formula, taking into account
its various forms for different types of particles and
their energies (see [1] and references threin).
In [2] it was shown for the first time that ioniza-
tion loss of a system of particles moving close to each
other differs from the plain sum of independent par-
ticles losses. Here the ionization loss of a high-energy
electron-positron pair was calculated in the vicinity
of its creation point in substance. It was shown that
in this region the pair loss is significantly suppressed
comparing to the sum
∑±
of independent electron
and positron losses, which became known as Chu-
dakov effect. With the increase of the distance from
the creation point the ionization loss monotonically
grows reaching finally the limit corresponding to this
sum. The reason of the Chudakov effect is the mu-
tual screening (destructive interference) of the elec-
tron’s and the positron’s Coulomb fields which dis-
appears when the particles fly apart from each other
on large distance. The typical dependence of the pair
loss on the distance z from the creation point is pre-
sented on Fig.1. The dashed line here corresponds
to the value of
∑±
, ωp is the plasma frequency of
the substance, γ is a characteristic value of electron
or positron Lorenz-factor. Let us note that in [2],
as well as in further papers reporting theoretical and
experimental investigations of e−e+ pairs ionization
loss (for detailed list of references see, e.g. [3]), this
quantity was studied in the same substance in which
the pairs were created. The substance was considered
as homogeneous and boundless.
Fig.1. Typical dependence of e−e+ pair ionization
loss in the same substance in which the pair is
created on distance from the creation point
In [4] and [5] the influence of boundary effects
upon e+e− pair ionization loss was investigated. It
∗Corresponding author E-mail address: trofymenko@kipt.kharkov.ua
60 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2017, N3(109).
Series: Nuclear Physics Investigations (68), p.60-66.
was shown that such effects are significant if an ul-
trarelativistic pair loses energy moving through a suf-
ficiently thin layer of substance (thin plate). It was
assumed that the pair is created in a target situated
in front of the plate on some distance from it. The
boundary effects in this case are caused by the trans-
formation of the electromagnetic field around the pair
after its emission from the target in which it is cre-
ated. Such transformation leads to generation of the
so-called transition radiation which affects upon the
pair loss in thin plate. As shown, the pair ionization
loss in this case increases within much larger distance
from its creation point than the one ∼ γ/ωp
1, on
which the Chudakov effect takes place in a boundless
medium. Moreover the possibility of existence of the
effect opposite to the Cudakov one (anti-Chudakov
effect) was indicated in this case. It is the exceeding
by the value of the pair ionization loss of the sum
of independent electron and positron losses on some
distance from the pair creation point.
Let us note that a single experiment devoted to
the study of e+e− pairs ionization loss, in which the
pair creation and ionization loss processes occurred
in different targets, was reported in [6]. However, the
investigation of boundary effects was not the aim here
and relatively thick targets were used.
In the papers [4] and [5] a special simplifying ap-
proximation was used for calculation of the pair ion-
ization loss. Its essence is the following. Let the
transverse (with respect to the average direction of
the pair motion) distance between the electron and
the positron equal s on some distance z ∼ γs from
the pair creation point. In the considered approxi-
mation the pair ionization loss for such z is assumed
to equal the ionization loss of the electron and the
positron which constantly move with parallel veloc-
ities separated by the distance s in the transverse
direction. At high energies of the particles (γ ≫ 1)
such approximation is validated by the small value of
the pair divergence angle and has always been used
for theoretical considerations of pairs ionization loss
(see Ref. in [3]). We will further call it the parallel
approximation.
In [7] and [8] it was shown that under some condi-
tions the parallel approximation is not strictly valid
for consideration of pairs ionization loss in thin tar-
gets. More precise study without the use of such
approximation, presented in these papers, revealed
some new features of such loss concerning, particu-
larly, its asymptotic (for z → ∞) behaviour at ultra-
high energies of the pair.
The present paper is devoted to theoretical study
of the anti-Chudakov effect in ionization loss of an
ultrarelativistic e+e− pair in thin target without the
use of parallel approximation. Such generalized ap-
proach (comparing to the one applied in [5], where
the discussed effect was predicted with the use of such
approximation) allows to investigate the dependence
of the value of the effect on various parameters as-
sociated with the pair, such as its divergence angle,
the position of its creation point inside the target etc.
