Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation
The areas of applicability of kinematic and dynamic theories of parametric X-ray radiation (PXR) of relativistic particles in a crystal are considered. It is shown that dynamic diffraction of the PXR is possible in crystallographic planes with low Miller indexes in the central part of the PXR reflec...
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| Cite this: | Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation / A.V. Shchagin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 86-88. — Бібліогр.: 20 назв. — англ. |
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| description | The areas of applicability of kinematic and dynamic theories of parametric X-ray radiation (PXR) of relativistic particles in a crystal are considered. It is shown that dynamic diffraction of the PXR is possible in crystallographic planes with low Miller indexes in the central part of the PXR reflection only, where the PXR yield is weak and masked by other kinds of diffracted radiation. Therefore the properties of the PXR reflection are well described by the kinematic PXR theory. Results of the analysis are compared to experimental data.
Аналізуються області застосовності кінематичної та динамічної теорій параметричного рентгенівського випромінювання (ПРВ) релятивістських частинок у кристалі. Показано, що динамічна дифракція ПРВ можлива тільки на кристалографічних площинах з малими індексами Міллера і тільки в центральній області рефлексу ПРВ, де вихід ПРВ малий і маскується іншими видами, що піддалися дифракції, випромінювань. Тому властивості випромінювання в рефлексі ПРВ добре описуються кінематичною теорією ПРВ. Результати аналізу порівнюються з експериментальними даними.
Анализируются области применимости кинематической и динамической теорий параметрического рентгеновского излучения (ПРИ) релятивистских частиц в кристалле. Показано, что динамическая дифракция ПРИ возможна только на кристаллографических плоскостях с малыми индексами Миллера и только в центральной области рефлекса ПРИ, где выход ПРИ мал и маскируется другими видами подвергшихся дифракции излучений. Поэтому свойства излучения в рефлексе ПРИ хорошо описываются кинематической теорией ПРИ. Результаты анализа сравниваются с экспериментальными данными.
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ISSN 1562-6016. ВАНТ. 2015. №4(98) 86
AREAS OF APPLICABILITY OF KINEMATIC AND DYNAMIC
THEORIES OF PARAMETRIC X-RAY RADIATION
A.V. Shchagin1,2
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2Belgorod State University, Belgorod, Russia
E-mail: shchagin@kipt.kharkov.ua
The areas of applicability of kinematic and dynamic theories of parametric X-ray radiation (PXR) of relativistic
particles in a crystal are considered. It is shown that dynamic diffraction of the PXR is possible in crystallographic
planes with low Miller indexes in the central part of the PXR reflection only, where the PXR yield is weak and
masked by other kinds of diffracted radiation. Therefore the properties of the PXR reflection are well described by
the kinematic PXR theory. Results of the analysis are compared to experimental data.
PACS: 41.60.-m, 61.80.Cb
INTRODUCTION
Radiation of fast particles in a medium with periodi-
cally varying permittivity was first considered in [1].
Later, the kinematic theory of such radiation produced
by relativistic particles moving in a crystal was pro-
posed in [2]. In modern literature this radiation is names
as parametric X-ray radiation (PXR). Therefore we will
use this term in the present paper. The PXR reflection is
emitted by relativistic charged particles moving inside a
crystal, the energy and the direction of the PXR propa-
gation in the PXR reflection are close to conditions of
the X-rays diffraction. Therefore shortly after [2], the
dynamic PXR theory that takes into account the diffrac-
tion of the PXR in a crystal was proposed in refs. [3, 4].
According to the dynamic PXR theory, the PXR under-
goes to diffraction at the same family of crystallograph-
ic planes were it is was produced and can be emitted in
the vicinity of the velocity vector of incident particle. In
the experimental researches (see reviews [5, 6] and also
[7 - 12]), the main properties of observed PXR in the
PXR reflection were well described by the kinematic
PXR theory, meanwhile the dynamic PXR in forward
direction was not observed. However, discussions about
existence and possibilities of observation of the dynam-
ic PXR were continued, see e.g. [13 - 16] and brief re-
view of the discussions in [17]. At last, weak spectral
peaks of diffracted PXR were observed in narrow angu-
lar range in the vicinity of the incident particles velocity
vector V
. Below we will discuss conditions, when the
dynamic PXR diffraction is sufficient and when the kin-
ematic approach is enough for description of properties
of radiation in the PXR reflection.
