SmallAngle Scattering of XRays by NeutronIrradiated GermaniumG. den Ouden Citation: Journal of Applied Physics 39, 4509 (1968); doi: 10.1063/1.1655794 View online: http://dx.doi.org/10.1063/1.1655794 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/39/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Agglomeration dynamics of germanium islands on a silicon oxide substrate: A grazing incidence small-anglex-ray scattering study Appl. Phys. Lett. 102, 161603 (2013); 10.1063/1.4802843 Grazing incidence small angle x-ray scattering from free-standing nanostructures J. Appl. Phys. 86, 6763 (1999); 10.1063/1.371724 Self-assembled carbon-induced germanium quantum dots studied by grazing-incidence small-angle x-rayscattering Appl. Phys. Lett. 74, 3785 (1999); 10.1063/1.124179 SmallAngle Scattering of XRays from Neutron Irradiated Copper J. Appl. Phys. 30, 646 (1959); 10.1063/1.1735208 SmallAngle XRay Scattering J. Appl. Phys. 27, 620 (1956); 10.1063/1.1722443

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JOURNAL OF APPLIED PHYSICS VOLUME 39, NUMBER 10 SEPTEMBER 1968

Small-Angle Scattering of X-Rays by Neutron-Irradiated Germanium*

G. DEN OunENt

Materials Research Laboratory and Department of Physics, University of IUinois, Urbana, Illinois 61801

(Received 14 March 1968; in final form 23 May 1968)

A study has been made of neutron-irradiated germanium single crystals by measuring the small-angle scattering of x-rays. The magnitude of the observed scattering is found to be strongly dependent on the crystallographic orientation of the specimen with respect to the direction of the incident x-ray beam. The scattering can be explained in terms of double-Bragg scattering and is believed to be due to lattice strain associated with the radiation-induced defects. The results imply that the damaged regions (spikes) which in addition to point defects are produced by the bombarding neutrons have a disordered (possibly amorphous) structure with a density which differs from that of the surrounding matrix by at most a few percent. Anneal­ing measurements reveal a distinct annealing stage centered at about 160°C in which approximately 60% of the small-angle scattering recovers. The activation energy of this stage is found to be 1.2 eV and is as­sociated with the annihilation of either vacancies or divacancies. The annealing of the remainder of the small-angle scattering in the temperature range above 350°C is attributed to the recovery of the damaged regions.

I. INTRODUCTION

In recent· years considerable attention has been devoted to the theoretical and experimental study of germanium irradiated with heavy particles such as protons, deuterons, and fast neutrons. Irradiation with heavy particles is believed to produce two types of damage: (1) isolated point defects (vacancies, in­terstitials, Frenkel pairs), and (2) spike damage (regions produced by temperature spikes and/or short­range displacement cascades).

Experimentally, irradiated germanium has been in­vestigated by measuring the change upon irradiation of properties such as electrical resistivity, Hall co­efficient, photoconductivity, thermal conductivity, density, length, and lattice parameter. 1 Since these properties are affected by both types of damage an unambiguous interpretation of the experimental results is often very difficult. Indirect experimental evidence of the existence of spike regions has been given by Craw­ford and Cleland2 and by Stein,3 who measured Hall mobility and electrical conductivity of neutron­irradiated germanium and were able to explain the results in terms of disordered regions surrounded by a double space-charge layer, as proposed by Gossick.4

A more direct method of studying specifically the spike damage is electron microscopy. Parsons et al.s studied neutron-irradiated germanium in the electron micro-

* Work supported by the U.S. Atomic Energy Commission under Contract AT(11-1)-1198 and by the National Science Foundation.

t Present address: Philips Research Laboratories, Eindhoven, The Netherlands.

1 See, for a review, J. W. Corbett, in Solid State Physics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1966), Suppl. 7.

2 J. H. Crawford, Jr. and J. W. Cleland, J. Appl. Phys. 30, 1204 (1959).

3H. J. Stein, J. Appl. Phys. 31, 1309 (1960). • B. R. Gossick, J. Appl. Phys. 30, 1214 (1959). 5 J. R. Parsons, R. W. Balluffi., and J. S. Koehler, Appl. Phys.

Letters 1, 57 (1962).

scope and thus confirmed the existence of spike damage. Recently, den Ouden6 in a similar investigation observed damaged regions which exhibited both strain contrast and structure factor contrast and which were found to be thermally stable up to 350°C. Bertolotti et al} Chang,S and Noggle and Stiegler9 studied the etch behavior of neutron-irradiated germanium and in this way also confirmed the existence of spike damage.

Another useful method of studying the spike damage is measuring the small-angle scattering of x-rays. Fujita and GonserlO applied this technique to deuteron­irradiated germainium at low temperature and were able to explain their measurements in terms of regions with an electronic density different from the surrounding matrix. In the experiment described in this paper the small-angle scattering of x-rays by germanium single crystals irradiated with fast neutrons was measured in order to obtain a better understanding of the nature and the annealing behavior of the damage produced by the irradiation.

