Physica status solidi / B.: Volume 133, Number 1 January 1 [Reprint 2021 ed.] 9783112495483, 9783112495476

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physica status solidi (b) basic research

Volume 133

Number 1

January 1986

AKADEMIE-VERLAG BERLIN VCH PUBLISHERS INC. • DEERFIELD BEACH/FLORIDA I S S N 0370-1972

phys. stat. sol. (b), Berlin 133 (1986) 1, 1 - 4 3 2 , K l — K 7 6 , A l — A

International Classification System for Physics*) 60. Condensed matter: structure, mechanical and thermal properties (see also 68.20. Solid surface structure, 71. Electron states) 61. Structure of liquids and solids; crystallography (see also 68.20. Solid surface structure, 71. Electron states) 62. Mechanical and acoustical properties of condensed matter (see also 61.70. Defects in crystals, 68.30. Surfaces and interfaces) 63. Lattice dynamics and crystal statistics (see also 65. Thermal properties, 66.70. Thermal conduction, 68.30. Dynamics of surface and interface vibrations, 78.30. Infrared and Raman spectra) 64. Equations of state, phase equilibria, and phase transitions 65. Thermal properties of condensed matter (see also 63. Lattice dynamics; for thermodynamic properties of quantum fluids, see 67.40; for thermal properties of solid helium, see 67.80) 66. Transport properties of condensed matter (nonelectronic) 67. Quantum fluids and solids; liquid and solid helium 68. Surfaces and interfaces; thin films and whiskers (for impact phenomena, see 79; for crystal growth, see 61.50) 70. Condensed matter: electronic structure; electrical, magnetic, and optical properties 71. Electronic states (see also 63. Lattice dynamics, 73. Electronic structure and electrical properties of surfaces, interfaces, and thin films) 72. Electronic transport in condensed matter (for surfaces, interfaces, and thin films, see 73) 73. Electronic and electrical properties of surfaces, interfaces, and thin films 74. Superconductivity 75. Magnetic properties and materials 76. Magnetic resonances and relaxation in condensed matter; Mossbauer effect 77. Dielectric properties and materials (for conductivity phenomena, see 72.20 and 72.80) 78. Optical properties and condensed-matter spectroscopy and other interactions of matter with particles and radiation (for phonon spectra, see 63) 79. Electron and ion emission by liquids and solids; impact phenomena

*) Excerpt; reproduced with permission of International Council for Scientific and Technical Information (ICSTI).

(The Substance Classification is given on cover three)

phys. stat. sol. (b) 133 (1986)

Author Index S . G . ABDULLAEVA

171

G . AMBRAZEVICIUS

K45

A . ANDERSON

0 . V . FARBEROVICH

601

F.FERMI

379

.

.

.

475

R . FERRER

K61

G . FISCHER

157

505

G . C. F L E T C H E R

241

E . ANTIC-FIDANCEV

345

N . P . FOKINA

127

M . ANTON

563

T . FRAUENHEIM

755

K . I . ANDRONIK Y U . M . ANNENKOV

E . ARUSHANOV

K1

S . ASANO

333

G . BABONAS

K45

V . N . BAIER

211,583

0 . G. BALEV

;

V . F . BANAR R . S . BANSAL V . G. BARYSHEVSKII V . K . BASHENOV . N . N . BELEVICH A. A. BOBYSHEV 1 . 1 . BOIKO

:

H . BÖTTGER M . G . E . BRAND W . BRAUNE A . BREZINI L , A . BUGAEV L . L . BUISHVILI H . BURGHARDT R. A. K. L. A. A. R. A. L. M. H.

N. V. R. B. J. A.

.

DASS N. DATSYUK J . DERI T . DESHMUKH T . DEVREESE DHOBLE

721

K61 297 75 285 K49, K53 621 721

.

K143 119 K149 463, KL29 195 127 755

CARLES V . DE CARVALHO A . CHAO T . CHELIDZE E . CHEREDNICHENKO P . CHERNYAKOVA M . CHORBADJIAN B . CHORMONOV CIANCHI COLOCCI CONZELMANN

L . DASS

K103

KL K95 143 127 573 K49 K103 505 363 379 655 ,

701

:

.

