DEFECT PROPERTIES FROM X-RAY SCATTERING EXPERIMENTS

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DEFECT PROPERTIES FROM X-RAY SCATTERING

EXPERIMENTS

H. Peisl

To cite this version:

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Abstract. — Lattice distortions due to defects in crystals can be studied most directly by elastic X-ray or neutron scattering experiments. The size of the defects can be determined from the shift of the Bragg reflections. Defect induced diffuse scattering intensity close to and between Bragg reflec-tions gives information on the strength and symmetry of the distortion fields and yields the atomic structure of point defects (interstitials, vacancies, small aggregates). Diffuse scattering is a very sensitive method to decide whether defects are present as isolated point defects or have formed aggregates. X-ray scattering has been used to study defects produced in various ionic crystals by 7- and neutron irradiation. After an introduction to the principles of the method the experimental results will be reviewed and discussed in some detail.

1. Introduction. — X-ray scattering has been very 2. Principle of X-ray scattering from lattice dis-successfull in revealing the arrangement of atoms and tortions. — 2.1 DEFECTS AS ELASTIC DIPOLES. — The molecules in crystal lattices during the past half century, additional volume Av needed by a defect can be deter-It was only during the last decade that X-ray scattering mined from the lattice parameter change Aa/a. The techniques have been developed to study crystal lattice s o.c ane d s i z e f a c t o r X =a~l ^- is related to Av = 3 X.

defects [1-5]. High power X-ray generators, position . AC sensitive detectors together with computerized electro- C1S t h e d e f e c t concentration.

nics and scattering setups allow to get information on I n 8e n e r a l t h e displacements u are anisotropic and a X

defect structures and defect correlations as well in ^n s o r i s necessary to describe the defect's strain field

irradiated materials as in doped ones. eu [11 ]• For a defect described by Xvu various equivalent

For the understanding of defect structures and defect orientations v may exist in the crystal. E0- is then

mobilities in metals this was a breakthrough [3, 6] obtained by summing over all nd orientations,

mainly because no other methods exist which give very -M

detailed information on the defects. For ionic crystals Sy = £ Kj Cv (1)

such other methods like optical absorption and emis- v _ 1

sion studies, EPR, ENDOR, etc. do exist. X-ray and Cv is the atomic fraction of defects having the

orien-neutron scattering experiments can still give additional tation v. For some later formula we prefer to use the information which is not obtainable by these so-called double-force tensor, which is directly related techniques. to the X tensor,

The advantage of having these other methods on _ 1 „

hand was very helpful during the pioneering period of iJ ~ fi fa 'Jk' kl *• '

developing these X-ray techniques [7-101. ~ . , . , „ , , , . , ... . e . „ . . . ,. ... . Q is the mean atomic volume and Siik, are the elastic

Most lattice defects, as well extrinsic (impurities) as .. . . , ,Jkl .

• . . • , . . . ..i. , ii J * J j compliances in the tour index notation,

intrinsic (vacancies, interstitials, small and extended _r . . , _. , , , . . ,

j c t i * \ • i j - * u *u i I J F °r defects (concentration C) randomly distributed

defect clusters), in general disturb the close packed , . . . , , , / . }

, e c . * i J t u r J and oriented in a crystal the volume change as measured

order of a perfect crystal and therefore need more or , , J . b

, , • tl_ . , v ... . by lattice parameter change is given by

less volume in the crystal. X-ray scattering gives J K & 6 J

detailed information on the displacements of the AVjV = 3 Aaja = £ tit = C trace Xi}

lattice atoms due to defects and from these informa- ' tion the nature, structure and defect strength can be _ C_ trace Py .

deduced. - Q CX1 + 2 C1 2 * ^ '

