TEXTURE PROBLEMS

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TEXTURE PROBLEMS

U. Gonser, H.-D. Pfannes

To cite this version:

U. Gonser, H.-D. Pfannes. TEXTURE PROBLEMS. Journal de Physique Colloques, 1974, 35 (C6), pp.C6-113-C6-120. �10.1051/jphyscol:1974610�. �jpa-00215710�

JOURNAL DE PHYSIQUE Colloque C6, suppliment au no 12, Tome 35, Ddcembre 1974, page C6-113

TEXTURE PROBLEMS

U. GONSER and H.-D. PFANNES

Fachbereich Angewandte Physik, Werkstoffphysik und Werkstofftechnologie Universitat des Saarlandes, Saarbriicken, Germany

Rhumb. - L'orientation prkfkrentielle d'une assemblee est appelee texture, I'assemblke pouvant comporter des cristaux, des molecules, des spins, les axes principaux du GCE A des sites specifiques, etc. On analyse l'influence de la texture en tant qu'effet extrinshque (ou accidentel) sur les intensitks relatives d'un spectre hyperfin et compare a des effets intrinshques (caracteristiques du matbiau) telles la dependance angulaire des spectres hyperfins et l'anisotropie des vibrations du rkseau donnent lieu a 1'Effet Boldanskii-Karyagin EGK. On discute la corrklation entre la texture et les paramhtres Mossbauer prksentant une dkpendance angulaire. 1) L'orientation des spins dans un matQiau magnetique ou des axes principaux du GCE peut &re dkterminke en corrklant les textures obtenues par RX aux intensites relatives des spectres hyperfins. 2) Inversement, on peut determiner la texture en analysant les intensites relatives des spectres hyperfins en fonction de la direction de propagation des rayonnements y. 3) L'attention principale est consacrke a la corrklation entre l'EGK et la texture. L'effet GK est devenu une mkthode importante de dktermination de I'anisotropie des vibra- tions du reseau ; il est cependant difficile de distinguer les deux effets en competition. Quelques exemples dkmontrent l'influence sensible de la texture sur les spectres hyperfins en particulier dans des cas ou une faible orientation produit sur les intensitks relatives des raies un effet identique a celui d'une anisotropie relativement elevke des vibrations thermiques du reseau.

Abstract. - Texture is the preferred orientation of an assembly. The assembly may consist of crystals, molecules, spins, principal axes of EFG's at specific sites, etc. The influence of texture as an extrinsic (accidental) effect on the relative line intensities of a hyperfine pattern will be analysed and compared with the intrinsic effects (characteristic of the material) like angular dependence of the hyperfine interaction and lattice vibrational anisotropy causing the Goldanskii-Karyagin effect, GKE. Correlation between texture and angular dependent Mossbauer parameters will be discussed : 1) The orientation of the spins in a magnetic material or the principal axes of the EFG can be determined by correlating X-ray textures with the relative line intensities of the hyperfine pattern.

2) The reverse : the texture can be determined by analysing relative line intensities of the hyperfine pattern by varying the propagation direction of the y-radiation. 3) The main interest is focused on the correlation of GKE vs. texture. The GKE has become an important tool to determine lattice vibrational anisotropy ; however, it is difficult to distinguish between these two competing effects.

Some examples will demonstrate the sensitive influence in the hyperfine pattern by texture, parti- cularly, cases where a relatively small deviation from randomness produces the same effect on the relative line intensities as a relatively large lattice vibrational anisotropy.

1. What is texture ? - Three states concerning the orientation of an assembly can be distinguished : random orientation, preferred orientation, unique orientation. Normally random or unique orientation (single crystal) are the desired states for an investiga- tion. However, usually the material provided by nature (mineralogy, biology, physical metallurgy, etc.) is in the intermediate state of preferred orientation. The following analogy shown in figure 1 seems appro- priate : from the three states of aggregation the gaseous (random motion) and the solid state (unique atomic arrangement) are reasonable well understood and the bulk of the investigations deal with these states. The intermediate liquid state (preferred mean near neighbor configuration) is the least investigated state, however, the bulk of our planet is in this state (core, ocean, biology).

The preferred orientation of an assembly we will call texture where the assembly may consist of

1. crystals, 2. molecules, 3. spins,

4. principal axes of electric field gradients (EFG) at specific sites, etc.

