Characterization of the perfection of large single crystals by means of γ-ray diffractometry

Prog. Crystal Growth and Charact. 1987, Vol. 14, pp. 315--365 Printed in Great Britain. All rights reserved

0146-3835/87 $0.00 + .50 Copyright ~) 1987 Pergamon Journals Ltd

CHARACTERIZATION OF THE PERFECTION OF LARGE SINGLE CRYSTALS BY MEANS OF 3-RAY DIFFRACTOMETRY Jochen R. Schneider and Hans A. Graf Hehn-Meitner-lnstitut f o r Kernforschung, Glienicker Str. 100, D-1000 Berlin 39, F.R.G.

Bragg diffraction experiments wlth y-radiatlon of energies of the or-

der of 400 key allow for high resolution studies of bulk properties of large single crystals which are of relevance for the characterization of as grown single crystals as well as for the investigation of structural phase transitions. The absorption of this radiation In matter Is -1 very weak, as an example, the mean free path in copper is po = I cm. Therefore samples can be mounted in any cryostat, furnace or high pressure device without causing window problems. The Bragg angles are only of the order of I" and thus the shape of the measured diffraction profile Is mainly affected by lattice tilts, y-ray dlffractometry Is complementary to back scattering techniques which are most suitable for lattice parameter measurements. Using a double crystal setting diffraction patterns are recorded wlth an angular resolution of I second of arc. The integrated reflecting power Is measured absolutely with an accuracy of II or better and by using various wave lengths in the range between 0.02 and 0.04 A, highly accurate, model independent structure factors can be determined from imperfect single crystals by means of wave length extrapolation techniques. Because of the short wave length polarization effects are negllblble. The full width at half maximum of the diffraction pattern of a perfect crystal as pre-

315

316

J.R. Schneiderand H.A. Graf

dicted by dynamical diffraction theory is generally less than 0.5 seconds of arc and the extinction length is of the order of 0.5 mm. Therefore small distortions from a perfect l a t t i c e cause rather large changes in the measured integrated reflecting power. Recently Pendel16sung intensity beats could be measured in Si with 316 and 468 keV y-radiation allowing to determine the 220 structure factor with an accuracy of ± 0.1% . In addition a surprisingly anisotropic strain f i e l d has been observed in floating-zone grown Si crystals.

I.

INTRODUCTION For the measurement of an X-ray single crystal diffraction profile

one normally needs an X-ray tube, a monochromator, the sample, various collimators and a detector. In y-ray dlffractometry the X-ray tube is replaced by a radioactive y-ray source (~200 Curies), which emits one or several strong y-lines in the energy range between I00 and 700 keV. y-radiation is highly monochromatic by nature (AE/E=lO -6

at room

temperature) and no monochromator is needed on a y-ray diffractometer. With the aid of a Ge solid state detector the y-llne in question is easily separated from Compton background and other lines by electronic pulse hight discrimination leading to a background in Bragg diffraction experiments of the order of 0.05 cps. Because of the small values of AE/E the resolution of the dlffractometer is determined by the angular divergence of the incident y-ray beam as defined by collimators behind the source and in front of the sample. Typical distances between the pair of collimators vary between 2 and 4 m, the best angular resolution obtained so far in a one crystal y-ray diffractometer is I0". The wave length of the incident beam is typically 0.03 A, which leads to small Bragg angles of eB~l °. As a consequence crystals are studied in transmission (laue) geometry. Furthermore distances of the order of 2 m are needed between sample and detector in order to separate the Bragg spots from different reflexlons in the detector plane. Fig. i shows a schematic drawing of the y-ray diffractometer installed at the Hahn-Meltner-lnstltut in Berlin which is operated either wlth a I98Au or a Ig21r y-ray source. The interaction of 400 keV y-radlation wlth matter is weak, the mean free path is of the order of cm and thus more than 2 orders of

317

•y-ray d i f f r a c t o m e t r y Au -source in

leod shielding ]r-source in

somple mounted

Oe solid stote

leod shielding

on o x-Ira circle

detector

I

r-. . . . '~

m 4mrn

m I

- .,,,

x,, 4mm

t

o'-I

,.

I

_ 0ooo

NoIITL) scintillotion counter

11

~

~T

Loser

[

J L

Flg.l. Schematic drawing of the y-ray dlffraclxMneter at the HahnHeltner-lnstltut in Berlin, which can be operated either with 411.79 keY radiation from 19eAu or with 316.5, 468.06 or 604.4 key radiation from z921r.

magnitude larger than for X-rays, therefore large samples can be studied. In addition the mounting of samples within cryostats, furnaces or high pressure devices does not cause experimental difficulties. As an example one could refer to the combined light optical and y-ray diffraction study of the ferroelectrlc-ferroelastlc phase transition In KH2PO 4 and RbH2P04 [i], where the temperature close to the critical temperature was varied in steps of approximately 0.01K with a stability of ±2.10-3

over three hours. Due to the small Bragg angles the

shape of the measured y-ray rocking curves is mainly caused by lattice tilts and the technique is thus complementary to backscatterlng experlments which are most suitable for lattice parameter measurements [2). It has been demonstrated that the diffraction of 0.03 A yradiation in imperfect single crystals can be interpreted within the kinematical theory of diffraction for most cases, or better, by going to higher orders of reflexlon, which involves only minor changes in the shape of the irradiated crystal volume element, the kinematical limit can always be reached [3]. In this limit the measured y-ray rocking curve is directly proportional to the so called mosaic distribution function W(.), provided the instrumental resolution function is narrower than the full width at half maximum (FW~) of W(.) and its fine structure, respectlvely. As a consequence, in the past most y-ray diffraction experiments were undertaken in order to study the mosalclty of large single crystals, especially of crystals intended for use

J. R Schneiderand H.A. Graf

318

in neutron scattering experiments either as monochromators or samples [4-5]. More recently interesting crystal growth studies have been performed on line with y-ray diffraction experiments [6]. y-ray diffractometry has been used to study changes in texture as caused by structural phase transitions [1, 7], as well as for accurate absolute measurements of structure factors [8-12].

The f i r s t y-ray dlffractometer was suggested by H. Maier-Leibnitz and buildt by one of the authors (J.R.S.) at the Instltut Laue-Langevin in Grenoble in 1970 [13], a f a c i l i t y largely extended since then by A. Freund and collaborators. Similar instruments are now operated at the Technlcal University of Munich, the Kernforschungsanlage in JUlich, the Rutherford Laboratory near Oxford (U.K.), and the MaxPlanck-lnstltut fur Metallforschung in Stuttgart. Dlffractometers especial|y designed for structure factor measurements are in operation at the Hahn-Meitner-lnstitut In Berlin and at the Research Reactor Facility in Columbia, Missouri, U.S.A.

As far as the characterization of the perfection of large stngle crystals

by y-ray dlffractometry is concerned the state of affairs

has been summarized in a contribution to the proceedings of a Nato Advanced Study Institute on the characterization of crystal growth defects by X-rays methods which was held In Durham in summer 1979 [14]. In the following we will frequently refer to this paper and omlt the explanation of those quantities which were already discussed in ref. 14, instead, the emphasis is on the more recent developments in y-ray dlffractometry.

After a survey of the different y-ray sources actually

in use, a double crystal y-ray dlffractometor is described which Improves the instrumental resolution by at least one order of magnitude to I second of arc. Examples of high resolutlon studies of crystal mosalclty are presented which demonstrate the power of the technique and i t s enlarged range of application due to the gain In instrumental resolution. So far y-ray dlffractometry was applied to investigate imperfect single crystals, however, from measurements of Pendell~sung intensity beats in perfect silicon crystals wlth 316 and 468 keV yradiation the 220 structure factor was determined with an accuracy of 0.1% and comparison could be made with the corresponding X-ray

Pendell~sung value. In addition anisotroptes in the Pendell~sung I n t e n s i t y beats, which have been observed in floating-zone grown St

~-ray diffra~om~ crystals are presented and the potential of l-ray dtffractometry for the characterization of nearly perfect single crystals ts discussed on the basis of Kate's s t a t i s t i c a l dynamical theory of crystal d i f f r a c tion. II.

GAHi'4A-RAYSOURCES In the past the standard source in y-ray dlffractemetry yes neu-

tron activated radioactive gold. The natural abundance of 197Au is nearly I00%, its cross section for thermal neutron capture Is of the order of I00 barn and sources of high specific activity can be produced wlth neutron fluxes easily available in research reactors. In Grenoble a gold foll of 0.2xlOx4 ~

in dimension was irradiated wlth

a thermal neutron flux of 1.6.1014 cr2sec -I for four days t e ~ t a l n an activity of about 75 Curie [13]. The distance between the source and the collimator in front of the sample was 4 m leading t~an angular resolution of the dlffractometer of I0" in the scattering plane and of 8' vertical. In spite of the good instrumental resolution the gold source still provided a maximum intensity of approximately 50(0) cps over the cross section of 0.2xlO ~

measured in the photo peak

of a 2 inches thick NaI(TI) scintillation counter. At least g~l& of the Y-quanta emitted from the gold source are due to the 411.79 keV line. The two additional lines emitted by I98Au at 675.87 and 1087.66 keV contribute only about I% to the total y-ray flux [15). Further lines of significant intensity at energies of 158.37 and 208.19 keV are emitted from IggAu which is produced by a second neutron capture. In absolute measurements of Bragg diffraction patterns, which is one of the main goals in y-ray dlffractometry, the direct beam intensity has to be determined accurately. Part of the emitted y-radiation has been Compton scattered in the source material which leads to a significant low ener~ contamination of the Intrlnslcly monochromatic x4ine [16]. In order to avoid corrections for this effect the use of a solid state detector with hlgh ener~ resolution is preferable. On the other hand these detectors are reliable only for relatively low count rates. Therefore it is an advantage of the Au source that about 90% of the y-quanta whlch enter the detector in direct beam measurements also contribute to the Bragg scattered intensity. The important disadvantage of the Au source is its short half-llfe of only 2.7 d which requires source exchange once a week and makes that this source can be used only on sites In the close neighbourhood of research reactors.