1. THE PXR FREQUENCY
AND THE BRAGG FREQUENCY
The frequency of the PXR spectral peak PXRω [2, 7]
1
sin 1 cosPXR
Vg V
c
εω φ θ
−
⋅
= ⋅ ⋅ ⋅ −
(1)
is some different from the Bragg frequency Bω in the
crystal in the same direction
2
2B
cg
g
ω
ε
=
Ω⋅
, (2)
where g is the module of the reciprocal lattice vector; ε
is the permittivity in the crystal; φ is the angle between
the particle velocity vector V
and the crystallographic
planes; θ is the angle between the observation direction
and the particle velocity vector V
; Ω
is the unit vector
in the direction of the PXR emission. Explicit expres-
sion for normalized difference of Bragg (2) and PXR (1)
frequencies in approximation of small angles
||, 1δ δ⊥ << in the vicinity of the PXR reflection center
has been derived in [19]
2 22 2
0
2 24sin 4sin
effB PXR
PXR
γ ηγ χ ηω ω
ω φ φ
−− ++ +−
≈ = , (3)
where 2 2
||η δ δ⊥= + is the angle of declining from the
center of the PXR reflection in arbitrary direction. One
can see from (3) that the PXR frequency is always be-
low of the Bragg frequency and the difference of fre-
quencies is minimal in the area of the center of the PXR
reflection at η = 0, where the PXR yield is close to zero.
In the area of the maximal PXR yield at 1
effη γ −= , the
difference of frequencies (3) is
2
22sin
effγ
φ
−
. The difference
of frequencies (3) quickly increases at further increasing
of the angle η of declining from the center of the PXR
reflection.
2. CONDITIONS FOR DYNAMIC
DIFFRACTION OF THE X-RAY RADIATION
It is known that the effective dynamic X-ray diffrac-
tion is possible in the case when the monochromatic
X-ray beam declines from the exact Bragg direction
within the angular limits
sin 2
gC χ
ϑ
φ
∆ = ± , (4)
where C is the coefficient equal to unity for perpendicu-
lar ( )σ polarization and cos2φ for parallel ( )π po-
larization of radiation, gχ is the Fourier component of
dielectric susceptibility, and quickly reduces outside the
limits (4), see [20]. In our case of fixed direction of ra-
diation corresponding limits of the deviation of X-ray
radiation frequency can be found by the differentiation
of the Bragg law. They are
ISSN 1562-6016. ВАНТ. 2015. №4(98) 87
22sin
gB
B
C χω
ω φ
∆
= ± . (5)
3. CONDITIONS FOR DIFFRACTION
OF THE PXR
Comparing (3) and (5), let us estimate conditions
when the effective manifestations of the PXR diffraction
are possible. They are possible in the case when the devi-
ation of the PXR frequency (3) are within limits (5), at
1
B PXR
PXR
B
B
ω ω
ω
ω
ω
−
≤
∆
. (6)
Inserting (3) and (5) into (6), we obtain estimation of
the angular size of the central part of the PXR reflection
Dσ , where effective manifestations of the PXR dynam-
ic diffraction are possible
1
2
2
1g
D eff
eff
C χ
η σ γ
γ
−
−≤ = − . (7)
The condition for existence of such angular area is
the positive sign of the radicand in eq. (7)
22 g effC χ γ −> . (8)
It is interesting that both the condition for existence
of the PXR diffraction (8) and the angular size of the
central part in the PXR reflection where the diffraction
is possible (7) are independent of the direction of the
PXR reflection emission. Below we will consider the
most favorable for existence of the dynamic diffraction
case of high enough incident particles energy, when one
can neglect the value 2γ − in comparison to 0χ , in
other words at brightly expressed longitudinal Ter-
Mikaelian density effect [2, 8, 13]
2
0γ χ− << . (9)
At condition (9), the angular size of the PXR reflec-
tion 1
0effγ χ− = does not depend on the incident parti-
cle energy, the estimation of the angular size of the cen-
tral area σ (7) depends on the crystal properties and
polarization only
0
0
2
1gC χ
η σ χ
χ
≤ = − , (10)
and condition (8) takes the shape
02 gC χ χ> . (11)
Note that always Dσ σ> . The angular size of the
area (10) can be expressed (normalized) in units of the
angular size of the PXR reflection 0χ .
00
2
1g
N
C χσσ
χχ
= = − . (12)
One can consider the inequalities (8) and (11) as an
estimations criterion for choose of the crystallographic
plane where the PXR dynamical diffraction is possible.
One can use the expressions (7), (10), (12) for estima-
tion of the angular size of the central part in the PXR
reflection, where the dynamical diffraction is possible.
Note, that less expressed effects of dynamical diffrac-
tion and increased radiation attenuation are possible in
the vicinity of the merges of the angular area (7), (10),
(12) as well as the manifestations of the secondary max-
ima of pendulous solution in the vicinity of the area in
thin crystals.