.

101 195 301 . K125 229 491

A . M . GABOVICH

135

S . G . GAGARIN

499

SADHANA GARG

K73

N . Z . GASANOV

K25

F . M . GASHIMZADE

179

V . I . GATALSKAYA 1. I . GEGUSIN K . P . GHATAK B . H . GHIYA

K31, K137 195 K67 K125

C. TRALLERO G I N E R M. GLINSKI M . GOIRAN D . J . GONZALEZ V . E . GORBACHEV J . GOSLAR G. H. V. G.

GROSSO C. GUPTA P . GUPTA D . GUSEINOV

563 K57 K1 277 285 321 K107 IC73 249, K 1 7 K25

S . G . GUSEINOV N . GUSKOS

K25 K133

D . HAJDUKOVI

where R is the gas constant. To obtain an expression for the change in compressibility A%, we calculated dxjdj) and get m m yum exp [1 + exp (E/kT)]* p ,

X l

W

3. Calculation and Results For t h e calculation of thermodynamic properties from t h e above expressions, t h e volume change at pt may be taken from the d a t a given by Olijnyk et al. [7] and t h e volume fraction of defects / may be taken from the table given in paper I, depending upon t h e structure/coordination number of the new phase after transition. W i t h this, one can calculate the activation energy, A oc, A O p , and A^. I t is easy to consider t h e pressure induced structural transition in Ge from phase I —• I I at pt = 11.4 GPa at which the transition to phase I I ((3-Sn) becomes complete. In case of Si the situation is not so easy. Due to lack of d a t a of various thermodynamic quantities near the transition, in different phases, it is not simple to justify t h e validity of calculations of this paper. However, from t h e d a t a of F / F 0 versus p [7], it is possible to extract the values of compressibility near t h e transition I —• V. This requirement guides us to choose in Si the transition I —• V having coordination n u m b e r s 4 and 8, respectively. Moreover, the phase structure of Si-V has the lowest energy [7] in t h e observed range of interest. Thus mainly t h e energy considerations, so f a r as t h e stability of a particular phase at a particular pressure is concerned, have been the guiding factor in choosing the transformation Si I —>• Si V in this paper. A F / F 0 can be directly taken f r o m t h e d a t a for Ge and Si. Depending on the coordination number of the new phase, we can, in some approximation at least, have an idea of t h e volume fraction of defects / from paper I. The relevant values of A F / F 0 , pt, / , and x1 and other calculated quantities of EjkT and of thermodynamic properties are presented in Table 1. 4. Discussion We observe from Table 1 t h a t the calculated values of A^ at transition under pressure are a t least in order-of-magnitude agreement with t h e data. I t is possible to calculate t h e change in entropy, once t h e change in compressibility and t h e ( d p j d T ) _ 1 d a t a are available. However, in absence of detailed d a t a we are unable to compare t h e calculations except t h a t of compressibility with those extracted from the F/ F 0 versus p curves. I t can be f u r t h e r observed t h a t A oc and A C p are of the same order as those

P. K.

74

DIXIT

et al.: The Structural Disorder Model

Table 1 Thermodynamic properties of Si and Ge near the pressure-induced transition at pressure pt and at room temperature T ele- transition ment Si

Ge

AF/F0

f

ElkT

I ^ V -0.22 (data for comparison)

- 0 . 5 0 0.240

I ^ II -0.16 (data for comparison)

- 0 . 3 5 0.172

exp (EjkT)

pt (GPa)

x1 (l• r + Ci and t —• t + T. Therefore, according to the Bloch theorem for the y-quantum incident wave packet E0(r; ) exp {i(kr — wt)} with the wave vector fc and the frequency a> the solution of (12) should be found in the form of an expansion in Bloch waves, oo - ojt)} £ £ Er+P„ T P= ~ OO

E(r; t) = exp {i(kr

X exp {i{t + px)r)

{r; oj + P&) X

exp {— ipQt)

.

