Physica status solidi / A.: Volume 63, Number 1 January 16 [Reprint 2021 ed.] 9783112501849, 9783112501832

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plxysica status solidi (a)

ISSN0931*8965 * V O L . « . NO. 1 . JANUARY 1981

Classification Scheme 1. Structure of Crystalline Solids 1.1 Perfectly Periodic Structure 1.2 Solid-State Phase Transformations 1.3 Alloys. Metallurgy 1.4 Microstructure (Magnetic Domains See 18; Ferroelectric Domains See 14.4.1) 1.5 Films 1.6 Surfaces 2. Non-Crystalline State 3. Crystal Growth 4. Bonding Properties 5. Mössbauer Spectroscopy 6. Lattice Dynamics. Phonons 7. Acoustic Properties 8. Thermal Properties 9. Diffusion 10. Defect Properties (Irradiation Defects See 11) 10.1 Metals 10.2 Non-Metals 11. Irradiation Effects (X-Ray Diffraction Investigations See 1 a n d 10) 12. Mechanical Properties (Plastic Deformations See 10) 12.1 Metals 12.2 Non-Metals 13. Electron States 13.1 Band Structure 13.2 Fermi Surfaces 13.3 Surface and Interface States 13.4 I m p u r i t y and Defect States 13.5 Elementary Excitations (Phonons See 6) 13.5.1 Excitons 13.5.2 Plasmons 13.5.3 Polarons 13.5.4 Magnons 14. Electrical Properties. Transport Phenomena 14.1 Metals. Semi-Metals 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Films 14.3.2 Surfaces a n d Interfaces 14.3.3 Devices. Junctions (Contact Problems See 14.3.4) 14.3.4 High-Field Phenomena, Space-Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence See 20.3; Junctions See 14.3.3) 14.4 Dielectrics 14.4.1 Ferroelectrics 15. Thermoelectric and Thermomagnetic Properties 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions 17.1 Field Emission Microscope Investigations 18. Magnetic Properties 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.2.1 Ferromagnetic Films 18.3 Ferrimagnetic Properties 18.4 Antiferromagnetic Properties ( Continued on cover three)

phys. stat. sol. (a) 68 (1981)

Author Index H . A . AARNA M . S. ABRAHAMS A . A D AU D . K . AGRAWAL QU. A . ALATIEF S. ALEXANDROVA M . H . AMABAL J . K . A . AMUZU M . A . ANGADI G . ABLT H . ARNDT P . V . ASHRIT

389 K3 K179 K189 313 371 605 K7 K77 475 501 K77

Y . BADR V . E . BAGIEV I . L . BAGINSKII A . A . BAHGAT S . R . BAKHCHIEVA V . A . BALASHOV J . J . BARA B . DI BARTOLO R . BASU . G . L . BELENKII Y . N . BELOGUROV G . V . BERENSHTEIN P . K H . BEZIRGANYAN G . K . BHAGAVAT S . V . BHORASKAR Z . BÌLEK V . A . BILINKIN J . J . M . BINSMA A . B . BISWAS G . BLASSE V . N . BLIZNYUK J . BLOEM H . J . BLYTHE M . BOHL E . BONETTI G . R . BOOKER G . BOBCHARDT K . BORMAN O . BRÜMMER J . BUCHAR I . BUNGET A . J . BTTRGGRAAF H . - J . BURMEISTER

355,699 K19 271 K39 23 K19 119 K31 K153 97 45 K27 91 K15 K103 259 45 595 K123 157,569 K43 595 K195 529 645 K3 501 501 KL 15 259 K55 229 213

C. B . CARTER L . M . CASPERS A . CAST ALDINI A . CAVALLINI J . CERVENAK G . S . CHADHA D . K . CHAKRABARTY

335 K183 143 143 K163 625 K123

A . N . CHAKRAVARTI M . R . CHAVES E . R . CHENETTE H . * R . CHILD A . K . CHOWDHURY A . CHRISTOPH J . VON CIEMINSKI N . COWLAM A . F . CRAIEVIOH C. CRUZ F . CUMBRERA P . DANESH J . DANKO M . P . DARIEL A . DARLII&SKI L . I . DATSENKO H . A . DAVIES L . M . DEDUKH I . Y A . DEKHTYAR B . DIETRICH T . VAN D I J K A . E . DJAKOV G . DLUBEK V . K . DOLGANOV K H . R . DRMEYAN M. DRYS A . V . DVURECHENSKII F . DWORSCHAK A.M.DYACHENKO S . A . DYOMIN T . D . DZHAFAROV A . EBERT C. ECOLIVET SH. M . EFENDIEV F . EL-KABBANY E . EVANGELISTA

K97 605 445 31 K97 511 501 625 321 487 631

.

743 K31 329 663 439 625 K63 K147 511 229 743 K115 265 91 693 K203 K195 . K27 K93 431 209 K107 K19 355, 699 645

C. FAINSTEIN D . FAINSTEIN-PEDRAZA R . G . FEDCHENKO N . F . FEDOROV V . D . FEDOTOV T . G . FINSTAD H . - J . FITTING Z . FRAIT J . FRANKE

417 723 K147 K83 209 223 K47 669 K U

R . T S . GABRIELYAN H . L E GALL L U . GANGHUA H . - D . GEILER K . P . GHATAK L . J . GILING

345 247 705 K203 K97 595

760 M . D . GLINCHUK J . GODLEWSKI P . GONDI J . GONG V . S . GORNAKOV G . GÖTZ V . G . GRIBULYA K . A . GSCHNEIDNER, J E SHEN. G U A N G J U N P . GÜNTER R . K . GUPTA E . Y U . GUTMANAS M . GUTOWSKI

Author Index K43 K199 143, 6 4 5 445 K63 K203 K67 329 705 KILL 313 193 103

P . HAASEN H . HAEFNEE P . HANIC J . HABT WIG W . HÄSSLEE E . HEGENBARTH C. H E I D E N H . W . HELBERG J . HENKE J . L . HEBBEBA P S . VAN H E U S D E N W . HINZ C.HOLSTE G . HÖLZEB J . HOKÄK J . HOBVÄTH F . H . HSU J . L . HUTCHISON

193 495 79 511 K175 K175 K83 203 487 K35 137 K217 213 511 407 687 31 K3

V . A . ISTJPOV L . V . IVAEVA K . IWASAKI

501 K127 K137

J. JÄÄSKELÄINEN

K . JAGANNADHAM U . JAHN S . C. J A I N U . A . JAYASOORIYA J . P . JOG I . D . JOSHI Y U . P . KABANOV J . KACZEB M . G . KALITZOVA W. KÄPPLEB G. J . KEABLEY M. G. KEKUA S. F . A. KETTLE H . E . KHODENKOV V . I . KHEUPA S . V . KHUDYAKOV H . - H . KIRCHNEB

241

299 K203 163 169 K103 K23 K63 K87 743 475 169 23 169 461 439 431 481

K . KIBOV K . I . KIBOV N . N . KISELEVA E . N . KISLOVSKII V . G . KOHN G . J . VAN DEE K O L K A . I . KOLYUBAKIN A . S . KOMALOV R . V . KONAKOVA S . KOÑCZAK P . KOBDOS E . G . KOSTSOV R . KOZUBSKI L . KBAUS A. KBELL J . KBEMPASKY V . D . KBEVCHIK N . KROÓ Z . KBUCZYÑSKI A . KBUMINS M. KBYSTEK V . S . KSHNYAKIN D . KUCHABCZYK P. L. KUKK A . I . KUPCHISHIN A . I . KUZNETSOV H. M. F. H. L. H. L. P. V. V. E. P. P. E.

K . LACHOWICZ J . J . LAMMERS LANGE J . LAUTER H. N. LEE LEMKE A . LITVINOV LOBOTKA M . LOMAKO A . LOMONOV LÓPEZ-CBUZ LOSTÁK LUKÁ(f>y3H0HHMM OT>KIirOM (1200 °C HJIH KpeMHHH H 870 °C HJIH TepMaHHH), JIh60 npH BblTHrHBaHHH KpHCTajuioB H3 pacnjiaBa. Ilpn KOHueHTpauHH 0Ji0Ba MeHee 2 x 10 1 9 c m - 3 OJIOBO BXOHHT B pemeTKy MaTpnq Kan npimecb 3aMeiueHHH c o6pa30BaHHeM sp3 riigpHHHOii CHCTeMbi XHMHiecKHX CBH3eii. PaccMaTpHBaeTCH Moaejib, onHCbiBaroman noBeneHne npHMecHbix aTOMOB ojioBa Kan H30T0nHHecK0ii npHMecH. IIpn KOHueHTpauHHx, npeBbimaromnx yKa3aHHyio uejiMMHHy, B cTpymype jiernpoBaHHbix nojiynpoBOHHHKOB o6pa3yi0TCH accounaTH npHMecHbix aTOMOB ojioBa.

1. Introduction In recent years much attention has been paid to the investigations of the state of tin impurity atoms in silicon and germanium by means of Mossbauer spectroscopy [1 to 6], The interest is supported by the idea that tin must substitute for the matrix atoms and utterly pronounced isotopic effects must be observed. Moreover, silicon and germanium doped with tin, may serve as model objects for a study of point defects (radiation defects in particular) by Mossbauer spectroscopy. The first attempts to introduce tin in germanium by fast cooling of melts, containing 1 to 75% tin resulted in an emergence of a Mossbauer spectrum of 119 Sn substitutional atoms on the background of an intense spectrum of metallic tin. This, of course, embarassed the evaluation of parameters of the Mossbauer spectra and led to contradictory results [1 to 4], The present paper is devoted to studies by Mossbauer spectroscopy of the state of 119 Sn impurity atoms in silicon, germanium, and in silicon-germanium solid solutions as well, the method of sample doping allowed to control tin impurity atoms precipitated as a phase of (3-Sn [5, 6]. J 2

) Pavlova 15, 380060 Tblissi, USSR. ) Politekhnicheskaya 26, 194021 Leningrad, USSR.

24

S. R. Bakhchieva, M. G. Kekua, and P. P. Sekeght

2. Experimental Method Tin was introduced into the samples by two methods — by diffusion doping and in the course of crystal pulling from a melt. For diffusion doping chemically etched monocrystalline silicon samples (undoped p = 2 X 10 13 c m - 3 , phosphorus-doped n = = 1.5 X 10 20 c m - 3 , and boron-doped p = 1 X 10 20 cm - 3 ) and germanium samples (undoped n = 3.4 X 10 13 cm - 3 , arsenic-doped n = 9 X 10 19 cm - 3 , and gallium-doped p = 6.5 X 10 19 c m - 3 ) were used. Oxygen concentration in the samples was 10 15 c m - 3 . The samples were both dislocation free and with dislocation density as high as 107 c m - 2 . The diffusion was carried out from a deposited metallic tin layer ( 118 Sn, 119 Sn, and 119m Sn) in evacuated quartz ampoules at 1200 °C (for silicon) and 870 °C (for germanium) during 4 to 10 h. After the diffusion annealing the samples were rinsed in hot HC1 + H F mixture to remove the residual tin from the surface. The tin concentration profiles in silicon and germanium were measured by solving thin layers in an etchant composed of H F and HN0 3 , with the following measurement of the residual activity. The samples of silicon, germanium, and silicon-germanium solid solutions tin ( 118 Sn and 119 Sn) doped in the course of pulling the crystals from large melt volumes, were polycrystalline. Tin contents varied from 6 X 1018 to 6 X 10 20 c m - 3 . In the study of after-effects of the 118 Sn (n, y) 119 Sn reaction in silicon and germanium, Si and Ge samples tin U 8 Sn doped were irradiated in water-cooled reactor channels with thermal neutron flux as high as 2 x 10 20 neutrons c m - 2 . I n order to consider the effect of the fast neutrons and of the reactor y-ray background on the state of tin impurity atoms in silicon and germanium, reference Si and Ge samples, 119 Sn and ii9mgn ¿oped, were irradiated simultaneously. Tin concentration in the samples was « 10 19 cm" 3 . The Mossbauer spectra were recorded with an electrodynamic-type spectrometer over the temperature range 80 to 1000 K . Mg 2 Sn was used as standard source and absorber. The spectra were treated using a standard program with BESM-4 computer. The isomer shifts are given relative to Mg 2 Sn. 3. Experimental Results 3.1 Samples

doped

by means

of

diffusion

The tin concentration profile in silicon and germanium is well described by an erfc function and no near-surface part was observed, such as had been noted in [7]. The diffusion data are the following: diffusion coefficient D = 2 x 10~13 cm2 s _ 1 , surface concentration N0 = 7 X 1019 cm" 3 for silicon; D = 7.8 X 10-13 cm 2 s -1 , N0 = = 2 X 1020 c m - 3 for germanium. Table 1 Parameters of the Mossbauer spectra of 1 1 9 Sn in silicon and germanium (diffusion doping) matrix

type of spectrum

silicon

absorption emission absorption emission

germanium

errors

r

«5 (mm/s)

