TWO-PHONON BOUND STATES IN AMMONIUM CHLORIDE AT FINITE TEMPERATURES

HAL Id: jpa-00221582

https://hal.archives-ouvertes.fr/jpa-00221582

Submitted on 1 Jan 1981

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

TWO-PHONON BOUND STATES IN AMMONIUM CHLORIDE AT FINITE TEMPERATURES

A. Anikiev, B. Umarov, V. Gorelik, J. Vetelino

To cite this version:

A. Anikiev, B. Umarov, V. Gorelik, J. Vetelino. TWO-PHONON BOUND STATES IN AMMONIUM CHLORIDE AT FINITE TEMPERATURES. Journal de Physique Colloques, 1981, 42 (C6), pp.C6- 152-C6-154. �10.1051/jphyscol:1981646�. �jpa-00221582�

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZe'ment au n012, Tome 42, de'cembre 1981 page c6-152

TWO-PHONON BOUND STATES I N AMMONIUM C H L O R I D E A T F I N I T E TEMPERATURES

A.A. Anikiev, B.S. Umarov, V.S. ~orelik* and J.F. ~etelino**

S. Umarov PhysicotechnicaZ i n s t i t u t e , Dushanbe, USSR

*P. Lebedev PhysicaZ i n s t i t u t e , Moscow, USSR

*Y Department of EZectricaZ Engineering, University of Maine, Orono, Maine 04469, U . S.A.

Abstract.- Two-phonon d e n s i t y of s t a t e s in 23 band of ammoni- um chloride c r y s t a l was c a l c u l a t e d by t h e metkod of Green's temperature f u n c t i o n s considering anharmonic phonon-phonon in- t e r a c t i o n . T h e o r e t i c a l l y c a l c u l a t e d temperature dependence of quasibound two-phonon s t a t e is compared t o t h e R a m a n s c a t t e r - m g experiments.

1. Introduction. - Along with t h e usual phonon s h i f t and widening t h e anharmonic phonon-phonon i n t e r a c t i o n leads t o t h e appearance of ano- malous s t r u c t u r e in c r y s t a l o p t i c a l spectra. The e x h i b i t i o n of t h e s e p e c u l i a r i t i e s may be explained by considering t h e processes of reso- nance o r bound phonon s t a t e s a s a p a r t i c u l a r case of t h e i r s c a t t e r - ing on each o t h e r and decay. The d e t a i l e d a n a l y s i s of conditions of resonaqce m d bound phonon s t a t e s formation was c a r r i e d out i n ^VIOP!~

/I/ by t h e Green's function formalism a t f=O.

Our approach i s s i m i l a r t o t h e formalism used i n work /I/. How- ever, we apply t h e temperature Greent s functions and consider t h e temperature e f f e c t s on t h e o p t i c a l phonon l i f e t i m e in a self-consis- t e n t manner.

2. Results.- The two-phonon temperature Green1 s function has a form

r e p r e s e n t a t i o n depending on "imaginary" time - temperature; fi i s a EIamiltonian o p e r a t o r containing t h e fourth-order anhamuonic term; fi

i s a "time1'-ordering operator.

In accordance with t h e r u l e s of t h e a n a l y t i c a l c a l c u l a t i o n of t h e diagrams in technique atr#O vre c o n t r a s t one loop i n Bethe-Solpe- t e r equation with t h e f u n c t i o n : n ( ~ - ~ ~ ~ - ~ ) = i ~ ) ( ~ - ~ ' ~ r ' ~ ~ ] % t h e momen-

( 0 )

-

turn representatior- a sbgle-phonon propagator Gi ( k , i a) has t h e following form:

( 0 )

+ -1

6, ( k , i ~ n ) = [fib ^-q + i f r ] ^{- [ d n} +q

-'!

