Observation of diffraction radiation of oscillator on angular distribution measinrement

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Observation of diffraction radiation of oscillator on angular distribution measinrement

A. Grubich, O. Lugovskaya, S. Cherkas

To cite this version:

A. Grubich, O. Lugovskaya, S. Cherkas. Observation of diffraction radiation of oscillator on angu- lar distribution measinrement. Journal de Physique I, EDP Sciences, 1993, 3 (11), pp.2139-2149.

�10.1051/jp1:1993236�. �jpa-00246858�

Classification Physics Abstracts

07.85 29.90

Observation of diffraction radiation of oscillator

_on

angular

distribution measurement

A-O-

Grubich,

O-M-

Lugovskaya

and S.L. Cherkas

Institute of Nuclear Problems, Minsk 220050, Republic Belarus (Received 13 January 1993, accepted in final form 23 July

1993)

Abstract The expression ^for angular distribution of diffraction radiation of oscillator

(DRO

in single crystals is derived. The effect of the beam parameters _on angular distribution density and oscillator kinetics _are taken into account. The analysis of the conditions for DRO observation

on the parametric radiation and bremsstrahlung "background" is made. Some experimental procedures for detection DRO _are suggested.

1 Introduction.

For _a

charged particle

channeled

along

_a

crystal

axis _or

plane

the

projectile path

is _a result of correlated collisions with

crystal

atoms. The

particle

moves in _an effective

potential

obtained

by smehring

the

crystal potential along

the axis _or

plane iii.

This motion is

accompanied by

_a

special

type of radiation, so-called

channeling

radiation [2,3].

X-ray

radiation of _a relativistic

oscillator in _a

crystal

is

essentially

modified under diffraction conditions for emitted

photons.

A

new diffraction radiation is _a result of coherent summation of two processes

photon

radiation and

photon

diffraction. It has been called diffraction radiation of oscillator

(DRO)

_[4].

Though predicted

back in 1977 [5], DRO has not _so far been observed

experimentally.

Its

study

is of considerable

interest,

since DRO may find

application

in

treating

different effects in the

optics

of _a relativistic emitter

moving

in

refracting

media [6].

Besides,

DRO

angular

distribution is _a strong ^function of the

charged particle

beam

characteristics,

which _may be used for

monitoring high-quality

beam parameters.

In this _{paper we}

analyze

conditions of DRO observation in

experiments

_on measurement of

angular

distribution of the radiation emitted towards the lateral diffraction maximum

by

rel- ativistic electrons

(positrons) passing through perfect single crystals

under

planar channeling.

DRO should be detected

against

a

"background"

of parametric X-radiation

(PXR) iii

and

bremsstrahlung

diffracted

by

the

crystalline planes.

2

Spectral-angular

distribution.

Suppose

that

charged particles

_{cross a}

plane-parallel single crystal plate along

_a direction

specified by

the unit vector _e3 which does _not coincide in the

general

_case with _a unit vector of the normal _to the surface N directed inwards. Let _us further _{assume that the axis}

given by

the vector _{e3 crosses a}

family

of

crystallographic planes corresponding

to the

reciprocal

lattice _{vector r}

((r(

₌

27r/d,

d is the

interplanar distance).

In this _case the _wave

length

of

photons

with the _wave vector kB ₌

wBe3/c

and

frequency

_wB> determined from the

equality

[kB +

r[

= [kB( and satisfies the

Bragg

condition AB " 2dsin@B, where sin _@B

=

-e~.r/[r[.