4. RESULTS AND DISCUSSION
Below we will consider the case of perpendicular
polarization at 1C = . In this case the condition for ex-
istence of the PXR diffraction (11) and angular size of
the central part (12) depend on crystal properties only,
02 gχ χ> , (13)
00
2
1g
N
χσσ
χχ
= = − . (14)
The most popular crystal in experimental studies of
the PXR is the Si singlecrystal. Therefore let us perform
estimations for this crystal. Calculations show that at
PXR frequencies exceeding atomic frequencies and out
of resonance frequencies the condition (13) is fulfilled
for three crystallographic planes with non-zero structure
factors − (111), (220) and (400) only, and the angular
size of the area Nσ (14) is 0.275, 0.45 and 0.077 of the
angular part of the PXR reflection angular size respec-
tively. This means that small central angular part of the
PXR reflection only, where the PXR yield is minimal,
can be diffracted. Similar situation is realized for Ge
singlecrystal, where the angular size of the field Nσ
(14) is 0.47, 0.67, 0.45 and 0.18 for crystallographic
planes with non-zero structure factors satisfying to con-
dition (13) – they are crystallographic planes (111),
(220), (400) and (422) respectively. One can expect the
most prominent manifestations of the PXR diffraction
are possible from crystallographic planes (220) in both
crystals. The situation with other crystals demands of
separate analysis.
Experiments for observation of two-dimensional an-
gular distribution of the yield in the PXR reflection
from Si crystal (220) crystallographic plane were per-
formed in refs. [10, 12]. In both research main proper-
ties of the PXR reflection are well described by the kin-
ematic PXR theory and any manifestations of the dy-
namic phenomena were not found. However, this does
not exclude the possibility of diffraction in forward di-
rection of some part of radiation from the central area of
the PXR reflection.
The diffracted PXR from the Si (111) crystallo-
graphic plane has been observed in ref. [17]. The dif-
fracted PXR has been observed in forward direction in
the angular range about 00.36 χ± relative to the inci-
dent particles velocity vector (see Fig. 3 in ref. [17]).
These angles are close to above obtained estimation of
the angular range for Si (111) plane 00.275 χ± .
It is evident that the kinematic theory can not de-
scribe properties of the PXR that is diffracted in forward
ISSN 1562-6016. ВАНТ. 2015. №4(98) 88
direction and has been observed in refs. [17, 18]. The
manifestations of the dynamical effects in the central
area of the PXR reflection are unnoticeable because of
the low PXR intensity in this area and masking influ-
ence of other kinds of diffracted radiation, like as transi-
tion radiation at entrance surface of the crystal and
bremsstrahlung radiation in the crystal. Therefore the
effects of diffracted PXR should be weakly expressed in
the experiments on observation of the PXR reflection
and properties of the PXR reflection are well described
by the kinematic PXR theory. Note, that the PXR fre-
quency and the frequency of other kinds of diffracted
radiations are close one to another but different (see eq.
(3)) that gives the possibility for experimental resolving
of their yields by frequency [19].
CONCLUSIONS
The estimations presented in the paper show, that
conditions for effective manifestations of the dynamical
PXR diffraction in the considered crystals are executed
for low-indexed crystallographic planes in the central
area of the PXR reflection only, where the PXR yield is
close to zero and masked by other types of radiation of
incident particles. Therefore the influence of this phe-
nomenon on the yield in the PXR reflection is insignifi-
cant and the main properties of the PXR in the PXR
reflection are very well described by the kinematic PXR
theory. This circumstance does to exclude the existence
of the diffraction of some part of the central area in the
PXR reflection in forward direction that has been ob-
served in papers [17, 18].
ACKNOWLEDGEMENTS
Author is thankful to A.P. Potylitsyn for discussion
of the work. The research has been performed due to the
partial support by the Ministry of education and science
of the Russian Federation under project 3.2009.2014/K.
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Article received 02.06.2015
ОБЛАСТИ ПРИМЕНИМОСТИ КИНЕМАТИЧЕСКОЙ И ДИНАМИЧЕСКОЙ ТЕОРИЙ
ПАРАМЕТРИЧЕСКОГО РЕНТГЕНОВСКОГО ИЗЛУЧЕНИЯ
А.В. Щагин
Анализируются области применимости кинематической и динамической теорий параметрического рент-
геновского излучения (ПРИ) релятивистских частиц в кристалле. Показано, что динамическая дифракция
ПРИ возможна только на кристаллографических плоскостях с малыми индексами Миллера и только в цен-
тральной области рефлекса ПРИ, где выход ПРИ мал и маскируется другими видами подвергшихся дифрак-
ции излучений. Поэтому свойства излучения в рефлексе ПРИ хорошо описываются кинематической теорией
ПРИ. Результаты анализа сравниваются с экспериментальными данными.