319

320

J.R. Schneiderand H.A. G~f In addition to the gold source the x-ray dtffractometors at the I n s t t t u t Laue-Langevfn have been equtped recently with a second source of 700 Curie radioactive Cs and the 661.63 key line from 137Cs is used to measure y-ray rocking curves in large imperfect stngle crystals [17]. The h a l f - l i f e

of 137Cs ts 30 years and thus the decay of this x -

ray source is negligible, The specific a c t i v i t y of the CsC1 source used at the ILL is of the order of 50 Curie ca "3 which has to be compared with an average specific a c t i v i t y of t h e i r gold source of t y p i c a l l y 5000 Curie cm-3, The useful depth of the gold source is 1fatted to about 4 mm because of the rather strong self-absorption of the 411.79 key x - l i n e in gold, The mean free path of 661.63 key yradiation in CsC1, however, is an order of magnitude larger so that the useful length of the source is of the order of 4 cmwhich partly compensates for the r e l a t i v e l y low specific a c t i v i t y of the CsCl source. At the I n s t i t u t Laue-Langevin the 137Cs source is presently used for the investigation of large as grown Cu stngle crystals with diameters of the order of 6 Cmo Although the Bragg scattered intensity is roughly proportional

to X2, which means a reduction by a fator of

about 2.5 compared with the r e f l e c t i v i t y

for 411.79 keV y-radiation,

the Cs source is of advantage for samples larger than 4 cm as shown in Fig.2 for the Cu 111 reflexton. A typical rocking curvemeasured with 661.63 keY y-radiation at a 6 cm thick Cu crystal is shown in Fig.3. The r e l a t i v e l y high background of about 6 cps is probably due to Compton scattering in the large sample. The peak r e f l e c t i v i t y at a temperature of T=1357 K is also given in Ftg.2 because at the ILL l a r ge Cu single crystals are grown in a Bridgeman furnace mounted on the x-ray diffractometer which allows for simultaneous investigations of crystal mosalclty. First results obtained on the JUlich y-ray dlffractometer using 468.06 keV radiation from 1921r are described in ref.6. The natural abundance of 1921r is 38 % and its cross section for thermal neutron capture is of the order of 1000 barn. The h a l f - l i f e is 74.2 days and sources with activities of the order of 200 Curie are commercially available. Strong y-lines are found at energies of 295.44 {28}, 308.44 {27}, 316.5 {76}, 468.06 {4g} and 604.40 {8} keV. The numbers in brackets represent the number of emitted y quanta per 100 decays [151. Using a Ge solid state detector these lines can be well separated from each other and from further weak lines not mentioned here. I f the dlffractometer operates with a scintillation counter of much lower energy resolution in general one can use only

321

~-ray diffractometry

.~u- i= source,

/

300K

~/Au-source,

//

~ ~

T=1357K

ICs-s~rCe, T"300K

{ e

/ rI

/Cs-sO..~ce,

T-1357 K _

oD.

COPPER (1113~ ' ~ ' ~ y=o.167.

1

2

3

4 .5 thickness [cm)

6

Ftg.2. R e f l e c t f v t t t e s tn the peak of the 111Bragg retrlexion of a copper single crystal with a mosaic spread of ~.10 minutes of arc calculated as a function of sample thickness. The calculations are performed for two d i f f e r e n t temperatures of the crystal and for 411.79 keV ~-radlatton from 198Au or 661.63 keV radiation from 137Cs, respectively (from Freund [17]).

the 468.06 key l i n e . A comparison of the I n t e n s i t y of the 411.79 keY 11ne from XSSAu wtth the I n t e n s i t i e s of the I r 1tries for the sources used at the Hahn-Meftner-Instttut tn Berltn ts presented fn ref.18. For a gtven I r l i n e the dtrect beam measurement 1s more d i f f i c u l t than tn the case of the 411.79 keY ltne from 198Au, because the cont r t b u t t o n of the Compton part of the pulse htght spectra from higher energy l i n e s has to be subtracted. However, t t has been demonstrated recently [19] that d i r e c t beam measurements for the 316.5 keV I r l i n e can be perfomed wtth an accuracy of ~ 0.5% assuming that the c o n t r i bution of the higher energy l i n e s ts constant over the narrow window of the electronic discriminator.

322

J. R. Schneider and H. A. Graf

counts] 10 secJ 800

"O

600

(~ 400 :I::: 0 qJ

o~ 200 0

0 0

, m J m I ,~ 5 10 15 20 25 30 crystal rotation angle in minutes of arc

Ftg.3. 111 rocking curve of a 6 cm thick copper crystal measured at the I n s t t t u t Laue-Langevin in Grenoble with 661.63 keV y-radiation from t37Cs (from Freund [17]).

Besides the r e l a t i v e l y long h a l f - l i f e

of the I r source i t s attrac-

tion is due to the fact that one now has 4 y-1tnes with wave lengths of 0.0206, 0.0265, 0.0301 (from t9BAu) and 0.0392 A available at the HMI y-Pay diffractometer which allow for wave length extrapolation in accurate structure factor measurements [20]. An example is given in Fig. 4. The logarithm of absolute structure factors measured by means of y-radiation on vanadium stngle crystals is plotted as a function of X2. I f primary extinction is negligible,

following Darwin's extinc-

tion theory, the data points should be on a straight l l n e , i t s i n t e r section with the ordinate corresponds to the logarithm of the extinction free structure factor. Measurements were done on two d i f f e r e n t samples. Both data sets l i e on s t r a i g h t ltnes and provide identical values for the extinction free structure factors. This confirms the a p p l i c a b i l i t y of Darwin's e x t i n c t i o n theory tn v-ray dtffractometry and proves the r e l i a b i l i t y

of the d i r e c t beam measurements. From the

slope of the dotted line for the 110 ref3exton a mosaic spread of 17" was estimated which is much smaller than the instrumental resolution of 2 . 6 ' . In this case the

wave

length extrapolation was the only direct

way to correct for e x t i n c t i o n . Compared with y-radiation from 192Ir, the Au source is of advantage for the measurement of very weak Bragg reflexlons, again, because

323

-y-raydiffractometry In IF;I 110

34

zoo~ J~

220 ~l

"~ 321

"

:

|

!

|

I

!

I

I

I

400

.'~ 330 '~ 420 .~E

,

4

I

.

I

I

422 '~ i

l

l

,

l

,

,

,

,

.

0 2 & 6 8 ~ ]2 14 16 18 square of wavelength: ~z. 10-4 [~2]

Flg.4. Determination of extinction free structure factors of vanadium by means of wave length extrapolation. The individual measurements have been performed on an absolute scale using y-radiatlon wlth energies of 604.4, 458.06, 411.79 and 316.5 keV. The structure factors for the reflexlons 110, 200 and 211 have been measured in two different vanadium single crystals.

there are no x-ltnes of s i g n i f i c a n t i n t e n s i t y with energies higher than 411.79 keY. For example, i f one wants to measure a weak superl a t t i c e 11ne 1/2(hk]) of a strong reflexton hk] with 0.0392 A radiation from 192Ir, the measured intensity may contain a certain amount

of indirect X/2 contamination, i . e . 0.0206 A radiation Is Bragg scattered from reflexion hkl and the Compton part of the corresponding pulse hight spectrum produced by the solld state detector overlaps with the electronic discriminator window set to select the photo peak of the 0.0392 A llne. In many cases this problem can be avoided by selectlng a sultable reflexlon hk1. Otherwise the intensity of the Bragg scattered 0.0206 A y-radiation has to be determined from a s i multaneously measured pulse hight spectrum whtch then allows for a correction of the i n d i r e c t X/2 contamination.

324

J. R. Schneider and H. A. Graf

At the Research Reactor F a c i l i t y of the University of Missouri in Columbia, U.S.A., a f u l l y automated 4-circle y-ray diffractometer is operated with 103.18 keV x-radiation from 153Sm [21]. The source mat e r i a l is 99 % enriched 152Sm03 powder which is pressed into a rectangular well 12.7 x 4.3 x 1.3 mm3 of a high-purity aluminium holder. The ensemble is irradiated for one week in a neutron flux of ~ 4.10 IW cm-2 sec-z, producing approximately 1000 Ci of IS3sm. The h a l f - l i f e of the source is only 46 hours so that i t has to be replaced after approximately 6 days. Because the source spectrum contains y-lines as close as 5.7 keV to the main line the use of a high resolution solid state detector is compulsory. Because of dead-time problems in these detectors the direct beam intensity is determined at the end of a measuring period only, and the data set is normalized according to the decay constant of the source. The Missouri y-ray d i f f r a c tometer allows the measurement of a whole set of structure factors for charge density studies in about 2 or 3 weeks on samples which can be used in the complementary neutron diffraction experiments. Unfortunately the longer wave length of 0.12A can cause extinction problems.

Ill.

DOUBLECRYSTALGAMMA-RAYDIFFRACTOMETER

The overall length of a one crystal y-ray diffractometer of the type shown in Fig.l is about 6 m. Within this frame and for suitable cross sections of the incident beam the attainable resolution is limited to about I0" and the application of y-ray diffraci~metry is bound to investigations of heavily distorted single crystals. The FWHM of the intrinsic diffraction pattern of perfect crystals is proportional to the wave length and only of the order of 0.I" for O.03A radiation. Studies of nearly perfect crystals with short wave length is of interest because in principle deviations of the order of 0.I" from perfect behavlour become visible in rocking curve measurements. The first double crystal y-ray dlffractometer was built in 1981 by Merlam Abdul Gani et al. at the Rutherford Laboratory in the U.K. using O.0301A y-radiation from tgSAu [22]. The authors report on IIi double crystal rocking curves taken through a pair of SI crystals each 30mm thick which showed FWHM = i" but, as far as we know, the work was not

pursued further.

~-~y diffra~om~w

The one crystal y-ray dlffractometer operated at the Hahn-MeitnerI n s t i t u t was designed for accurate structure factor measurements, i.e. the goal was to obtain reasonable intensity in a direct beam of compact cross section, so that the measurement averages over a small sample volume element which is important for crystals with inhomogeneous mosaic structure. Normally a cross section of 2 mmdiameter is chosen which leads to a relatively large angular divergence of the direct beam of about 2.5' for the arrangement shown in Fig.1. Therefore, in many cases one does not gain information on the shape of the intrinsic diffraction pattern from the measured rocking curve. The structure factor measurements on vanadium mentioned above represent a typical example. Because knowledge of the intrinsic diffraction pattern is crucial for a direct extinction correction this situation is dissatisfying. By adding a second axis to the HMI diffractometer and using an internally strained Si single crystal as f i r s t crystal a double crystal y-ray dlffractometer was realized with an angular resolution of approximately 1" [23]. I t is now possible to measure in a given sample volume element absolute integrated reflecting powers with 4 wave lengths in the range between 0.02 and 0.04 A under optimum conditions, as well as the shape of the intrinsic diffraction pattern with I" resolution. In addition, this double crystal y-ray diffractometer opens new possibilities for the characterization of large as grown single crystals. In the following we will briefly discuss the experimental set-up, the instrumental resolution, the diffraction properties of the f i r s t crystal which are crucial for the successful operation of the diffractometer and, finally, present some examples for high resolution studies of crystal mosaicity. A more detailed discussion of the spectrometer is given In ref.23.

III.l.

