Coherent X-ray radiation along the velocity of relativistic electron crossing a periodic layered medium Текст научной статьи по специальности «Физика»

ФИЗИКА

УДК 537.8

COHERENT X-RAY RADIATION ALONG THE VELOCITY OF RELATIVISTIC ELECTRON CROSSING A PERIODIC LAYERED MEDIUM

S.V. Blazhevich, D.N. Efimtceva, K.S. Lyushina, O.Yu. Shevchuk, A.V. Noskov

Belgorod National Research University, 85 Pobedy St, Belgorod, 308015, Russia

E-mail: [email protected]

Abstract. In the present work, the dynamic theory of coherent X-radiation generated by a relativistic electron crossing the multilayered medium in the direction of the electron velocity vector is developed in Bragg scattering geometry for general case of asymmetric reflection. The comparison of the analogous radiation generated by a relativistic electron in the single crystal medium is made. The developed theory predicts the existence condition for the radiation and describes its spectral and angular characteristics.

Keywords: Relativistic electron; periodic layered target; coherent X-radiation dynamic theory

1. Introduction

Traditionally, the relativistic particle radiation in a periodic lamellar structure was considered as resonance transition radiation (RTR) [1, 2]. Significant contribution to the study of X-ray transition radiation was made by a group of physicists from Japan [3-5].

In the work [4] for the first time the periodic target consisting of plates of several hundred nanometers thick was used and the energy of the radiated photons was of 2-4 keV for the fundamental harmonic. The authors of [4] assert that the radiation intensity obtained in this experiment exceeds the synchrotron radiation intensity achievable on the modern electron accelerators. A theoretical description of RTR in the above-mentioned media was presented in the work [6]. Subsequently in the work [7] the RTR of the relativistic electron in a layered medium was considered together with parametric X-radiation (PXR). In the work [8] analogous with the coherent X-radiation in a single crystal medium [9-12], the radiation in a multilayered periodic structure was considered in the dynamic approximation as the scattering of the pseudo photons of the relativistic electron coulomb field in the amorphous layers of the structure. In [8] the coherent radiation in a periodic multilayered structure was considered for the first time as the result of the contribution of two radiation mechanisms, namely PXR and diffracted transition radiation (DTR).

The theory of the radiation of the relativistic electron in layered periodic media [8] describes properly the experimental data presented in the same work [13]. The data were obtained in the experiment, where the structure was used with the layers of about 1 nanometer thick, and the photons were generated with energy of about 15 keV. A detailed comparison of the theory [8] and the experiment [13] are presented in the work [14].

It's necessary to note that in all the cited works the radiation process in a layered medium was considered in Bragg geometry only for the case of symmetric reflection, where the angle between target surface and reflecting planes/ (layers) is equal to zero. Later, in our works [15-16] the dynamic theory of coherent X-radiation by the relativistic electron, crossing a layered medium in Laue geometry for the general case of asymmetric reflection of the relativistic electron coulomb field in respect to the entrance surface of the target was built. It was clarified in these works that the radiation yield in a periodic layered target significantly exceeds the yield in the crystal and the additional opportunity of the yield increase by means of the choice of optimal reflection asymmetry was shown. It was revealed in these works that the radiation yield in the periodic layered target significantly exceeds the yield in the crystalline one, and the additional opportunity of the yield increase by the choice of the optimal reflection asymmetry was proved.

The theory of parametric X-radiation (PXR) of a relativistic charged particle in a single crystal medium forecasts the radiation not only in the Bragg direction, but also in the direction along the particle velocity vector

(FPXR) [17-19]. FPXR is a result of the dynamic diffraction of pseudo photons of the particle coulomb field on the atomic planes in the crystal. The attempts of the experimental study of FPXR are known [20-24], but the first report about FPXR detection in the crystal target was recently made in the work [24]. The detailed theoretical description of the dynamic effect of FPXR and accompanying background of transition radiation (TR) in a crystal in the case of symmetric reflection was provided in the works [25-27].

