EVALUATION OF DISLOCATION PINNING EFFECTS FROM SIMULTANEOUS MEASUREMENTS OF ATTENUATION AND SOUND VELOCITY IN COPPER

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EVALUATION OF DISLOCATION PINNING

EFFECTS FROM SIMULTANEOUS

MEASUREMENTS OF ATTENUATION AND SOUND

VELOCITY IN COPPER

D. Lenz, K. Lücke, H. Schmidt

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C S , suppZe'ment au nOIO, Tome 4 2 , octobre 2981 page C5-381

EVALUATION O F DISLOCATION PINNING EFFECTS FROM SIMULTANEOUS

MEASUREMENTS O F ATTENUATION AND S O U N D VELOCITY IN COPPER

D. Lenz, K. Liicke and H. Schmidt

Ins ti t u t fur A2 Zgemeine Metal Zkunde und MetaZZphysik, RWTH Aachen, F. R. G.

Abstract.- The frequency dependence00 to 200 M H Z ~ O ~ the sound velocity and of the ultrasonic attenuation a has been measured on high purity Cu during 3 MeV y-irradiation at room temperature. The results are evaluated with respect to dislocation effects which under the present conditions cause the major contributions to the frequency dependent reduction of sound velocity AvD!v0 and attenuation aD. The y-dose dependence of AvD/v0 and a~ is In com- plete accordance with the Granato-Lucke theory of dislocation resonance damping. It is shown that simultaneous measurements of AvD/vo and aD strongly improve the accuracy of MHz-investigations of dislocation damping effects.

1 . Introduction.- In high purity single crystalline £cc metals and in

dilute alloy crystals (typically < 102 ppm) the ultrasonic attenuation a in the MHz f-range is mainly caused by anelastic dislocation effects

attributed to overdamped dislocation resonance /l/. The dislocation decrement 6 is accompanied by the dislocation modulus defect MD which is measured as a reduction and dispersion of sound velocity. Anelasticity due to dislocation resonance has been successfully described on the ba- sis of Koehlers vibrating string model /2/ by the Granato-Lucke theory

/ 3 / . This theory yields from 6 ( f ) - and MD(£)-data quantitative information about interesting dislocation properties (e.g. dislocation density, dislocation loop length, dislocation damping force factor).

Simultaneous measurements of damping

and

modulus are a standard technique in the Hz- and kHz-ranges whereas in most MHz investigations only the attenuation is measured. One reason is that the dislocation attenuation effects are large and easy to be measured as function of f by the pulse-echo technique whereas the related AV-effects are small and not easily measured as function of frequency. By the present investigation we try to find out whether simultaneous 6(f)- and MD(f)-measurements improve quantitative evaluation of dislocation damping phe- nomena.

2. Experimental.- The investigated sample was of high purity Cu (c5ppm unknown impurities, residual resistivity ratio RR~=2200), spark-erosion cut from a <Ill>-oriented Bridgeman crystal, carefully lapped (plan-

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C5-382 JOURNAL DE PHYSIQUE

parallelism and flatness of endfaces < 10-4 cm/cm, crystallografic or-

ientation difference < 0.25 degrees from <Ill>). After this preparation

the sample was pre-annealed (4h, 650°C, 3.3~1 o - ~ P ~ )

,

deformed to increa-

se the dislocation density (0.1% compression along <Ill>) and finally

annealed (4h, 650°C, 1.3~1 o - ~ P ~ ) in order to reduce and stabilize the

dislocation density and structure respectively /4/.

The ultrasonic phase velocity V was determined from v =n-2s/t with

s=sample length in direction of sound propagation, t-the travel time and n the number of round trips of the sound pulse between the end-

faces of the sample. The time t was measured at 25

+

0.05OC between 10

and 200 MHz by the pulse-overlap technique /5/ (absolute accuracy

5x1

o - ~

and a sensitivity of 1x1 o - ~ )

.

The attenuation was simultaneously

measured by the standard pulse-echo technique /6/. Matec equipment was used. For generation and detection of the sound pulses a compressional quartz-transducer (X-cut, 10 MHz fundamental frequency) was bonded onto the sample by "Nonaq". The y-irradiation was produced by a Van De Graaff electron accellerator in anAu-target (Frenkel defect production rate

5.5~10' FD/cm3 sec /7/)

.

(90 MHz). Because of too high initial 5155i

0 50 100 150 200

attenuation no measurements were possi- Frequency 1 [MHZI

J

3. Results.- Fig.1 shows the frequency

dependence of sound velocity v (f) before

ble above 90 MHz in the unirradiated

(open circles 0) and after (crossed

state. Before irradiation the velocity FIG 1 Dislocation contribut-

increases with increasing frequency in ion to frequency dependence

of sound velocity. the sense of "normal" dispersion. How-

Cu,longw~ve,q~~[ll1]~298t0OSK

ever, after irradiation the velocity decreases from 10 to 70 MHz and becomes f-independent at f>70 MHz. This "anomalous" dispersion behavi- our is due to phase effects caused by sound diffraction in the sound-

field of the transducer / 8 / . After correction for this effect and in-

fluences of the transducer-bond /9/ ( - 1 pm Nonaq thickness) shown in

fig.1 by the solid circles and crosses the velocity of the irradiated sample (crosses) is frequency independent within experimental accuracy

(ideal elastic body). This velocity v, is the reference velocity for

circles +B) heavy y-irradiation

(0

= 2500 pAh). This y-dose comple-

Y

-

21 "0

I

tely suppresses dislocation damping 2.10.~

T due to dislocation pinning by the

irradiation induced point defects. unlrradlated

Fig.1 shows that this pinning increases

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the separation of the wanted dislocation contribution AvD (f) from the measured

velocity v (f) according to AvD (f) =

v

-

v(£).

