on the ratio of the relativistic electron pxr in the bragg...

ISSN 1027�4510, Journal of Surface Investigation. X�ray, Synchrotron and Neutron Techniques, 2010, Vol. 4, No. 2, pp. 303–314. © Pleiades Publishing, Ltd., 2010.Original Russian Text © S.V. Blazhevich, A.V. Noskov, 2010, published in Poverkhnost’. Rentgenovskie, Sinkhrotronnye i Neitronnye Issledovaniya, No. 4, pp. 40–51.

303

INTRODUCTION

When a fast charged particle crosses a single crystal,its Coulomb field is scattered on the system of parallelatomic crystal planes, generating parametric X�rayradiation (PXR) [1–3]. The theory of parametricX�ray radiation (PXR) of a relativistic particle in thecrystal predicts radiation not only in the Bragg scatter�ing direction, but also along the emitting particlevelocity (FPXR) [4–6]. This radiation results fromdynamical diffraction effects in PXR. There have beenattempts to experimentally study FPXR [7–11]. Onlyin the experiment in [11] were X�rays of relativisticelectrons from a thick absorbing single�crystal targetdetected under conditions of FPXR generation; how�ever, the sought reflection was very weak against thebackground of radiation generated by electrons onstructural elements of the experimental setup. Thus,the theoretical study of PXR properties in the direc�tion of the relativistic electron velocity and the searchfor conditions for more prominent experimentalobservations of this dynamic effect remain urgent.

The dynamic FPXR effect was theoreticallydescribed in detail in the symmetric case in [12–14].PXR and transition radiation (TR) in the general caseof asymmetric reflection and FPXR in the Bragggeometry were theoretically described in [15–17] and[18], respectively. In these papers, it was shown thatthe spectral–angular density of these emission mech�anisms depends strongly on reflection asymmetry andthe effects associated with this density were deter�mined.

In this paper, based on the two�wave approxima�tion of the dynamic diffraction theory [19], analyticalexpressions for PXR and FPXR amplitudes are derivedin the general case of asymmetric reflection, as well asexpressions describing the spectral–angular density of

radiations. Based on the derived expressions, the ratioof radiation yields is studied depending on reflectionasymmetry.

RADIATION AMPLITUDE

Let us consider radiation of a fast charged particlecrossing a single�crystal plate with constant velocity V(Fig. 1). In this case, we will consider equations for the

On the Ratio of the Relativistic Electron PXR in the Bragg Direction and in the Forward Direction in Laue Geometry

S. V. Blazhevicha and A. V. Noskovb

aBelgorod State University, ul. Pobedy 85, Belgorod, 308015 RussiabBelgorod University of Consumers’ Cooperatives, ul. Sadovaya 116a, Belgorod, 308023 Russia

e�mail: [email protected] August 10, 2009

Abstract—Parametric X�ray radiation and parametric X�ray radiation at a small angle with respect to thevelocity of a relativistic electron crossing a single crystal plate in Laue scattering geometry are studied basedon the dynamic X�ray diffraction theory. Analytical expressions for the spectral–angular density of radiationsare derived in the general case of asymmetric reflection. The ratio of the contributions of these radiationmechanisms is considered.

DOI: 10.1134/S1027451010020230

µ

kωV

V2g

θ

θB

ψ0

ψg

kg

θ'

ωVV2 + g

n x

L

δ

µ

Fig. 1. Emission geometry and the system of notations ofthe used quantities: θ and θ' are emission angles, θB is theBragg angle (the angle between the electron velocity V andatomic planes), δ is the angle between the surface andatomic planes under consideration, and k and kg are thewave vectors of incident and diffracted photons.

304

JOURNAL OF SURFACE INVESTIGATION. X�RAY, SYNCHROTRON AND NEUTRON TECHNIQUES Vol. 4 No. 2 2010

BLAZHEVICH, NOSKOV

Fourier transform of the electromagnetic field

. (1)

Since the relativistic particle field can be consid�ered as transverse, the incident E0(k, ω) and diffractedEg(k,ω) electromagnetic waves are defined by twoamplitudes with different transverse polarizations,

(2)

Polarization unit vectors , , , and are cho�

sen such that and are perpendicular to the vec�

tor , and the vectors and are perpendicular to

the vector . The vectors and lie in theplane of vectors and ( polarization), and vectors

and are perpendicular to it ( polarization);is the reciprocal lattice vector which defines the sys�

tem of reflecting atomic crystal planes.

The system of equations for the Fourier transform ofthe electromagnetic field in the two�wave approxima�tion in the dynamic diffraction theory is written as [20]

(3)

where and are the coefficients of the Fourierexpansion of the dielectric susceptibility of the crystalin reciprocal lattice vectors ,

(4)

Let us consider a crystal with central symmetry. In expression (4), the quantities and

are given by

(5)

where is the average dielectric susceptibil�ity, F(g) is the form factor of an atom containing Z elec�trons, is the structure factor of the unit cell con�taining atoms, and is the root�mean�squareamplitude of thermal vibrations of crystal atoms. TheX�ray frequency region , is considered.

( )ω = ω −∫ 3( , ) ( , )expdt d t i t iE k r E r kr

ω = ω + ω

ω = ω + ω

(1) (1) (2) (2)0 0 0 0 0

(1) (1) (2) (2)1 1

( , ) ( , ) ( , ) ,

( , ) ( , ) ( , ) .

k E E

k E Eg g g

E k k ee

E k e k e

(1)0e (2)

0e (1)1e (2)

1e(1)0e (2)

0e

k (1)1e (2)

1e

= +gk k g (2)0e (2)

1ek gk π

(1)0e (1)

1e σ

g

−

⎧ ω + χ − + ω χ⎪⎪= π ωθ δ ω −⎨⎪ω χ + ω + χ − =⎪⎩

2 2 ( ) 2 ( ) ( )0 0

2 ( )

2 ( ) ( ) 2 2 ( )0 0

( (1 ) )

8 ( ),

( (1 ) ) 0,

s s s

s

s s s

k E C E

ie VP

C E k E

g g

g g g

kV

χg −

χ g

g

( )

χ ω = χ ω

= χ ω + χ ω

∑

∑

( , ) ( )exp( )

' ''( ) ( ) exp( ).

i

i i

g

g

g g

g

r gr

gr

( )−

χ = χg g 'χ g ''χg

( )( ) ( )( )

/ / 2 20 0

2 20

1' ' ( ) ( ) exp ,2

1'' '' exp ,2

F g Z S N g u

g u

τ

τ

χ = χ −

χ = χ −

g

g

g

0 0 0' ''iχ = χ + χ

( )S g

0N uτ

'( 0χ <g 0' 0)χ <

The quantities and in system (3) aredefined as

, , ,

, , , (6)

where is the component of the virtualphoton momentum, perpendicular to the particlevelocity ( , where � 1 is the anglebetween the vectors and ); is the angle betweenthe electron velocity and the system of crystallo�graphic planes (Bragg angle), is the azimuthal angleof radiation, measured from the plane formed by vec�tors and , the reciprocal lattice vector is given by

, and ωB is the Bragg frequency. The

angle between the vector and the incident wave

vector is denoted by , and the angle between the

vector and the diffracted wave vector is

denoted by . System of equations (3) at the parame�ter and 2 describes the and �polarized fields.

