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THE AMPLITUDE DEPENDENT DISLOCATION DAMPING IN HIGH PURITY MOLYBDENUM
CRYSTALS
R. Grau, H. Schultz
To cite this version:
R. Grau, H. Schultz. THE AMPLITUDE DEPENDENT DISLOCATION DAMPING IN HIGH
PURITY MOLYBDENUM CRYSTALS. Journal de Physique Colloques, 1983, 44 (C9), pp.C9-705-
C9-709. �10.1051/jphyscol:19839106�. �jpa-00223340�
Colloque C9, supplgment au n012, Tome 44, dgcembre 1983 page C9-705
THE AMPLITUDE DEPENDENT DISLOCATION DAMPING IN HIGH PURITY MOLYBDENUM CRYSTALS
R . ~ r a u * and H. Schultz
Max-PZanck-Institut far MetaZZforschung, Institut far Physik,
Heisenbergstrasse 1, 0-7000 Stuttgart and Institut far fieoretische und Angewandte Physik der Universitllt Stuttgart, F. R. G.
*iZrr AnZagenbau G.m.b.H., 0-7000 Stuttgart 40, F.R.G.
Abstract
-
The amplitude dependence of the y peak shows anomalous behaviour f o r low ampl i t u d e s , and p a r t l y i r r e v e r s i b l e e f f e c t s a t l a r g e r amplitudes.This hinders a d i r e c t determination of the a c t i v a t i o n volume. However, an estimate of the actual activation volume can be obtained from the observed peak broadening. The r e s u l t i s compatible with a kink-pair formation process.
Dislocation damping peaks i n ul tra-high purity molybdenum c r y s t a l s (residual r e s i s t - ance r a t i o 230 000) were investigated by torsion-pendulum technique. The y peak (400-490K, 1Hz) shows recovery a f t e r heating u p t o 600 K, compare curve 1 and 2 in f i g . 1 , ( 1 ) ( 2 ) . The amplitude dependent damping was obtained during one heating up by online data processing of t h e o s c i l l a t i o n amplitudes ( 2 ) .
Fip. 1:
Damping (om') vs. T (temperature), ul tra-hi gh purity Mo-single crystal (RRR=32.000, spec. axis < I l l > ) . ( 1 ) following 1% tension (200K)+0.1%
torsion (293K); ( 2 ) 2nd run following ( 1 ) a f t e r heating t o 600K ( f = 0 . 5 Hz, Amp1 i tude 1,.5x10-5)
Fig.2 shows t h e amplitude dependence of the y peak f o r two frequencies i n a range of low amplitudes (2xl0-6-1,2xl0-5 r e l a t i v e surface s t r a i n ) . The behaviour observed i n t h i s amplitude range i s anomalous: t h e damping peak becomes s m l l e r , and s h i f t s t o higher temperaturzs with increasing amplitudes (curves 1...6, in f i g . 2 ) . For l a r g e r ampl i tudes (1x10- - 1 . 2 x 1 0 ~ p l i tude dependence becomes normal ( f i g.3). The dependence Q - ~ vs. amplitude ( A ) f o r the y peak i s shown i n f i g . 4 f o r two amplitude ranges: ~<1.5x10-4 and ~ ~ 1 . 5 ~ 1 0 - 5 . The curve obtained from t h e large s t a r t i n g ampl i - tude ( ~ ~ " 1 . 5 ~ 1 0 - ~ ) mets exactly toggther w i t h t h e damping c u r v a b t a i n e d from a lower s t a r t i n g amplitude (Ao=1.5x10- ) as long as the temperatures a r e below 460 K.
For higher temperatures t h i s t e s t shows i r r e v e r s i b l e changes i n the specimen s t a t e f o r the l a r g e r amplitudes i n a time s c a l e of one decay.
In the following we r e s t r i c t ourselves t o the amplitude range below 1 . 5 ~ 1 0 - ~ and temperatures, where i r r e v e r s i b l e e f f e c t s a r e small i n the time scale of one decay, and consider the broadening of the y peak due t o internal s t r e s s e s .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19839106
C9-706 JOURNAL DE PHYSIQUE
F i g . 2 : T h e y m a x i m u m : A m p l i t u d e d e p e n - F i g . 3 : A s f i g . 2 , b u t f o r l a r g e r a m p l ~ t u d e s -for small amplitudes (A) a t 2 fre- 0 . 1 ) ~ = 1 x 1 0 - ~ , ( 2 ) - ( 6 ) AA=2xlO-
6
quenci es
.