The general case of asymmetric reflection was presented in FPXR theory built in Laue scattering geometry [28] and in a theory of transition radiation (TR) built in Bragg geometry [29].

In the present work the dynamic theory of the coherent radiation along the velocity of the relativ-istic electron crossing the periodic layered medium in Bragg scattering geometry was built for general case of asymmetric reflection, when the reflecting layers in the target are situated under a free angle relative to the target surface (symmetric reflection is a special case of the reflection). The expressions for spectral-angular characteristics of FPXR and TR in a periodic layered medium are derived on the basis of two wave approximation of the dynamic diffraction theory.

2. Radiation amplitude

Let us assume that a relativistic electron with the velocity V passes through the multilayered structure, which consists of periodically situated amorphous layers of the thicknesses a and b (T = a + b - the structure period). The substance of the layers a and b have the dielectric susceptibility %a andxb correspondently (fig.i).

Fig. 1. The radiation process geometry and table of notations for the using magnitudes.

0 and 0' are the radiation angles, 0B is the Bragg angle (the angle between the electron velocity V and the layers of the target), 5 is the angle between the target surface and the layers of the target, k and k are the

wave vectors of the incident and diffracted photons

For studying the electromagnetic radiation accompanying this process we will use the two-wave approximation of the dynamic diffraction theory. Let us consider the Fourier image of the electromagnetic field

E(k,0) = Jdt d3r E(r,t)exp(i0t - ikr) . (1)

As the relativistic particle coulomb field could be represented practically as transverse, the incident E0(k,0) and the diffracted Eg (k, 0) electromagnetic waves are determined by two amplitudes with different values of transverse polarization:

-

- E (D

E0(k, () - E0(1) (k, ()e0U + E0 (k, ()e

Eg (k, () - Eg1) (k, ()e((1) + Eg2) (k, ()e(

42)/

»(2) 0

(2),

where the vectors e and e 02) are perpendicular to the vector k = k + g . The vectors e 02), e(2)

lie in the plane of vectors k and kg ( n -polarization) and the vectors e 01) and e(1) are normal to it ( u -polarization). The vector g is analogous to the reciprocal lattice vector in the crystal, and it is perpen-

2n

dicular to the layers and its length is g = n, n = 0,± 1,±2,...

The system of equations for the Fourier image of the electromagnetic field in two-wave approximation of the dynamic diffraction theory is as follows:

J(q2(1 + i 0) - k 2)E0s) + q2 i-g C (s'T) Egs ) = 8n2ie<aQVP(s) ô(q-kV),

[o2X g C(s,T) E0s) + (q2(1 + i o) - k g2) Egs ) = 0, (3) where i , i are the coefficients of Fourier expansion of dielectric susceptibility in the reciprocal lattice vector g :

1(q, r) = (Q)exp(/gr) g (q) + ii g (Q))exp(/gr). (4)

g g

The quantities C( s'T) and P( s ) are defined in (3) as follows:

C(s,x) = e0s)e(s) = (-1)xC(s), C(1) = 1, C(2) = |cos20B\,

P( s ) = e 0s / ^ ), P(1) = sin 9, P(2) = cos 9, (5)

where ^ = k - qV / V2 is a component of the momentum of the virtual photon, which is perpendicular to the particle velocity V ( ^ = / V, where 0 <<1 is the angle between vectors k and V ), 0 B is the Bragg angle, 9 is azimuthal radiation angle, counted from the plane formed by the velocity vector V and g vector perpendicular to the reflecting layers. The magnitude of the vector g can be expressed by the Bragg angle 0B and the Bragg frequency qb : g = 2qb sin0B / V. The angle between vec-

qV qV

tor —— and the incident wave vector k is notated as 0, the angle between vector —— + g and the

V2 V2

wave vector of the diffracted wave k g is notated as 0'. The system of equations (3) with s = 1 and

n

T = 2 describes the n -polarized fields. In this case, t = 2 if 20B < —, otherwise t = 1.