(mm/s)

-0.062 +0.062 -0.060 +0.060

0.85 0.87 0.85 0.87

±0.015

±0.02

Tin Impurity Atoms in Si, Ge, and Si-Ge Solid Solutions

I

25

0 -7 7 — i/lirai/s)

-2

vlmm/s) Fig. 2

Fig. 1

Fjg. 1. Absorption (a), (c), (d), (e) and emission (b), (f), (g) Mossbauer spectra of 119 Sn impurity atoms in silicon. The spectra were recorded at 80 K. (a), (b), (f), (g) diffusion doped samples (maximum tin concentration 8 X 1019 cm - 3 ); (c), (d), (e) samples doped in the process of pulling from the melt (tin concentration 1 X 1019 (c), 2 X 1019 (d), and 5 X 1019 cm"3 (e); (a), (c), (d), (e) samples doped with 119 Sn; (b), (f) samples doped with 1 1 9 m Sn; (g) sample doped with 118 Sn; (f), (g) samples exposed to a flux of thermal neutrons 2 x 1020 neutrons c m - 2 Fig. 2. Absorption Mossbauer spectra of 119 Sn impurity atoms in germanium at 80 K. (a) diffusion doped sample (maximum tin concentration 2.0 x 1020 cm - 3 ); (b), (c), (d), (e) samples doped in the process of pulling from the melt (tin concentrations 1 X 1019 (b), 2.0 X 1019 (c), 2.5 X 1019 (d), 2.0 X 1020 cm' 3 (e))

The typical 119Sn Mossbauer spectra (emission and absorption ones) in silicon and germanium are shown in Fig. 1 and 2; in all cases the spectra consist of single bands with isomer shifts similar to those of the grey tin Mossbauer spectrum (see Table 1). The spectrum parameters in practice do not depend on the conductivity type or on the carrier concentration of the starting samples and or on dislocation density. This type of spectra will be noted as type I spectra. 3.2 Samples

prepared

by pulling 119

the crystal

from

a

melt

The pattern of Mossbauer spectra of Sn impurity atoms in silicon, germanium, and silicon-germanium solid solutions depends on tin concentration. For tin concentrations lower than 2 x 1019 c m - 3 the spectra are single bands with parameters close to those of type I spectra. For tin concentrations exceeding this value the Mossbauer spectra represent a superposition of two poorly resolved bands (see Fig. 1, 2), their relative intensity being a function of tin concentration. The spectra were treated assuming a superposition of a symmetric quadrupole doublet (type II spectrum) and a single band, located around either the left or right component of the quadrupole doublet (see Fig. 1,2). The type II spectrum parameters are brought together in Table 2. In the samples, heavily doped with tin, higher than 5 X 1020 cm - 3 , in which a microstructure analysis has revealed a precipitation (1 to 2%), the spectra consisted of single bands, corresponding to metallic tin. The 119 Sn spectrum for silicon-germanium solid solutions was similar to that of 119Sn spectra for silicon and germanium.

26

S. R . B a k h c h i e v a , M. G. K e k u a , and P . P . S e r e g i n

Table 2 Parameters of the Mossbauer spectra of 119 Sn in silicon, germanium, and silicon-germanium solid solutions (doping during pulling from the melt) matrix

tin concentration (10« cm" 3 )

spectrum, corresponding to isolated atoms 6 (mm/s)

silicon

< > germanium < > solid solution*)


= 4 ( c i - P12) > S3 = p33 — pzl,

The calculated free energy of the Heusler alloys was analysed in the previous papers [5 to 9]. I t was indicated that two succeeding order-disorder transitions are possible in such alloys: 1. The transition from the state of complete disorder of atoms (structure of the type A2) to the state of partial order where S2 =|= 0, $ 4 =)= 0, = 0, Sa = 0 (structure of the type B2). This transition occurs at temperature Tv 2. At temperature T2 Tx there occurs the second transition leading from the structure of the type B2 to the state of higher order of atoms where all long-range order parameters have non-zero values (structure of the type L2X). There is full accordance between the above expectations and the experimental results [11 to 15]. I t may be also indicated that if the interaction energies of atoms fulfil some special requirements only one order-disorder transition may occur. I t

Order-Disorder Phase Transitions in Ternary and Heusler Alloys Table 1 The co-ordination numbers Z^q^) and in the structure of the type ~L2l Zp,

Qi

02

Z11 Z12

0 8 0 4 0 4 0 8 0

0 0 6 0 6 0 6 0 0

Z

13

Z21 z22 Z23

Z31 Z

3i

Z33

39

Zflr(Q2)

leads straight from the structure of the type A2 to the structure of the type L2 X [6 to 8]. However, up to now, there have been no experiments recording such a phenomenon. The temperatures T 1 and T 2 are given by the following formulae: T

=

*

T ^ -

3Tf 13 (e 2 ) -

Ax = {2[4TF 1 2 ( e i ) +

±WLS(QL) -

X [4TF 23 ( ei ) T

*

=

4 T F

+

13(A) + SW23(QL) -

6Tf 23 (e 2 ) + 3W13(E2)}* -

+ (1 + S2I>) (2 -

3FM(ft)] +

32 [ 4 W 1 2 ( E I ) -

3TF23(e2)] + M [ W

(1 +

+

-

,

3TT M ( f t )] + 2[4TF23({?1) -

3[Tf 12 (g 2 ) + W23(Q2) -

m{(1

1 MHz) and high density O 107 bits/cm2) current-access bubble circuits has been introduced by Bobeck in two quite comprehensive papers [1, 2]. Among problems concerned with this new type of bubble circuits the problem of the analytical description of the driving fields and dynamic behaviour of bubbles in magnetostatic traps in the current-access circuits seems to be most important for the computer simulation of the operating margins of bubble devices. It is well known that a rigorous determination of the operating margins for permalloy bubble circuits implies that the "bubble-permalloy" coupled system should be taken into account. In contrast to the unsaturated permalloy patterns, the current-access bubble circuits reveal a negligible interfluence of the bubble stray fields and the magnetic fields generated by the conductor patterns. This means that the magnetostatic traps of the current-access circuits are "rigid" as compared to the "soft" traps of the permalloy circuits. Thus, the problem of the analytical description of the bubble behaviour in the current-access circuits disintegrates into two independent problems, the first being the analysis and calculation of the conductor-pattern driving fields and the second being the solution of the dynamic bubble equations using these fields along with the bubble effective propagating and radial fields. In the first part of the paper the problem of the quasi-stationary current distribution over an infinite conducting sheet perforated with holes of mostly circular aperture is considered. An alternative approach to the problem of the conductorpattern magnetic field calculations based on the Stokes' theorem is presented. The analytical expressions for the round hole magnetic fields are given in closed form. In the second part of the paper the dynamic bubble behaviour in the conductorpattern circuits is described and the general equations of the bubble propagation are J)

Vavilova 24, Moscow 117812, USSR.

56

E . P . LYASHENKO a n d V . K . RAEV

presented. I n conclusion the HF-screening and the crossing skin effect arising at high operating frequencies are also discussed. 2. The Vector Potential and the Magnetic Fields of the Distributed Currents 2.1 Quasi-stationary

current

distribution

The quasi-stationary current distribution over t h e infinite conductor overlay with a hole (Fig. 1) is t h e well-known two-dimensional Laplace problem (1)

V*Y=0,

for t h e E field sources f a r away (div i = 0). One defines t h e current density and direction at infinity lim ix = i0,

x,y—*oo

lim iy = 0 .

(2)

x,y—*

Then the solution of the Laplace equation (1) with t h e boundary condition (2) determines t h e potential W and t h e current vector i(x, y) in t h e 8 region in a unique way. This solution for t h e 8 region bounded by t h e r contour (Fig. 1) is given by t h e conformal mapping method of t h e theory of complex variables [3, 4]. The point is t o find a n analytical function W = U + jV, which transforms t h e S region of t h e z-plane into t h e simple S' region of the TF-plane, where t h e solution of (1) is well known. Then t h e complex current-vector components can be found from . _ dU 1 8F ~ 0r ~ 7 '

. _ ** ~

lr

8F_ 1 W 0r ~ r

fy'

{ )

If t h e r contour of t h e 8 region is a circle with radius a, t h e corresponding analytical function W(z) = U + jV is determined b y t h e known Zhuckovsky function [4, 5] W(z) = C(z + a 2 / 2 ) •

(4)

The factor C is obtained from t h e boundary condition (2), i.e. C — i0. The polar components of t h e current density i are as follows: i, = i0 cos = arctan (x/y), and describes t h e necessary quasi-stationary current distribution for t h e circular hole. The problem of t h e current distribution at t h e r contour is fully identical to t h e one for the laminar stream of a liquid around /"-contoured body without circulation. I n t h e considered case t h e complex potential function (4) evidently is t h e superposition : W = + Wt of t h e potential of uniform current W0 ~ z and t h e one of the electric dipole Wx ~ 1/z, vanishing in the infinity. This assertion is available for t h e / 1 contour of any (noncircular) configuration as well, t h e current distribution around r P(R,e,Z)

Fig. 1. The principal geometry of the distributed current problem under consideration. S is an infinite conductor sheet with a hole, bounded by the r contour

57

On the Analysis of Current-Access Bubble Circuits

which has to be the superposition of the uniform distribution (as z) and some perturbation that can be represented by the space arrangement of dipoles («5 ±1/2) and multipoles ( « + l/z"). By the choice of the different combinations of the simple dipole and multipole functions one can pick out the conformal mapping for any practically interesting form of the J1 contour.

2.2 Calculation of the magnetic fields of the distributed

current

The calculation of the magnetic fields of the distributed current i(Q) are based on the solution of Poisson's equation for the vector potential A [6],

where P is a considered point, Q is a variable point of integration, and RPQ = \rP — r Q (. The magnetic field H is obtained by H = V X A , H(P)

= Vp x ^ I dt> . 471 I Rpq

(7)

For an analytical calculation of the field H, created by the distributed current i(x, y), a direct usage of Biot-Savart's relation is often too complicated even for the case of linear currents. For the surface current the calculation by (7) is even more complicated. According to the plane geometry of the problem under consideration the magnetic field Hz (Fig. 1) can be calculated by a much less tedious approach using Stokes' theorem [11]. Namely, neglecting the thickness of the conducting overlay we rewrite (7) in the form " ^ ^ ¿ f j g l d S ,

(8)

where i(Q) is a two-dimensional current density now. Let us now multiply in a scalar way both sides of the vector relation (8) by the unit vector n along the S surface normal (along the z-axis): n-H(P)

= HZ{P) = ¿ 1 ^

X ^ ) ^

(9)

s Applying Stokes's theorem on the flow of curl of the i(Q) RPQ vector through the S surface, we get

where is(Q) is the current component tangent to the F contour, and the Q point of the contour integration relates to the r contour now. Applying relation (10) instead of (6), (7) simplifies the problem in a giant way since it reduces the double integration over the S surface to the single contour integration along the r contour, bounding the S surface. To illustrate this approach we shall find the Hz field expression for the circular hole.

58

E . P . LYASHENKO a n d V . K .

RAEV

Evidently, the current vector i is always tangent to the r contour, as it also follows from (5) i {r = a) = 0 , i = i (r = a) = — 2i sin cp , (11) and r

t

Rpq{r

=

a)

=

(/a2 +

v

R2

0

+

a

2

-

2Ra

c o s (

and also the anisotropy parameter p were treated as the fitting parameters.

70 72 74 76 78 80 82

70 72 74 76 78 80 82 84

Co concentration (ai%)—

Fig. 1

Co concentration iat%}—-

Fig. 2

Fig. 1. Experimental points and the best fitted curve for the composition dependence of the perpendicular anisotropy K a Fig. 2. Experimental points and the best fitted curve for the composition dependence of the linear magnetostriction (Ai/ijg^x

K. Twabowski, H. K. Laohowicz, M. Gutowski, and H. Szymczak

106

The values of the remaining parameters, zi}, y \

X

Fig. 1. A disclination of Frank vector Si = tat + cu2 + to3 near a free surface S

111

Straight Disclinations near a Free Surface (I)

Here D = Oj2jt(\ — v), x± = x + d, r\ = xf + V2> @ is the shear modulus, v the Poisson ratio. If S is a boundary of a half-space, the following boundary condition must hold: o"xj = 0 ,

(6)

where atj are the stress fields of the considered disclinations in the half-space. The Greek letter a — 1, 2, 3 is used for designation of corresponding disclinations. The general way of solving such problems consists in defining stress tensors a which summed up with gxy — zy(%I — y2)/4, g% = (1 - 2v) In r2 + y2/rI ,

gil = 2vzx2jr\ , = —x2y\r\, Oij = Dœ2gli- , „»2 gxx = -zy(y2 + Zxl)/4 ¿2 9zz = —2vzyjr\ ,

gfy = zy{x\

gyz = —(1 — 2v) In r2 — xljr\ ,

gtx = x2yjrl ,

o§ = Dm3gI)iz f gil = - In r, - f\r\ gfz = -

-

2v In r2 - v/( 1 -

- y2)l4,

g% = zx2(x 1 --

v\{\ 2v) ,

2v) ,

(9)

(10)

y2)l4

g% = - In r2 - xijrl

- v/(l -

2v) ,

gfz = x2y/4 , (H)

where x2 = x — d, r\ = x\ -)- y2. Thus for wedge disclinations ct>3 and twist disclinations oj1 the images are disclinations of opposite sign at the point (d, 0) (Fig. 2 a, c), but for twist disclinations a>2 we choose the image as the same disclination at the point (d, 0) (Fig. 2 b).

