Investigation of radiation damage by X-ray diffraction

ELSEVIER

Journal

of Nuclear

Materials

216 (1994) 170-198

Investigation of radiation damage by X-ray diffraction Peter Ehrhart Institut ftir Festkiirperforschung,

Forschungszentrum

Jiilich, D-52425 Jiilich, Germany

Abstract X-ray diffraction methods allow the investigation of point defects and small defect agglomerates or precipitates, i.e. defect sizes ranging from the atomic level up to large dislocation loops that are well above the visibility limit of the transmission electron microscope. The discussion includes measurements of the change of the lattice parameter and differential dilatometry as well as the diffuse scattering. Especially the diffuse scattering at small q-values, i.e. close to Bragg reflections (Huang diffuse scattering) and at small angles (small angle X-ray scattering), allows a rather straightforward interpretation and is discussed in some detail. The power and the limitations of the methods are illustrated through examples of defects in metals and semiconductors. Finally the application of the anomalous scattering and the investigation of the extended X-ray absorption fine structure are discussed, as synchrotron radiation makes these techniques generally available.

1. Introduction Scattering experiments with radiation of a wavelength similar to the atomic distances of u 1 A are the appropriate tools for the investigation of the microscopic structure of defects in crystalline solids. Quite different types of radiation like X-rays, thermal neutrons or electrons have been used for such structural analysis. However, the transmission electron microscope (TEM) has become the major tool for the investigation of the microstructure of radiation defects and this fact is reflected by a large number of dedicated lectures in this school. This dominance of the TEM is due to the availability of electromagnetic lenses for electrons that allow the direct imaging of rather small defects (d r 1.0 nm). X-ray or neutron topography that use the local variation of the Bragg condition around defects yield also direct images, however, topography is usually only applicable for low defects concentrations in nearly perfect single crystalline samples. In contrast to the case of the direct imaging, X-rays have advantages in the evaluation of the details of the diffraction pattern, i.e. higher resolution and lower inelastic scattering background. The defect induced changes in a diffraction pattern are schematically shown 0022-3115/94/$07.00 0 1994 Elsevier SSDZ 0022-3115(94)00302-S

Science

in Fig. 1. Compared to the “perfect crystal” we observe essentially three changes: (a) a shift of the position of the Bragg peaks corresponding to the average change of the lattice parameter. (b) An attenuation of the intensity of the Bragg peaks which can be quantitatively described by a “static Debye-Waller Factor” (DWF). This value yields the average number of atoms that are no longer contributing to the Bragg scattering due to the defect induced displacements of the lattice atoms. As the attenuation of the Bragg peaks is generally small for the conditions of interest there are no examples of the application of these measurements to the investigation of radiation damage, however, similar information can be deduced from diffuse scattering. (c) A diffuse X-ray scattering (DXS) background, that reflects the no longer perfectly destructive interferences, is observed between the Bragg peaks. The distribution of this diffuse scattering intensity contains very detailed information on the defects. Two special regions where the intensity is often relatively high and where analytical approximations for the scattering law can be derived are often distinguished: close to the Bragg reflection (Huang Diffuse Scattering, I-IDS) and close to the forward direction (Small Angle X-ray Scattering, SAXS). The X-ray methods reach the sensi-

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171

P. Ehrhart /Journal of Nuclear Materials216 (1994) 170-198

2. Interaction

of X-rays with matter

X-rays are electromagnetic waved lengths, A, between 10 and 0.1 A quantum energies, E, between 1 and convenient units the transformations, energy is given by h[A] .E[keV]

1

4

tivity to observe the small diffuse intensity of typically lo-100 ppm of atomic size defects. Therefore the determination of the structural properties of self-interstitial atoms @IA) and vacancies (V) has become a unique field of application for the X-ray diffraction technique (XRD). The additional characteristic feature of the DXS is the increase of the scattering intensity when defects form agglomerates. Therefore clustering can be very sensitively detected. It is demonstrated, that valuable complementary results to the TEM investigations are obtained, as the method is nondestructive and directly yields statistically relevant averages over defect populations. In the following chapters we will first give a short introduction to the interaction of X-rays with matter [l] and then introduce the basic theory of diffuse scattering. After some remarks about the experimental techniques in Section 4 we will demonstrate the applicability of the method by different examples in the field of radiation defects.

(1)

The most fundamental scattering process of these electromagnetic waves is by the interaction with the electric field of the electron (magnetic scattering can be generally neglected by X-rays). The details of this scattering process, especially the separation of the elastic and inelastic (Compton-1 scattering contribution at large scattering vectors, must be calculated by quantum mechanics. However, the elastic scattering cross section, which is the process of interest here, can be quantitatively understood by a classical calculation of the induced dipole radiation of the electron, that yields the Thomson scattering cross section: du -=d&.,,

Fig. 1. Schematics of the scattering experiment and of the change of the diffraction pattern due to point defects: (a) Scattering triangle: k,, k’ = wave vector of the incoming and scattered waves; k = k’ - R, is the scattering vector and 26 the scattering angle. (1k I = 4rr/h sin 19). (b) Diffraction pattern of the “ideal” crystal - essentially &functions at k = G - and of the distorted crystal. (c) Arrangement of atoms in a perfect crystal and in a crystal containing point defects with distortion fields.

= 12.39.

with typical wavecorresponding to 100 keV; in these of wavelength to

e2

2 1 + cos2 26

I mc2 I

2

e2/mc2

is the classical radius of the electron and its square (r,” = 7.94 x 1O-26 cm2 = 0.0794 barn) is generally used as a reference unit (“electron unit”) for X-ray scattering cross sections. The second factor of Eq. (2) considers the polarization of the incoming wave (see Fig. 2) and is given for the case of a nonpolarized incoming wave. If there are several scattering centers, there is a phase shift between the different scattered waves that has to be considered in the calculation of the total scattering amplitudes. This phase factor can be understood from Fig. 3 which shows the scattering of an incoming wave characterized by a wave vector k, of

el=

e' - e,, ~0.~21) /I

Fig. 2. Scattering of an electromagnetic wave by an electron. For an electric field vector perpendicular to the scattering plane (0) the cross section is independent of the scattering angle whereas for E within the plane the cross section is zero at 26 = 90”.

P. Ehrhart /Journal

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of Nuclear Materials 216 (1994) 170-198

generally well localized we obtain the scattering cross sections for an ensemble of N atoms, that are located at the positions r,,

k’ Fig. 3. Schematic illustration of the phase difference of the scattered waves from two scattering centers.

magnitude 2~/h and a scattered wave k’; as we consider only elastic scattering k’ has the same magnitude as k,. The resulting amplitude, A, from the two scattering centers is A = A, + A, eik’,

(3)

and the scattering intensity or the scattering cross section is given by ( A 1’. This scattering amplitude depends only on the scattering vector k = k’ - k, whose magnitude is: 1k 1= (41r sin 4)/A. In order to calculate the scattering amplitude of an atom with N electrons we consider that the electrons are not located at fixed points but distributed according to their wave function + and obtain the so called atomic form factor, f: f =

E [a,bieikr@

i=l’

d3r = (p(r) ,

IfO12$_

Th.

S(k)

= i

2

f,, e”‘n 2.

I n=l

(7)

I

From this basic formula we can directly deduce the Bragg scattering: we expect a maximum intensity when all scattering contributions are in phase or kr, = 2~. This means that the projection of r,, on k is a multiple of 2~ or the atoms can be considered as arranged on planes of separation d or nh = 2d

sin 8.

(8)

In order to deduce the details of the diffuse scattering from Eq. (7) we need an appropriate description of the

eikr d3r, 10

i.e. the Fourier transform of the electron density distribution. The atomic scattering cross section is g(k)=

This result corresponds to the so-called kinematic approximation that neglects multiple scattering, but is sufficient for the discussion of defect scattering. The attenuation of the scattering due to the thermal motion of the atoms can be considered by a thermal Debye Waller Factor that is neglected (or might be considered to be included into the actual value of f) for simplicity. In the following we will discuss the scattering function S(k), i.e. the scattering cross section per atom that is given in units of the Thomson cross section:

(5)

As shown in Fig. 4 the interference of the scattered waves (Eq. (3)) yields a decrease of f for large values of the scattering vector that must be considered in X-ray scattering cross sections; this decrease depends of course also on the size of the atom, i.e. there is a constant value for a “point charge” - such a constant value is observed for the nuclear scattering of neutrons - and the decrease of f is faster for a lower charge of the nucleus and a corresponding weaker attraction of the electrons. In order to understand the scattering of an array of atoms or a periodic crystal we proceed in a similar way as in Fig. 3 and consider the atoms with their form factor f as the scattering centers. As these atoms are

9

f 8 7

6 5 4 3 2 1 0

0.2

0.4

0.6

0.8

1.0

sin*/A Fig. 4. Atomic scattering factor for six ions having a total of ten electrons surrounding the nucleus. The atomic scattering factor for neon (Z= 10) and a “point atom” containing ten electrons are shown for comparison.

P. Ehrhart /Journal

of Nuclear Materials 216 (1994) 170-198

173 K olrctron

Fig. 5. Energy shifts of the inelastic scattering contributions relative to the elastic line (schematic).

of all atoms in a defective crystal, and the models that are based on elasticity theory are discussed in the next chapter. The additional inelastic scattering processes that contribute to a scattering background are summarized in Fig. 5. (a) Very close to the elastic line we observe the thermal or phonon scattering. This thermal diffuse scattering (TDS) is usually not separated experimen-

Fig. 7. Schematic view of the generation of fluorescence radiation. After a K-shell electron has been ejected by an X-ray quantum an L-shell electron jumps into its place and emits fluorescence radiation.

positions

Aluminum i I i.

-1

0

! !