II. EXPERIMENTAL PROCEDURE

The specimens, which are specified in Table I, were cut in the required orientation from large germanium single crystals into slices of dimensions 3X 1XO.05 cm. They were irradiated in the CP-5 reactor of Argonne National Laboratory with an estimated total dose of 5X 1019 fast neutrons cm-2 at an ambient temperature of approximately 40°C. Mter irradiation the specimens, which had become heavily radioactive, were stored for several months at room temperature before being used. Each irradiated specimen, together with a nonirradiated

6 G. den Ouden (unpublished). 7 M. Bertolotti, T. Papa, D. Sette, B. Grasso, and G. Vitali

Nuovo Cimento 29, 1200 (1963). ' 8 R. Chang, J. Appl. Phys. 28, 384 (1957)1 9 T. S. Noggle and J. O. Stiegler, J. Appl. Phys. 30, 1279 (1959). 10 F. E. Fujita and U. Gonser, J. Phys. Soc. Japan 13, 1068

(1958) . 4509

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4510 G. DEN OUDEN

TABLE I. Specimens.

Specimen

2

3

4

5

6

Type

n-type

intrinsic

n-type

n-type

n-type

intrinsic

Electrical resistivity

(n·cm)

0.043

>30

0.163

0.043

0.163

>30

reference specimen of same type, resistivity, and orientation, was then thinned to a final thickness of about 0.005 cm by grinding, using successively finer carbon-silicon paper on a glass plate, followed by mechanical polishing of both sides with the help of polishing cloths with successively finer pastes (down to

d

9 1 ____ _

n

a

10 em

a

i k

b

FIG. 1. Schematic top view of symmetrical four-slit small­angle x-ray scattering system. (a) 10 in. steel I-beam bench; (b) journals supporting stainless steel rods (c); (d) x-ray tube (8° takeoff angle); (e) tube carriage; (f) aluminum cars; (g) slit housings of collimating section; (h) brass bellows; (i) goniometer which supports analyzing sections; (j) vernier reading to l' of arc' (k) specimen chamber; (1) axis of rotation of goniometer and location of sample; (m) stainless steel rods connected to goniometer via aluminum frames (n); (0) slit housings of analy­zing section; (p) proportional counter; (q) aluminum carriage; (r) radiation filter. Not shown are the goniometer drive mechan­ism, clutch, and counterweights.

Specimen surface Type of measurement

(331) isochronal annealing

(100) isochronal annealing

(521) isochronal annealing

(331) isothermal annealing

(521) isothermal annealing orientation dependence

(100) orientation dependence

1 p, particle size) on a flat brass disk. The specimen to be measured was then cleaned with acetone ar.d glued onto the aluminum specimen holder with Sauereisen cement.

The apparatus used for the small-angle x-ray scatter­ing measurements, a schematic top view of which is given in Fig. 1, will be described in detail elsewhere.ll It consists essentially of four identical slits which are separated from each other by a distance of 10 cm. The first two slit housings (collimating section) define the incident x-ray beam, the last two slit housings (analyz­ing section) are connected via an aluminum frame with a goniometer and can rotate round a vertical axis midway between the second and third slit. The specimen holder can be mounted in such a way as to locate the specimen exactly and reproducibly on this axis of rotation, perpendicular to the primary beam. The slits had a height of 1 cm and were set for a width of 0.030 cm. As x-ray source a copper tube of General Electric Co. (type CA-7H) was used, operated at 3S kV and 23 rnA. Scattering was measured between 20' of arc and 3°40' arc, in most cases on one side of the incident x-ray beam.

o SPECIMEN 1 '" SPECIMEN 2 o SPECIMEN 3

i 10 ~~0~3~0~~~~~5~0~6do~~m~=w~LJ

_ SCATTERING ANGLE (MINUTES OF ARC)

FIG. 2. Small-angle scattering from three germanium single crystals irradiated with fast neutrons.

11 S. Diamond, G. den Ouden, and R. O. Simmons (unpub­lished) .

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x _ RAY seA T T E R IN G B Y N E U T RON - I R R A D I ATE D G E R MAN I U M 4511

To obtain the scattering due to the irradiation effects, the scattering of the nonirradiated reference specimen was subtracted from that of the irradiated specimen. All small-angle scattering measurements were carried out in vacuum at room temperature. Corrections for radio­activity of the specimens were made. Before and after measuring the scattering curve of a specimen the scattering of a standard scatterer at a fixed angle was measured. For this purpose a quenched foil, 0.0075 cm thick, of AI-Ag (10% by weight) was used. The scattering of this standard scatterer had been calibrated against the intensity of the incident x-ray beam. Thus all scattering measurements could be normalized to a common value of the intensity of the incident x-ray beam.

Annealing of the specimens took place outside the x -ray apparatus either in a furnace (isochronal annealing) or in oil baths (isothermal annealing). The furnace annealing was carried out in helium atmosphere and the temperature in the furnace could be kept constant within 2 degrees centigrade. For the oil baths, Dow-Corning oil was used and the temperature in the baths was regulated within 0.5°C of the required temperature.