DEFECT PROPERTIES FROM X-RAY SCATTERING EXPERIMENTS

H. PEISL

Sektion Physik der Ludwig-Maximilians-Universitat, Miinchen, FRG

Résumé. — La déformation du réseau cristallin due aux défauts peut être étudiée le plus

direc-tement par des expériences de diffusion de rayons X ou par la diffusion des neutrons. Il est possible de déterminer la dimension des défauts par le déplacement des reflets de Bragg. L'intensité de la diffusion tout près et entre les reflets de Bragg, due aux défauts, donne des informations sur la symétrie et sur l'intensité des champs de déformations. Elle indique aussi la structure atomique des défauts ponctuels (interstices atomiques, lacunes réticulaires, petits agrégats). La diffusion près des reflets de Bragg est une méthode très sensible pour déterminer si les défauts ont un caractère ponc-tuel isolé ou s'ils sont formés d'agrégats. On a utilisé la diffusion des rayons X pour l'étude des défauts produits dans une variété des cristaux ioniques par irradiation avec rayons y et avec des neutrons. Ayant donné une introduction aux principes de la méthode, les résultats des expériences sont résumés et discutés en détail.

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C7-48 H. PEISL

For preferred orientation of the defects Aala is different for different crystal directions ei [l21 :

2 . 2 X-RAY SCATTERING FROM A CRYSTAL LATTICE WITH DEFECTS. - The principle of an X-ray scattering experiment is shown in figure 1. k, and k are the wave vectors of the incoming and scattered X-ray wave, respectively ; for elastic scattering the X-ray wavelength

A is unchanged

I

k,

I

=

I

k

I

= 2

~ 1 % .

The important magnitude is the scattering vector K = k, - k,

I

K

I

= 4 n/A.sin 8, 2 8 is the scattering angle.

defects the lattice atoms are shifted to new sites rm = rm

+

U,. All defects with scattering amplitude fD(K) contribute to the displacements U, by their individual displacement field U:. The scattered X-ray intensity is now given by

Figure 2 shows in a schematical manner the essential changes :

i) The Braggpeaks are shified due to scattering from an average, changed lattice. This shift gives the lattice parameter change

ii) Deviations from this average lattice gives rise to a diffuse scattering close to the Bragg peaks and between the Bragg peaks.

K = k - k,

\ '\ a>

I ( E )

T T b>

h.-

FIG. 1.

-

Elastic X-ray scattering

a ) Definition of the scattering vector :

K = k - k o , \ K

1

-4n/l,.sin6

I

k

I

= Iko

!

= 2 x/lr wave vector

I, X-ray wave length, 2 6 scattering angle.

6 ) Scattering from a crystal lattice :

f(K) scattering amplitude

rm position vector.

The X-ray wave scattered from an individual atom m has the amplitude fm(K). A wave scattered from an atom at a lattice site r, is shifted in phase by exp(iK. r,) compared with a wave scattered from an atom at rm = 0. The total scattering intensity I(K) in electron

units is obtained by summing a11 scattered amplitudes from the m atoms in a crystal with the proper phase factor and square the sum :

For an ideal periodic arrangement of the atoms in a crystal lattice the scattered intensity is zero except for all atoms scattering in phase K.rm = 2 m, n = 1, 2 , 3

...

This happens whenever the scattering vector is equal to a reciprocal lattice vector G.K = G is the vector notation of Bragg's law. Due to the introduction of

K

FIG. 2. -X-ray scattering from a crystal lattice with defects.

The diffuse scattering intensity for a random distribution of defects (low concentration n) is obtained by subtraction of the Bragg scattering intensity (eq. (5)) from the defect scattering intensity (eq. ( 6 ) ) .

In most cases e. g. for interstitial atoms, vacancies or impurities the displacements are small and the phase factor can be expanded.

Within this approximation the diffuse scattering intensity gives the Fourier transform

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The square of the first term is the Laue scattering from the defect and eventually from its immediate neighbourhood. The scattering amplitude of the defect is

fD =

f

for a self-interstitial,

fD

=

-

f for a vacancy and fD

=f,

-

f for an impurity in the most simple case.

In ionic crystals defects may exist in various charge states and this changes

f,

accordingly.