Experimental set-up

In general the dlffractometer is operated with 0.0392~ y-radiation from z921r. The cross section of the direct beam is 2 x 4 mm 2 and the distance between source and f i r s t crystal is 2500 mm. The other characteristic dimensions of the diffractometor are shown in Flg.5. The f i r s t and the Second crystal are oriented to about I ' by operating the dlffractometer in the one crystal mode. This precision is sufficient because of the small Bragg angles. For operation in the double crystal

325

J.R. Schneiderand H.A. Graf

326

mode the whole x- ¢ - circle carrying the second crystal (sample) is shifted in the scattering plane perpendicular to the direct beam by Ax=16.34 mmwhich is determined by the Bragg angle of the f i r s t and the distance between the two crystals. In spite of the long beam paths and the lack of sophisticated temperature control in the laboratory the position of the diffraction pattern is reproducible to withln ±I" and its FWHMto ±0.1". At present the dlffractometer is operated with a resolution of not better than I", the mechanical reproducibility is considered sufficient.

111.2.

Instrumental resolution

In the case of small Bragg angles (cos eB=l) the geometrical broadening of a rocking curve measured in the dls~rslve (++) setting is equal to the wave vector spread Ak/k o of the direct beam times sin ) B " In y-ray dlffractometry sin 0 B Is typically of the order of I0 ~ and the wave vector or energy spread Ak/k o = 10 -~ is by about 2 orders of magnitude smaller than for a characteristic X-ray line. Therefore, compared to X-ray diffraction, the geometrical broadening of a double crystal y-ray rocking curve measured in the dispersive mode is by about 3 orders of magnitude smaller and typically of the order of 10 .3 seconds of arc which is negligible for the present purpose. Therefore, the HMI double crystal y-ray dlffractometer is operated in the dlspersive mode, i.e. the beam is diffracted away from the direct beam direction twice, which leads to a background in the measured rocklng curve of not more than 0.I cps.

111.3.

Diffraction propertles of the flrst cr@stal

As discussed in more d e t a i l

in a subsequent section of the present

paper the integrated r e f l e c t i n g power measured with short wave length y - r a d i a t i o n for varying r a t i o s of sample thickness and e x t i n c t i o n length exhtbtts o s c i l l a t i o n s around a constantmean value with amplitudes of the order of ±10%, the so c a l l e d PendellSsung i n t e n s i t y beats. The integrated r e f l e c t i n g power has been measured with 0.0392A y-rad i a t i o n In various volume elements of a disc shaped Si crystal 80 mm in diameter and 3.74 mm t h i c k , which was cut parallel to the (001)

327

~-ray diffractometry

first crystol

gN

second crystot

Si 220

•l,dX =

--2500

-,-

800

16.3Z,mrn

zlX

1880

2680

FIg.5. Schematic lay-out of the double crystal y-ray diffractometer installed at the Hahn-Meitner-lnstltut In Berlin. planes from a <001> Czochralski-grown crystal. I f the sample is oriented such that the <001> growth direction lies tn the scattering plane the measured values of the integrated reflecting power correspond to the theoretical ones calculated from dynamical diffraction theory, no significant fluctuations could be observed for measurements in different volume elements. However, i f the crystal Is rotated around the <220> scattering vector so that the <001> growth direction is turned out of the scattering plane the measured integrated reflecting power, Rmeas, increases dramatically. In Fig.6 Rmeas is plotted for various t i l t angles ~ and positions in the sample. The data were corrected for absorption and tlme decay of IS21r. Close to the center of the crystal Rmeas increases by a factor of 20. Qualitatively this observation was explained in [23] by assuming that the (220) l a t t i c e planes show an

328

J. R. S c h n e i d e r and H. A. G r a f

~,°,

5 .1= 35 .c

,= 2 C U O m 10 O~ O

0 -30 -20 -10 0 10 20 30 [ram] location of the meosured crystal volume elements

Flg.6.

Integrated reflecting power measured with 0.0392 X radiation in different volume elements of the first crystal of the H I double crystal y-ray dlffractometer which is a disc of 80 m diameter and 3.74 mm thickness. Its center Is labeled by 0 and the investigated volume elements are 10 mm apart. The sample has been rotated around the <2ZO> scattering vector. The t i l t

angle Y, which varied between 0 and 75°, is a fur-

ther parameter.

effective cylindrical curvature about the <111> axis. In addition to the integrated reflecting powers, double crystal rocking curves have been measured close to the center of the SI disc. The results are presented in Flg.7. The FWHM of the double crystal rocking curvemeasured for ~=5 ° is considered as the intrinsic FWHM of the diffraction profile of the second crystal. Obviously, by tilting the first crystal

~-ray diffra~om~w

329

around its scattering vector one gains substantially in intensity on the expense of the instrumental resolution. The maximum intensity of a 2 x 4 mm 2 monochromatic y-ray beam incident onto the second crystal, l.e. the sample to be studied, Is about 500 cps for a 200 Cl source and an angular resolution of 2". The hlgh resolution studies of crystal mosalclty, which are presented next, have been performed wlth 0.0392A

3OO

200

100

0 100

o o o o

Ftg.7.

o

1 2 3 4 crystal rotation angle in seconds of arc

5

Double c r y s t a l y - r a y rocktng curves measured with 0.0392A r a d i a t i o n a t Si 220 by rocking a second St c r y s t a l w i t h an

intrinsic diffraction profile of approximately 0.5% The first crystal was tilt around the scattering vector in steps of 15" up to 75 ° . The tllt angle Y and the full width at half maxlmum (FWHM) are presented for each rocking curve.

330

J.R. Schneiderand H.A. Graf

y-radiatlon from a 1921r source with an activity of approximately 50 Ci The angular resolution was chosen between I and 2 seconds of arc.

III.4.

Mosaicity of vanadium single crj/stals

Double crystal y-ray rocking curves were measured in symmetrical Laue geometry at the 110 and 002 reflexion of a disc shaped vanadium crystal with surfaces parallel to the (110) lattice planes. The results are presented in Fig.8 and 9. In the center of the crystal at the 110 reflexion the rocking curve shows a regular shape with FWHM= 19". In contrary, at the 002 reflexion, i.e. after rotation of the disc around 4110> by 90", a very irregularly shaped rocking curve was

u~ t"

._c "10

P 0 U u~ O~ 2 mm

rn mm

crystal rotation angle in seconds of arc

Fig.8.

Series of double crystal y-ray rocking curves measured at the 110 reflexlon of a disc shaped <110~ vanadium single crystal. The diameter of the circular cross section of the incident beam was 2 mm, the wave length X=0.0392 A, and the angular resolution AmAp = 1.8".

~/-raydiffractometry

331

6°' I counts

&

g

~X = 7 mm

0

1.0

80

120

160

200

2t,O

280 u

crystal rotation angle in seconds of arc

Ftg.9.

Series of double crystal y-ray rocking curves measured at the 002 reflexton of the vanadium crystal which showed the 110 d i f f r a c t i o n patterns plotted in Ftg.8. The diameter of the c i r c u l a r cross section of the incident beam was 2 m ,

the

wave length X=0.0392 k, and the angular resolution A=Ap= 1.8".

observed. The crystal has been displaced perpendicular to the direction of the direct beam in steps of 1 mm and for both reflexions one finds strong fluctuations in the shape of the measured double crystal rocking curves. This rather tnhomogeneous mosatc structure ts not unusual for metal crystals and has been found frequently in studies by means of y-ray dtffractometry, even at lower angular resolution. For posstble interpretations of these experimental

results one may consult

ref.14. III.5.

Diffraction pattern of a Czochralskt-grown Copper single crystal

Fig.lO shows the double crystal rocking curves measured at the {220} planes of a <111> Czochralskt-grown dtsc shaped copper crystal provided by W.Uelhoff from the KFA JUlich. The sample was 3 m thick and 10 mm tn diameter, The d i f f r a c t i o n patterns show no substructures

332

J. R. Schneider and H. A. Graf counts¢ 90 sec |

°°°F 500I 400

i

ii i

~,2o

1°I

0

10 20 30 40 crystol rotetion ongle in seconds of Qrc

u

Fig.t0. 220 double crystal y-ray rocking curves measured on a <111> C zochralski-grown copper single crystal provided by W.Uelhoff, KFA-JUIich. Cross section of the incident beam 2x4 m 2, wave length ~=0.0392 A, angular resolution AmAp = 1".

and their FWHMvaries between 3.8" and 4.5". The relatively long t a i l s may be due to some remaining surface damage because the incident y-ray beam is higher than the hight of the sample so that the surfaces f u l l y contribute to the diffraction pattern. The crystal was cut by spark erosion and chemically etched. III.6.

Mosalcl.t~of a .FeS2 (pyrlt~) st.ngle, crystal

Double crystal y-ray rocking curves were measured on a FeS2 single crystal produced by means of the chemical vapour transport technique [24]. The volume of the sample was of the order of 40 nu3. The FWHM

of the {220} rocking curves presented in F t g . l l varies between 3.2"

333

~-ray d i f f r a = o m ~ w

and 5.2" and they show some indications of a weak substructure. In contrast, the 111 diffraction profiles exhibit a pronounced substructure. Because the FWHMof the subpeaks ts vew close to the instrumental resolution of 1.5" their shape provides no infomatton about the intrinsic diffraction pattern. The subpeaks are separated by approximately 5". These f i r s t results suggest that the growth perpendicular to the (111) l a t t i c e planes ts perturbed, however, more systematic studies are needed. 111.7.

Diffraction pattern of a bent silicon wafer

The 111 double crystal rocking curve measured at a standard 4111> n-type St-wafer of 100 mm diameter and 0.4 mm thickness Is shown in coun~ 180sec~

count 180sac

2°2

oolf

/

600

/~

go0f '"

,~

FWHM= ~2"

600

111

400 p

W = 120°

o. _

|

202 i

tO

400

220

~00

FWHM = 32"

200 0 ,-'--

0

10

20

-

w-. . . . . . .

30

crystal rotation angle in seconds of arc

L

111

W = 2~0 °

O.

0

10

20

30

crystal rotation angle in seconds of arc

Ftg.11. Double crystal rocking curves measured at the {220} and (111) l a t t i c e planes of a FeS2 (pyrite) single crystal provided by Ennaout et al [24]. Cross section of the incident beam 2x4 mm2, wave length x=0.0392 k, angular resolution A,,,Ap = 1.5". The curves through the 111 data points have to be considered as a guide to the eye supported by the known shape of the Instrumental resolution function.

334

J. R. Schneider and H. A. Graf

Fig.12. Because of the small Bragg angle of eB=0.358° a 4 mm high section of the wafer is f u l l y bathed in the direct beam of 2 mmwidth during the whole measurement. The angular resolution was approximately 1" and the measured FWHMof 44" can be attributed to a curvature of the (111) planes with a radius of curvature of p = 460 m. With the resolution of 1" one would have been able to resolve an i n t r i n s i c d i f fraction pattern of FWHM = 4" which corresponds to a radius of curvature of p ~ 5000 m. This result indicates the possibility to study weak distortions of Si wafers by means of x-ray diffractometry. IV.