Such results may be valuable in connection with the
problem of experimental study of such effect.
2. ASYMPTOTIC VALUE OF THE
EFFECT
Let us consider ionization loss of an ultrarelativis-
tic e+e− pair in the plate (thin target) situated in vac-
uum on distance z1 from the layer of substance (thick
target) in which the pair is created, for instance, by a
high-energy photon (Fig.2). In the paper [8] the ex-
pression defining the value of such loss per unit path
was obtained. Let ηp and ωp be the plasma frequen-
cies of the thin and the thick targets respectively. Let
us require the thickness a of the thin target to satisfy
the condition a ≼ η2p/I, where I is the mean ion-
ization potential of its atoms. The thickness of the
other target is supposed to significantly exceed the
value ω2
p/I. By z0 we denote the distance between
the downstream surface of the thick target and the
pair creation point inside it. We will study the sim-
plest case corresponding to equal energy distribution
between the particles of the pair. By γ we will de-
note the Lorenz-factor of a single particle. In this
case the discussed expression can be presented in the
next form:
dE
dz
= 2η2pe
2{Ec + Etr + Eint}. (1)
Here the term proportional to
Ec = lnκ0 −
1
2
(2)
describes the ionization produced in the plate by
the proper Coulomb fields of the electron and the
positron. The appearance of the value κ0 in (2) is as-
sociated with the fact that we consider the so-called
Fig.2. Traversal of the thin target by the pair
created in the thick target
restricted ionization loss. It is caused by interactions
of charged particles (e+ and e− in our case) with the
atomic electrons which are accompanied by the mo-
mentum transfer less than q0 = κ0I/γ. The restricted
loss is often more preferable for experimental mea-
surement than the value of the total ionization loss.
1It comes from the characteristic value 1/γ of the pair divergence angle and transversal (with respect to the particle velocity
direction) size 1/ωp of a charged particle field in a polarizable medium. Here and further we take to speed of light to equal
unit.
61
The term in (1) containing
Etr = ln(ωpγ/I)− 1 (3)
describes the contribution to the ionization of the
plate from the transition radiation generated during
the pair emission from the target in which it is cre-
ated. Here we assume that the condition
q0 ≫ ωp ≫ I/γ (4)
is fulfilled.
The term associated with Eint represents the in-
fluence of interference of all the fields taking place in
the considered process (proper fields of the particles
and transition radiation fields generated by them)
with each other upon the pair ionization loss in the
plate. It has the following form:
Eint = G(ψ, γ) + F (z1, ψ, γ), (5)
where ψ is half of the pair divergence angle (see Fig.2)
and
G = −W 4
p
κ0∫
0
dκκ
2π
(κ2 − α2)
2π∫
0
dφ
cos[4καZ0 cosφ]
K+K−Ω−Ω+
(6)
and
F = − 1
2π
κ0∫
0
dκ
2π∫
0
dφ(f1 + f2 + f3) (7)
with
f1 = κ(κ2 − α2) cos[4κα(Z1 + Z0) cosφ]/(K+K−),
f2 = 2W 2
p κ(K+ − 1) cos[Z1(1 + κ2+
+2κα cosφ)]/(K2
+Ω+),
f3 = −2W 2
p κ(κ
2 − α2) cos[2κα(Z1 + 2Z0) cosφ+
+Z1(1 + κ2)]/(K+K−Ω−)
and K± = κ2 + α2 ± 2κα cosφ+ 1, Ω± = K± +W 2
p ,
Wp = ωpγ/I, α = ψγ, Z1,0 = z1,0I/(2γ
2).
The first two terms in (1) equal the asymptotic
(z1 → ∞) value of the sum of independent losses of
the electron and the positron in the thin target in the
considered process, which we will denote as S±. Each
particle in this case is assumed to move from z = −∞
to z = +∞ normally traversing the targets. The
quantity S± differs from the corresponding quantity
Σ± in a boundless medium. On the one hand, such
difference is associated with the contribution of tran-
sition radiation generated during the particle traver-
sal of the thick target. On the other hand – with the
fact that in the considered thin target the particle
ionization loss occurs without the so-called density
effect2, which is significant in boundless medium (for
details see [11]).