The quantities 10 and i g in this periodic structure are nominated as follows:

. . a b

1 0(q) = T1 a + T1 b , (6a)

1 g (Q) = eXP(-iga)- 1 (1 b -1 a ). (6b)

igT

The following expressions will be obtained from (6) and will be used further:

, = a , b ,

Xo = T1 a + T1 b (7a)

1 0 = T 1 a + T 1 b , (7b)

2 sin i ga

X g X-g =-—-(x 'b -l'a ), (7c)

2 sin I ^

X gX -g

gT

(%ь Xij)

(7d)

Solving the following dispersion equation (8) which is obtained from system (3) by means of standard methods of the dynamical theory of X-ray scattering of waves in a crystal

(02(1 + Xo) - k 2)(02(1 + Xo) - kg2)-04x-g xgcw = 0,

*0J JV" VA ' A0) kg ,

we will obtain projections of the wave vectors k and k g . Let us search projections of the wave vectors k and kg in the following form:

(8)

kx = ш cos у 0 +

шХ 0

+

X

0

2 cos у 0 cos у 0

kgx =шcos у g +-

шХ 0

+

2cos yg cos yg

We will use the known expression connected to the dynamic additions X0 and X

where P = a -x (

1 -It

Y

1

X g = 2

a = —- (kg - k2), Y0 = cosy0, у = cos у , у 0 is the angle be-

шр+х 0 ^

Y 0

(9a) (9b)

(10)

0

ш

tween the incident wave vector k and the normal to the plate (target) surface n, y - the angle between the wave vector kg and vector n (see fig.1).

Let us find the wave vectors k and k

g

k = 0/1+ X0 + X 0, kg =0/1+ X0 +Xg. Taking into account that k|| « 0 sin y 0 , kg|| « 0 sin y g , we will obtain:

(11)

X(12) =

ш

P±.

P2 + 4XgX-gC(')2 ^

Y 0

X(0'2) =ш

4y a

-P±

P2 + 4XgX-gC(')2 ^

Y 0

л

J

(12a)

(12b)

Since the dynamic additions are small (|X0| << 0 , |Xg| << 0), it can be shown that 0 « 0' (see fig.1) and we will further designate these both angles as 0 .

It is convenient to represent the solution of the system (3) for the incident field of the periodic structure in the following form:

E

(s)medium = 2 ieVQP(s)

Y

-ш2р-2ш-^ X 0

Y 0

ш

+

4 "yJL(x0 -X(j))0 -X<02)) y0

E0s)(1) 8(X 0-X?) + E0s)(2) 8(X 0-X(2))

-5(X0 -x*0) +

2

4

0

where A0 = ш

V +92-x0 n 2

, у=1/VT — V2 is Lorentz factor of a particle, £0s) and E,

4 s) 0

(2)

V J

are free incident fields in the concerned media.

For the field in a vacuum before the periodic structure, the solution of system (3) can be represented in form:

E0 =

8л ieV9P

( s )

ш

-s( — A0).

— Xo —А

ш

The expression for the field in the vacuum behind the target can be written as:

(14)

s )vacll _ E0 =

2ieV9P( s)

-S(A 0 — A*0) + £0s) Rad S

ш

2A 0

А o +

шХ o

X o

(15)

ш

where £0(^Rad is the amplitude of the coherent radiation field along the velocity of the electron.

From the second equation of the system (3) the expression relating the incident and diffracted field in the crystal will follow:

£ (s)medium =

2шА,

ш2хeC~*

£ ( s) medium

(16)

Using the usual boundary conditions on the input and the exit surface of target:

J E(0s)vacldX = J E(0s)mediumdX , J Eg)mediumdA = J Egs)vac dA

j e*s)medium exp

g j g s f A Л iL У

g_

v ' g J

dA g = 0,

we will obtain the expression for the amplitude of the radiation field.

