Localized modes of ammonium ion in KCl lattice

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Pramana, Vol. 17, No. 3, September 1981, pp. 257-270. © Printed in India.

Localized modes of ammonium ion in KCI lattice

R K E S A V A M O O R T H Y , V U M A D E V I and D E B E N D R A N A T H SAHOO,

Reactor Research Centre, Kalpakkam 603 102, India MS received 14 March 1981 ; revised 18 July 1981

Abstract. Following the standard Green's functions matrix partitioning technique, the force constant changes needed to explain the translational (6.8 THz) and torsional (10.1 THz) modes occurring in the KCI:NH~i system are calculated. Three different defect site symmetries are considered for the ammonium ion impurity. These are (i) Oh, in which the ammonium ion is a free rotor, (ii) Ta, in which it is a hindered rotor and (iii) C4v, in which it rotates freely about a N -- H ... C1 axis and librates around the other two crystallographic axes. Oh defect symmetry explains only the translational mode, while in the other two symmetries both the modes are explained with reasonable changes in the force constants. It is also shown that the same set of force constant changes explains the local modes in the deuterated sample as well.

Keywords. Local modes; molecular impurity; Green's function technique; force constant changes.

1. Introduction

The effects o f substitutional molecular impurity on the lattice vibrational modes of alkali halides have been havestigated in the past b o t h from theoretical and experimental viewpoints. Occurrence o f local and resonant modes is well established in these systems. Smith et al (1972) performed neutron inelastic scattering measure- ment on KC1 : N H + system and observed two local m o d e s - - a translational mode at 6.8 T H z and a torsional one at 10,1 THz. They also observed a broad peak near 7 T H z in the deuterated sample. When a tetrahedral ammonium ion substitutes a potassium ion at the octahedral site, the defect point group can either be of Oh symmetry or lower; if the ammonium ion is assumed to undergo free rotations about all the three axes, it conforms to Oh symmetry whereas if one allows for hindered rotations a b o u t one or more axes, the symmetry is lower. Kaplan and Mostoller (1974a, b) have used the coherent potential approximation (CPA) incorporating the mass and force constant changes to explain the local modes of KCI : NH~- system.

CPA with mass disorder alone predicts a quasi-resonant mode at 5.80 T H z while CPA with mass defect and force constant change accounts for a mode at 6.8 THz.

In both these calculations, the point group o f the defect was taken to be Oh and the high frequency torsional mode o f the ammonium ion remained unexplained. Re- garding the possibility o f considering defect site symmetries lower than Oh, we have the following experimental results. Infrared measurements of Plumb and Hornig (1953) suggest a uniaxial rotation for the NH~ ion in some alkali halides. (See however, G o y a l and Dasannacharya 1979). In this paper we consider in addition to the 257

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258 R Kesavamoorthy, V Umadevi and Debendranath Sahoo

Oh defect symmetry, the possibility of the defect site acquiring Ta and C4v symmetries.

In the case of Td symmetry, the NH + ion librates around all the three axes as a hindered rotor whereas, in the case of C~o, it rotates freely about N--H...CI axis while librat- ing around the other two crystallographic axes. We follow the standard Green's function matrix-partitioning technique under the purview of the rigid-molecule approximation. Our analysis indicates that when the structure of the ammonium impurity is taken into account, both the translational and torsional modes in the KCI:NH~ + system can be explained. Also, the same set of force constant changes explains a translational mode at 6.4 THz and a torsional mode at 7 THz in the KC1 : ND~ system. A brief account of this work was reported earlier (Kesava- moorthy et al 1980). The theory pertinent to our analysis is presented in § 2 ; § 3 contains a description of various models and the results; conclusions based upon our results are discussed in § 4.

2. Theory

The time-independent equations of motion for a crystal containing a substitutional molecular impurity in the harmonic approximation can be written in the matrix form as

(oJ ~ M - - 0 - - ~ L ) U = 0 ,

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where M and ~ are the mass and force constant matrices of the perfect crystal and 8L is the perturbation matrix arising due to the presence of the impurity. The elements of 8L are given by

8L=~ (lx, l'x'; o, 8) = w= AM 8u' 8xK' 8/~ + k ~b~ (Ix, l'g'). (2) We use the notation of Maradudin et al (1971) --1, l' denote lattice points; K, x' are sublattice indices; a,/3 are cartesian components (x, y, z); oJ is the phonon frequency;

AM is the difference in mass between the defect and the host atom and Aft is the change in the force constants.

