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A DYNAMICAL BEHAVIOUR OF THE FERROELECTRIC SPINS
L. Novakovij
To cite this version:
L. Novakovij. A DYNAMICAL BEHAVIOUR OF THE FERROELECTRIC SPINS. Journal de Physique Colloques, 1972, 33 (C2), pp.C2-73-C2-74. �10.1051/jphyscol:1972220�. �jpa-00214956�
JOURNAL DE PHYSIQUE Colloque C2, supplkment au no 4, Tome 33, AvriZ 1972, page C2-73
A DYNAMICAL BEHAVIOUR OF THE FERROELECTRIC SPINS
The Boris Kidrich Institute of Nucl. Sci.
Theoretical Physics Group, POBox 522, Belgrade, Yugoslavia
RBsumB. - Nous introduisons un modkle general par 1'6tude du comportement dynamique des spins du systeme ferroelectrique au voisinage de la temperature de transition pour les cristaux ferroelectriques du type du phosphate diacide de potassium. L'hamiltonien gkn6ral du modele est diagonalis6 par I'introduction d'un ensemble d'operateurs de spin. Les zhangements de frkquences et les constantes sont calcul6s et introduits dans les expressions conventionnelles pour la partie coherente de la section differentielle dans le cas de la diffusion de neutrons par les composks deute- rCs.
Abstract. - A general model is introduced to study a dynamical behaviour of the system of ferroelectric spins in the neighbourhood of the transition temperature in hydrogen-bonded ferroelectric crystals of the potassium dihydrogen phosphate type. A general hamiltonian of the model is diagonalized by introducing a set of bosonic creation and annihilation operators in order to represent the spin variables. The frequency shifts and damping-constants are calculated and incorporated into the conventional expressions for the coherent part of the differential cross section for neutrons scattered by the deuterated compounds.
An effective hamiltonian was derived in the previous papers [I], [2] to describe the system of protons in hydrogen-bonded ferroelectric crystals such as dihy- drogen phosphate (KH2P0,) and its isomorphous substances assuming that each proton has two equili- brium positions. Including kinetic and potential energy of the protons the hamiltonian is expressed in the language of second quantization as follows
where the summations .are taken over all lattice sites (f, g denote the unit cells, while a,
P
denote the proton sites in the unit cells). The various coupling constants contained in the above hamiltonian are explained in detail in the cited references. The fictitious spin compo- nents appearing in eq. (1) have two distinct values, i. e. S,,, =4,
or -3.
T o analyse the spectrum of elementary excitations characterizing a given ferro- electric crystal we may use the method of second quantization in which the various spin components are replaced by a number of bosonic creation and annihilation operators. There are at present three such representations, one due to Holstein-Prima- koff [3], the other ude to Dyson-Malejev [4], [S], and finally the third due to AgranoviLToSi6 [6].Using any one representation the hamiltonian (1) may be expressed as a harmonic part and anharmonic part. The harmonic part is the same in all three representations and has the form
where B& and B,, are bosonic creation and annihila- tion operators, respectively ; q denotes a reciprocal lattice vector, a denotes the type of elementary excita- tions, while j = 1, 2, 3, 4 labels the frequencies.
In the low temperature limit all four frequencies are equal to each other, i. e.
where w and K are certain energy parameters. In the high temperature limit all frequencies are different, i. e.
where T, denotes the phase transition temperature, and Pj and Q j are certain parameters. The lowest energy frequency mode is characterized by
so it behaves above Tc like
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972220
This frequency often called the (( ferroelectric )) mode part of a neutron differential scattering cross section is closely connected with the dynamical susceptibility. has the form
The anharmonic part of the hamiltonian (1) needs d2gcoh - b2
5
S ( I ( , W ) ,a special treatment. We use the method due to Mara- dQ ds h k , (6) dudin and Fein [7] to study a scattering of neutrons
by an anharmonic ferroelectric crystal. The coherent where the scattering amplitude S(K, o) is defined by
Using the method of Green's functions we obtain for every elementary excitation a frequency shift A(qaa ; o) and a damping constant r ( q a a ; w ) in the high temperature region as follows
where J . . =
-
v J . . - K . . = v J . .-
?J JI '
The present method is applied to the ferroelectric mode to obtain
The present results can be tested by measuring an Such behaviour has already been seen in a deute- intensity of the coherent scattering. Our theory pre- rated single crystal KD2P04 [ 8 ] , so the present diets for such an intensity to be proportional to theory is a t least qualitatively in agreement with
12,
c~ T(T - T,)-',
T > Tc.
( 1 1 ) experiment.References
[I] NOVAKOVIC (L.), J. Phys. Chem. Solids, 1970, 31, 431. 1010 ; English tfansl. : Soviet Phys., JETP, 1958, NOVAKOVI~ (L.), STAMENKOVIC (S.) and VLAHOV (A.), 6, 776.
ibid., 1971, 32, 487. [6] AGRANOVICH (V. M.) and Tosrd (B. S.), Zh. Eksperim.
[2] NOVAKOVIC (L.), Introduction to Quantum Theory of i Teov. Fiz., 1967, 53, 149.
Solid State, Chapter V, lecture notes, unpublished. L71 MARADuDIN (A. A.) and FEIN (A. E.)y Phys.
1962, 128, 2589.
r31 HOLSTEIN (T.) and PRIMAKOFF
m.1,
P ~ Y S . ~ e v . , 1940, [81 UyER (W. J. L.), COwLEY (R. PAUL (G. L.l,58, 1098. and COCHRAN (W.), in Neutron Inelastic Scatter-
[4] DYSON (F. J.), Phys. Rev., 1955, 102, 1217, 1230. ing, Vol. 1 (International Atomic Energy Agency, [5] MALEEV (S. V.), Zh. Ekspevim. i Teor. Fiz., 1957, 33, Vienna, 1968), 257.
