TEXTURE PROBLEMS AND THE GOLDANSKII-KARAGIN EFFECT TREATED BY THE INTENSITY MATRIX METHOD

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TEXTURE PROBLEMS AND THE

GOLDANSKII-KARAGIN EFFECT TREATED BY THE INTENSITY MATRIX METHOD

H. Spiering, H. Vogel

To cite this version:

H. Spiering, H. Vogel. TEXTURE PROBLEMS AND THE GOLDANSKII-KARAGIN EFFECT

TREATED BY THE INTENSITY MATRIX METHOD. Journal de Physique Colloques, 1979, 40

(C2), pp.C2-50-C2-52. �10.1051/jphyscol:1979218�. �jpa-00218556�

JOURNAL DE PHYSIQUE Collogue C2, supplement au n° 3, Tome 40, mars 1979, page C2-50

TEXTURE PROBLEMS AND THE GOLDANSKII-KARAGIN EFFECT TREATED BY THE INTENSITY MATRIX METHOD

H. Spiering and H. Vagel

Institut fiir Anorganisahe und Analytisohe Chemie der Johannes Gutenberg-Univevsitat, D-6S00 Mains, B.R.D.

+Physikalliaohes Institut der Vnivevsitdt Erlangen-Nihmberg, D-8520 Evlangen, B.R.D.

Résumé.-Les problèmes de textures et d'effet Goldanskii-Karyagin sont facilement traités si les intensités de transition sont écrites dans le formalisme des matrices densitées. On donne ici explicitement les équations correspondant aux cas d'interactions magnétique ou quadrupolaire pu- res.

Abstract.- The texture problem and the Goldanskii-Karyagin effect are easy to survey if the inten- s i t i e s of the transitions are written in terms of the intensity matrix. For pure magnetic and qua- drupole interaction the equations are explicitly given.

I. Introduction.- I t i s often useful to rewrite well known relations in a new fashion, which threws light upon the problems from another direction. One should look upon the intensity matrix, previously introduced / l / , in this way.

If the eigenstates of the nuclear spin Hamiltonian are

11 g0> = m g 8 11 m > and l l e > = m e | l m >

1 gBB g 68 ' g g ' e a e a ' e e (0 where |I m> are eigenstates of the S operator re- ferred to a system S, the transition amplitudes for a pure multipole transition (L) are proportional to (p=+.l for left and right circular polarized light) :

c.,0 _ Z a.B L

ap M VLM UMp (a,6,Y) (2)

The rotation (a,B,y) rotates the system S to the system S' (e , e , e |j y ), in which the z-axis is parallel to the y-direction. VT' are tensors of rank L and depend on the eigenstates (1) :

..a, 6 ,2L+1 .2 1 , e.* g . . V ' = (-T , , j m ,m (e ) gD C(I L I ,m Mm )

LM 21 +1 e' g a &B g e' g e

6 (3)

For the incoherent absorption process of randomly distributed nuclei the products of the am- plitudes are summed up not the amplitudes themsel- ves, The absorber matrix

a,6_ aa,g ^ c B j * ( 4 )

pq p q

completely describes the absorption properties of the nuclei. The absorption of a y-beam with the density matrix p is proportional to Tr (p y ' ) . a B

ft R

Usually the y ' are summed-up over all nu- clei in the crystal for the direction under consi- deration. Equation (2) allows us to first take the

sum over all nuclei and then to transform it to te desired direction of observation. For this purpose the intensity matrix I is defined :

,a,B = ,va,8.« a,B t h a,B = fIa,B^ , „ W CVLM ' ^VLM' ' S O ttlat Ypq Up q ; W

It has the transformation property according to equation (2)

Ka.B.y) = DL(a,S,y)* I DL(a,B,y) (6) It completely characterizes the absorption

of the crystal of the whole Mossbauer absorber. It can be calculated in an arbitrary coordinate system, for example the coordinate system of the absorber S . The result of aMossbauer experiment therefore A contains the positions of the lines and its inten- sity matrices. The interpretation of this result deals with the sum over all nuclei I = SI and the N intensity matrix of each nuclei I . The sum may be N carried out over equivalent sites of the crystal, then I is the matrix of a single crystal, or a dou- ble sum over the crystals in a powder absorber.

To calculate the macroscopic intensity ma- trix of an magnetically ordered single crystal the transformation property under time inversion 0 is very useful. It becomes

<9 * e_1)MM' " (-'>*M' W - M (7>

All components of the matrix I can be measu- red if circular polarized -quanta are used. With linear or unpolarized -quanta only the combination

M+M1

h*

^{+ (}

-

^])

W - M

^{( 8}

>

can be obtained. In the following the I-matrix for a textured powder absorber is written down. These equation can easily be used to look at the conse-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979218

quences of an anisotropic Lamb Mzssbauer factor f for an ideal powder absorber, the well known Gol- danskii-Karyagin effect.