The quantity Eint, which describes the difference
of the pair ionization loss value (1) from S± consists
of two terms. The term F depends on the distance z1
between the plates and tends to zero at large values
of this quantity. It happens due to quick oscillations
of the cosines in the integrand in (7) at large z1. In
[4] and [8] it was shown that the pair ionization loss
should cease to depend on z1 and become constant
at z1 ≫ LI ∼ 2γ2/I. The quantity LI is of the order
of so-called formation length of transition radiation
waves with frequencies close to the value of I. It is
these frequencies which make the main contribution
to ionization loss.
The term G in (5) does not depend on z1 and re-
mains even in the case of rather large separation be-
tween the targets. It defines the difference between
the asymptotic values of the pair ionization loss and
the sum S± of independent e+ and e− losses in the
considered process. The existence of such difference
is one of the new characteristic features of ionization
loss in the discussed situation which differs it from
the case of ”classical” Chudakov effect in a boundless
medium where such difference is absent. The appear-
ance of the term G in the expression for pair ioniza-
tion loss is associated with going beyond the parallel
approximation in our consideration. In [8] such term
was investigated in a simplified case corresponding to
pair creation in the immediate vicinity of the down-
stream surface of the thick target (z0 → 0). It was
shown that in this case the value of G is negative and
slowly grows in absolute value with the increase of the
energy of the pair. At the energies of about 100 GeV
the relative value of such asymptotic suppression of
ionization loss may exceed 50%. The approximation
z0 ≈ 0 works well at rather high energies of the pair
at which the condition z0/LI ≪ 1 is fulfilled by wide
margin. Let us note that the asymptotic difference
between dE/dz of the pair and S± in the considerd
case is caused by the mutual interference of the tran-
sition radiation fields generated by the electron and
the positron upon their emission from the thick tar-
get. Generally speaking, such interference does not
vanish even on rather large distance from this target.
Let us consider more general case of z0 ̸= 0 and
investigate the dependence of the value of G on z0.
It is natural to expect such dependence to be most
pronounced at rather low (however, ultrarelativistic)
energies of the pair when the condition z0/LI ≪ 1
has the chance to be violated.
Fig.3 shows the ratio G/S± as a function of z0
for the energy of the pair equal 100 MeV. It is ob-
tained on the basis of numerical calculation of the
integrals in the expression (6). The calculation is
presented for three values of the pair divergence an-
gle ϑ = 2ψ. Here and further we use the values
ωp ≈ 30 eV and I = 175 eV (as for silicon) of the
corresponding quantities. The figure demonstrates
that in the case z0 → 0 the value of G is negative
and the asymptotic suppression of dE/dz comparing
2It was predicted by Fermi [9] and thoroughly studied by Stermheimer [10].
62
to S± takes place. It grows with the decrease of ϑ and
for the most probable value ϑ0 = 2/γ of this angle
becomes rather significant.
Fig.3. Dependence of the relative value of asymp-
totic difference between dE/dz and S± on z0 for
different pair divergence angles ϑ
However, with the slight increase of z0 (up to sev-
eral microns, depending on ϑ) the sign of G changes
and it becomes positive. This means asymptotic ex-
ceeding by the magnitude of the pair loss of the value
S±. The maximum positive value of G/S±, on the
contrary (comparing to the maximum negative one),
is larger for the values of the pair divergence angle
exceeding the most probable one. For the considered
value of the pair energy it can reach the value of about
4%.