2

4

6

A-

Fig. 6. Comparison of the scattering cross sections of SIA and vacancies in Al with Compton scattering and thermal diffuse scattering at 4 K. Calculations assume CuK,-radiation (A = 1.54 AI 151.

tally from the elastic line and rather the “energy” integral of the line is measured. The “broad” integral over A E corresponds to a short time average or a snap shot of the atomic arrangement in the sample and the final scattering function corresponds therefore to a superposition of many snap shots. (b) The Compton scattering can be understood by a collision of the X-ray quanta with nearly free electrons. Conservation of energy and momentum yield an energy shift that depends on the scattering vector. This energy shift is large enough to allow for the elimination of this background by crystal monochromators. Fig. 6 compares the scattering cross sections of TDS and Compton scattering to that of a low concentration of Frenkel defects (see Section 3). Only close to the Bragg reflections is this defect scattering much higher than the background. Nevertheless as the Compton scattering is incoherent and independent of the atomic structure it is usually not a severe limitation to structure investigations. (c) Fluorescence radiation is the recombination radiation after the ejection of an inner shell electron (see Fig. 7) and is therefore closely related to the absorption coefficient of the X-rays that is shown in Fig. 8. Most characteristic are the absorption edges that are observed when the energies are just high enough to eject a corresponding electron. As this resonance process can depend on the surrounding of the excited atom the so-called EXAFS (extended X-ray absorption fine structure) oscillations are observed above the absorption edge (see Section 6). This resonance absorption has also effects on the total form factor of the atom. These changes are considered by the so-called Hiin1 corrections to the scattering factor f0 that is valid far from resonance. f=fo

+ Af’ + iAf”.

(9)

Far from resonance these corrections are small, however, very close to the resonance energy they can be quite large (corresponding to several e.u. [2,31). There-

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of Nuclear Materials 216 (1994) 170-198

lengths. For a quantitative description of the distribution of the scattering intensity from a crystal that contains point defects (Fig. 1) we start with the basic Eq. (7), that presents a Fourier transform of the atomic arrangement in the crystal. This atomic arrangement is a superposition of the local distribution of the defects and of their individual displacement fields. Krivoglaz [6] has shown that these contributions can quite generally be factorized and that the diffuse scattering cross section S from the point defects (single defects or clusters) can be expressed as the product of the square of the Fourier transform of the concentration fluctuation of the defects, E(q), and the square of the scattering amplitude, A(k), due to one defect

S(k) = ( I E(q) I *> I A(k) I 2,

WAVELENGTH A, It Fig. 8. Mass absorption coefficients of several elements. Superimposed on an increase of p/p _ h3 absorption edges are observed if the energy is high enough to eject an electron out of a deeper shell.

(10)

where k is the scattering vector and q the distance of k from the nearest reciprocal lattice vector G. If correlations between the defects are important these are manifested in modulations of 1c’(q) 1’. For a random distribution of defects ( 1c(q) 1’) is given by 41 - cl which can be approximated by c for c =Z 1 (c is the defect concentration). Consequently the scattering pattern is determined by the scattering due to one defect S(k) = c 1A(k) ) ‘. Within this so-called single defect approximation the theory of the diffuse scattering from defects is well developed [7-91. We will start with an introduction to this theory and will discuss the modulation of the results due to defect correlations in the final Section 3.3. The scattering amplitude of the individual defect is given by a coherent superposition of the scattering amplitudes resulting from the defect atoms and the displaced lattice atoms in their neighborhood. Accordingly the diffuse-scattering intensity arising from defects can be written as

S(k)

=

c(f,,+ cf,

-

eik'm(eiksm 1)

I*,

(11)

m

fore this “anomalous scattering” can be used in order to discriminate different scattering contributions (see Section 61.

3. Theoretical

background

of the diff’use scattering

technique

The theory of the diffuse scattering applies to the elastic scattering of X-rays as well as neutrons, however, we apply the X-ray notation, using the atomic form factors and not the nuclear neutron scattering

where fo represents the scattering amplitude from the additional or missing atoms which define the lattice defect (“Laue scattering”). The sum in Eq. (11) describes the diffuse scattering amplitude (total amplitude minus the Bragg reflection amplitude) from the lattice atoms m displaced from their ideal positions rm by s, due to the presence of the defects (“distortion scattering”). Some examples of the Laue scattering amplitude, fo, are given in Eq. (12). For a vacancy we obtain a value of -f. This is not zero because the lattice sum in Eq. (11) includes all lattice sites and we have to compensate for this atom. For an interstitial atom in a dumbbell configuration that is centered at a

P. Ehrhart /Journal of Nuclear Materials 216 (1994) 170-198

175

For cubic crystals the relaxation volume of the defect as well as the measured change of the lattice parameter, Au/a, are directly related to the trace of Pij. Vre’ = Tr P/(C,,

Aa

1 AV

-=_-=_-=a 3v

Kanzaki

forces

lattice site we obtain a phase factor, where 1 is half of the distance between the dumbbell atoms.

=

vacancy,

f(2 cos kf - 1)

split interstitial

ic,,

cluster of n interstitial atoms.

e”‘d

1

atom,

As a next step we need an adequate description of the displacement field and we can use the results of elasticity theory that yields a description of the displacements by only a few parameters, the so-called Kanzkai forces (Fig. 9). These forces act on the neighbouring atoms of the defect and this force field decays generally very fast with distance such that only few forces are necessary. Using these forces 6, the static lattice displacements can be calculated:

m,j

and

Si(q) = cdij(q)e(q),

(13)

i

where G is the harmonic lattice Greens function of the ideal lattice that can be deduced from the dispersion curves as measured by neutron scattering. F” is the Kanzaki force acting on atom m. These Kanzaki forces can be understood by replacing the defect by an arrangement of forces Fim which produce - in the ideal lattice - the same displacements as the defect in the real lattice. For large distances r” from the defect lattice statics go over into continuum theory and we can describe the asymptotic displacements by the first moment of the force field, the so-called dipole (force) tensor 110,341: Pij = adds”‘. m

Tr P

30 (C,, + 2%)

.

(16)

S(f) = ifC’.(Cij,

r^,

pij)t

(17a)

where fct.(Cij, i, Pii) is an angularly dependent function. The characteristic decrease proportional to r-* yields a q-l behavior of the Fourier transform: 6, Pij)

(1%)

impurity,

(12)

s/ = c G:“em

c

30

s’(q) = $fC’.(Cij,

-f f’-f

substitutional

c V=’

(15)

CLj are elastic constants and R is the atomic volume. The long range displacement field can generally be described by

S(L)

Fig. 9. Description of the atomic displacement field by Kanzaki forces acting on near neighbors of the defect.

f,(k)

+ 2C,,),

(14)

The size of the displacements is given by an analytical function that depends on the elastic constants, the direction within the crystal (4, i refer to unit vectors) and the dipole tensor (for details see Refs. [7,8]). Using these expressions for s(r) or i(q) we can look on the scattering cross section S in more detail. In principle we can numerically calculate the exact scattering starting from the basic equation (11). The remaining problem is the poor convergence of the lattice sum due to the slow l/r* decrease of the displacements. This problem can be reduced by a splitting of the calculation into a numerical part that considers a limited number N’ of atoms that are close to the defect and have larger displacements and an analytical approximation for the contributions from larger distances. As the displacements are very small at larger distances the exponential in Eq. (11) can be expanded and we obtain Z exp(ikr,)(iks,). Due to the periodicity of the Fourier transform of the displacement field: f(k) =f(q), we can write S(k)

= c fD + E f, eik’n(eiksn- 1 - iks,) n=l 2 +iN(q>

,

(18)

with f= F/n,,,. In the last term of Eq. (18) the discrete lattice sum is replaced by the continuum expression, and the atomic scattering factors fi are replaced by their average value f. If there are several atoms in the unit cell this average is given by the structure factor of the unit cell,

P. Ehrhart/Joumal of Nuclear Materials216 (1994) 170-198

176

F, divided by the number n,,,, of atoms in the cell. The term (ib,) in the sum considers that this contribution is already contained in the analytical expression for k-i(q) that is discussed in detail below (HDS). Using this formalism we can calculate the diffuse scattering for all values of k and for defects of any size. 3.1. Point defects 3.1.1. DXS for different defect structures An example of the calculated scattering

cross sections for different SIA configurations in Al is shown in Fig. 10. In this model simple central forces to the nearest neighbours are assumed and therefore only one parameter (F,) can be changed to fit this model to the experiment for the case of the octahedral and tetrahedral positions, the dumbbell distance 1 is an additional parameter for this case. We observe quite different scattering patterns for the different configurations, that should allow a discrimination by comparison to the experiment. In addition, after having determined the defect structure due to the scattering pat-

tern, the defect concentration can be obtained the absolute scattering intensity [4,5].

from

3.1.2. Scattering at small values of q HDS: Due to the q-’ behaviour of S(q) (Eq. (17)) the term k?(q) is dominating close to the Bragg reflections (Eq. (18)) and as the values of the Fourier transform at small q image the displacements at larger distance this scattering intensity images the long range part of the displacement field. Considering in addition only the leading term remaining after further approximations for the contributions to the sum in Eq. (18), this equation yields

S(k)=clf,-f(L/c)+i~(q)l*,

(19)

with L/c=

g

(l-casks,).

n=l

L is identical

to the exponent of the static Debye Waller factor which therefore can be determined from

(100) Split

1

Octahedral

d

Tetrahedral

3 5

20.

Scattering

angle 26

Scattering

angle 2*

Scattering

angle 26

Fig. 10. Force model for three different configurations of the SIA in a fee lattice and the corresponding diffuse scattering intensity profiles 141.

P. Ehrhart /Journal of Nuclear Material 216 (1994) 170-198

the diffuse scattering. For defects with large displacement fields L is larger than f,, and L is always positive and nearly constant over the range of small q-variations. The most characteristic features of the HDS can be demonstrated for the simplest case of an isotropic displacement field in an isotropic crystal: V’

S(+Cr=__ r3 v=’

q

Yfl

4

that the next important term arises from the interference of L/c with this term. As L is positive the sign of this mixed term depends on the sign of the displacement field (or V”‘) and changes with the direction of

4

r (20)

4a-y r3’

177

’ .