112

A . E . ROMANOV a n d V . I . VLADIMIROV

• •

V

V

•A

H

• • •

H

I'

d-

A •

Fig. 2. The image disclinations and surface distributions of twist disclinations and edge dislocations chosen to satisfy the free surface boundary conditions; a) for twist disclination with aix; b) for twist disclination with o>2; c) for wedge disclination with to3 3. The Distribution of Surface Disclinations and Dislocations According to (2) to (4) and (9) to (11) the following forces normal to the free surface are acting on t h e surface S: el _r_ Â1 = - 2Do\zd(d2 - y2)l(d2 + y^ , (12) ™ZZ I ' JZ e2 , i2 = - 2Dco2zy(y2 + 3d2)j(d2 + if f , (13) ° XX ~T "xx 2 (14) o'4 + Oxy = —2Dco3dyl(d + if The remaining forces which are normal to t h e surface are compensated by image disclination fields. We have to add the new tensors a^f to the fields cr™ and off for a complete solution of the problem. These tensors compensate the residual forces (12) to (14), b u t do not influence the other boundary conditions. We shall not define stress functions, b u t we shall introduce t h e distributions of surface disclinations and dislocations for determining a F o r twist disclinations they will be arrays of twist disclinations having F r a n k vector at = u>ey (Fig. 2 a, b). For a wedge disclination the surface defects will be an array of edge dislocations with b = bev (Fig. 2 c). The surface arrays of chosen twist disclinations have the component Oxx on the surface, b u t do not change t h e other forces normal to the free surface. I n its t u r n t h e array of surface dislocations eliminates a% + a%, having no effect on the remaining components. The stress tensors of such disclinations and dislocations situated on the surface S in the point p can be written in the form [22] erf.A = DœgP.à(x, y, z, p) , 9pxè = - zvv(vl + gpj = _ 2vzyplr2p ,

,

gPj = zx(x2 -

y*p)l4 ,

(15)

5>PJ = xVv!rl. 9PJ = - ( ! - 2") ln rv ~ *2l4 , or,L DbgP.-i-(x, y, z, p) v (16) gp± = x(x* + 3y2p)l4 , gp± = x(x* - y%)/rJ , Sff± = 2vx/ff , 2 where yp = y — p, 20 K (Fig. 5). These are long values for the molybdate group. The temperature dependence of r could only be fitted to a three-level system in an approximative way. The real situation is probably even more complicated. An approximate fit yields e = 5 cm - 1 , p21 = 1.5 m s _1 , and pal = 1.0 m s _ 1 . Let us now turn to the doped samples. The composition Na 2 Wi.9Moo.i0 7 shows the same luminescence characteristics as Na 2 W 2 0 7 if excitation occurs below 280 nm (i.e. in the tungstate groups). The emission band is somewhat broader on the long-wavelength side indicating (inefficient) energy transfer from the tungstate lattice to the molybdate groups. For excitation wavelengths longer than 320 nm the luminescence characteristics are the same as inNa 2 Mo 2 0 7 .The molybdate emission of Na 2 Wi.gMoo.i0 7 shows the same temperature dependence of its decay time as Na 2 Mo 2 0 7 . The luminescence characteristics of "Na2Mi.98Uo.o207" are rather complicated. Using data obtained on Na 2 U 2 0 7 it was possible to describe these by assuming a two-phase system, viz. Na 2 W 2 0 7 (with practically no uranium) and a uranate compound the composition of which could not be determined. This means that the solubility of Na 2 U 2 0 7 in Na 2 W 2 0 7 is very low. Radiative energy transfer occurs from the tungstate phase to the uranate phase. The latter emits green at helium temperature, but red at higher temperature. This is due to rapid energy migration feeding traps [11]. As a consequence Kroger's data on Na 2 W 2 0 7 -U [5] are not reliable. The spinel Na 2 W0 4 shows weak luminescence below 120 K. The maximum of the emission band was situated at 480 nm, the maximum of the corresponding excitation band at about 250 nm. In view of the low efficiency of the luminescence we abandoned further measurements. 4. Discussion The present results show that an interpretation in terms of tetrahedral and octahedral groups will not be very useful. The undoped samples show only one emission and excitation band. Energy transfer at low temperatures between the crystallographically different groups does not seem very probable [12]. Further the molybdate emission from Na 2 Mo 2 0 7 and Na 2 Wi.9Moo.i0 7 is the same. Since the octahedral species usually

160

M. J. J. Lammebs and G. Blasse

have energy levels at lower energy [5, 13], emission from Na 2 Mo 2 0 7 should come from the octahedral site. In Na 2 Wi. 9 Mo 0 .i0 7 , however, molybdenum will be preferentially in tetrahedral sites. We have to conclude that the luminescence of the undoped compounds originates from a site which cannot be characterized as typically tetrahedral or octahedral. This is not too surprising in view of the large variation in distances ( N a 2 W 2 0 7 : W ( l ) - 0 (octahedral) 1.725Á ( 2 x ) , 1.921 Á ( 2 x ) , 2.252 Á ( 2 x ) and W ( 2 ) - 0 (tetrahedral) 1.759 A, 1.762 Á, 1.807 A ( 2 x ) ) [2], Since energy migration seems to be of no importance in these compounds (see [12] and the results for Na2Wi.9Moo.i07), we prefer to consider the [ W 2 0 7 ] 2 - chain as the luminescent centre, without specifying whether the emission originates from the W(l) or the W(2) site. In this connection it may be questioned whether the assignment of the vibrational spectra in terms of W 0 4 and W 0 6 vibrational modes [4] is to be preferred to an assignment in terms of W - 0 and W - O - W vibrational modes [3], although the latter authors start from an incorrect crystal structure. The decay measurements reveal a remarkable difference between the luminescence of Na 2 W 2 0 7 and Na 2 Mo 2 0 7 . In the case of the tungstate the decay time decreases with increasing temperature, whereas in that of the molybdate the decay time increases with increasing temperature. For isomorphous tungstates and molybdates this has never been observed, as far as we are aware. I t means that the splitting of the emitting levels in the tungstate and the molybdate must be different. This phenomenon has been discussed before [7, 9, 14], In the case of isolated tetrahedra or octahedra the emitting levels have designation 3 T [9], i.e. are spin as well as orbit degenerate. Splitting of this level has been observed. Por the lighter elements it amounts to a few wave numbers and is due to the crystal field lower than cubic. For the heavier elements it is somewhat larger and has been ascribed to spin-orbit interaction. The different behaviour of t(T) indicates that the splitting in the case of Na 2 Mo 2 0 7 and Na 2 W 2 0 7 is due to different origins. Although the exact nature of the emitting level is not known, we propose that the splitting in the case of the molybdate is due to the crystal field. Its value (5 cm - 1 ) is in good agreement with values observed for other molybdates. In view of the site symmetry of the Mo 6+ ions in Na 2 Mo 2 0 7 (C2 and Cs) the 3 T level will split into three sublevels. This explains why the three-level scheme yields only approximative agreement. In Na 2 Mo 2 0 7 the lower sublevel has the higher transition probability in contrast with the situation in CaMo0 4 [7], In view of the isomorphy between Na 2 Mo 2 0 7 and Na 2 W 2 0 7 the different behaviour of the tungstate must have a different origin, viz. splitting of the 3 T level by spin-orbit coupling as proposed before for other tungstates in view of the large value of the spinorbit constant of tungsten [9], I t seems very improbable that the crystal field in Na 2 Mo 2 0 7 would be greatly different from that in Na 2 W 2 0 7 . In Na 2 W 2 0 7 the lower sublevel has now the lower transition probability. Also the splitting is much larger. Further discussion of these effects does not seem possible as long as the exact nature of the emitting state is not more precisely known. The same is true for a striking similarity between both compounds, viz. the value of the quenching temperature of the luminescence of Na 2 Mo 2 0 7 and Na 2 W 2 0 7 . This lies in both compounds around room temperature, the molybdate value being only slightly lower than that of the tungstate (see also [5]). Usually molybdates quench at considerably lower temperatures than tungstates [5]. Note, finally, that Na 2 W0 4 (spinel structure with isolated W 0 4 tetrahedra) does luminesce. I t has been argued elsewhere that a requirement for efficient tungstate luminescence is a coordination around the tungstate group consisting of small, highly charged cations which counteract the expansion of the W - 0 bonds upon excitation [15, 16]. This is clearly not fulfilled by the Na + ion. As a consequence the W - 0 bonds

161

Luminescence of Sodium Molybdate and Sodium Tungstate

expand relatively strongly resulting in inefficient luminescence down to very low temperatures. This explanation is corroborated by the relatively large Stokes shift, viz. some 20000 c m - 1 (compare, for example, C a W 0 4 with 16000 c m - 1 [5]). References [1] I. LINDQTTIST, Acta chem. Scand. 4, 1066 (1950). [2] K . OKADA, H . MORIKAWA, F . MABUMO, a n d S. IWAI, A c t a c r y s t . B 3 1 , 1 2 0 0 ( 1 9 7 5 ) . [3] T . DUPITTS a n d M . VILTANGE, M i k r o c h i m . A c t a ( W i e n ) 2, 2 3 2 ( 1 9 6 3 ) . [4] F . K N E E a n d R . A . COKDBATE, SR., J . P h y s . C h e m . S o l i d s 4 0 , 1 1 4 5 ( 1 9 7 9 ) .

[5] F. A. KRÖGER, Some Aspects of the Luminescence of Solids, Elsevier Publ. Co., Amsterdam 1948. [6] R. D. SHANNON and C. T. PREWITT, Acta cryst. B26, 1076 (1970). [7] J . A . GROENINK, C. HAT?FOOT, a n d G. BLASSE, p h y s . s t a t . sol. (a) 5 4 , 3 2 9 ( 1 9 7 9 ) . [8] W . VAN LOO a n d D . J . WOI/TERINK, P h y s . L e t t e r s A 4 7 , 8 3 (1974). [9] G. BLASSE, S t r u c t u r e a n d B o n d i n g 4 2 , 1 (1980).

[10] W. VAN Loo, J . Lum. 10, 221 (1975). [11] D . M . KROL a n d G . BLASSE, J . c h e m . P h y s . 6 9 , 3 1 2 4 (1978).

D. M. KROL, Thesis, University of Utrecht, 1980. [12] R . C. POWELL a n d G. BLASSE, S t r u c t u r e a n d B o n d i n g 4 2 , 4 3 (1980).

[13] G. BLASSE and A. BRIL, Z. phys. Chem. (Frankfurt/M.) 57, 187 (1968). [14] H . RONDE a n d G. BLASSE, J . i n o r g . n u c l e a r C h e m . 4 0 , 2 1 5 ( 1 9 7 8 ) .

[15] G. BLASSE, J. chem. Phys. 51, 3529 (1969). [ 1 6 ] G. BLASSE a n d G. J . DIRKSEN, J . i n o r g . n u c l e a r C h e m . 4 3 , 1 8 4 (1981). (Received

11

physica (a) 03/1

August

11,

1980)

S. K.

TRIKHA

and S. C.

JAIN:

Computer Simulation of the Phase Transition

163

phys. stat. sol. (a) 68, 163 (1981) Subject classification: 2 Department of Physics and Astrophysics Hindu College, Delhi (bJ1)

University

of Delhi (a) and

Computer Simulation of the Phase Transition in Ice By S. K . TRIKHA (a) a n d S. C. J A I N

(b)

A computer simulation study of ice at low temperatures is made using the Lennard- Jones potential as the representative interaction between ice molecules. The open cubic structure of the ice molecule is considered. This study reveals the occurrence of a phase transition in ice around 100 K. Mit dem Lennard-Jones-Potential als die wesentlichste Wechselwirkung zwischen den Eismolekülen wird eine Computersimulationsuntersuchung für Eismoleküle bei tiefen Temperaturen durchgeführt. Die offene kubische Struktur des Eismoleküls wird berücksichtigt. Die Untersuchung zeigt das Auftreten eines Phasenübergangs in Eis bei etwa 100 K.

