Radiation from relativistic positrons during planar channeling in crystals
We present a detailed calculation of channeling radiation of planar-channeled positrons from crystal targets in the framework of our approach, which was proposed recently.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2004
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | Radiation from relativistic positrons during planar channeling in crystals / V.F. Boldyshev, M.G. Shatnev // Вопросы атомной науки и техники. — 2004. — № 5. — С. 117-119. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-805512015-04-19T03:02:47Z Radiation from relativistic positrons during planar channeling in crystals Boldyshev, V.F. Shatnev, M.G. Взаимодействие релятивистских частиц с кристаллами и веществом We present a detailed calculation of channeling radiation of planar-channeled positrons from crystal targets in the framework of our approach, which was proposed recently. Представлено докладні обчислювання випромінювання при каналіруванні площинно-каналіруючих позітронів із кристаличної мішені за нашим підхідом, який було запропоновано раніше. Представлены подробные вычисления излучения позитронов при плоскостном каналировании в кристаллической мишени в рамках предложенного ранее подхода. 2004 Article Radiation from relativistic positrons during planar channeling in crystals / V.F. Boldyshev, M.G. Shatnev // Вопросы атомной науки и техники. — 2004. — № 5. — С. 117-119. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 61.85.+p; 41.60.-m http://dspace.nbuv.gov.ua/handle/123456789/80551 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Взаимодействие релятивистских частиц с кристаллами и веществом Взаимодействие релятивистских частиц с кристаллами и веществом |
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Взаимодействие релятивистских частиц с кристаллами и веществом Взаимодействие релятивистских частиц с кристаллами и веществом Boldyshev, V.F. Shatnev, M.G. Radiation from relativistic positrons during planar channeling in crystals Вопросы атомной науки и техники |
| description |
We present a detailed calculation of channeling radiation of planar-channeled positrons from crystal targets in
the framework of our approach, which was proposed recently. |
| format |
Article |
| author |
Boldyshev, V.F. Shatnev, M.G. |
| author_facet |
Boldyshev, V.F. Shatnev, M.G. |
| author_sort |
Boldyshev, V.F. |
| title |
Radiation from relativistic positrons during planar channeling in crystals |
| title_short |
Radiation from relativistic positrons during planar channeling in crystals |
| title_full |
Radiation from relativistic positrons during planar channeling in crystals |
| title_fullStr |
Radiation from relativistic positrons during planar channeling in crystals |
| title_full_unstemmed |
Radiation from relativistic positrons during planar channeling in crystals |
| title_sort |
radiation from relativistic positrons during planar channeling in crystals |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2004 |
| topic_facet |
Взаимодействие релятивистских частиц с кристаллами и веществом |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80551 |
| citation_txt |
Radiation from relativistic positrons during planar channeling in crystals / V.F. Boldyshev, M.G. Shatnev // Вопросы атомной науки и техники. — 2004. — № 5. — С. 117-119. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT boldyshevvf radiationfromrelativisticpositronsduringplanarchannelingincrystals AT shatnevmg radiationfromrelativisticpositronsduringplanarchannelingincrystals |
| first_indexed |
2025-07-06T04:33:49Z |
| last_indexed |
2025-07-06T04:33:49Z |
| _version_ |
1836870721880457216 |
| fulltext |
RADIATION FROM RELATIVISTIC POSITRONS
DURING PLANAR CHANNELING IN CRYSTALS
V.F. Boldyshev, M.G. Shatnev
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: shatnev@ kipt.kharkov.ua
We present a detailed calculation of channeling radiation of planar-channeled positrons from crystal targets in
the framework of our approach, which was proposed recently.
PACS: 61.85.+p; 41.60.-m
1. INTRODUCTION
Theoretical studies of the radiation from planar-
channeled electrons and positrons are due to
M.A. Kumakhov and R. Wedell [1], N.K. Zhevago [2],
A.I. Akhiezer, I.A. Akhiezer and N.F. Shul`ga [3,4].
Experimentally the channeling radiation (CR) of
positrons was observed by different groups [5,6]
demonstrating strong and sharp peaks in the spectrum.
The purpose of the present work is to calculate the
spectral-angular distribution of the channeling radiation
intensity emitted from positrons in the framework of
approach, which was proposed recently [7].