8n 2 ieVOP(s) 1

E,

( s )Rad _ 0 = '

ш

A(i)

exp

f A(2) A* N

i A 0 — A 0 l

У 0

— A(s2)

exp

a 0 — a 0

L

У 0

(17)

A(g2)

ш

+ ■

ш

Y

— шХ 0 — 2A0 2(A*0 — A(02))

1 — exp

f A(2) A* Y

-L

exp

f A(i) A* N

A0 — A0l

/J

— A(P

ш

+

ш

— шх 0 — 2A*0 2(A*0 —A«) j

Nf f A(1) — A* NN . 1 — exp i A° A° L

exp

f A(2) A* N

• A 0 —A 0 L

JJ

exp

A0 +■

L

(18)

Before the analysis of spectral-angular characteristics of the radiation, it is necessary to note that three mechanisms of the radiation make contributions to the total radiation yield: bremsstrahlung, transition radiation (TR) and parametric radiation in forward direction (FPXR). The amplitude Eos)Rad contains the contributions of the radiations analogous to FPXR and TR in the crystal.

Let us represent the expression for the radiation field (8) in the following form:

s)rad _ r(s) , r(s) 0 " FPXR TR ,

(19a)

1

1

2

X

X

У

У

0

0

0

2

У

У

У

0

0

0

1* , шХ0 X0 +— 2 J

Е

FPXR

8n2ieVQP(s) ш

ш

2X

V

X(g2) X(02)

Y 0 j

A i.x(2)-x*n ЛЛ '

X*0 -X(2) X® X®

X* -X<1)

00

X(1 exp

1 - exp

v v

f f

1 - exp

г x(2) x

x 0 -x 0

* Л

i

L

-X(2)exp

x 0 -x 0

* Л L

y0

J

0 0- L

Y 0

exp

JJ

Xa> -X* Л

/ъ0 /ъ0 L

v

Y0

J

X?-X*0 ЛЛ r.x(2)-X*0 Л

L

Y0

X0 +-

JJ

E (s) = 2ieV9P(s) У0 etr = e

ш

exp

ш

^——0 L

Y0 J

Л

(19b)

ш

VшX 0 + 2x0 2x0 J

X

1 -■

^(X? -X(02))e y0

i x(02) +x(}) -2x*0 l y0

Xg exp

Г X(02) -X

* л

i 0 L

y0

- X^ exp

r i^i* Л

i

(1)

.X0 -X0

L

y0 J

(19c)

J

The summands in the square brackets of the expression (19b) represent two branches of the dispersion equation solution corresponding to the two X-ray waves excited in the periodical medium.

For further analysis of the radiation, we will represent the dynamic additions (10) and (11) in the following view:

X1.2) = X0 ='

ш

T( s)

x: c

2s

Г^(s) - ip(s)(1 + s)

+

+

s) -s-ip{s)((1 + s)£( s) -2к{s)s)-p

(s)

(s) (s)

(s)

2 Г (1 + s)2

(20а)

s)2s

X(gu) =■

ш

Xg C(s)

ip (1 + s) 2

1

s) - s - ip(s) ((1 + s)£( s) - 2k( s )s) - p

(s)

(s) (s)

(s)

2 Г (1 + s)2

(20b)

s)2s

where

1 + s

2v(

х)(ш) =

a

sin2 0

B

gT

v (s) =

IC(^ V2C(') |xb -Xa

с (s)R^vxix~;

sin

1 -

ш(1 -9 cos ф cot 9 B)

Л

ш ,

X0

2C(s) ■ Г ga Л sinI — 1 X b X a

V 2 J

g aX'a + bXb

e

x

X

X

4

2

4

P(s) =

Хо

axa +bxb

g

R^V^ C(s) |xb -xa|C

(s)

2

sin

к( s) =

x 0

2C

(s)

sin

ga

g

x ъ x

axa+bxb

v 2 ,

s =

Ï0

(21)