The

spectral-angular

distribution of the number of

photons

with the _{wave vector}

k([k[

=

w/c)

and

polarization s(a,

_7r) emitted from the

crystal

towards the lateral maximum

given by

the

vector kBr ₌ kB _{+ r can} be found

by

the

general

method described in reference [8].

where _a

=

1/137,

fl is the oscillation

frequency

in the

laboratory

coordinate system,

Lo

₌

L/~to,

^L

being

the

plate thickness,

_~to

= e3N,

rot

(rot> ro2)

is the transverse radius vector of

particle

at the moment it _enters the

crystal (ri (t

=

o)

₌

roil,

era _ii

[k, r],

_e~~

ii

[k

_r,

e~~] polarization

unit vectors,

Rj~~~

^{is the} coefficient

defining

the

efficiency

of emitted

photon

diffraction in

crystal

for the Laue (~ti =

kBrN/ [kBr(

>

o)

and

Bragg

_(~ti <

o)

diffraction

geometry

~ ~ ~~ ²

~~ ~

2

(n2s nis)

^{' ~~~}

(for

_an

explicit

form of the coefficient

R[

the reader should refer to

(8]), Ca

= I, C«

= cos 2@B> n~~ = I + dps + al

/2

is the index of X-radiation refraction in _a

crystalline plate

in the _case of two-wave diffraction of

photons by

the atomic

lattice,

_{p =} 1, 2

di(2)s

=

k

-ai~ti + xo (~to +

~ti)

+

I(ai~ti

_{+ xo (~to ~ti}

_))~

₊

^~to~tir~1 ₍₃₎

al "

((k r[~

_[k[~)

/[k[2

is the deviation from the _exact

Bragg condition,

_rs

=

r[

+

irl'

XrX-rC),

=

Xo =

Xi

+

iXi>

_{XT, X-r are} the

crystal complex susceptibilities, k~s

= k _{r +}

"~"~~

is the

photon

_wave vector

propagating

in _a

crystal

at _a small

angle

with the vector

kBr,

c~to

fl ^xIQ

~"~ "

$ /

_d~

_~(l)

-«IQ

exp

ikpseiu / _{dt'@i (t')}

_inflt

+

ikpse3.u/2. / _{dt'@~(t') Hit}

,

~ ~ (4)

is the coefficient of Fourier

expansion

^of the function

periodic

in t with

period

27r

In,

v(t)

₌

ucos(@(t))e3

+

u@(t)

is the

particle velocity,

= [@(@1, @2)( « 1, @1(t +

27r/fl)

₌

@i(t),

_{@i(o) =} Hoi, _{@2 =} @02, Ho = (Hoi> _{@02) is} the

particle

entrance

angle

with respect to the axis

given by

vector _e3

(we

assume the

particle

_goes

through

the

crystal along

the

periodic trajectory

in the

plane specified by

the vectors _ei and (@o2e2 + cos (Ho)

e3)

_{ei, e2, e3} stand for the basis vectors of the Cartesian coordinate system

originating

at the

crystal

surface irradiated with

charged particles),

lpn =

~~

~t~~

+

d~

_{+ 2dps}

al +

Hi ~~°~~e2(k r)

^~~~

~

(5)

w w w

is the

complex

coherent

length

^of

radiation,

n ^2«/n

~~

27r ~~~~~~~' ~~~

In

equation (ii [(k- r)rot(

_£

wdroi/c, Hi

is of the _same order of

magnitude

_as the maximum value of the _square of the

angle

_Ho at which the

particle

penetrates the

crystal,

the modulus

~~°~~e2(k r)

_{£ 2@o2d,} where d is the

angle

between the _wave vector of the emitted

w

photon

k and the vector kBr

(the polar angle

of

radiation).

For the

following

consideration

we suppose that

wdroi/c

« I,

Hi

_« ~tQ~, 2@o2d « ~tQ~, ^where ~ttr =

w/wL,

_wL

being

the

Langmure frequency

of the

crystal,

are fulfilled.

3.

Angular

distribution.

The scalar

product

_ens.an

(the

index _p is

dropped

here due _{to a}

negligible dependence

of the Fourier component on the

dispersion

branch

number)

will further be written _{as a sum} ens

at

_{+ ens}

al,

_{the first term of which describes the}

oscillatory

motion of the

particle

(at

= ant

ei)

and the second its linear motion in the

crystal (al

₌

(ant, an2))

^{In this case}

~

the part ^of

equation ii, proportional

to the modulus

squared ens al

,

should be

interpreted

as a

spectral-angular

distribution of PXR emitted

by

the oscillator and

corresponding

to the constant component ^{of its}

velocity.