ОБЛАСТІ ЗАСТОСОВНОСТІ КІНЕМАТИЧНОЇ ТА ДИНАМІЧНОЇ ТЕОРІЙ
ПАРАМЕТРИЧНОГО РЕНТГЕНІВСЬКОГО ВИПРОМІНЮВАННЯ
А.В. Щагин
Аналізуються області застосовності кінематичної та динамічної теорій параметричного рентгенівського
випромінювання (ПРВ) релятивістських частинок у кристалі. Показано, що динамічна дифракція ПРВ мож-
лива тільки на кристалографічних площинах з малими індексами Міллера і тільки в центральній області ре-
флексу ПРВ, де вихід ПРВ малий і маскується іншими видами, що піддалися дифракції, випромінювань.
Тому властивості випромінювання в рефлексі ПРВ добре описуються кінематичною теорією ПРВ. Результа-
ти аналізу порівнюються з експериментальними даними.
E-mail: shchagin@kipt.kharkov.ua
А.В. Щагин
А.В. Щагин
|
| id | nasplib_isofts_kiev_ua-123456789-112229 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:53:11Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Shchagin, A.V. 2017-01-18T19:51:53Z 2017-01-18T19:51:53Z 2015 Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation / A.V. Shchagin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 86-88. — Бібліогр.: 20 назв. — англ. 1562-6016 PACS: 41.60.-m, 61.80.Cb https://nasplib.isofts.kiev.ua/handle/123456789/112229 The areas of applicability of kinematic and dynamic theories of parametric X-ray radiation (PXR) of relativistic particles in a crystal are considered. It is shown that dynamic diffraction of the PXR is possible in crystallographic planes with low Miller indexes in the central part of the PXR reflection only, where the PXR yield is weak and masked by other kinds of diffracted radiation. Therefore the properties of the PXR reflection are well described by the kinematic PXR theory. Results of the analysis are compared to experimental data. Аналізуються області застосовності кінематичної та динамічної теорій параметричного рентгенівського випромінювання (ПРВ) релятивістських частинок у кристалі. Показано, що динамічна дифракція ПРВ можлива тільки на кристалографічних площинах з малими індексами Міллера і тільки в центральній області рефлексу ПРВ, де вихід ПРВ малий і маскується іншими видами, що піддалися дифракції, випромінювань. Тому властивості випромінювання в рефлексі ПРВ добре описуються кінематичною теорією ПРВ. Результати аналізу порівнюються з експериментальними даними. Анализируются области применимости кинематической и динамической теорий параметрического рентгеновского излучения (ПРИ) релятивистских частиц в кристалле. Показано, что динамическая дифракция ПРИ возможна только на кристаллографических плоскостях с малыми индексами Миллера и только в центральной области рефлекса ПРИ, где выход ПРИ мал и маскируется другими видами подвергшихся дифракции излучений. Поэтому свойства излучения в рефлексе ПРИ хорошо описываются кинематической теорией ПРИ. Результаты анализа сравниваются с экспериментальными данными. Author is thankful to A.P. Potylitsyn for discussion of the work. The research has been performed due to the partial support by the Ministry of education and science of the Russian Federation under project 3.2009.2014/K en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Релятивистская электроника Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation Області застосовності кінематичної та динамічної теорій параметричного рентгенівського випромінювання Области применимости кинематической и динамической теорий параметрического рентгеновского излучения Article published earlier |
| spellingShingle | Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation Shchagin, A.V. Релятивистская электроника |
| title | Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation |
| title_alt | Області застосовності кінематичної та динамічної теорій параметричного рентгенівського випромінювання Области применимости кинематической и динамической теорий параметрического рентгеновского излучения |
| title_full | Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation |
| title_fullStr | Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation |
| title_full_unstemmed | Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation |
| title_short | Areas of applicability of kinematic and dynamic theories of parametric X-ray radiation |
| title_sort | areas of applicability of kinematic and dynamic theories of parametric x-ray radiation |
| topic | Релятивистская электроника |
| topic_facet | Релятивистская электроника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112229 |
| work_keys_str_mv | AT shchaginav areasofapplicabilityofkinematicanddynamictheoriesofparametricxrayradiation AT shchaginav oblastízastosovnostíkínematičnoítadinamíčnoíteoríiparametričnogorentgenívsʹkogovipromínûvannâ AT shchaginav oblastiprimenimostikinematičeskoiidinamičeskoiteoriiparametričeskogorentgenovskogoizlučeniâ |