The non-vanishing part of 8L is of dimension (3p + 6) where p is the number of host atoms with which the rigid molecular impurity interacts, and the other six degrees of freedom come from the translation and rotation of the impurity.

Following Sahoo and Venkataraman (1975), the perturbed frequencies can be obtained from the secular equation

det [1 -- g" (co 2) 4 " (oJ2)] = 0, (3)

where 8i" = 81't + 81" G" 81't;

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the superscripts t and r refer to translation and rotation respectively. It may be noted that 81 tt and gtt are (3p + 3) × (3p + 3) matrices and G" is a 3 × 3 matrix. Group theoretic considerations permit us to work with $i ;t and g" matrices of lower dimensions. When 81 ;t and gtt are projected onto the irreducible representations in

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Localized modes of NH~ in KCI 259 which the impurity atom moves, these have a dimension a where a is the number of times that particular representation occurs in the character decomposition of the defect point group. Therefore, our analysis consists essentially in,

(i) finding out the character decomposition for the defect point group, (ii) obtaining the symmetry adapted vectors,

(iii) projecting g and ~! in the appropriate irreducible representations, and (iv) evaluating the necessary force constant changes required to produce the

observed local mode frequencies.

3. Models and results

When a tetrahedral ammonium ion substitutes a potassium ion occupying an octahedral site in the KCI lattice, several defect point groups are possible (Goyal and Dasannacharya 1979). In the following analysis, we assume that the ammonium ion is rigid and interacts only with its six first nearest neighbours. The defect point group is then of twenty-four dimensions. Before considering the various possible models of the defect-host system, We outline the procedure adopted for the computation of the perfect lattice Green's function. First the phonon polarization vectors and frequencies were computed by the standard Gilat-Raubenheimer method (1966) using the shell model parameters (model V of Copley et al 1969). For this calculation, 182 points were chosen in the irreducible part of the Brillouin zone. Green's functions upto four neighbours were calculated using the standard expression (see p. 66 of Maradudin et al 1971). The symmetry forms of the Green's function matrices are given in appendix 1. The independent elements were calculated by reducing the standard expression to the appropriate symmetrised forms. The imaginary parts of these elements were first calculated by a discrete lattice summation. The real parts were obtained from these by Kramers-Kronig relation.

We consider below three possible models of the defect point group of the KCI : NH + system.

3.1 O n defect symmetry

We imagine that in figure 1 the ammonium ion can freely rotate about all the three axes. Such a model conforms to the defect point group Oh. Using the sum rules operating between the force constant changes A¢ tt, A¢ "t and A¢ "' (Sahoo and Venkataraman 1975) we find that

a i ; ' = 81" as 81'~---- ~1~'= 0. (5)

This result is to be expected since in this model, the ammonium ion rotates freely about all the three axes and consequently, there is no coupling between rotation of NH + and translations of its nearest neighbours. Therefore, in this model we can expect to explain only the translational mode.

The different vibrational modes of this point group, when character analysed, decompose into the various in-educible representations as

I ~t°t (Oh) = Az. -~ E. -+- 3F1. + Flo + Va. + F20. (6)

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z I 13

I 11 18 16 • CI

I -I- - - -a6ff9

I J " I .,.." 6 I / " -

I..."- & " i . /

12

Figure 1. Ammonium impurity in potassium chloride lattice.

The symmetry adapted basis vectors calculated using projection operator technique indicate that the impurity ion translates in the F 1, representation. Making a trans- formation to this irreducible representation ~ " and g" become 3 × 3 matrices listed in appendix 2. Solving equation (3) in the F lu representation, we find that AE is a function of A;, (equation (A. 1) of appendix 2). Here A¢ = ~ - E* and

Ay = 9' -- y*, the starred quantities correspond to the crystal with defects and the unstarred quantities refer to the perfect crystal. These are presented in figure 2a in which the nature of these force constants is self-explanatory. Figure 3 shows a plot of AE vs Ag,. From this figure it can be seen that the translational mode in the KCI : NH~- system at 6.8 THz can be explained with A¢/¢ = -- 0.197, i.e. a 19.7%

increase in the radial force constant (when the tangential force constant change (A~,) is set equal to zero). This compares well with a 23 % increase obtained in the CPA calculation (Kaplan and Mostoller 1974). With a finite decrease in ~,, the value of

A~ required to explain the mode decreases.