OBSERVATIONOF PENDELLOSUNGINTENSITY BEATS

Recently progress has been made in charge density studies of simple metals using absolute structure factors measured by means of y-ray d~ffractometry. Comparison of the experimental data with the results f~om various modern bandstructure calculations reveals that the absolute scale of the measured structure factors should be known with

counts 330 sec 1500 C

1000

500

100

r

0

30

60 90 120 150 180 210 2/,0 270 crystal rototion ongle in seconds of orc

300

Flg.12. III double crystal y-ray rocking curve of a standard n-type s111con wafer of I00 mm dla~ter and 0.4 n

thickness.

The wafer Is bathed In the y-ray beam of 2x4 ae~ cross section, wave length ~=0.0392 A, angular resolutlon AmAp = I"

335

~-ray diffra~om~w

an accuracy of better than 0.5 % [20]. In order to check the reliability of the Thomson cross section used In the interpretation of the Bragg scattered y-ray intensities, for 316.5 and 468.06 keV y-radlatlon the silicon 220 structure factor has been determined from Pendell~sung intensity beat measurements wlth an accuracy of better than 0.I % [25] and comparison was made with X-ray Pendell6sung data of corresponding accuracy [26,27]. We had some difficulties to flnd a sufficiently perfect silicon crystal for this experiment, on the other hand some new possibilities for the characterization of the degree of perfection of nearly perfect single crystals by means of y-ray dlffractometry became aparent and will be discussed later.

IV.I.

Theoretical expressto ~ to deScrltb~ Pendell~sun9 i n t e n s t t ~ beats

The integrated reflecting power for diffraction of X-rays in transmission geometry through a plane parallel perfect single crystal Is given within dynamical diffraction theory (here we follow ref.28) by !

dyn

ro

IFHI e"w x2 . . . . Kn'p Xn,P = ~ • Vcell • V - - ~ . stn 2eB

• R y (Laue) n,p

(1)

n and p denote the two polarization states of the incident X-ray beam, l.e. electric field vector E normal or parallel to the scatteI

ring plane, r ° Is the classical electron radius. F H is the real part of the structure factor, e -w the Debye-Haller factor and Vcell the volume of the unlt cell. x represents the wavelength, Kn, p Is the polarization factor and e B Is the Bragg angle, b represents the ratio of the direction cosines Yo and YH and Is equal to I for symetrlc Laue geometry. R n,p y (Laue) is well approximated by

R y (Laue)- ~ n,p - ~e

2A),p

-Po" t/Yo

2 , " [Jo (2iAn,p " ~ 2 + gn,p) .

Jo (x) dx -I ]

0

(2)

336

J.R. Schneider and H. A. Graf

with

An, p .

.

ro

•

.

.

I • e - w " x • Kn, P • t . . . . V c e l l • /J YoYH J'

II

II

FH k=

(3)

Ir~

~Fo -

I

(41

I

FH

FH

(l-b) gn,p = -

• Uo " Vcell

4 V~-

" ~

(5)

r ° . I F H I e "w • X • Kn, p

provided gn,p <41 and k <<1. Because in X-ray diffraction experiments photoelectric absorption dominates the total attenuation coefficient i t is Justified to put Po equal to the measured linear absorption coefficient, t represents the thickness of the crystal plate. Jo(X) and Jo(iX) are the zero order Bessel functions of the f i r s t kind for real and imaginary arguments, respectively. Yo and YH are again the direction cosines for stal surface. I FH

the incident and emergent beams relatlve to the cryII

FH is the imaginary part of the structure factor

i FH), =J

(FH = + and c the ratio of the imaginary part of the structure factor for a given Bragg peak relative to i t s value in forward direction. Equation (2) has been derived by dropping terms of the order of k2 and g2 compared with unity. For symmetrical transmission geometry ( i . e . b = I and thus g = O) a f i r s t order correction to equation (2) results in a change of scale of R y (Laue) by a factor of n,p (I + 7 k2/16). In the following I t will be shown that this correction is negligible for the diffraction of 316.5 and 468.05 keV w-radiation in silicon. F " in equation (4) is related to ~o by the optical theorem 0

F

ii

~o = 2 r 0 ~, o

Vce11

(6)

337

~-ray diffra~om~w The total attenuation c o e f f i c i e n t measured in s i l i c o n with 316,5 key ( i . e . ~ = 0.0392A) y-radiation i s pEO.25 om-1 and thus several orders of magnitude smaller than the value observed in X-ray d i f f r a c t i o n exI

perlments. With Po = 0.25 cm" I , FH = 67.08 from [26], and wlth the usual assumption that c ts close to unity one calculates from equa-

tions (6) and (4) k= 0.003 and hence a correction to the scale of equation (2) of the order of 5.10 -6 , which i s negligible. For an unpolarlzed incident beam as used in y-ray diffraction experiments both polarization states are equally l i k e l y . There Is superposition of the waveflelds of both polarization states which leads to minima of v i s i b i l i t y of the PendelIBsung intensity beats [29, 30]. These "fading regions" occur at intervals of 0.5-(2n+1)N PendelIBsung oscillations, where n Is a positive integer or zero and

N=

1 + I cos 20B | 2 ( 1 - I cos ~eB I)

(7)

For diffraction of 0.0392A y-radlatlon at the 220 reflexlon of silicon the Bragg angle is small, o B = 0.58", so that N is of the order of 5000. Thus the first minimum of visibility of the Pendell~sung Intensity beats (i.e. n=0) is expected at values of A of the order of 7500. In Pendell~sung intensity beat studies wlth x-radlation the maximum A value is of the order of 200 and therefore the fading will not be visible, The integrated reflecting power for an

unpolarized beam is given by

RdYn : I- (RndYn + Rpdyn) 2

(8)

. dyn _ dyn where Kn and Xp are calculated separately from equation (I) using

Kn=l and Kp = Jcos2eBI, respectively. In Ftg.13 Rdyn is plotted as a function of

1 A :- ~, (An + Ap)

(9)

up to values of A = 120. The difference between A and An, Ap i s only of the order of 10-~.

338

J.R. Schneider and H. A. Graf

0-

8.6 ._c ,J

=

I.

o

/o

6o

80 A

=

Ftg.13. Integrated reflecting power for diffraction of 0.0392 A y-radiation at silicon 220 calculated from dynamical theory as a function of the parameter A, which ts the ratio of sample thickness and extinction length.

In order to observe such Pende11~sung intensity beats experimentally, one can measure the integrated reflecting power as a function of A. For a gtven Bragg reflexton this has been done either by varying the wave length of the incident beam, or by varying the thickness of the sample. The calculations shown in Fig.13 are made for St 220 and y-radiation of fixed wave length, x = 0.0392A. The variation of A from 0 to 120 corresponds to a variation of the crystal thickness from 0 to about 2 cm. I f one is not especially interested in PendellSsung oscillations for values of A smaller than about 20 the variation of sample thickness can be realized stmply by t t l t t n g a plate shaped crystal of thickness of a couple of mm around the scattering vector. For successive values ¥t of the t t l t

angle ¥, the integrated r e f l e c t t v t t y is

then determined from ~-step scans tn symmetrical Laue geometry. I f ¥=0 corresponds to the setting where the sample surfaces are perpendicular to the incident beam direction, the corresponding sample thick. ness ( t in equations 2 and 3) is calculated according to

339

-t-ray diffractometry

(lO)

t = T / cos Yt

w i t h T equal to the thickness of the perfect part of the c r y s t a l plate. Because of surface damage, T may be s l i g h t l y smaller than the thickness To measured nomal to the crystal surfaces. To f i r s t approximation one can assume t h a t the damaged surface layers w i l l not take part in dynamical d i f f r a c t i o n , but w i l l scatter the i n c i d e n t r a d i a t i o n according to the kinematical d i f f r a c t i o n theory. Therefore, a kinematical scatt e r i n g term was included in the formular for the integrated r e f l e c t i n g power. • Rcalc= q

t ktn Rdyn cosB---~ +

'

(11)

with Rdyn from equation (8) and

Q __

r o2 .IFH 12 . e-2W • X3

1+ cos22eB

Vcell sin 2eB

2

(12)

The thickness of the two damaged surface layers ts denoted Tktn, so t h a t i t s c o n t r i b u t i o n to the scattering f o r a given angle of i n c l i nation, Tt, ts proportional to

t k i n = Tkj n / cos~ I .

(13)

Xn most cases where surface damage has to be taken into account i t w i l l be necessary to derive values for Tkt n by a f i t

to the experimental i n -

t e n s i t y data. Thus, the number of adjustable parameters needed to descrlbe a series of r e f l e c t i v i t y measurements at various t i l t angles T i , w t l l usually be three : Tkt n, T and the s t r u c t u r e factor FH ( i n c l u d i n g the Debye-Waller f a c t o r ) , whereby T and FH occur as a product. In the experiments presented t n t h e following section, these q u a n t i t i e s have been determined by minimizing the function

N[ ]2

X _ 2 =4- I

Robs - Rcalc --i" Oobs

(;4)

340

J.R. Schneiderand H.A. Graf

using the standard f i t program MINUIT [31]. The index i labels the independent measurements of the integrated reflecting power, 1 Robs, for sample setting Yi" °obsl is the standard deviation of Ribs. The number of observations included in the various f i t s varied between 20 and 120.

IV.2.

y-ray diffraction experiments - the silicon 220 s t r u c t u r e factor

Rocking curves were measured in symmetrical Laue geometry at the

220 reflexlon of a large floatlng-zone <001) grown silicon single crystal provided by Wacker Chemltronlc, Burghausen, FRG. The sample was a plate I0 cm in diameter and about I cm thick with chemically polished surfaces. The diffractometer was operated in the one crystal mode using 0.0392 and 0.0265 A y-radiation from Ig21r, respectively. The crystal surface was put perpendicular to the incident y-ray beam with an accuracy of approximately ± 0.02" using a laser beam which is in line wlth the incident y-ray beam. After orientation of the sample with 0.0392 A y-radiatlon to within ± 0.01 ° the crystal miscut In the scattering plane turned out to be less than 0.5 ° . The resulting deviation from symmetrical Laue geometry is negligible because the deviation of the parameter b from unity is only of the order of 10 .4 . The integrated r e f l e c t i n g power was measured with a s t a t i s t i c a l accuracy of the order of ± l i

for d i f f e r e n t angles of i n c l i n a t i o n ,

T, of the sample with respect to the incident beam direction. The measured integrated r e f l e c t i n g powers were corrected for time decay of the source and for absorption. A f t e r scaling the whole data set to an absolute measurement at T = 0 the experimental data could be compared with the theoretical integrated r e f l e c t i n g power calculated from equation (1). The absolute measurement involves a measurement of the i n t e n s i t y of the d i r e c t beam which had to be attenuated by a l a r ger amount in order to operate the solid state detector in i t s l i n e a r range. Therefore the s t a t i s t i c a l

accuracy of t h i s measurement is only

of the order of 2 % which l i m i t s the precision of the absolute scale

to ± 2%. The scaled experimental integrated reflecting powers oscillate around a mean value which is (5 ± 2) % larger than the theoretical value.