Let us solve the dispersion relation following fromsystem (3) for X�ray waves in the crystal,

(7)

by standard methods of the dynamic theory [19].

Let us search for the projections of wave vectors and onto the X axis coinciding with the vector (Fig. 1) in the form

(8)

Let us use the known expression relating thedynamic components and for X�ray waves [19]

, (9)

where ; ;

; ; is the angle between thewave vector of the incident wave and the normal vec�tor to the plate surface, and is the angle betweenthe wave vector and the vector (Fig. 1). The vector

and magnitudes are given by

, . (10)

С( )s ( )sP

( ) ( ) ( )0 1

s s sC = e e С (1) 1= BС(2) cos2= θ

( ) ( )0 ( / )s sP = µe µ

(1) sinP = ϕ

(2) cosP = ϕ

/ 2V= − ωk Vµ

V /Vμ = ωθ θ

k V Bθ

ϕ

V g

B B/2 sing V= ω θ

2VωV

k θ

2Vω

+V g gk

'θ1s = σ π

2 2 2 2 4 ( )20 0( (1 ) )( (1 ) ) 0sk k C

−

ω + χ − ω + χ − − ω χ χ =g g g

k

gk n

0 00

0 0

0

cos ,2cos cos

cos .2cos cos

x

x

k

k

ωχ λ= ω ψ + +

ψ ψ

λωχ= ω ψ + +

ψ ψ

gg g

g g

0λ λg

002

γωβλ = + λ

γ

gg

00

1γ⎛ ⎞β = α − χ −⎜ ⎟γ⎝ ⎠

g 2 22

1 ( )k kα = −

ω

g

0 0cosγ = ψ cosγ = ψg g 0ψ

kn ψg

gk nk gk

0 01k = ω + χ + λ 01k = ω + χ + λg g


ON THE RATIO OF THE RELATIVISTIC ELECTRON PXR IN THE BRAGG DIRECTION 305

Substituting (8) into (7) and taking into account

(9), , and , we obtain theexpressions for dynamic components

(11)

Since and , it can be shown that

(Fig. 1); therefore, in what follows, we denotethe angle as .

We write the solution to the system of equations (3)for incident and diffracted fields in the crystal as

(12a)

(12b)

respectively, where ;

; γ = is the particle Lorentz

factor; and and , are the free incidentand diffracted fields in the crystal, respectively.

For the field in vacuum in front of the crystal, thesolution to system (3) is given by

(13)

where the relation is used.

|| 0sink ≈ ω ψ | | sink ≈ ω ψg g

2

2

(1,2) 2 ( )

0

(1,2) 2 ( )00

0

4 ,4

4 .4

s

s

C

C

−

−

⎛ ⎞γωλ = β ± β + χ χ⎜ ⎟γ⎝ ⎠⎛ ⎞γγλ = ω −β ± β + χ χ⎜ ⎟γ γ⎝ ⎠

gg g g

gg g

g

0λ ω� λ ωg �

'θ ≈ θ

'θ θ

( )( )

γ−ω β − ω λ

γπ θ=

γω λ − λ λ − λγ

×δ λ − λ + δ λ − λ + δ λ − λ

cr

(1) (2)

202 ( )

( ) 00

(1) (2)0 0 0 0

0

( ) (1) ( ) (2)0 0 0 0 0 0 0 0

28

4

*( ) ( ) ( ),

ss

s s

ieV PE

E E

g

g

( )( )

( )

ω χπ θ= −

ω γλ − λ λ − λ

γ

×δ λ − λ + δ λ − λ + δ λ − λ

cr

(1) (2)

2 ( )2 ( )( )

2(1) (2)0

2

( ) (1) ( ) (2)

8

4

* ( ) ( ),

sss

s s

CieV PE

E E

gg

g g g gg

g g g g g g g g

2 20

0*2

−⎛ ⎞γ + θ − χλ = ω⎜ ⎟⎝ ⎠

γωβλ = + λ

γ0

0

* *2

gg

21 V−

(1)( )0

sE(2)( )

0sE

(1)( )sEg

( )

( )

π θ= δ λ − λω −χ − λ

ωπ θ= δ λ − λ

⎛ ⎞ω γ γ γ−χ − λ + β⎜ ⎟γ ωγ γ⎝ ⎠

vac I2 ( )

( )0 0 0

0 0

2 ( )

0 0 00

8 1 *2

8 1 * .2

ss

s

ieV PE

ieV Pg g

gg g g

( ) ( )0 00

* *γδ λ − λ = δ λ − λ

γ

gg g

The incident and diffracted fields in vacuumbehind the crystal are given by

(14)

where е and are the fields of coherentradiations along the electron velocity and near theBragg direction, respectively.

From the second equation of system (3), theexpression relating the diffracted and incident fields inthe crystal follows,

. (15)

To determine the incident and diffracted

fields, we use ordinary boundary conditions onthe input and output surfaces of the crystal plate,respectively,

(16a)

(16b)

and obtain the expression for radiation fields,

(17a)

( )( )

π θ= δ λ − λ

λω −χ −ω

ωχ+ δ λ +

ωχ= δ λ +

vac II

Rad

vac Rad

2 ( )( )0 0 0

00

( ) 00 0

( ) ( ) 0

8 1 *( )2

,2

,2

ss

s

s s

ieV PE

E

E Eg g g

Rad( )0

sE Rad( )sEg

cr cr( ) ( )0 2 ( )

2s ss

g

E EC

ωλ=ω χ

gg

Rad( )0

sERad( )sEg

vac I cr cr

cr vac II

( ) ( ) ( )0 0 0 0 0

( ) ( )0 00 0 0 0

0 0

, 0,

exp exp ,

s s s

s s

E d E d E d

E i L d E i L d

λ = λ λ =

⎛ ⎞ ⎛ ⎞λ λλ = λ⎜ ⎟ ⎜ ⎟γ γ⎝ ⎠ ⎝ ⎠

∫ ∫ ∫

∫ ∫

g

λ = λ λ =

⎛ ⎞ ⎛ ⎞λ λλ = λ⎜ ⎟ ⎜ ⎟γ γ⎝ ⎠ ⎝ ⎠

∫ ∫ ∫

∫ ∫

vac I cr cr

cr vac

( ) ( ) ( )0 0

( ) ( )

, 0,

exp exp .

s s s

s s

E d E d E d

E i L d E i L d

g g g g

g gg g g g

g g

( )( )

( )

( )

⎡ ⎤ωχω χ + λ⎢ ⎥γπ θ ⎣ ⎦=γω ω λ − λγ

⎡⎛ ⎞⎜ ⎟⎢ ω ω× +⎜ ⎟⎢ γ−χ ω − λ⎜ ⎟⎢ λ − λ⎜ ⎟γ⎢⎣⎝ ⎠

⎛ ⎛ ⎞⎞ ⎛λ − λ ω× − − −⎜ ⎜ ⎟⎟ ⎜⎜⎜ ⎜ ⎟⎟γ −χ ω − λ⎝⎝ ⎝ ⎠⎠

ω ⎞ λ − λ⎟γ+ − −λ − λ ⎟ γγ ⎠

Rad

2 ( ) 02 ( )

( )

(1) (2)0

(2)00 0

(2)

0 0

(1)

(1)0

*exp28

2

*2 *2

*1 exp

*2

*1 exp*2

sgs

s g

LC iieV PE

i L

i

g

g

g gg

g gg

g g

g

g g

g gg

⎤⎥⎛ ⎛ ⎞⎞⎥⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟ ⎥⎝ ⎝ ⎠⎠⎥⎦

,Lg

306


BLAZHEVICH, NOSKOV

(17b)

The bracketed terms in expressions (17a) and (17b)correspond to two different branches of X�ray wavesexcited in the crystal.

