ANOMALOUS MODULUS DEFECT DUE TO DISLOCATION PINNING

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ANOMALOUS MODULUS DEFECT DUE TO DISLOCATION PINNING

D. Lenz, H. Schmidt

To cite this version:

D. Lenz, H. Schmidt. ANOMALOUS MODULUS DEFECT DUE TO DISLOCATION PINNING.

Journal de Physique Colloques, 1983, 44 (C9), pp.C9-717-C9-721. �10.1051/jphyscol:19839108�. �jpa-

00223342�

ANOMALOUS MODULUS DEFECT DUE TO D I S L O C A T I O N P I N N I N G

D. Lenz and H. Schmidt

Institut

far

AZZgemeine MetaZZkunde und MetuZZphysik, RWTH Aaehen, F.R.G.

A b s t r a c t

-

The f r e q u e n c y dependence (10

-

100 MHz) of t h e d i s l o - c a t i o n modulus d e f e c t ( M D ) and damping Q-I h a s been measured a t RT d u r i n g y - i r r a d i a t i o n . I n v e r y h i g h p u r i t y s i n g l e c r y s t a l s

( r e s i s t i v i t y r a t i o RRR 2 10.000) d i s l o c a t i o n p i n n i n g a t low d o s e s l e a d s t o an i n c r e a s e of MD a t high f r e q u e n c i e s . Depinning e f f e c t s can b e excluded b e c a u s e MD a t low f r e q u e n c i e s and t h e s i m u l t a n e o u s l y measured Q-1 a t a l l f r e q u e n c i e s show normal d i s - l o c a t i o n p i n n i n g b e h a v i o u r f o r a l l y-doses. The p r e s e n t l y r e - p o r t e d anomalous MD-behaviour i s p r e d i c t e d by t h e t h e o r y and t h u s a s t r o n g proof f o r t h e d i s l o c a t i o n r e s o n a n c e mechanism.

I - INTRODUCTION

Damping Q-' and modulus d e f e c t MD d u e t o overdamped d i s l o c a t i o n r e s o - n a n c e of t h e Kohler-Granato-Lucke (KGL) t y p e / 1 , 2 / c a n b e d e s c r i b e d by

w i t h A = G ~ ~ A L ~ / I ~ c ; T = BL /12C 2 ( 3 a ) ; ( 3 b )

where 6 i s t h e l o g a r i t h m i c d e c r e m e n t , G = s h e a r modulus, b = Burgers v e c t o r , L = 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 , A = d i s l o c a t i o n d e n s i t y , C = l i n e t e n s i o n , f = w / 2 ~ = f r e q u e n c y of measurement and B = damping c o e f f i c i e n t . E q u f l - 3 ) a r e v a l i d f o r a d e l t a - f u n c t i o n d i s t r i b u t i o n o f L.

The l e n g t h L i s d e t e r m i n e d by p i n n i n g p o i n t s ( n o d a l p o i n t s of t h e d i s l o c a t i o n n e t work, i m p u r i t y atoms, i r r a d i a t i o n i n d u c e d p o i n t de- f e c t s e t c ) and i n c r e a s i n g t h e number of p i n n i n g p o i n t s n ( e . g . by i r r a d i a t i o n ) s h o r t e n s L = l / n . I r r a d i a t i o n p i n n i n g h a s been f r e - q u e n t l y used f o r s t u d i e s o f p o i n t d e f e c t k i n e t i c s . I n a l l known c a s e s b o t h Q-I and MD d e c r e a s e d u r i n g a c c u m u l a t i o n of immobile d i s l o c a t i o n p i n n i n g p o i n t s . S p e c i a l c a s e s / 2 / where mobile p i n n i n g p o i n t s a r e i n - v o l v e d a r e n o t c o n s i d e r e d h e r e . However, i t h a s been p o i n t e d o u t / 3 / t h a t a t h i g h f r e q u e n c i e s ( U T >> 1 ) t h e r e s o n a n c e damping t h e o r y p r e - d i c t s an i n c r e a s e of MD d u r i n g p i n n i n g . From e q u . ( l ) , ( 2 ) t h e change o f Q-1 and MD w i t h n i s g i v e n by

