Inelastic scattering cross section of neutrons in ferroelectrics

Physim

44 (1969) 69-80

INELASTIC

o North-Holland

SCATTERING

Publishing

CROSS

Co., Amsterdam

SECTION

OF NEUTRONS

IN FERROELECTRICS V. K. JAISWAL Physics

Department,

and P. K. SHARMA

University

of Allahabad,

Allahabad,

India

Keceived 31 October 1968

synopsis An expression is obtained for the one-phonon differential scattering cross section of neutrons coherently scattered from a ferroelectric crystal using the method of double-time Green functions. This has been derived from a Hamiltonian propounded by Nettleton which considers fourth order anharmonic terms in the lattice energy. It is shown that, due to the introduction of higher order anharmonic terms, the scattering function is no longer a two-delta function but has a Lorentzian shape. The width and the shift in the frequency are obtained.

1. I&Yx&~~ovz. The scattering of thermal neutrons by crystals has been a subject of considerable theoretical and experimental literature. As the thermal neutrons have energies of the order of the energies of the lattice vibrations, the measurements of the energies of neutrons coherently scattered by a crystal as a function of the scattering angle can provide fruitful information about the frequencies of the lattice vibrations for a given polarization and wave vector. In recent years most detailed information about the lattice dynamics of a crystal has been obtained from the inelastic neutron scattering experiments. The first theoretical treatment of inelastic scattering of neutrons was carried out by Weinstocki). He obtained expressions for the inelastic scattering cross sections by considering processes in which a neutron either gains energy from or loses energy to the crystal. His work concerns mainly one-phonon processes, i.e. processes in which a neutron either creates or absorbs a single phonon. It has been extended to multi-phonon processes by Sjolanderz). Multi-phonon processes become important at temperatures well above the Debye temperature only and, therefore, under ordinary circumstances they are less important than the one-phonon processes. For a particular angle of scattering the coherent one-phonon inelastic spectrum in the harmonic approximation consists of a number of deltafunction peakss). The peaks correspond to the one-phonon processes in which a neutron excites or deexcites a single phonon in passing through the 69

V. K. JAISWAL

70

crystal. The location

AND P. K. SHARMA

of the peaks is determined

by the conservation

rules

for the energy and momentum. The one-phonon processes, and therefore the one-phonon peaks, are particularly important from the point of view of crystal dynamics49 5). When anharmonic

forces are present, they manifest themselves in a most

significant way in the one-phonon peaks. Anharmonic forces cause an interaction between phonons. As a result of the phonon-phonon interaction the phonons get a finite life time, while the energies are shifted with respect to the harmonic values. As a consequence the inelastic spectrum now exhibits finite peaks rather than delta-function singularities, which are shifted relative to those predicted by the harmonic approximation. The frequency shift and the width are both temperature dependent. During the last few years, the coherent scattering of neutrons by anharmonic crystals has been the subject of investigation by many workers+la). The earliest work using time-independent perturbation theory is due to Van Hoveil) at zero temperature. This work has been extended to finite temperatures by Kokkedees) who obtained an expression for the one-phonon scattering cross section by means of time-dependent many-particle perturbation theory. Baymis) has given a treatment of the neutron scattering from an anharmonic crystal which involves the evaluation of the Fourier transform in space and time of the time-relaxed displacement correlation function. Luttinger and Ward’sis) technique runs parallel to this with the evaluation of the phonon propagator. Silverman and Joseph 14) have proposed a Hamiltonian to describe displacive ferroelectrics in the paraelectric phase by making use of available information about the lowest transverse optic mode from the observations of dispersion curves and the dielectric constants. The Hamiltonian was constructed by modifying an expression used earlier by Szigetirs) in connection with the temperature dependence of the dielectric constant of alkali halides. The major modification is that all the long-wavelength transverse optic modes, which are unstable in the harmonic approximation, are collectively labelled with a zero wave vector and are assigned an imaginary frequency. The Hamiltonian was used by them to obtain expressions for the linear and nonlinear response to the dielectric constant in the paraelectric phase of a ferroelectric material. In a recent study16) we used this Hamiltonian to investigate the electric-field dependence of the Curie temperature. Nettletonl7) has modified the Silverman- Joseph Hamiltonian by considering fourthorder anharmonic terms in the discussion of four-phonon interactions among the acoustic and optic modes in strontium titanate. In this paper we have calculated the differential cross section for the coherent inelastic scattering of neutrons from a ferroelectric crystal by onephonon processes. The principle advanced is to use the Hamiltonian proposed by Nettletoni7) and the method of double-time Green functions given

NEUTRON

by Zubarevrs). Hamiltonian scattering

SCATTERING

It is shown

that the higher-order

may be held responsible

71

IN FERROELECTRICS

terms in the crystal

for the observed

behaviour

of the

spectrum.