With the increase of the energy of the pair for
the fixed value of α = ψγ the scale of the distance
z0 on which the quantity G noticeably changes grows
proportional to the square of the energy. Indeed, the
analysis of the integrand in (6) shows that for the
fixed values of α and Z0 the single quantity here de-
pending on the energy isWp = ωpγ/I, which is a part
of expressions for Ω± as well. However, taking into
account the condition (4), we can approximately set
Ω± ≈ W 2
p , which leads to disappearance of Wp from
(6) at all. Let us note that such simplification is valid
only for z0 ̸= 0 when it does not affect upon the con-
vergence of the integrals in (6). In this case the values
of G for different γ become equal for a fixed value
of Z0 = z0I/(2γ
2). Therefore, with the increase of
the pair energy the asymptotic anti-Chudakov effect
takes place for much larger values of z0 comparing to
the considered case of E = 100 MeV.
Fig.4. Dependence of the maximum positive relative
magnitude of asymptotic difference between dE/dz
and S± on the pair divergence angle for two values
of its energy
Fig.4 shows the dependence of the maximum posi-
tive relative magnitude (G/S±)max of the asymptotic
anti-Chudakov effect on the angle of the pair diver-
gence for two values of its energy E. Here we see that
the considered magnitude increases with the decrease
the energy. It happens due to logarithmic decrease
of the value S± with the decrease of E, which fol-
lows from (2) and (3). Therefore low (however, still
ultrarelativistic) energies of e+e− pairs seem more
preferable for the experimental search of manifesta-
tion of the considered effect. Further for certainty we
we will concentrate on the energy E = 100 MeV, for
which the formation length is LI ∼ 20 µm. The point
is that the measurements at lower energies might be
technically very difficult due to small size of LI in
this case.
To conclude this section it is necessary to note
that the consideration presented here is restricted by
the values of distances between the targets for which
the thin target can be considered as having unlim-
ited transversal size. It is the case if the conditions
z1 ≪ dγ and d ≫ γ/I (where d is the transversal
linear size of the target) are fulfilled. For instance, if
E = 100 MeV and d = 1 cm, the presented consider-
ation works for z1 ≪ 100 cm (the second condition is
fulfilled by wide margin).
3. ESTIMATION OF THE MAXIMUM
VALUE OF THE ANTI-CHUDAKOV
EFFECT
In the previous section the pair ionization loss
was considered for asymptotically large separations
between the targets (z1 ≫ 2γ2/I). It was shown
that under some conditions the value of such loss may
slightly exceed the sum S± of independent losses of
e+ and e− traversing the same system of two targets
and the value of such effect was estimated.
The maximum value of the considered effect, how-
ever, can be expected to take place for smaller values
of z1. Therefore, in order to define the optimal pa-
rameters for its observation it is necessary to consider
the pair ionization loss for arbitrary separations be-
tween the targets. For very small values of such sep-
aration the pair ionization loss is significantly sup-
pressed and grows with the increase of z1. It is the
Chudakov effect which takes place in this case. It is
natural to expect the maximum value of the discussed
anti-effect for z1 ∼ LI at which dE/dz approaches its
asymptotic value. Let us derive a simplified analyti-
cal expression for the term F defining the dependence
of the pair ionization loss on z1 in this case.
Let us begin with the term proportional to f2 in
the expression (7) for F denoting it as F2. For the
considered values of z1 it is possible to use the sta-
tionary phase method [12] for calculation of one of the
integrals (let it be the one with respect to φ) here.
The stationary phase points φ0 here are defined from
the equation
∂
∂φ
(1 + κ2 + 2κα cosφ) = 0,
63
which has three solutions within the integration in-
terval: φ0 = 0, π, 2π. Since the points φ0 = 0, 2π lie
on the edges of this interval their contributions are
half as large as the one from a point situated inside
it. However, the values of the integrand f2 in these
points are equal and we can substitute their contri-
bution by a complete (not half of it) contribution,
e.g. from the point φ0 = 0. After some transforma-
tions the contributions from the points φ0 = 0, π can
be united into one integral which has the following
form:
F2 ≈ 2W 2
p
√
π
αZ
Re
+∞∫
−∞
dκQ(κ)eiZq(κ)−iπ/4, (8)
where
Q(κ) =
√
κ(κ− α)2
[(κ− α)2 + 1]2[(κ− α)2 + 1 +W 2
p ]
and q(κ) = 2ακ− κ2 − 1. Here
√
κ is a single-valued
branch of the analytic function w =
√
κ with the ar-
gument value arg(w) = arg(κ)/2. It equals |
√
κ| for
κ > 0 and i|
√
κ| for κ < 0.