E q u . ( 4 ) shows t h a t i n c r e a s i n g n ( i - e . p i n n i n g ) always r e s u l t s i n de- c r e a s i n g Q-I which i s t h e normal p i n n i n g b e h a v i o u r . However, equ. ( 5 )

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19839108

C9-718 JOURNAL DE PHYSIQUE

shows t h a t MD d e c r e a s e s o n l y w i t h i n c r e a s i n g n i f w~

<

1 b u t i n c r e a s e s i f U T

>

1. T h i s l a t t e r MD b e h a v i o u r h a s b e e n named "anomalous" i n s p i t e o f t h e f a c t t h a t it i s a d i r e c t c o n s e q u e n c e o f d i s l o c a t i o n r e s o n a n c e b e h a v i o u r . To o u r knowledge e x p e r i m e n t a l e v i d e n c e f o r t h i s e f f e c t h a s n e v e r b e e n r e p o r t e d . N e v e r t h e l e s s , s u c h o b s e r v a t i o n s would s t r o n g l y c o n f i r m d i s l o c a t i o n r e s o n a n c e b e h a v i o u r .

I1

-

- EXPERIMENTAL

The t e c h n i q u e s f o r a t t e n u a t i o n a and sound v e l o c i t y v measurements and t h e p r o c e d u r e s t o s e p a r a t e t h e d i s l o c a t i o n e f f e c t s by y - i r r a d i a t i o n h a v e b e e n d e s c r i b e d e a r l i e r / 4 / . The u l t r a p u r e C u - c r y s t a l ( < I l l >

1 1

sound d i r e c t i o n ) was p r e p a r e d by e l e c t r o l y t i c a l (HN03-) p u r i f i c a t i o n o f ASARCO-Cu f o l l o w e d by vacuum f u s i o n and c r y s t a l growth i n a h i g h p u r i t y (ULTRA-CARBON) g r a p h i t e c r u c i b l e / 5 / . The c r y s t a l h a s a resi- d u a l r e s i s t i v i t y of p0(4.2K) = 0.17 nficm c o r r e s p o n d i n g t o t h e r e s i s t i - v i t y r a t i o RRR = p(293K)/po(4.2K) Z 10.000. A f t e r p r e p a r a t i o n / 6 / t h e s a m p l e was deformed 0.02% I n c o m p r e s s i o n a l o n g < I l l > and a n n e a l e d 1 h a t 373 K. The y - i r r a d i a t i o n and t h e u l t r a s o n i c measurements were p e r - formed a t room t e m p e r a t u r e . A f t e r e a c h i r r a d i a t i o n r u n t h e i n i t i a l

( u n i r r a d i a t e d ) sample s t a t e was r e - e s t a b l i s h e d by a n n e a l i n g t h e sample 2 h a t 773 K / 7 / b e f o r e t h e a- and v-measurements were r e p e a t e d a t a n o t h e r f r e q u e n c y a s f u n c t i o n of i r r a d i a t i o n d o s e .

I11

-

RESULTS

F i g . 1 shows t h e f r e q u e n c y dependence of d i s l o c a t i o n d e c r e m e n t 6 ( = 0.115 a / £ ) a n d modulus d e f e c t MD ( = AM/Mo = 2 ( v o - v ) / v o ) f o r t h e u n i r r a d i a t e d sample

(Gy

= 0) and a f t e r

Gy

= 10 fiAh 3 MeV y-dose.