’

‘4

S(q)=i-,. The magnitude

of s(r) (or the defect strength, C), can be expressed by the relaxation volume and the Eshelby constant y, and R is the atomic volume. Inserting this result yields:

s=c f&L,c-fp(i.,)

*.

(21)

This equation shows again that the leading term of the amplitude increases proportional to q-l and shows

Fig. 11. (a) Displacement field s(r) and (b) the intensity distribution k.i(q), that is characterized by the Huang spheres. For Q perpendicular to G the scattering vector k is perpendicular to i(q).

Table 1 Correlation of defect symmetries, the characteristic forms of the dipole tensor Pij in the cubic system and the shape of isointensity lines around highly symmetric Bragg reflections in cubic lattices

Interstitial

positions

in bee - lattices

(Ill}-split

Cmwdioo

F?o

Interstitial

positions

in fee - Lattices

{ill}-split

Symmetry

of

Curves Of iso-intensity in (110) plane of rec. lattice

P. Ehrhart /Journal

178

of Nuclear Materials 216 (1994) 170-198

4. This leads to an asymmetry of the HDS comparing the scattering at &-:q close to a Bragg peak (e.g. for positive Ye’ as expected for SIA the scattering at positive 4 will be higher than that at negative q-values). From the measurements at kq these symmetric and antisymmetric contributions can be separated and we obtain the leading term separated; i.e. the HDS in its narrow sense [ll]: Mq)=(&(q+)+&%(q-))/2

0.0.

defect clusters

--I 0

0

0

.

.

0

0.

.=

0

.

2

rel =ClJ12

Single defects

5:

cos(kq)

(22)

.

S”

After determining Vre’ from Eq. (22) the Debye Waller factor I, can be determined from the antisymmetric part

s’

: q-4

q-2j (23)

s,,,i(4)=(s(q+)-s(q-))/2'

g.

-V~-

The distribution of the intensity around a Bragg reflection is given by the cos2(k. q) distribution shown in Fig. 11, the so-called Huang spheres. If q is perpendicular to k = G there is zero intensity. Due to the inversion symmetry of the displacement field of point defects, s( -r) = --s(r), there is necessarily such a zero-plane, even in the general case of an anisotropic defect in an elastically anisotropic crystal. However, if there are shear components in S(I) these planes are inclined and by averaging over all equivalent orientations of the defect in the crystal these planes may degenerate to lines and are finally lost depending on the symmetry of the defect. Hence, the observation of zero intensity lines can be used to determine the defect symmetry (see Table 1). For cubic crystals the quantitative description of the HDS intensity can be expressed in terms of three quadratic forms of the components of Pii:

--logq Fig. 12. Schematics of the formation of defect clusters by the agglomeration of primary point defects and the corresponding change of the scattering close to a Bragg peak.

formalism is valid, however, the results are different as for this special case k = q. Therefore Eq. (21) leads to

(25) i.e. there is no qd2 behaviour anymore and fD cannot be neglected: The scattering intensity can be understood by the local density change at the position of the defect that might be partly compensated by the displacement field. Taking typical values for vre’ in metals [lo] we obtain

Vacancy: S, = c 1f 12$

$

+yC3)c Pi i>j

-/‘)(Tr P)2 + r”‘C

1.

i>j

(Pii -4j)’

SIA

f&f=

f&f=

+l;

I/=‘/(

-1;

Vrel/( ~0)

ya)

= 0 + SAXS,

= 1 --) cancellation.

(24)

The y(‘) depend on the elastic constants Cij and the directions of q and k; the general expression is given in Refs. [7,8]. Hence from a fit of Eq. (24) to the intensity distribution around a suitable Bragg reflection the three parameters (Tr PI*, ZPii - P,,)2, and EPz = can be determined. If the defect concentration is not known we obtain, however, only the product of c and the quadratic tensor parameters. In this case we can combine this result with the measured Au/a that yields c . Tr P (Eq. (16)) and can determine c and Tr P separately. SAXS: For the small angle scattering the same

This simple estimate demonstrates, that vacancies dominate the SAXS whereas SIA dominated the HDS because of their larger P’. 3.2. Dense defect clusters DXS: In the most simple approximation we can consider a defect cluster as a new point defect (Fig. 12). We start with a random distribution of defects for which the HDS is given by SA m c(Pe’j2. If it is assumed that upon clustering the long-range strain fields of point defects (that part which gives rise to HDS) superimpose linearly, then for clusters containing n defects, HDS is given by the relation

s;, _ (c/2)(

nL+

u ns:, .

(26)

P. Ehrhart /Journal of Nuclear MateriaLF216 (1994) 170-198

179

This result can be simply understood as the defects that were scattering independently (and incoherent due to the random distribution) now scatter coherently. This coherence yields a factor of n in the intensity. As the lattice parameter (- c * TrP) is not affected by this process eventual losses of defects during agglomeration can be detected, and the combination of S, and Au/a yields the increased defect size and the concentration of clusters. The approximations in the calculations of the Huang scattering are valid only in the region q -=sz l/R,, where R,, is the distance from the centre of the defect cluster to the location where the displacements s(R,,) have fallen to l/G. Thus for larger clusters, the regions for measuring SHuang and Santi are closer to the Bragg peaks. For larger values of q the diffuse scattering from an isotropic displacement field can be described by the asymptotic expression [9] S..s=Cli12-g+J,

(27)

where y is the Eshelby constant and 4 is a function of the angle between G and q. Characteristics of the asymptotic scattering are its linear dependence on the defect concentration, proportionality to qm4 and independence to cluster size. Although Eq. (27) is derived for isotropic defects these characteristics of the asymptotic scattering are quite generally valid. For quantitative results, however, numerical calculations of SAoS are necessary. As the displacement field of a cluster, e.g. a dislocation loop can be calculated by elastic continuum theory [12], the scattering can be calculated in a straightforward manner by the use of Eq. (18) [13]. These calculations reveal characteristic differences in the asymmetry of the asymptotic scattering for vacancy and interstitial loops (see Section 5.3) and in the scattering for different loop habit planes and Burgers vectors. In addition very characteristic scattering patterns due to stacking faults can be expected (Fig. 13) 1131. SAYS: In this region we obtain a similar increase of the intensity as in the HDS:

Fig. 13. General features of the scattering pattern in the (011) plane of the reciprocal lattice of Al. Isointensity lines are shown for interstitial loops on all {ill) planes: upper part for Frank or faulted loops and lower part for perfect loops [13].

Integrating the Laue term (see Eq. (12)) of Eq. (28) and assuming that the inclusion has a different scattering factor fA and number of atoms, or density nA than the matrix (f,, n,) we obtain S=

II

The scattering effect bles (as compared to we can even neglect (28) and discuss the within the so-called widely used for the

(28)

is again largest for voids or bubloops). For these kinds of defects the distortion scattering in Eq. intensity distribution of larger q two phase model that has been description of precipitates 1141.

’ jqrdr[. cV2(nAfA- n,f,> 1v/e

(29)

The first bracket represents the difference in the scattering power, or diffraction contrast between the precipitate and the matrix (e.g. fA = 0 for voids). The second term of this equation represents the form factor of the precipitate or void and determines the q-dependence of the scattering intensity. For the simplest example of a sphere of radius R we obtain 1 _ V I/

2

eiqr

=

I

3 [

sin qR - qr cos qR ’ (qR13

’

(30)

P. Ehrhart /Journal

180

of Nuclear Materials 216 (1994) 170-198

This function is shown in Fig. 14. At small values of qR the function can be approximated by exp(-q2R,.3); where R, is the Guinier radius. The relation of R, to the correct radius depends on the shape of the particle and for the sphere we obtain R, = @. For the application it is important that R, is directly obtained from a logarithmic plot of S against q2. At larger q-values we observe oscillations and a decrease of the envelope _ qm4. This region is called the Porod region and the intensity is proportional to the surface area of the precipitate. Remarkably these approximations remain valid for all shapes of particles. In this special region of small angle scattering, neutron scattering has some advantages (larger penetration and therefore thicker samples, wavelengths can be chosen large enough in order to allow no Bragg scattering and therefore no background due to “double Bragg” scattering) and has been widely used for the investigation of radiation induced defects in structural reactor materials [15,16].

cl

3.3. Defect correlations

k

Starting from Eq. (10) we have discussed the scattering from well defined and randomly distributed defects. However, non random defect correlations, that are considered in E(q), might build up especially at higher concentrations. As E(q) depends only on q it is strictly periodic in the reciprocal lattice and this scattering contribution can therefore be distinguished and separated from the single defect scattering A(k) if measurements around different reciprocal lattice points are available. As an example for the change of the scattering pattern due to fluctuations the ordering or the decomposition of an alloy is shown in Fig. 15. These fluctuations start from a random pair correlation at the level of the solute atoms. However, there may be correlations between finally formed clusters: e.g., during the precipitation process there is a dilution of the

k

Fig. 15. Schematic presentation of the ordering and the decomposition of an initially randomly distributed alloy and the corresponding change of the diffuse scattering intensity. (a) Random solid solution, the decrease of the Laue scattering is determined by the decrease of f(k). This reference line is shown in all figures. (b) and (c) Different states of the decomposition of a clustering alloy; (d) and (e) short range order and final long range order of an ordering alloy.

matrix close to a precipitate such that no precipitates grow in close vicinity. As the effects of these correlations are most prominent at very small q-values they are generally observed in SAXS and can be considered by the particle interference function [14]. Both ordering as well as clustering may be observed during irradiation e.g. the ordering of voids or bubbles or the agglomeration of smaller defect clusters within the volume of a collision cascade. In order to understand the effects of such correlations we split the total pair correlation function into the random part considered so far and the deviation of the pair correlation function from a random one, E(P): I(E(q))I’=c(l

-c)

+ CE(P)

eiqp.

(31)

P

For a simple model of N defects that are correlated within a region with a concentration c,,(r) we obtain 1171

10'

q-R IO’

Fig. 14. Form factor of a spherical precipitate.

t#?=

/c,,,(r)

eiqr dr//c,,,,(r)

dr.