Texture is a common phenomenon in nature. Pro- cesses like growth, sedimentation, precipitation, crys- tallization, recrystallization, plastic deformation, etc., result in products which have usually a pronounced texture : The sky oriented plants - like a tree

-

document macroscopically the microscopic texture consisting of the alignment of molecules, in physical metallurgy nearly any fabrication process leads to metals with a preferred orientation of the crystallites,

9

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974610

C6-114 U. GONSER AND H.-D. PFANNES states of aggregation

gaseous state Itquid state solid state

random motion preferred mean unique atomic near-neghbor confiouratlon arrangement and

long-range order (lattice)

>fates" concerning orientation of associations

randomness texture single crystal

t t t t t t t t t t t t t t t t t t t t t f t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t

random orientation ,~fefefred orientation uniaue orientation

FIG. 1. - Schematic representation of the states of aggrega- tions and the states concerning orientation of associations.

FIG. 2. - Texture of a champagne cork Veuve Cliquot.

and the spin textures of sedimentations are important in determining the location of the magnetic pole in prehistoric times.

Here in France we are remined that the cork texture is tested as an indicator concerning the quality of the wine. Figure 2 shows the cut of a champagne cork and its texture. The wider and significant aspects of this topic might be treated by the after-dinner speaker.

2. Defining texture.

-

Problems concerning texture are most significant in physical metallurgy from a fundamental as well as from a technological point of view, thus, the systematic studies on texture pro- blems originates in this discipline [I]. Every textbook on physical metallurgy has a chapter devoted to texture. By the various processes in metallurgy the individual grains become oriented toward a certain direction where this alignment is usually intimately connected with the main direction of the flow of the material and the temperature (gradient). According to the various treatments involved one distinguishes between : deformation texture, hot and cold rolling texture, recrystallization texture, annealing texture, tension, compression and torsion texture, surface texture from machining and polishing, etc. One might state, that it is difficult to produce a piece of metal without texture, that is random orientation of the crystallites. As an example the columnar grains of cast pure aluminium extending from the surface inward along the greatest temperature gradient are

shown in figure 3. Of course, also cubic metals like Cu, Ag, Ni, Pb, Pt, y-Fe (fcc) and a-Fe, Nb, Mo (bcc) are usually strongly textured after plastic defor- mation [I]. In many cases textures with their aniso- tropic properties may be troublesome, however, in modern metallurgy for certain applications one is interested in fabricating alloys with extreme high textures (faceted eutectics) [2] also, the texture of the crystallites influences the desired magnetic anisotropy (spin texture).

In the old days of Mossbauer spectroscopy Sn119 sources were made from metallic tetragonal P-tin.

It was realized that the resonance line of Snllg-@-tin was broadened by an unresolved quadrupole split- ting [3], thus, the angular dependence of the hyperfine interaction becomes significant. The texture of a tin foil (Fig. 4) used in such experiments should serve as an example [4]. In this pole figure the relative probabilities of finding grains with certain orientations are indicated by different hatchings and also the rolling direction producing the foil is marked. From this figure it is quite evident that texture is a difficult problem applied to Mossbauer spectroscopy because the intensities in a Mossbauer hyperfine spectrum reflect the texture of the sample in the sense of an integration over all orientations.

Being concerned with Mossbauer spectroscopy we are mainly interested in the texture of the spins and axes of the EFG which are usually again related to the

TEXTURE PROBLEMS

FIG. 3. - Texture of cast pure aluminium.

a3

E l 4 0 5 0 6 ' j g 7

me

Rolling D ~ r e c t i o n

FIG. 4. - Pole figure for the texture of a cold-rolled tin foil.

Relative probabilities of finding grains with certain orientations and the rolling direction are indicated.

texture of the crystallites. The orientations and angles of interest are shown in figure 5a.

Here, we only consider textures which are axial symmetric in respect to the y-ray propagation direc- tion (2-direction in Fig. 5a). 6, and 6, represent the angle between the propagation direction of the y-radiation and the Vzz-(main direction of the EFG,

FIG. 5. - a) Orientation relationship. b) Random texture distribution. The letters are explained in the text.

with the assumption q = 0) or spin-direction, respec- tively. The general case of simultaneous presence of magnetic dipole and electric quadrupole interaction is rather complex [5]. We imagine the sample being in the centre of a sphere with a radius of unity length.

The texture will be characterized by a function DE(fIq,d which represents the relative number of V,,-or spin- directions per surface element dl2 oriented towards this surface element of an area dl2 = sin O,,, dBq,, d@

D, is only dependent on O,,, because of our assumption of axial symmetric textures. As an example DE(Oq,d for the case of random distribution, that means in fact no preferred orientation, is shown in figure 5b (DE(fIq,",> = 1).