The square of the second term is the distortion scattering. Close to the Bragg peaks, K z G the scatter- ing intensity is given by the Fourier transform of the distortion field [13]. This Huang dgfjse scattering (HDS) intensity corresponds to the one-phonon scattering from thermal vibrations (TDS) :

Interference between Laue and Huang scattering amplitudes gives rise to an asymmetry of the scattering intensity distribution with respect to the Bragg peak.

_..,

This asymmetry depends on the sign of

fD

and U, thus in~mediately can tell whether a measured scattering intensity distribution is mainly due to interstitial or vacancy type defects.

.4way from the Bragg peaks (Zwischenreflex- keuung [14]) both scattering amplitudes are of the same order of magnitude. The scattering intensity here mainly depends on the relative phase factor between scattering from the defect and scattering from the displaced neighbouring lattice atoms. Information on the defect site and symmetry can be obtained from the

Zwischenrefiexstreuung.

2 . 3 DEFECT PROPERTIES FROM DISTORTION SCAT- TERING. - In order to gain a rough idea of what one expects experimentally the displacement field and its Fourier transform has to be considered in some detail. In some distance from a point defect (Fernfeld) the displacements fall off like

and yield a Fourier transform centered close to the reciprocal lattice point

For an isotropic defect in an isotropic medium the situation is schetched in figure 3. The displacement field is isotropic as well as the Fourier transform. As the scattered intensity depends on the product K.; we

expect zero intensity whenever K I

u".

In the special case of an isotropic defect this happens on a plane I G

through the reciprocal lattice point. Equal scattering intensity is expected on spheres touching the reciprocal lattice point (Huang spheres). Maximum scattering intensity is expected in the direction of GO(//

u").

The Huang scattering intensity is proportional to the square of the scattering vector and falls off like l/g2, where g is the distance from a Bragg peak.

II~DS W K2/g2

.

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For an anisotropic displacement field also the Fourier transform becomes anisotropic. For this reason in general the zero intensity plane no longer is perpendicular to the reciprocal lattice vector. An anisotropic displacement field in general is produced by an anisotropic defect. Anisotropic defects can have different orientations in the lattice. For a random distribution of the defect orientations the diffuse scattering intensity is averaged. Each defect orientation yields a different zero intensity plane. By the averageing over different defect orientations this zero intensity planes may coincide, so that a zero intensity plane still exists or the zero intensity planes intersect in a way that a zero intensity line appears. For low symmetry defects this averageing process yields scattering intensity in all

FIG. 3. - Isotropic displacement field u(r) in real space and Fourier-transformed $K) in reciprocal space.

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0 - 5 0 H. PEISL

directions without zero intensity planes or lines, respective1 y

.

The results of such averageing processes for various defect symmetries 12, 131 are shown schematically in figure 4. Tn a cubic crystal a tetragonal defect gives rise to a diffuse scattering intensity distribution with a zero intensity plane perpendicular to the reciprocal lattice vector around 00 h, a zero intensity line perpendicular to the reciprocal lattice vector around 220 and full scattering intensity all around 222. Measuring the diffuse scattering intensity around some high symmetry reciprocal lattice points in some low index directions immediately gives the symmetry of the displacement field which resembles the symmetry of the defect. From measurements of the absolute scattering intensity the three quadratic tensor parameters

can be obtained. Figure 5 shows which combinations of the xs can be measured for certain reciprocal lattice points and directions.

FIG. 5. -Information from Huang diffuse scattering for

several reciprocal lattice points and directions :

2.4. ABSOLUTE DEFECT CONCENTRATION A N D DEFECT PAIRS. - The Huang scattering intensity I,, (eq. (1 1))

and the relative lattice parameter change Aala (eq. (3)) depend in different ways on the dipole tensor :

Measuring IHDS and Aala simultaneously gives the defect concentration C.