The s o l i d and t h e dashed c u r v e s f i t t e d t o t h e measured d a t a a r e t h e t h e o r e t i c a l m a s t e r c u r v e s g i v e n by t h e KGL-theory f o r a n e x p o n e n t i a l L - d i s t r i b u t i o n / I / . The l a r g e open c i r c l e reminds t h a t t h e 6 - and MD- m a s t e r c u r v e s a r e n o t t o b e f i t t e d i n d e p e n d e n t l y b u t o n l y a s a f i x e d s e t / 4 / . Values f o r A and L a r e o b t a i n e d by t h e u s u a l e v a l u a t i o n pro- c e d u r e / 4 / e . g . from t h e c o o r d i n a t e s of t h e d e c r e m e n t maximum

f,, = 0.113 C/L B 2 2 2 6

,

, = 0.565 RGb AL / C

With f m X = 5 . 3 MHz, 6 , ~ = 0.01 Np ( F i g . 1 ) and C = 5 . 5 . 1 0 - ~ ~ N ,

G = 4 . 0 8 - 1010 N / m 2 , B = 6 . 5 - 10-5 NS/m2,

R

= 0.088 ( o r i e n t a t i o n f a c t o r ) , b = 2.55.1 0-A m w e g e t t h e i n i t i a l l o o p - l e n g t h Lo = L(yy=O) =

4.2- 10-5 cm and t h e d i s l o c a t i o n d e n s i t y A = 2 . 3 - 1

o7

cm-

.

A f t e r

gy

= 10 J L A ~ i r r a d i a t i o n ( c f . F i g . 1 ) t h e l o o p l e n g t h i s r e d u c e d t o L(Gy = 10 fiAh) = 2 . 6 . 1 0 - ~ cm.

F i g . 2 compares t h e change o f d e c r e m e n t 6 (Gy)-6($Zjy=0f measured a t 1 0 , 50 and 70 MHz d u r i n g y - i r r a d i a t i o n (open s y m b o l s , r i g h t s c a l e ) w i t h t h e s i m u l t a n e o u s l y measured change of modulus d e f e c t { M ( @ Y = o )

-

M

( G y )

}/Mo = 2 {v

( G y )

^-V

(Gy

= 0 ) }/vo ( s o l i d symbols, l e f t s c a l e )

.

^Whereas

t h e d e c r e m e n t a t a l l f r e q u e n c i e s d e c r e a s e s w i t h Qy t h e modulus d e f e c t dogs s o o n l y a t 10 MHz. A t 50 and 70 MHz, however, MD f i r s t i n c r e a s e s (by 2 . 1

o - ~

a t 50 MHz and by 3 - a t 70 MHz) and s t a r t s t o d e c r e a s e a t

Gy

? 10 fiAh.

I V

-

DISCUSSION

-

The q u a l i t a t i v e l y d i f f e r e n t MD(py)-behaviour a t t h e low and t h e h i g h e r

duced from the measured &(%)-data by shifting the KGL-master curves along the slope-I line (dotted line in Fig.1). The agreement between measured and predicted MD(gy)-behaviour is in accordance with the

6

(f) and MD

(f) measurements (Fig.

I

) .

and with their theoretically pre- dicted changes during irradiation and furthermore it justifies the use of the exponential L-distribution in Fig.1. The anomalous MD increase is only observed as long as during irradiation the condition f > 8 f m x is fulfilled. This is seen in Fig.3 which shows the irradiation in- duced change of the decrement maximum fmX(gy) and the additional number of pinning points per initial loop length p (gy)

= { L ~ - L

(Gy) I / L(gy)

=

~ ' f ~ ~ ( @ ~ ) / f ~ ~ ( O ) -

^1.

By comparison with Fig.2 it is seen that the modulus starts to decrease with 0 (i.e. shows the normal behaviour) as f

A

approaches the frequency f * (fX 10 MHz for f -50 MHz; f*

Z

14 MHz

for !!=TO MHz). From the dose dependence of the number of additional

pinning points ~ ( $ 8 ~ ) in Fig.3 it is found that MD turns to normal pinning behaviour at p

1 1.

It is pointed out that the tbtal number of pinning points involved in the experiment is very small. With the above values for A and Lo the Y-dose of gy

=

15 pAh (which acc. to Fig.3 induces about one pinner per initial loop length) results in pA/Lo

=

5.5.1011 pinners/cm3 which corresponds to about ppm and shows the high sensitivity of MHz dislocation pinning measurements.