In the considered case Z ∼ 1 and it is κ ∼ 1 that
make the main contribution to the integral in (8).
Thus, using the condition Wp ≫ 1, which follows
from (4), and taking into account that α ∼ 1, we can
make the following simplification: (κ−α)2+1+W 2
p ≈
W 2
p . Let us also make the substitution κ− α = x in
(8). In the result we obtain the expression for F2 in
the following form:
F2 ≈ 2
√
π
αZ
Re
{
eiχ(Z,α)
+∞∫
−∞
dx
x2
√
x+ α
(x2 + 1)2
e−iZx2
}
,
(9)
where χ = Z(α2 − 1)− π/4.
As shown in the previous section for the case
Z ≫ 1, the highest magnitudes of the considered anti-
effect should be expected for the values of ϑ exceeding
the most probable one (which corresponds to α > 1).
Therefore, taking into account that it is x ≤ 1 that
make the main contribution to the integral in (9), we
will neglect here the value of x comparing to α and
write:
√
x+ α ≈
√
α. After such simplification the
obtained integral can be analytically calculated (see
[13]) and we finally obtain:
F2 ≈ −
√
π
Z
Re
{
eiχ(Z,α)
[√
2πZ(1 + i)− (10)
−πeiZ(1 + 2iZ)[1− Φ(
√
iZ)]
]}
.
Here
Φ(
√
iZ) = eiπ/4
2√
π
√
Z∫
0
e−it2dt
is the error function. The analogous procedure per-
formed for the terms in (7) associated with f1 and
f3 shows that for the considered values of Z they are
small compared to the term F2. Therefore we will
neglect them and write F ≈ F2.
It is natural to expect that the manifestation of
the anti-Chudakov effect will be mostly pronounced
at z1 ∼ 2γ2/I for the values of z0 for which it has
the largest value at z1 → ∞ (see previous section).
Therefore it is such values of z0 (which depend on E
and ϑ) which we will consider further. For instance,
in the case 100 MeV pair with the divergence angle
ϑ = 3ϑ0 (see Fig.3) it is z0 ≈ 1 µm.
Fig.5 shows the example of the dependence of pair
ionization loss in thin target on distance z1 between
the targets in the region z1 ≤ 2γ2/I. Here we see
that there exists a maximum (at the point z1 = z1m)
in z1-dependence of dE/dz in the considered case. It
is another new feature which differs the pair loss in a
thin plate from its loss in a boundless medium, where
such dependence is monotone (see Fig.1)3. In the
considered case the magnitude of anti-Chudakov ef-
fect at the position of dE/dz maximum is more than
two times larger than its asymptotic magnitude at
z1 ≫ 2γ2/I.
Fig.5. Dependence of pair ionization loss in
thin target on separation between the targets for
E = 100MeV, ϑ = 3ϑ0 and z1 = 1µm
The figure shows that the approximation (10) for
the term F in the interference factor (5) works rather
well in the considered range of distances z1. How-
ever, it gives slightly smaller value of ionization loss
in the maximum position. Therefore, in the present
work we will also use the exact expression (7) in our
investigation of the maximum possible values of the
anti-Chudakov effect for a single pair.
The approximation (10) may be of high impor-
tance for application of the present theory for real
experimental situation. In this case in order to es-
timate the possible value of the considered effect it
is necessary to make the procedure of averaging of
the expression (1) with respect to various parame-
ters, such as ϑ, z0 etc.4 The use of expression (10)
3In fact, nearly monotone - for details see [5].
4It is necessary to note that in this case it is also desirable to take into account the unevenness of the initial photon energy
distribution between the particles in the pair.
64
instead of (7) for such estimation might significantly
reduce the calculation time.