An important parameter in expression (20) is the parameter s , which we will rewrite as

s =

sin(9B - Ô)

(22)

sin(9 B + 5)

Parameter s defines the degree of the reflection asymmetry of the field relative to the target entrance surface. Here 0B is the angle between the electron velocity and the reflected layers, 5 is the angle

between the target entrance surface and the reflecting layers. For the fixed value of 0B the parameter s defines the entrance surface orientation relative to the reflecting layers (fig.2). When the incident angle of the electron on the target (0B + 5) decreases the parameter 5 becomes negative and then its absolute value increases, (in the extreme case 5 ^ —0b ), which leads to the increase of s. On the contrary, when the incident angle increases, the parameter s decreases. In the case of symmetric reflection when 5 = 0, the asymmetry parameter s = 1.

2

Fig. 2. Asymmetric ( S > 1, S < 1) reflections of the radiation from the periodic layered structure

2. Spectral-angular density of the radiation

Let us consider the u - polarized waves (s=1). Substituting (19b) and (19c) into the well-known expression for spectral angular density of X-ray radiation

" , (23)

d2 N 2

= ю2(2л)-6У £0 dbdQ 1

s=1

we will find the expressions which describe spectral angular density of FPXR and TR mechanisms.

d2 N

ш-

fpxr

dodQ. ж Г

92 +Y-2 +

\аХ'а + bXb

2 rfpxr :

(24a)

T

R

FPXR

K - i

;Р(1 + s) Л

2

У

r K л r

exp - iB — - E + K

V s J V

,p(1+s) W..D K Л

exp

iB —

s J

1 - exp - iB a

V V V

r E-k+j P(1 -g)

s

2s

a f-K + . p(1 -s)

a---+1-

exp

f-k Л

V

iB—

sJ

s

2s

f r

1 - exp - iB

V V

a--

a E + K + , p(1 -s)

a---+ i-

s

2s

i

exp

■ + i-

-iB K

vs

2s

(24b)

ш

d2 N

-— 61 dшdQ ж2

TR

1

92 + y

9 + y2 +

(25a)

ах'а + bXb'

T

R

TR

RtTR

(

2K exp

1 + -

- iBl a - — + V s

E ip(1 -s)

2s

K - i ^ Л expi-„ЗД + K - i^) expB f,

(25b)

The following notations are put into the formulas (27) and (28):

9)=

f

V

92 +y-2 +

л\

K =

-s-ip(s)((1 + s)Es) - 2ls)s) -p

( s ) s )«

,( s )2

(1 + s)2

-Is )2s

Е(ш) =

gT sin2 9^

sin

lX b X a

ш

1 -v ш B J

+

1 + s

2V«

, p =

ax""+ bXb

g

X X a j

2

sin

ga 2

i = ■

sin

g

Xb Xa

ax!+ bxX

2ш,

B =

sin

X a

gT sin(9 B + 5)

-L, 92 = 9sinф.

(26)

2

2

9

e

2

1

X

2

2

X

2

s

2

1

2

2

4

2

In accordance with (24b), two waves that contribute to the FPXR can exist in a periodic layered medium.

The contribution of the first or second wave could be significant if, respectively, the first or the second of the next equations has a solution:

Re

a —

ï —к ,p(1—£

■ +j

£

28

a —

ï —V?

—£

= 0,

Re

a — -

ï + к o(1 — ££

■ + j

£

2£

a—

ï+të

—£

= 0.

(27a) (27b)

£

1

Since a > 1 it can be shown that equation (27b) has a solution under the condition s > —2 , and

a 2

1

the equation (27a) is solvable on the condition that s < ——. Thus, under different values of asymmetry

a

parameter, the first or the second of x-ray waves can contribute in the FPXR.