Let _{us use} the harmonic oscillator with the transverse

velocity

vi

=u.@oi

_.ei

.cos(fl.t+~2o)+u.@o2e2 (7)

as a model of _a relativistic oscillator. We choose _@02

= o, _{~2o "} o _as initial conditions. It is

further assumed that the transverse

velocity

perturbatior~,

^determined

by particle

transverse oscillations in the channel, is

small,

I,e, the condition ^~ ~~°~

< l is met.

Therefore,

expansion 8fl

can be carried out in this small parameter. ^{Then for the}

Fourier-components

_an we obtain

a~

"

(~01e3 ^{~n+I (ill}

^~

^~n-I(il) )~) i~n-i(il)

~

~n-3(il) ~n+i(il)

_~

~n+3(il)j

,

~~

_~~

~ ~~

~"~~~

~~ _~

~

~~ ~"~~~~~ ~~ _~ ~~ "+~~~~)

' ~~~

where Jn(1~) ^{is the n-th order Bessel}

function,

1~ ⁼

kei ~(~

^Consider

^dipole approximation.

In this _case the condition

1~ < should be

fulfilled,

then

at

= Hoi ei

~

al

= e3

~

2 _{c c}

It follows from

(I)

that the

magnitudes

of

frequencies

_w and

angles d,

~g (~g is the a2imuth

angle

^measured from diffraction

plane specified by

the vectors kBr and

r)

of the emitted

photon

with the _wave vector k _are close to the values _at which the real part of the coherent

length

of

radiation

lpn

- oc. This condition _can be

conveniently

rewritten _as follows

ai = D

xi(fl II fin/D. (9)

Here

fl

=

~to/~ti,

^D

(wB, d)

= ~t~~ + ~t£~ ⁺ d~

2fl/wB.

To

simplify

the treatment, _we have

dropped xi, rl'. Equation (9)

satisfies the condition I

lip»

= o either _on the first diffraction branch if P ₌ D ₊

fir[ ID

_>

o,

_{or on} the second _one

provided

P _< o. For Laue

diffraction,

the number of _a branch where DRO is emitted is determined

by

the

sign

of the

quantity

D

(since fl >o).

Considering

the

foregoing

and

confining

ourselves to the Laue diffraction _case,

provided

that the radiation maxima _on different harmonics do not

overlap,

and

assuming

the

frequency

_w

found from I

lip»

₌ 0 to be

approximately equal

to wB, we obtain the DRO

angular

distribution

(for

_{n =}

I) (see

^Ref. _[9])

N]~

~~o ⁼

°

"~~°~

~~~~~~~ j~ _~~~~ ~~'~~

£ _{L$( (@(D)d~i}

₊

_{(I @(D))d~2) (lo)}

'

~

167rc

~n B

^{~ ~}

i

where all the

frequency-dependent

terms in the

right-hand

side of

equation (lo)

_are taken for

w = wB,

L$(

₌

L$(~ ii

_exp

(-Lo/L[(~)),

_L$(~ is the

photon absorption length

~Ps ^C ~~ _{+ fl~~}

j~ ~j

~~~ L°B

Xi fl (D

_{+ ds)~ +}

r(-d~

ds ₌

r$/2Xi,

~~~~

~~~

~ ~~~) ~

~

~~

~

~~~/~)~

^~~~~

If the condition

Lo

< Labs" ^~ is

fulfilled,

the

dependence

of

(lo)

_on d is

completely

w

[x[[ fl

described

by

the

angle

function

(12).

Consider the behavior of equation

(lo)

at different values of the

Bragg frequency

_wB. We

assume to have _a beam of

charged particles

of _a fixed _energy, then

varying

the value of _@B,

I-e-

rotating

the

crystal

with respect to the beam incidence direction, _we shall obtain different patterns of DRO

angular

distributions

(in

_{[10] we} described DRO

angular

distribution _{as a} function of oscillator

energy).

4. DRO

angular

distribution _{as a} function of

bragg angle

at a fixed

particle

_energy.

Note that the

equation

D

= o defines the relation between the

photon

emission

angle

d and the

photon frequency

for the

complex Doppler

effect. It _can be rewritten _as

w =

~~~ (13)

'f~~

+'ftr + ~~

We would like to recall the basic parameters ^{of the}

complex Doppler

effect

following

from

(13).

