Polarizational Bremsstrahlung of Atomic Particles In The
Relativistic And Ultra-Relativistic Regimes
A. V. Korol∗ , O. I. Obolensky† , A. V. Solov’yov† and I. A. Solovjev∗
∗
Russian Maritime Technical University, Leninskii prospect 101, St.Petersburg 198262, Russia
A.F. Ioffe Physical-Technical Institute, Polytechnicheskaya 26, St.Petersburg 194021, Russia
†
Abstract. The features of polarizational bremsstrahlung emitted in an elastic collision of a charged particle with a manyelectron target are studied for the relativistic and ultra-relativistic collision regimes. The numerical data on the cross
sections and angular distributions of polarizational bremsstrahlung are presented. It is demonstrated that the polarizational
bremsstrahlung cross section increases logarithmically with the energy of incident particle. The relativistic effects were shown
to result in substantial asymmetry of the angular distribution of emitted photons.
INTRODUCTION
Ordinary BrS
1
The relativistic formalism for the bremsstrahlung process
has been recently developed [1]. It accounts for two leading mechanisms of the radiation production, the ordinary
and polarizational bremsstrahlung. In this work we study
the features of polarizational bremsstrahlung emitted in
an elastic collision of a charged particle with a manyelectron target for the relativistic and ultra-relativistic
collision regimes. We present the results of the calculations of the cross sections and angular distributions of polarizational bremsstrahlung. We demonstrate that the polarizational bremsstrahlung cross section increases logarithmically with the energy of incident particle. We show
that the relativistic effects result in substantial asymmetry of the angular distribution of emitted photons.
The Feynman diagrams corresponding to the elastic
bremsstrahlung process of a structureless charged projectile scattered by many-electron atom are shown in Figure
1.
The first diagram describes ordinary bremsstrahlung,
i.e. the process of photon emission by a charged projectile accelerated in the static field of a target. This is a well
known quantum mechanical process the basic description of which can be found in textbooks (see, e.g. [2]).
For a review of ordinary bremsstrahlung see [3]-[5] and
references therein.
Two other diagrams in Figure 1 represent polarizational bremsstrahlung [6]. Here, the photon emission occurs due to the virtual excitations (polarization) of the
target electrons under the action of the field created by a
charged projectile. This mechanism of the radiation production was recognized relatively recently. A comprehensive review of theoretical methods and experimen-
2
(ω,k,e)
Polarizational BrS
2
1
1
0
n
(ω,k,e)
0
0
2
n
0
(ω,k,e)
FIGURE 1. Feynman diagrams corresponding to the elastic bremsstrahlung process of a structureless charged projectile
scattered by many-electron atom. The first diagram represents
ordinary bremstrahlung, two other diagrams describe polarizational bremsstrahlung. The solid lines correspond to the projectile wavefunctions satisfying the Dirac equation which accounts for the central field of the target. The initial (index ’1’)
and the final (index ’2’) states of the projectile are characterized
by the momenta p1,2 . The double lines denote the states of
the target: the index ’0’ marks the initial and the final states,
the index ’n’ corresponds to the intermediate virtual state. The
dashed lines designate the emitted (real) photon of energy ω ,
momentum k and the polarizational vector e. The dotted lines
stand for the virtual photon, the energy of which equals to ω
while the momentum q is not fixed by any kinematic relations.
tal data on polarizational bremsstrahlung can be found
in [6]-[9] and references therein.
Unlike ordinary bremsstrahlung, the cross section of
which is proportial to the factor 1/M 2 (M is the mass
of a projectile), the polarizational bremsstrahlung cross
section is almost insensitive to the variations of M [6].
Therefore, in order to reveal the features of the relativistic and ultra-relativistic regimes of polarizational
bremsstrahlung and take away the features of ordinary
bremsstrahlung we consider in this paper collisions involving heavy projectiles. We use the relativistic Born
approximation which is valid for the relativistic and
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
111
ultra-relativistic collisions of heavy particles. Atomic
units h̄ = me = |e| = 1 are used throughout the paper.
In this section we give the resulting expressions for the
polarizational bremsstrahlung cross section within the
relativistic Born approximation [10].