341

~-ran/diffra~om~ Following the arguments presented at the end of the preceedin9 sect i o n t h i s s h i f t of (5 ± 2) % in the measured i n t e n s i t y was a t t r i b u t e d to surface damage. Within the f i t t i n g

procedure using MINUIT the

average thickness o f , t h e damaged layers was determined to (5 ± 2) .m. The t o t a l

thickness of the sample was measured at the Bundesanstalt

f u r MaterialprUfung in B e r l i n to To = 9.4871 ± 0.0006 mm using the d i gital,

o p t o - e l e c t r o n i c c a l i p e r MT 30 of the company Hetdenhain. In the

area of i n t e r e s t the crystal

surfaces showed a r i p p l e of ± lpm which

was determined at the Bundesanstalt Fdr Materialp~fung in Berlin, using a "Perthometer S6P" of the company FeinprOf. I t is i n t e r e s t i n g to note that the uncertainty in the thickness of the undamaged bulk material of the order of ± 2 .m introduces an e r r o r of only ± 2.10 -4 in the determination of the structure factor FH.

Fig.14 shows the integrated reflecting power measured with 0.0392 A radiation as a function of the parameter A as well as of the t i l t angle Y. The solid llne represents the final result from the f i t inclu-

A1

I

A2

I

A3

I

A4

I

A5

I

sections for pQrtiGI fits

R~

5.0

4.5 t 4.0

i

43.84

i

i

4675 I

I

i

i

49.66 I

I

i

i

52.57 I

i

i

55.48 I

I

i

H

I

58.39 I

A -~

14.56 20.32 2&57 28.02 30.95 33.50 3578 37.80 39.65 41.34 V/ crystol tilt ~gle in degree

t

34.21

Flg.14. Integrated reflecting power measured with 0.0392 A y-radiation at the 220 reflexion of a 9.487 mmthick silicon single crystal (Wacker code no. 29 784/10). The experimental data are p l o t t e d as a function of the parameter A and the t i l t ttvely.

angle Y, respec-

The s o l i d l i n e represents the f i n a l r e s u l t from the f i t

of expression (11) to 64 data points up t o y

= 34.21 °.

342

J.R. Schneiderand H.A. Graf dtng 64 data points up to ~ = 34.21 °, The data measured at higher svalues show a systematic deviation from the solid l i n e . For a detailed analysis of t h i s observation the whole data set was devlded into 5 sections A1 to A5 as indicated in Fig.14. The f i t t i n g

procedure was

applied to each subset of data and the calculated values of FH.T and Tki n are gtven in Table 1, in addition the values of FH.T are p l o t t e d in Fig.15. The values f o r sections A1 to A3 do agree within t h e i r standard deviation, whereas the two remaining values are system a t i c a l l y higher. For t h i s reason only the data up to a t i l t Y = 34.21 ° were included in the f i n a l

angle of

fit.

The parameter A as defined by equations (3) and (9) is proportional to the wave length of the diffracted radiation and therefore the startlng point for Y = 0 of the PendellSsung intensity beats observed with 0.0265 A y-radlatlon is at A = 29.65 compared to A = 43.84 for 0.0392 A. In order to obtain an overlap of the two data sets on the A scale the sample was t i l t e d up to T = 52.76° for the measurements wlth 0.0265 A radiation, the resulting Pende115sung intensity beats are shown in Fig.16. The solid line represents the final f l t result including the

~-T

F..T

~. = 0.0392 A

645

645

640

640

635 I

~

~

~

I

I

|

e I

635

I

A1 A2 A3 A4 A5 ""3&~ °

~. = 0.0265 A

~ i

I

I

I

I

I

I

B1 B2 B3 B4 B5 B6-"34.32 °

Ftg.15. The product of s t r u c t u r e f a c t o r , FH, and thickness of the perf e c t part of the sample, T, as obtained by f i t t i n g

expression

(11) to the various sections of the measured PendellSsung Int e n s i t y beats indicated tn Fig.14 and 16.

~ray diffractometry

+'

I

B2

I

m

I

343

B+.

I

Bs

J

Be

I

sections for partial fits

R

+

+ ++

~E~o

++

25 2965

I

I

.I

I

|

I

I

I

I

I

I

44.53 I

I

I

I

I

I

I

I

I

47.38 i

I

m

A ..-

17.78 2/,66 2964 33161 3693 39.77 42.27 /J,.48 46.&6 4&25 4989 51.38 5226 V/ crystal tilt ongCein degree t 34.32

Fig.16. Integrated r e f l e c t i n g power measured with 0.0265 A y-radiation at the 220 reflexton of the s i l i c o n crystal which was used for the measurement of the Pendell~sung intensity beats with 0.0392 A radiation shown in Ftg.14. The experimental data are plotted as a function of the parameter A and the t i l t

angle T,

respectively. The solid l i n e represents the final r e s u l t from the f t t

first

of expression (11) to 43 data potnts up to 34.32 °.

43 data points up to Y - 34.3Z °. At t t l t

angles Y of the order

of 50 ° the deviation of the measured data from the theoretical prediction based on the f i r s t

43 data points becomes rather dramatic. This

is obvious from Fig.15 where we plotted the FH,T values detemtned from f i t s to sections B1 to B6 of the measured data shown In Ftg.16. Details of the results from the partial f i t s are presented in Table 2. From Tables 1 and 2 i t is s t r a i g h t forward to calculate the 220 structure factor, F22O, for s i l i c o n . For the reasons mentioned above we have used the results for the f i t s including data up to Y = 34.21 ° f o r 0.0392 A and up to 34.32 ° f o r 0.0265 A radiation. Tkt n n s subtracted from the measured sample thickness, To, to obtain the thickness, T, relevant for the observed Pendel16sung intensity beats. From FH • T values provided by the f i t t i n g

procedure one gets :

344

J R Schneiderand H A Graf F = 67,03 '+ 0.02 f o r X = 0.0392A 220 F = 67.08 -+ 0.03 f o r X = 0.0265A 220 The result obtained for the two different y - ray lines agree reasonably well . The errors quoted in Tables I and 2 are calculated ignoring the uncertainty of about ,+ 2% in the absolute scale of the experimental data which gives rise to an uncertainty in Tki n of ,+ 2gm and which is much larger than the standard deviation for Tkl n as calculated by the f i t t i n g procedure. On the other hand a slight change in the scale of the measured integrated reflecting power has no significant effect on the values of FH • T obtained from the f i t . The uncertainty in the thickness T dominates the error in F220 for X = O.0392A, whereas for x = 0.0265A the standard deviation of FH.T makes the largest contribution to the final error in F220. The two Fz20-values from y-ray Pende11Bsung i n t e n s i t y beat measurements should be compared with the r e s u l t obtained by Aldred and Hart [26] from X-ray PendellBsung f r i n g e studies on s i l i c o n : F220 = 67.08 -+ 0.06 for X-rays. Performing a d i f f e r e n t type of X-ray Pende115sung experiments, rec e n t l y , Teworte and Bonse [27] obtained s i l i c o n structure factors with an even smaller e r r o r bar. Both X-ray data sets agree p e r f e c t l y w e l l . In order to s i m p l i f y the comparison between X-ray and y-ray structure f a c t o r s , Aldred and Hart's values f o r the l a t t i c e parameter (instead of the recently reported [32] more accurate value) and the Debye-Waller f a c t o r were used in the i n t e r p r e t a t i o n of the y-ray measurements. The c o n t r i b u t i o n of nuclear Thomson s c a t t e r i n g is included in both, X-ray and y-ray data. y-ray and X-ray 220 structure factors f o r s i l i c o n agree extremely w e l l , the agreement is of the order of ,+ 0.5°/oo. One can therefore conclude that Bragg d i f f r a c t i o n is very well described by means of the classical Thomson cross section up to photon energies of the order of 450 keY, at l e a s t f o r the low order reflexions tn matter not much heavier than s t l i c o n . This r e s u l t is of more general i n t e r e s t because the y-1tnes used in the present experiments are at the short wave iength edge of the spectral d i s t r i b u t i o n of presently available Synchrotron r a d i a t i o n .

345

~-ray diffra=om~w

Table I. Results obtained by f i t t i n g

expression (11) to the i n t e -

grated r e f l e c t i n g power measured with 0.0392 A v - r a d i a t i o n as a function of crystal

ttlt

angle ~ at the 220 retlexton

of the <001> f l o a t i n g - z o n e grown s t l t c o n crystal with Wacker code no. 29784/10. The various ranges of the t i l t considered in the p a r t i a l

fits

angle

correspond to the regions A1

to A5 indicated in Ftg.14. N i s the number of data included tn the p a r t i a l

f t t s and GOF and R represent the usual quan-

t l t l e s to describe the quality of a f l t . FH-T is the product of structure factor and thickness of the perfect part of the crystal; Tkl n represents the thickness of the two damaged surface layers.

Table 1.

DIFFRACTION OF 0.0392A y-RADIATION AT Sl 220 ttlt

0

N

range

< ~ < 20.32

FH.T

Tkin[p]

GOF

R[%]

21

635.75 ± 0.23

6.42 ± 0.22

1.01

0.85

20.80 < Y •

28.33

21

635.51 ± 0.23

6.11 ± 0.20

2.08

1.23

28.64 < Y •

34.21

22

635.51 ± 0.22

4.90 ± 0.19

2.45

1.35

34.44 < ¥ •

38.37

20

636.03 ± 0.22

4.51 ± 0.19

1.17

0.92

38.56 • Y •

42.14

22

636.74 ± 0.19

4.84 ± 0.17

5.33

1.98

0

34.21

64

635.58 ± 0.13

5.73 ± 0.12

2.29

1.34

< Y <

2

GOF =

X n-p

2

R=

_X N

t1=1(Rtobs • wi ) n = number of observations p = number of parameter~

t

Robs = measured integrated r e f l e c t t n g power wt

L~:G -

L

= statistical

weight

346

J.R. Schneiderand H.A. Graf Table 2.

Results obtained by f i t t i n g

expression (11) to the i n t e -

grated r e f l e c t i n g power measured with 0.0265 A y - r a d i a t i o n as function of c r y s t a l t i l t

angle v at the 220 r e f l e x i o n

of the <001> f l o a t i n g - z o n e grown s i l i c o n c r y s t a l with Wacker code no. 29784/10. The various ranges of the t i l t angle ¥ considered in the p a r t i a l f i t s correspond to the regions B1 to B6 indicated in Fig.16.

Table 2.