2. Texture.- In this section we assume that the f factor is isotropic. The absorber consists of small single crystals, which have the known I-matrix I C described in a coordinate SC fixed to the crystal.

The system

sA

is fixed to the absorber and is ro- tated by the Euler angles (a,B,y) to S (a,

c

B,y) .The relative rob ability to find a single crystal with theorientation

sC

(a,B,y) shall be T (a,B,y), the texture function of the absorber being ₃₀ and nor- malized to _:

This texture function is more general than that used in /2/ where only axial symmetries are consi- dered, so that T depends on only two angles. With the transformation property (6) we have in the sys- tem S A :

IA = DL (@,@,Y) 1' D~(~,B,Y)+

:lo)

The weighted average over all I-matrices I f

I

'

- , ^,

T(CX,B,Y) I~(~,B,Y) d~ (11) can be easily carried out, if the texture function is developed in a series of rotation matrices.

C L' L'

T(a,B,y) = L,, m, m1 t m r m Dmlrn(a,B,~) (12) Then the components of are given by

I P = C k L ' L L

jt ' 1 - 1

Iml

_I ^{j TLlrn'}

with the tensors of rank L' I

The components of of the powder absorber 1Tk are measured quantities. With the known I matrix of the single crystal (Ili), whose distribution is C under study, we have a set of linear equations for the texture coefficients tL' L' is 6 2 for the

mm' '

dipole transition. The ( 2 ~ + 1 ) ~ equations are in this case :

111 P = -I/& Too -I/& Tlo -l/&0 T20 100 P =

1 1 6

Too +2/&0 TzO P -I/& Too +I/& Tlo -1/&0 TzO 110 P = -I/& Tl-1 -1/To T2-l

P

10-1 =

-114

+1/fio T2-1

P

I01 = I/& T11 +l/Ji-0 Tpl P

1-10 = I/& T11 -1/Ko T2l P

11-1 = - 1 T2-2

I

! I

I = -I/& T22

If I is known the TLIm P are uniquely deter- mined. The equations for the texture parameters are:

T~~ =

1/J5

sp (1') t80, (t:o = 1 )

Two examples :

i) For a pure magnetic hyperfine interaction in a ferromagnetic material we can study the domain dis- tribution of the absorber. The I matrix of a do- c main becomes diagonal if the z axis of the coordi- nate system SC is chosen to be parallel to the ma- gnetic field at the nucleus. For two of the non zero transitions (a,B) we have for example :

The equations (16) reduce to

which determine the 9 texture parameters tmo, L L42.

ii) The intensity matricesof the quadrupole transi- tions at E+ and E- referred to the principal axis system of the EFG are /3/ :

I 1 l 1

with y,6 =

JZ

(1 7 (3/(3+~l~))~)~. ll is the asyuune- try parameter.

Then the tensors are :

3. ~oldanskii-Karyagin-Ef fect (GKE)

.-

^{On the f ol-}

lowing an ideal texture free powder is assumed, but the Lam ~Essbauer factor f is anisotropic. As has been done by Goldanskii et al. / 4 / , f=expE(T

$)27

itself is developed in a series of spherical har- monics instead of (y kI2. We can use the equations + + above if f is written as :

C L

f(f3.y) = L,m fzm DOm (0,B.y)

,

^{L = 0,2} (21) 0 = f3 and $=II-y are the polar angles of the y-direc- tion in the system S (a,B,y). The y-direction is C parallel to the z-axis of the absorber system

sA.

JOURNAL DE PHYSIQUE

The equations (12,13) reduce to p C k L ' L L

Ikk = L' (-1) (o i -k) TLvo (22) and

The I matrix is diagonal as a result of the rota- tional symmetry of an ideal powder.

If T20 vanishes, the anisotropic f factor does not change the absorption area of that transition. For a quadrupole doublet one obtains :

In cartesian coordinates the condition is

but the orientation of the f-tensor is arbitrary with respect to the principal axes of the electric field gradient.

References

/I/ Spiering, H., Hyperfine Interactions

3

⁽¹⁹⁷⁷⁾

213.

/2/ Pfannes, H.D. and Gonser, U., Appl. Phys.

1

(1973) 93; Pfannes, H.D. and Fischer, H., J.

Physique Colloq.

2

(1976) C6-45.

/3/ Spiering, H., Vogel, H., Hyperfine Interactions 4 (1977) 229.

-

141 Goldanskii, V.I., Makarov, E.F., Khrapov, V.V., Phys. Lett.

3

(1963) 334.