1. Introduction I n recent years computer simulation studies [1 to 4] of the motion of atoms and molecules have become a powerful technique in understanding the various physical properties of condensed matter at low temperature. Earlier we have reported on the phase transition in ammonium chloride using this approach. The purpose of the present communication is to predict the occurrence of a phase transition in ice at low temperature using the method of computer simulation. Giauque and Stout [5] carried out specific heat measurements in ice in the temperature range 15 to 273 K and the results showed an anomaly around 100 K . Later on Pick [6] also obtained a similar transition in ice in the temperature range 100 to 120 K. The experimental studies of the dielectric behaviour and dynamic stiffness constants by Dengel et al. [7] and Helmreich [8], respectively, indicated also the occurrence of an anomalous behaviour in ice below 150 K . 2. Crystal Structure of Ice Barnes [9] used X-ray methods to study single crystals of ice and showed t h a t the oxygen atoms are arranged in a tetrahedral pattern with an 0 - 0 distance of 0.276 nm. Later on the position of hydrogen atoms in the ice structure was located with the help of neutron diffraction studies by Peterson and Levy [10]. The nearest O - H distance is given as 0.096 nm. I n the present calculations we have considered the open structure of ice as shown in Fig. 1. The side of the unit cell is taken as 0.3175 nm [11]. An oxygen atom having two hydrogen atoms as its satellites (at a distance of 0.096 nm) is considered at the centre of a body-centred lattice having four oxygen atoms placed at its vertices exhibiting a tetrahedral coordination. The moment of inertia of the ice molecule is 3.084 X 10~47 kg m 2 . I t belongs to space group (I 4X md). Delhi 110007, India. Ii»

S. K. Trikha and S. C. Jain

164 3. Interacting Potential

In order to compute the potential energy of the dynamical system we have preferred to use the well-known Lennard-Jones (6-12) potential [12], (1)

between the central ice molecule with other oxygen atoms placed at the vertices. a represents the distance of closest approach ( = 0.2725 nm). I n our further study we intend to use a better form of potential which includes the interaction of the polar ice molecule. 4. Algorithm for the Rotational Motion of Molecules Let l^t), l2(t), and l3(t) be the directions of the principal axes of the molecule at time t and /¡¡, and I3 the principal moments of inertia. The angular velocity W and the angular momentum Q of the molecule is given as [13] W(t) = Q(t)

Wl(i)

= Il(Dx(t)

IS)

(2)

+ w2(i) l2(t) + co3(i) l3(t) ,

h(t) + l2(02(t) l2(t) + Iscos(t) ls(t) .

(3)

«!(£), co2(t), and co3(t) are the components of the angular velocity W(t). The equations of motion to be solved are 1.(0 = W(t) fi(0

(4)

X lx(t) ,

(5)

= T(i) ,

where r is the torque exerted on each molecule by its environment, a ( = 1, 2, 3) refer to the direction of the principal axes. The computer code consists of mainly three parts ; a routine to set up the initial positions and orientation of molecules from the crystal structure, a routine to advance the time step with the help of an algorithm, and the routine to calculate the forces and torque exerted on each molecule by its neighbouring molecules. I n the present case we use an algorithm which is very similar to the conventional molecular dynamics study in liquids [2], The position r(t), velocity v(t), and acceleration a(t) of a particle in the algorithm are replaced by the parameters IJt), lx(t), and lx(t). The corresponding equations for the rotational motion of molecules are written as la(t + M) = la(t) + lx(t) At + \ [Ux(t) Q(t + At) = Q(t)

-

la(t -

+ | [2t(t + A i ) + 5 r ( t ) -

Ai)] Ai2,

r(t -

Ai)] Ai

(6)

(7)

where lx(t) and lx(t) are the first and second time derivatives of the direction cosine, respectively. The only complication in the present case is that we have the additional

-0.31?5nm

Fig. 1. Open cubic structure of ice. The large circles represent oxygen atoms; small solid circles refer to hydrogen atoms

Computer Simulation of the Phase Transition in Ice

165

constraint that la(t) are orthogonal unit vectors. Also the axes fixed in space are used to solve the rotational equations of motion of the molecules. Further one writes W(t) = I " 1 • Q(t),

(8)

where I denotes the inertia tensor and the components of W(t) are ^ ( o = Q(t) • m i i co2(t) = Q(t) • h(t)!h

1

, .

(9)

co3(i) = Q(t) • l3(t)II3 . Using the above equations, we get 4 ( 0 = «3(t) l2(t) 4 ( 0 = coS) IS)

co2{t) IS) ,

~ co3(t) WO .

i3(t) = o>.2(t) WO From (9) we get

(1°)

coS) WO •

¿>iM = {*(t) • h(t) + o)2(t) « 3 ( 0 [h ~ hDlh (b2(t) = {r(t) . l2{t) + co3(i) a>S) lh ~ hDlh

¿ 3 (0 = MO • l3(t) + "MO o>2(t)

, .

(11)

- /2]}//3 •

Differentiating equations (10) we obtain

IS) = -

{«1(0 + c o ! « } IS)

+

+ i t w ) • is) + cos) «2« tA - h + 4]} wo - I?{t(t) IS)

= -

. IS) - coS) coS) [A + h ~ 4 ] } WO .

\wl(t) + cof(0} I S )

(12 a)

+

+ I?{x{t) • IS) + coS) coS) Ih - h + A]} W0 -

J ? 1 M 0 • W 0 - coS) « i ( 0 i h + h ~ ^i]} W 0 ,

j3(o = - {tof(o + «1(0} w o + + I?{T(t) • IS) + COS) «l(0 [I3 - h + hi) W0 - nl{r(t) • IS) - COS) «2(0 lh + I i - 4]} W0 •

(12 b)

(12c)

The algorithm given above has been found to give both excellent stability and longtime energy conservation which are the necessary conditions for such a type of computation. I n the present analysis the angular momentum of the central ice molecule (Fig. 1) is gradually increased to study its rotational motion (libration) in the field of four neighbouring ice molecules. The rotational kinetic energy of the system is computed for different values of angular momenta. 5. Discussion and Results I n Fig. 2 the variation of the average total energy (E') is shown as a function of average rotational kinetic energy 3 ion substitution the effects are small and so cannot be unambiguously explained. However, the general trends, particularly on 16 N substitution at the nitrate ion, indicate the heteroionic coupling is involved in this region of the infrared spectrum but not to the same extent as in the corresponding region of the I I N S spectrum. 3.4

Mechanism

It is clear that the juxtaposition of the I I N S spectra alongside the infrared and Raman drastically modifies the interpretation of the vibrational spectra in the 450 to 1000 c m - 1 region. Whilst heteroionic vibrational coupling can be ignored, as a first approximation, for the infrared and Raman, it is imperative that it be included in order to explain the IINS. Although this much is clear, the details are more obscure. There is no doubt that the internal motions of the ammonium ion are not involved in the heteroionic coupling — they all lie at too great a frequency — so that, acoustic, translational, and librational motions are the only ones available and these external modes couple into the internal by a ¿-dependent multiphonon mechanism. I t seems highly probable that hydrogen bonding is important for the coupling because it is a hydrogen-bond network which links the ammonium and nitrate ions together. Certainly, it is clear that the proposed combination bands, discussed earlier, involving translational and librational modes of the ammonium ion, will contain a considerable motion of those hydrogen atoms which are strongly bonded to the oxygen atoms of the nitrate ion [5]. I t is perhaps pertinent to point out t h a t combination modes i>6 + v's and 2v'a have been proposed to explain bands at 515 and 640 c m - 1 in the I I N S spectrum of NH 4 Br [11]. Additionally, the symmetric hydrogen-bond stretch for the H ( N 0 3 ) i ion has been assigned at 516 c m - 1 in the I I N S spectrum of CsH(N0 3 ) 2 [12]. Clearly, ammonium nitrate could have modes related to those found in these compounds. Further, the frequency similarities in this region between these compounds and ammonium nitrate make coupling of the type we have observed entirely reasonable. 4. Conclusions In their vibrational studies of crystals, chemists have often centred their discussion on the relative applicability of site- and factor-group models. The former is, in principle, applicable throughout fc-space whilst the latter is valid only at k = 0. I n the extension of the factor-group model, away from the zone centre, the full space group has to be considered. Two questions at once arise: (i) Does dispersion occur ? and (ii) Does the mixing between different modes vary throughout fc-space ? To date, these questions have

Inelastic Neutron Scattering and Vibrational Spectral Data of Ammonium Nitrate

177

been explored in rather simple systems and the extension of the answers to complicated solids is rather tentative. Whilst it appears generally supposed t h a t in such crystals internal modes show rather little dispersion, the question of t h e ¿-dependence of interaction terms has been neglected. The results presented in t h e present paper suggest t h a t such a ¿-dependence can exist and is a subject meriting f u r t h e r investigation. Certainly, there seems to be no feature of t h e present system which might make the phenomena encountered in the I I N S spectra unique. One u n f o r t u n a t e aspect of the present work is the uncertainty currently associated with the crystal structure of phase V of ammonium nitrate. Whilst there is nothing in t h e results presented in t h e present paper which appears inconsistent with t h e reported structure, knowledge of the hydrogen a t o m positions might have provided a deeper insight into the origin of some of our results. Acknowledgements

We would like to t h a n k t h e Science Research Council, Fisons, Ltd., and t h e Royal Society for financial support. References [1] G . J . KEABLEY, S. F . A . KETTLE, a n d J . S. INGMAN, J . c h e m . P h y s . , i n t h e press. [2] U . A . JAYASOORIYA, G. J . KEABLEY, a n d S. F . A . KETTLE, J . C. S. C h e m . C o m m u n .

745

(1980). [3] J . L. AMOBOS, F . AREESE, a n d M. CANUT, Z. K r i s t . 1 1 7 , 9 2 (1962). [4] I . A . OXTON, O. KNOP, a n d M . FALK, J . p h y s . C h e m . 8 0 , 1 2 1 2 ( 1 9 7 6 ) . [5] G . J . KEABLEY, S. F . A . KETTLE, a n d I . A. OXTON, S p e c t r o c h i m . A c t a 3 6 A , 4 1 9 ( 1 9 8 0 ) .

[6] B. W. LUCAS, private communication. [7] H. C. TANG and B. H. TOBBIE, J. Phys. Chem. Solids 39, 845 (1978). [8] Z. IQBAL, Chem. Phys. Letters 40, 1, 41 (1976). [9] E . L . WAGNER a n d D . F . HORNIG, J . c h e m . P h y s . 1 8 , 2 9 6 (1950). [10] G . J . KEABLEY, S. F . A . KETTLE, a n d I . A . OXTON, S p e c t r o c h i m . A c t a 3 6 A , 5 0 7 ( 1 9 8 0 ) .

[11] K . MIKKE and A. KBOH, Chalk River Symp. Inelastic Scattering of Neutrons in Solids and Liquids, 1962, Vol. II, IAEA, Vienna 1963 (p. 273). [12] K . P . BBIEBLEY, J . HOWARD, C. J . LUDMAN, K . ROBSON, T . C. WADDINGTON, a n d J . TOMPKINSON, C h e m . P h y s . L e t t e r s 5 9 , 4 6 7 (1978). (Received

12

physica (a) 63/1

August

4,

1980)

OM. PRAKASH and G. SHINWASAN : Permeability Components and Anisotropy in Y I G

179

phys. stat. sol. (a) 63, 179 (1981) Subject classification: 18.3, 22.8.2 Physics Department, Indian Institute of Technology, Bombay1)

Initial Permeability Components and Anisotropy in YIG By OM. PRAKASH a n d G . SEINIVASAN From the study of variation of initial permeability, with temperature in polycrystalline YIG, the value of anisotropy constant (K) is estimated and compared with the single crystal value of first order anisotropy constant. Also the contributions to initial permeability due to domain wall displacement and domain rotation are isolated. Further it is observed that in case of large grain size sample the initial permeability nearly varies as K ^ 1 ! 2 for low temperature, and as K ' 1 for high temperature. Aus einer Untersuchung der Änderung der Anfangspermeabilität und der Temperatur in polykristallinem Y I G wird der Wert der Anisotropiekonstante (K) bestimmt und mit dem Einkristallwert der Anisotropiekonstanten erster Ordnung verglichen. Auch die Beiträge zur Anfangspermeabilität infolge von Domänenwandverschiebung und Domänendrehung werden separiert. Ferner wird beobachtet, daß im Fall von Proben mit großen Kornabmessungen die Anfangspermeabilität fast wie K-1/2 für tiefe Temperaturen und wie K'1 für hohe Temperaturen variiert.