3.2 Td defect symmetry

In this model it is assumed that the three-fold axes of the impurity molecule coincide with the three-fold axes of the defect site and that the ammonium ion librates about all the three crystallographic axes. From the character table, the total representation of this group is found to be

r t°t ( r 9 = A1 + E + 2Fz + 4 F~, (7)

and it can also be noted that the impurity ion can translate in the F~ representation which appears four times in the total representation and can rotate in the F 1 representation that appears twice.

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Localised modes of N H + in KCI 261 Z

£ ~ ~ £ ~ ~ E 3 ~

o - ~ ~ ~ p ~ Y

X

t L, t

t.--IIIIII-o"' "t et Td

O h C4v

• NH~,

• C!-

(a)

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Figure 2. Pictorial representation of the force constants in the various defect p.oint group symmetries. Here c refers to the radial spring and y, 8 the tangential springs.

Arrows indicate displacements of ions.

,_' ,_, , _o, , %

ZI~ ( 10 4 dynes/cm )

- 4

- 8

-12

m

I I

~ 2

o~

0 ) t -

O

,0

Figure 3. Av vs AE in the Flu representation of Oh symmetry.

3.2a F~-representation: First let us try to explain the translational mode occurring at 6.8 THz in the KCI : NH~ system. Making use of the sum rules as before it can be shown that this model has three independent force constant changes denoted by AE, A7 and Ab (Sahoo and Sahu 1981). Here, Ae refers to the change in the radial force constant connecting the translations of the impurity and its nearest neighbours along the interatomic direction; A~, is the change in the tangential force constant connecting the translations of the impurity with its nearest neighbours transverse to the interatomic direction and parallel to each other; and A8 is the change in the tangential force constant connecting t~e translations of the impurity and its nearest neighbours transverse to the interatomic direction and perpendicular to each other.

A pictorial illustration of these force constants is given in figure 2b.

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Solving the 4 × 4 determinantal equation (3) corresponding to the F 2 representation, one obtains A~ as a function of A~, and A3 (see equation (A.2) in appendix 2).

Let us first allow for a change in the radial force constant only i.e. set Ag, --- A3 = 0.

This means that the coupling between the rotation of the impurity ion with the translations of its nearest neighbours vanishes. In this simplified picture, a translational mode at 6.8 THz as observed by the neutron scattering experiment (Smith et al 1972) can be explained with A~ = -- 4508.4 dynes/cm that corresponds to 19"7~o increase in ~. The same change in ~ explains a translational mode at 6.4 THz in the KC1 : N D ] system. We next consider the case where AE, A~, ~ 0 and for simplicity set A3 = 0 . Figure 4 shows the variation of A~ as a function of A~ for 6-8 THz. Allowing for a change in 3, as much as 20 ~o of E, the curve in figure 4 is not altered significantly.

3.2b Fl-representation: As noted earlier, the torsional mode of Tn symmetry belongs to this representation. Making use of the projected g,t and 81" matrices (given in appendix 2) and setting the tangential force constant change A3 to be zero we arrive at a third degree equation in A7 (equation (A.3) of appendix 2). Substituting the observed frequency of the torsional mode viz. 10.1 THz for the ammonium ion in the KC1 lattice, we find that the only physically reasonable solution of equation (A.3) corresponds to A v = 1962.08 dynes/cm --- 0.086~ i.e. the tangential spring decreases by 8.6 ~o of the host crystal radial spring. It may also be noted that the s~.me change in the force constant explains a torsional frequency at 7.0 THz in the deuterated sample.

3.3 C4v defect symmetry

We consider now in figure 1 that one of the N - - H bonds points always towards a chlorine ion. In this case, the impurity NH~- is allowed to rotate freely about this N - - H bond and librate about the other two axes. The defect point group of such a configuration is C4~, The total representation for this point group decomposes as

F '°' (C4J = 5A1 -k- A~ q- 2B~ + B 2 q-- 6E.

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I -12

~ - 4

2

<3 --8

D

I 0

I I I I I

4 8 12

~1~ (104dynes/era)

Figure 4. A~' vs A~ in the F2 representation of Td symmetry with A3 ~ 0.

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Localized modes of N H + in KCI 263 From the character table it can be seen that the symmetry modes in which the impurity ion translates can belong to the A~ or E irreducible representations. Similarly the torsional mode may belong to either As or E species. To find the irreducible representation to which the translational and torsional mode of N H + belong, let us first project ~tt on to the A~-representation. We find that ~tt (As) = 0, implying that no local mode can belong to this species. Therefore, the torsional mode belongs to the E-representation. Let us suppose now that both translational and torsional modes are contained in the E-species. For simplicity, we set the changes in the two tangential springs (indicated as 71 and ~'9. in figure 2c) equal. Then, we find that the conditions obtained from the determinantal equation cannot be satisfied consistently by both the frequencies. Hence, both the translational and torsional modes cannot belong to the E-type of vibration. Therefore, we assign the translational mode to the A 1 - representation.