∞
l(l + 1)
×
l=1 2l + 1
2
qmax
q − ω2
2
+ 2 (p1 ·nq ) (p2 ·nq )
dq
2
qmin
2
αl (ω , q, k) +
∑
q2 − ω 2 + 2 p21 sin2 θq
l(l + 1) ×
2 (q2 − ω 2 )2
2
(λ )
∑ βl (ω , q, k) .
(1)
λ =0,1
Here Z p denotes the charge of the projectile, α is the fine
structure constant, q is the transferred momentum, q min
and qmax are the minimal and maximal values of q.
The partial generalized polarizabilities are defined as
follows:
αl (ω , q, k) =
2l + 1 2
e ×
q
∑
ε0 j0 l0
εn jn ln
(λ )
βl
(−1)
(1)
(−1)
2C0n (l) f0n (k; l) fn0 (q; l)
(2)
2 − ω2
ωn0
ε0 j0 l0
εn jn ln
(λ )
(λ )
(λ )
2 ωn0 C0n (l) f0n (k; l) fn0 (q; l)
,
2 − ω2
ωn0
λ = 0, 1
,
κ = l(l + 1) − j( j + 1) −
(4)
(3)
Quantities with the indexes "0" and "n" refer to the
initial/final and intermediate states of the target, re(λ )
spectively; ωn0 = εn − ε0 . The radial integrals f ba (s; l)
(λ = −1, 0, 1, (b, a) = (0, n), x = (k, q)) are defined as
follows:
∞ (−1)
dr g∗b (r)ga (r) + fb∗ (r) fa (r) jl (xr)
fba (x; l)=
0 ∞ (0)
dr g∗b (r) fa (r) + fb∗ (r) ga (r) jl (xr)
fba (x; l)=
0
∞ j (xr)
(1)
l
fba (x; l)=
dr g∗b (r) fa (r) − fb∗ (r) ga (r)
xr
0
κb − κa ∗
g (r) fa (r) + fb∗ (r)ga (r) ×
−
l(l + 1) b
112
1
4
(5)
is the relativistic quantum number. The angular coeffi(λ )
cients C0n (l) can be found in [10].
A logarithmic growth with the energy of the incident
particle is an important characteristic of the polarizational bremsstrahlung cross section in the relativistic and
ultra-relativistic regimes [6]. This striking feature was
theoretically predicted and discussed for the dipole case
in [11]-[14]. The full multipole analysis and the first numerical results were presented in [10]. It can be shown
[10] that if ε1,2 ω , the polarizational bremsstrahlung
cross section
ω3
q0
dσ
≈ Zp2 α 2 H0A (ω , qmin , k) ln
dω
qmin
v1
q0 γ
1
+ 2 2 H0B (ω , qmin , k) ln
qmin
2 ε1 ω
p2 2
+ 12
lnγ − 1
.
(6)
ω
v21
Here γ = ε1 /Mc2 is the relativistic factor, q 0 ∼ 1/Rat
is the cut-off parameter (R at is the characteristic atomic
radius), qmax q0 qmin ,
∞
H0A (ω , q, k)= ∑
l=1
∞
(ω , q, k) = (2l + 1) e2 ×
∑
1 d jl (xr) jl (xr)
+
x dr
xr
where jn (z) is the spherical Bessel function,
THEORY
dσ
ω
= Zp2 α 2
dω
2p1
H0B (ω , q, k)= ∑
l=1
2
2l(l + 1) αl (ω , q, k) 2l + 1
2
2
l (l + 1)2
(λ )
βl (ω , q, k) .(7)
∑
2l + 1 λ =0,1
Expressions (6) and (7) show that the behavior of the polarizational bremsstrahlung cross section at ε 1,2 ω is
determined by the terms proportional to ln(q −1
min Rat ) and
lnγ . The former term corresponds to the contribution of
the longitudinal (Coulomb) part of the electromagnetic
interaction between an incident particle and a target. The
latter term corresponds to the contribution of the transverse (vector) part of the interaction. The contribution of
the term with the lnγ increases as the energy of the incident particle grows and becomes predominant at γ 1.