1. Introduction The initial permeability, ¡xv and its variation with temperature in Y I G has been studied extensively. Contributions to lui arise both from the domain wall displacement and the domain rotation. Only a few attempts [1 to 3] have been made so far to estimate the two contributions from the experimental data on /x¡. In the present work we have isolated the two contributions to ^ by studying the variation of ¿«j with temperatures in polycrystalline yttrium iron garnets. We have also estimated the anisotropy constant in this system. 2. Experimental Polycrystalline samples of pure YIG were prepared by the standard ceramic techniques of normal sintering (NS) and hot pressing (HP). High sintered density (98.5% of X-ray density) and uniform microstructure were achieved. The average grain size of the NS and HP YIG samples was 12 and 3.5 ¡I,m, respectively. The details of the microstructure have been discussed elsewhere [3], The initial permeability ¡ii and its variation with temperature have been studied on toroidal samples at 110 kHz using a russian LC-meter (Model E I 2-1). 3. Results and Discussion The variation of real part of the initial permeability, fi{, in case of HP and NS YIG's is shown in Fig. 1. /ij for the NS sample is observed to be considerably higher than for the HP sample. The fall in ^ is extremely sharp at the Curie temperature, T c . J)

12•

Bombay 400076, India.

Om. Prakash and G. Srinivasan

180

Fig. 1. The variation of initial permeability, [i\, with temperature in normal sintered (A) and hot pressed (o) YIG samples

600

too

200

200

hOO T(Kh

600

The contributions to fa are from domain wall displacement and domain rotation. The initial susceptibility due to domain wall displacement, is given by [4] 3 DMl X&

=

(1)

16

m

where Y is a geometrical factor. Evidently, the fracture strength can be enhanced by increasing K\ c or lowering the flaw size. I n conventionally cold-pressed and sintered alumina specimens (as-sintered state) fracture surface topography reveals surface flaws with a0 as 50 [im and ac 150 fxm, corresponding to a usual bending strength of about 300 MPa. Without applying special technologies (e.g. extrusion, hot pressing) it is rather difficult — if not impossible — to reduce the initial flaw depth further in order to increase the fracture strength. Therefore, an enhancement of the fracture toughness by forming a suitable microstructure remains the only possible way. Claussen [2] found a significant increase in Ki c by adding unstabilized Zr0 2 to alumina. Cooling the structure after hot pressing the phase transformation of zirconia initiates high residual stresses which result in microcracking of the alumina matrix. Pompe et al. [3] have shown theoretically that such a mechanism really could be the cause of the observed increase in K j c . Whereas Claussen reported a slight drop of the fracture strength of hot pressed A1203 ceramics if Zr0 2 was added, Lange [4] observed a parallel improvement of at and K\t. in the same system up to Kic = 8 MPa m 1 ' 2 . Considering the homogenization difficulties for two-phase ceramic materials, the temperature dependence of theZr0 2 transformation (its temperature range corresponds to about t h a t reached in the working zone of a cutting tool), and the price of ZrO ? it seems interesting to look for a single-phase structure capable to reveal a similar !) Postfach 19, D D E - 8 0 2 7 Dresden, D D R .

184

A . KBELL

inicrocrack mechanism. As pointed out previously [5] only second-order residual stresses resulting from thermal expansion anisotropy can affect the fracture process of polycrystalline A1203 significantly: for the largest grains of a single-phase alumina structure the associated stored elastic energy is sufficiently high to initiate microcracks just prior to the unstable propagation of the macrocrack. This is in accordance with Lange's [6] criterion t h a t a2D must reach a critical value for the initiation of a microcrack (a local tensile stress, D grain size).2) I n this way the residual tensile stresses in these largest grains relax just before the specimen fractures, and spalling of them occurs along the lattice planes having the lowest specific fracture energy as to be expected for the stress free state [8]. Trying to use this mechanism to improve the fracture behaviour of conventionally sintered single-phase A1203 it is clear that (i) to avoid an overall coarsening of the grain structure the quantity of large grains must be restricted to an optimum concentration, (ii) the dispersed large grains must not be admitted to grow too large, because the aim is to generate stable microcracks but not flaws and the microcrack length increases with grain size. Therefore, a bidisperse structure consisting of isolated large grains embedded in a homogeneous, sufficiently fine-grained matrix of the same material could be expected to have better fracture properties than monodisperse alumina. It is the aim of the present work to test this possibility. 2. Experimental Fine-grained A1203 powder having a medium grain size of 0.3 ¡j.m, a specific surface (BET) of 21 m 2 /g, and an MgO content of 0.5 wt% 3 ) had been granulated adding 0.5 to 5.0% of an organic binder, cold-pressed, and sintered in hydrogen to bars of 6 X 6 X 50 mm 3 with a density of 3.91 to 3.95 g/cm 3 . Varying the sintering temperature in the range from 1550 to 1750 °C and the isothermal sintering time between 30 and 120 min different structures could be manufactured. Bending strength and fracture toughness were determined for the as-sintered specimens. For three-point bending tests the cross-head speed and span length were 0.5 mm/min and 30 mm, respectively. K J c was measured with an indentation technique based on the calibration curve published by Evans and Charles [9], the indenter load being 98.1 N. I n this way fracture strength and toughness could be measured on the same specimen. Young's modulus was determined from the measured resonance frequency of the bars. _ Besides the medium grain size D measured on SEM and C-Pt replica electron micrographs of the fracture surface using the usual intercept method [10], the grain size distribution of characteristic structures was determined (see Table 1). As the replica technique had proved suitable to image microcracks, the microcrack density q was determined from the measured maximum caliper length l t of the observed systems according to the definition given in [3] 21?

2 ) It is not justified to use criteria obtained from a simple comparison of the elastic energy stored before the initiation of microfracture with the surface energy consumed after the microcrack has propagated to its final length [7, 8]. Instead of this Lange's treatment [6] involves the infinitesimal change of the two energies in the course of crack propagation. 3 ) CT 8000 FS, Giulini, FRG.

185

Alumina Structure with Improved Fracture Properties

where A is the analyzed area. 4 ) Due to the definition of l,- the microcrack density o is very sensitively affected by the transition from a structure of isolated cracks to longer, branched microcrack systems. Unfortunately, it is only possible to determine an average crack density for the whole fracture surface (i.e. not exclusively for the region where the macrocrack becomes unstable). Furthermore, there are no informations about the density of microcracks oriented parallel to the macrocrack plane (i.e. perpendicular to the main tensile stress). Therefore, the physical substance of the absolute microcrack density values presented here should not be overestimated, and no further attempt to improve the analysis (e.g. by using a three-dimensional definition of o) was made. 3. Resulting Structures and Properties Experimental results are presented in Table 1. The structural development of the densified ceramic body is as follows: Reaching the final density of about 9 8 % the structure is very fine grained (1) = 1 to 2 ¡j.m) and homogeneous in grain size (Fig. 1). For the discussion presented here this is called stage 1. Further sintering results in a regular grain growth (stage 2). Fig. 2 Table 1 Stages of structure development in single-phase A1203 structure

properties

D ((xm) Q

400 K the P P E almost disappears and there is no difference between illuminated and nonilluminated states anymore). AT in (11) was obtained from the additional loading due to thermal expansion during pulse heating (see Fig. l b ) [11]. For this method AT = AaaTSI&eLocT

,

(12)

where AffaT is a stress increase due to thermal expansion obtained by extrapolation (dashed line in Fig. l b ) of the initial slope ff,, to the time Ai, ocT being the coefficient of the thermal expansion. AcraT was also taken into account in determining the slope of the A-t curve in the point OT+AT, is measured, were the same (within the experimental error) as those obtained using «Tt+at.o+ao- The reproducibility of AH corresponds to an accuracy better than + 5 % in AT. 3. Experimental Results and Discussion 3.1 Dependence

of Aa and AH on plastic

deformation

and

temperature

Fig. 2 a presents the dependence of the activation area, Aa, on plastic deformation for the temperatures 400, 300, and 220 K. All points were obtained by extrapolating Aa(Ar) to AT = 0, where At is the pulse in the cr-jump method. The extrapolation is shown schematically by dashed lines in Fig. 2 a. Fig. 2 b shows the dependence of the enthalpy of activation, AH, on plastic deformation. All points were obtained by extrapolation of AH(AT) to AT = 0. (AT = 5 to 20 K were used in the experiments.) All curves in Fig. 2 start at stresses slightly higher than the yield stress r y (defined as in [15]) at the corresponding temperature. As can be seen from Fig. 2a and b there is practically no influence of plastic deformation on A a and AH. Neither

Thermal Activation Analysis of Dislocation Motion in CdTe

197

tvj fa»100-

0.4-

«I

0

L

Fig. Pig. 2. The dependence a) of the activation area A a (in 62 units) and b) of the enthalpy of activa10"5 s"1. (O dark, tion AH on the applied shear stress at (1) T = 400, (2) 300, and (3) 220 K, i • light) Fig. 3. The dependence a) of the activation area A a (in 62 units), and b) of the enthalpy of activation AH on temperature is there an influence of illumination on A a and AH. (The points for the illuminated state were obtained during steady-state plastic deformation, e, = e d , where 1 = light, d = dark.) We conclude that the defects introduced by plastic deformation are not the obstacles determining thermally activated dislocation motion in CdTe, at least in the range of strains and temperatures investigated. From this it also follows that the increase of applied stress, r, during plastic deformation, i.e. work hardening, produces practically no change in the effective stress, Te, responsible for thermally activated dislocation motion. Fig. 3 a and b present the dependences of A a and A H on temperature. There is only a weak increase in AH with temperature at T > 400 K , where the P P E almost disappears. F o r T 400 K AH decreases with decreasing temperature proportional to kT. A decrease of AH in proportion to that of kT is normal for experiments using a limited range of ¿(see (1) and (4)) and means that at lower temperatures a higher stress r e is helping the same dislocations to overcome only just the top of the activation barriers. If there were more than one type of obstacles present the weaker ones are thermally overcome at low T, the stronger ones at higher T. Then the change AH(T) — under the assumption q = const — would indicate a change in deformation mechanism. Since AH(T) in Fig. 3 b is smooth we do not consider this to happen below 400 K . 3.2 Obstacle profile

analysis

Since during plastic deformation there appears to be no change in the effective stress, we can assume as a first approximation that r e is proportional (or equal) to r y + At, where Ty is the yield stress and At the loading pulse. I n Fig. 4 data obtained for A A at various temperatures in the nonilluminated and illuminated states are plotted versus (RY + At). This is the obstacle profile. D a t a for the illuminated state were obtained at small plastic strains where the P P E has its maximum value [15]. As may be seen from Fig. 4 at 450 and 400 K all experimental points lie on the same curve as the experimental points obtained for the illuminated state at 300 and 220 K . The

198

E . Y . GUTMANAS and P . HAASEN

m riK)Fig. 4

Fig. 5

Fig. 4. The dependence of the activation area AA (in b2 units) on the effective stress re = ry + AT at various temperatures in the illuminated and nonilluminated states. r y is the yield stress. A, A T = 450; O 400; • 300; T, V 220 K ; full symbols for light, open symbols for dark Fig. 5. The dependence of the yield stress, r y , on temperature in the illuminated (1) and nonilluminated (d) states

experimental points for the nonilluminated state at 300 and 220 K indicate a smaller activation area at the same stress compared with 450 and 400 K or a lower stress for the same Aa. From this it may be concluded that the barrier becomes lower with a temperature decrease for dislocation motion in the nonilluminated state. The points for the nonilluminated state do, however, fit the curve for the illuminated state if they are shifted to higher stresses according to the difference in the yield stress, r y , for both states. The dependence of r y on temperature for both illuminated and nonilluminated states is shown in Fig. 5. An attempt was made to obtain the obstacle -profile in the illuminated state through A a and A H during the transient region of the P P E . Fig. 6 shows an example for the

Fig. 6. Activation analysis in the illuminated state for a positive P P E in CdTe at 300 K . a) The stress-time curve illustrating the experimental procedure; b) the dependence of the activation area AA (in 62 units) on time t • c) schematic drawing of the effective stress r e vs. activation volume b A a, where stages 1, 2, 3, and 4 shown by arrows correspond to the same numbers in b)

199

Thermal Activation Analysis of Dislocation Motion in CdTe

Fig. 7. Curve 1 is a plot of the effective stress re vs. activation volume b A a. The solid line corresponds to the experimental data of Fig. 4 for the illuminated state and the dashed line is an extrapoation based on curve 2, where the data of Fig. 4 are plotted as b A a vs. 1/T and on the value AO = 0.46 eV at 300 K.