Fig.6. Estimation of the maximum value of the
anti-Chudakov effect at E = 100MeV for the given
parameters of the targets (ωp = 30 eV , I = 175 eV )
as a function of the pair divergence angle. Solid
line – calculation with the use of (7), dashed line –
calculation based on (10)
The calculation results for the relative value of
the considered effect for distances z1 = z1m corre-
sponding to the maximum of dE/dz dependence are
presented on Fig.6 for differnt values of ϑ. It shows
the results obtained with the use of both the expres-
sions (7) and (10). Here we see that the magnitude of
the anti-Chadakov effect at z1 = z1m may be several
times larger than its asymptotic one (see Fig.3).
Fig.7. Dependence of the separation between the
targets, corresponding to the maximun value of
the effect, on the pair divergence angle. Solid line
- calculation with the use of (7), dashed line -
calculation based on (10)
In conclusion let us also present a figure (Fig.7)
which demonstrates the value z1m of separation be-
tween the targets, corresponding to the maximum
magnitude of the considerd effect, as a function of
the angle ϑ of the pair divergence. The figure in-
dicates significant nonmonotonic dependence of the
separation on this angle.
4. CONCLUSIONS
In the present paper the ionization loss of an ultra-
relativistic e+e− pair in thin target was considered.
The pair was supposed to be created in the second
target situated in front of the first one on some dis-
tance from it. The main attention was drawn to the
study of the so-called anti-Chudakov effect in the pair
ionization loss. The effect was theoretically consid-
ered without the use of the approximation of parallel
electron and positron velocities, which had been ap-
plied in the earlier paper [5] where the possibility of
such effect was predicted. With use of such more gen-
eral approach it was shown that under some condi-
tions the discussed effect should be manifested even in
the asymptotic case of large separations between the
targets. The maximum value of the effect is achieved
for the value of such separation which is proportional
to the square of the energy of the pair and depends
on its divergence angle as well. The effect is more
pronounced at small (however, ultrarelativistic) en-
ergies of the pair for the values of such angle which
exceed the most probable one.
ACKNOWLEDGMENTS
The work was partially supported by the project no.
0117U004866 of the Ministry of Education and Sci-
ence of Ukraine, by the project CO-1-8/2017 of the
National Academy of Sciences of Ukraine and by the