Conditions for observation of the MD anomaly:

Several conditions must be fulfilled in order to observe the anomalous MD-effect. We believe that the difficulties involved with these con- ditions explain why the effect has not been observed hitherto. The conditions are

(i) In the case of a delta-function L-distribution the effect occurs for f > fMAX

= 12

C / ~ ? T B L ~ (cf. equ. (3b)

)

. However, in reallity L is the mean loop length of an exponential L-distribution (cf. the fits in Fig.1). The shaded area in Fig.1 shows that for the exponential L- distribution the MD-effect occurs at very high frequencies (and is of very small magnitude (AM/M in the 10-5-rang

) ) .

In this case the above condition is changed to f 2 ^8fmX

⁼

0.9 C/L B (cf. equ.(6)). The factor 5

8 is derived experimentally by comparing the MD(gy)-behaviour in Fig.2 with fmX (Gy) in Fig. 3.

(ii) Since the dislocation line tension C

( -

G) slightly decreases with increasing T and the phonon viscosity B

( -

T, for T > 0.1

8 ~ ,

eD

=

345 K, Debeye temperature of Cu) increases with T the experiments are to be performed at temperatures as high as possible for stable dislocation pinning by irradiation. For Cu this temperature is about 400 K

/7/.

(iii) Most important, however, is the fmX - ^L~ dependence (equ.(6)) calling for as long as possible mean free loop length Lo. Since Lo is determined by the residual impurities only crystals of very high puri- ty are useful. Standard high purity Cu (RRR

2

1000) as has been used in dislocation resonance experiments before

/6/

exhibits the 6(f)- maximum in the range 30 < fMAX < 50 MHz. Since the pulse echo frequency

range is limited to f

5

200 MHz the anomalous MD-effect cannot be ob- served with such crystals because of condition (i). Only in crystals of super high purity (RRR

1

10.000) f m x is lowered to about

5

MHz

(cf. Fig.1) and the MD-effect becomes observable at f > 40 MHz.

(iv) The highest attenuation which can be measured by the pulse echo technique is a < a ~ ~ M . 1 4 dB/bsec for observation of at least 2 echos.

The corresponding maxlmum decrement GLIM which can be measured is shown

in Fig.1. Since 6 -

^L~

^{for f}

²

fmX the increased damping due to the

samples high purity (long loop length Lo) has to be com~ensated either

by careful annealing of predeformed samples in order to sufficiently

C9-720 JOURNAL DE PHYSIQUE

reduce A or by microplastic deformation of a well annealed sample ha- ving very low initial dislocation density. The latter technique was chosen in the present experiment (deformation 0.02% in compression).

Such preparation techniques ensure that the dislocation density re- mains low enough to fulfill the condition 6

<

dLIM but is as high as possible in order to provide measurable MD-effects. As seen in Fig.2 this ~ o i n t is very critical since the anomaly is very small (AM/Mo

1

3-10- , cf. also Fig.1) whereas pulse echo overlap measurements resolve only MD-changes in the range AM/Mo

2 1 - 1 0 - ~

under high attenuation con- ditions i.e. where only two echoes are available for overlap.

Origin of anomaly: Fig.4 explains the physical origin of the MD anomaly.

In the upper part of Fig.4 the stress wave cr(wt) forces the dislocation segment of length L to oscillate. Due to dissipative forces (phonon viscosity B) the displacement 5 is phase shifted b y y with respect to

U.

The damping Q-I and the modulus defect AM/& are proportional to 5 "

resp.C1(i.e. the 90'-out-of-phase and the in-phase components of dis- placement). In the lower part of Fig.4 c" ^and c' are depicted for the high frequency case f

=

w / 2 ~ > > fmX of overdamped resonance (cf. equ.