,,,

where u z is a law of phonon d-ispcrsion, and r i s a small but f i n i t e phonon width a t low temperatures. The t o t a l two-phonon Green1 f unc-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981646

t i o n , obtained by summation of geometrical progression s e r i e s /I/

v ~ i l l be w r i t t e n a s follows:

~ , ( ( i w ~ ) = f l ( ~ ' , W ~ ) / ~ - Y ~ U ( ~ , ( 3 ) where Je is anharmonic coupling constant and

F

n (K, L W ~ L ' ~ ) =&j2 ^G ;P ⁽ k- E, L ^{~ f}- iuo,,.J ⁱ 6, (k, i @nj (4) In a s e l f - c o n s i s t e n t t r e . ~:icn% phonon damping $ p i n Lqs - . (2) t o (4) should be replaced by r m 2 ^(K, i ~ n ) The self-energy 2 ( K,t~n)was evalua- t e d a s t h e sum of t h e diagram ( a t lorn temperature approxknation) :

Z =

o-e

n n a l y t i c a l expression f o r f(KliO,) has a f o m :

AKT - + A ^{4 . - '}

~ ( ~ i ~ n ) - - ~ ~ j d J k ~ ~ ~ ( ~ c . d n ~ n . ) ~ , ( l ( ' * k , ~ ~ f l + ' ~ ' ) ^nl- - ^c@ (5) In o r d e r -to e v a l u a t e t h e i n t e g r a l s in Eqs (4.) , (5) we consider t h e

first term of propagator ( 2 ) and t h e i n t e g r a t i o n i n t e r v a l Pmm Ic)o t o

uC+@ where p i s one-phonon bandwidth. F.loreover, t o compare t h e c a l - culated d e n s i t y of s t a t e s t o t h e experimental Raman s c a t t e r i n g expe- riments we assumed t h e phonon p a i r momentum .-t K=O and make t h e replace- ment &*&= 1400cm-~ f o r NHLC1. The self-energy p a r t i s t h e n r e w r i t -

~ ( @ ~ ) & ~ ~ ( ~ ? b n ( ~ + i l (c)

~ - 2 ( & ~ + * - I c l " / t (d'

defined by Eq. (3) a t low tempera- se-f s c t o r . h i c t i o n ff(i0) expressed through t h e temperature depending phonon damping (6) i s given:

( 7 )

i n i n t e g r a l form obtained i.n work /5/ f o r t h e d i s i e r s i o n l a w R 2 d = pw:+

ypr(i

= 32cni-', A s a second s t e p , we c a l c u l a t e d two-phonon spectrum a t f i x e d value YL, ⁼ 32cmm1 and v a r i o u s temperatures. P ( W ) s p e c t r a , p l o t - t e d a s t h e f u n c t i o n of energy and temperature, a r e 2 shown in Fig.2.

When t h e t temperature increases, t h e two-phonob continuum i n t e n s i t y i n c r e a s e s t o o , and quasibound s t a t e i n t e n s i t y decreases. The decay of t h e quasibound s t a t e occurs a t T=280K in a small disagreement w i t h t h e experimental d a t a /4/. It should be noted t h a t t h e calcula-

C6- 154 JOURNAL DE PHYSIQUE

Fig. 1. Spectral function lotted a s a func- t i o n of frequency f o r d i f f e * ent values of anharmonic coupling constant.

The Raman spectrum studied in work /3/ i s shown by dotted l i n e .

t e d value of a coupling constant i s overstated a s we d i d not take i n t o consideration t h e hybridization process between t h e two-phonon continuum 2v4 and closely located one-phonon st8Be Vl= 3045 cm-I.

Fig. 2. The two-phonon spect-

ra &(L~,T) a s function of

frequency for various tempe- ratures. Two-phonon quasibo- un6. skate i s appeared a t temperature 1~2801:.

P3FEL3.&JC.t;S

I. 3.Ruvalds and B.Zawadowski, ;'hys.;:ev. BJ2, N4, 1172, 1370.

2. B..A.Abrikosov, L.F.Gorkov, B.?Z.Dzyaloshinski. Iethods of f i e l d quantum theory in s t - a l i s t i c a l physics. Physr~atgiz , ^I?oscov~, 1962.

3 . I:~.V.Belousov and D.F.Pogarev in "Oxidic L a t t i c e Vibr6i;ionst', '%a- uka" , Leningrad ( 1981, p. 265.

4. G. G.:',iit-in, V.S. Gorelik, I.l.d.Sushchinski. Sol.St.Plqs. (Soviet), 16, 10, 2956, 1974.

-

5. iV.A.Bowers and K.B.Rossnstock. J.Chem.Pliys. , 2 , ^10%, 1953.