The

frequency

_wD

"

ml /(nfl)

^{and the} Lorentz-factor _~tD

=

wL/(nfl) (below

_we shall consider

only

the first harmonic _n

=

ii

determine the

point

of the

complex Doppler

effect

birth,

at _{~t =} ~tD the radiation

frequency

_w

= wB, the radiation

angle

d

= o. As _~t

increases,

the _upper and lower branches of the

complex Doppler

effect

diverge

from the point _wD. When

~t » _'iD the minimum

frequency

of the lower branch _wrnin

=

wD/2,

the maximum

frequency

of the _upper branch _wrnax

= 2fl~t~. We _assume that the

Bragg frequency

is _{wB > wrnin.} If wB > wrnax, then Do ₌

'f~~

_+~tQ~

+d~ 2fl/wB

> o and the latter

inequality being

fulfilled

only

at p = I. The DRO

angular

distribution is _a

single

maximum

strictly aligned

with the vector

kBr

As _wB

approaches

_wrnax,

F(d)

is of the order of

,

the

angular

width m 2

).

_{In the}

2flr(

^~

region

_wB < wrnax the form of the DRO

angular

distribution

density

is

considerably different,

I-e- _a

single

_narrow maximum at the

angle

d

= o transforms into two maxima

(the

a2imuth

angle

_~g is assumed _to be _a fixed

one).

The

approximate position

of the mentioned maxima in the

angular

distribution

corresponds

to the

angle dD

obtained from _a

complex Doppler

effect equation D [w=w~ = o. One of them

corresponds

to the second

dispersion

branch and is at

angles

^d <

dD(D

<

o),

the other to the branch _p

= I and lies at

angles

d >

dD(D)

>

o),

j

j2+fit@

1,2 D _S

fl/

_L°B

(i~)

The maximum value of the

angular

function is still of the order of

2flr(

5. Effect of the beam parameters on

angular

distribution

density.

The

foregoing

is valid

only

under ideal conditions when _a

spread

of

charged particles

_over

energy ~h~t/~t ^and an efficient

angular spread

@~R =

l~

@( + 2

in

e2(@o2d

(15)

(n

₌

kBr / [kBr satisfy

the

inequality

)

+ 2(@eR'f)~

/3 ~~'~~@ (~6)

For experiments on the

majority

of accelerators

~'~

> 2 (@~R~t)~

/3 (for

_~t of the order of

100),

'f

therefore,

to evaluate

angular

distribution

density

when

inequality (16)

is invalid, _one should average

(lo)

_over the beam _energy

spread.

In _our further calculations _we assumed the beam to have normal _energy distribution with

dispersion

_{ab =}

jh

and that it

averaged

~t In 4

~~~~ ~~ /~ab

~~~

~~~b~

^~

~' ~~

~

~~~

6. Oscillator kinetics.

As far _as the DRO maximum

intensity

is

concerned,

_one should conduct _an

experiment

at beam

energies

of I loo MeV. At such beam

energies

and at X-radiation

energies

limited _to at most several tens of

KeV,

the PXR is attenuated due _to its threshold character. Below _~ttr the PXR

intensity drops

with

decreasing

_{energy as} (~t/~ttr)~

Therefore,

DRO _can be observed

on the PXR

"background".