project F64/17-2017 of State Fund for Fundamental
Research of Ukraine.
References
1. P. Sigmund. Particle penetration and radiation
effects. Berlin-Heidelberg: ”Springer”, 2006.
2. A.E.Chudakov. On an ionization effect associ-
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3. K.K.Andersen, J. Esberg, K.R.Hansen, et al.
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ÎÁ ÀÍÒÈÝÔÔÅÊÒÅ ×ÓÄÀÊÎÂÀ Â ÈÎÍÈÇÀÖÈÎÍÍÛÕ ÏÎÒÅÐßÕ
ÓËÜÒÐÀÐÅËßÒÈÂÈÑÒÑÊÎÉ ÝËÅÊÒÐÎÍ-ÏÎÇÈÒÐÎÍÍÎÉ ÏÀÐÛ Â ÒÎÍÊÎÉ
ÌÈØÅÍÈ
Ñ.Â.Òðîôèìåíêî
Ðàññìîòðåíû èîíèçàöèîííûå ïîòåðè óëüòðàðåëÿòèâèñòñêîé e+e−-ïàðû â ñèñòåìå èç äâóõ ìèøåíåé �
òîëñòîé, â êîòîðîé ïàðà ðîæäàåòñÿ, è òîíêîé, ðàñïîëîæåííûõ îäíà çà äðóãîé íà íåêîòîðîì ïðîìåæóò-
êå. Îñíîâíîå âíèìàíèå óäåëåíî èññëåäîâàíèþ ýôôåêòà óâåëè÷åíèÿ ïîòåðü ïàðû â òîíêîé ìèøåíè ïî
ñðàâíåíèþ ñ ñóììîé íåçàâèñèìûõ ïîòåðü ýëåêòðîíà è ïîçèòðîíà (êîòîðûé ÿâëÿåòñÿ îáðàòíûì ýôôåêòó
×óäàêîâà ïîäàâëåíèÿ èîíèçàöèîííûõ ïîòåðü ïàðû). Çàäà÷à ðåøåíà áåç èñïîëüçîâàíèÿ ïðèáëèæåíèÿ
ïàðàëëåëüíûõ ñêîðîñòåé ýëåêòðîíà è ïîçèòðîíà, êîòîðîå îáû÷íî ïðèìåíÿåòñÿ ïðè òåîðåòè÷åñêîì ðàñ-
ñìîòðåíèè èîíèçàöèîííûõ ïîòåðü ïàð. Èçó÷åíà çàâèñèìîñòü âåëè÷èíû äàííîãî ýôôåêòà îò ýíåðãèè è
óãëà ðàçëåòà ïàðû. Ïîêàçàíà âîçìîæíîñòü ñóùåñòâîâàíèÿ òàêîãî ýôôåêòà â àñèìïòîòè÷åñêîì ñëó÷àå
ïðè áîëüøèõ ðàññòîÿíèÿõ ìåæäó ìèøåíÿìè. Ïîëó÷åíî óïðîùåííîå àíàëèòè÷åñêîå âûðàæåíèå, îïèñû-
âàþùåå çàâèñèìîñòü èîíèçàöèîííûõ ïîòåðü îò ðàññòîÿíèÿ ìåæäó ìèøåíÿìè â óñëîâèÿõ ïðîÿâëåíèÿ
ðàññìàòðèâàåìîãî ýôôåêòà.
ÏÐÎ ÀÍÒÈÅÔÅÊÒ ×ÓÄÀÊÎÂÀ Â IÎÍIÇÀÖIÉÍÈÕ ÂÒÐÀÒÀÕ
ÓËÜÒÐÀÐÅËßÒÈÂIÑÒÑÜÊÎ� ÅËÅÊÒÐÎÍ-ÏÎÇÈÒÐÎÍÍÎ� ÏÀÐÈ Â ÒÎÍÊIÉ
ÌIØÅÍI
Ñ.Â.Òðîôèìåíêî
Ðîçãëÿíóòî iîíiçàöiéíi âòðàòè óëüòðàðåëÿòèâiñòñüêî¨ e+e−-ïàðè â ñèñòåìi ç äâîõ ìiøåíåé � òîâñòî¨, â
ÿêié ïàðà íàðîäæó¹òüñÿ, i òîíêî¨, ùî ðîçòàøîâàíi îäíà çà îäíîþ íà ïåâíîìó ïðîìiæêó. Îñíîâíó óâàãó
ïðèäiëåíî äîñëiäæåííþ åôåêòà çáiëüøåííÿ âòðàò ïàðè â òîíêié ìiøåíi ïîðiâíÿíî ç ñóìîþ íåçàëåæ-
íèõ âòðàò åëåêòðîíà i ïîçèòðîíà (ÿêèé ¹ ïðîòèëåæíèì äî åôåêòó ×óäàêîâà çìåíøåííÿ iîíiçàöiéíèõ
âòðàò ïàðè). Çàäà÷ó ðîçâ'ÿçàíî áåç âèêîðèñòàííÿ íàáëèæåííÿ ïàðàëåëüíèõ øâèäêîñòåé åëåêòðîíà i
ïîçèòðîíà, ÿêå çàçâè÷àé çàñòîñîâó¹òüñÿ ïðè òåîðåòè÷íîìó ðîçãëÿäi iîíiçàöiéíèõ âòðàò ïàð. Âèâ÷åíî
çàëåæíiñòü âåëè÷èíè äàíîãî åôåêòó âiä åíåðãi¨ i êóòà ðîçëüîòó ïàðè. Ïîêàçàíà ìîæëèâiñòü iñíóâàííÿ
òàêîãî åôåêòó â àñèìïòîòè÷íîìó âèïàäêó ïðè âåëèêèõ âiäñòàíÿõ ìiæ ìiøåíÿìè. Îòðèìàíî ñïðîùåíèé
àíàëiòè÷íèé âèðàç, ùî îïèñó¹ çàëåæíiñòü iîíiçàöiéíèõ âòðàò ïàðè âiä âiäñòàíi ìiæ ìiøåíÿìè â óìîâàõ
ïðîÿâó åôåêòó, ùî ðîçãëÿäà¹òüñÿ.
66
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