For q --) 0 we obtain with C#J’= 1 the same scattering

P. Ehrhart/Joumal of Nuclear Materials 216 (1994) 170-198

log1 t

r

Fig. 13, i.e. there is a strong decrease of the intensity around q *R = 1 and finally at larger q the unmodified scattering of the “single entities”, that might be atoms or smaller clusters, is observed (Fig. 16).

“9 -2

1

4. Experimental

-2 /q“single”

-m

Fig. 16. Schematics of the intensity distribution for defects that are correlated within a region of radius R. The limit for q -+ 0 corresponds to the HDS of a dense cluster and the scattering at q z+ l/R corresponds to that of the underlying defect.

intensity as for a dense cluster of N atoms. If we assume further that the defects are contained within a sphere, the scattering is modified by the function of _&sitii

181

witive

techniques

In order to exploit the information contained in the scattering function S(k) we have to measure the distribution of the scattered intensity as a function of k, i.e. as a function of the scattering angle 26( m I k I) as well as of the direction within the reciprocal lattice of the crystal (4 or o-rotation) and single crystal samples are therefore required. Measurements of the change of the lattice parameter as well as of the HDS have been achieved by conventional laboratory techniques (Fig. 17a). The characteristic radiation from an X-ray tube is selected by a bent focussing crystal monochromator, which is necessary in order to obtain sufficient intensity. The scattered radiation is detected at the scattering angle 26. A monitor detector samples the incoming

Pt.

l2 kW Rotating Anode ttoco1 spot oAn=1+1

no&

nonochromalor

Fig. 17. Experimental setup of an X-ray diffraction experiment. (a) laboratory system, (b) beam line at a synchrotron.

P. Ehrhart /Journal

182

of Nuclear Materials 216 (1994) 170-198

beam and allows corrections for beam fluctuations. The two major technical advances that yield an order of magnitude improvement in the measuring time are: the rotating anode X-ray generator and at the other end of the system the position sensitive detector that allows the simultaneous measurement within a certain angular region. Fig. 17b shows a typical setup at a synchrotron. Due to the high intensity of the beam

focussing is generally not used and a more convenient flat crystal monochromator arrangement is applied. By rotation of the crystal to the appropriate Bragg angle any wavelength can be selected out of the continuum radiation of the synchrotron, and by using a two crystal arrangement the beam can be kept at a fixed exit position for all wavelengths. The figure shows in addition the different rotation axes that are necessary to align a single crystal. The 9 (or 01 rotation of the sample and the 24-rotation of the detector are used in order to achieve scans in a plane of the reciprocal lattice. A radial scan in the reciprocal lattice is achieved by a 1: 2 coupled motion of sample and detector (a--28-scan) and is schematically illustrated in Fig. 18a. The figure presents the physics of the scan, however, in order not to be confused it should be kept in mind that this most convenient illustration by the stretching of the scattering triangle does not exactly represent the motion in real space, because the direction of the incoming beam is fixed. The same rotations are shown in Fig. 18b: a symmetric motion of tube and detector with a fixed sample; such so-called gravimetric systems are used for fluids. A scan in a perpendicular direction (angular scan or rocking curve) is obtained with a fixed scattering triangle by a rotation of the sample by an angle w. In addition to the endpoints of the wave vectors the figure illustrates the area that is scanned by a linear position sensitive detector, which registers the intensity along an arc on the Ewald circle. The “point resolution” in reciprocal space is illustrated in Fig. 19; the resolution element within the scattering plane is determined by the divergence of the incoming beam, li, and the reflected beam, Ed. This resolution element might be further increased when the wavelength or the lengths of R, and k’ are not very sharp. In order to obtain the full resolution volume (Aq13 the divergence of the beam perpendicular to the diffraction plane 6’ must be included. As the finally observed diffuse scattering intensity is proportional to the size of this resolution element the optimization of the system is always a compromise between intensity and resolution. This scattered intensity can be written: quanta

I I

z-

S

=Z,,Ad,,N

dR,pS(k),

where Z,, is the incoming intensity

1quanta s-l cm-* 1,

Ewold

ce ii i

Edion \

I Bragg osit ion

j \;’

stcirt osition Y

i

Fig. 18. (a) Illustration of a 6-26 scan in the reciprocal space. The scan starts with an angular deviation - A from the Bragg angle and stops at a deviation + A. The sections of the Ewald circles present the angular range of the position sensitive detector at the corresponding angular positions. (b) Corresponding “gravimetric” rotations in real space, that keep the sample orientation fiied.

A

is the illuminated area of the sample, d,, the effective thickness of the sample that is determined essentially by the absorption within the sample (for reflection at a thick sample d,, = 1/(2~.); for other

scattering

geometries

see Ref. [2]), and N is the num-

ber of atoms per cm3. Hence A *d,, . N is the number of scattering atoms. dR, is size of the solid angle of the scattered beam (= EWE!= area of the detector/distance to the sample squared). p is the polarization factor that depends on the polarization of the incoming beam and the scattering angle. For the determination of S(k), I, and the geometrical parameters (A, d&l must be known. However, I,

183

P. Ehrhart /Journal of Nuclear MateriaLF216 (1994) 170-198

(= 108-10’ q/s for rotating anodes and even higher for synchrotrons) is too high to be measured by quantum detectors. Therefore a convenient technique is the use of a well known reference scatterer as a calibration sample (e.g. polystyrene) that eliminates the geometrical parameters as well and S(k) is given by [18]

S(k) = r

z

(4ffWcal Peal -y-W,,,. d

cd

N

(34)

eff

For a further discussion of the experimental techniques we refer to the literature, see e.g.: diffuse scattering far off Bragg reflections [19], HDS from point defects [20] and HDS from large clusters that require the higher resolution of three-crystal diffractometers [21,22]. Instrumentation for synchrotron beam lines is discussed in Ref. [23].

5. Example applications 5.1. Lattice parameter

and differential dilatometry

Measurements of the change of the lattice parameter have the highest sensitivity for large Bragg angles as can be seen by differentiation of Bragg’s law (Eq. (8)) Ad/d

= A6 cot 8.

(35)

Measurements can be done with powder samples (typical precision: Ad/d = 10m4), however, with single crystal methods (e.g. the Bond method 1241) higher preci-

Fig. 20. Formation and elimination surface as the final source and sink.

of a vacancy with the

sion can be achieved for nearly perfect crystals (Ad/d = 10-6). If the defect concentration is known (e.g. from chemical analysis) we can directly deduce Tr(P) or the relaxation volume Vre’ from the measured lattice parameter (Eq. (15)). Hence, Fe’ has been determined for solute gases (H, 0, N, Cl in metals or many solute atoms in dilute alloys. The classic way to separate c and Vre’ and hence to determine the vacancy concentration at high temperatures is the combination of As/a with length or density measurements, the differential dilatometry [25]. To understand this measurement we have to consider the defect formation volume VF in addition. As shown in Fig. 20 we finally form a vacancy by taking an atom out of the crystal and placing it at the surface; if there is no local relaxation (V”’ = 0) we increase the crystal by one atomic volume, i.e. the formation volume VF = 1. With Vre’ = 0 there is no change of the “microscopic length” Aa but there is a change of the “macroscopic length” AL or the volume. Or generally for cubic crystals vrel

3!LCn,

(3%

a VF

WI

3+=cy* For a vacancy we can easily see that vr = R + P,

(37) and for an interstitial we take one atom from the surface and squeeze it into the crystal: vr = -a

+ vre’.

(38)

If we consider both defects we obtain 3($-.5)=c,-ci.

Fig. 19. Schematics of the “point resolution” in the diffraction plane.

(39)

Hence, we determine the change of the number of lattice sites and hence the concentration of the dominating defect. As the relaxation- and formation volumes cancel in Eq. (39) the result is independent of the

184

P. Ehrhart /Journal of Nuclear Materials 216 (1994) 170-198

structure of the defects; remarkably, this means also that it is independent of a possible aggregation of the defects into voids or loops, as long as these defects are smaller than the extinction length of the X-rays, i.e. the X-rays yield a proper average over the displacement field. Only if defects are lost into larger sinks like line dislocations or networks are they no longer detected. However, by irradiation we produce Frenkel pairs and as long as these pairs are frozen in, we expect c, - ci = 0 or Au/a = AL/L and this observation was one of the experimental proofs for the production of Frenkel pairs. Under these conditions either Au/a or AL/L have been used for estimates of V$’ in combination with defect concentrations as determined from resistivity measurements: c = Ap/pFP; pFP = specific resistivity change per Frenkel pair. The separation of vacancy and interstitial contributions will be discussed in Section 5.2. Differential dilatometry will yield nonzero results only for higher annealing temperatures where defects are lost to sinks. As there are generally only a few defects left no detailed results have been reported. This situation changes for high dose irradiations at high temperatures where SIA form large loops that interact with dislocations and finally form a dislocation network. Fig. 21 shows an example of a 100°C irradiation of Cu foils with 3 MeV protons. At very low doses Au/a = AL/L, while at higher doses we observe a further increase of AL and even small negative values of Au. This behaviour indicates that at high doses most of the SIA are absorbed in dislocation networks and vacancies are the dominant defect type with an “excess concentration” of 7 x 10m3 at 2 dpa. The negative value of Aa needs a more detailed discussion and we start with the limiting configurations of completely relaxed vacancy agglomerates (loops, or stacking fault tetrahedra, SF’T) or unrelaxed agglomer-

1

DIdpal 2 Fig. 21. Fractional change of the length, AL/L and of the

lattice parameter, ha/a

as a function of irradiation fluence:

3.4 MeV proton irradiation of copper at 300°C 1261.

I

OO

I

1

0.2

c

1

0.4 C,, lat%l

’