III. EXPERIMENTAL RESULTS

Small-angle scattering measurements were carried out successively on all specimens. In Fig. 2 the scattered intensity of specimens 1, 2, and 3 (Table I) is plotted as a function of the scattering angle. To eliminate the difference in scattered intensity of the various speci­mens due to their unequal thickness all scattering curves were normalized to the thickness of specimen 1 (i.e., 0.0046 cm). Figure 2 shows that there is a significant difference in the magnitude of the small-angle scattering of the various specimens. In Fig. 3 the logarithm of the

100.---r---.---,--~-.---r-~

50 a § (f)

Ck:

~2

~ z ::::> a ~10 ~ tii z

~ 5

f

° SPECIMEN 1 A SPECIMEN 2

o SPECIMEN 3

FIG. 3. Plot of log! vs 02 for three different specimens.

° SPECIMEN 1 h SPECIMEN 2 o SPECIMEN 3

FIG. 4. Plot of log! vs logO for three different specimens.

scattered intensity of the same specimens is plotted vs the squared value of the scattering angle. The curves are similar in shape and are linear only for small values of the abscissa. Figure 4 gives a plot of the logarithm of the scattered intensity vs the logarithm of the scattering angle resulting in straight lines for all three specimens over a large angular range. The slope of the straight line has a value of approximately minus two.

In view of the significant difference in magnitude of the small-angle scattering of the three specimens an attempt was made to investigate a possible relationship between the crystallographic orientation of the speci­men and the scattered intensity. To this purpose some specimens were rotated around an axis going through the center of the specimen and parallel to the slits of the system. The scattered intensity at a fixed scattering angle was then measured as a function of the angle of rotation. The results of such a measurment on specimen 5 are shown in Fig. 5. Also plotted in Fig. 5 are the data obtained by the same measurement on a quenched foil of polycrystalline AI-Ag (10% by weight) of thickness 0.0075 cm. The intensities are given in fractions of the scattered intensity at rotation angle zero. Figure 5 shows the strong dependence of the scattered intensity on the crystallographic orientation of the irradiated german­ium specimen with respect to the direction of the in­cident x-ray beam.

Isochronal annealing measurements were carried out on specimen 1 (annealing time 30 min), specimen 2 (annealing time 20 min) and specimen 3 (annealing time 20 min) in the temperature interval from 40°C up to 600°C. The complete scattering curve was measured immediately after the anneal at every annealing tem­perature. The measured isochronal annealing curves of the three different specimens are similar in shape. In Fig. 6 some of the results for specimen 3 are shown. Here the scattered intensity is plotted vs the annealing temperature for 4 of the 29 measured scattering angles

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4512 G. DEN DUDEN

Ul z ~ 5 u

~4 '--> I-

!£ 3 w l-

1:

o SPECIMEN 5 '" Al-Ag(10% by weight)

20

FIG. 5. Scattered intensity at fixed scattering angle (30 min of arc) as a function of angle of rotation for neutron-irradiated germanium and quenched AI-Ag (10% by weight).

(30',45',60', and 75'). Figure 7 gives the average of the scattered intensity over all measured scattering angles as a function of annealing temperature. The scattered intensity is given in fractions of the average scattered intensity before annealing. Approximately 60% of the small-angle scattering recovers in a stage centered at about 160°C. The remainder anneals in the temperature interval 350°-500°C. There is an increase in scattered intensity in the temperature range just before the two stages.

Figure 8 gives the results of isothermal annealing measurements carried out on specimens 5 (annealing temperature 160°C) and specimen 6 (annealing tem­perature 170°C). Immediately after each anneal a complete scattering curve was measured. The average of the scattered intensity over all measured scattering angles is plotted as a function of annealing time. The average of the scattered intensity is again given in fractions of the value before annealing.

IV. THEORY OF RADIATION DAMAGE

Before discussing the experimental results it is useful to review some basic elements of the theory of radiation damage. 12 When a neutron traveling through a crystal lattice with energy E strikes a lattice atom, the energy T transferred to that atom is given by the formula

T=[4M/(1+M)2JE sin28/2, (1)

where M is the mass number of the lattice atom and 8 the deflection angle of the colliding neutron. Assuming isotropic scattering, the differential cross section for elastic scattering can be expressed as

(2)

Here (fT is the total cross section for elastic scattering and Tm is the maximal transferred energy. When T is larger

12 See, for instance, G. J. Dienes 3;nd G. H. Vineyard, Radiation Effects in Solids (Interscience Publishers, Inc., New York, 1957).

than Ed, the threshold energy for displacement, the struck atom will be displaced. This primary knock-on then can move through the lattice and interact with other lattice atoms by screened Coulomb forces.