0

10

20 bAoilO ml-

0

A a change in this region at 300 K. A measurable change of Aa was observed in crystals with a strong P P E after switching the light on or off. This change disappears with time when e reaches the steady state. Fig. 6 c shows schematically the possible changes of the obstacle profile in the transient region of the positive P P E . No difference was measured at 300 K within the experimental accuracy between the free energy of activation in the illuminated (AG,) and nonilluminated (AG or AGd) states. This may be easily understood, since even for a strong P P E in the crystals investigated ¿ ¿ / ¿ i ^ 4 at 300 K and AG, - AG i = kT In e d /e, « 0.035 eV is much smaller t h a n AG (or AGd). An increase of A a observed in the illuminated state immediately after switching on the light must mean, however, that there is a change in AG for dislocation motion, not a change in the discrete obstacle concentration, c. This is because for discrete obstacles the effective length of dislocation segment I changes more rapidly with c at the same r t h a n the activation distance Ay, ans Aa = I Ay should decrease as c increases which is not observed. The obstacle profild is built for the illuminated state in Fig. 7, the solid line corresponding to the experie mental curve in Fig. 4. The dashed line is an extrapolation based on curve 2 in Fig. 7, where the data of the solid line in Fig. 4 are plotted empirically as b A a versus 1/T, and on the values obtained for AG at 300 K. 3 ) 3.3 The mechanism effect in CdTe

of dislocation

motion

and the nature

of the

photoplastic

The values obtained for the thermal activation parameters of dislocation motion in CdTe, a relatively small activation area and a relatively high free energy of activation AH are strongly supporting the mechanism of formation of double kinks as controlling process of plastic deformation. Indeed, if we assumed t h a t discrete obstacles were determining the dislocation mobility in CdTe and if we took the activation distance Ay ^ b, then for TE = TY + AT = 3 MN/m 2 we obtain from Fig. 4 I ss 140 b for the illuminated state. A rough estimate then gives a bow-out amplitude ~ b for such segments. This appears to be too small for a dislocation that has to overcome a strong obstacle (with AG = 0.46 eV at 300 K). The stress level indicated on the obstacle profile means that the dislocation velocity is not high and dislocation motion is thermally activated. If the mechanism controlling the dislocation motion in CdTe 3

) It should be noted that an activation analysis of plastic deformation of CdTe using stress relaxation tests [7] yields a big difference between AH (or AG) calculated according to (6) and values obtained from the dependence of b A a onr e . The 14*

Fig. 3. The temperature dependence of the relaxation times T % in PE. o first series, at a proton resonance frequency of 21.5 MHz, A second series, at a proton resonance frequency of 90 MHz

212

V. D.

FEDOTOV

et al.: Nuclear Spin-Lattice Relaxation in Polyethylene

periments in P E the spin diffusion can average out even the relaxation times between all the three phases. In Fig. 3 the temperature dependences of the spin-lattice relaxation times T ^ j of the crystalline, of the intermediate, and of the amorphous phases of the P E are represented. Each of these times is completely determined by the molecular motions and the structure of the corresponding polymer phase. Eig. 3 shows five minima in the temperature dependence of the relaxation times. These minima were related to the different relaxation processes in consistency with Tlt Tle, and T2 results and with results from mechanical and dielectric relaxation investigations summarized in [4], too. Accordingly the T ^ and TjU minima at low and high temperatures are caused by torsional oscillations of methylene groups and rotations of methylene groups in defects, respectively, whereas the minimum in is caused by anisotropic rotations of methylene groups in the amorphous regions. Reierences [1]

[2] [3]

[4] [5]

[6] [7] [8] [9]

[10] [11]

M.

and L . P . I N G M A N , phys. stat. sol. (a) 4 6 , 2 1 3 ( 1 9 7 8 ) . and N. A . A B D E A S H I T O V A , Abstr. X X . Congr. Ampere, Tallinn 1 9 7 8 (p. A 2 3 1 1 ) . V . D . F E D O T O V , J . K . OVCHINNIKOV, N . A . A B D R A S H I T O V A , and N . N . K U Z M I N , Vysokomol. Soed. All), 3 2 7 ( 1 9 7 7 ) . V. D . F E D O T O V and N. A . A B D R A S H I T O V A , Vysokomol. Soed. A 2 1 , 1 5 8 ( 1 9 7 9 ) . V. D . F E D O T O V and N. A . A B D R A S H I T O V A , Vysokomol. Soed. A 2 1 , 2 2 7 5 ( 1 9 7 9 ) . J . J E E N E R and P. B R O E K A E R T , Phys. Rev. 157, 232 (1967). G . M . K A D I E V S K Y , V . M . C H E R N O V , A . S H . A G I S H E V , and V . D . F E D O T O V , in: Nekotorye voprosy fiziki zhidkosti, Vol. 5, Kazan 1974 (p. 73). V. D. F E D O T O V and N. A . A B D R A S H I T O V A , Vysokomol. Soed. A19, 2811 (1977). A . A B R A G A M , The Principles of Nuclear Magnetism, Clarendon Press, Oxford 1 9 6 1 (p. 1 2 0 ) . G . M. K A D I E V S K Y , V. D. F E D O T O V , and R. G . G A F I Y A T U L L I N , Dokl. Akad. Nauk SSSR 2 1 0 , 140 (1973). V. D . F E D O T O V and A. N. T E M N I K O V , Vysokomol. Soed. B21, 512 (1979). PUSTKKINEN

V . D . FEDOTOV

(Received August 21, 1980)

C.

HOLSTE

e t al.: Change of A c t i v a t i o n Area during Cyclic D e f o r m a t i o n

(I)

213

phya. s t a t . sol. (a) 63, 213 (1981) S u b j e c t classification: 10.1, 21.1 Sektion

Physik

der Pädagogischen

Hochschule

Dresden1)

Change of Activation Area during Cyclic Deformation I. Experimental Results

By C. HOLSTE, F . LANGE, a n d H . - J . BURMEISTER

E x p e r i m e n t a l results a r e p r e s e n t e d f r o m room t e m p e r a t u r e stress r e l a x a t i o n t e s t s on polycrystalline p u r e nickel cyclically d e f o r m e d a t c o n s t a n t t o t a l s t r a i n a m p l i t u d e s . Mainly t h e r e l a x a t i o n b e h a v i o u r is investigated a t d i f f e r e n t stress levels along t h e stabilized hysteresis loops. T w o experim e n t a l l y observable a c t i v a t i o n a r e a s (the operational a n d t h e corrected a c t i v a t i o n a r e a Ae a n d Aec, respectively) a r e defined, which can b e d e t e r m i n e d f r o m t h e r e l a x a t i o n curves. T h e definition of Aec t a k e s work h a r d e n i n g effects i n t o a c c o u n t which m a y occur d u r i n g r e l a x a t i o n . W h e r e a s Ae d e p e n d s strongly as well on t h e mode of t h e relaxation t e s t as on t h e plastic s t r a i n a m p l i t u d e a n d on t h e plastic s t r a i n w i t h i n a loading cycle, Aec is n e a r l y i n d e p e n d e n t of t h e s e p a r a m e t e r s . A m o r e detailed discussion of t h e e x p e r i m e n t a l results will be given in P a r t I I of this w o r k . E s w e r d e n experimentelle Ergebnisse v o n R a u m t e m p e r a t u r - S p a n n u n g s r e l a x a t i o n s v e r s u c h e n a n vielkristallinem reinem Nickel mitgeteilt, das bei k o n s t a n t e r G e s a m t d e h n u n g s a m p l i t u d e zyklisch v e r f o r m t wird. I m B l i c k p u n k t des Interesses steht das R e l a x a t i o n s v e r h a l t e n bei unterschiedlichen S p a n n u n g s n i v e a u s entlang stabilisierter Hystereseschleifen. E s w e r d e n zwei experimentell beoba c h t b a r e Aktivierungsflächen (die experimentell direkt m e ß b a r e u n d die korrigierte Aktivierungsfläche Ae bzw. Aec) definiert, die aus d e n R e l a x a t i o n s k u r v e n b e s t i m m t w e r d e n k ö n n e n . Bei d e r Definition v o n A e o ist berücksichtigt, d a ß w ä h r e n d d e r R e l a x a t i o n Verfestigungseffekte a u f t r e t e n k ö n n e n . W ä h r e n d A e sowohl s t a r k v o n der A r t des R e l a x a t i o n s v e r s u c h e s als a u c h v o n der plastischen D e h n u n g s a m p l i t u d e u n d v o n der plastischen D e h n u n g innerhalb eines L a s t z y k l u s a b h ä n g t , i s t ^ e c nahezu u n a b h ä n g i g von diesen P a r a m e t e r n . E i n e ausführlichere Diskussion der experimentellen Ergebnisse wird in Teil I I dieser A r b e i t gegeben.

1. Introduction In recent years the research of the stress-strain response and the nature of microscopic mechanisms in the saturation stage of cyclically deformed single-phase metals has been intensified [1 to 5]. Thereby the interest in the dislocation behaviour and in the kind of accomodation of plastic strain during a single stabilized loading cycle has increased, too [6 to 10]. For a basic study of the dislocation mechanisms responsible for the plastic deformation the analysis of thermally activated processes was shown to be a valuable tool [11]. While for many years thermally activated processes have been examined for unidirectionally strained metals, only in recent time more detailed investigations on rate-dependent effects in cyclically strained materials were performed [12 to 16], In investigations on hardening mechanisms connected with thermally activated slip in f.c.c. metals the activation area is often used as a characteristic parameter. Up to the present except for the early work of Wadsworth and Hutchings [17] and few experimental results given in [18], no measurements of the change of activation area in cyclic deformation of f.c.c. metals are known to the authors. In the last few years frequently the stress relaxation technique was preferred for the determination of activation parameters [19], mainly, because microstructural !) W i g a r d s t r . 17, D D R - 8 0 6 0 D r e s d e n , D D R .

214

C. Holste, F. Langb, and H.-J. Burmeister

changes are assumed to be minimum or non-existent during the stress relaxation test [11]. If the flow stress a is proved experimentally to decrease linearly during relaxation with In (—a), the dependence of the flow stress on the plastic strain rate e p during the test can be described in the Arrhenius form with a stress-independent activation area. This was found often to be possible for plastically deformed pure f.c.c. metals [20, 21]. The experimentally measurable activation area Ae directly correlated to the slope S of the In (-a)-a diagram by the relation Ae = SkT/bX will be termed in this work the operational activation area, analogous to the terminology in [11]. I n this definition b is the absolute value of the Burgers vector, A is a mean orientation factor relating the tensile stress a to the resolved shear stress in the active slip planes, and the other quantities have the usual meaning. In Part I of this paper results are presented of room temperature stress relaxation experiments on polycrystalline pure nickel cyclically deformed under ambient temperature at medium constant total strain amplitudes. The main work was done in determining the operational activation area for different stress values along the stabilized hysteresis loops. So far there is no common basis for a quantitative interpretation of the characteristic cyclic change of the operational activation area observed in this work. I n Part I I of the present paper a model is discussed quantitatively which allows to explain the specific dependence of A e on the plastic strain in a single loading cycle in a rather easy way. Starting from the basic assumptions in [10] the specimen of a cyclically deformed metal is interpreted as a plasticalty heterogeneous body in a mesoscopic scale (of order of 1 urn). Finally it may be noted that in this work no attempt is made to reveal the elementary dislocation-obstacle mechanisms responsible for strain-rate effects in cyclically deformed nickel. A t present this seems successful only with additional information from strain-rate experiments at different temperatures. 2. Experimental and Evaluation Procedure