3.3a Al-representation: Applying the sum rules on force constants we find that the 5 x 5 secular equation obtained from the ~tt and gtt given in appendix 2, contains three independent force constant changes. We denote them by AEa, AE x and A~z, and these are pictorially represented in figure 2c. A~ 3 is the change in the radial force constant connecting the translations of the impurity and its nearest neighbours in the plane perpendicular to N - - H . . . CI bond along the interatomic direction and Aq,z are the changes in the radial force constants connecting the translations of the impurity and its nearest neighbours in the positive and negative N - - H . . . C1 direction. To simplify our calculations, we set Aq : A~ = Ac. Substituting the observed frequency of 6.8 THz in the secular determinant, we obtain a second degree equation in A~. One of the solutions of this equation relates AE to A~ 3 (see equation (A.5) of appendix 2). This solution is not acceptable as it yields unphysically large values of A~ (as much as 3E) for all values of A¢ 3. The other solution is AE= -- 4505.4 dynes/cm;

this corresponds to a 19"7~o increase in the host lattice radial force constant. The same value of AE also explains a local translational mode at 6-4 THz in the KC1 : ND~

system.

3.3b E-representation: Once again, application of the sum rules on the force constants shows that the 6 × 6 secular equation of this species has A~z, A71 and A72 as independent force constant changes, where AEz is the same as in case (i) and, A9'1 and A72 are the changes in the tangential force constants connecting the translations of the impurity and its nearest neighbours in the positive and negative N--H...CI direction transverse to the interatomic direction and parallel to each other. These springs are indicated in figure 2c. The secular determinant yields, in this case, a third degree equation in AT. Two of the roots of this equation are not acceptable as these imply unphysically large force constant changes. The third root gives an increase in the tangential spring amounting to 4.4% of the host radial spring i.e. A 7 = -- 1006.7 dynes/era = -- 0.044 ~. The same analysis in the deuterated system shows that a torsional mode at 7 THz can be explained with the same value of AT.

4. Conclusions

In order to explain the two local mode frequencies observed in the KC1 : NH~- system (Smith et al 1972) we considered three models for the motion of the ammonium ion

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belonging to the Oh, Td and C4v defect point groups. The model with Oh symmetry does not allow for a coupling between the rotation of N H + and the translations of its nearest neighbour CI and so cannot explain the torsional mode at 10.1 THz. Thus the free rotation model for the impurity ion is ruled out. When the structure of the impurity is taken into account, i.e. in the T~ and C4v defect point groups, both the translational and torsional local modes of NH~- in KCl lattice can be explained with reasonable changes in force constants. Unlike the Oh symmetry, as these models contain the moment of inertia as a parameter, the local modes in the deuterated sample can also be explained. Assuming that the same set of force constant changes apply in the isotopic case, we find that in the KCI : ND + system, a torsional mode at 7 THz and a translational mode at 6.4 THz are produced. In the neutron scattering experiment, Smith et al (1972) observed a broad band of low intensity in the vicinity of 7 THz. The resolution in their experiment was poor to show the torsional and the translational local modes separated clearly. From our analysis, we have shown that the structure of the impurity cannot be neglected when one is attempting to explain both the local modes. An unambiguous choice between the other two models is not possible with the available experimental data on KCI : NH~- system. From the character table, it can be seen that in Td symmetry the torsional mode is only infrared active while in C4v symmetry it is both Raman and infrared active. Therefore, the appearance of a local mode at 10.1 THz (,-~ 300 cm -I) in a Raman scattering experiment will favour C~ symmetry while its absence will favour Td symmetry for the impurity ion. However, a form factor analysis from the quasi-elastic neutron scattering experiment on KCI : NH + system, similar to those on KI : N H + and KBr : N H + (Goyal and Dasannacharya 1979) may lead one to choose the correct model for the impurity ion.

Appendix 1

We write below the Green's function matrices g(m, n), m, n = 0, 1... 6 relevant to our first neighbour defect space model. Here m and n label the atoms in the perfect

crystal (see figure 1).