At such

energies

_a

particle

is _a quantum oscillator. That is

why

in further calculations _we should allow for the coefficient of

population Qn

of the _n th energy level with _{a transverse} motion energy

£ni determining

^{the chosen}

frequency

of transition An =

£ni £mi

between the levels _n and _m. The value of

@(1/4

in

(lo)

_can be treated in

terms of the

quasiclassical theory

of the quantum ^oscillator _{as an} estimate of the value of

[Xnm[~

^fl~

/c~,

where

[Xnm[~

is the

dipole

momentum of the radiative transition _{n - m.}

In what follows _we shall consider DRO emitted due to the radiative transition between the levels 1 _~ o, then the DRO

intensity

is

proportional

to

f/° _{Qi(z)dz, Qi being population}

_of

the first level of the transverse motion energy.

The relation between the

population Qi

and the

crystal

thickness is described

by

the system of

equations:

~~ (

~"

~"

_~

~

_~~"

_~~'

where

Wnm

is the

probability

of _a non-radiation transition from the level _{m to} the level

n, rn

=

~jwnm;

m

Wmn =

27r/h~j

< O

[<

_{po (Vnm[ p}

>[

N _>< N

[<

_{p [Vmn[ po}

>[

O >.

N,p

(EN ^Eo

+

£jj(p)

_£jj

(po

+

£n £m)

Here _< O[ ^and < N[ are

ground

and excited states of the

crystal,

Eo and EN _{are corre-}

sponding energies,

_{p, po} are the

starting

and final _momentum of the

electron, parallel

to the

channeling plane,

V is the

potential

of electron

crystal interaction,

the indices _m and _n

imply

that the matrix element is taken _over the _wave

eigenfunction

of the electron transverse motion in the channel. Summation is made _over the

crystal

excited states < N[ ^{and the} momenta p.