The fatigue tests and stress relaxation experiments were performed with a motordriven push-pull Zwick machine. Flat specimens (generally of cross-section area 50 mm 2 , free specimen length between the grips 35 mm) of annealed pure (99.99 w t % ) polycrystalline nickel (mean grain diameter 20 ¡j.m) were used. The strain was measured directly on the specimen with an inductive extensometer (gauge length 20 mm) and could be controlled by a closed-loop system. With the whole electronic equipment strains down to 10 -6 could be resolved. The non-predeformed specimens were cyclically strained (strain rate e = 8 X 10~4 s - 1 ) at constant total strain amplitudes ea (1 X 10 -3 ea N i S i . The Ni atoms that are products of the first reaction are assumed to diffuse in the NiSi layer and react with Si at the NiSi-Si interface according to the second reaction. We cannot calculate the free energy change for these two reactions as the physical state and the activity of the Ni are unknown. Thus it would be incorrect, in the calculation of the enthalpy change of the reactions, to use published values of the heat of formation where the standard states of the elements are assumed. Calculations of the entropy for this system will also be difficult. I t may be mentioned that the enthalpy change for the first reaction is positive (10.9 kcal/mol of NiSi) assuming the reactants and the products are in their standard states. The combination of the first and the second reactions yields a negative enthalpy change 4.9 kcal/mol of NiSi) under the same assumptions. In understanding this system one has to consider that during the growth of NiSi there exists a concentration gradient of Ni in the system. Ni atoms that are products of the first reaction will move away from the reaction zone by diffusion. The lowering of the free energy by reaction with Si according to the second reaction may be viewed as a driving force. So in trying to reach equilibrium between the reactants and products the first reaction will proceed to supply Ni. In this system it seems likely to assume that the diffusion of Ni through the NiSi layer is not the rate controlling step in the growth of NiSi as it has been reported that the growth rate of this phase is linear in time [5], This indicates that the growth is reaction controlled. The reaction at either of the phase boundaries of NiSi could be the rate limiting step and the diffusion rate of Ni in NiSi may be very fast in comparison to the rate of these reactions. I t is interesting to look at these findings in the light of the considerations stated in the paper of Tu [12]. I t is pointed out that the Si atoms in the single crystal are rather strongly bonded. This means that these atoms require a high thermal energy to be "released" from the lattice. The energy required is higher than the thermal energy found at a temperature where many silicides grow. This suggests that the metal atoms would be the dominant moving species during the growth of many silicides at low temperatures unless Si could be "released" from the substrate by metal atoms penetrating the rather open crystal lattice and thereby modify the bonds of Si atoms. Whether one will have any supply of Si that could grow in the silicide layer would then depend upon the relative magnitude of the rate of this "releasing" mechanism and the reaction rate with Si. One cannot, from our measurements, make any conclusions regarding the mobility of Si in a NiSi layer. The value of the mobility of Si might well be comparable or larger than that of Ni. I t may be just that there is no supply of "free" Si atoms available. Previous experiments [8] have indicated that the growth mechanism proposed in this work is possible. The previous work made use of a P t silicide marker. That marker did, however, influence the speed of the growth [13], so it could not be concluded that the proposed growth mechanism was the dominating one without the marker present. In the present study the marker did not influence the speed of growth indicating the proposed process to be the dominating one in the Si-Ni thin film system. 5. Summary A marker experiment with implanted X e in Ni 2 Si has been performed to study the movement of the atom species during the growth of NiSi. The experiment is in reasonable agreement with a model where Ni is the dominant moving species. 15*

T. G. FINSTAD : Transformation of NÙS to Ni Si in Thin Films

228 Acknowledgement

T h e a u t h o r t h a n k s J . R o t h for d o i n g t h e i m p l a n t a t i o n . T h i s work w a s p a r t i a l l y s u p p o r t e d b y t h e R o y a l N o r w e g i a n Council for S c i e n t i f i c a n d I n d u s t r i a l R e s e a r c h , a n d by NAVF. References [1] K . N. T u and J . W. MAYER, in: Thin Films. Interdiffusion and Reactions, Ed. J . M. POATE, K . N. Tu, and F. W. MAYER, Intersoience, New York 1978 (p. 359). [ 2 ] M . WITTMEB, D . L . SMITH, P . W . L E W , a n d M . - A . NICOLET, S o l i d S t a t e E l e c t r o n i c s 2 1 , 5 7 3 (1978). [3] H . B . GHOZLENE, P . BEAUFRÈRE, a n d A . AUTHIER, J . a p p i . P h y s . 4 9 , 3 9 9 8 ( 1 9 7 8 ) .

[4] J . 0 . OLOWOLAFE, M.-A. NICOLET, and F. W. MAYER, Thin Solid Films 38, 143 (1976). [5] D. J . COE and E. H. RHODRICK, J . Phys. D 9, 965 (1976). [ 6 ] G . J . VAN GUYS, W . F . VAN DER W E G , a n d D . SIGURD, J . a p p i . P h y s . 4 9 , 4 0 1 1 ( 1 9 7 8 ) . [7] R . PRETORIUS, C. L . RAMILLER, S. S . LAU, a n d M . - A . NICOLET, A p p i . P h y s . L e t t e r s 3 0 , 5 0 1

(1977). [8] T . G . FINSTAD, J . W . MAYER, a n d M . - A . NICOLET, T h i n S o l i d F i l m s 5 1 , 3 9 1 ( 1 9 7 8 ) .

[9] W.-K. CHU, J . W. MAYER, and M.-A. NIOCLET (Bd.), Backscattering Spectroscopy, Academic Press, New York 1978. [10] S. T. PICAUX, E. P. EERNISSE, and F. L. VOOK (Ed.), Application of Ion Beams to Metals, Plenum Press, New York 1974. [11] Z. L. LIAU and T. T. SHENG, Appi. Phys. Letters 32, 716 (1978). [12] K. N. Tu, Appi. Phys. Letters 27, 221 (1975). [13] T. G. FINSTAD, Thin Solid Films 51, 411 (1978). (Received August 8, 1980)

229

T. VAN DIJK and A. J. BTXRGGRAAF : Grain Boundary Effects phys. stat. sol. (a) 63, 229 (1981) Subject classification: 1.4 and 14.4; 22.6.1 Laboratory of Inorganic Chemistry and Materials Science, Engineering, Twente University of Technology, Enschede1)

Department

of

Chemical

Grain Boundary Effects on Ionic Conductivity in Ceramic GcLZri-xOa-^/a) Solid Solutions By T . V A N D I J K a n d A . J . BURGGBAAF

Complex admittance measurements are performed on high-purity ceramics prepared by means of the alkoxide synthesis and on less pure ceramics obtained from the citrate synthesis. The results on ceramic materials with grain sizes ranging from 0.4 to 20 ¡xm are compared with those from a single crystal. The activation enthalpy for grain boundary conductivity A-FFGH = (118 2) kJ/mol for the samples studied, is independent of composition, grain size, and preparation method. Grain boundary conductivity values and consequently the relevant pre-exponential factors are an order of magnitude smaller for the alkoxide materials than for the citrate materials. The ratio of grain bulk and grain boundary conductivity (ffb/orgb) for alkoxide materials with grain-sizes 0.4 to 0.8 ¡xm varies from 8.5 to 1.0 in the temperature range 500 to 700 °C. Messungen der komplexen Admittanz werden an hochreinen keramischen Materialien, die mittels Alkoxidsynthese hergestellt wurden, und an weniger reinen Keramiken aus der Zitratsynthese gemessen. Die Ergebnisse an keramischen Materialien mit Korngrößen im Bereich von 0,4 bis 20 [im werden mit denen an einem Einkristall verglichen. Die Aktivierungsenthalpie für Korngrenzenleitfähigkeit beträgt A-ffgb = (118 ± 2) kJ/Mol für die untersuchten Proben, unabhängig von der Zusammensetzung, Korngröße und Präparationsmethode. Die Werte der Korngrenzenleitfähigkeit und dementsprechend die relevanten Präexponentialfaktoren sind für Alkoxid-Materialien eine Größenordnung geringer als für Zitrat-Materialien. Das Verhältnis der Kornvolumen- und Korngrenzenleitfähigkeit (tr^/cgb) für Alkoxidmaterialien mit Korngrößen zwischen 0,4 und 0,8 [xm variiert von 8,5 bis 1,0 im Temperaturbereich 500 bis 700 °C. 1. Introduction The ionic conductivity of ceramic cubic stabilized zirconias depends on physical and chemical properties of the ceramic samples such as composition, ordering and ageing effects, porosity, grain size, and purity. Separation of grain bulk and grain boundary conductivity can be performed by complex plane analysis (see Section 3.1). Ac conductivity investigations on Z r 0 2 - C a 0 and Z r 0 2 - Y 2 0 3 ceramics [1 to 7] revealed a large scatter in the grain boundary conductivity. The ratio of grain bulk and grain boundary conductivity at 500 °C, for example, varies from about 2.6 [3] to about 0.07 [4], although the same composition (0.91 Zr0 2 -0.09 Y 2 0 3 ) was studied. This ratio varies at 550 °C from 4.0 to 0.14 for 0.9 Zr0 2 -0.1 Sc 2 0 3 ceramic specimens with grain sizes 5 to 75 ¡j.m sintered at 1800 to 2200 °C during 4 h, but scatters between 0.5 and 0.2 for specimens with grain sizes 6 to 35 ¡i.m sintered at 2000 °C for 0.25 to 5.0 h [7], This illustrates the influence of sample preparation conditions. I t appears that both grain bulk and grain boundary conductivities are decreased by adding impurities, frequently used as sintering aids [6, 8]. Mechanisms proposed to account for these effects are dissolution of impurities in the grains, segregation of them in the grain boundaries and, if present in higher concentrations, the formation of thin continuous or discontinuous layers of a second phase. !) P.O. Box 217, 7500 AE Enschede, The Netherlands.

230

T . V A N DIJK a n d A . J . B U K G G R A A F

Impurities of 1 mol% Fe 2 0 3 , MgO, Ti0 2 , and A1203 negatively influence the ionic conductivities of both the grains and the grain boundaries if compared with purer materials in the same study [6]. However, grain boundary conductivities of these very impure materials [6] far exceed those of samples containing only minor impurities [3]. A recent study on the influence of some mol% Ti0 2 and A1203 as sintering aids in calcia-stabilized zirconia shows a strong deterioration of the performance of the impure materials compared with analytically or technically pure materials, even up to 1000 °C [8]. I t seems likely t h a t conflicting results obtained by various authors largely originate from preparation methods of the materials. Analysis at 450 °C of grain boundary conductivity ogb as a function of grain size of yttria-stabilized zirconia prepared by means of the alkoxide synthesis [5] showed a linear increase of 1 0 6 H z are observed. These can be attributed to electrode interface polarization phenomena and the geometrical capacitance of the sample, respectively. This indicates that the polarization phenomena observed for the ceramic samples are due to the ceramic microstructures of the samples, essentially the presence of grain boundaries. From Fig. 5 a and b it is evident that the grain boundary dispersion is much larger for the alkoxide than for the citrate material. For both types of ceramic materials the centres of the semicircles are located only slightly below the horizontal cr'-axis. This probably indicates uniform grain boundary properties throughout the samples, whereas a wide distribution of these properties causes a large dispersion in relaxation times and consequently makes the semicircle centre fall below the horizontal axis [1, 16]. The availability of two types of ceramic materials with approximately the same composition exhibiting apparently widely differing grain boundary properties, as well as single crystalline material, enables us to demonstrate both the justification of the electrical equivalent circuit (Fig. 1) and the independence of grain bulk conductivity (the high frequency intercept of the grain boundary polarization semicircle with the horizontal axis) on microstructure. From Table 2 it follows that the bulk conductivity of materials with identical compositions is, within experimental error, independent of material preparation.

236

T . v a n D i j k and A . J. B t j b g g r a a f

Fig. 5. a) Complex admittance diagram a) at 500 °C of Gd0.495Zr0.505O!.753 (alkoxide synthesis), • sintered at 1400 °C, grain size 0.4 ¡j.m; o sintered at 1430 °C, grain size 0.8 [im; b) at 500 °C of Gd0.4q8Zr0.502O1.751 (citrate synthesis), • sintered at 1700 °C, grain size 5.0 [Jim, O sintered at 1550 °C, grain size 0 94 jxm; c) at 500 °C of single crystalline Gdo.52Zro.48O1.74

Conductivity values of the single crystalline material coincide with values obtained from previous ceramic data [13]. Table 2 Comparison of bulk conductivity values for citrate, alkoxide, and single crystalline (SC) materials GdOi.s (mol%)

method

phase

bulk ionic conductivity (10~3 ( n m)" 1 ) 700 °C 500 °C 600 °C

44.8 49.5 49.8 52 52 53.2

citrate alkoxide citrate SC citrate [13] citrate

F P P P P P

4.1 24 31 9.5 9.0 4.1

28 110 140 52 44 25

102 267 345 173 150 95

4.2.2 The grain boundary conductivity of alkoxide and citrate material Grain boundary conductivities of several samples prepared by means of the citrate and alkoxide synthesis are plotted as a function of temperature in Fig. 6 together with results of Schouler et al. [3] for 0.91 Zr0 2 -0.09 Y 2 0 3 and of Chu and Seitz [2] for 0.05 Zr0 2 -0.15 CaO. I t appears from Fig. 6 that the grain boundary conductivities of the citrate materials exceed those of the alkoxide materials by more than one order of magnitude. Besides

Grain Boundary Effects on Ionic Conductivity in Gd I Zr 1 _ a ; 02-( a ;/2)

237

this it appears t h a t t h e grain b o u n d a r y conductivities of t h e citrate samples are larger a n d those of t h e alkoxide samples are smaller t h a n t h e literature values mentioned for structurally related systems (however with a strongly different l a n t h a n i d e concentration) . F o r t e m p e r a t u r e s above 650 °C, grain b o u n d a r y conductivities of citrate materials seem t o increase f a s t e r with increasing t e m p e r a t u r e t h a n is t o be expected f r o m t h e Arrhenius behaviour below 650 °C. However, as t h e semicircles are v e r y small a n d bulk conductivities are relatively high a t higher t e m p e r a t u r e s , large errors are involved in estimating t h e grain b o u n d a r y conductivities a t t e m p e r a t u r e s higher t h a n 650 °C. T h e ratios of grain bulk a n d grain b o u n d a r y conductivity of some samples are shown as a f u n c t i o n of t e m p e r a t u r e in Fig. 7. I n t h e t e m p e r a t u r e range measured, crb was in all cases larger t h a n crgb for t h e alkoxide material a n d practically always smaller t h a n for t h e citrate material. This cannot be explained b y differences in grain size (cf. A a n d • in Fig. 7). a h ja g b values were f o u n d for alkoxide materials a t 500 °C which were larger t h a n those reported in literature [1 t o 7] a n d which range f r o m about 4 t o 0.1. The bends in t h e curves of Fig. 7 arise a t least partially f r o m t h e grain bulk conductivity behaviour, because in t h e t e m p e r a t u r e range investigated a bend in t h e log crb versus 1/ T curve is often observed for stabilized zirconias. F r o m Fig. 6 a n d 7 it appears t h a t t h e grain b o u n d a r y conductivity depends strongly on material p r e p a r a t i o n and/or p u r i t y . Surprisingly, t h e smaller grain size alkoxide sample has a larger a g b t h a n t h e larger grain size alkoxide. The difference is, however, too small t o be conclusive. More detailed studies of this grain size range, t a k i n g into account grain size distributions are a t present u n d e r investigation.