(6)). Consider the unpinned (p=O) loop: it is seen that the dislocation moves 90'-out-of-phase as a rigid rod over most of its length whereas only close to the pinning yoints small in-phase contributions

are ob-

served. This means that Q- is much higher than AM/Mo(see also Fig.1).

Now consider the pinned (p=l) loop: ~t is seen that the out-of-phase displacement (i.e. the damping) is somewhat reduced. However, the total in-phase displacement is increased since the added pinning point gives rise to two additional small in-phase contributions in its vici- nity, i.e.to the "anomalous" MD increase.

V - CONCLUSIONS

The present measurements show for the first time that the dislocation modulus defect at very high frequencies increases with dislocation pinning in the overdamped state. This observation proves the mechanism of dislocation resonance. The magnitude of the effect is in accordance with the exponential distribution of dislocation loop length. The paper discusses the narrow experimental conditions for the observation of the dislocation modulus defect "anomaly" which can be measured only in crystals of super high purity after careful treatment for optimal dislocation density.

ACKNOWLEDGEMENTS

Work supported by the Minister fur Wissenschaft und Porschunq NEW, West-Germany .

REFERENCES

/1/ J.S.Kohler: "Imperfections in Nearly Perfect Crystals"

J.

Wiley, New York 1952.

A.V.

Granato, K.Lucke:

J.

Appl. Phys. 27 (1956) 583.

/2/

D.Lenz, K.Lucke: in "Proc. 5th ICIFUA" (D.Lenz,K.Lucke Eds.) Vol.11, p.48, Springer Berlin, Heidelberg, New York, 1974.

/3/ R. Truell, A.V.Granato:

J.

Phys. Soc. Jap.

18

Suppl.1 (1963)- /4/ D.Lenz, K.Lucke, H.schmidt:

J.

de Physique, Colloque

C 5

(19811381.

/5/

H-Briining: Thesis RWTH Aachen, 1982.

/6/

P.Winterhager, K.LUcke: J. Appl. Phys. 44 (1973)

4855.

/7/ H.Inagaki, F. Hultgren, K-LGcke: Acta Met. 18 (1970) 713.

d u l u s d e f e c t AM/& f o r t h e un- i r r a d i a t e d s t a t e and a f t e r a 3 MeV y-dose of gy = 10 L L A ~ . The shaded a r e a shows t h e "ano- malous" i n c r e a s e of AM/M, a t h i g h f r e q u e n c i e s . The s o l i d and t h e dashed c u r v e s f o l l o w from t h e KGL-theory f o r a n , e x p o n e n - t i a l l o o p l e n g t h d i s t r i b u t i o n .

F i g . 3 : y-dose dependence o f fMAX ( = f r e q u e n c y o f t h e de- crement maximum, c f . F i g . 1 ) and p ( = number of i r r a d i a t i o n induced p i n n i n g p o i n t s p e r i n i t i a l l o o p l e n g t h L o ) .

y - i r r a d i a t i o n . The decrement a t a l l f r e q u e n c i e s and t h e modulus a t 10 MHz show normal Oy-depen- dence. A t 5 0 and 7 0 MHz t h e mo- d u l u s b e h a v i o u r i n i t i a l l y ( f o r Oy S 7 kAh) i s anomalous ( i - e . t h e modulus d e c r e a s e s w i t h p i n n i n g )

.

cos W +

2-

^ela ~ ~ ~ ( w t - v l

stresrmve -.? --.--- --'L = 3 ' r o r w t + ~ ~ r i n w t

Oatoratton ~ m p

-= ⁷

w = ? r f , p=O p=l

L LIZ

Fig.4: Schematic e x p l a n a t i o n o f t h e modulus anomaly: Changes of damping ( - 90'-out-of-phase d i s - placement

5")

and modulus d e f e c t

-

in-phase-displacement

5'

) of a d i s l o c a t i o n l o o p L i n t h e over- damped s t a t e a t h i g h f r e q u e n c i e s

( W T

>>

I , i . e . f

>>

f , c f . F i g . I ) . due t o one a d d i t i o n % X p i n n i n g p o i n t .