3. Relative line intensities. - Considering Moss- bauer spectroscopy the texture is reflected in the intensities of the Mossbauer spectrum. But texture is not the only effect which can influence the intensi- ties. Generally speaking, the relative line intensities in the hyperfine pattern are governed by :

1. Intrinsic effects (characteristic of the material) : a) angular dependence of the hyperfine interaction, 6 ) lattice vibrational anisotropy (Goldanskii-Karya- gin effect).

C6-116 U. GONSER AND H.-D. PFANNES

2. Extrinsic effects (accidental) : a) preferred orientation (texture), b) saturation effects including dichroism.

In this article we will adopt the thin absorber approximation, thus, neglecting saturation and pola- rization effects. Also, we will not consider relaxation effects.

3 . 1 The angular dependence of the hyperfine interaction in a monocrystal or uniformely magne- tized sample [6] will be expressed by the ratio Rq or R, of the intensities of the ^f 3 ^-t

+

3 and

+

^{% -+ f}

+

^lines

or Am = 0 lines and the Am = f 1 lines respectively

These ratios are plotted in figure 6. For randomly oriented or randomly magnetized samples the ratios R, = 1 and Rm = 2.

0 15 30 L 5 60 7 5 90

[degrees]

FIG. 6.

-

Ratio of the line intensities Rq and R m vs. the angle

Qq and Om, respectively, between the propagation direction of the y-ray and the principal axis of the EFG.

3 . 2 In regard to the Mossbauer effect, the relative number of particles oriented within the cone shell indicated in figure 5a is important, because all these give the same relative line intensities in a Mossbauer spectrum, thus, the R,,, of hyperfine splitted spectra caused by texture and denoted in this case by R,,,,,

can be evaluated by weighting the monocrystal inten- sities (1) with D, dS2 and integrating :

) J

⁽¹

+

cos2 8,) DE(Bq) sin 0, dBq d@

0

R q ~ e x =

fr ^{lfl (i}

^- cos2 19, DE(oq) sin 0, dBq d@

and (2)

4 2

fr ¹

^{(1 - cos2 8.) DE(Bm)} sin 8, dB, d@

0 R m ~ e x =

1; a ⁽

¹

⁺

cos2 8,) DE(Bm) sin 8, dB, d@

where u = cos O,,,. Both RqTex and RmTex can be expressed by

where

For elucidating an application of eq. (3) in figure 7a we show texture distributions of Lorentzian shape with different half widths

r

^[7]. In figure 7b the

FIG. 7. - Lorentzian distribution and corresponding RQ,m- values as function of r/2.

corresponding R,,,-values are shown. They are ranging from Rq = 3 to R, = 1 corresponding to the monocrystal and random values and Rm = 0 to R,,, = 1 in the same sense.

TEXTURE PROBLEMS C6-117

3.3 The vibrational anisotropy (VA) of the mean square displacement (MSD) leads to the following expression for the Debye-Waller factor f in the case of axial symmetric MSD's (MSD in V,, or spin- direction is denoted by

<

^{x i}

>,

MSD perpendicular to V,, or spin by

<

^x:

>,

which is independent of special directions within the plane incident to V,, or spin because of axial symmetry) :

with

&

MSD =

<

^xf

> +

^{- cos2 8,,,}

k2 where the lattice anisotropy parameter

and k represents the magnitude of the y-wave vector.

In the case of a random distribution it follows for R,,, of hyperfine splitted spectra a deviation from the values R, = 1, R, ⁼ 2. This effect is called the Goldanskii-Karyagin effect (GKE) [8-101. The ratios Rq,mGKE can be calculated by weighting the integrants of (2) with thef-factor (4) and substituting DE(8,,,) = 1 because of random distribution :

5

f (8,) (1

+

cos2 8,) DE(Oq) sin 8, dB, R,GKE = 0 n,2

1

^{f (8,) (j} 5 ^{- COS'} 6,) DE(8,) sin 6, d6,

0

and (5)

For a physical interpretation of the VA or GKE it seems, however, more appropriate to define RqSmGKE not as a function of the lattice anisotropy parameter but as a function of the ratio of the MSD's e. g.

Rq,mG~~(Wl.,II) where

< x i

>

w,

^{EZ -}

_<

_{x l}

_>

^and

TI =

^---

< _< "'

_xg

^> _> ^.