For further analysis of radiation, we write thedynamic components in (11) as follows:

(18a)

(18b)

where

(19)

Since the inequality is valid

in the X�ray frequency range, is a rapid func�

tion of frequency ; therefore, for further analysis ofFPXR and TR spectra, it is very convenient to con�

sider as a spectral variable characterizing thefrequency . An important parameter in expression(19) is the parameter , which defines the degree ofasymmetry of field reflection with respect to the platesurface. We note that wave vectors of incident and dif�fracted photons in the symmetric case form equalangles with the surface plate (Fig. 2); in the case ofasymmetric reflection, the angles are unequal. In thesymmetric case, and ; in the asymmetriccase, and .

Let us write the asymmetry parameter in the form

, (20)

( )( )

⎡ ⎤ωχ + λ⎢ ⎥ ⎛ ⎞γ γ⎡⎛ ⎞π θ ω ω⎣ ⎦= × −ω β − ω λ +⎜ ⎟⎜ ⎟⎢ ⎜ ⎟γω γ⎣⎝ ⎠ −ωχ − λ λ − λ⎝ ⎠ω λ − λγ

⎛ ⎛ ⎞ ⎞ γ⎛ ⎞λ − λ ω ω× − − −ω β − ω λ × +⎜ ⎜ ⎟ ⎟ ⎜ ⎟⎜ ⎜ ⎟ ⎟γ γ⎝ ⎠ −ωχ − λ λ − λ⎝ ⎝ ⎠ ⎠

Rad

02 ( )

( ) 2 (2)0 0 (2)(1) (2) 0 0 0 0 00 0

0

(2)2 (1)0 0

0 (1)0 0 0 0 0 0

*exp28 2

* *2 2( )2

*exp 1 2

* *2 2( )

gss g

LiieV PE

i L

g

g

g ⎛ ⎞⎛ ⎛ ⎞ ⎞⎤λ − λ −⎜ ⎟⎜ ⎜ ⎟ ⎟⎥⎜ ⎟⎜ ⎜ ⎟ ⎟γ⎝ ⎠⎝ ⎝ ⎠ ⎠⎦

(1)0 0

0

*exp 1 .i L

( )ωχ ⎛ ⎞⎛ ⎞ρ − ε − ε − ελ = −ξ + ± ξ + ε − ρ ξ + κ ε − ρ + κ ε⎜ ⎟⎜ ⎟ε ⎝ ⎠⎝ ⎠

2 2 2

( )( ) 2

(1,2) ( ) ( ) ( ) ( ) ( ) ( ) ( )0

' (1 ) (1 ) (1 )2 ,2 2 2 4

ss

s s s s s s sC i i

g

( )ωχ ⎞⎛ ⎛ ⎞ρ − ε − ε − ελ = ξ − ± ξ + ε − ρ ξ + κ ε − ρ + κ ε ⎟⎜ ⎟⎜

⎝ ⎠⎝ ⎠

2 2 2

( )( ) 2

(1,2) ( ) ( ) ( ) ( ) ( ) ( ) ( )' (1 ) (1 ) (1 )2 ,2 2 2 4

ss

s s s s s s sg

C i ig

( )

BB

B

tg

( ) ( )( )

2( )

( ) 2 ( )

( ) ( )( ) ( ) ( )0

( )0 0 0 0

1( ) ,2

(1 cos c )2sin( ) 1 ,' '2

' ''cos '', , , .

cos ' ' ''

s ss

s

s s

s ss s s

s

C V C

C C

C

− εξ ω = η ω +ν

⎛ ⎞ω − θ ϕ θθαη ω = = −⎜ ⎟ω⎝ ⎠χ χ

γ ψ χ χχε = = ν = ρ = κ =γ ψ χ χ χ

g g

g g g g

g

B/2 2 ( )'2sin 1sV Cθ χg �

( )( )sη ω

ω

( )( )sη ω

ω

ε

1ε = /2δ = π

1ε ≠ /2δ ≠ π

δ + θε =

δ − θ

B

B

sin( )sin( )

ε > 1FPXR

k

θB

δ PXR

kg

δ�θB

δ�θBδ�θB

kgkg

PXR

k k

FPXR FPXR

ε = 1 ε < 1

PXR

Fig. 2. Symmetric (ε = 1) and asymmetric (ε > 1, ε < 1) reflections of the particle field.



where θB is the angle between the electron velocity andthe system of crystallographic planes. We note that the

angle of electron incidence onto the plate sur�face increases with decreasing parameter and viceversa (Fig. 2).

SPECTRAL�ANGULAR RADIATION DENSITY IN THE BRAGG SCATTERING DIRECTION

Let us write expression (17a) for the radiation fieldamplitude in the Bragg direction in the form of two terms,

, (21a)

(21b)

(21c)

Expression (21b) describes the PXR field ampli�

tude, and (21c) describes the amplitude of the DTR

field induced due to diffraction of transition radia�

tion generated on the input surface at the system of

atomic planes of the crystal.

Substituting expressions (18) into (21b) and (21c),

we write them in the form

(22a)

(22b)

Bδ − θ

ε

= +

RadPXR DTR

( ) ( ) ( )s s sE E Eg

−

−

−

⎛ ⎛ ⎞⎞λ − λ− −⎜ ⎜ ⎟⎟⎜ ⎟γ⎡⎛ ⎞ω χ γ ⎜ ⎟π θ ⎝ ⎠= − × β + β + χ χ⎜ ⎟⎢ ⎜ ⎟γω γγ λ λ − λ⎣⎝ ⎠⎜ ⎟β + χ χ

γ γ ⎜ ⎟⎝ ⎠

⎛ ⎛ ⎞λ − λ− −⎜ ⎟⎜ ⎟γ⎛ ⎞γ ⎝ ⎠− β − β + χ χ⎜ ⎟γ λ − λ⎝ ⎠

PXR

2

2

2

(2)

2 ( )2 ( )( ) 2 ( )

(2)2 ( ) 00 0

0

(1)

2 ( )

(1)0

*1 exp

8 1 4* *

8 4

*1 exp

4*

sss s

s

s

i LCieV PE C

C

i L

C

g g

gg gg g

g g gg g

g

g g

ggg g

g g( )

⎞⎤⎜ ⎟⎥

⎡ ⎤⎜ ⎟⎥ ωχ× + λ⎢ ⎥⎜ ⎟⎥ γ⎣ ⎦⎜ ⎟⎥⎜ ⎟⎥⎝ ⎠⎦

0 *exp ,2

gg

Li

( )