(13)

Substituting (13) into (12) and expanding the Bloch wave amplitudes in the integrand of (12) in a Taylor series, one obtains [1] the following set of equations for the dynamical diffraction of y-quanta by a crystal, in the presence of a shift wave (3) caused by the variable external field ^U+r+px)

I _

OO = L 2 r' p' = — oo

i

j

+*

m + p Q )

+ Px,k

+r

+r

_ ^ (

{ k

+

Et+Pi,(v,

r +

+p'x;Co)Et+r'+p-*(r;o>

oj + pQ)

+p'Q).

=

(14)

The tensor £(fc + r + px; fc + * ' + p'x; co) may be interpreted as the tensor amplitude of the scattering of an incident y-quantum with wave vector (fc + r ' + p'x) and frequency (oj -\-p'Q) into a y-quantum with wave vector (fc + t + px) and frequency (OJ + pQ) by a crystal element and is determined by %(k + r + pk, fc + t + t' + P'x; to) = =

{-1)(P~P')

+

£

V_p)(r'a)

£

j M0,M S

+

+ r + px, k + r + r ' + p'x;

(»772) (-!)-?•> e~Hp-p')v J p+ra ((fc

n--»

a> -

M0M)

+ r

X

) « ) J w+p -((fc + r + r') a)

[oj^M, M0) + nO] +

ir\2

where the tensor ji- is determined by ¿t?(fc + r + px, fc + r + r ' + p'x;

= w f S + 1 )

ex P

M0, M)

{ - Y z > {k

}

x

(x) (M\ J%(fc + r + r ' +p'x)

\M0)

(16)

80

V . G . B A K Y S H E V S K I I a n d V . V . SKADOROV

5. The Modulation ol the Mossbauer Radiation by a Variable External Field In this section we will discuss the diffraction b y a moving grating induced b y the variable external field in a crystal. Assuming that the crystal is f a r f r o m the exact Bragg condition the diffraction b y the crystal lattice m a y be neglected. I t follows f r o m (14) that the diffraction b y the moving grating obeys the set of equations co +

ocpEpx{r;

pQ)

2i

—

(nPV) E

p x

{ r ; co +

pQ)

=

oo =

where

p'=

2 x{knp, — CO

knp.;

co) Ep.x(r\co

+ p ' Q )

(17)

,

is the normal t o the phase surface of the Bloch w a v e px) r — (co + pQ) t]} in a crystal. T h e quantities ocp = 2 p (x\k) c o s y + p2(x/k)2, where y> is the angle between the vectors x and k, determine the number of the Bloch waves which are excited in the crystal in the presence of a variable external field. Since |Jp(fco)| 1, then \%^(knv, knp>; m)\ sS \jupy\ where yj^ are the matrix elements of the polarizability tensor of the crystal's nuclear subsystem and (j,py = £ M) are the matrix elements of the tensor fx — E

p x

n

p

=

(k

+

px)lk

( r ; co - { - p Q ) exip

{i[(k

3,

=

2 3

2

M0, M)

MM

that is defined b y (15) f o r r = %' = 0 and p = p' = 0. Con-

M0,M

sequently, the y-quanta with w a v e vectors (k + px) and frequencies (co + pQ) interact with the moving grating only when the condition \rxp\ sg \fipy\ is satisfied. F o r |«p| > \[ipY\ the interaction between the moving grating and the y-quanta w i t h w a v e vectors ( k + px) and frequencies (co + pQ) m a y be neglected. A s a result, the y-quantum field in the crystal consists of the (2m + 1) Bloch waves that interact with the moving grating, where the integer m m a y be defined by the condition «

= 1 ^ 2 ^ 1 0 0 8 ^ 1 ,

(18)

and an infinite set of the Bloch waves that do not interact with the nuclear subsystem of the crystal. Since ( n p n ) = 1 + 2 p ( x j k ) c o s y , where n = kfk is the normal t o the phase surface of the Bloch w a v e E(r; co) exp {i(kr — cot)}, then nv m a y be considered t o be equal t o n f o r \p\ ^ m and the (2m 1) Bloch waves interacting with the m o v ing grating are orthonormal t o the phase normal n — kjk up t o \jUpn\2. The Bloch w a v e amplitudes are described b y the equations co + p Q )

co -

[co'(Jf, M0)