2. GENERAL REMARKS ON CR AND
CALCULATION
We consider a relativistic charged particle incident
onto a crystal at a small angle to a crystal planes. In the
planar channeling case for positively charged positrons,
the channel is between the crystal planes. This channel
is the source of a potential well in the direction
transverse to particle motion giving rise to transversely
bound states for the particle. Transitions to lower-
energy states lead to the phenomenon known as
channeling radiation (CR). Calculations of the CR
process are carried out by using the rules of quantum
electrodynamics [1,2]. The Doppler formula for the
energy of photon emitted is derived using the energy
and momentum conservation laws. For this energy one
gets
θβ
ω
ω
cos//1 −
′= nn (1)
where nnnn ′−=′ εεω ; nε and n′ε are the discrete
energy levels of the transverse oscillations of the
positron in the channel before and after radiation,
respectively;
E
p//
//
=β , E and //p are the energy
and the longitudinal momentum of the positron, and θ
is the angle of radiation emission relative to the
direction of motion of the channeled positron. For
positrons of not too high transverse energies, a good
approximation (see e.g., Ref. [1,2]) is the harmonic
potential leading to equidistant energy levels
)2/1( +Ω= nnε (2)
where ,022
E
U
pd
=Ω pd is the distance between
planes in the corresponding units, and 0U is the depth
of the potential well. Since )2
2
1
1(// −−≈ γ
E
p
(where
m
E
=γ ), )2
2
1
1(cos θθ −≈ and taking into account
Eq. (2), Eq. (1) can be expressed as
)221(
)(22
γθ
γω
+
′−Ω
=
nn
(3)
It follows from Eq. (3) that the radiation of a maximum
frequency
)(22 nn ′−Ω= γω (4)
is emitted in the forward direction (at 0=θ ). The case
1=′− nn corresponds to the peak values of the
experimental channeling radiation spectra [6], being the
first harmonic with the photon energy Ω= 22γω . As it
follows from Eq. (3), photons emitted via positron
transition from any initial level n to the final level 1−n
are identical (i.e., have the same energies for the same
emission angles). This means that the resulting
amplitude should be given by an additive superposition
of amplitudes of all such transitions. The positron state
outside the crystal )0( <z is a plane wave, whereas
inside the crystal )0( >z , the part of its wave function
corresponding to the transverse motion is a
superposition of the harmonic oscillator eigenvectors.
Factors nc describing transitions from the initial state
to states with the transverse energy levels n can be
found using boundary conditions set upon the wave
function at the crystal boundary )0( =z . Then, a
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 117-119. 117
transition to the closest lower level 1−n occurs with
emission of a photon having energy ω . One may expect
that the total amplitude of the transition from the initial
to final state accompanied by the photon emission is
determined by products of the amplitudes nc and
1, −nnM . Following the rules of the quantum mecha-
nics, we express this amplitude as
∑ −∝
n nnMncA 1, , (5)
were summation is performed over all the harmonic
oscillator levels. Also we must find an additive
superposition of amplitudes of all transitions
.,..3,2 −→−→ nnnn etc. Taking into account
these considerations, we can write the transition matrix
element in the form
∑ ∑ −=
j n jnnMncifM ,
2
,
2
(6)
where nnj ′−= . It is well known from the quantum
electrodynamics that the transition matrix element is
given by
,
2222
λω
π
J
V
e
ifM
⋅
⋅
= (7)
where ,
137
12 =e 3LV = is the normalization volume,
2,1=λ indicates the linear photon polarization, and
, 3†
∫ Ψ⋅−Ψ ′= rdrkieJ
λαλ (8)
with *
λεαλα
⋅= . The prime indicates the final state.
As we mentioned above, the wave function is the
solution of the time-independent Dirac equation for a
relativistic particle moving with momentum
),,0(// zpypp =
in a one-dimensional planar
potential )(xU periodic in the x direction (which is
normal to the channeling planes)
Ψ=Ψ−+∇⋅ )()( xUmEi βα
, (9)
where m and E are the particle’s mass and energy, α
and β are the Dirac matrices. Separating the wave
function Ψ into large and small components,
Ψ
Ψ
=Ψ
b
a (10)
and using the standard representation for the Dirac
matrices, leads to a Pauli-type equation for the large
components,
0))((1))(( =Ψ−−+Ψ∇⋅−+−∇⋅ amxUEamxUE
σσ
Since a potential )(xU is independent of y and z , the
solution of last equation is a plane wave in the yz plane
χϕ )()](exp[ xyypzzpia +∝Ψ . (11)
This allows us to transform a Pauli-type equation into a
one-dimensional, relativistic Schrödinger equation for
the transverse motion
)()()(2
)(2
2
1
xxxU
dx
xd
E
ε ϕϕ
ϕ
=+− , (11a)
where
E
ypzpmE
2
2222 −−−
=ε . (12)
In a given potential, the latter will assume certain
bound-state eigenvalues ,...)2,1,0(0 =< nnε , with
corresponding eigenfunctions )(xnϕ . The wave
function of Eq.(10) is finally obtained in the form