In the

expression

for the function argument one can

neglect

the differences EN

Eo

and

£n

-£m, expand

the quantity _£jj

(pi -£jj (po

_{over parameter}

[p

_po

,

assuming

the momentum transfers _to be small

~((~P) £jj

(p~j

_~ ^{(P2 p~~)2}

2m~

^{+ (P3 po31 c}

and

ignore

the term

~~~j$°~~~

Eventually

_we have for

wmn

~t

~'~"

~j~~ / _{~~l ~}

_fifmn

_{(~l Tl/2)}

_'fifmn

_{(Tl/2 ~l)}

'~

(~l

_Tl

/2,

_Tl

/2

_~l

,

~

r

the function

fl

_is

_expressed

_{in terms of the electron}

_density

_{form factor}

F(q)

and the

amplitude

of

crystal

atoms thermal oscillations _{~, na} is the

crystal

atoms

density,

_e is _an electron

charge, fl(q, q')

=

/ _{dq jz2}

_z

_F(q)

_z

_{F(q') Fiq) F(q'i)}

(e-(~-~')~U~/2 ^e-~~U~/2 e-~'~U~/2)

+(F(q

₊

q') 1/Z F(q) F(q'))e~(~+~')~~~/~

Here _q is the vector with the coordinates

(qi,

_q2,

o),

and

q'

is the _one with

(q[,

_-q~,

o),

Z is the nucleus

charge number, Mmn(k)

_are the matrix elements:

Mmn(k)

=

#n(x)

e~~~~ ~fim(x) dx.

The _wave

eigenfunctions

of the

Schr6dinger equation

for the

potential U(x)

₌

Uo/ch~(ax)

_are

taken _{as wave} functions of transverse motion. The _constants Uo ^and o are chosen _so that the

potential

should be close to _an

interplanar potential

obtained from the Moliere model.

In

figure

I the

populations Qn

_are

plotted against

^the

depth

of

penetration

into the

crystal.

The calculations _were done for the 50 MeV electrons channeled in _a Si

single crystal along

the

(loo) planes.

The

starting populations

_were calculated for _a beam

penetrating

the

crystal

at an

angle

of 2.3 _x lo~~ _{rad and}

an

angular spread

of I _x lo~~ rad.

20 o,

oo

to

3

o,

O-.~...j...:....j...:..j

...j ..j

O 2 4 6 8 to

L.

(cm) ^Xlo'

Fig. I. Level population _{as a} function of depth of electron penetration into the crystal, the beam

angular spread 10~~ _{rad, the} angular at which the beam penetrates the crystal 2.3 _x 10~~ _rad.

Qi Population of the first level, Q2- Population of the second level et al.

7.

Experimental procedure.

Except

for DRO parametric X-radiation and

bremsstrahlung

diffracted

by

atomic

planes

are

also emitted into diffraction maximum. For _wB 2 wrnax the DRO intensity ^will be

compared

to the

angular density

of

bremsstrahlung

at the

angle

d

= o, for _wB < wrnax one should try to discover _a

relatively

_narrow DRO maximum _on the PXR

"background",

where the distribution

maximum is attained _{at an}

angle dph

Ci

'f~~ +~tQ~.

The calculations _are meant for _an accelerator whose basic

properties

_are

given

in reference

ill]

the electron _energy 50

MeV,

the beam

angular spead ^lo~~ rad,

the _energy

spread

for 50 MeV

~'~

= o.5il. We suggest ^the 'f

following experimental procedure:

the beam _moves under

planar channeling conditions,

_e-g- between the

(loo) planes,

the radiation

being

recorded at _an

angle equal

to the two

Bragg angles

^relative to the

longitudinal velocity

of e~ The X-radiation is diffracted

by

the

(llo) planes (see Fig. 2).

The

Bragg angle

is

changed by

the

crystal

rotation in the diffraction

plane.

One measurement will be made at wB "

2fl~t~,

two others at _w <

2fl~t~.

In the

experimental

geometry under consideration

(earei)~

~ l,

(e«rei)~

~ d~sin~~gTg~@B> ^{I-e- the emitted DRO} will

actually

be _a-

polarized,

the

bremsstrahlung

has _a mixed

polarization.

The calculation is

done for

~g ⁼

7r/2,

the PXR

acquires

_a

polarization (see

Ref.

[12]).

^The

crystal

thickness should be chosen _so

that,

_on the _one

hand,

the ratio between DRO and

bremsstrahlung angular

densities _as well _as between DRO and PXR should be

maximum,

and _on the other

hand,

the

crystal

thickness should be greater than the extinction

length

to fulfill the

dynamic

diffraction condition. The intensities of

bremsstrahlung,

PXR and DRO _are

respectively proportional

to

L(,

_Lo and

f/° _Qi(z)dz.

_The enhancement of the DRO

intensity

with

increasing

L is less

pronounced

than that of the

bremsstrahlung

and PXR intensities since

Qi(x)

is _a

decreasing

function. The

crystal

thickness _was chosen to be I _x lo~~cm.

fi

'

(220) '

'

I,

~

)

,~

~ (loo)

_

kB~

Fig. 2. Geometry of _an experiment to observe the diffraction radiation of oscillator.

The basic parameters ^for the

corresponding

measurements and maximum values of DRO, PXR and

bremsstrahlung

are tabulated in table I. The calculations _were done for _a silicon

single crystal.

The

averaging

over the beam

spread

_was

performed

for the _case when the

inequality (16)

does not

hold,

therefore the table contains the column with the value

2~t~/fl/3.

The

angular

distributions

corresponding

to the first and second _rows of _a table _are

pictured

in

figures

3a, ^{3b. As} a result of beam energy

spread

for _{case wB < wrnax} two

peaks

_merge

together.

^The

bremsstrahlung

_was calculated from the formula [12]

Table I. Basic parameters and maximum values of

DRO, Bremsstrahlung

and PXR

angular

distributions for three different

experimental procedures.

fifa fifa

~B L°B~*~

(X~~

Labs @

'ftr ~D

~i~~~

_X10~

~i~~

(°)

^~~~ _X IQ ^~~ X10~ ~~d _d

= dD d

= o d

=

dph

2,1 87.51 o.126 9.16 4.82 o 5.74 o.287 o.233

2 5 37.05 0.704 o.69 26.4 l192. I-ii 0.333 0.539

3 lo 18.60 2.80 o.09 lo8 598.6 o.248 o.438 0.988

(*)

The

frequencies

_were calculated for difraction

plane (220).

~~d,Br

"

)~~~~~~~

^~~~

~

~ ~

+ 167rcsin _@B

((d2

^{+ ~t~2}

xi

₊

flr()