In the casezof ionic crystals and especially alkali halides where other methods exist to determine the defect concentration C of defect pairs (e. g. radiation induced Frenkel pairs) with different elastic dipole tensors for the interstitial P, and the vacancy P,, P,

and P, can be determined separately. Again the diffe- rent dependence of I,,, and Aala on C, P , and P , I,,, C(P:

+

P:), Aala

-

C(P,

+

P,) (16) gives two independant equations for P, and P,.

2 . 5 HUANG SCATERING FROM DEFECT CLUSTERS.

-

Huang scattering intensity is very sensitively influenced when defects agglomerate. This is schematically demonstrated in figure 6. The scattered intensity from a random distribution of isolated point defects (concen- tration n) is IHDs

-

n p 2 . If the n defects form n,, clusters each containing an average of Z point defects : n = n c , . Z . Now for the Huang diffuse scattering intensity from defect clusters the same relation as above holds 1:;s

-

n , - , . ~ . ? ~ . Under the assumption of linear superposition of the defect strength P, P,, = 2 . P

Cl

one gets IHDs = Z.IHDS. The scattered intensity is enhanced by a factor Z if defects form clusters. More detailed considerations allow to get information about the shape of the clusters [IS].

K -

FIG. 6.

-

Huang diffuse scattering from defect clusters (expla- nation see text).

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The asymptotic scattering intensity is linearly proportional to the elastic dipole tensor and the scattering vector K and it falls off more rapidly with the distance g from a Bragg peak, as l /g4.

The asymptotic scattering intensity depends linearly on the elastic dipole tensor as does the relative lattice parameter change. There is one difference, IAs depends on the absolute value of P, whereas in Aa/a also the sign of P enters. In a case where different defects are present in the same concentration e. g. vacancies and interstitials, where one type of defects expand the lattice and the other type contracts the lattice by the same amount one would get Aa/a = 0 and still could get information from IAs # 0. The asymptotic scattering intensity has for certain directions superimposed on the l/g4 dependance oscillations [19]. These oscilla- tions also contain information on the defect strength, defect type and defect orientation, especially for defect clusters which have condensed into dislocation loops [20].

2.7 ZW~SCHENREFLIIXSTREUUNG.

-

Huang diffuse scattering intensity is centered close to the Bragg reflexions and gives information on the long ranged displacement field (Fernfeld). Information on the displacements in the immediate vicinity of a defect (Nahfeld) can be obtained from the scattered intensity farther away and between the Bragg Peaks [14].

2.8. SMALL ANGLE SCATTERING.

-

Scattering for K z G = 0 is only observed if a defect and/or its displacement field is connected with a change of the local electron density. This makes small angle scattering to a powerful tool to detect vacancies, large vacancy clusters and eventually large interstitial clusters in diatomic solids.

3. Experimental results. - As well defect concentrations (c

5 10-3) as defect strengths (AV

x

-

I R

...

+

3 R) are of such orders of magnitude that the X-ray scattering experiments described in the previous chapter must be performed with high sensitivity and very accurately in order to detect the small changes to be expected.

3.1 LATTICE PARAMETER CHANGE.

-

Figure 7 shows the results of an experiment [21] where the relative lattice parameter change of KC1 was determined as a function of the F center concentration. For additively coloured crystals AV =

+

0.6 Q means that an F center

needs 0.6 atomic volumes of extra space in the crystal. In an X-irradiated crystal AV =

+

1,3 R is greater. The explanation for this is that F centers are formed during irradiation as parts of Frenkel pairs [22, 231 and the interstitial defect needs an additional extra volume.

3 . 2 HUANG DIFFUSE SCATTERING FROM POINT DEFECTS.

-

Point defects in KBr after X-irradiation

FIG. 7.

-

Relative lattice parameter change versus F-centre concentration in KC1 [21] : 0 X-irradiated

;A

additively coloured.

at 6 K were investigated by Huang diffuse scattering of

X-rays [9, 101. Figure 8 shows a typical result. The X-ray scattering intensity was measured close to the 600 Bragg reflection peak in [l001 and [010] direction, respectively. The crosses give the background scattering due to thermal diffuse scattering, Compton scattering and stray radiation of the unirradiated crystal, measured at 6 K. After irradiation at 6 K an increase of the scattering intensity is observed in [100], whereas in [OlO] the intensity stays constant. The latter fact tells us immediately (see Fig. 5) that n, = 0.