−

χπ θ=γω γ β + χ χ

γ γ

⎡ ⎛ ⎞⎛ ⎞ λ − λω ω× + −⎜ ⎟⎢⎜ ⎟⎜ ⎟ ⎜ ⎟γ−ωχ − λ λ ⎢⎝ ⎠⎣ ⎝ ⎠⎛ ⎞⎤ ⎡ ⎤λ − λ ωχ− − + λ⎜ ⎟⎥ ⎢ ⎥⎜ ⎟γ γ⎣ ⎦⎥⎝ ⎠⎦

DTR2

( )2 ( )( )

2 ( )0

0

(1)

0 0 0

(2)0

8

4

*exp

* *2 2

**exp exp .

2

sss

s

gg

CieV PE

C

i L

Li L i

g

gg g

g

g g

g

g g

g

−

⎛ ⎡ ⎛ ⎞ ⎤ξ + ξ + ε⎜ ⎜ ⎟− − σ + − ρ Δ⎢ ⎥⎜ ⎟ε⎜ ⎢ ⎥ξ + ξ + επ θ ⎣ ⎝ ⎠ ⎦= ⎜

ω θ + γ − χ ⎜ ξ + ε ξ + ξ + εσ + − ρ Δ⎜ ε⎝⎛ ⎞ξ − ξ + ε⎜ ⎟− − σ + −⎜ ⎟εξ − ξ + ε ⎝ ⎠−

ξ + ε

PXR

2

2

2 2

2

2

2

( ) ( )( ) ( ) ( ) ( ) (2)

( ) ( )2 ( )( )

2 2( ) ( ) ( )

0 ( ) ( ) (2)

( ) ( )( ) ( )

( ) ( )

( )

1 exp4

1 exp

s ss s s s

s sss

s s ss s

s ss s

s s

s

ib bieV PE

i

ib b

( )⎡ ⎤⎞

⎟ρ Δ⎢ ⎥⎟⎢ ⎥ ⎡ ⎤ωχ⎣ ⎦ + λ⎟ ⎢ ⎥γ⎣ ⎦⎟ξ − ξ + εσ + − ρ Δ ⎟ε ⎠

2

( ) ( ) (1)

0

( ) ( )( ) ( ) (1)

*exp ,2

s s

gs s gs s

Li

i

( )

− −

⎛ ⎡ ⎛ ⎞ ⎤⎛ ⎞ ξ − ξ + επ ε ⎜ ⎜ ⎟= θ − − σ + − ρ Δ⎢ ⎥⎜ ⎟ ⎜ ⎜ ⎟ω εθ + γ θ + γ − χ⎝ ⎠ ⎢ ⎥ξ + ε⎝ ⎣ ⎝ ⎠ ⎦×

⎡ ⎛ ⎞ ⎤⎞ ⎡ ⎤ωχξ + ξ + ε⎜ ⎟ ⎟− − σ + − ρ Δ + λ⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎟ε γ⎢ ⎥ ⎣ ⎦⎣ ⎝ ⎠ ⎦⎠

DTR

2

2

2

( ) ( )2( ) ( ) ( ) ( ) ( ) ( ) (1)

2 2 2 2( )

0

( ) ( )( ) ( ) ( ) ( ) (2) 0

4 1 1 exp

*exp exp ,2

s ss s s s s s

s

s ss s s s

gg

ieVE P ib b

Lib b i

308


BLAZHEVICH, NOSKOV

where

(23)

The parameter can be written as

, (24)

from which it follows that is equal to half the elec�tron path in the plate, expressed in terms of the extinc�

tion length .

The PXR yield is formed for the most part by onlyone branch, i.e., the first one in expression (18a), cor�responding to the second term in (22a). As is easy tocheck, the real denominator part vanishes only in thisterm. The solution to the corresponding equation,

(25)

defines the frequency in the vicinity of which thespectrum of PXR photons emitted at a fixed observa�tion angle is concentrated.

Substituting (21a), (22a), and (22b) into the well�known [20] expression for the spectral�angular densityof X�ray radiation,

, (26)

we find the expressions describing the contributions ofPXR and DTR mechanisms to the spectral�angularradiation density and the term resulting from interfer�ence of these mechanisms,

(27а)

(27b)

(28а)

(28b)

(29а)

(29b)

Let us consider the case of a thin target

, where the absorbance can beneglected. To exhibit the dynamic effects, we will con�sider a crystal of such a thickness that the electron path

( )

2 2

2 2

( ) ( )(2)

( ) ( )

( ) ( )(1)

( ) ( )

2( ) 2 2

0 ( )( ) 20 0

( )

( )

0

1 1 ,2 2

1 1 ,2 2

1 1 1' 1 ,' ' '

'.

2

s s

s s

s s

s s

sss

s

s

C

C Lb

−

ξε + − ε κΔ = + +ε ε ξ + ε ξ + ε

ξε + − ε κΔ = − −ε ε ξ + ε ξ + ε

⎛ ⎞θ⎜ ⎟σ = θ + γ − χ ≡ + +

⎜ ⎟ν ⎜ ⎟χ χ γ χ⎝ ⎠

ωχ=

γ

g

g

( )sb

( ) ext

( )( )

12sin

ss

B

LbL

=δ − θ

( )sb

=

ωχext( )

( )

1

'

s

sL

Cg

ξ − ξ + εσ + =

ε

2( ) ( )( ) 0

s ss

*ω

Rad22 22 6 ( )

1

(2 ) s

s

d N Ed d

−

=

ω = ω πω Ω

∑

−

θω =

′ω Ω π θ + γ − χPXR

PXR

( )22 2 2

( ) ( )2 2 2 2

0

,4 ( )

s

s sd N e P Rd d

( ) ( ) ⎛ ⎛ ⎞⎞ξ − ξ + ε+ − ρ Δ − − ρ Δ σ +⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎛ ⎞ εξ ⎝ ⎝ ⎠⎠= −⎜ ⎟⎜ ⎟ ⎛ ⎞ξ + ε⎝ ⎠ ξ − ξ + εσ + + ρ Δ⎜ ⎟⎜ ⎟ε⎝ ⎠

ПРИ2 2

2( ) ( ) (1) ( ) ( ) (1) ( ) ( )

2

( )22 2

( ) ( ) (1)

1 exp 2 2exp cos

1 ,

s s s s s s

s

s s

b b b

R

− −

ω = θω Ω π

⎛ ⎞× −⎜ ⎟′θ + γ θ + γ − χ⎝ ⎠

DTR

DTR

( )22 2

( ) 22

2( )

2 2 2 20

4

1 1 ,

s

s

s

d N e Pd d

R

( )

( )

ε + ε= − ρεξ + ε

⎡ ⎛ ⎞ ⎤ξ + ε ⎛ ⎞− ε ξ + εκ⎜ ⎟⎢ ⎥× + ρ⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥ε⎜ ⎟ ε ξ + ε⎝ ⎠⎣ ⎝ ⎠ ⎦

DTR

sh

2( ) ( ) ( )

2

2( ) ( )

2 ( ) 2 ( ) ( )

2

4 1exp

(1 ) 2sin ,2

s s s

s ss s s

R b

b b

−

− −

⎛ω = θ ⎜ω Ω π θ + γ⎝

⎞− ⎟⎟θ + γ − χ θ + γ − χ⎠

INT

INT

( )22 2

( ) 22 2 2

( )