Qualitatively, this means that relativistic particle, unlike its non-relativistic counterpart, interacts with the
target not only via the Coulomb field but also (and in
the ultra-relativistic case, predominantly) via the field
of transverse virtual photons. The effective radius of
this field increases as the energy of the incident particle
grows, almost to infinity in the ultra-relativistic case. As
σ on incident proton relativistic γ factor in collisions
FIGURE 2. Dependence of the polarizational bremsstrahlung cross section ddω
+12
+46
+78
+91
for emitted photon energies ω = (a, c, e, and g) 1.5I and
with (a and b) Al , (c and d) Ag , (e and f) Au , and (g and h) U
(b, d, f, and h) 4I (where I is the ionization potential of the 1s target subshell, which approximately equals 2.3, 31, 93.5 keV, and
132 keV for Al+12 , Ag+46 , Au+78 , and U+91 , respectively). The thick solid line represents relativistic cross section (1). The thin
solid line corresponds to the logarithmic approximation (6). The behavior of the terms proportional to the squares of the αl (ω , q, k),
(0)
(1)
βl (ω , q, k), and βl (ω , q, k) is shown by dashed, dotted, and dot-and-dash lines, respectively.
σ
FIGURE 3. Angular distributions dωd dΩ
for polarizational bremsstrahlung emitted in collisions of 3 GeV protons with (a and
k
+12
+46
+78
b) Al , (c and d) Ag , (e and f) Au , and (g and h) U+91 ions for two emitted photon energies of (a, c, e, and g) 1.5I and
(b, d, f, and h) 4I (see Figure 2). The thick solid line represents the relativistic cross section (1); the thin solid line was obtained in
the nonrelativistic dipole approximation. The dashed, dotted, and dot-and-dash lines correspond to the contributions of the terms
(0)
(1)
proportional to the squares of the moduli of the αl (ω , q, k), βl (ω , q, k), and βl (ω , q, k) polarizabilities, respectively.
2
113
a result, the distances at which an incident particle can
effectively polarize a target increase, and therefore the
polarizational bremsstrahlung cross section increases.
NUMERICAL RESULTS AND
DISCUSSION
The results described in this section refer to proton collisions with the Al+12 , Ag+46 , Au+78 and U+91 hydrogenlike ions. These results can easily be generalized to neutral Al, Ag, Au, and U atoms by multiplying the cross
sections obtained for the hydrogen-like ions by a factor
of 4. The generalization is possible due to the fact that the
inner shell electrons make the main contribution to the
total polarizational bremsstrahlung cross section in the
region of photon frequencies higher than the K-shell ionization threshold (e.g., see [6]). The factor 4 accounts for
a twofold increase in the polarizabilities of filled atomic
K-shells.
The dependence of the polarizational bremsstrahlung
cross section on the relativistic factor γ of the incident
proton for two emitted photon energies is shown in Figure 2. These results visually demonstrate the logarithmic
growth of the polarizational bremsstrahlung cross section
with increasing the energy of the incident particle.
The logarithmic growth of the cross section is not the
only feature of the relativistic regime for polarizational
bremsstrahlung. Another interesting consequence of radiation lag is the substantial differences between the angular distributions calculated in the relativistic and nonrelativistic approaches. The importance of taking relativistic effects into account in considering polarizational
bremsstrahlung is clear from Figure 3. The length of the
segment connecting the origin and a curve point is equal
to the differential polarizational bremsstrahlung cross
section (in millibarns) in the corresponding direction.
The direction along the horizontal axis is the direction of
incident particle motion. The curves describing the contributions of cross section components proportional to
(0)
the squares of the moduli of the α l (ω , q, k), βl (ω , q, k),
(1)
and βl (ω , q, k) polarizabilities are also shown in the
figure. Note, that the sum of these curves does not correspond to the total cross section, which also takes into
account the cross terms. The results obtained in this
work show that accounting for the relativistic effects and
the effects related to the radiation of high-multipolarity
photons noticeably change the angular distributions of
emitted photons and make these distributions appreciably asymmetric compared with the distributions of the
nonrelativistic dipole approximation, which are symmetrical with respect to the θk → π − θk operation. The relativistic angular distributions are shifted in the direction in
which the incident particle moves, and the distributions
114
become more asymmetric as the emitted photon energy
increases.
We note that when considering relativistic and ultrarelativistic collision regimes one should also take into
account interesting features due to the Doppler and the
aberration of light effects [11]-[14].
This work was partially supported by the Russian
Academy of Sciences (Grant 44).
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