X

J

p + n

(ka)

J

+ nQ\ + iT\2

n + p

.(ka)

'

1

'

Mössbauer Radiation Dynamical Diffraction in Crystals

81

where the tensor fj?(k; M0, M) is defined by

ff(h;M0

. W - ^ v t Z + l ) * x exp

{

-z

j { k ) }

W W )

1M)

W

1 ¡2

,

(21)

F o r Bloch waves which do not interact with the nuclear subsystem of the crystal the equations are given b y 2i - - (n P V) EP«(r;

co + pQ) = xl(a>) Ep,(r;

oj + pQ) .

(22)

The q u a n t i t y (x/fc) varies from 1 0 " 7 to 10~ 4 under ultrasonic pumping. A s follows from (18), the number of interacting Bloch waves varies a s a function of the y - q u a n t u m wavelength and the concentration of Mossbauer isotopes from a few tens t o three. The number of interacting Bloch waves increases sharply a t the angles ip for which |(jr/2) — f \ gi 10~ 2 . Thus, under ultrasonic pumping the y - q u a n t u m diffraction b y the moving grating is a multiwave one. A similar result can be obtained for radiofrequency and ultrahigh frequency fields and light waves. T o write the solution of the set (19) it is convenient to introduce a 2(2m + 1) X X 2 (2m + 1) matrix £ N with the matrix elements

-y (X^P,P'

2

=

S j

V

rt

£

Z

(iri2)(-iyr-P'>e-«*-p')9 W _ [a)1{Mf M g )

o> M)

M

M„M

{d'fr;

M))PiP,

+

n Q ]

+

3 (ka)Jn+Pika)

p+n i r j 2

- V ( « W ,

(23)

where ¡i,y = 1, 2 ; p = m,..., — m; /j,piV = {e*ftev) are m a t r i x elements of the tensor p?; are orthonormal vectors on the plane which is orthogonal to the p h a s e normal n = k/k. The solution of the equation system (19) can be written in the following form:

E^r;

w + pQ) = exp {i i X L y =1, 2

t{o>) kn(r - r0)} X m £ (exp {i\x^kn{r p' = —m x

X E;.„(r,co where E$K(r; a> pQ) = (e*Ep„(r; The solution of (22) is given b y

Epx(r;co

+pQ)

- r 0 ) })£;£. x

+p'Q),

(24)

co -\-pQ)) is the projection of Ep„ on the vector e^.

= exp { » i * § ( < » ) (k + ptt) (r - r 0 ) } Ep,(r0-,

o> + pQ)

(25)

for \p\ > m. The expressions (24) and (25) connect the s t a t e of the y - q u a n t u m electromagnetic field a t the crystal point r0 with t h a t a t the point r. In order to obtain the y-quantum field in the crystal in the case when the y-quantum wave packet E0(r; a>) exp {i(kr — cat)} is incident on the incoming crystal surface it is necessary to supplement (19) and (22) b y boundary conditions. In the present p a p e r we suppose t h a t the angle between the wave vector k and the normal N to the incoming surface of a crystal is small. Since the elements of the matrix have values of the order of 6

physica (b) 133/1

82

V . G . B A B Y S H E V S K I I a n d V . V . SKADOROV

10-4 in this case, the boundary condition is given by E1 = En ,

(26)

E1

where is the incident y-quantum field on the incoming surface of the crystal; E11 is the y-quantum field inside the crystal on its incoming surface. Since the shift wave (3) caused by the variable external field is propagating into the crystal, then the crystal incoming surface at t is defined by the radius vector r(t) =r0

+ a sin (xr0 — Qt + ) n(r — r 0 )} x p,p' = —m — oo X (S(r -

r0;

e'»

W))PTP-

X exp {i[kr0 -

JPika) E0(r0,

to(t — i 0 )]} ~

OJ)

X

+

CO

+

J exp {ikne(a>) n(r — r 0 )} exp {ikq)(ry r0;t)} — 00

X

X E°(r0; CO) exp {»[fcr0 - aj(t -