By applying the formulas given above to neutron­irradiated germanium, the following results are obtained. On the assumption that the average energy of the neutrons striking the germanium specimen is 1 MeV the maximum energy of the primary knock-ons is found to be 54 keV. The mean energy of the primary knock-ons will be half this value, i.e., 27 keY if the scattering would be completely isotropic and elastic. This as­sumption, however, is not correct. It is therefore necessary to make use of a correction factor, which accounts for the anisotropy and inelasticity of the scattering. This correction factor is estimated to be about i for germanium, resulting in a mean energy of about 18 keV for the primary knock-ons.

According to Seitz and Koehler13 a large portion of the energy of the bombarding particle, however, will be transferred to the lattice in the form of lattice vibra­tions. These sudden lattice excitations are called temperature spikes and the regions where this thermal energy is dissipated can be considered as regions where the temperature has reached a very high value, possibly well above the melting point, during a very short time (of the order of 10-11 sec.) The local heating within the spike regions may stimulate activated processes during the time they are at high temperature. It might be possible, for example, that Frenkel defects are generated, part of which are subsequently annealed. Temperature spikes might also produce regions with a frozen-in liquid structure. In this case the concept of point defects within the spike consequently has become inadequate. Several calculations, based on different assumptions, have been made of the total number of atoms displaced by fast incident particles.

8 1.4,----,---,--)---,r---,.---,)-..., ~ L51.2r-z z ~ 1.Gr- • :::l

~ Q.8r-

IDMr-i: OAt­Vi a:io.r­I-

B 6 v ~ ~ Os

"g 0 0

§oo

" 0

° SCATTERING ANGLE 30' -o SCATTERING ANGLE 45' II SCATTERING ANGLE 60' -'l SCATTERING ANGLE 75' • = O,O,A,V -

-11 o

g A

-~ a !;----:;i;I;;---r;iv;-1--:;ii\;;------d;.~:;__~c::iI*-l~ .... a 100 200 300 400 500

_ ANNEALING TEMPERATURE (oc)

FIG. 6. Isochronal annealing. Scattered intensity at four scattering angles as a function of annealing temperature for specimen 3.

13 F. Seitz and J. S. Koehler, in Solid State Phyics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1956), Vol. II, p. 305.

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x -RAY S CAT T E R I N G BY N E U T RON - I R R A D I ATE D G E R MAN I U M 4513

Whereas the results of these calculations are essentially in agreement there exists a significant discrepancy con­cerning the fate of the displaced atoms. Seitz and Koehler!3 calculated the mean distance between second­ary displacements along the' path of a primary knock-on for typical cases and found values large enough to justify the conclusion that most displaced atoms will become interstitial atoms, leaving their original lattice sites as vacancies. Brinkman,!4 on the other hand, found much smaller values for this mean distance. In his model a knock-on with energy lying between two critical values El and E2 will produce a displacement spike. This Brinkman displacement spike in its intial unstable stage consists essentially of a hole (multiple vacancy of high order) around which the associated interstitials form a shell. Due to the high temperature of the shell this configuration will collapse to form a disordered region.

Seeger!5 took a somewhat different view and intro­duced the concept of the depleted zone. According to him a substantial part of the interstitials rejected from the region where the displacement cascade took place are carried away over large distances by the propagation of dynamic crowdions. This will result in the production of regions characterized by a high density of vacancies and surrounded by a few nearby interstitials and many more distant interstitials. It must be noted here that the crowdion propagation mechanism was proposed to explain the radiation damage in fcc metals. It is ques­tionable whether in a material such as germanium, which does not have close-packed rows of atoms, this mechanism plays a role of importance.

In the case of neutron irradiation the possibility of neutron capture must also be considered. The absorption of thermal neutrons by lattice atoms and the subsequent radioactive decay results in the production of impurity atoms. This can be a complicating factor, especially

14.r----,r----.-----.----~----~-

FIG. 7. Isochronal annealing. Scattered intensity (average over all measured scattering angles) as a function of annealing temperature for specimen 3.

14 J. A. Brinkman, Am. J. Phys. 24, 246 (1956). 15 A. Seeger, Pl'oceedinKs of the Second International Conference

on the Peaceful Uses of Atomic Energy (United Nations, New York, 1958), Vol. VI, p. 250.

~2,~--~--~~--~--~--~h_--~--~ o w ~ 00 00 m ~ ~ -ANNEALING TIME (MINUTES)

FIG. 8. Isothermal annealing. Scattered intensity (average over all measured scattering angles) as a function of annealing time for specimens 5 and 6.

when properties related to the density of charge carriers are measured. The 'Y rays and electrons generated by the process of absorption and decay are expected to produce isolated point defects while the recoiling atoms can produce small defect clusters. Neutron capture in germanium leads to the production of the acceptor impurity 71Ga (30.4%) and the donor impurities 75As (9.8%) and 77Se (1.2%). Schweinler!6 has estimated the average kinetic energy of the recoiling germanium atom to be 182 eV, a value sufficiently large to expect the production of small defect clusters. On the basis of an estimate of the thermal neutron flux encountered in the present experiment it must be expected that the con­tribution of the damage due to recoil is negligibly small, while the amount of impurities produced is of the order of or larger than the amount originally present.