⁽⁶⁾

In general, W, # W i l as can be seen from the following more precise definition of the MSD's ratios.

The MSD's shall be calculated by a Debye model thus resulting in an additional dependency of W,,,,

from the Debye temperature 8,,,or 6Dx1, as parameter.

This we indicate e. g. for a Debye temperature of 300 K by

where k2

<

^X:

>

⁼ 0.465 because of

and k = 7.28

k 1

for the 14.4 keV y-rays of 57Fe.

Analogously we get

where again k2

<

^{x i}

>

⁼ 0.465 corresponding to a Debye temperature OD,,, = 300 K in V,, or spin- direction.

In figure 8 R,,, as function of ^E or W,,Il is shown.

Note that for example in the cases, where W, = 0.5 and

TI

⁼ 2, i. e.

<

^{x i}

>

half as large as

<

^x:

>,

FIG, 8. - Ratio of the line intensities ^&GXE and R ~ G K E VS. the lattice anisotropy parameter 8 and the anisotropy ratios W,

and WII for a Debye temperature of 300 K.

the %,,,, values are not far from the corresponding vaIues R, = I, Rm = 2 of a sample without GKE, namely

C6-118 U. GONSER A N D H.-D. PFANNES 4. Correlations between texture and angular depen-

dent Rlossbauer parameters. - 4.1 ORIENTATION

DETERMINATION OF THE PRINCIPAL AXES OF THE EFG

OR OF THE SPIN BY CORRELATING X-RAY TEXTURES WITH RELATIVE HYPERFINE LINE INTENSITIES. - This method is particularly useful if single crystals are not available. The method was applied in the spin orien- tation determination of ferromagnetically ordered cementite, Fe,C. The eqs. (2) or (3) cannot be used directly in this case because X-ray texture measure- ments yield only the crystallographic texture which must not necessarily coincide with spin or V,, texture.

As additional parameter the angles combining both types of textures have to be introduced in (2) or (3).

For more details see ref. [ll].

4.2 TEXTURE DETERMINATION FROM HYPERFINE INTERACTION. - The texture of a sample can be determined by comparing the relative line intensities of the hyperfine pattern by varying the propagation direction of the y-radiation [12]. This method might be regarded as the reverse of 4.1. The method can be improved in accuracy by using polarized y-radiation.

Of course, this method can not compete with the common X-ray texture determination, however, there are some cases where this method might be useful, because X-rays cannot be applied, for instance, if one is interested in the preferred orientation of the heme planes in organic tissues like the liver or the heart.

An approach to evaluate the texture by means of Mossbauer technique can be made in the following way : as is shown in figure 9 for the case of a Lorent-

FIG. 9. - Decomposing of a Lorentzian texture distribution in a step sum function. The letters are explained in the text.

zian texture distribution it can be decomposed in a sum of step functions with equal width A0. The R,,,-values can be calculated from eq. (2) or (3), for a fixed number n of step functions (for example n = 10 as shown in Fig. 9, A0 = n/20) and known D,(0,) values. Reversely one can determine experi- mentally n Rq,,-values by turning the absorber by ten

different angles with respect to the y-ray. These known angles have to be introduced in the formulae for the R,,, of the step sum function texture distri- bution as parameter. One gets an equation system to determine DE(B,). This method is only an approxima- tive one, but the exact method consists of solving a system of integral equations of the type (2), (3) which is very complicated.

4.3 TEXTURE AND GOLDANSKII-KARYAGIN EFFECT.

-

This correlation is of great interest because the GKE has become an important tool in the determination of the vibrational anisotropy (VA).

It is difficult to distinguish between these two competing effects and it is our aim in this chapter to compare effects on the hyperfine pattern due to texture and vibrational anisotropy, particularly, we want to demonstrate on examples where the sensitive influence in the hyperfine pattern by texture produces the same effect on the relative line intensities as a relatively large lattice vibrational anisotropy.

It is very easy to construct a texture which yields the same Rq,,Te, as Rq,,GKE produced by GKE. It follows by comparison of eq. (2) and (5) under the condition Rq,m~ex = R q , m ~ ~ ~

a texture distribution of the type of a Gaussian error function, which is often realized approximately in real textures. Figure 10 shows the corresponding texture

o.o! ^i-, / , ,

.

^{, , r , ,}

.

^{, 3}

.

^{, , r I}

0 15 30 L5

-

6 0 7 5 ^[degrees] 90 Frci. 10.