0.6

0

1’ 0.8

Fig. 22. Differential dilatometry similar to Fig. 21 after He implantation into copper foils at T = 50°C. For a homogeneous implantation a variable energy of O-6 MeV was used [27]. Lower part: ratio of the concentration of the He atoms and vacancies that characterizes the density of the He atoms in small bubbles.

ates (voids or bubbles), respectively. For a vacancy within a loop V*“‘/a = - 1 and we expect a large negative change Aa and no change in length (Eqs. (36b) and (37)). For a void Vre’/R = 0, we expect no change in Aa but a large change in length. Hence, Fig. 21 could be understood by a mixture of 20% of vacancies contained in loops and 80% contained in voids. TEM shows that these agglomerates are rather small and only = 10% of the total vacancies are contained in visible loops or WT. This indicates that the invisible smaller defect agglomerates have only partly relaxed and the total average relaxation is V rel/.fI = - 0.2. On the other hand, this figure indicates that for irradiation conditions where larger voids are the dominant defect type the change in Aa can be neglected and only the measurement of the swelling is sufficient to characterize the vacancy concentration. The basic Eq. (39) is not changed if we consider the implantation of He-gas into a solid because the gas atoms are implanted from the outside and do not change the number of lattice sites (independent of their final position): for an atom remaining on an interstitial site: Vg$ = V&,i and for a substitutional site there is only a change of the relaxation volume of a vacancy. Hence, the changes after the implantation [27], shown in Fig. 22, are similar to proton irradiation. Eq. (39) yields the “excess” number of vacancies and with the independently determined He concentration the “average density” of the He can be determined

185

P. Ehrhart /Journal of Nuclear Materials216 (1994) 170-198

scattering is appreciably higher than the low temperature thermal diffuse scattering which contributes the main background scattering in this region; background problems arise only when there is interference with the Bragg peak. Therefore good resolution is required in this region. Examples of defect induced scattering, obtained by subtraction of the background are given in Fig. 24. Defect concentrations were between 300 and 500 ppm. The left-hand side shows the measured intensities along four different Ewald circles in the (110) plane of the reciprocal lattice, corresponding to four different orientations of the sample around its [liO]axis. The full curve gives a quantitative fit of the calculated curve for the (lOO)-split interstitial to the data. A comparison with the curves of Fig. 10 shows clearly that the data cannot be fit by octahedral or tetrahedral positions of the interstitial. The fit procedure was started in the region between the Bragg reflections. Only two parameters were used to describe the interstitial structure: the spacing between the atoms in the dumbbell 1, and the magnitude of the forces F between the dumbbell atoms and their nearest neighbors, which were assumed to be central forces. For the scattering of the vacancies, which is only a correction in this region, Vre’/a = 0 was assumed. Initially only the shape of the intensity profile was fitted to the experimental data. In spite of the restriction to only two parameters a good fit was obtained. In a second step, from the absolute magnitude of the intensity, the

(Fig. 22). These He atoms are contained in small bubbles; only the largest bubbles (d 5: 1.5 nm), that represent about 10% of the total number of vacancies are visible in the TEM [281. The average densities given, cue/cv, show that the pressure within the bubbles is above the equilibrium pressure but far below the pressure necessary to initiate the punching out of interstitial loops. Similar results have been obtained for Ni [29]. 5.2. Diffuse scattering from point defects 5.2.1. Metals

Measurements of the diffuse scattering intensity over a large area of the reciprocal lattice have been reported for FPs in Al and Cu. Fig. 23 shows the typical intensities as compared to the scattering background for a measurement along an Ewald circle between Bragg reflections [241 and close to a Bragg peak [30]. Measured curves for Al are shown before and after irradiation with 3 MeV electrons at 5 K yielding about 300 ppm of Frenkel defects. Between the Bragg peaks the background (Compton and thermal diffuse scattering) is between 20 and 100 times higher than the defect scattering. This ratio is more favorable for materials with higher atomic number. Nevertheless, very accurate measurements are necessary for the evaluation of the small difference between the irradiated and unirradiated sample. Near Bragg reflections the defect

100 Irrodlotcd

sample

onncolcd

somplc

10

43

3.0 scottwing

ongh

26

6b

40

42

4

Fig. 23. Typical experimental results of the scattering background and the scattering due to 300 ppm of Frenkel defects in electron irradiated Al. L&I: scattering far off Bragg reflections along an Ewald circle (as indicated in the insert) 1321. Right: Huang scattering close to a (200) reflection in [lOa] direction (see insert) [30].

186

P. Ehrhart /Journal

of Nuclear Materials 216 (1994) 170-I 98

scattering

angle

26

Fig. 24. Defect induced scattering intensity in Al electron irradiated at 5 K. Left: Comparison of an optimum fit for (lOO)-split interstitials and vacancies to experimental data points (Ap,, = 160 nn cm) [4]. Right: Huang scattering intensity close to a (200) reflection in [lo01 direction (0: Apa = 192 nn cm; 0: Ap, = 120 nR cm) [30]. Insert shows the location of the measuring points in reciprocal space.

interstitial concentration was determined and thus also pF = resistivity per Frenkel pair. With this known concentration, the scattering at small angles (which is mainly due to vacancies) could be calculated taking the volume relaxation V[$’ of the vacancy as a fitting parameter. The results using this procedure are summarized in Table 2. The right-hand side of Fig. 24 shows, with much enlarged scales, the diffuse scattering close to the (200) reflection. The predicted q-’ decrease of the intensity

y--‘/f2 VF’r’L/O

1.9 1.9

-

0.05 > = 0.0

1.9 - 0.05

5

2-

E

Q

-,7 L.>

cu

=

3l-

2-

Table 2 Results for Frenkel defects in Al and Cu Al

E

G

1.45 > 1.70

1.55

l-------J O-

I 0

- 0.22

-0.1

I -0.2

I -0.3

-0.4

rel

PFD/pncm

4.2 > 3.9

4.0

%I /n

2.5

Fig. 25. Dependence of the values of Vire’ and pF determined a Upper value from DXS [4,31], lower value from HDS+ Aa /a [30,32] and in between the resulting average value.

from S, and ha/a ted Al.

on the assumption of VT’ for e--irradia-

P. Ehrhart /Journal

of Nuclear Materials 216 (1994) 170-198

is clearly shown by a slope of -2 in the double logarithmic plot. Secondly, it can be seen that the intensities are proportional to the defect density measured by Aps. In the direction normal to the scattering vector, i.e. for q parallel [Oil], no intensity was observed; this indicates (see Table 1) a cubic or tetragonal symmetry of the long-range displacement field of the defects in agreement with the (lOO)-split configuration of the interstitial. By combination of the absolute HDS intensity and the results of the measurements of the change of the lattice parameter an independent determination of c and TrP (or Vre’) was achieved (Eqs. (15) and (24)). This evaluation is straightforward only as long as the relaxation of the vacancies can be neglected. For the general solution we have only two equations for three unknown quantities:

For metals, where v?’ is generally very small, we may start by neglecting the vacancies and calculate the change of ViTe’ and c (or pF = Ap/c) and then calculate the possible changes of the results using VI’ as a parameter. The results for Al are plotted in Fig. 25. There are two branches of solutions for the quadratic equations (40) and a limit for the relaxation of the vacancies at -0.39 0. This limit for the real solution of the system can be easily understood because the measured value of As/a is positive. The lower branch of solutions seems unlikely for small VT’ as it corresponds to c + a. However, the optimum set of data I@ Vre’, pF can only be found by comparison to add;tioial data. For the data given in Table 2 the value of Vvre’ is from DXS, while for other metals agreement with pF values from other methods has been used. The agreement of the results of Table 2 shows that the relaxation volumes can be determined to a precision of * 10%. The sources of uncertainties arise mainly from variations of the background intensity, the errors of the calibration of the intensities and on the homogeneities and calibration of the irradiation dose when we compare different experiments. The general applicability of the methods of combining HDS and the change of the lattice parameter is demonstrated by the results for FDs in many other metals: fee (Ni, Pt), bee (MO, Fe) and hcp (Zn, Co, Zr), for review see e.g. Refs. [10,32-351. However, results for isolated defects can only be obtained as long as these defects are frozen in at 4 K and as long as the concentration is low enough that there is no defect interaction. As shown in Section 3.2 we expect a strong increase of the scattering intensity when interstitials form agglomerates. Therefore if clus-

187

tering occurs at high irradiation doses, we expect a nonlinear increase of the scattering intensity with the defect concentration (as measured by Ap or As/a), and in addition a restriction of Huang scattering to smaller q-values. Therefore measurements of the concentration dependence of Huang scattering should sensitively indicate the beginning of defect agglomeration. Fig. 26 shows a plot of the scattering intensity in copper over a wide dose range [32]. It can be seen that in the dose range corresponding to Ap = 120 nfI cm, in which the structure determinations were made, there is a linear increase of the intensity with the defect concentration, as measured by resistivity. At higher doses there is a nonlinear increase, which indicates the beginning of defect agglomeration. This is also very sensitively indicated by the symmetry parameter UC’), which is zero within the errors as long as there are single defects and which increases rapidly with Ap as soon as defect agglomeration begins. A mean number, II, of about 1.4 interstitials per agglomerate is observed at 220 nQ cm corresponding to about 1000 ppm of Frenkel defects. This interstitial clustering is explained by a combination of the statistics of the spontaneous recombination processes (Luck-Sizmann effect), radiation induced diffusion and the influence of collision sequences. These effects cannot be predicted quantitatively and in addition depend on the irradiation conditions. Therefore for every metal the results of diffuse scattering experiments must be tested carefully for the possible influence of defect agglomeration in the dose range that is investigated. Fig. 27 shows scattering intensities as observed for Au where these tests obviously show clustering during irradiation at 5 K down to the lowest concentrations that were investigated.

0

With increasing

100

dose there is an

200 !&nQcml

Fig. 26. Variation of the defect strength cn(” = cf(TrP)’

and of the symmetry parameter cZIs) = ci&, jPi$ with the defect density as measured by ApO during electron irradiation of Cu at 4.7 K [32].