V. DISCUSSION OF EXPERIMENTAL RESULTS

A. Irradiation

Small-angle scattering of x-rays can be produced by particles or regions which have an electronic density different from that of the surrounding matrix. The theory of this type of scattering has been developed by Guinier!7 (see Appendix). It is interesting to see whether it is possible to explain the small-angle scattering data obtained in this experiment in terms of this theory. Figure 3, in which log! is plotted vs (}2, shows concave curves for the three different specimens. In terms of Guinier scattering this could mean that the scattering particles have a size .. distribution with a radius of gyration varying from, about 24 A to about 40 A. This interpretation, however, cannot be correct since the Guinier approximation is only valid «(}R<0.2X) for

16 H. C. Schweinler, J. Appl. Phys. 30, 1125 (1959). 17 A. Guinier and G. Fournet, Small-Angle Scattering of X-rays

(John Wiley & Sons, Inc., New York, 1955).

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4514 G. DEN OUDEN

scattering angles smaller than 8X 10-3 rad, where the slope gives a radius of gyration of about 40 A.

The fact that the Guinier approximation does not hold over the whole angular range is illustrated also by Fig. 4, in which log! is plotted vs loge. This plot gives straight lines for angles larger than 10-2 rad with a slope of approximately minus two. In this angular range the (1-4 approximation apparently does not hold either, since in that case, due to the dimensions of the incident beam, one would expect the slope to have the value minus three.

Furthermore, it is seen from Figs. 2, 3, and 4 that the scattering curves of the three specimens are similar in shape, but that the magnitude of the scattering of the different specimens differs significantly. Since all specimens had been irradiated with the same neutron dose this difference must be due to the difference in orientation of the specimens with respect to the direction of the incident x-r9-Y beam. One possible explanation could be that the scattering is produced by nonspherical particles with a preferred orientation. This explanation, however, does not seem right since the extrapolated value of the scattered intensity to zero scattering angle differs substantially for the various specimens.

That the small-angle scattering of neutron-irradiated germanium is indeed strongly dependent on the orienta­tion of the specimen with respect to the direction of the incident x-ray beam is illustrated by Fig. S. Measure­ments revealed that the intensity of the transmitted beam is independent of the orientation. With the help of stereographic projection or an Ewald sphere con­struction it can be shown that the peaks in the scattered intensity occur for rotation angles where a low-order Bragg reflection is expected. This can be illustrated more easily with the relatively simple case of specimen 6 I specimen surface (100); rotation around [llOJ I· The measured curve, not shown here, is symmetric with two peaks centered in the rotation angles ± 100 and with a half-width of approximately 50. The peaks cover an angular range within which (311), (331), and (422) reflections occur. Although the orientation dependence was not investigated in detail, these preliminary results indicate that the small-angle scattering is associated with the occurrence of Bragg reflections.

Beeman et al.18 discussed the mechanism of double­Bragg scattering, which they believe to be responsible for the small-angle scattering from cold-worked metals. Double-Bragg scattering of x-rays takes place when the once-reflected x-ray beam is again reflected in the for­ward direction by the same set of crystal planes. When the once-reflected beam is broadened by nonuniform strain the twice-reflected beam will be broadened also, giving rise to small-angle scattering. Another possibility is that the once-reflected beam is reflected again by the same crystal planes, which are rotated around the

18 W. W. Beeman,P. Kaesberg, J. W. Anderegg, and M. B.Webb, in Hand-buch der Physik, S. Fliigge, Ed. (Springer-Verlag, Berlin, 1957), Vol. XXXII, p. 362.

direction of the once-reflected beam, again producing scattering at small angles.

These considerations suggest that the major part of the small-angle scattering as measured in this experi­ment should be explained in terms of double-Bragg scattering rather than in terms of the Guinier theory. The presence of strain in the irradiated specimens, required for this scattering mechanism to occur, has been confirmed by x-ray diffraction analysis and elec­tron microscopic observation.6 The strain is presumably associated with the defects produced in the crystal by the bombarding neutrons. This conclusion about strain is in agreement with the results of recent work by Baldwin and Thomas,19 These authors. measured the anomalous x-ray transmission of neutron-irradiated germanium and attributed the decrease in transmitted intensity to the strains in the neighborhood of the disordered regions produced by the irradiation.

The fact that the Guinier scattering is absent or at least limited yields a maximum value for the product of No and n2, where No represents the number of scattering particles or regions per cm3 and n the difference in the number of electrons in the region and in the same volume of the surrounding matrix (see Appendix). Taking the values 1017 cm-3 for No and 4SX 10-8 cm for the mean diameter of the region, as found in the electron microscopic study, this leads to the conclusion that the difference in mass density between region and sur­rounding is at most 5%. It must be noted that this percentage might well be too high since the value chosen for No is believed to be underestimated.