DIFFRACTION OF 0.0265A y-RADIATION AT Si 220

tilt range

N

FH.T

Tkin[U]

~ = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

0

GOF

= = = = ~ = = = ~ = = = ~ = = =

R[%] s ===

< v < 29.64

31

635.74 ± 0.34

7.50 ± 0.37

1.12

1.30

30.07

< T < 37.53

22

636.73 ± 0.39

7.34 ± 0.38

0.87

1.12

37.82

• ~ < 42.50

18

637.52 ± 0.37

5.29 ± 0.38

0.86

1.10

42.73

< V • 46.65

20

640.06 ± 0.36

4,47 ± 0.34

1.23

1.34

46.83

< ~ • 49,73

18

643,42 ± 0.34

5,92 ± 0,33

1.43

1.40

49.89

• ~ < 52.76

21

648.44 ± 0.38

8.35 ± 0.29

4.50

2.48

< ~ < 34.32

43

635.90 ± 0.28

7.75 ± 0.29

1.11

1.30

0

GOF =

2 X n-p

2 X

R= N

i__Z1(Riobs • wi ) n = number of observations

p = number of parameters

t

Robs = measured integrated r e f l e c t i n g power I w

: s t a t i s t i c a l weight

~==

=~

===

z =

347

~-raydiffra~om~w IV.3.

Anlsotroplc strain in lar9e floatlng-zone

¢001> 9town

s i l i c o n stngle crystals Kato developed a theory to describe Pende116sung fringes tn d i s t o r ted crystals and applied t t to Laue case d i f f r a c t i o n of X-rays tn homogeneously bent crystals [33, 34]. He found that for small bending Pendell~sung fringes are s t i l l

observable but the fringe distances de-

crease with Increasing the curvature. For strong bendtn9 the fringes fade out. Hart [35] defomed e l a s t i c a l l y a wedge shaped specimen of e f f e c t i v e l y perfect s t l i c o n by applying a temperature gradtent normal to the Brag9 planes and recorded both direct and diffracted beam sect i o n patterns on photographic plates for various sample deformations. Both section patterns contain a set of Pende118sung fringes which move towards the thin end of the wedge as the elasttc deformation is fncreased. As a consequence of the fringe movement, the local fringe spacing is observed to decrease when the deformation is increased, In the present, exploratory stage of Pendell~sung i n t e n s i t y beat studies with y-rays it is tempting to use these X-ray diffraction results for a qualitative discussion of the discrepancy between the Pendell~sung intensity beats measured with 0.0265 A y-radiatlon and those calculated for a perfect single crystal shown in Flg.16. Wfth increasing t t l t

angle T the observed Pende11~sung i n t e n s i t y

beats s h i f t to smaller values of A. Using short wave length Synchrotron radiation one could measure Pende115sung i n t e n s i t y beats by tuning i t s wave length tn ftne steps for a fixed t i l t angle T. As a res u l t we would expect for increasing values of T a decrease of the pertod of the i n t e n s i t y o s c i l l a t i o n s (Pende115sung length), which corresponds to an increase of the tnternal s t r a i n . In thts sense the r e s u l t presented in Fig.16 would indicate an antsotroptc strain tn the studied s t l i c o n c r y s t a l , which ts a floating-zone <001> grown crystal plate, boron doped wtth an average e l e c t r i c r e s i s t i v i t y of 20 ncm. After rotating the crystal plate by 90" around <001% again Pendell(Jsung i n t e n s i t y beats were measured with 0.0392 and 0.0265 A y-radiat i o n , but now at the symmetry related reflexton 220. The results are shown in Ftg.17 and 18 together wtth the 220 data already discussed. At zero t t l t angle one obtains the same value for the integrated re-

348

J.R. Schneider and H. A. Graf

R 8

~,

reflexion 220

7

t

x

0

cL 6

/

C L.)

reflexion 220

"o5 e

4 I

/.3.8/.

/.6.75

i

i

/.9.66

I

i

52.57

i

i

55./.8

i

i

58.39

A

Ftg.17. PendellSsung intensity beats measured with 0.0392 A y-radiatlon at reflexlons 220 and 220 of the <001> floatlng-zone grown silicon single crystal with the Wacker code no. 19 784/10.

flecttng power, however, with increasing t i l t

angle a continuous In-

crease of the integrated reflecting power is observed. For 0.0392 A radiation this continuous increase in intensity tsl~dulated by Pendell~sung ostllattons with amplitudes and periods of about half the value found for the oscillations at 2~0. One would conclude from Fig.17 that the diffraction at the 220 planes ts affected by a stronger internal strain f i e l d as tn the case of the 2~Oplanes, which is another indication for an anlsotroplc strain field in our silicon crystal. This result is perhaps surprising because In a naive picture one could expect similar diffraction properties for both reflexlons which are perpendicular to <001>, the growth direction of the crystal. Flg.18 shows a similar result obtained wlth 0.0265 A y-radlation. However, the Pendell~sung oscillations are further reduced in amplitude and their period is only about 1/3 of the PendellSsung length observed at 220. In other words the interference effects in a strained crystal are increasingly smeared out with decreasing wave length of the dif-

~-ray diffractometry

~ o

349

7

f

x

o

6

/

o~

f g~

t

/" i

reflexion 220 ,

2965

N¢ 4 ~

.

. 32_63 . .

.

35.60

3 ; .58

.

. 41.56 . .

.

W+¢+'~

4/~.53

A

Fig.lB. Pendell0sung intensity beats measured with 0.0265 A y-radiatlon at reflexions 220 and 2~0 of the same silicon crystal used to measure the data plotted in Fig.17.

fracted radiation. This is in llne with the finding that a given crystal, which shows dynamical diffraction behavlour for X-rays, may behave llke a rather imperfect crystal in diffraction experiments with shorter wave length y-radiation (3). A much stronger increase of the measured integrated reflecting power as a function of t i l t

angle had been found at the 220 reflexion

of a Czochralski <001> grown silicon crystal which is used as f i r s t crystal in our double crystal y-ray dlffractometer. As shown in Fig.6 the difference for measurements at Y = 0° and T = 75°, respectively, is of the order of a factor of 20 close to the center of the Crystal. On one side of the crystal, about 30 mm apart from its center the measured integrated reflecting power increased only by a factor of about 2. Hoping to find a more perfect region similar measurements have been performed with 0.0392 A y-radlatlon on the Wacker Chemitronic floating-zone <001> grown silicon crystal (sample no. 29784/10) on which

350

J.R. Schneiderand

H.A.

Graf

the PendellSsung intensity beat studies discussed above had been carried out. The sample has been t i l t e d clockwise and counter-clockwise in steps of AT = 15° up to a maximum t t l t

angle of v = 75 °, The re-

sults obtained for the measurements at the reflextons 400, 220, and 220 are presented in Fig.19 to 21. After correction of the measured integrated i n t e n s i t y for absorption and time decay of the 192Ir y-ray source, for a perfect crystal one would expect to obtain values of the integrated r e f l e c t i n g power which agree to within ± 10 %, the amplitude of the PendellSsung o s c i l l a t i o n s in the relevant range of the parameter A, see Fig.13. The measurements at reflextons 400 and 2~0 displayed in Fig.19 and 20, indeed, show t h i s expected behaviour. Differences are found with the results of the 220measurements presented in Fig.21. There is a systematic increase in the integrated r e f l e c t i n g power with t i l t

angle ~, which corresponds to the r e s u l t

Silicon 400 (sample 29784/10) integrated r e f l e c t i n g power in a r b i t r a r y units

!

/ / 75 °

-..~,

,- .

-,soN~'-: 1~" \

/

.

; .~'f oo

OO l',J 6

4

2

2

4

6

30 °

/V'

8

crystal volume elements lcm apart

Fig.19. Integrated reflecting power measured at the 400 reflection of a <001> floating-zone grown silicon crystal (Wacker code no. 29 784/10), 100 mm In diameter and approxlmetely 10 mm thick. g volume elements on a l l n e across the crystal center have been studied in intervals of 10 mm. The sa~le was t i l t e d in steps of 15° from I=O, for the crystal surface perpendicular to the incident beam, clockwise up to Y=75° and counter-clockwlse up to T=-75° . The measurements were performed on the HMI y-ray dlffractometer operated in the one-crystal mode using a 0.0392 A beam of cross section 2x4 mm2. The experimental data were corrected for absorption and time decay of the 1921r y-ray source.

351

•y - r a y d i f f r a c t o m e t r y

Silicon 2~0 (sample 2978M10) integrated

reflecting

power

75 ° - 4 - .o_ -~--- - - -~- . . _. " . .,. ~ 5I

-

-

-

j

,

,

,

~

-

.

.

=

~

30 o ~ J ..... . ' . . . . . . ~ . E ' . -15

c

.

. .

.

.

.

. .

8

.

.

.

.

.

o

.

.

.

2

.

2

.

.

1,

v=

L~ =

.

t,

...-,.,:~.~_ . . . - ~.. ~

.

.

6

-_

~....JJlj

4L5

3oo/ 15 °

.

6

8

crystal volume elements l cm apart Fig.20. Integrated reflecting power measured wlth 0.0392 A y-radlatlon at the ~ 0 reflexlon of the <001> floating-zone grown silicon crystal wlth Wacker code no. 29 784/10. The scanning mode is described in the capture of Flg.19.

Silicon 220 (sample 29784 / 10) integrated

reflecting

in a r b i t r a r y

power

units

7s~ .

L .

.

, .

.

" .

. . . . . . B 6 4

.

,

'

.

~ T > v 2

.

, .

2

.

~

"

.

j'j"

.

4

.

-" -

¢, r"

6

I /.Z

i ~,.~_-

.'~_~

.

0o

tso

~u I . /

/

8

crystal volume elements lcm aport Fig.21. Integratd reflecting power measured with 0.0392 A y-radlatlon at the 220 reflexlon of the <001> floatlng-zone grown sillcon crystal wlth Wacker code no. 29 784/10. The scannlng mode is described In the capture of Fig.19.

of the much more detailed study depicted fn Fig.18 and 19, but also a systematic dependence on the position of the investigated volume element in the crystal plate is found. The specimen discussed so far (Wacker code no. 29 784/I0) was the most perfect s111con crystal which we investigated so far by means of

352

J.R. Schneiderand H.A. Graf y-ray d t f f r a c t e m e t r y .

I t was produced ustng an tl~oroved growth tech-

nique in which the parameters a f f e c t i n g the convection tn the melt had been changed with respect to the standard Wacker f]oattng-zone growth technique f o r which the crystal with code no. 29 878/03 is an example. In Fig. 22 - 24 the integrated r e f l e c t i n g power measured with 0.0392 A y - r a d i a t i o n at r e f l e x t o n s 400, 250 and 220 are p]otted as a function of t i l t

angle ¥ and position in the crystal plate, which

again was about 1 cm thick and 10 cm in diameter with chemically polished surfaces. For the central region the 400 data of Fig. 22 show f a i r l y

constant values as expected for perfect c r y s t a l s , but

there is a systematic increase of i n t e n s i t y in the regions close to the edge of the sample. The 220 results plotted in Fig. 23 show increasing values of the integrated r e f l e c t i n g power for clockwise t i l t i n g and a marked jump in i n t e n s i t y f o r one side of the crystal at T = 75° . The f a c t t h a t t h i s feature is not observed f o r counter-c]ockwise t i l t i n g the i n t r i n s i c

may be regarded as an i n d i c a t i o n of inhomogeneittes fn lateral

s t r a i n f i e l d along the <001> growth d i r e c t i o n ,

however, at present some local surface damages cannot be excluded. The most s t r i k i n g r e s u l t , however, is shown in Fig.24. The integrated ref l e c t i n g powers measured at r e f l e x i o n 220 show a strong increase with tilt

angle, by about a f a c t o r of 5 in the central region of the samp-

l e . Again, on one side of the crystal one observes a marked difference Silicon z,O0 (sample 29878103) integrated reflecting power in arbitrary units

J .6o" 8

6

z,

2

2

z,

6

75°

8

crystal volume elements l cm aport Flg.22. Integrated reflecting power measured with 0.0392 A y-radlation at the 400 reflexion of a <001> floating-zone grown silicon crystal (Wacker code no. 29 878/03). The scanning mode is described in the capture of Fig. 19.