2. General formzdation. The differential scattering cross section per unit solid angle d1;2and per unit interval of out going energy ds of the neutron for coherent scattering in the first Born approximation can be written asrs)

d20,,, = 6-l dL? da

-!!LS(Q, Lo), I!701

where S(Q, co) =

&-z K%,bKbK*exP(-iQ+-(~K)

- VWI~

X

co

x

dt eimt
u(ZK; t] exp[iQ*u(Z’K’;

O)]).

(2)

s -ca Here qo and q are the initial and final wave vectors of the neutron, fiQ = fi(qo -

and

q)

%Ll = (7%2/2m)(q; -

42)

are the momentum and energy transferred from the neutron to the crystal due to scattering, bK is the scattering length of the nuclei, r(K) is the rigid lattice equilibrium position vector of the Kth atom in the Zth unit cell whose mass is ?%K and u(X) is its displacement from the equilibrium position. The vector r(H) can be expressed as r(W

= ~(4 + r(K),

(3)

where x(Z) is a crystal translation vector which can be written as an integral linear function of three primitive translation vectors ~1, us and as: x(l) = liar +

J2a2

+

l3a3,

(4

where Ii, 1s and 1s are any positive or negative integers which we refer to collectively as 1. The vector r(K) gives the position of atom K in the unit cell; K running from 1 to n, the number of atoms per unit cell; u(ZK; t) is the Heisenberg operator : u(ZK; t) = eiHti””u(ZK) epiHt’li, while the angular bracket denotes the canonical-ensemble the expectation value of the operator: = Tr (e-@HO)/Tr (e--OH),

(5) average of

(6)

where Tr denotes the trace of the expression and j3 = l/k~T, kg being the Boltzmann constant and T the absolute temperature. The expression for

72

V. K. JAISWAL

AND P. K. SHARMA

S(Q, W) can be expanded in powers of atomic displacements two terms in the phonon expansion are S(Q, m) = So(Q, w) + Sl(O> 4

+

and the first

-.*,

(7)

where

z by e-wcK)

WC)) = N‘Q(co) d(Q)1

so(Q,

e-iQ’r(K)12;

(8)

K s1 (Q,w) = &,C, l& bKbKf e-[w(K)+w(K’)l

exp{-iQ.[r(ZK)

- r(l’K’)]}

X

co

s

dt eicut
x

(9)

--m

in which W(K) is the Debye-Waller factor and A(Q) equals unity if Q is a reciprocal lattice vector and vanishes otherwise. The first term in eq. (7) describes the coherent elastic scattering involving no phonons and the second term corresponds to coherent inelastic one-phonon scattering of neutrons by the crystal. For temperatures not much larger than the Debye temperature, one-phonon processes give dominant contribution to the scattering cross section. To evaluate the scattering factor Sr(Q, o), we express the atomic displacement vectors u(ZK) in terms of phonon annihilation and creation operators a,5 and a&. as

Substituting

the expression for u(ZK) from eq. (10) into eq. (9) we obtain

Sr(Q, W) = &(%cN)-~ x

x

c

2 bKbK, e-(w(K)+w(K’))

1K

l’K*

kj

k’j’

X

(m@&‘)-’

[Q l e(K, kj) Q l e(K’, k’j’)]

(4mkjwk’j’)-*

x exp(-i(Q

-

X

k) l [r(ZK)

-

r(Z’K’)]}

x

rdt e’“t
(11)

where Ak* = a,j + a?,. Here and in what follows we use only one index k to denote kj. Using the cyclic boundary conditions, eq. (11) can be written as Sl(Q, co) = ;

x

,I;,, d(Q -

(4wk~k’)-*

k) d(Q -

dt eimt
k’) F*(Q,

AZ(O)>,

k) F(Q> k’) x

(14

NEUTRON SCATTERING

IN FERROELECTRICS

73

where F(Q, k) = g bk

’ (@e(K

x exp(-i(Q

- k) *r(K)}.

k)) exp{-W(K))

X (13)

The problem is thus reduced to evaluating the Fourier transform of the correlation function . This can be done by several techniques. Here we use the method of the double-time Green function as illustrated by Zubarev 1s) and described in a previous paper la) to calculate this correlation function. 3. Greeti fwzctions and the Hamiltonian. We define the retarded function of the system for the acoustical mode as

Green

G&(t) =
([A;(t), A:+(O)]>.