B. Annealing

The results of the annealing measurements must now be considered. As shown in Figs. 6 and 7, isochronal annealing reveals a stage between the annealing tem­peratures 120°C and 190°C, centered at about 160°C in which approximately 60% of the small-angle scattering recovers. The remainder disappears in the temperature range above 350°C.

Figure 8 shows the measured small-angle scattering as a function of annealing time at a fixed temperature. Meechan and Brinkman20 indicated a method to obtain the activation energy of migration of the defects moving in a certain annealing stage by combining the isochronal and isothermal annealing curve. In this method one determines the annealing temperature T; at which a certain portion ilP; of the measured property anneals from the isochronal annealing curve and the annealing time ilTi necessary for anneal of the same portion from the isothermal annealing curve. When the annealing is associated with a unique activation energy, the plot 10gilTi vs liT; will result in a straight line, the slope of

19 T. O. Baldwin and J. E. Thomas, J. Appl. Phys. 39, 4391 (1968). The author is grateful to these authors for an account of their work, prior to its publication.

20 C. J. Meechan and J. A. Brinkman, Phys. Rev. 103, 1193 (1956) .

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x -RAY S CA T T E R I N G B Y N E U T RON - I R R A D lATE D GERM ANI U M 4515

which yields the activation energy. Generally, the two specimens used for the isochronal a?,d is.otherI?al ~n­nealing measurements must have Identical hlstones. This condition is not necessary, however, when the annealing obeys a chemical rate equation of the form

dn/dt= -cron"Y exp( -Em/kT) , (3)

where 1t represents the defect concentration, c a numerical constant depending on the geometry of the lattice Yo the effective jump frequency, 'Y the order of reacti~n and Em the activation energy of migration.

The ~ethod of Meechan and Brinkman was applied to the annealing data obtained in this experiment in order to check whether the annealing in the state at 160°C can be considered as due to defect migration and, in that case, to determine the activation energy of migration of these defects. Figure 9 s.ho~s a plot ~f log.1Ti as a function of I/Tdor a combmatlOn of sl?ecI­mens 3 and 5. The values of 10gAT i and 1/ T i are obtamed from the isochronal and isothermal curves of Figs. 7 and 8. From the slope of the straight line drawn through the points of Fig. 9 an activation energy of migration is calculated at approximately 1.2 eV. .

When an annealing process is governed by a chemical rate equation the order of reaction can be determined from the isothermal annealing curve. This method, however can only be used when the relationship be­tween the measured physical property and the annealing defects is known; for example, when point-defect annealing is studied by means of electrical resistivity measurements, in which case the relationship is taken to be linear. In Fig. 10 the logarithm of the average of the scattered intensity over all measured scattering angles is plotted as a function of annealing time for specimen 6. The average intensity is given in unannealed fractions of the average intensity annealing in the stage at 1600C. Except for the initial part the curve is linear in shape. With the assumption that a chemical rate equation governs the annealing, this leads to the con­clusion that the kinetics of the process become first order after a higher-order initial stage.

100

50

'iii' ~ 20 :::> Z

~10 \,;' q

t 5

2

1 2.15 2.~

FIG. 9. Determination of activation energy from Figs. 7 and 8.

3' >-~ f-.<{

Vi ~ 0.5 Zz ~<{ ~5 wz IDO ~i= ~~0.2

t ~ !----,,.J;;--*-+n-~;__-;;;;;--___;;;;;;_ 0.10 20 4!J 60 80 100 120

- ANNEALING TIME (MINUTES)

FIG. 10. Plot of the logarithm of the average scattered intensity as a function of annealing time for specimen 6.

At this point it is interesting to consider the results of some other studies dealing with the annealing behavior above room temperature of irradiated germanium. Annealing stages very similar to those obtained in the present work have been reported by Baldwin and Thomas.19 These authors measured the anomalous x-ray transmission of neutron.-irradiated germanium ~nd found that annealing occurs m two temperature regIOns centered at about 150° and 400°C. They assume that the annealing is caused by the breakup of the damaged regions into two kinds of defects which have very different migration activation energies. Eldros,21 i~ an infrared-absorption study of pile-irradiated germamum, found an annealing stage at about 150°C only in the specimens irradiated with a prevailing amount of thermal neutrons and concluded that the stage is due to the migration of point defects or small defect clusters. Konopleva et al.,22 measuring electrical conductivity a~d Hall coefficient, found two distinct annealing stages m neutron-irradiated germanium. One stage is centered around 175°C with an activation energy of 1.25 eV; the other one lies above 250°C with an activation energy of 2.6 eV. The authors associated the first stage with the migration of vacancies, the second on~ with the mig.ra­tion of divacancies. Ishino et al.23 studied the anneahng behavior of 'Y-irradiated n-type germanium by measuring Hall mobility and elecrrical conductivity and were able to distinguish three annealing stages above room temperature. The largest of these, which is located at about 160°C, has an activation energy of 1.2-1.3 eV and was also associated with vacancy migration.