~-raydiffractometw

353

Silicon 220 (samp~ 29878103) integrated reflecting power in arbitrary units

4

#

75 °

....... 8

6

.~/ ,~

2

.

2

.

.

.

.

4

.

L~_oo

.

6

/

v,

8

crystal volume elementslcm aport Ftg.23. Integrated r e f l e c t i n g power measured wtth 0.0392 A y - r a d i a t i o n at the 2~0 reflexton of the <001> floating-zone grown s i l i c o n crystal wtth Wacker code no. 29 878/03. The scanning mode ts described tn the capture of Ftg.19.

between the data obtained for clockwise and counter-clockwise t i l t i n g , respectively. In spite of the fact that more systematic investigations are needed the following observations may stimulate future studies. For zero t t l t angle, t . e . when the <001> growth direction was placed in the scattering plane, a l l measurements of the integrated r e f l e c t i n g power tn the Czochralskt as well as tn both the floating-zone grown stltcon crys t a l s yielded the value predicted by dynamical d i f f r a c t i o n theory for perfect c r y s t a l s . Therefore we do not believe that surface damage causes the strong deviations from dynamical behavtour observed for Increastng t t l t angles. The strongest effect was found for one of the {220} planes which we labeled (220). The data taken at reflexton 400 showed the smallest deviation from the value predicted by dynamical theory, t f any. The tnternal s t r a i n seems to be strongly antsotroptc and is much larger for the crystal produced by means of the so far standard Wacker floating-zone technique than tn the specimen froa the improved growth technique. - T,*

J. R. Schneiderand H. A. Graf

354

Silicon 220 (sample 29878/03) integroted reflecting power in orbitrory units

<

_,so ~o~

~-...... ,, ~

,

~

,,,\

..........'.~... ... ...'

i

" " ' " ' > ' " ' ? ' ~ " " ~~ " ~"

~

o

o 8

~ 6

m

-

~

,

s

o

~ /-,

2

/

oo 2

4

6

~6oo '5°

J~

8

crystal volume elements lcm aport Fig.24. Integrated reflecting power measured with 0.0392 A y - r a d i a t i o n at the 220 reflexlon of the <001> floating-zone grown silicon crystal with Wacker code no. 29 878/03. The scanning mode is described in the capture of Fig. 19.

IV.4.

Potential of Kato's statistical d~namlcal theor:/ for the interpretation of y-ray diffraction data

In general X-ray and neutron d i f f r a c t i o n data are analysed using two d i f f e r e n t theoretical approaches. The kinematical theory i s v a l i d f o r i d e a l l y imperfect single c r y s t a l s and assumes t h a t each i n d i v i dual atom contributes equally well to the s c a t t e r i n g and t h a t m u l t t p l e s c a t t e r i n g events are n e g l i g i b l e , i t is a s c a t t e r i n g theory w i t h i n 1. order Born approximation. Diffraction in ideally perfect slngle crystals is described by means of the dynamical diffraction theory where the coherent interaction of all waves propagating In the solid is considered. For most diffraction experiments in real crystals the values of the integrated reflecting power calculated by means of these theories, Rkt n and Rdyn, r e s p e c t i v e l y , d i f f e r by some orders of magnitude and the measured value, Rmeas, is somewhere in between Rdyn <

Rmeas < Rktn

(15)

355

~-ray diffra=ometw The difference between _Rmeas and Rkin. is called extinction and ts discussed in terms of the extinction c o e f f i c i e n t

y -

Rmea-~s < I .

(16)

Rkin In the limit of very thin crystals both diffraction theories yield the same value for the integrated reflecting power llm t+O

Rdy n : Rkl n

•

(17)

I f one confines the discussion to d i f f r a c t i o n experiments in Laue geometry for negligible absorption, one can conclude that a defect free

crystal plate w111 show dynamlcal diffraction behavlour only i f i t s thickness is larger than a f i n i t e minimum value which is of the order of the so called extinction length, text .

=

text

Vcell ro • I F H I e "w . ~ "Kn, p

(18)

The parameter A defined in equation (3) can then be expressed as A

=

t

(lg)

text and is equal to unity for a sample of thickness t = tex t . For A << 1 the d i f f r a c t i o n tn a defect free crystal can be described by means of kinematical theory and no extinction occurs. Wtth increasing values of A a difference between Rmeas and Rkt n butlds up whtch is called primary e x t i n c t i o n . Because of lack of a better model the defect structure of real crystals used as samples tn d i f f r a c t i o n experiments ts described by means of Darwin's mosaic model [36]. In t h i s simplifying picture the imperfect crystal of thickness T ts assumed to be an aggregate of a great number of independently scattering perfect crystal blocks of thickness t << T. The absorption tn a perfect block ts assumed to be negligible. The deviation of the l a t t i c e plane orientation of one blnck from the mean l a t t i c e plane orientation for the whole crystal is described by

356

J. R. Schneiderand H. A. Graf

means of the so-called mosaic distribution function W(w). On the basis of this model Darwin formulatd the well-known intensity transfer equations (Laue geometry) : dP° dT

-

Uo Po - o Po + o PH

(20)

dPH dT

-

~o PH + o Po - o PH

which are the starting point also for more recent extinction theories [37]. Po(T) and PH(T) represent the power of the incident and diffracted beams, respectively, at depth T; Po is the total linear absorption coefficient. I f the mean thickness of the perfect blocks is much smaller than the extinction length, primary extinction is negligible and the coupling coefficient o is proportional to the mosaic distribution function, W(~):

o(~)

= W(~)

•

Q

(21)

cos BB

with Q defined in equation (12). The energy t r a n s f e r equations are solved assuming that o(~) is constant a l l over the sample volume, the e f f e c t of tnhomogenetties in the mosaic structure has been discussed recently [38]. The coupling between the i n t e n s i t i e s in the d i r e c t i o n of the transmitted and the d i f f r a c t e d beam leads to a reduction of the integrated r e f l e c t i n g power compared with Rkt n and t h i s e f f e c t is called secondary e x t i n c t i o n . For wave lengths of the order of O.03A the e x t i n c t i o n length is generally much larger than the mean block seize ( t ~ lOpm). Because o is proportional to X2 also secondary ext i n c t i o n is weak and can be described r e l i a b l y within Darwin's ext f n c t t o n theory [ 3 ] . In the case of X-ray d i f f r a c t i o n often both e x t i n c t i o n e f f e c t s are important, so that the d i f f r a c t i o n theory has to t r e a t the coexistence of amplitude and i n t e n s i t y coupling in the sample. There seems to e x i s t no rigorous solution of t h i s problem based on Darwin's concepts. I t ts e s s e n t i a l l y the success of the presently available, approximate e x t i n c t i o n theories tn the refinement of large X-ray and neutron d i f f r a c t i o n data sets , which j u s t i f i e s t h e i r use.

357

~-~y diffra=om=w In a series of papers Kato has developed a s t a t i s t i c a l d i f f r a c t i o n theory [39-41] which covers the f u l l

dynamical

t r a n s i t i o n from dy-

namical to kinematical d i f f r a c t i o n behavtour ( t . e . from coherent to f u l l y incoherent scattering) as a function of two correlation parameters ~ and E, the f i r s t one describing short range and the second one describing long range correlation. Starting from wave equations of the Takagt-Tauptn type the sample properties are taken into account via a phase factor

÷ ÷

G = exp 2~t (H.u)

(22)

÷

where H is the r e f l e c t i o n vector of a (hypothetical) perfect crystal and ~ is the displacement of the l a t t i c e point from the position in the perfect state. I f G(o) and G(z) are the phase factors corresponding to the points A and B in the crystal which are seperated by the distance z a correlation function f ( z ) is defined by the ensemble average of t h e i r product f ( z ) = and can be written in the general form f ( z ) = E2 + (1-E 2) g(z). E =

(22) (23) (24)

is the average of the l a t t i c e phase and thus represents the " s t a t i c Debye-Waller factor", g(z) is called the " i n t r i n s i c correlation function" from which an i n t r i n s i c correlation length T can be calculated T = ~ g(z) dz.

(25)

0

The general shape of the correlation function f(z) is shown in Fig. 25. In order to relate this new theoretical approach to the conventional extinction theories Kato investigated secondary extinction on the basis of the mosaic model [42]. The numerical analysis was done for the special case of a plane parallel crystal plate in symmetrical Laue geometry. I t turns out that the conventional theory underestimates extinction by about 4% and 30% for y = 0.5 and 0.15, respectively. For y • 0.7 the conventional extinctlon theory is practically correct. This result ~ustfles the approach presently adopted for the extinction correction of y-ray data from imperfect single crystals because the experiments are in general designed to assure y > o.g. In the above example i t was assumed that the crystal consists of crystallltes with a mlsorlentatlon larger than the width of the diffraction pattern of

358

J.R. Schneiderend H. A. Graf

1.0

L. 0 U

2

0 Z

Flg.25. Correlation function f(z) which, in the frame of Kato's statistical dynamical diffraction theory, connects the degree of perfection of a given sample to its diffraction properties. is a correlation length and E the static Debye-Waller factor (from Kato [42]).

each c r y s t a l l t t e and the parameter E was set zero. The neglect of p r t mary e x t i n c t i o n demands • ~

text/COSeB. In the following we consider

the integrated rflectlng power calculated by means of Kate's statistical dynamical theory as a function of the parameter A for diffraction of 0.0392 k y-radiation at reflection 220 of a nearly perfect Si single crystal, i.e. for values of the parameter E close to unity.

Kato calculates the integrated reflection power as the sum of a coherent, an incoherent and a mixed term:

R = Rcoh + Rmtx + Rtncoh.

(Z6)

In the l t m t t of an t d e a l l y perfect c r y s t a l (E = 1) one gets R = Rcoh, which ts equal to the value Rdyn calculated from c l a s s i c a l dynamical

diffraction theory. For an Ideally imperfect single crystal the coherent and mixed terms vanish and one obtains the result which is usually determined from kinematical diffraction theory.