-

'Texture - and VA - distributions, D~(8q.m) and

C . M S D ( ~ , , ~ ) yielding the same R q , r n ~ ~ ~ . ~ e x

TEXTURE PROBLEMS C6-119 function D,(@,,Q) for ^& =

+

0.465, W , ⁼ 2, (solid

line) and ^E = - 0.465, W , = 0 (solid dashed line) both normalized to the random distribution (dashed line). These texture distributions should be compared with the corresponding MSD from eq. (4) where we take into account a constant scale factor c to get comparable ordinate values of c. MSD with DE :

(4) + k2 MSD ⁼ k2

<

^X:

> +

^{& cos2 @q,m}

or with

kZ

<

x:

>

⁼ 0.465 for OD,, = 300 K and 14.4 keV y-radiation

c.MSD = 1 $ cos2 O,,, with c = 2.151 k2

.

The c.MSD curves belonging to ^E =

+

0.465 or

E = - 0.465 are indicated by a solid line and solid- dashed line, respectively, in figure 10. The correspond- ing R,,,,,, values are marked on the curves. Thus, we have to compare both the solid and solid-dashed lines, for texture and VA, respectively, to appreciate the sensitivity of both effects to R,,,. We see that already a rather small texture corresponds to a relati- vely big VA.

FIG. 11. - Polarcoordinate representations of DE and c. MSD for &=0.465, ^{W l} =2, & T ~ ~ , G K E =O.943 0. Rm~ex.~~~=2.182 ^2.

Another representation of texture and VA may be still more informative, namely the polarcoordinate representation of both. Figure 11 shows the magnitude of D,(O,,,) and c.MSD(O,,,) as function of the polar angle O,,, for E = 0.465, W, = 2. Figure 12 represents the hypothetical case of these functions : ^E = - 0.465, W , = 0, i. e. rigourously fixed in VZz -or spin- direction concerning VA. In both cases we see that the tangerine (Fig. 11) and the squashed quasi-ellipsoid (Fig. 12) -texture distributions approach much closer the random sphere as the tall quasi-ellipsoid (Fig. 11) - and toroid (without inner hole) (Fig. 12) - MSD distributions.

itipsoid

-

texture

RG. 12. - Polarcoordinate representations of DE and ^c. MSD for &=--0.465, WL=O, R9~ex,~~~=1.067, Rm~ex,~~~=1.812.

Note, if both texture and VA are present in one sample, the tangerine texture from figure 11 together with the toroid VA from figure 12 or the quasi-ellipsoid VA from figure 11 together with the quasi-ellipsoid texture from figure 12 as the two competing effects (texture and VA) just compensating each other. The resulting values R, = 1, R, = 2 making believe that no VA and no texture being present.

Acknowledgment. -The support of the ^(< Deutsche Forschungsgemeinschaft )) and IRSID ^>) is grate- fully acknowledged.

U. GONSER AND H.-D. PFANNES

References

[I] BARRETT, C. S., Structure of Metals (McGraw-Hill Book Comp., Inc. New York), 1952.

[2] SAHM, P. R. and LORENZ, M., J. Materials Science 7 (1972) 793.

131 MEECHAN, C. J. and MUIR, A. H., Jr., Review of Modern Physics 36 (1964) 438.

[4] MENGELBERG, H.-D., unpublished.

[5] KREBER, E. and GONSER, U., Nuel. Instrum. Meth. (in press).

161 FRAUENFELDER, H., NAGLE, D. E., TAYLOR, R. D., COCHRAN, D. R. F. and VISSCHER, W. M., Phys. Rev.

126 (1962) 1065.

[7] PFANNES, H.-D. and GONSER, U., Appl. Phys. 1 (1973) 93.

[8] GOLDANSKII, V. I., MAKAROV, E. F. and KHRAPOV, V. V., Phys. Lett. 3 (1963) 334.

[9] KARYAGIN, S. V., Dokl. Akad. Nauk. SSSR 148 (1963) 1102.

[lo] FLINN, P. A., RUBY, S. L. and KEHL, W. L., Science 143 (1964) 1434.

[ l l ] GONSER, U., RON, M., RUPPERSBERG, H., KEUNE, W. and TRAUTWEIN, A., Phys. Stat. Sol. 10 (1972) 493.

[I21 PFANNES, H.-D. and GONSER, U., Proceedings International Conference on Mossbauer Spectroscopy, Bratislava, Czechoslovakia (1973).