P. Ehrhart /Journal

188

of Nuclear Materials 216 (1994) 170-198

Au 12005

IO111 I loL &bo:102nkm)

:

/

(bga =2lnh

102 /

10’ LJ!_J 1

01

301

-

q

WI

10’ LJ----k 1

01

001 -

q WI

Fig. 27. Diffuse scattering close to the (400) and (200) reflection of Au as observed after e--irradiation at 4.7 K to different doses [32]. The nonvanishing intensity at the (200) reflection indicates non-zero values for the off diagonal elements of the dipole tensor of the defects (see Table 1).

obvious change of the intensity distribution in all directions in reciprocal space: at large q a steeper decrease that can be approximated by a qe4 law, and at small q the intensity increases faster than the concentration, which is proportional to Ap. Similar results have been

4t

( 10”e-/cm2 )

observed for bee Nb and hcp Cd, and indicate SIA migration during irradiation [32]. The determination of the defect structure in alloys is complicated by the fact that many different defect configurations might be present: e.g. for an AB alloy

4t

( 10"

e-/cm’ )

Fig. 28. Change of the HDS and of the lattice parameter as a function of the irradiation dose for GaAs and InP [38]. (a) The scattering cross section S, is multiplied by q* in order to compensate for the average decrease of SH with q-’ and to allow for the calculation of a reasonable average value over a larger region of q-values. (b) Change of the lattice parameter.

P. Ehrhart /Journal of Nuclear MateriaLF216 (1994) 170-198

189

Table 3 Relaxation volumes for defects in some fee alloys

CussNi,, Ni,Fe

solid sol. s = 0.7

Cu,Au

s > 0.9 s = 0.4

s = 0.0

Vir=’/ n

V”Y’/ R

pro / uLn cm

1.6 1.5 1.4 I 0.8

=O -0.15

13.2 1.2

WI

5.7

[371

=0

Ref. [361

00 there might be A-A, A-B and B-B dumbbells present

simultaneously. Nevertheless the scattering yields an average value over all defects and in fortuitous situations a separation may be possible. The results for some fee alloys are summarized in Table 3. For alloys of atoms of similar atomic size and mass the results are the same as for pure metals independent of the type of alloy: CuNi is a solid solution with a tendency for short range decomposition and Ni,Fe is a long-range ordered alloy [36]. In contrast to that there is a much smaller Fre’ for Cu,Au that is independent of the degree of long-range order. This low value can be rationalized if essentially only the smaller Cu atoms are stable at the interstitial sites [37]. 5.2.2. Semiconductors and insulators Due to the possibility of different charge states for SIAs and vacancies that might be connected with different defect configurations the number of possible defect structures is once more increased as compared to the metallic alloys. Nevertheless, the DXS investigations seem to show some characteristic results. Although there might be ionization induced migration processes [lo] during irradiation Fig. 28 shows a linear increase of SH with irradiation dose for GaAs and InP [38], that indicates the dominance of stable defects even at high electron doses. In contrast to the increasing S, there is no increase in As/a for InP. This implies that Vire’= - VVre’.As similar compensations are observed for Si and Ge, it appears that this result might be characteristic for some semiconductors; this observation might be rationalized by the more open diamond or zinc blende structure that allows for much space for SIA whereas the missing atom creates a large hole that is prone for larger inward relaxations than in metals. A second complication compared to metals is shown in Fig. 29. The scattering cross section is multiplied here with q2 in order to compensate for the q-* decrease and we expect a constant value. Instead we observe a decrease at small values of q that is most prominent for InP. As this decrease seems not to depend on the defect concentration it indicates a deviation from the l/r2 decrease of the elastic displace-

0.01

0.02

0.03

0.04

0.05

0.06

g/G Fig. 29. Dependence

of SH on the distance q in the [Sll] direction from the (511) reflection for GaAs and InP. As S, is multiplied by q2 the HDS should yield a constant value [381.

ment of point defects. All details of the scattering including the asymmetric part (Eq. (23)) can be explained by the assumption of close FPs. Fig. 30 shows the corresponding superposition of the displacements that yields large displacements between the vacancyinterstitial pair and a cancellation at larger distances. Similar FPs have been observed also for Ge. This behaviour shows that close FPs are much more stable in these crystals than in metals. In GaAs the annealing of these FP’s on the Ga sublattice occurs around room temperature whereas the FP’s on the As sublattice anneal around 500 K. As there is no indication of defect agglomeration during the 300 K annealing the defects on the two sublattices can be separated in this fortuitous example. The relaxation volume of the Ga interstitial is rather large:

Fig. 30. Schematic view of a closely correlated Frenkel pair and the atomic displacements of the neighboring atoms. The preferred orientation of the pair is close to (111).

P. Ehrhart /Journal

190

of Nuclear Materials 216 (1994) 170-198

V,f,$ = 1.90 (even after considering V<& = -0.60) as compared to the interstitial atom on the As sublattice of = 1.00 (see Ref. [38] for a discussion including the possibility of the production of di-interstitials). Similarly large differences between the two sublattices have been observed for KBr where a separation was possible by the combination of optical absorption spectroscopy, EPR, Au/a and HDS 1391. However, more investigations are necessary for a systematic understanding of the defect properties similar to that in metals. 5.3. Diffuse scattering from clusters 5.3.1. Dense clusters HDS from small SL4 clusters: During the annealing

in stage I [lo] SIA become mobile and recombine with vacancies and if, as in a pure metal, there are no other traps available the only way to avoid complete annealing is by the formation of small immobile SIA clusters. As there is nearly never complete annealing in stage I, SIA clustering is a general phenomenon and the sizes and structures of the clusters have been systematically investigated. Fig. 31 shows the change of the HDS close to the (4OOkreflection of e--irradiated Ni [40] along with the annealing to the end of stage II, and demonstrates again the characteristic changes of Fig. 12. As long as the clusters are small and a well defined

Fig. 32. Isochronal annealing of the HDS intensity and of the change of the lattice parameter of e--irradiated Ni; c,(5 K) = 300 ppm. At some points the average number N of SIA per cluster is indicated as obtained from Eq. (41).

-’ region can be observed the average number (N) of SIA in a cluster can be deduced directly from Eq.

4

(26) after considering the change of the total concentration by the decrease of Aa or Ap.

(N) =

SH(T) +Aa(5 K) SH(5 K)Aa(T)

With the experimental sH [a.u.] t

‘\

104

resolution

selected for the in-

I coCL.5Ki = 300PPm

2

i

Lq-4

‘t?

:

10'

40

300K

i. \

0 :

l-

’ ”

t

(41)

’

I

1 OS0i

6K

A’:

’

+

2 ‘i

$1 q-2 $1

6

‘

% 10'

i

1

G:(400)

‘;

q:[lOOl 10

1 0

1

I

II

5

10

20

-

q/G

50

100

150

200

250

300

T/K -

_I_ 30

lo3

Fig. 31. Change of the distribution of the HDS intensity close to a @OO)_reflectionof e--irradiated and annealed Ni; co” (5 K)= 500 ppm. There is a q-’ behaviour at 6 K and an increase of the intensity after annealing at 300 K.

Fig. 33. Comparison of the growth of the cluster sizes (N > in pure Ni and in dilute Nisi,,,, and NiGeo,O, alloys. Initial concentration of Frenkel defects after 3 MeV e--irradiation: ~(4.5 K) = 300 ppm. The smallest values of (N) for NiS&, show that there is a mixture of different defect solute complexes containing one or two interstitial atoms; the trapped di-interstitials are immobile up to T = 250 K [41].

191

P. Ehrhart /Journal of Nuclear Materials216 (1994) 170-198

Lo

0

50

100

150 TM -

200

250

Fig. 34. Comparison of the growth of the sizes of clusters of interstitial atoms (N) in pure Ni and the concentrated Ni-base alloys Ni,Fe and NiCu. The initial concentration of Frenkel defects after electron irradiation at 4 K was: c,(4 K) = 500 at ppm. Di-interstitials are formed in Ni,Fe during annealing

stage I and are immobile up to room temperature [36].

vestigation of single defects the approach can be used typically up to (N) = 50. However, it must be kept in mind, that a linear superposition of the displacement

field is assumed and more detailed models are necessary if there are additional relaxations during clustering. For typical metals the maximum error might be estimated as I/Ie’ changes from a starting value of = 1.70 to an asymptotic value of 10 for a very large dislocation loop [41]. Fig. 32 shows the continuous cluster growth during annealing of pure Ni and Fig. 33 shows how a small amount of solute atoms can dramatically suppress the cluster mobility and the consequent cluster growth [41]. Similar differences are observed with concentrated Nibase alloys in Fig. 34. A comparison to the cluster growth in some other metals is discussed in Refs. [34,35]. HDS from SL4 and vacancy loops: Dependent on the metal there is a transformation of the clusters to the form of well defined loops for (N) between 10 and 30. Therefore the scattering of these larger clusters can be compared to model calculations for dislocation loops that yield quantitative results from the asymptotic (or qw4) scattering. By selecting the appropriate Bragg reflections this scattering can be very sensitive to the vacancy or interstitial type of the loops: e.g. Fig. 35 shows that there is a strong asymmetry of the intensity at the (222) reflection that varies with the loop type [13,42]. In addition the position of the strong maximum scales with the loop size, i.e. q,,,;R = const. Hence the size distribution of both types of loops can be deduced from a fit of calculated distributions to the

I

I

-I

C

b

Fig. 35. Diffuse scattering calculated for vacancy (dashed) and interstitial loops (solid) in copper. The intensity is scaled by q4/R2 and is shown for the [ill] direction close to a (222) reflection [42]. The calculation presents an average over the cubic equivalent orientations of a faulted or Frank loop. (habit plane: {ill) and Burgers vector a/3 (111)). For this special direction the scattering from a perfect loop is very similar (Fig. 13). (a) loop radius R = 1 nm, (b) R = 2 nm and (c) R = 4 nm.