FIg. 26 shows the

three contributions to the integrated reflecting power for E = o.go and ~ cos6)B/text=O.l as a function of A. For samll values of A, Rcoh

~-ray diffra~om~w

359

dominates and decreases to about 20% of the mean value Rdyn calculated from dynamical theory at A = 40, where the Pendell~sun9 o s c i l l a t i o n have faded almost completely. The incoherent t e m increases proportional to A up to A = 1 where a cross over to a constant level o f ~

10% of Rdyn

is observed. For A > 15, Rmix becomes the leading term and dominates the r e f l e c t i n g power at high values of A. Fig. 27 again shows the 3 components of the integrated r e f l e c t i n g power as a function of A but now f o r E = 0.98, i . e . f o r a more perfect c r y s t a l . The incoherent c o n t r i b u t i o n is very small. The coherent term is s i g n i f i c a n t up to A = 120, so t h a t Pendell6sung o s c i l l a t i o n s are observed over the whole range of A values. I t is i n t e r e s t i n g to note that the value of the integrated reflecting power at A = 120, R(E = 0.98, A = 120) = 34.6, is much larger than the corresponding value in Fig. 26, which is R(E = 0.9, A = 120) = 22.8. Because for E = 1 the integrated reflecting power at A = 120 is about 4.5, strong variation of R(A = 120) as a function of the parameter E is expected for nearly perfect crystals. This behaviour is demonstrated in Fig. 28 where R(A) is plotted for E values varying between 0.99 and I. Inspection of the expression for Rcoh [44] reveals that the PendellSsungs-length increases proportional to 1/E. In Fig. 26-28 the parameter ¢ was fixed according to coseB/text = 0.1, which is considered as a reasonable value for a nearly perfect crystal. Changing ¢ does not alter the general appearence of the family of curves plotted In Fig. 28. With decreasing value of ~ the curves are shifted towards the result obtained for an ideally perfect crystal. Taking the result for E = 0.996 as reference one can show that a slmillar R(A)-curve is obtained for z coseB/tex t = 0.05 and E = 0.992 or • cosBB/text = 0.15 and E = 0.997. For nearly perfect crystals variations of the parameter E therefore affect the

R(A)-

curves much stronger than does a change in the correlation length z. I t is very i n t e r e s t i n g to t e s t Kato's s t a t i s t i c a l by experiment, however, as far as we know only l f t t l y

dynamical theory work has been

done along t h i s l i n e . There is a paper by Baruchel et al. [43] where the integrated r e f l e c t t n 9 power is measured with neutrons on a YIG stngle c r y s t a l . The neutron wave length was varied between 0.4 and 1.1 A and approximately 1 1/2 Pende115sung o s c i l l a t i o n s were observed. With a suttable choice of the parameters E and z the experimental data could be well described by means of Kato's theory. A series of systematic i n v e s t i g a t i o n s by means of y-ray d i f f r a c t i o n

is conceivable

360

J. R, Schneiderand H. A. Graf

321

~ cos~=01

E=OgO

:I ,el

J:!i 4: ~ 0

.

. 10

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

2

............... 20

"Rc~h "Rw~oh 30

/.0

50 A

60

Flg.26. The coherent, incoherent and mixed term of the integrated reflecting power, R, calculated within Kato's s t a t i s t i c a l dynamical diffraction theory as a function of the parameter A for diffraction of 0.0392 A y-radiation at silicon 220. The correlation length T was fixed so that ~.coseB/text=0.1, the static Debye-Waller factor was chosen to E=o.g0.

starting from a highly perfect Si crystal. The concentration of s t a t i s t i c a l l y distributed defects can then be increased by means of neutron transmutation doping [44] which Is based on the nuclear reaction 30Si(n ' y ) 3 1 S i

+ 31 p + 6-.

The production rate f o r phosphorous is proportional to the thermal neutron fluence which is the integral of the neutron f l u x over neutron energy and i r r a d i a t i o n time. Most of the induced i r r a d i a t i o n damage is due to y- and G-emission r e c o i l and f a s t neutron knock-on displacements. Various annealing procedures have been studied v i a r e s i s t i v i t y

~t-ray d i f f r a c t o m e t r y

R

24

~16

J ~11" I¸

J

12

eo

361

j.<~ f o''I''I +

R=Rc~ Rawx+R ~ , , %

ao

12o

):i 12-

81 O) O

2O

30

~0

50 A

6O -

Flg.2~ The coherent, incoherent and mixed term of the integrated reflecting power, R, calculated within Kato's statistical dynamical diffraction theory as a function of the parameter A for diffraction of 0.0392 A y-radiation at silicon 120. The correlation length • was fixed so that ~,cosBB/text=O.l,

the

static Debye-Waller factor was chosen to E=0.98.

measurements. The integrated reflecting power, R, measured as a function of the parameter A in samples of varying defect concentrations could provide a test on Kate's statistical dynamical diffraction theory. Diffraction experiments with y-radiatlon can be performed on large samples, which are relatively easy to handle, and s t i l l reach lower A values, so that coherent scattering effects can be studied In even more heavily distorted crystals. From such experiments one may be able to learn more about the effect of defects induced by neutron transmutation doping on bulk properties of SI llke strain. Perhaps static Debye-Waller factors can be determined thisway in very dilute systems.

362

J.

R.

Schneider

and

H.

A.

Graf

3O

5.

"(" COS @I~ _-0.1

E =0990

/

E = 0992

20' E = 099/, t~

15-

E -- 0996

10-

E : 0998

"0 O~

E=I.0 ,

0

l

,

.

i

20

.

.

.

.

!

/,0

.

.

.

.

i

.

.

.

.

60

!

-

•

•

80

•

!

.

X~O A

.

.

.

120 =

Fig.28. Integrated r e f l e c t i n g power calculated within Kato's s t a t i s t i c a l dynamical d i f f r a c t i o n theory as a function of the parameter A f o r d i f f r a c t i o n of 0.0392 A y - r a d i a t i o n at s i l i c o n 220. The c o r r e l a t l o n length z was fixed so that ¢.coseB/text=O.1. The value f o r the s t a t i c Debye-Naller factor was varied tn steps of 2,10 -3 in the range of 0.99 < E < 1.

References

[1]

P. Bastte, J, LaJzerowtcz and J.R. Schneider (1978) J. Phys, C 11, 1203.

[2]

A. Freund and J.R. Schneider (1972) J. CrTst. Growth 13/14, 247.

[3]

J.R. Schneider, in: Nuclear Science ApplJcat|ons A, Vol. 2. Ed, R. Klaptsch (Harwood, U.K., 1981) pp. 227-276.

[4]

J.R. Schneider (1975) J. Appl. Cryst. 8_, 195.

~ray diffra~om~w

Is]

A. Freund and J.B. Forsyth, tn: Neutron scattering, Ed. G. Kostorz, Vol. 15 of the series: Treatise onM atertals Science and Technology, Ed. H. Herman (Academic Press, New York, 1979) pp. 471-489.

[6]

J.M. Welter, F.J. Bremer and H. Wenzel (1983) J. tryst. Growth

63, 171. [7]

P. Bastte and P. Becker (1984) J. Phys. C 17, 193.

[8]

J.R. Schneider, N.K. Hansen and H. Kretschmer (1981) Acta Cryst.

3T, 711.

[9]

R.W. Alklre, W.B. Yelon and J.R. schneider (1982) Phys. Rev. B 2_66, 3097.

[10] W. Jauch, J.R. Schneider and H. Dachs (1983) Solid State Communtc. 4_.8.8, 907. [11] N.K. Hansen, J.R. Schneider and F.K. Larsen (1904) Phys. Rev. B

2_9, 917. [12] M.C. Schmidt, R. Collela and D.R. Yoder-Short (1985) Acta Cryst. A 4!, 171. [13] J.R. schneider (1974) J. App1. Cryst. 7_, 741. [14] J.R. Schneider, in: Characterization of crystal growth defects by X-ray methods. Eds. B.K. Tanner and O.K. Bowen (Plenum Press, New York, 1980) pp. 186-215. [15] Chr. Melxner (1974), Berlchte der Kernforschungsanlage JOllch Nr. 1087-RX. [161J.R. Schneider (1976) J. Appl. Cryst. 9, 394. [17] A. Freund (1985) private communication. [18] J.R. Schneider (1983) j . Cryst. Growth 6_55,660.

363

364

J. R. Schneiderand H. A. Graf [19] H.R. Kretschmer (1985) dissertation. Technlsche Untverstt~t Berl tn. [20] J.R. Schneider and H.R. Kretschmer (1985) Naturwissenschaften 7_22, 249. [21] R.W. Alktre and W.B. Yelon (1981) J. Appl. Cryst. 14__,362. [22] S. Mertam Abdul Gant, G.F. Clark and B.K. Tanner (1981) Inst. Phys. Conf. Ser. No. 60: Sectton 5, 259. [23] J.R. Schneider and H.A. Graf (1985) J. Cryst. Growth, tn prtnt. [24] A. Ennaout, S. Flechter, W. Jaeger~nn and H. Trtbutsch (1985) J. Electrochem. Soc., fn prtnt. [25] H.A. Graf and J.R. Schneider, tn preparation. [26] P.J.E. Aldred and H. Hart (1973) Proc. Roy. Soc. Lond. A 332, 223, tbtd. A 332, 239. [27] R. Teworte and U. Bonse (1984) Phys. Rev. B 29, 2102.

[28] J.J. DeMarco and R.J. Wetss (1965) Acta Cryst. 19, 68. [29] M. Hart and A.R. Lang (1965) Acta Cryst. 1_99, 73.

[30] H. Hattort, R. Kurtyama and N. Kato (1965) J. Phys. Soc. Japan 2o, lO47. [31] F. James amd H. Roos (1975) Computer Phystcs Conwuntc. 10_, 343. [32] P. Becker, P. Seyfrted and H. Stegert (1982) Z. Phys. B 48, 17. [33] N. Kato (1963) J. Phys. Soc. Japan 1_88, 1785. [34] N. Kato (1964) J. Phys. Soc. Japan 19, 67; ibid. gl , [35] H. Hart (1966) Z. Phys. 189, 269.

971.

~-ray diffm=om~w

[36] C.G. Darwin (1922) Phil. Rag. 43, 800. [37] P.J. Becket and P. Coppens (1974) Acta Cryst, A 30, 129. [38] G. Mazzone (1981) Acta Cryst. A 37, 391. [39] N. Kato (1976) Acta t r y s t . A 32, 453; tbtd A32, 458. [40] N. Kato (1979) Acta Cryst. A 35, 9. [41] N. Kato (1980) Acta Cryst. A 36, 171; tbtd A36, 763; tbid A36, 770, [4Z] N. Kato (1982) Z, Naturforsch. A 37a, 485.

[43] J, Baruchel, J.P. Gutgay, C. Mazur~-EspeJo, M. Schlenker and J. Schwetzer (1982), J. de Physique, Colloque C7, 107. [44] J.M. Meese, D.L. Cown and M. Chandrasekhar (1979) IEEEE Transactions on Nuclear Sctence, Vol. NS-26, No. 6, 4858.

365