ANALYSIS OF TEXTURE AND VIBRATIONAL ANISOTROPY BY MÖSSBAUER SPECTROSCOPY

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ANALYSIS OF TEXTURE AND VIBRATIONAL

ANISOTROPY BY MÖSSBAUER SPECTROSCOPY

H.-D. Pfannes, H. Fischer

To cite this version:

ANALYSIS OF TEXTURE AND VIBRATIONAL ANISOTROPY

BY

MOSSBAUER SPECTROSCOPY

H.-D. PFANNES

F B 6, Laboratorium fiir Angewandte Physik Gesamthochschule Duisburg, 4100 Duisburg, Germany

and H. FISCHER

FB 12.1, Angewandte Physik, Universitat des Saarlandes, 6600 Saarbriicken, Germany

Rksum6. - L'ktude des intensites des lignes dans un spectre hyperfin permet l'analyse des orientations pr6f6rentielles (textures) ou d'anisotropie des vibrations du rkseau. I1 est possible de developper en skrie des fonctions sphkriques, d'une part, la texture inconnue et, d'autre part, la dependance angulaire qui apparait dans les integrales de Pintensite. Ensuite les intkgrales sont simplifiks par une somme limitke des fonctions sphkriques avec coefficients inconnus. Quand on tient compte des relations qui existent pour les rotations des fonctions spheriques on peut calculer les coefficients par une methode de moindre carre sur la base des intensites relatives mesurks en positions differentes de source et d'6chantillon. L'application de cette m6thode aux probl5mes d'anisotropie de vibration permet de determiner le paramMre d'anisotropie et de distinguer la texture de I'effet Goldanskii-Karyagin.

Abstract. - The study of the line intensities in a hyperfine pattern makes possible an analysis of the texture or VA. The basic principles of the method are : The (unknown) texture distribution is expanded in a series of spherical harmonics and also the angular dependencies of the resonant absor- bed y-radiation which appear in the intensity integrals. These integrals thus are simplified to a k i t e sum of spherical harmonics with the unknown expansion coefficients of the texture function as factors. Respecting the relations for rotation of spherical harmonics a set of intensity ratio measurements for different angle positions of source (polarized) and absorber can be solved for the texture expansion coefficients by a least square fit procedure. Application of this method to the vibrational anisotropy problem makes possible the determination of the anisotropy parameter and allows to distinguish between texture and Goldanskii-Karyagin effect.

1. Basic concept of texture investigation by Moss- bauer effect.

-

The spatial orientational distribution of many physical parameters such as grain distribution inia polycrystalline metal or distribution of the spins in a magnetic material, are important as well for techni- cal and metallurgical applications as - especially in respect to spin structure problems - in physical research. For the orientation of an assembly we will adopt the expression texture, where the degree of orientation may range from ideal, complete orienta- tion (e. g. monocrystal) over preferred orientation (texture in its proper sense) until random orientation. The physical parameters whose texture we will deal with here are the principal axis of the electric field gradient

(Vz',,,

q = 0) or the internal magnetic field on the nucleus site. There are several possibilities of repre- senting a texture. We characterize it here by a function

D (8, cp) where D represents the probability to find

spins or VZzfs directed with the polar and azimuthal angles (8, cp) towards a surface element dQ (cf. [I]) of an unit sphere. For instance a random texture will be

where we respected the normalization condition

J

D(8, cp) dQ = 1

.

In a polar coordinate representation the random texture is represented by a sphere with radius 1/(4 z). The relative line intensities R in a hypertine splitted Mossbauer pattern will depend on D (neglecting relaxation-, saturation- and vibrational anisotropy- effects). The question arises which information can be obtained on the texture function D by examination

C6-46 H.-D. PFANNES AND H. FISCHER

of the relative line intensities. For unpolarized source source radiation considered here the clrm = 0, we radiation e. g. the ratio RBia of the /3 and cc line inten- obtain 6 expansion coefficients. More information

sities in a magnetically splitted hyperfine pattern is about texture cannot be obtained from Mossbauer investigation with unpolarized y-rays. The resulting

4 u

RBla =

-

--- texture

3 2 - u

I

where

The integration ranges over the whole sphere. It is clear, that the measurement of only one intensity ratio cannot give much information about the texture function D. Therefore one is led to rotate the absorber and determine more RBIa(o) where w describes the rotation w = (@, O, Y ) in terms of Euler angles of the absorber. In the integral the weight function sin2 0 then

must be correspondingly transformed. Since we will not restrict on unpolarized source radiation later on and in this case other weight functions than sin2 8 arise, it may be better to express the rotation in a general form. Regarding the weight functions for polarized source and absorber (cf. [2] and section 2 ) one notices that they are all composed of spherical surface harmonics. Thus the integral u can be expressed

by

where

Cl.m = 0 except for

The rotation is affected by

we call minimum-texture. It is compatible with the experimentally determined intensity ratios. The D'J" can be obtained from the experimental data by least squares fit of the intensity ratios with the known rotation angles corresponding to (5).