( 6 ) ~=1.2x10-4. "Normal"
amp1 i tude depen- ( 1 ) ~ = 2 x 1 0 - ~ , ( 2 ) - ( 6 ) ~ ~ = 2 x 1 0 - ~ dence.( 6 ) A=I. 2x10-5. "Anomalous" ampl i tude dependence.
2 -
Fig.4: Damping
( 0 - l )
vs. amplitude ( A ) Each curve i s obtainedby
decay, s t a a t i ~ g from A, ' 1 . 5 ~ 1 0 - ~ ( l a r g e amplitude range) and from Ao2:l.Bx10- , (low ampiitude range). A smooth overlap of both curves indicates s t a b l e dislocation arrangements- " I r r e v e r s i b i l i t y t e s t " -
Fig.5 shows the shape of the y maximum in a plot Q-' vs. 1/T in comparison with a single Debye maximum. The background damping was approximated by a l i n e a r curve.
The high temperature part of the experimental curve ( a f t e r background correction) drops below the theoretical Debye curve. This can be explained by the beginning of annealing of the y process during the measurements. In f i g . 6 we compare the experi- mental peak shape with theoretical curves, which are obtained by a superposition of Debye curves (see ( 3 ) ) , d i s t r i b u t e d according t o a Gaussian function:
For t h e low temperature side of the maximum we can disregard annealing e f f e c t s and the f i t t i n g of t h i s part of the experimental curve resulted in kF1.5. The experimen- t a l curve ( f i g . 6 ) i s s l i g h t l y asymmetric, due t o annealing e f f e c t s . We assume t h a t the "true1' curve i s symmetrical against 1 / T , and t h a t the width of the peak i s given by 2b. The asymmetry of the experimental curve '(a-b)/(a+b) was in a l l cases below 0.1.
Fig.6: The experimental y peak (heavy t o a s i n g l e Debye peak.
drawn)
compared w i t h t h e o r e t i c a l. . .
exp. data p o i n t s curves, c a l c u l a t e d f o r d i f f e r e n t Gauss-
background s u b t r a c t e d parameters B---
Debye Peak c a l c u l a t e d w i t h H=1.09 eV, f =2.5x101*0
The B parameter obtained i n t h i s manner was p l o t t e d a g a i n s t 1/T ( f i g . 7 ) . As discussed i n (3), B=Bo=constant, independent o f T, r e s u l t s from a d i s t r i b u t i o n i n T~
Fig.7: The Gauss parameter B vs. 111 f o r several specimens.
B = Bo+BH/kT RH= 0.05 eV
F o l l o w i n g p a r t l y d i f f e r e n t p l a s t i c ten- s i o n deformations, a l l specimens ob- t a i n e d 0.1% p l a s t i c t o r s i o n a t 293 K, which creates a s i m i l a r i n t e r n a l s t r e s s l e v e l . Therefore BH equal f o r a1 1 specimens.
( o r f r e s p e c t i v e l y ) , and B - T - ~ r e s u l t s from a d i s t r i b u t i o n i n H. As shown i n f i g . 7 we have i n o u r case a combination o f both d i s t r i b u t i o n :
This i s expected i n the k i n k - p a i r f o r m a t i o n (KPF) model, where a d i s t r i b u t i o n o f f r e e d i s l o c a t i o n l o o p l e n g t h s should determine t h e d i s t r i b u t i o n o f T
,
and t h e r e f o r e R,
and where a d i s t r i b u t i o n o f i n t e r n a l stresses and free l o o p l e n g t h s should determ?ne t h e d i s t r i b u t i o n o f H and BH. Here we observe ( f i g . 7 ) t h a t BH=0.05 comes o u t n e a r l y equal f o r specimens o f d i f f e r e n t degree of p l a s t i c deformation. This can be under- stood by t h e f a c t t h a t a l l specimens probably have t h e same i n t e r n a l s t r e s s l e v e l , due t o a small p l a s t i c t o r s i o n s t r a i n (0.1%), which was a p p l i e d t o a l l specimens, f o l l o w i n g d i f f e r e n t degrees o f p l a s t i c t e n s i l e deformation, see ( 2 ) ( 9 ) .