2 2 2 20 0

14

1 1 ,' '

s

s

s

d N e Pd d

R

( )(ε= − ξ − ξ + εξ + ε

⎡ ⎛ ⎞ ⎤ξ − ξ + ε⎜ ⎟− − σ + − ρ Δ⎢ ⎥⎜ ⎟ε⎢ ⎥⎣ ⎝ ⎠ ⎦×ξ − ξ + εσ + − ρ Δ

ε⎛ ⎡ ⎛ ⎞ ⎤ξ − ξ + ε⎜ ⎜ ⎟× σ + − ρ Δ⎢ ⎥⎜ ⎜ ⎟ε⎢ ⎥⎝ ⎣ ⎝ ⎠ ⎦

ξ + ξ− σ +

INT

2

2

2

2

2

( ) ( ) ( )

( )

( ) ( )( ) ( ) ( ) ( ) (1)

( ) ( )( ) ( ) (1)

( ) ( )( ) ( ) ( ) ( ) (1)

( ) (( ) ( )

2 Re

1 exp

exp

exp

s s s

s

s ss s s s

s ss s

s ss s s s

ss s

R

ib b

i

ib b

ib⎡ ⎛ ⎞ ⎤⎞⎞+ ε⎜ ⎟ ⎟⎟− ρ Δ⎢ ⎥

⎜ ⎟ ⎟⎟ε⎢ ⎥⎣ ⎝ ⎠ ⎦⎠⎠

2)( ) ( ) (2) .

ss sb

( )( ) ( ) 1s sb ρ �( )s

ρ



length in the plate would be much

longer than the extinction length of

X�ray waves in the crystal, i.e., . In this case,the expressions describing the spectra of PXR, DTRradiations, and their interference take the form

(30)

(31)

(32)

The expressions derived make it possible to studyall PXR spectral�angular characteristics.

SPECTRAL�ANGULAR RADIATION DENSITY ALONG THE EMITTING PARTICLE VELOCITY

From the expression for the radiation field ampli�tude along the emitting particle velocity, we separatethe two terms

, (33a)

(33b)

(33c)

Expressions (33b) and (33c) describe the FPXRand TR field amplitudes, respectively.

The appearance of the FPXR reflection requiressatisfying at least one of the following equalities

, , (34)

i. e., the denominator real part of at least one of brack�eted terms in expression (33b) should be zero.

Substituting (18a) into (33b) and (33c), we writethe expressions for the amplitude of PXR along theelectron velocity and TR in the form

(35а)

( )B/ sinL δ − θ

=

ωχext( )

( )

1

'

s

sL

Cg

( ) 1sb �

⎛ ⎞ξ⎜ ⎟= −⎜ ⎟ξ + ε⎝ ⎠

⎛ ⎛ ⎞⎞ξ − ξ + ε⎜ ⎜ ⎟⎟σ +⎜ ⎜ ⎟⎟ε⎝ ⎝ ⎠⎠×⎛ ⎞ξ − ξ + ε⎜ ⎟σ +⎜ ⎟ε⎝ ⎠

PXR 2

2

2

2( )

( )

( )

( ) ( )( )2 ( )

2( ) ( )

( )

4 1

sin2

,

ss

s

s sss

s ss

R

b

⎛ ⎞ξ + εε ⎜ ⎟=⎜ ⎟εξ + ε ⎝ ⎠

DTR

2

2

( )2( ) 2 ( )

( )

4 sin ,s

s s

sR b

⎛ ⎞ξ − ξ + ε ξ + ε⎜ ⎟= − ε⎜ ⎟εξ + ε ⎝ ⎠

⎛ ⎞⎛ ⎛ ⎞⎞ξ ξ + ε⎜ ⎟σ + −⎜ ⎜ ⎟⎟ ⎜ ⎟ε ε⎝ ⎝ ⎠⎠ ⎝ ⎠×ξ − ξ + εσ +

ε

INT

2 2

2

2

2

( ) ( ) ( ) ( )( )

( )

( ) ( ) ( )( ) ( )

( ) ( )( )

4 sin

sin sin

.

s s s ss

s

s s ss s

s ss

bR

bb

= +

RadFPXR TR

( ) ( ) ( )0

s s sE E E

( )

−

−

ω χ χπ θ=γω λβ + χ χ γ

⎛ ⎛ ⎞ ⎛ ⎞⎞λ − λ λ − λ− −⎜ ⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟γγ⎜ ⎟⎝ ⎠ ⎝ ⎠× −⎜ ⎟λ − λ λ − λ⎜ ⎟⎜ ⎟⎝ ⎠

⎡ ⎤ωχ× + λ⎢ ⎥γ⎣ ⎦

FPXR

2

2

2 ( )2 ( )( )

2 ( ) 0

0

(2) (1)0 0 0 0

00

(2) (1)0 0 0 0

00

0

4 1*

2 4

* *1 exp 1 exp

* *

*exp ,2

sss

s

CieV PE

C

i i

Li

g g

gg g

−

−

⎛ ⎞π θ ω ω= −⎜ ⎟⎜ ⎟ω ωχ + λ λ⎝ ⎠⎡⎛ ⎞⎜ ⎟⎢ ⎛ ⎛ ⎞⎞λ − λβ⎜ ⎟⎢× − −⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟γ γ⎜ ⎟⎢ ⎝ ⎝ ⎠⎠β + χ χ⎜ ⎟⎢ γ⎣⎝ ⎠⎛ ⎞ ⎤⎜ ⎟ ⎥⎛ ⎛ ⎞⎞λ − λβ⎜ ⎟ ⎥+ + −⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟γ γ⎜ ⎟ ⎥⎝ ⎝ ⎠⎠β + χ χ⎜ ⎟ ⎥γ⎝ ⎠ ⎦

ωχ× +

TR

2

2

2 ( )( )

0 0 0

(2)0 0

2 ( ) 0

0

(1)0 0

2 ( ) 0

0

0

4* *2 2

*1 1 exp

4

*1 1 exp

4

exp2

ss

s

s

ieV PE

i L

C

i L

C

i

gg g

gg g

( )⎡ ⎤λ⎢ ⎥γ⎣ ⎦

00

* .L

( )(1)0 0*Re 0λ − λ = ( )(2)

0 0*Re 0λ − λ =

−

⎛ ⎡ ⎛ ⎞ ⎤ξ + ξ + ε⎜ ⎜ ⎟− − σ + − ρ Δ⎢ ⎥⎜ ⎟ε⎜ ⎢ ⎥π θ ⎣ ⎝ ⎠ ⎦= × −⎜

ω γ + θ − χ ⎜ξ + ε ξ + ξ + εσ + − ρ Δ⎜ ε⎝⎡ ⎛ ⎞ ⎤ξ − ξ + ε⎜ ⎟− − σ + − ρ Δ⎢ ⎥

⎜ ⎟ε⎢ ⎥⎣ ⎝ ⎠ ⎦−ξσ +

FPXR

2

2 2

2

( ) ( )( ) ( ) ( ) ( ) (2)

2 ( )( )

2 2( ) ( ) ( )

0 ( ) ( ) (2)

( ) ( )( ) ( ) ( ) ( ) (1)