192

P. Ehrhart /Journal

of Nuclear Materials 216 (1994) 170-I 98

experimental data. An example for loops in Ni-ion irradiated Cu is shown in Fig. 36 and the corresponding size distribution of loops in Fig. 37 [42]. As TEM cannot distinguish the type of these small loops a direct comparison is only possible for the total number of loops; the results show good agreement for larger loops, however, there is a discrepancy at small loops. As these small loops are close to the visibility limit of the TEM the difference might be explained by a loss of sensitivity. Similar differences are observed also for loops observed after e--irradiation and neutron irradiation [43]. It is also remarkable that the total number of SIAs and vacancies obtained from the fit are nearly equal. SAXS from voids, bubbles and precipitates: Similar to the HDS from SIA the SAKS from vacancies increases during agglomeration (Eq. (28)). A direct application of this increase is shown in Fig. 38, which shows the increase of SAXS during annealing of Cu in stage III. The size of the clusters corresponds to an average number of = 4 at 260 K and = 30 vacancies at 300 K. For larger clusters the form factor can be determined in addition (Eq. (29)) and an example for the scattering of large voids is shown in Fig. 39. By the combination of measurements of SAKS with SANS a large q region was investigtted [44] and a perfect Guinier plot yields R, = 215 A. At larger q-values, i.e. in the Porod region, an anisotropic scattering was observed that was explained by facetted voids again in agreement with the TEM. An example of the growth of bubbles during implantation of He into Ni is shown in Fig. 40 [45]. An increase of the Guinier radius with dose is directly observed and deviations from the Guinier law at small q indicate bubble ordering. From the anisotropy of this intensity the pressure within these bubbles was deter-

-460246

L

R lnml Fig. 37. Dislocation loop size distribution obtained by fitting the diffuse scattering of Fig. 36. (a) Vacancy loops, (b) interstitial loops and (c) combined vacancy and interstitial loops (log. scale!); the full circles show the corresponding TEM results.

q hm-9 Fig. 36. Diffuse scattering intensity multiplied by q4 as observed after irradiation of Cu with 60 MeV Ni ions at room temperature (& = 1.2~ lOI3 ions/cm*). Measurements at the (222) reflection in [ill] direction of q [42].

mined in addition and a corresponding “density” of C&C, = 1.5-2.0. Although the implantation conditions are assumed to be very similar to those discussed in Section 5.1 this density is higher than the results of the differential dilatometry (c&c, = OS-l.0 for similar cue [29]) and the bubbles seem larger than those

193

P. Ehrhart /Journal of Nuclear Materials 216 (1994) 170-198

1

U.&i OR

1

1.0

I

1.2

k IA? -

0.1 ' ' 0 0.4 0.6

1 08

I 1.0

' ' 1.2

k IA%+

0.1 ' ' 0 0.4 06

_R

Fig. 38. Guinier plot of the SAKS during annealing of e--irradiated copper [31]. The data are normalized by the electrical resistivity in order to compensate for the total loss of defects. Therefore the scattering due to isolated vacancies is nearly constant up to 200 K and an increase of the SAKS is observed during stage III.

observed ferential indicated

by TEM for the samples investigated by difdilatometty. SANS investigations [46] have in addition possible contributions of defects

to the scattering; this indicates that very careful control

of all implantation parameters is necessary ducible results on these parameters.

for repro-

In addition to voids, the formation of precipitates is a major microstructural change in high temperature

Id -_ 5-

Of A 02 l 03

0

2-

IO2 I, 5-

T E "

?

c q2 9 !O' L 8' = cc 5 -lC5

2-

400

0

x2tA

cmool

o-2

1

Fig. 39. Guinier plot of the small angle scattering of neutron irradiated AI (c#t= 3.9 X 102’ n/cm2 at 50°C). The slope yields a Guinier radius of 215 A [44].

194

P. Ehrhart /Journal of Nuclear Materials216 (1994) 170-198

irradiated alloys. Depending on the composition of precipitate and matrix (Eq. (29)) the SAXS or SANS can very sensitively detect these inclusions, and especially SANS is routinely used for the characterization of reactor vessel steels (for example see Refs. [15,16]). 5.3.2. Defect correlations within displacement cascades After irradiation of aluminum with neutrons at 5 K, v. Guerard et al. [47] first observed an increase of the scattering intensity at small q (Fig. 41) that could be attributed to defect correlations. In addition the scattering intensity was higher than expected from the data for single FPs (Table 2). According to the introduction in Section 3.2.2 these observations yield a model for the cascade damage in Al: the cascade in Al is very dilute but within a spherical region of R = 50 A there is a high correlation of small SIA agglomerates containing on the average about 3 SIA. A refinement of the model considering the PKA spectrum of the reactor has been given in Ref. [48]. A more complete insight into the primary cascade

0.10

structure can be obtained from the combination of HDS and SAXS that yields the additional correlation of the small vacancy agglomerates. Fig. 42 shows such measurements for Fe [17]. The average cascaOderadius for the SIA is determined to be (R,)i = 36 A and the SAXS yields (R,), = 15 A; i.e. the vacancies (dominated by single and divacancies) are lying in the center of the cascade. Hence, these scattering effects seem to open a way to investigate the primary cascade structure within dilute cascades that cannot be resolved by TEM. After annealing through stage I these correlation effects are no longer visible for the SIA, however, the vacancy correlations are visible up to the end of stage II.

6. Special applications of synchrotron radiation Synchrotrons supply a “white” spectrum of radiation and therefore the wavelength used for the experiment can be continuously changed without loss of

ku-2)

0.15

Fig. 40. Guinier plot of the SAXS of Ni after implantation of He at 50°C. The deviation from the Guinier line at small 4 indicates the beginning of bubble ordering [45].

P. Ehrhart /Journal of Nuclear Materials216 (1994) 170-198

195

intensity. Two applications of this property that have a great potential for the application to irradiation defects will be introduced in this final section. As there are not yet many examples of application, this introduction will be very short and qualitative. 6.1. Anomalous scattering Close to absorption edges the scattering amplitudes of the different atoms are strongly changed (up to about 20%) by the dispersion corrections (Eq. (9)). This change can be used in order to discriminate different contributions to the scattering and/or to increase the contrast of a precipitate [49]. These changes are illustrated in Fig. 43 for the example of the SAXS of precipitates and voids. Comparable or even larger changes of the scattering factors can be achieved for neutron scattering by an exchange of suitable isotropes; however this requires different samples. Measurements at one sample can be achieved for magnetic samples

01 .OO

.lO q

(l/Al

“’

m. These changes of the scattering contrast are large as long as the scattering of the defect itself - the Laue scattering as introduced above - is important and small angle scattering is an obvious application. In contrast to that the HDS is dominated by the scattering from the displacement field of the matrix that scales with f (Eq. (24)) and no additional information is expected for the HDS in its narrow sense. The scattering at larger values of q is determined by the interference of all terms and it depends on the details of the samples

I.q2 lo.u.1

Fig. 42. Diffuse scattering after 5 K neutron irradiation b#d5 2.1 x lOi n/cm’) of Fe. The diffuse scattering intensity close to a (440) reflection (upper part) is fitted lay a model of SIA correlations within a sphere of R= 36 A. For the SAXS (lower part) a fit assuming randomly distributed cascades (full line) as well as a better fit with highly correlated cascades is shown [ 171.

10 8 6

O”l.Jh+_ _._____________ _-----

____..

whether the application useful.

of the anomalous

scattering is

6.2. EX4FS 005

0.10

0.15 _ 0.20

-9

IA-‘1

Fig. 41. Diffuse scattering multiplied by q* as observed after neutron irradiation at 4 K (c#tI: 1.0 X 10” n/cm’). The dotted line indicates the corresponding scattering intensity for single FPs and the full line a model that fssumes high defect correlations within a sphere of R = 50 A, i.e. an increased density of small clusters containing (N > = 3 SIA 1471.

For energies slightly higher than an absorption edge the absorption coefficient p(E) of a crystalline solid shows oscillations superimposed on the overall A3 behaviour. These oscillations arise from interference effects as the wave that describes the ejected photoelectron can be scattered by the neighbouring atoms (Fig. 44). Dependent on E or k of the photoelectron there

P. Ehrhart /Journal

196

of Nuclear Materials 216 (1994) 170-198

may be constructive

or destructive interference with the backscattered wave. The relative change of p(E) compared to the undisturbed value ps(E) is given by (for details see Ref. [23] and references therein) x(k)

= {(P(E) = 7

.............

a

B

- /-%(‘9)l/PCLo(~)

l

:::::..o:::::

0 . . . . . . .

A;

:::: :::: . . . . . . . . . . . . . . . . . . . . . . .o.o.

E -

E.

__---

............. ............. .... .H..

....

............. .............

(42)

E E.

E-E.

k=

. . .

. . . . . . . . . ...* . . . . . . . . . . 5..

... ..........

E-E

+ Qj(k)]

-*.

l

.,.s . . . . . . . . .

*... . . . . .

$F,(k)D,(k) exp( -2a2k2)

Xsin[2krj

l

:;-;:““::;::

. . . .

I

;(E-E,).

