NGUYEN VAN TRONG: Neutron Scattering on Anomalous Fluctuations of Phonons 146

phys. stat. sol. (b) 88, 145 (1978)

Subject classification: 6 ; 11; 13.5.2

Laboratory of Theoretical Physics, Institute of Physics, State Committee for Science and Technology, Hanoi')

Neutron Scattering on Anomalous Fluctuations of Phonons BY

NGTJYEN VAN TRONG

The process of inelastic neutron scattering in crystal is investigated in the cam, where the crystal is suggested to be an electron-phonon system placed in an external radiation field. The cross-section of one-phonon scattering process is calculated by using the two-particle correlation function of phonon. It is shown that near the borderline of parametric instability region the cor- relation function of phonon and the cross-section of neutron consequently take anomalously large values.

Inelastische Neutronenstreuung in Kristallen wird fur den Fall untersucht, wo der Kristall als Elektron-Phonon-System betrachtet wird, das sich in einem IuBeren Strahlungsfeld befindet. Der Streuquerschnitt der Ein-Phononenstreuung wird mit der Zweiteilchenkorrelationsfunktion der Phononen berechnet. Es wird gezeigt, daB in der NIhe der Grenzlinie des parametrischen Instabilitatsbereiches die Korrelationsfunktion der Phononen und der Streuquerschnitt der Neutronen anomal groSe Werte annehmen.

1. Introduction The problem of neutron scattering in solids has caused attention of physicists for

a long time. In the classical work of Van Hove [I] the cross-section of neutron scattering is found to be expressed via the correlation function of phonons which are then calculated on the base of the fluctuation-dissipation theorem.

The fluctuation-dissipation theorem is not valid for non-equilibrium systems when the external field is presented. I n the present work the neutron scattering on fluc- tuations of phonon is studied in the case where the crystal is suggested to be an elec- tron-phonon system placed in an external radiation field. The phonon correlation function entering into the expression of the cross-section of the one-phonon scattering process is calculated by using the formalism proposed in our previous paper [2] . The phonon may be excited by the radiation field because of the electron-phonon inter- action. It is shown that the cross-section for inelastic scattering of neutrons by one phonon takes anomalously large values near the borderline of the parametric insta- bility region.

2. Cross-Section of Neutron Scattering

We start from the following Hamiltonian:

where V ( r ) = @r8 S(r - Rl,) is the potential of neutron-crystal interaction, @ls

the nuclear scattering amplitude Rl, the coordinate vector of atom s in unit cell I ; mN, p , and r are mass, momentum, and coordinate vector of the neutron, respec- tively, H ( t ) is the Hamiltonian of the crystal placed in an radiation field.

b

l ) Hanoi, Vietnam. 10 physica (b) 88/1

146 NQUYEN VAN TRONG

Using the Born approximation we may calculate the transition probability of a neu- tron from the state with momentum p and energy E = p2/2m, to the state with p' and E' = pt2/2mN (see I, 3) .

The transition probability turns out to be proportional to the correlation function (A;8r(t') Azs(t)), where

S( t ) = T exp { --i dt' H(t ' ) } , T is the Dyson chronological operator Uz8 = RZ8 - RE is the displacement from the equilibrium position Re, of the atom s in unit cell I , k = = p' - p with h = 1 the symbol (...) means the statistical averaging over states of the crystal.

For the one-phonon scattering process the correlation function (A;s,(t') Az8(t)) is approximately equal

Az8(t) = S-l(t) exp (ikUi,) S ( t ) . (2 )

t o

(A:e*(t') Aza(t)> x ( k * U$s,(t') k * Uzs(t)) . As the Hamiltonian H ( t ) of the crystal depends explicitly on the time due to the

presence of the radiation field, the correlation function (U$s,(t') UZ,(t)) depends in general on the times t and t' separately, but not on their difference t - t'. Therefore, its Fourier transformation can be written

However, the calculation (Section 3) shows that the spectral distribution of the cor- relation function approximately can be represented in the form

(u;s'(w) vZa(0')) = 2n 6(w' - w) (u;',', ZS)@ . (4)

In this case, the cross-section for inelastic scattering of neutrons by one phonon is found to be

and

The treatment for obtaining the above expression is based on the expansion of the displacement Vz,(t) in normal coordinates

and on the following notations:

B8 = ( @ I S > , dZ8 @)la - < @ I S > , <AhAl*.S') = & ~ Z Z & d Y

eqj is the phonon polarization vector of mode j and wave vector q, K the lattice recip- rocal wave vector, Rt the position vector of atom s in each unit cell, and dl2 the solid angle.

Neutron Scattering on Anomalous Fluctuations of Phonons 147

The cross-sections (6 a, b) contain the phonon correlation function (QEj), which must now be calculated. For simplicity we restrict ourselves to consider the one-phonon mode so the index j will be omitted.

3. Correlation Function of Phonons

the Hamiltonian H(t) in the form of second-quantization [2] For calculating the phonon correlation function (Qi), it is convenient t o represent

H ( t ) = Hoit) + Hint 2 ( 7)

Here ap,d(ai,8) is the annihilation (creation) operator of an electron with momentum p and spin 8, bk(b+) the phonon annihilation (creation) operator with wave vector k, Ao(t) = (cEo/wo) cos mot the vector potential of the radiation field, cp9 = 4ne2/qz and V k represent the Coulomb and the electron-phonon interactions, respectively. The normal coordinate Q, is defined by the relation Q, = b, + b t , .

Using the formalism proposed in [2] one may calculate the correlation function of phonons <Qi(t') Qq(t)) in the framework of the random-phase approximation.

In the two-mode approximation, the expression of the spectral distribution of the phonon fluctuations is expressed, in the following, as

(Q&(w') Qq(w)> = 2~ 6(0' - 0) <QE>, 9 (8)

x @(w - 0 1 ) + 6(w - w2) + 6(w + 01) + 6 ( 0 + cop)} , (9) where

Pp(w) = (0 - wq + i z g y - (0 + wp + i z ; y , EQ(W) = 1 - cp9P*(w) ,

148 NQUYEN VAN TRONG: Neutron Scattering on Anomalous Fluctuations of Phonons

are quadratic frequencies of plasmon-phonon coupled modes, ,u is the chemical poten- tjal of electrons, J,(& the Bessel function of the first kind, t,, and z, are formally introduced relaxation times of phonon and electron, respectively. In the following theory zel and zpl are assumed as infinitesimal.

In the absence of a radiation field (I2 = 0) the formula (5 a, b) yields the well- known results of the equilibrium case [3], where when T = 0 the spectral distribution of phonon fluctuations vanishes for positive values of w.

This fact means that the scattering process of a neutron by a created phonon is only possible in this case.

In the presence of a radiation field, the correlation function (Q",>, does not vanish for w > 0 in the T = 0 limit. Consequently the scattering process of a neutron by a phonon with both created and annihilated states is possible. One notes also that the intensity for the phonon annihilation process is proportional to the intensity of the radiation field.

Near the borderline of the parametric instability region in which 1 - I2 = 0 the function F ( q , o, T, P ) increases greatly as compared with the equilibrium function

- l)-l. Therefore, the neutron scattering cross-section (5) also takes anomalously large values.

[l] L. VAN HOVE, Phys. Rev. 95, 249 (1954). [2] NGUYEN VAN TRONQ and HA VINH TAN, phys. stat. sol. (b) 84, 345 (1977). [3] D. PINES, Elementary Excitations in Solids, New York/Amsterdam 1963.

References

(Received January 27, 1978)