FIG. 8.

-

Huang diffuse scattering intensity of KBr a t 6 K close to a 600 Bragg reflection in [l001 and [010] directions [24].

+

unirradiated ; 0 X-irradiated.

The defect induced scattering intensity distribution in (100) direction shows an asymmetry with respect to the Bragg peak. The additional scattering intensity is higher on the high angle side. This tells immediately that the scattering is dominated by defects which expand the lattice.

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C7-52 H. PEISL

log-log scale versus g/G in figure 9. The experimental points fit straight lines with slope- 2 which is expected for Huang scattering (see eq. (14)).

FIG. 9.

-

Huang diffuse scattering intensity as a function of the

relative distance from reciprocal lattice points [IO].

From the absolute magnitude of the scattering intensity n, and R, can be determined. The authors were able to separate the contributions of the various defects which are created during irradiation (see [24]) and could determine the elastic dipole tensors for the charged Frenkel pair, the anion vacancy and the interstitial anion.

3 . 3 DIFFUSE SCATTERING FROM DEFECT CLUSTERS.

-

Defect clusters are formed in a secundary defect reac- tion in irradiated crystals whenever the temperature is high enough that at least one defect species becomes mobile. We expect cluster formation either during annealing of a crystal irradiated at low temperature or during irradiation at elevated temperatures. Both has been studied by X-ray scattering.

Cluster formation during annealing of KBr which had been irradiated at 6 K was also studied by Spalt et al. [10]. After warming up to 30 K diffuse scattering intensity was detected in [010] direction close to the 600 Bragg reflection, where none had been detected

right after irradiation. The scattering intensity further increased with increasing annealing temperature. This is caused by a change of the defect symmetry. Defect pairs, triplets, etc. have a different symmetry. At temperatures above 60 K the scattering intensity distribution in addition to the l/g2 dependence showed the l/g4 dependence expected for asymptotic scattering. The crystal lattice now is heavily distorted by the defect clusters which grow with increasing temperature.

Cluster formation during irradiation at elevated temperatures was observed in LiF after y-irradiation a t room temperature [7]. Figure 10 shows the diffuse scattering intensity which was observed close to the 400 Bragg reflection after increasing irradiation times. In the log-log scale as well the l/g2 as the l/g4 dependence is observed. Two important changes occur after additional irradiations. The intensity increases faster in the Huang (l/g2) region than in the asymptotic (l/g4) region. This indicates that z, the number of defects in a cluster, increases. Further more the g-value where l/g2 goes over to ]/g4 shifts towards smaller g. From this an increase of the cluster size can be deducted. The cluster radius determined in this way is given in figure 10 besides the irradiation time.

FIG. 10. - Diffuse scattering from y-irradiated LiF after

various irradiation times at room temperature 371.

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References

[l] PEISL, H. and TRINKAUS, H., Comments on Solid State

Phys. 5 (1 973) 167.

[2] DEDERICHS, P., J. Phys. F. Metal Phys. 3 (1 973) 471.

[3] EHRHART, P., SCHILLING, W. and HAUBOLD, H.-G., Adv.

Solid State Phys. 14 (1 974) 87.

[4] Proceedings of the International Discussion Meeting on

Studies of Lattice Distortions on Local Atomic Arran- gements by X-ray, Neutron and Electron Diffraction in J. Appl. Crystullogr. 8 (1975) 79.

[S] HAUBOLD, H.-G., Revrre Phys. Appl. 11 (1976) 73.

[6] HAUBOLD, H.-G., Proc. International Conf. on Funda-

mental Aspects o f Radiation Damage in Metals,

Gatlinburg, USA (1975).