The same annealing stage has been found by Pigg and Crawford24 in a similar study. These authors, however, attribute the annealing in this stage to the dissociation of vacancy-donor pairs with subsequent migration of the liberated vacancies. In their model the apparent

21 R. Eldros, Arkiv Fysik 28, 447 (1965). • 22 R. F. Konopleva, S. R. Novikov, and S. M. Ryvkin, Sov.

Phys.-Solid State 6, 2610 (1965). " 23 S. Ishino, F. Nakazawa, and R. R. Haslguti, J. P~ys. <:hem.

Solids 24, 1033 (1963); F. Nakazawa, S. Ishino, J. HIgashimak­agawa, and R. R. Hasaguti, J. Phys. Chem. Solids 26, 1895 (1965). ! 41

24 J. C. Pigg and J. H. Crawford, Jr., Phys. Rev. 135, All (1964) .

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4516 G. DEN DUDEN

activation energy (1.2 eV) is the sum of two energies, EB and EM, EB (",,0.4 eV) being the binding energy of the vacancy-donor pair and EM (",,0.8 eV) the energy for vacancy migration. Evidence of the dependence of the annealing behavior on the nature and concentration of the impurity content has also been given by Brown et aP"

Although the association of the annealing stage at 160°C and the corresponding apparent activation energy of 1.2 eV with vacancy migration seems to be in agreement with the results of quenching and self­diffusion experiment,26 there is nevertheless serious doubt concerning the validity of this interpretation.

Recently, Whan,27 measuring infrared absorption of oxygen-doped germanium, has shown that vacancy migration takes place at a much lower temperature than hitherto assumed. Also, Scholz and Seeger28 have suggested that the annealing stage at 160°C with the activation energy of 1.2 eV is due to the migration of divacancies rather than of vacancies. In the present experiment no clear distinction can be made between these two types of defects. The annealing process in the stage centered at 160°C should therefore be regarded as a process in which either isolated vacancies or di­vacancies are annihilated, giving rise to a reduction of lattice strain.

The annealing of the small-angle scattering in the temperature range above 350°C is assumed to be due to the disappearance of the damaged regions or spikes produced by the neutron bombardment. Electron microscopic observation of the same specimens as used in this experiment6 justifies this assumption. The damaged regions, which are visible by both strain contrast and structure factor contrast, are found to be thermally stable up to 350°C and disappear at higher temperatures.

VI. CONCLUSIONS

The results obtained in the present investigation lead to the following conclusions:

(1) Neutron-irradiated germanium produces small­angle scattering of x-rays. The magnitude of this scattering is strongly dependent on the orientation of the specimen with respect to the direction of the in­cident x-ray beam.

(2) The small-angle scattering should be explained in terms of double Bragg scattering rather than in terms of the Guinier theory and is assumed to be due to lattice strain associated with the radiation-induced defects.

(3) The damaged regions (spikes) should be re-

20 W. L. Brown, W. M. Augustyniak, and T. R. Waite, J. Appl. Phys. 30, 1258 (1959).

26 A. Hiraki and T. Suita, J. Phys. Soc. Japan 18, Suppl. III, 254 (1963).

27 R. E. Whan, Appl. Phys. Letters 6, 221 (1965); J. Appl. Phys. 37, 2435 (1966).

28 A. Scholz and A. Seeger, Proceedings of the 7th InternationaJ Conference~onlthe.Physics of Semiconductors (Dunod Cie., Paris, J965) ,_Vol. III, p. 93.

garded as disordered (possibly amorphous) regions surrounded by a strain field and having a mass density which differs from that of the surrounding lattice by at most a few percent.

(4) The annealing of the small-angle scattering is characterized by a distinct stage centered at about 160°C, where approximately 60% of the effect recovers. A value of 1.2 eV has been found as the apparent activation energy of the defects migrating in this stage. These defects are believed to be either vacancies or divacancies.

(5) The annealing of the remainder of the small­angle scattering in the temperature range above 350°C is attributed to the recovery of the damaged regions.

ACKNOWLEDGMENTS

The author wishes to thank Professors J. S. Koehler and R. O. Simmons for much support and encourage­ment in this work. The Organization for Economic Cooperation and Development (O.E.C.D.) is gratefully acknowledged for a travel grant, obtained through the courtesy of the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).