Pig. 6

Fig. 7

Fig. 6. Arrhenius plots of grain boundary conductivity of various citrate and alcoxide samples, x Gd0.498Zr0.502O1.751 (citrate synthesis) sintered at 1700 °C, grain size 5 ¡¿m; o Gdo.49sZro.502O1.751 (citrate synthesis) sintered at 1550 °C, grain size 0.94 ¡im; A Gdo.44sZro.552O1.776 (citrate synthesis) sintered at 1550 °C, grain size 0.8 (xm; • Gdo.532Zro.467O!.734 (citrate synthesis) sintered at 1700 °C, grain size 20 ¡im; v Gdo.495Zro.505O1.753 (alkoxide synthesis), sintered at 1400 °C, grain size 0.4 ¡xm; • Gd0.495Zr0.505O!.753 (alkoxide synthesis), sintered at 1430 °C, grain size 0.8 ¡xm Fig. 7. R a t i o of grain bulk 0^ and grain boundary conductivity rfgb as a function of temperature (symbols see Fig. 6)

238

T . VAN D I J K a n d A . J . BURGGRAAF

The values of the activation enthalpy AH and the pre-exponential factor cr0 for grain bulk and grain boundary conductivity obtained from Arrhenius plots are shown in Table 3. Table 3 Activation enthalpy and pre-exponential factors for grain bulk and grain boundary conductivity for citrate, alkoxide, and single crystalline materials GdOi.5 prepara(mol%) tion method

(°C)

44.8 49.8 49.8 53.2

C C C C

1550 0.8 98 1550 0.94 (0.35) 96 1700 5.0 (2.0) 98.5 1700 20 95

49.5 49.5

A A

1400 1430

52

S (CP-Phase)

dg

Ts

—

(firn)

0.4 0.8 -

"Ob (IO3 (Q m)- 1 )

Aflgb (kJ/mol)

Q-fmac) 0Bb (IO6 (Ci m)' 1 )

107.3 79.8 79.8 99.6

70.2 7.9 7.9 21.0

116.8 117.5 118.1 119.6

1.22 2.4 4.6 8.6

80.2 80.2

5.6 5.6

116.5 120.2

0.28 0.36

97.2

34.2

ele th Ai? b (%) (kJ/mol)

93 94 -

—

-

The independence of material preparation for AH h and cr0b is again illustrated together with the minimum values for AH h and

m

(n)

B)

H TE

with 1 3 [ n = 3 fc< >

(n) (n)

U t e

= a. e

«TM

; a„, bn being real coefficients.

2.2 Electromagnetic

fields in anisotropic

medium

No. 2

Applying Maxwell's equations to this anisotropic and bigyrotropic medium, and using general expressions of ¡jLij tensors, we find in the coordinate system of Fig. 1 «11

£

12

e

e

21

e

22

e

31

e

e

0

32

0

13

K

23

K ky k0

e

33

ky

k0

ky k0

0

0

0

0

Ex

= 0,

k0

k0

fti

P12

i"l3

Hx

K k0

0

0

M 21

^22

¡J-23

Hy

ky k0

0

0

31

¡¿33

Ht

where k0 = 2jr/A, A is the optical wavelength. To solve this system we cancel out its principal determinant and obtain the following fourth-degree equation : Akj + Bk$ + Ckvkj

+ Dksykz + Ek\k\

where the expressions for A, B, The solutions of this equation optical (MO) phenomena, such from this equation when eq and teristic equation of medium No.

+ Fkf + Gk% + Hkyk,

+ K = 0 ,

G, D, E, F, G, H, and K are given in the Appendix. correspond to eigenmodes of the medium. Magnetoas Faraday rotation and Voigt effect, are studied ¡x^ are MO coefficient functions. This is the charac2.

M. T o r f e h and H. Le G a l l

250

Pig. 3. Eigen wave vectors of the propagation in the anisotropic guide and the isotropic adjacent media

The wave propagating a t the same time in t h e three media must have t h e same ky component to satisfy t h e boundary conditions. The characteristic equation shows that for a given ky there are four solutions for kz in medium No. 2 corresponding to four different eigenmodes as shown in Fig. 3. Any mode of this medium will be a linear combination of these eigenmodes: E H


>

, j

+

• 7

ae

+ b e

—ik22 z Hf)

Ef

+ c e

+

Hf

— ik7 z E

, 6, and q> (Fig. 4) :

R(ip,B,q>) = Rz(4, _B, 0 , D, E, F, G, H, a n d X c o e f f i c i e n t s are real. References [1] P . K . TIEN, R . J . MARTIN, R . C. L E CRAW, a n d S. L . BLANK, A p p l . P h y s . L e t t e r s 21, 3 9 4 [2]

(1972). P. K. TIEN,

[3]

S . YAMAMOTO, Y .

D.

P.

S C H I N K E , a n d S . L. B L A N K , J . appl. P h y s . 4 5 , 3 0 5 9 ( 1 9 7 4 ) . K o YAM AD A, a n d T. MAKIMOTO, J . appl. Phys. 43, 5090 (1972).

[4] S. WANG, M. SHAH, a n d J . D . CROW, J . a p p l . P h y s . 4 3 , 1 8 6 1 (1972).

Trans. Microwave Theory a n d Tech. 2 1 , 7 6 9 ( 1 9 7 3 ) . J . WARNER, I E E E Trans. Microwave Theory a n d Tech. 23, 70 (1975). S. YAMAMOTO a n d T. MAKIMOTO, J . appl. P h y s . 45, 882 (1974). J . P . C A S T E R A a n d G. H E P N E R , I E E E Trans. Magnetics 13, 1583 (1977). K . K I T A Y A M A a n d N. K U M A G A I , I E E E Trans. Microwave Theory a n d Tech. 25, 282 (1977). A . Y A R I V , I E E E J . Q u a n t u m Electronics 9 , 9 1 9 ( 1 9 7 3 ) . M . T O R F E H , L . COURTOIS, L . SMOCZYNSKI, H . L E G A L L , a n d J . M . D E S V I G N E S , Physica

[5] J . W A R N E R , I E E E

[6] [7] [8] [9] [10] [11]

( U t r e c h t ) 8 9 , 2 5 5 (1977). (Received

October 29,

1980)

259

J„ B u c h a r and Z. B i l e k : Plastic Flow of Irradiated Steel phys. stat. sol. (a) 68, 259 (1981) Subject classification: 10.1; 11, 21.1.1 Institute of Physical

Metallurgy,

Czechoslovak Academy

of Sciences,

Brno1)

Plastic Flow of Irradiated Steel at High Strain Rates By J . B u c h a b and Z. B i l e k Results are given of an investigation of the influence of neutron irradiation on mechanical properties of a low alloy structural weldable steel under high strain rates 10 3 s - 1 ) produced b y the Hopkinson split bar technique (HSBT) method. Particular attention is paid to the influence of post-irradiation annealing on the recovery of the material. The results are interpreted in terms of dislocation dynamics. I t is shown that the neutron irradiation does not influence the mechanism of plastic deformation at high strain rates produced b y H S B T . Der Einfluß einer Neutronenbestrahlung auf die mechanischen Eigenschaften v o n kohlenstoffarmen Schweißstahl bei hohen, durch die Methode der Hopkinson-Meßstange hervorgerufenen Verformungsgeschwindigkeiten (zx 10 3 s"1) wird untersucht. Besondere Aufmerksamkeit wird dem Einfluß einer Nachbestrahlungstemperung auf die Erholung des Materials gewidmet. Die Ergebnisse werden mit der Versetzungsdynamik analysiert. Es wird gezeigt, daß die Neutronenbestrahlung keine Wirkung auf den Mechanismus der plastischen Verformung bei hohen Verformungsgeschwindigkeiten hat.

1. Introduction Neutron irradiation has an adverse effect on the mechanical properties of low alloy steels. Much has been written in recent years about neutron embrittlement of structural steels. These reports have clearly demonstrated trends in embrittlement with neutron exposure which involve a large increase in the brittle-ductile transition temperature. One of the ways how to establish this transition is based on the knowledge of the dependence of the flow stress at upon temperature T and strain rate e in nonirradiated and irradiated conditions [1]. This way enables one to interpret the radiation embrittlement in terms of the plastic deformation mechanism defined by T and e. Several investigators [2, 3] have shown that in the temperature-strain-rate spectrum of low alloy steels we can observe several regions which reflect different mechanisms of plastic deformation. In the case of irradiated materials, the question arises of what kind is the influence of neutron irradiation on the mechanisms of plastic flow in the entire spectrum of strain rate and temperature. Literature reviews [1, 2] show that the effect of strain rate on the post-irradiation plastic deformation of steels has not been investigated systematically of extensively at high strain rates. Especially for irradiated steels tested at strain rates higher than 10 s - 1 sufficient data are lacking. One of methods for determining the flow stress a t at high strain rates is represented by the Hopkinson split bar technique (HSBT) (cf. [4]). The method uses high strain rates produced by stress-pulse loading. Not considering a wave character of the loading, the results can be described by the stress-strain relationship at a given value of e. Recently, HSBT has been applied more often for the research of materials used for fast reactor pressure vessels [5]. However, as .it.is stated by the review in [4], more detailed fundamental studies are urgently needed. !) Ziikova 22, 6 1 6 6 2 Brno, CSSR. 17*

J. Buchar and Z. Bîlek

260

The present paper studies the influence of irradiation on the dependence o t {s) with the aim of ascertaining the incidental change in the mechanism of plastic flow. Contrary to studies made hitherto, our investigation was carried out during recovering, which has been realized so far only for low values of è [6]. 2. Experimental Details For the experiment, low alloy structural weldable steel with chemical composition 0 . 2 2 % C, 0 . 9 % Mn, 0 . 2 4 % Si, 0 . 0 1 5 % P , 0 . 0 8 % Cu, 0 . 4 2 % Ni, 0 . 2 7 % Cr was chosen. The steel was heat-treated by quenching at 900 °C (water) and by annealing at 720 °C (air). From the steel penny-shaped specimens, 1.4 X 1 0 - 2 m in diameter and l 0 = 2 X 10~ 3 m thick, were prepared. Some specimens were irradiated by thermal and fast neutrons at 150 °C in the Y V R - S reactor with thermal-neutron exposure 0 t h = 1-2 X 1 0 2 1 neutrons m~ 2 and fast-neutron exposure 0 f n = 3.5 X 1 0 2 3 neutrons m - 2 (energy E > 0.5 MeV). The variation of the neutron exposure along the height of the irradiation capsule did not exceed 1 0 % . Non-irradiated and irradiated specimens were loaded by the H S B T in pressure. A detailed description of our experimental set-up and the method of data evaluation is given in [7]. Stress pulses with duration of 30 X 1 0 - 6 s and amplitudes within the range 900 to 1500 MPa were used. Such amplitudes guarantee a strain rate of the order ,10 3 s" 1 . Experiments were run in three stages : (i) a control group of non-irradiated specimens was loaded; (ii) specimens after irradiation were tested; (iii) irradiated specimens, annealed at T = 200, 250, 300, and 350 °C were tested. For each annealing temperature, various times of annealing were taken: £a = 16, 48, 72, 96, and 168 h. - All experiments were carried out at room temperature. From the experimental results the dependences a t {e) for a permanent strain of 0 . 1 % were constructed. 3. Experimental Results The course of the flow stress (Xo.i versus s for our steel in both non-irradiated and irradiated conditions is plotted in Fig. 1. I t turns out that the neutron irradiation results in a considerable deformation hardening which depends upon the strain rate. Both curves in Fig. 1 can be described as follows : oo.i =