(( )

1 exp4 1

1 exp

s ss s s s

ss

s s ss s

s ss s s s

ss

ib bieV PE

i

ib b−

⎞⎟⎟ ⎡ ⎛ ⎞ ⎤γ + θ× ω⎟ ⎜ ⎟⎢ ⎥γ⎣ ⎝ ⎠ ⎦⎟− ξ + ε − ρ Δ ⎟ε ⎠

2

2 2

) ( ) 0( ) (1)

exp ,2s

s

Li

i

310


BLAZHEVICH, NOSKOV

(35b)

As for PXR, the FPXR yield is formed for the mostpart only due to one of the branches, which corre�sponds to the second term (35a), since the denomina�tor real part vanishes only in this term,

. (36)

Substituting (33a), (35a), and (35b) into (26), weobtain expressions describing the contributions of

FPXR and TR mechanisms to the spectral�angularradiation density and the term resulting from inter�ference of these mechanisms,

, (37а)

(37b)

(38а)

(38b)

(39a)

− −

⎛⎛ ⎞⎛ ⎡ ⎛ ⎞⎛ ⎞ ξ ξ + ξ + επ θ ⎜⎜ ⎟⎜ ⎜ ⎟= − − − − σ +⎢⎜ ⎟⎜⎜ ⎟⎜ ⎜ ⎟ω εγ + θ γ + θ − χ⎝ ⎠ ⎢ξ + ε⎝⎝ ⎠⎝ ⎣ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎤⎞ ξ ξ − ξ + ε⎜ ⎟ ⎜ ⎟− ρ Δ + + − − σ + − ρ⎟⎥⎟ ⎜ ⎟ ⎜ ⎟ε⎥ ξ + ε⎦⎠ ⎝ ⎠ ⎝ ⎠

TR

2

2

2

2

( ) ( ) ( )2 ( )( ) ( ) ( )

2 2 2 2( )

0

( ) ( ) ( )( ) ( ) (2) ( ) ( ) ( ) (

( )

4 1 1 1 1 exp

1 1 exp

s s sss s s

s

s s ss s s s s

s

ieV PE ib

b ib b−⎛ ⎡ ⎤⎞⎞ ⎡ ⎛ ⎞ ⎤γ + θ⎜ ⎟⎟Δ ω⎢ ⎥ ⎜ ⎟⎢ ⎥⎜ ⎟⎟ γ⎣ ⎝ ⎠ ⎦⎢ ⎥⎝ ⎣ ⎦⎠⎠

2 2) (1)

0

exp .2

s Li

ξ − ξ + εσ + =

ε

2( ) ( )( ) 0

s ss

−

θω =

ω Ω π θ + γ − χ

FPXRFPXR

22 ( ) 2 2( ) ( )

2 2 2 20

4 '( )

ss sd N e P R

d d

( ) ( )⎛ ⎛ ⎞⎞ξ − ξ + ε⎜ ⎜ ⎟⎟+ − ρ Δ − − ρ Δ σ +⎜ ⎜ ⎟⎟ε⎝ ⎝ ⎠⎠=

⎛ ⎞ξ + ε ξ − ξ + ε⎜ ⎟σ + + ρ Δ⎜ ⎟ε⎝ ⎠

FPXR

2

22

2 2

( ) ( )( ) ( ) (1) ( ) ( ) (1) ( ) ( )

( )2( )

( ) ( )( ) ( ) (1)

1 exp 2 2exp cos1 ,

s ss s s s s s

s

ss s

s s

b b b

R

−

−

⎛ ⎞ω = θ × −⎜ ⎟⎜ ⎟ω Ω π θ + γ θ + γ − χ⎝ ⎠

TRTR

2

22 ( ) 2

( ) 2 ( )2 2 2 2 2

0

1 1 ,4 '

ss sd N e P R

d d

( )(

( ) ( )(

( )

⎛ ⎞ξ⎜ ⎟= − + − ρ Δ⎜ ⎟ξ + ε⎝ ⎠

⎛ ⎛ ⎞⎞⎞ ⎛ ⎞ξ + ξ + ε ξ⎜ ⎜ ⎟⎟⎟ ⎜ ⎟− − ρ Δ σ + + + + − ρ Δ⎜ ⎜ ⎟⎟⎟ ⎜ ⎟ε ξ + ε⎝ ⎝ ⎠⎠⎠ ⎝ ⎠

⎛ ξ − ξ + ε⎜− − ρ Δ σ +⎜ ε⎝

TR 2

2

2

2

2( )

( ) ( ) ( ) (2)

( )

2( ) ( ) ( )

( ) ( ) (2) ( ) ( ) ( ) ( ) (1)

( )

( ) ( )( ) ( ) (1) ( ) ( )

1 1 exp 2

2exp cos 1 1 exp 2

2exp cos

ss s s

s

s s ss s s s s s

s

s ss s s s

R b

b b b

b b⎛ ⎞⎞⎞⎜ ⎟⎟⎟⎜ ⎟⎟⎟⎝ ⎠⎠⎠

( ) ( )

( )

⎛ ⎛ ⎞⎞⎛ ⎛ ⎞ξ + ε ξ − ξ + εε ε + ⎜ ⎜ ⎟⎟+ + − ρ − − ρ Δ × σ +⎜ ⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎜ ⎟⎟ε ε εξ + ε⎝ ⎝ ⎠ ⎝ ⎝ ⎠⎠

⎛ ⎛ ⎞⎞⎞ξ − ξ + ε⎜ ⎜ ⎟⎟⎟− − ρ Δ σ +⎜ ⎜ ⎟⎟⎟ε⎝ ⎝ ⎠⎠⎠

2

2

2

( ) 2 ( ) ( )( ) ( ) ( ) ( ) (2) ( ) ( )

( )

( ) ( )( ) ( ) (1) ( ) ( )

22 11 exp cos exp cos

exp cos ,

s s ss s s s s s

s

s ss s s s

bb b b

b b

−

− −

⎞⎛ω = θ − ⎟⎜ ⎟ω Ω π θ + γ⎝ θ + γ − χ θ + γ − χ⎠

INTINT

22 ( ) 2( ) 2 ( )

2 2 2 2 2 2 20 0

1 1 1 ,4 ' '

ss sd N e P R

d d



(39b)

In the case of a thin nonabsorbing target, expres�sions (37b), (38b), and (39b) describing the spectratake the form

(40)

(41)

(42)

RATIO OF PXR AND FPXR YIELDS

Since the PXR and FPXR spectra described by the

expressions and depend on the parameter, it is of interest to consider the effect of asymmetry

on the ratio of yields of these radiations derived fromexpressions (30) and (40),

. (43)

It follows from (43) that as the asymmetry parame�ter decreases, i.e., the angle of particle inci�dence on the target surface increases (Fig. 2), theintensity of PXR at small angles to the particle velocity