APPENDIX

The small-angle scattering of x-rays caused by particles or regions that have a different electronic density with respect to the surrounding matrix can be adequately described by the approximation formula of Guinier17

1(0) =I.Nn2 exp( -4rR~02/3}'.2), (Al)

where 1(0) represents the measured intensity as a function of the scattering angle 0 in radians, I. the intensity scattered by one electron, N the number of scattering particles, n the difference in the number of electrons in the particle and in the same volume of the surrounding matrix, R the radius of gyration of the particle and A the wavelength of the radiation used. The radius of gyration is defined by

R2= f r2p(r)dVrl f p(r)dVr• (A2)

Here per) is the electronic density as a function of r, the distance from the center of mass of the particle, and dVr

a volume element of the particle. R can be obtained experimentally from the slope of the plot 10gI(O) vs 02

•

The radius of gyration is characteristic for the size of the scattering particle but does not unambiguously deter­mine its shape. Particles of different shape can have the same radius of gyration. The Guinier approximation is only valid for sufficiently small angles, OR<0.2A in case of spherical particles, for example. For large angles the scattered intensity is given by the equation

1(0) =f(j-4. (A3)

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x - RAY S CAT T E R I N G BY N E U T RON - I R R A D I ATE D G E R MAN I U M 4517

The proportionality factor f is linearly dependent on the surface of the scattering particle. When the scattering particles have a shape which deviates from a sphere then there is an intermediate angular range where neither the Guinier approximation nor the fr4 dependence holds. Scattering formulas for this intermediate range have been calculated and tested for various particle shapes. In this range, for example, the scattered intensity from cylinders will decrease as (]-1, from disks as (]-2. In the foregoing it is assumed that the scattering particles are randomly distributed and widely dispersed. For dense systems the situation is quite different, since in that case interference effects should be taken into account.

It must be remembered, finally, that in experiments where the primary x-ray beam is defined by slits the measured scattering curve is different from the theoreti­cal curve. Corrections for the beam dimensions, there­fore, should be made. Important facts in this respect are that, when the theoretical scattering curve is Gaussian, as in the case of the Guinier approximation, the meas­ured curve is proportional to the theoretical curve and that in the angular region where the theoretical scattering is proportional to (]-4 the observed scattered intensity should decrease as (]-3. Several methods of correction have been developed for the more general case.

JOURNAL OF APPLIED PHYSICS VOLUME 39, NUMBER 10 SEPTEMBER 1968

Temperature Dependence of the Widths and Positions of the R Lines in Li2Ge1015:CrH

RICHARD C. POWELL

Air Force Cambridge Research Laboratories, Bedford, Massachusetts

(Received 22 January 1968; in final form 7 May 1968)

The widths and positions of the Rl fluorescence lines of trivalent chromium ions in nonequivalent lattice sites in lithium germanate crystals have been measured at temperatures from 24°K up to 200oK. The temperature dependences of the linewidths are explained in terms of microscopic strains, direct phonon processes, and two-phonon Raman scattering processes. The temperature dependences of the lineshifts are attributed to two-phonon virtual processes and direct phonon processes. The theraml changes in the widths and positions are found to be of the same order of magnitude. These results are compared with those from similar measurements on ruby- and chromium-doped MgO.

I. INTRODUCTION

The gross features of the optical absorption and emission spectra of Li2Ge,015: Cr3-t crystals are similar to those of trivalent chromium in other host lattices such as Al20 3 and MgO.I.2 The four most intense sharp lines in the spectra appear to be two sets of R lines associated with chromium ions situated in none­equivalent sites in the crystal. Results are reported here of measurements of the temperature dependence of the widths and positions of these four lines in the fluorescence spectrum of lithium germanate at temper­atures between 24° and 200°K. These results were obtained from crystals containing 0.038,0.12, and 0.18 wt% chromium and are compared with similar measure­ments made on ruby and MgO:Cr3+. Theoretical analysis of the data indicates that microscopic strains, direct phonon processes, and the Raman scattering of phonons by the impurity ions contribute to the broad­ening of these lines. The shift in the positions of the lines with temperature can be attributed to two-phonon virtual processes and direct phonon processes.

1 C. A. Pitha and B. DiBartolo, Bull. Am. Phys. Soc. 12, 293 (1967) .

2 H. G. Lipson and R. C. Powell, J. Appl. Phys. 38, 5409 (1967).

II. EXPERIMENTAL

The lithium germanate samples used for this in­vestigation were cut and polished with faces per­pendicular to the a, b, and c axes of the crystal and had approximate dimensions of 8X5X2 mm. They were mounted on the cold finger of a Janis model 8DT cryostat capable of continuous temperature variation from 4.2° up to 300oK. Excitation was provided by an air-cooled General Electric BH6 1000 W high-pressure mercury lamp, filtered by 4 em of saturated CUS04. The fluoresecnece observed at 90° to the direction of excitation was chopped and focused onto the entrance slit of a McPherson model 213 one-meter scanning motlochromator set to achieve a resolution of 0.6 em-I. The signal was detected by an RCA 7102 photo­multiplier tube cooled by liquid nitrogen. The modu­lated signal was then amplified by a P.A.R. JB-5Iock-in amplifier and displayed on a strip chart recorder.

The unpolarized fluorescence spectrum between 6500 and 8000 A is shown in Fig. 1 at three temper­atures for a lithium germanate sample with 0.18% chromium. The spectrum consists of the sharp no­phonon lines and broad vibronic sidebands. As temper­ature is decreased, more of the emission occurs in the no-phonon lines and less in the vibronics. Below

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