Let us consider the direction of the energy transfer of the two waves responsible for the formation of the FPXR. For this purpose, we will consider the group velocities of the radiation waves along the OX axis, neglecting absorption. The projections of the wave vectors of the waves along the OX axis (9a) in the periodic layered structure in the case of non-absorbing targets are as follows:

k((1'2) = rnsin(9B +5) +

шХ0

- + -

ш

sin

ga

Xb —X'a\C

(s)

V =

' гр

2sin(9B +5) 2 sin(9B —5)

fak(12) N

gT

ïH+ï)

The group velocities of these X-ray waves (31) have the form:

( -.2- (

v j

sin(9 B + S) —

sin2 0

B

sin(9 B — S)

1 +

ï(œ)

ЛУ

— 8 (28)

(29)

— £JJ

It can be shown that the group velocity of the waves which correspond to the first branch of the dispersion relation solution is positive (at®/ cto)—1 > 0 and the energy of the wave is transferred from the input surface to the output surface of the target. The group velocity of the second wave is always negative

(ck(2') / a©) < 0, consequently, the energy of the wave transfers from the output to the input surface of the

target. This fact leads to the suppression of the second wave of the FPRX in a periodic layered medium in the case of a crystal of a considerable thickness when the transmitted energy is completely absorbed.

Thus, for a sufficiently large thickness of the crystal, FPXR corresponding to the second branch of

1

the dispersion relation solution is suppressed. However, on the conditions s < —2, the FPXR which cora 2

responds to the first of generated x-ray waves in periodic layered medium will be material.

Let us demonstrate this claim by numerical calculations performed by the formulas (24) and (25). In Fig. 3 and Fig. 4 the curves are constructed describing the spectral and angular density FPXR and TR of the relativistic electron of energy E=200MeV which crosses the periodic layered structure C-W, that consists of the layers of carbon and tungsten. Furthermore, the curves in Figure 3 are constructed for the

1

case where the asymmetry parameter s < —- and contribution comes from the first branch FPXR with

a

1

positive group velocity of the X-ray waves. In Figure 4 the curves are plotted for the case s > —2 where

a

the contribution of the first branch is absent, and the contribution of the second one is suppressed because of the negative group velocity of the correspondent waves.

£

2

It is necessary to note that in Figures 3 and Fig. 4 the curves are constructed with the same path length L 0.3 jum and the photonsLe « Lph in the target. In this case, the length

of the electrons L =

T

6.2 /m is much more then L

' ph '

sin(#B + 5) of the photon absorption in the structure Labs =

®\axl+ bxb\

In this case, the transition radiation consists of the radiation produced at the exit surface of the target. It should be noted that the width of the peak FPXR in this case, as it follows from Figure 3, is about 25eV, which is much wider than in the crystalline medium (crystal 1-2 eV). This fact will ease the experimental research and the identification of FPXR in a periodic layered structure.

Fig. 3. Spectral-angular density of FPXR and TR of the relativistic electron in the periodic layered medium consisting

of carbon and tungsten layers (the case of S < 1)

Fig 4. Spectral-angular density of FPXR and TR of the relativistic electron in the periodic layered medium consisting

of carbon and tungsten layers (the case of S > 1)

3. Conclusion

In the present work a dynamic theory of coherent X-rays along the velocity of the relativistic particle in a periodic layered structure in Bragg scattering geometry is built up for the general case of the asymmetric reflection of the particle field relative to the entrance surface of the target.

On the basis of the two-wave approximation of the dynamical theory of diffraction, the expressions which describe the spectral and angular characteristics of the radiation from the two radiation mechanisms FPXR and TR are obtained.

The very existence of the dynamic effect of the FPXR in a periodic layered structure is shown for the first time. It is also shown that the spectral-angular density of the FPXR considerably depends on the asymmetry of the electron field reflection relative to the surface of the target under fixed path of the electron in the target. It is shown that the spectral peak of the relativistic electron parametric X-ray radiation

in the forward direction is many times larger than the peak of the emission spectrum of a single crystal, which may ease its experimental observation and investigation.

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