I n t h e f o l l o w i n g we use f o r t h e s t r e s s dependence o f H t h e f o r m u l a t i o n :
H(o) = ~ * - c . o ' - ~ ( 1 )
which was t e s t e d e x p e r i m e n t a l l y i n t h e range o f t h e a - r e l a x a t i o n , see ( 2 ) .
C9-708 JOURNAL DE PHYSIQUE
For t h e al peak i n Mo (KPF i n non-screw d i s l o c a t i o n s ) t h e e m p i r i c a l exponent K was o b t a i n e d e x p e r i m e n t a l l y ( 2 ) ( 9 ) as ~ = 0 . 5 t O . l , i n accordance w i t h t h e o r y (4). From (1) f o l l o w s f o r t h e a c t i v a t i o n volume:
ah c
~ ( 5 )
-
- = (1-K).-
ao oK ( 2 )
and from ( 1 ) and ( 2 ) :
Here a i s t h e l o c a l s t r e s s a c t i n on a d i s l o c a t i o n segment. I n o u r experiments we have as e x t e r n a l s t r e s s o ; I X ~ O - ~ G (G = shear modulus). The mean i n t e r n a l s t r e s s was obtained-from an at?alysis o f he amplitude dependence o f t h e r e l a x a t i o n
8
s t r e n g t h o f t h e a maximum t o ai:lO- G. I n t h e f o l l o w i n g we d i s r e g a r d o a g a i n s t oi.
The peak broadenihg o f t h e y maximum due t o a d i s t r i b u t i o n o f i n t e r n a l Btresses oi i s o b t a i n e d by averaging eq.3 over t h e specimen volume.
We i n t r o d u c e i n ( 4 ) t h e c o n d i t i o n o f Par&:
where ?WKis t h e KPF-energy, L t h e f r e e l o o p l e n g t h , and w r i t e v(o)=a-b-RK(o), where RK, i s t h e c r i t i c a l l e n g t h f o r KPF, i t f o l l o w s :
For molybdenum we determined 2WKzH
3
eV, and YI f we take ~ g 0 . 5 we o b t a i n f i n a l l y :
A s e t o f c o n s i s t e n t numerical values i s given b :
K = 0.5, L = 1000b, (1 = 25b, v = 2ib3, ~ . = 8 x l O - ~ G ( b = Burgers vector, G = shear modu- l u s ) . These values a r e compatible w i t h tAe kink-pair formation model. This i n t e r p r e - t a t i o n o f t h e y peak i s supported by f u r t h e r arguments, g i v e n i n ( 2 ) , ( 9 ) .
The anomalous ampl i t u d e dependence o f t h e y maximum i n t h e range o f low ampl i t u d e s appears t o be a general phenomenon ( 5 ) ( 6 ) . Anomalous ampl i tude dependences were ob- served i n another c o n t e x t by
KC?
e t a l . i n c o l d worked d i l u t e a l l o y s o f A1 ( 7 ) ( 8 ) . Here, i n b.c.c. metals t h e phenomenon may be r e l a t e d t o t h e i n s t a b l e d i s l o c a t i o n c o n f i g u r a t i o n s , c r e a t e d by low temperature deformation. This view i s supported by t h e f a c t t h a t i n t h e annealed specimen s t a t e , t h e a peak shows "normal" amplitude depen- dence. P o s s i b l y another e x p l a n a t i o n may be found i n a " c o u p l i n g " o f t h e a- and t h e y-processes, which operate simultaneously a t t h e temperature o f t h e y peak. A mored e t a i l e d r e p o r t wi 11 be given e l sewhere ( 9 ) .
( 1 ) Grau, R., and Schultz, H., J. Physique 42 (1981) C5-49 ( 2 ) Grau, R., D i s s e r t a t i o n U n i v e r s i t a t S t u t q a r t 1981
( 3 ) Nowick, A.S., and Berry, B.S., A n e l a s t i c R e l a x a t i o n i n C r y s t a l l i n e S o l i d s , Academic Press, New York 1972
( 4 ) Seeger, A., Z. Metallkde. 72 (1981) 369 ( 5 ) Rodrian, U., unpublished r e s u l t s on Ta ( 6 ) Matsui
,
H., unpublished r e s u l t s on Fe( 7 ) Kt, T.S., I n t e r n a l F r i c t i o n and U l t r a s o n i c A t t e n u a t i o n i n S o l i d s , p. 157, Pergamon Press, Oxford 1980, Ed. C.C. Smith
( 8 ) Pan, Z.L., Wang, Z.G., Kong, Q.H., and