The superscript + corresponds to the quantities calculated with positive ion at the origin (i.e. m ---- 0) while those with negative ion at the origin are given without any

m•

⁰ ¹ ² ³ ⁴ ⁵ ⁶

+

g+ o o g~+ o o g~+ o o g~+ o o g~ o o g~+ o o g+ o o

0 0 g+ 0 0 g+ 0 0 g+ 0 0 g+ 0 0 g+ 0 0 g+ 0 0 g+ 0

+ 0 0g~+ 0 0g~+ 0 0g,+ 0 0 e~ + 0 0g~+ o 0 g+~

0 0 g,

g+ 0 0 gl 0 0 g9 0 0 g , - g ~ 0 g4 g6 0 g4 0 - g 5 g4 0 g5 0 g+ 0 0 gl 0 0 glo 0 - g s g 4 0 g~ g, 0 0 ge 0 0 ge 0 0 0 g+ 0 0 gl 0 0 gxo 0 0 go 0 0 ge - g s O g, g~ 0 g t

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Localised modes of NH~ in KCI 265 superscripts. The values o f the independent elements occuring in the Green's function matrices corresponding to the frequencies of interest in this paper are given below:

Frequency(TH~

6.425 6.775 7.025 10.075

g+ 1.6237 E--05 1.3139 E--05 1.1645 E-05 4.4860 E--06 g+ --3.2043 E--06 --1.9940 E--06 --1.5559 E--06 --2.3147 E--07 g+ --1.5568 E-06 --7.5788 E--07 --5.4275 E--07 --6.2591 E--08 gx 1.8053 E--05 1.4573 E-05 1.2908 E--05 4.9597 E--06 g~ 7.7308 E-07 2.3747 E-07 1.3374 E--07 --7.1507 E--10 g5 --1.0051 E--06 --6.1730 E--07 --4.6861 E-07 -5.9931 E--08

g6 1.1604 E-06 4.8406 E-07 3.2651 E-07 2.9155 E-08

g9 --5.5575 E--08 -1.8843 E--07 --1.7093 E--07 --3.6374 E--08 gl0 7.0520 E-07 2.4239 E--07 1.5375 E--07 1.1401 E--08

In this table E-On means 10 -n

Appendix 2

We list here the projected perturbation matrix 8[ and Green's function matrix, g in the various representations considered in the text.

A2.1 Oh symmetry: Flu representation

(FI ) = - - A~, 0 2Ay )

0 - -

2Ay V'2A, b

where b = --2 (A~ + 2 A~,) + AM co~

AM : - M h ° s t - M impurity

= 21.063 a.m.u, for N H +

= 17.039 a.m.u, for ND~

and the force constants are as shown in figure 2a.

g =

2g6 + gx + glo 2a/if g4 2g + \ 2 a/2 g4 gl + g9 ~/if g+

)

2 g+ a/if g~ g~-

where the super- and sub-scripts on g are as defined in appendix 1.

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266 R Kesavamoorthy, V Umadevi and Debendranath Sahoo Setting det (1 - g

6i)

⁼0, we find,

where K, = (1 - g: AM u2), n = 1, 2, 3 (throughout this appendix)

A 2.2 T, symmetry: F2 -representation

The secular equation after some rearrangement yields, Aa = - Kl

+

^{B Ay}

+

^{D (Ay2}^-^AS2)

K'

+

^G^Ay^-L ^(Ay2^-^{~ 8 % )}^' ^(A.2)

where B = 2K1(g1 - g,

-t

^g6- 2 gl) -- 4 ~,(g,f - g:)

,

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Localized modes of N H ~ in KC1

267

Fl representation:

where ^p⁼ a2

021 - a2 ^{~ y '}

a is the K+-CI- distance = 3.12Ay

and I

is the moment of inertia of the impurity ion. For ammonium ion I = 4.915

x

104gm cm2 (calculated from the relations given in Parlinski 1969). I of NDf is taken to be 9.83

x

10-40gm cm2.

when AS is set equal to zero, the secular equation yields

A2.3 C4, symmetry: A , representation:

where c ⁼

-

(Aq

+

^AGJ

+

^wa^AM

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268 R Kesavamoorthy, V Umadevi and Debendranath Sahoo w h e r e d ---- g l - - g~ - - 2gs,

e = g x + g l o + 2 g 6.