~~~~°

_~~~~~

~

~ +

°~~~°

_~~~^~

~

~

(17)

7rsin~@B

/k _(d2

_{+ ~t~~}

_xi /k)

_7rsin~@B

/k _(d2

_{+ ~t~2}

_xi

₊

/k)

where @j is a root-mean-square

angle

of

multiple scattering

^of

charged particles

_per unit

length,

@j =

(Es / (mc~~t))~

,

Es =

@mc~,

_{LR is the radiation unit}

_length,

2LR

~eR

"

2Llis'

_l~

~lis/~0 _II

_~XP

(~L0/~lis)))

The first term in

(ii)

is

bremsstrahlung

emitted

by charged particle moving

with _a

velocity exceeding

that of the

generated radiation,

the second term is

bremsstrahlung

associated with

photons

whose refractive index is under the

unity

and

efficiency

of X-radiation reflection from atomic

planes

is at its maximum.

8. Conclusion.

As follows from table I and

figures

3a, 3b the DRO _can be detected in

experiments

_on the

measurement of the

angular

distribution. In

particular,

_{we suggest} that

keeping

_energy and

angle

_@B

unchanged,

_one should _vary the

charged particle

motion from

straightforward

to

oscillatory by rotating

the

crystal.

Then the

crystal

should be rotated in the diffraction

plane, thereby changing

the

angle

_@B. This results in different

angular

distributions.

One _may also conduct _an

experiment using

_{a narrow} slit

positioned along

the a2imuth

angle

~g ⁼

O(7r)

_as a radiation collimator. For _@B £

7r/4

the PXR becomes

linearly polarized (s

₌

a)

and is not emitted at the a2imuth

angles

_{~g =}

O(7r),

the DRO is also

linearly polarized (s

₌

a)

but is emitted at all values of _~g.

1

~ 8"

'

il

o

i

~

I z

2

a)

x

+*~

~ill

o t

j~

~

'

~

o

2i

§ o

Z to

~~

[3] KUMAKHOV M-A-, Phys. Lett. 57

(1976)

17.

[4] GRADOVBKY O-T-, Phys. Lett. 126A

(1988)

291.

[5] BARYBHEVBKY V-G-, DUBOVSKAYA I.Ya., Phys. Status Sofidi

(b)

82

(1977)

403.

[6] ^FRANK I.M., Vavilov-Cerencov Radiation. Theoretical Aspects.

(Moscow,

Nauka, Main Editorial Board for Physical ^and Mathematical Literature, 1988).

[7] BARYBHEVBKY V-G-, FERANCHUK I-D-, J. Phys. £Yonce 44

(1983)

913.

[8] BARYBHEVBKY V-G-, Channeling, Radiation and Reactions in Crystals at High Energy

(Minsk,

Belgosuniversitet, 1982).

[9] GRUBICH A-O-, LUGOVBKAYA O-M-, Vesti Akad. Navuk, BSSR Ser. Fiz. Energ. Navuk

(Byelorussian SSR)

No, 1

(1991)

61.

[10]GRUBICH A-O-, LUGOVBKAYAO.M., VestiAkad. Navuk, BSSR Ser. Fiz. Energ. Navuk

(Byelorussian SSR)

No. 3

(1991)

60.

[iii

KIM K-J-, BERZ M, et al., Nucl. Instrum. Methods A304

(1991)

223.

[12] BARYSHEVBKY V-G-, GRUBICH A-O- and LE TIEN HAT, Sov. Phys. JETP 94

(1988)

51.