)()////exp( xnrpi
mE
i
L
N
ϕ
χ
σ
χ
+
∇⋅−=Ψ , (13)
where 2L being the two-dimensional normalization
volume for the plane waves,
E
mE
N
2
+
= , and χ
being a two-component spinor which is
0
1
or
1
0
when the particle spin points in the z+ or the z−
direction in the rest frame, respectively. For the
harmonic potential 2
0)( xUxU = , it is well known that
the corresponding eigenfunctions being given by
),()2/2exp(
!2
14)( xEnHxE
nn
E
xn ΩΩ−
Ω
=
π
ϕ
where nH are the Hermite polynomials. According to
Eq. (8), we find, using Eq. (13) and last expression for
the wave functions, the first-order matrix element
corresponding to the jnn −→ transverse transition
,1])[(**
2)//////(2)2( χσλεχδπλ
BiANNkppJ +′−′−=
where
),
)
////(1//
,1)
)
11
(2
),
)
11
(//1//
),
)
11
(2
mE
p
mE
p
IB
I
mE
xk
mEmE
iIxB
mEmE
pIA
mEmE
iIxA
+−
′
−
+
=
+−
+
+
−
+−
=
+−
+
+
=/
+−
+
+
−=
ω
ωω
ω
ω
(14)
with
∫ −−=
∫ −−=
,
)(
)(*)exp(2
,)()(*)exp(1
dx
dx
xnd
xjnxxikI
dxxnxjnxxikI
ϕ
ϕ
ϕϕ
(15)
3
Then we calculate the matrix element and the
differential intensity, and after summation over
polarization of emitted photon and the positron we find
by integrating over //
2 pd ′
∑ ×∑ −=
j n jnnMnc
e
dd
Id 2
,2
222
π
ω
οω
),cos//( jnn −−−× ωθω βωδ (16)
where the factors nc are, in the case of the parabolic
potential and when the initial positron is a plane wave,
given by
)()
2
2
exp(4
!12 ΩΩ
−
Ω−
=
E
xp
nH
E
xp
Enn
ni
nc
π
. (17)
3. CONCLUSIONS
Following N.K. Zhevago [2], the spectral-angular
distribution of emitted photons is represented as
∑∝
f ifM
dd
Id 22
οω
(18)
The sum entering Eq. (18) is the one over the quantum
numbers f of the transverse motion of the particle. Then,
the probability of having a definite transverse energy is
taken into account by multiplying each term of this sum
by a corresponding factor. In our consideration, discrete
levels of the transverse motion in the harmonic
oscillator potential refer to the intermediate state of the
particle. Accordingly, the contribution to the intensity
due to transitions, e.g., between the closest levels is
determined by the square of the absolute value of
Eq. (5)
.
2
1,
)1(2
∑ −∝
n nnMnc
dd
Id
οω
(19)
In other words, unlike Ref. [2], we get an expression
that contains interference terms mixing amplitudes of
photon emission from different equidistant levels. We
would like to note that dynamics of channeling electron
in a crystal differs from that of the positron case. The
transverse potential well for the electron does not give
rise to equidistant energy levels for transverse particle
motion. Therefore, there are no interference
contributions to the photon emission intensity similar to
those present in Eq. (19). In our opinion, this could
explain the greater intensity in case of channeling
positron compared to that for the electron observed in
experiment [6]. Corresponding numerical calculations
will be given in a subsequent publication.
REFERENCES
1. M.A. Kumakhov and R. Wedell. Theory of
radiation of relativistic channeled particles // Phys.
Stat. Sol. 1977, v. 84, p. 581-593.
2. N.K. Zhevago. Emission of γ quanta by
channeled particles // Zh. Eksp. Teor. Fiz. 1978,
v. 75, p. 1389-1401 (in Russian).
3. A.I. Akhiezer, I.A. Akhiezer and
N.F. Shul`ga. To the classical theory of radiation of
fast particles in crystals // Zh. Eksp. Teor. Fiz.
1979, v. 76, p. 1244-1253 (in Russian).
4. A.I. Akhiezer and N.F. Shul`ga. Radiation of
relativistic particles in single crystals // Sov.
Phys.Usp. 1982, v. 25, p. 541-564.
5. R.O. Avakian, I.I. Miroshnichenko, J. Murrey
and T. Fieguth. Radiation emission by
ultrarelativistic positrons moving in a single crystal
near the crystallographic axec and planes // Zh.
Eksp. Teor. Fiz. 1982, v. 82, p. 1825-1832 (in
Russian).
6. J. Bak, J.A. Ellison, B. Marsh, F.E. Meyer,
O. Pedersen, J.B.B. Petersen, E. Uggerhøj and
K. Østergaard. Channeling radiation from 2-
55 GeV/c electrons and positrons // Nucl.Phys.
1985, v. 75, p. 491-527.
7. V.F. Boldyshev, M.G. Shatnev. On the
question of interference in radiation produced by
relativistic channeled particles. Proc of the First
Feynman Festival August, 2002, University of
Maryland, College Park, Maryland, USA, quant-
ph/0210203.
ИЗЛУЧЕНИЕ РЕЛЯТИВИСТСКИХ ПОЗИТРОНОВ
ПРИ ПЛОСКОСТНОМ КАНАЛИРОВАНИИ В КРИСТАЛЛАХ
В.Ф. Болдышев, М.Г. Шатнев
Представлены подробные вычисления излучения позитронов при плоскостном каналировании в
кристаллической мишени в рамках предложенного ранее подхода.
ВИПРОМІНЮВАННЯ ВІД РЕЛЯТІВІСТСЬКИХ ПОЗІТРОНІВ
ПРИ ПЛОЩИННОМУ КАНАЛІРУВАННІ В КРИСТАЛАХ
В.Ф. Болдишев, М.Г. Шатнєв
Представлено докладні обчислювання випромінювання при каналіруванні площинно-каналіруючих
позітронів із кристаличної мішені за нашим підхідом, який було запропоновано раніше.
3
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
3. CONCLUSIONS
REFERENCES
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