⎛ ⎡ ⎛ ⎞ ⎤ξ − ξ + ε⎜ ⎜ ⎟− − σ + − ρ Δ⎢ ⎥⎜ ⎟ε⎜ ⎢ ⎥− ⎣ ⎝ ⎠ ⎦= ⎜

⎜ξ + ε ξ − ξ + εσ + − ρ Δ⎜ ε⎝⎛⎛ ⎞⎛ ⎡ ⎛ ⎞ ⎤⎞ξ ξ + ξ + ε⎜ ⎟⎜ ⎜ ⎟ ⎟× − − σ + − ρ Δ⎢ ⎥⎜ ⎟⎜ ⎜ ⎟ ⎟ε⎢ ⎥ξ + ε⎝ ⎠⎝ ⎣ ⎝ ⎠ ⎦⎠

INT

2

2 2

2

2

( ) ( )( ) ( ) ( ) ( ) (1)

( )

( ) ( ) ( )( ) ( ) (1)

( ) ( ) ( )( ) ( ) ( ) ( ) (2)

( )

1 exp2 Re

1 1 exp

s ss s s s

s

s s ss s

s s ss s s s

s

ib b

R

i

ib b⎜⎜⎝⎛ ⎞⎛ ⎡ ⎛ ⎞ ⎤⎞⎞⎞ξ ξ − ξ + ε⎜ ⎟⎜ ⎜ ⎟ ⎟⎟⎟+ + − σ + − ρ Δ⎢ ⎥⎜ ⎟⎜ ⎜ ⎟ ⎟⎟⎟ε⎢ ⎥ξ + ε⎝ ⎠⎝ ⎣ ⎝ ⎠ ⎦⎠⎠⎠

2

2

( ) ( ) ( )( ) ( ) ( ) ( ) (1)

( )1 1 exp .

s s ss s s s

sib b

⎛ ⎛ ⎞⎞ξ − ξ + ε⎜ ⎜ ⎟⎟σ +⎜ ⎜ ⎟⎟ε⎝ ⎝ ⎠⎠=⎛ ⎞ξ + ε ξ − ξ + ε⎜ ⎟σ +⎜ ⎟ε⎝ ⎠

FPXR

2

22

( ) ( )( )2 ( )

( )2( )

( ) ( )( )

sin24 ,

s sss

s

ss s

s

b

R

⎛ ⎞ ⎛ ⎛ ⎞⎞ξ ξ + ξ + ε⎜ ⎟ ⎜ ⎜ ⎟⎟= − × σ +⎜ ⎟ ⎜ ⎜ ⎟⎟εξ + ε⎝ ⎠ ⎝ ⎝ ⎠⎠

⎛ ⎞ ⎛ ⎛ ⎞⎞ ⎛ ⎛ ⎞ξ ξ − ξ + ε ξ + εε⎜ ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎜ ⎟+ + σ + +⎜ ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎜ ⎟ε εξ + εξ + ε⎝ ⎠ ⎝ ⎝ ⎠⎠ ⎝ ⎝ ⎠

⎛ ξ− σ +⎜ ε⎝

TR

2

2

2 2

22

2( ) ( ) ( )( )

( ) 2 ( )

( )

2( ) ( ) ( ) ( ) ( )( )

2 ( ) 2

( )( )

( )( ) ( )

4 1 sin2

44 1 sin cos2

cos

s s sss s

s

s s s s sss

ss

ss s

bR

bb

b⎛ ⎛ ⎞⎞⎞⎛ ⎞⎞ ξ + ε⎜ ⎜ ⎟⎟⎟⎜ ⎟⎟ ⎜ ⎜ ⎟⎟⎟ε⎝ ⎠⎠ ⎝ ⎝ ⎠⎠⎠

2( )( )cos ,

ssb

⎡⎛ ⎞⎛ ⎛ ⎞ξ ξ + ε⎜ ⎟⎜ ⎜ ⎟= − × −⎢⎜ ⎟⎜ ⎜ ⎟⎛ ⎞ ε⎢ξ − ξ + ε ξ + ε⎣⎝ ⎠⎝ ⎝ ⎠⎜ ⎟ξ + ε σ +

⎜ ⎟ε⎝ ⎠⎛ ⎞ ⎞ ⎛ ⎞⎛ ⎛ ⎞⎞ξ + ε ξ ξ ξ − ξ⎜ ⎟ ⎟ ⎜ ⎟− σ + + + σ +⎜ ⎜ ⎟⎟⎜ ⎟ ⎟ ⎜ ⎟ε ε⎝ ⎝ ⎠⎠ ξ + ε⎝ ⎠ ⎠ ⎝ ⎠

INT

2

2 22

2

2

( ) ( ) ( )( ) 2

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) 2 ( )

( )

4 1 cos

cos cos 2 1 sin2

s s ss

s s ss s

s s s s s sss s s

s

bR

b bb⎛ ⎛ ⎞⎞⎤+ ε⎜ ⎜ ⎟⎟⎥⎜ ⎜ ⎟⎟ε ⎥⎝ ⎝ ⎠⎠⎦

2

.

PXR( )sR FPXR

( )sRε

( )= ξ + ε − ξ

PXR

FPXR

22( )

( ) ( )( )

ss s

sRR

ε Bδ − θ

312


BLAZHEVICH, NOSKOV

can significantly exceed the intensity of parametricradiation in the Bragg direction,

. (44)

Let us independently consider the variation of theintensity of each radiation. Figure 3 shows the curvesconstructed by formulas (30) and (40). These curvesshow the increase in the FPXR peak amplitude andthe decrease in the PXR peak amplitude with decreas�ing parameter . The curves were constructed for afixed observation angle, path length, and energy of theemitting particle.

Ratio (43) will also depend on the observationangle , the emitting particle energy defined by the

Lorentz factor , and the parameter . The lattercharacterizes the degree of wave reflection from thesystem of parallel atomic planes in the crystal, causedby interference features of waves reflected from atoms

of different planes (constructive ) or destructive

( ). The effects of dynamical diffraction appearonly in the case of constructive interference, i.e., forstrong wave reflections from the system of diffractingatomic planes of the crystal. Using Eq. (36) the solu�tion of which is the frequency in the vicinity of whichthe spectrum of parametric radiation photons concen�trated and which corresponds to the spectrum maxi�mum, we write ratio (43) in the form

. (45)

�PXR FPXR( ) ( )s sR R

ε

θ

γ( )s

ν

( ) 1sν ≈

( ) 0sν ≈

⎛ ⎞ε θ⎜ ⎟= + +

⎜ ⎟⎜ ⎟ν χ γ χ⎝ ⎠

PXR

FPXR

2

2

( ) 2 2

( ) ( ) 20 0

1 1' '

s

s s

R

R

Considering this relation at the maximum of the

angular density of parametric radiation ,

we obtain

. (46)

Since the parameter is always less than unity, itfollows from ratio (46) that the PXR yield will always

exceed the FPXR yield if If or even

slightly less than unity, the ratio of radiation yieldsdepends strongly on asymmetry.

Let us consider the effect of reflection asymmetryon the ratio of angular densities of radiations. To thisend, we integrate expressions (27a) and (37a) over the

frequency function ,

(47a)

; (47b)

, (48a)

. (48b)

Figures 4 and 5 show the curves constructed by for�mulas (47b) and (48b) and describing the PXR andFPXR angular densities, respectively, in the case when

. It follows from the figures that parametric

X�ray radiation along the emitting particle velocity atε � 1 is characterized by an angular density signifi�cantly higher than the PXR density.