T h e e q u a t i o n o b t a i n e d f r o m t h e secular e q u a t i o n is very lengthy a n d so f o r simplicity, we set A q = A ~ = At. This results in

{ [ 1 + ½A~ a (gl - - ga - - 2g5)] [1 + A , (gl - - ga)] - - 4 AE a A , g ~ }

× { [ K 1 + 2(g + - - g+) AE] [1 - - (2g + - - g l - - gg) A¢] - - [2(g~- - - g+) A¢]

× [g+ AMc°~ - - (2g + - - gl - - gg) A~]} = 0, (A.4) i.e. either

w h e r e

o r

AE = - - 1 + B ' A~ a A + A ' A~a' B' = ½ ( g l - - ga - - 2gs), ,4 = (g~ - g~),

,4' = ½ (g~ - g . - 2g~) (g~ - g~) - 4g~,

_ Kx

A , = K 1 [K 1 (2g + - - gx - - gg) - - 2Ks (g+ - - g + ) ] - I _ _ ~ "

E representation:

T h e (6 × 6) ~ / m a t r i x is given b y

q o o ° o

n r 0 0

~l (E) = AE 8 0 0 - - Ac s 0

0 0 0 0

(A.5)

w h e r e

(A.6)

p = - - ( A ~ I + Ay 2 + 2A%) + AMid" W (A~'2 - - Ayl) z a ~ x, q = - - A ? I + A y ~ a z x ,

r = - - A y g + A y ~ a 2 x ,

1 = Ay~ + Ag't (Ay2 - - Ayl) a 2 x, m = Ayz - - ATs (AYs - - Ayl) a2 x, n = - - ATx A y 2 a ~x.

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Localized modes of NH~ in KCI 269 Here a is the distance between K + and CI- as before and

x = l/[to' I - - (A~, i q- A~2 ) ag"].

g (E) =

g¢

0

o gi gto Vt2 g4 - - ~/2 gs

gto gt ~ 2 g~ V'2 g5

X/2 g4 V 2 g4 gl + g9 0 - - V 2 g~ ~/2 g6 0 gl --glo

V'2 g6 ~/2ge 2 g4 0

v5 d

V£ g.

2g~

0 gi - - gto

Here again we set Ayt = A~,~ = A7 in the secular equation. The resulting equation is

[(K 1 --}- K' A, a) q- (Q -}- Q' Aea) A),] [to' I q- {(to9 I) (gi - - gi0)

- - 2 a ' } Ay ~ 4a' (gi -- gto) AY ~] = 0,

(A.7)

where K, and K' are as before

Q = Kl(gx + gto -- 2g +) + 2 K a ( g ~ - g+),

Q' = K1 [ (gi + go - - 2g +) (gi + gto - - 2g +) - - 4 (g+ - - g4) (g+ -- g~)]

+ Ks [2(g + - - g+) (gi q- glo - - 2g +) + 4 (g+ -- g4) (g+ - - g+)]

q- K a [2 (g~ - - g+) (gi + g9 - - 2g +) q- 4 (g+ - - g+) (g2 + - - g4)], This leads to, either

or

A 7 -- K1 + K ' A ~3, (A. 8)

Q + Q ' A E 8

2a2--to s I (gi--gto) ~ [4a4+ to412 (gt--gio)2+ 12to' Ia 2 (gi--glo)] 1/' A T =

8aZ (gx--gi0)

(A. 9)

References

Copley J R D, Macpherson R W and Timusk T 1969 Phys. Rev. 182 965 Gilat G and Raubenheimer L J 1966 Phys. Rev. 144 390

Goyal P S and Dasannacharya B A 1979 J. Phys. C12 219 Kaplan T and Mostoller M 1974a Phys. Rev. 139 353 Kaplan T and Mostoller M 1974b Phys. Rev. B10 3610

Kesavamoorthy R, Umadevi V and Sahoo D 1980 Prec. NucL Phys. and Solid State Phys. Symposium (Delhi, Department of Atomic Energy) ( to be published)

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270 R Kesavamoorthy, V Umadevi and Debendranath Sahoo

Maradudin A A, Montroll E W, Weiss C H and Ipatova I P 1971 in Solid state physics (ods.) F Seitz and D Turnbull (New York: Academic Press) Supplement 3, Second Edition, Chap. 8 Parlinski K 1969 Acta Phys. Pol. 35 223

Plumb R C and Hornig D F 1953 J. Chem. Phys. 21 366 Sahoo D and Sahu H K 1981 (in preparation)

Sahoo D and Venkataraman G 1975 Pramana 5 175

Smith H G, Wakabayashi N and Nicklow R M 1972 Neutron inelastic scattering (Vienna: Inter- national Atomic Energy) p. 103