[7] PEISL, H., SPALT, H. and WAIDELICH, W., Phys. Status

Solidi 23 (1 967) K 75.

[g] SPALT, H., Z. Angew. Phys. 29 (1970) 269.

[9] LOHSTOTER, H., SPALT, H. and PEISL, H., Phys. Rev. Lert.

29 (1972) 224.

[l01 SPALT, H., LOHS~OTER, H. and PEISL, H., Phys. Status Solidi (b) 56 (1973) 469.

[Ill NOWICK, A. S. and BERRY, B. S., Anelastic relaxation in crystalline solids (Academic Press, New York-London) 1972.

[l21 PEISL, H., BALZER, R. and PETERS, H., Phys. k t t . 46A (1974) 263).

[l31 TRINKAUS, H., Phys. Status Solidi (b) 51 (1972) 307. [l41 HAUBOLD, H.-G., J. Appl. Crystallogr. 8 (1975) 175.

[l S] TRINKAUS, H., Phys. Status Solidi (b), 54 (1972) 209.

[l61 TRINKAUS, H., Z. A n ~ e w . Phys. 31 (1971) 229.

[l71 DEDERICHS, P. H., Phys. Rev. B 4 (1971) 1041.

[l81 TRINKAUS, H., 2. Naturforsch. 28a (1973) 980.

[l91 TKINKAUS, H., SPALT, H. and PEISL, H., Phys. Status Solidi (a) 2:(1970) K 97.

[20] SPALT, H., J. Physique Colloq. 37 (1976) C 7 this issue.

[21] BALZER, R., Diplomarbeit, TH Darmstadt 1967.

1221 PEISL, H., BALZER, R. and WAIDELICH, W., P&. Rev. Let!. 17 (1966) 1129.

1231 BALZER, R., Z. Phys. 234 (1970) 242.

[24] LOHSTOTER, H., Diplomarbeit, T H Darmstadt 1972.

DISCUSSION

C . JACCARD. - HOW does the analysis of diffuse scattering between the Bragg peaks compare with the

(< Extended

-

X

-

ray

-

Absorption

-

Fine

-

Structure >)

method with respect to sensitivity and structural information of the neighbourhood of a defect ?

H. PEISL. - In principle similar information could be obtained from EXAFS. So far no experimental results exist which show that EXAFS can be used for defect studies at all and have some doubts whether the structure of defects (c

<

10-3) can be deduced.

I see some chance to study impurity defect complexes by EXAFS.

W. KANZIG.

-

IS your theory too modified if you start out whith a perfect crystal to which you cannot apply the kinematical theory of X-ray diffraction ?

H. PEISL. - Diffuse scattering is observed in some distance from the Bragg peak, where one is allowed to use kinematical theory, even in perfect crystals. This has been shown as well theoretically as experimentally.

N. ITOH. - Could YOU comment on the effect of close-pair fo:m:tion on the diffuse X-ray scattering ?

For example, in the case when the U- and I-centers

form close pairs, it is still possible to separate the distortions of the ci- and I-center ?

H. PEISL.

-

If defect pairs are strongly correlated, i. e. if they have a fixed distance they would scatter like one defect and one could not separate the dis-

tortions of the ci- and I-center. Whether correlations

of close pairs have an influence on the diffuse scattering results could be decided experimentally if one looks in more detail in annealing experiments. There should be a change after close pairs have been annihi- lated. I agree that one should investigate this problem further.

F. BBNIERE. - Up to which temperature can the diffused intensity due to the defects beclearly separated from the thermal motion and Compton background in the alkali halides ?

H. PEISL.

-

Close to a Bragg peak Compton scattering is negligible. For the ratio of Huang diffuse scattering (HDS) to thermal diffuse scattering (TDS) one has for cubic crystals

It depends on the defect concentration c, the defect strength, given as the additional volume AV, the elastic constants C,

,,

C,,, the mean atomic volume Q and the measuring temperature T. For alkali halides we obtain

For typical defect concentrations C

<

10d4 and