GAPS IN PHONON DISPERSION CURVES FOR SUBSTITUTIONAL ALLOYS

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GAPS IN PHONON DISPERSION CURVES FOR SUBSTITUTIONAL ALLOYS

A. Czachor

To cite this version:

A. Czachor. GAPS IN PHONON DISPERSION CURVES FOR SUBSTITUTIONAL ALLOYS. Jour-

nal de Physique Colloques, 1981, 42 (C6), pp.C6-525-C6-527. �10.1051/jphyscol:19816153�. �jpa-

00221229�

JOURNAL DE PHYSIQUE

CoZZoque C6, supp26merzt au n o 12, Tome 42, de'cembre 1981

GAPS IN PHONON DISPERSION CURVES FOR SUBSTITUTIONAL ALLOYS

A. Czachor

Institute of NucZear Research, Swierk, Poland.

Abstract.

-

Neutron inelastic coherent scattering (NICS) crossection for alloy crystals often show multipeak structures and very low intensities between the peaks, the effects hardly within the reach of the C P A predic- tions. It i s shown. on the e x a m ~ l e of the force-defect linear chain. that the average local-information transfer approximation (ALIT A)

[ 31

leads to such effects.

In order that the potential energy be invariant with respect to an uniform transla-

U

tion of a solid, the self-force matrix

ell

^{must be}

CL

a 1

where

@li

i s the force matrix between lattice sites 1 and 1

.

Let us consider a substitutional alloy with two components A, B in concen- trations 1

-

c , c. Except in the additive limit [l],

5AB

⁼

⁽ 6AA ^{+ Q)}

-BB ^{)/2, it} i s difficult to fully incorporate the fundamental condition

(1)

into the CPA-type theories of disordered alloys

[z].

On the other hand the ALZTA

[3],

being rather crude in the decoupling procedure used to obtain the displacement-displacement Green S function Glt (t-3) =

((5

(t) 7?14(t')>>

,

accounts exactly for the transla- tional invariance. With the locator 4. L

a

where h$ i s a quasimass at the site 1 /in general complex/ and t) i s a frequency.

The time and space Fourier transform of the configuration average of the has here the form

[3,

47

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816153

C6-526 JOURNAL DE PHYSIQUE

The ALITA correctly predicts the dispersive local-mode branches and the in-band phonon frequency shifts for light impurities; for the heavy ones it leads to gaps in phonon spectra; usually the results appear in close analytic form. To give an example, Fig. 1 shows the mass-defect splitting in the

[oil]

T phonon branch for

96sAg. 035

L4I*

Fig. 1, Gap in the [001] T phonon branch of A1 e965Ag

.035 -

experimental data

[7l

and two ALITA fits [4]. Dashed Pine

-

dispersioa in pure

-

^#LIT*

----

^{*F.,nn "c.A A l .}

'

0 0.1 0 2 03 a4

The experimental NICS profiles usually show two peaks, but with a struc- ture superimposed, which may be not due to experimental scatter

-

see data for Rb,71K.29

[ 5 ]

; the valley between the peaks i s often deep and sharp

-

^{see data}

for N i Pt

[ G ] .

Qualitetively, such features a r e typical to the ALITA-plots;

,7

.3

the CPA-approaches so f a r gave at most two smooth peaks with a rather shallow minimum between them [l, 2, S].

To show it clearly, we shall examine within ALITA a simple system

-

^the

one-dimensional force-defect case; one can calculate here the ~ r e e n ' s function

(3)

analytically at arbitrary concentration c. It i s the two-species linear chain of identical masses and different coupling between different species (nearest neigh-

AA BA

bours only) :

(P

⁼ @BB = l

4

@AB =

4

^{= 1}

^{(1- A),}

M=l, where

I

^{i s the}

force defect. Eight 3-particle configurations pertinent to the case: AAA, AAB,

. . .

BBB, weighted: ( l - c )

3 ,

( I - C ) ~ C ,

. . .

c3, respectively, provide 8 contri- butions to the averages i n eqn

3 .

With the notation: r = (l-c) c,

1

⁼ M 3 / 2 1 , S 2

/ ,

the "dispersion relation1'

<c>-'

⁼ 0 takes the form

showing two force-defect gaps about the frequencies, at which there appear the poles of the locators, corresponding to the local configurations with one and two different neighbours. One should note, that with more distant interactions and with a higher dimensionality, a number of different local configurations and locators would be greater

-

there would be more gaps, and a multipeak structure in neutron scattering.

To introduce finite widths, let us put M-. M (l+i a)

.

The NICS cross sec- tion i s praportional to

h G

and we have

with B =

l,

sin 2 q / ~ 2 ( @ )

.

^{Fig. 2} shows this function for several wavevectors and 0( = .05, whereas Fig.

3

shows the effect of broadening of the NICS peaks, when o( increases. We can see that the broadening i s practically the one-side effect

-

in the gap regions the intensity is always extremally low.

Fig. 2 . Phonon dispersion in the force- Fig.

3.

Study of the peak broadening i n defect

( A =. ⁵⁾

linear chain at three the q=.4%ax NICS profile from Fig. 2, q-values, and the NlCS profiles i n the as a function of parameter

H

^{=. 1}

-

ALITA quasimass scheme : =ImM= .05. heavy line, .2

-

dashed,

.4 -

dotted.

The ALITA calculations for 3-dimensions clearly lead to similar features:

multipeak NZCS structures and low intensity between the peaks. Qualitative corre- lation of the predicted trends with the experimental findings emphasises the crucial

role of the condition

(I)

i n the theory of dynamics of disordered systems.

References

1. M. Elostoller and T . Kaplan, Phys. Rev. B16, 2350 ( 1 9 7 7 ) . 2 . G. Grunevald, J . Phys. F. : Metal Phys. 6 , 999 (1976).

3. A. Czachor, Report I N R No 173/II/PS/A, Warsaw, (1977).

4 . A. Czachor, Phys. Rev. B21, 4458 (1980).

5. W.A. Kamitakahara and S . R . D . Copley, Phys. Rev. B18, 3772 (1978) 6 . N . Kunimoti e t a l . , Y . Tsunoda, N . Wakabayashi, R.M. Nicklow and

H.G. Smith, S o l . S t . Comm. 25, 921 (1978).

7. A. Zinken e t a l . , U . Buchenau, H . J . Fenzl and H.R. Schober, S o l . S t . Comm. 22, 639 (1977).