The FPXR and PXR angular densities as functions

of the parameter are compared in Figs. 6 and 7,respectively. We can see that the FPXR angular densitydepends stronger on this parameter.

20'

−

θ = γ + χ

⎛ ⎞ε ⎜ ⎟= +

⎜ ⎟⎜ ⎟ν γ χ⎝ ⎠

PXR

FPXR

2

2

( ) 2

( ) ( ) 20

2 1 1'

s

s s

RR

( )sν

20' 1γ χ �

20' 1γ χ ≥

( )( )sη ω

=Ω π θ

PXRPXR

B

2( ) 2 ( )( )

2 2 ,8 sin

s ssdN e P F

d

+∞

−∞

θ

χ= ν η ω

⎛ ⎞θ⎜ ⎟+ +

⎜ ⎟⎜ ⎟γ χχ⎝ ⎠

∫PXR PXR

2

0( ) ( ) ( ) ( )

2

2

200

'( )

1 1''

s s s sF R d

=Ω π θ

FPXRFPXR

B

2( ) 2 ( )( )

2 28 sin

s ssdN e P F

d

+∞

−∞

θ

χ= ν η ω

⎛ ⎞θ⎜ ⎟+ +

⎜ ⎟⎜ ⎟γχ χ⎝ ⎠

∫FPXR FPXR

2

0( ) ( ) ( ) ( )

2

2

20 0

'( )

1 1' '

s s s sF R d

20' 1γ χ ≥

( )sν

Fig. 3. Comparison of the PXR and FPXR spectra at dif�ferent symmetries.

100

50

–0.5–1.0–1.50

ν(s) = 0.8θ

�χ10 �

= 1

�χ10 �

= 0.5γ

1

b(s) = 5

0.2

0.2

0.8

η(s)(ω)

ε = 0.8

R(s)PXR

R(s)FPXR



CONCLUSIONS

Let us formulate the main results obtained in thisstudy.

Analytical expressions for the spectral�angularradiation density along the emitting particle velocityand in the Bragg direction were derived in the generalcase of asymmetric reflection.

Analysis of the derived expressions showed that, atthe fixed angle between the electron velocity andthe system of diffracting atomic planes of the crystal

Bθ

and the constant electron path length in the crys�tal plate, the ratio of PXR and FPXR yields dependsstrongly on the angle between reflecting atomicplanes and the crystal plate surface and, hence, on thereflection asymmetry. It was shown that a decrease inthe parameter and an

increase in the angle ( ) of electron incidence onthe crystal plate results in a decrease in the PXR spec�tral�angular density and an increase in the FPXR den�sity that can even exceed the PXR density.

( )2 sb

δ

B B/sin( ) sin( )ε = δ + θ δ − θ

Bδ − θ

Fig. 4. Comparison of angular densities of radiations. Fig. 5. The same as in Fig. 4, but for another asymmetry.

Fig. 6. Comparison of FPXR angular densities for differentreflections.

Fig. 7. Comparison of PXR angular densities for differentreflections.

4

2

543210

F(s)PXR

F(s)FPXR

ν(s) = 0.8

�χ10 �

= 0.5γ

1

b(s) = 5

θ

�χ10 �

ε = 0.16

4

2

543210

�χ10 �

= 0.5γ

1ν(s) = 0.8

b(s) = 5

F(s)PXR

F(s)FPXR

ε = 0.8

θ

�χ10 �

8

6

4

2

543210

b(s) = 5

�χ10 �

= 0.5γ

1

F(s)FPXR

ε = 0.1

ν(s) = 0.9

θ

�χ10 �

ν(s) = 0.3

20

10

543210

�χ10 �

= 0.5γ

1ε = 2

ν(s) = 0.9

θ

�χ10 �

F(s)PXR

b(s) = 5

ν(s) = 0.3

314


BLAZHEVICH, NOSKOV

REFERENCES

1. M. L. Ter�Mikaelyan, The Influence of the Medium onHigh�Energy Electromagnetic Processes (AN ArmSSR,Yerevan, 1969) [in Russian].

2. G. M. Garibyan and Yan Shi, Zh. Eksp. Teor. Fiz. 61,930 (1971) [Sov. Phys. JETP 34, 495 (1971)].

3. V. G. Baryshevskii and I. D. Feranchuk, Zh. Eksp. Teor.Fiz. 61, 944 (1971) [Sov. Phys. JETP 34, 502 (1971)].

4. G. M. Garibyan and Sh. Yan, Zh. Eksp. Teor. Fiz. 63,1198 (1972) [Sov. Phys. JETP 36, 631 (1972)].

5. V. G. Baryshevsky and I. D. Feranchuk, Phys. Lett. A57, 183 (1976).

6. V. G. Baryshevsky and I. D. Feranchuk, J. Phys. (Paris)44, 913 (1983).

7. C. L. Yuan Luke, P. W. Alley, A. Bamberger, et al., Nucl.Instrum. Methods. Phys. Res. A 234, 426 (1985).

8. B. N. Kalinin, G. A. Naumenko, D. V. Padalko, et al.,Nucl. Instrum. Methods. Phys. Res. B 173, 253 (2001).

9. G. Kube, C. Ay, H. Backe, N. Clawiter, et al., in Proc.of the 5th Intern. Symp. on Radiation from RelativisticElectrons in Periodic Structures, Lake Aya, Altai Moun�tains, Russia, 2001, p. 25.

10. H. Backe, N. Clawiter, Th. Doerk, et al., in Proc. of theIntern. Symp. on Channeling�Bent Crystals�RadiationProcesses (Frankfurt am Main, Germany, 2003), p. 41.

11. A. N. Aleinik, A. N. Baldin, E. A. Bogomazova, et al.,Pis’ma Zh. Eksp. Teor. Fiz. 80, 447 (2004) [JETP Lett.80, 393 (2004)].

12. A. S. Kubankin, N. N. Nasonov, V. I. Sergienko, and I.E. Vnukov, Nucl. Instrum. Methods Phys. Res. A 201,97 (2003).

13. N. Nasonov and A. Noskov, Nucl. Instrum. Methods.Phys. Res. A 201, 67 (2003).

14. A. Kubankin, N. Nasonov, and A. Noskov, in Proc. ofthe 7th Intern. Russ.�Jpn. Symp. on Interaction of FastCharged Particles with Solids (Kyoto, Japan, 2002),p. 217.

15. S. Blazhevich and A. Noskov, Nucl. Instrum. Methods.Phys. Res. B 252, 69 (2006).

16. S. V. Blazhevich and A. V. Noskov, Poverkhnost’, No. 3,62 (2008) [J. Surf. Invest. 2, 225 (2008)].

17. S. V. Blazhevich and A. V. Noskov, Nucl. Instrum.Methods. Phys. Res. A 266, 3777 (2008).

18. S. V. Blazhevich and A. V. Noskov, Izv. Vyssh. Uchebn.Zaved., Fiz. 50, 48 (2007).

19. Z. G. Pinsker, Dynamical Scattering of X�rays in Crystals(Nauka, Moscow, 1974; Springer, Berlin, 1978).

20. V. A. Bazylev and N. K. Zhivago, Radiation from FastParticles in Matter and External Fields (Nauka, Mos�cow, 1987) [in Russian].

on the ratio of the relativistic electron pxr in the bragg...

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