2. Polarized y-rays.

-

If not a single line source is used the source radiation may be polarized (in the general case partially elliptically). The polarization can be created by magnetic dipole or electric quadrupole interaction of the source. The intensity ratios depend - besides on D - on (8,, qs) and the considered transition in source (S,,,) and absorber (a, b). The used geometry is shown in figure 1. Representing the

FIG. 1. - Orientation relationship of source - and absorber -

quantization axes, hs and h.

where the mi,") are connected with the reduced rotation ratio Rslals,b(8s, q,, o ) by the corresponding intensities matrix elements. The D'," introduced in (5) as _{Isla and Is,,}

J

(6)

1 s t a

D"'" = D(8, q) y,,, dQ R s ~ ~ / s ~ ~ ( ~ s Y V)SY = -

Is2b (9)

have a simple meaning : if the texture function D is we obtain analogously to (5) expanded in a series of spherical harmonics

I S ~ , ~ ; ~ , ~ ( ~ S ~ , w, = D(Q, m) =

C

D ~ ' " ~ ~ , ' " (7) 1' 1'

I,, =

2

c::;;~,~

C

D " ' ~

w!:

;

(@, @, y )

.

(10)

1' m r = - z ' m = -2'

the

D',"

are just the expansion coefficients. Since in (5)

the number 1 is restricted (e. g. to 1 = 2 in the case of The coefficients ck;;ia,b can be calculated by expressing

functions contained in (10) it is possible to extract 9 transitions. The evaluation of experimental data is expansion coefficients of the texture (for M 1 radia- then very simple, input parameters of the program tion). The whole procedure is analogous to the method are only the measured intensity ratios, the kind of the described in section 1. It is useful that a fit program for used transition in source and absorber and the posi-

FIG. 2. - Perspective representations of some minimum textures. The expansion coefficients D1sm for textures a)-f) are : a) D l 3 0 =

-

0.05, 0 2 . 0 =

-

0.08 ;

b) D l 3 1 = 0.1, 0 2 9 0 = - 0.1 ;

C) 0 2 9 0 = -0.1, D 2 , 2 = 0.1 ; d) 0 2 . 2 = 0.14 ;

e) D l . 0 =

-

0.15, 0 2 . 0 = 0.05, D 2 , 2 = 0.071 ;

C6-48 H.-D. PFANNES AND H. FISCHER

tions OS, qs and u of source and absorber, respectively. The resulting minimum-texture

may not be positive everywhere on the sphere. But because an addition of more expansion terms to Dm, does not change the intensity ratios we can always make D 2 0 on the whole sphere.

In figure 2 perspective polar coordinate representa- tions of some Dm, are shown. The progressive distor- tion of the random sphere by the influence of the

several Dl7" can be observed.

3. Relations to vibrational anisotropy (VA).

-

If the absorber has no preferred orientation, i. e. random distribution, but exhibits a VA this leads to the Gol- danskii-Karyagin effect [3]. Because in this case D = 1/(4 n), the effect of the anisotropy of the Debye- Waller factor can be considered as if one had the texture DvA (cf. [I], [3])

where E is the lattice anisotropy parameter and the

proportionality constants K+, K- follow from the normalization condition (2) as

K + = JE/(2 n

471

erf (Ji))

,

E

>

0 ;

L

erf (x) =

)

exp (- t2) dt (error function)

FIG. 3.

-

Expansion coefficient 0 2 9 0 of the texture DVA (see

text) vs. anisotropy parameter e.

For unpolarized source radiation and magnetic splitting of the absorber the expression (3) can be

written

or correspondingly for electric quadrupole splitting of the absorber

and _{In both cases r(@, O, Y ) is independent of @ and Y and}

K -

=

JT

exp(~)/(4 n

F(J--E))

,

E

<

0 ; is given by

F(x) = expf- x2)

S

:

exP(t2) dt (Dawson's integral). Since the DvA is independent of 9, the expansion coefficients Dl9" = 0 for m # 0 and I = 1, 2,

...

In addition also D~~~ vanishes as can be seen by direct calculation of from (6). For D2*0 results accordingly

0 2 9 0 =

1

DVA Y2.0 d o (14)

The function D ~ ' ~ ( E ) as calculated by numerical integration is shown in figure 3.

After fitting of the experimentally determined r(O) corresponding to (17) and comparison of the resulting D2>O with (14) or figure 3 the anisotropy parameter E

can be determined.

The accuracy of this method seems to be higher than the usual determination of E by only one measurement

because the whole angular dependency of the relative line intensities is taken into account. The simple test whether D'," = 0 except for I = 2, m = 0 allows to distinguish between VA and texture because a texture exactly of the form (12) seems unlikely.

References

[I] GONSER, U., PFANNES H.-D., J. Physique, Colloq. 35 (1974)

C 6-113.

[2] FRAUENFELDER, H., NAGLE, D. E., TAYLOR, R. D.,

COCHRAN, D. R., VISSCHER, W. M., Phys. Rev. 126

(1962) 1065.