The sum considers the different neighboring atoms. Most important here is the term sin(2krj + Qj(k)) that allows the determination of the distances r! of the neighbouring atoms from the EXAFS oscillations (the phase shift Gj is assumed to be known e.g. from reference systems). The amplitude of the EXAFS yields in addition the coordination number Nj or the number of neighbours in a shell. Fi is the backscattering amplitude, Dj considers inelastic losses and a Debye Waller factor considers static and dynamic displacements. As the concentration of radiation defects is low there is no measurable change of the EXAFS by intrinsic defects as this disturbed signal is superimposed on the signals of all unperturbed atoms. However, the EXAFS of solute atoms or probe atoms can be monitored if a large fraction of these probe atoms are perturbed by the trapping of a mobile defect. Such experiments have been performed for Al&) [51] and for an Al-1000 ppm Zn alloy [52]. The latter alloy has been irradiated at 80 K to obtain a total concentration of = 200 ppm of FDs. As the SIA are mobile during irradiation they can be trapped at the Zn atoms and Fig. 45 shows that there is a small change of the Zn absorption. The change is more

1 EN

EA

Energy

Fig. 43. Use of the anomalous scattering in order to increase the scattering contrast and to separate chemical and structural inhomogeneities. A precipitate of atoms (atomic number Z,, and scattering amplitude fA) within a matrix (atomic number Z,, and scattering amplitude fM) and a void ( fA = 0) are shown as examples. For close neighbours in the periodic table the scattering contrast I fA - fM I ’ is very small in the energy region where the normal scattering dominates (E,). However, for energies very close to the absorption edge the contrast is increased appreciably (by the decrease of the matrix scattering in the example). The void contrast scales with fill only and can therefore be distinguished.

clearly visible after a Fourier transformation of the signal. In modelling this change of the signal an octahedral position for the Zn atom yielded the best fit. However, this conclusion is at variance with the mixed

wave vector of

constructive

destructive interference

Fig. 44. Description of the EXAFS oscillations by the interference of the wave of the photoelectron with the backscattered waves.

197

P. Ehrhart /Journal of Nuclear Materials 216 (1994) 170-I 98

0

1

2

3

4

5

6

7

@

R(A)

9

Fig. 45. EXAFS oscillations at the Zn-K-edge of an Al-1000 ppm Zn alloy. Full lines before irradiation and dashed lines after electron irradiation at 80 K that is expected to yield a total concentration of = 200 ppm of FDs. (a) Absorption coefficient multiplied by k3 to compensate for the total decrease with k (see Fig. 8); (b) Fourier transform of (a) [52].

dumbbell structure deduced from mechanical relaxation and in addition a well established annealing step at 130 K is not visible in the EXAFS. Therefore, at present it is not clear that EXAFS has the sensitivity (due to the high concentration of defects and solute atoms required) to determine the detailed structure of SIA solute complexes. However, EXAFS might generally be considered to demonstrate the existence and the temperature range of defect trapping and detrapping reactions as nearly every element may be used as a probe atom. In addition EXAFS can be used to identify new phases if solutes migrate and segregate and precipitate at higher temperatures [53].

7. Concluding remarks I have tried to give an introduction to the basic principles of the XRD methods available for the investigation of radiation defects. The power as well as the limitation of the methods was demonstrated through a wide variety of examples. These examples should especially demonstrate the unique possibilities of the methods: (i) The investigation of intrinsic atomic defects that is generally applicable as there is no special probe atom necessary. (ii) The possibility to monitor the defect interaction and the related agglomeration in terms of agglomerate size and structure: Hence there is no gap in the cluster evolution starting from single SIAs and vacancies up to the direct visibility in the TEM. (iii) For the investigation of larger defect clusters, the diffuse scattering technique should be considered as complementary to TEM work. The largest advantages of diffuse scattering are the inherently good statistical sampling and its nondestructive character. Limits on its application are the necessity of singlecrystal samples with high perfection and the inability to

image individual defects when several different defect types are present in a sample.

References 111There are many textbooks on X-ray diffraction, e.g. L.V. Axaroff, R. Kaplow, N. Kato, R.J. Weiss, A.J.C. Wilson and R.A. Young, X-ray Diffraction (McGraw-Hill, New York, 1974). A. Guinier, X-ray Diffraction (Freeman, London, 1963). L.H. Schwartz and J.B. Cohen, Diffraction from Materials (Academic Press, New York, 1977). H. Schulz, Diffuse X-ray Diffraction and its Application to Materials Research in Current Topics in Materials Science, ed. E. Kaldis, vol. 8 (North-Holland, 1982). B.E. Warren, X-ray Diffraction (Addison-Wesley, London 1969). La International Tables of X-ray Crystallography, vol. I-IV (Kynoch Press, Birmingham, 1962). 131 D.T. Cromer and D. Liberman, J. Appl. Cryst. A 37 (1981) 267. 141 H.-G. Haubold, in Fundamental Aspects of Radiation Damage in Metals, eds. M.T. Robinson and F.W. Young, US-ERDA-Conf. 75 1806 (1975) p. 268. [51 P. Ehrhart, H.-G. Haubold and W. Schilling, Adv. Solid State Phys. 14 (1974) 87. b1 M.A. Krivoglax, Theory of X-ray and Thermal-Neutron Scattering by Real Crystals (Plenum, New York, 1969). [71 P.H. Dederichs, J. Phys. F3 (1973) 471. 181 H. Trinkaus, Phys. Status Solidi B 51 (1972) 307; 54 (1972) 209. [91 H. Trinkaus, Z. Naturforsch. 28a (1973) 980. [lOI W. Schilling, in these Proceedings. [ill K. Huang, Proc. Roy. Sot. A 190 (1947) 102. WI S.M. Ohr, Phys. Status Solidi B 64 (1974) 317. El31 P. Ehrhart, H. Trinkaus and B.C. Larson, Phys. Rev. B 25 (1982) 834.

198 [14] For special introductions

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of Nuclear Materials 216 (1994) 170-198

into the SAXS see: A. Guinier and G. Fournet, Small-Angle Scattering of X-rays (Wiley, New York, 1955); 0. Glatter and 0. Kratky feds.), Small Angle Scattering (Academic Press, London, 1982). [15] W.J. Phythian and CA. English, J. Nucl. Mater. 20.5 (1993) 162. [16] C.G. Windsor, J. Appl. Crystallogr. 21 (1988) 582. [17] J. Peisl, H. Franz, A. Schmalzbauer and G. Wallner, Mater. Res. Sot. Proc. 209 (1991) 271. (181 C.J. Sparks, in Anomalous Scattering, eds. R. Ramaseshan and SC. Abrahams (Mungsgaard, Copenhagen, 1975) p. 175. [19] H.-G. Haubold, J. Appl. Crystallogr. 8 (1975) 175. [20] H. Peisl, J. Appl. Crystallogr. 8 (1975) 143. [21] B.C. Larson, J. Appl. Crystallogr. 8 (1975) 150. [22] B.C. Larson and W. Schmatz, Phys. Rev. B 10 (1974) 2307. [23] Handbook on Synchrotron Radiation, vol. l-4 (Elsevier, New York, 1991). [24] W.L. Bond, Acta Crystallogr. 13 (1947) 814. [25] R.O. Simmons and R.W. Balluffi, Phys. Rev. 125 (1962) 862. [26] W. Jiiger, P. Ehrhart and W. Schilling, Solid State Phen. 3/4 (1988) 279. [27] A. Gaber and P. Ehrhart, Radiat. Eff. 78 (1983) 213. [28] P. Ehrhart, A. Gaber, A.A. Gadalla, W. JIger and N. Tsukuda, Nucl. Instr. and Meth. B 19/20 (1987) 180. [29] P. Ehrhart, A.A. Gadalla, W. Jiiger and N. Tsukuda, Acta Metall. 35 (1987) 1929 and 1942. [30] P. Ehrhart and W. Schilling, Phys. Rev. B8 (1973) 2604. [31] H.-G. Haubold and D. Martinsen, J. Nucl. Mater. 69/70 (1978) 644. [32] P. Ehrhart, J. Nucl. Mater. 69/70 (1978) 200. [33] P. Ehrhart, MRS Proc. vol. 41 (1985) 13. [34] P. Ehrhart, K.-H. Robrock and H.R. Schober, in Physics of Radiation Effects in Crystals, eds. R.A. Johnson and A.N. Orlov (Elsevier, 1986) p. 3. [35] P. Ehrhart and H. Schulz, in Landolt-Bdrnstein III/25, ed. H. Ullmaier, (1992) p. 88.

[36] 0. Bender and P. Ehrhart, in Point Defects and their Interactions in Metals, eds. J.I. Takamura, M. Doyama and M. Kiritani (University of Tokyo Press, 1982) p. 639. [37] R. Urban and P. Ehrhart, Mater. Sci. Forum 15-18 (1987) 1251. [38] P. Ehrhart, K. Karsten and A. Pillukat, MRS Proc. vol. 279 (1992) 75. [39] H. Spalt, H. Lohstijtter and H. Peisl, Phys. Status Solidi B 56 (1973) 469. [40] 0. Bender and P. Ehrhart, J. Phys. F 13 (1983) 911. [41] R.S. Averback and P. Ehrhart, J. Phys. F 14 (1984) 1347 and 1365. [42] B.C. Larson and F.W. Young, Phys. Status Solidi B 104 (1987) 273. [43] P. Ehrhart and R.S. Averback, Philos. Mag. A 60 (1989) 283. [44] J.E. Epperson, R. Hendricks and K. Farrell, Philos. Mag. 30 (1974) 803; R.W. Hendricks, J. Schelten and W. Schmatz, Philos. Mag. 30 (1974) 819. [45] H.-G. Haubold and J.S. Lin, J. Nucl. Mater. Ill/112 (1982) 709. [46] D. Schwahn, H. Ullmaier, J. Schelten and W. Kesternich, Acta Metall. 31 (1983) 2003. [47] B.v. Guerard, D. Grasse and J. Peisl, Phys. Rev. Lett. 4 (1980) 262. [48] R. Rauch, J. Peisl, H. Schmalzbauer and G. Wallner, J. Nucl. Mater. 168 (1989) 101. [49] Y. Waseda, Novel Applications of Anomalous X-ray Scattering for Structural Characterization of Disordered Materials, in Lecture Notes in Physics, ed. H. Araki (Springer, 1989). [50] P.A. Beaven, F. Frisius, R. Kampmann and R. Wagner, in Atomic Transport and Defects in Metals by Neutron Scattering (Springer, 1986) p. 228. [51] W. Weber and H. Peisl, see Ref. [36] p. 368. [52] T. Bolze and J. Peisl, Mater. Sci. Forum 15-18 (1987) 575. [53] F. Maury, N. Lorenzelli, M.H. Mathon, C.H. de Novion and P. Lagarde, J. Phys. C6 (1994) 569.