(14)

Similarly the Green function corresponding to the optical mode can be written. Here O(t) is the Heaviside step function. For the Hamiltonian, we adopt the expression given by Nettletonl7) in which higher order terms have been taken into account. This Hamiltonian in the absence of an external electric field can be written as

(15) Here N is the number of unit cells in the lattice; F(k) and G(k) describe the third- and fourth-order anharmonic couplings, respectively; oi and o: are the frequencies of the optical and acoustic modes for wave number k; qg and $0” are the normal coordinate and conjugate momentum of the longwavelength transverse mode. The frequency associated with this mode is purely imaginary in the harmonic approximation and is denoted by iw,O. The last two terms represent the fourth-order anharmonic contributions to the potential. The coefficients @ and Y, which determine the strength of these interactions, are Fourier transforms of the fourth derivatives of the lattice energy. The summations over k exclude the value Fz= 0.

V. K. JAISWAL

74

AND P. K. SHARMA

4. Scatterilzg

from acoustical $horzons. termining the Green function (14) is 1s) ih$G&&)

= fib(t) <[A:(t),

Using the commutation iF,$

G$(t)

The equation

of motion

for de-

A$(O)l>+ <[AZ(t), HI; Aif(O)>.

property

(16)

of the operators, eq. (16) can be written as

= ho.$((B$(t) ; A;?(O))),

where Bk is defined by Bk = ak -

(17)

a_ $. Differentiating

eq. (17) again, we

obtain -h$-

G&(t) = he@(t)

Now using the commutation [Bk,

Ak’]

=

2dk,

<[B:(t), relation

-k’,

which follows from the commutation operators. eq. (18) becomes -

-&

&:@)I> + o:W;, HJ; A;:(O))). (18)

relations for creation and annihilation

GEk,(t) = 2co;d(t) CC?,,/+ (cI$)~ G’&(t) +

x
(19)

If we assume that the expectation values may be factored acoustical parts, Fourier transforming this equation gives

co2- (w;)~ - ;

(q;)2 Ga(k)

G~k+o) =

0: 7c

in optical

and

&k’ +

1 + ;

,5 q@(kl, kz, -k) 1. *

which involves a summation of Green functions. It is to be noted from eq. (12) that the dominant terms in the summation are those with k = k’. For this case the Green functions with k2 # k on the right hand side of (20) contribute to higher order effectsa). Hence for calculating the one-phonon scattering cross section from eq. (12), the solution of eq. (20) may be written

NEUTRON

SCATTERING

75

IN FERKOELECTRICS

as G$&o) =

~kY43

= ~02-

(~$2 -

;

(c&)2@(k) - ;

z q,“@(kl,k, --K) (-$-)+
(21) Having formulated the Green function, we can now obtain density function J&(O) by using the relationi*) G&(o

+ i&)- G&(w - ie) = -i(esfi”

-

the spectral

1) J&(w).

(22)

If we set Ga(k) + ;

F q,o@(kl, k, -k) 1

and use the usual representation for the &function, spectral density function is given by

JEk44 =

[S(o 2(eez~vl) p$

The correlation spectral density

the expression

P;) - S(0 + PZ)].

can be expressed

function

for the

(24) in terms

of the

co fi
=

J

ea”“J&(co)

emimtdm =

--oo

= F

k

We now substitute &(Q,

0)

=

P$ -

{coth&%e)cos

i sin Pit}.

(25)

eq. (25) into eq. (12) and obtain for Si(Q, w) :

gF[ d(Q- ky(QJk)12 .

I

* (coth(@P~)

s

k

II.

dt cos PEtei”’ - i j dt sin P$! eiot

(26)

Finally we obtain the one-phonon differential cross section per unit solid angle and per unit interval of the outgoing energy of the neutron:

X [{coth(Q@P;)

+ l> 6(0 + P;) + {[coth(&‘3fiP,a) -

l> s(w - P;). (27)

76

V. K. JAISWAL

AND

P. K. SHARMA

5. Scattering from optical phonouts. In this case we start with the Green function for optical phonons and following the procedure as is used above, we obtain the equation of motion as -4 $

G&(t) = 2kB$qt)

6,,, + w;l<[B;,

H] ; A$(O))).

(28)

The commutation relation in the last term can be written as

If we substitute eq. (29) into eq. (28), the equation of motion for G&(t) becomes

-(-$+&) + $$

Gik,(t) = 2+3(t)

,% q;Wl, 1, *

6,,. + 2 T

kz, 4)

”

-

GO(h)G&(t) +

x ((A&4~,;A;f(O)).

(30)

b&&Q

The Green function

contained in the last term is a third-order Green function. To evaluate this Green function we introduce the following Green functions

(94) kxkak’ =

@O,,BO,,;

&c(O)X

These Green functions satisfy the following equations of motion

(32)

NEUTRON

SCATTERING

IN FERROELECTRICS

x G4fL+&p;*; Ago))).

77

(33)

The fourth-order Green functions can be decoupled as sums of products of second-order Green functions 20921).After such a decoupling has been done we obtain the following equations for the Fourier transforms of the Green functions &)2 -

(g

-

F

Go(k)]

Gftk’(~) =

where

and

w4

V. K. JAISWAL

78

Combining

AND P. K. SHARMA

eqs. (34) and (35a) we obtain

G;&o)

(36)

= W2 -

wp

-

G’(k) -

___ N

2o;M;(o)

’

where

An explicit

expression

for Mu

M;l(o + i&) = AZ(w) -

can be obtained

by writing

Z:(o).

(38)

The real part of Ml(w) gives rise to a shift of the frequency of the maximum response, while the imaginary part gives rise to a width to the response function. Finally we obtain from eqs. (37) and (38)

(39) and r;(o)

= s kl,kz

x {[ + ](&, + &) qm2 - (4il + 4iJ2) +

+ [- <~;:A;,>](& - WI&) qw2 - (4L - 4a2), where P denotes the principal E(O) =

1

F(O) = -1

value of the integral

for

co >

for

0 < 0.

(40)

and

0,

(41)

From eq. (36) we obtain for the correlation

function

as18)

-id

X

I

o2

-

(432

_

3g

G”(k) e- 2&A;(o)

+ 2io;f;l.(o)

E(W)) =

NEUTRON

emiwtr~(w)

X w2

_

((432

79

SCATTERING IN FERROELECTRICS

_

3$Y

GO(K) -

2@l;l(o)

1

1

(41)

2+ 4wfr;‘(o)

where Im stands for the imaginary part. With this result, Sr(Q, w) becomes 00

m4

X w2 _

(432

and the one-phonon

X

-

3g

GO(K) -

, (42) 20@;(o)

scattering cross section is given by

cc4 Go(K) -

* (43)

20;49(~0)

2+ 4(0$)2 Z’:(w)

1

Eq. (42) shows that the scattering function is no longer a two-delta function, but has a Lorentzian form. The effect of higher order anharmonic terms on the one-phonon peaks is to give rise to a frequency-dependent width and shift for the normal modes. Eqs. (39) and (40) give a direct measure of the shift and width of the phonon frequency. When the last two terms in eq. (15) are left out the Hamiltonian becomes the one given by Silverman and Josephl”). This Hamiltonian leads to infinitely sharp phonon peaks in the scattering spectrum. The effects of finite life times of the phonon states do not appear in this approximation. We are thankful Acknowledgement. Energy, Bombay, for financial assistance.

to the Department

of Atomic

80

NEUTRON

SCATTERING

IN FERROELECTRICS

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

Weinstock, R., Phys. Rev. 64 (1944) 1. Sjolander, A., Ark. Fys. 7 (1954) 375. Placzek, G. and Van Hove, L., Phys. Rev. 93 (1954) 1207. Maradudin, A. A., Montroll, E. W. and Weiss, G. H., Theory of Lattice Dynamics in Harmonic Approximation, Academic Press Inc. (New York, 1963). Egelstaff, P. A., Editor, Thermal Neutron Scattering, Academic Press Inc. (New York, 1965). Maradudin, A. A. and Fein, A. E., Phys. Rev. 12 (1962) 2589. Kashcheev, V. N. and Krivoglaz, M. A., Soviet Physics-Solid State 3 (1961) 1107. Kokkedee, J. J. J.. Physica 28 (1962) 374. Thompson, B. V., Phys. Rev. 131 (1963) 1420. Maradudin, A. A. and Ambegaokar, V., Phys. Rev. 135 (1964) A 1071. Van Hove, L., Hugenholtz, N. M. and Howland, L. P., Quantum Theory of ManyParticle Systems, W. A. Benjamin Inc. (New York, 1961). Baym, G., Phys. Rev. 121 (1961) 741. Luttinger, J. M. and Ward, J. C., Phys. Rev. 118 (1960) 1417. Silverman, B. D. and Joseph, R. I., Phys. Rev. 129 (1962) 2062. Szigeti, B., Proc. Roy. Sot. (London) A252 (1959) 217. Jaiswal, V. K. and Sharma, P. K., Physica 38 (1968) 409. Nettleton, R. E., Phys. Rev. 140 (1965) A 1453. Zubarev, D. N., Uspekhi fiz. Nauk 71 (1960) 71; (English transl. Soviet PhysicsUspekhi 3 (1960) 320). Van Hove, L., Phys. Rev. 95 (1954) 249. Bloch, C. and De Dominicis, C., Nuclear Phys. 7 (1958) 459. Gorkov, L. P., Soviet Physics - JETP 7 (1958) 505.