0

Fig.2a shows the frequency depen-

dence of attenuation a (f) for different

y-doses @ up to 2500 bAh. Since this

Y

dose completely suppresses the dislocation damping a is the background-

B

attenuation of the "dislocation free"

O 50 100 150 200

crystal / 4 / . Frequency t CMHZI

For the same y-doses as in fig.2a

Fig.2a

fig. 2b shows the v (f) -dependence (cor-

effect on sound velocity is only in the

1 O/oo range.

l 'i

rected for diffraction and bond effects).

4. Discussion.- Fig.3 shows the fre- 5 1 5 4

50 100 150 200

quency dependence of the dislocation Frequency f [ M H ~

Comparing fig.2a and 2b it is seen tnat

y-irradiation (i.e. pinning) decreases v 5165-

decrement (open symbols )

tqlong ~a~e,q~~[lt11, 298too5~

Fig. 2b the attenuation and increases the velo-

5

+ - . + - + - + - ,

city towards the limiting values aB and 550PAh

v

. resp.. In contrast to attenuation

e

5160 2.10-4

(showing

-

90% dislocation effect) the f

(a (f) -a, (f) ) [d~/bsecl /f [MHzI (1) _{FIG.2 Influence of y-irradi-}

ation on the frequency depen-

and dislocation modulus defect (solid dence of attenuation (fig.2a)

symbols) and sound velovity (fig. 2b)

.

MD: AM/Mo = [Mo-M(f)l/Mo = 2[v0-v(f)l/vo

Both quantities are compared w i t h the corresponding theoretical 6(f)-

and MD(£)-curve respectively given by the GL theory for an exponential

distribution of loop length / 3 / . With respect to the quality of the fit

we point out that the two theoretical curves used to fit the set of

measured data (6 (f) and MD (F)) for a given state of the sample (Qy) are

not independent. They must be thought as being linked together at one common point (large circles in fig.3) since dispersion and absorption in a dissipative medium are linked together by the Kramers-Kronig re- lation /10/. We mention furthermore that the irradiation effect is on-

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JOURNAL DE PHYSIQUE

D

Frequencv f [MHzl

FIG 3 y-dose dependence m ( £ ) - and MD(£)-

values in comparison with theoretical curves given by the GL- theory

duced defects is very low ( < 5 x 1 0 - ~ ppm) which excludes any measurable bulk effect on MD. Keeping both aspects in mind our irradiation ex- periment yields the strongest test of theory so far: our results (fig.3) are in excellent agreement with the GL-theory of dislocation resonance.

We now use the GL-theory for evaluation of dislocation pinning effects. The coordinates of G(£)-maxima (arrows in fig.3) are

2

fmax = 0.113 C/L B (4) (valid for the exponential loop length distribution, L = m e a n free loop length,

D

= orientation factor, G = shear modulus, b = Burgers vector, C = line tension, B = damping force factor /11/, A = dislocation density). The high frequency (f>>fmax) asymptote of the &(£)-curves

(dashed-double-aotted line of slope -1 in fig.3) is given by

The corresponding MD(£)-curves can be characterized by the coordinates of the point M D * = o . ~ MDLOW (arrows on dashed curves in fig.3) where MDLOW is the maximum modulus defect measured at f<<fMAX

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i s t h e same f o r a l l 6 ( f ) - c u r v e s i r r e s - p e c t i v e o f L . F i g . 3 p r o v e s t h e good

agreement between measured i r r a d i a t i o n

_-

;

i n f l u e n c e and t h e o r e t i c a l p r e d i c t i o n f o r

5

m s t a t i s t i c a l L-changes. h The number of a d d i t i o n a l i r r a d i a - a .$ t i o n i n d u c e d p i n n i n g p o i n t s p e r i n i t i a l

₂

l o o p l e n g t h P = ( L o / L ( g Y ) ) - 1 (L

(By)

=mean ' l o o p l e n g t h a f t e r i r r a d i a -

y-Irradiatlon Dose [pAh] t i o n w i t h d o s e

0 )

can b e c a l c u l a t e d a l -

t e r n a t i v e l y from t h e measured s h i f t s of FIG 4 y-dose dependence o f 6 ( f )

-

o r MD (f) - c u r v e s a c c o r d i n g t o 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 = J ( f m X ( @ Y ) / f m X ( 0 )

-

I = J f * ( g y ) / £ * ( 0 )

-

I ( 9 ) (C.£. equs. ( 4 ) and ( 7 ) )

.

We p o i n t o u t t h a t p i s o b t a i n e d w i t h o u t r e s o r t

t o t h e q u a n t i t i e s C , B , A e t c . i n equs. ( 3 1 , ( 4 ) , ( 6 ) , ( 7 ) . A s an example f i g . 4 shows t h e y-dose dependence o f p d e r i v e d from f i g . 3 and e q u . ( 9 ) . The r e s u l t i n g p ( g y ) b e h a v i o u r i s i n a c c o r d a n c e w i t h e a r l i e r r e s u l t s / 7 / by showing t h e unexpected p - s a t u r a t i o n a t r e l a t i v e l y low v a l u e s

( 2 1 . 5 ) .

A s mentioned above from p u r e l y t h e o r e t i c a l r e a s o n s t h e p r e s e n t t e c h n i q u e of s i m u l t a n e o u s 6 and MD measurements (which more t h a n doub- les t h e e x p e r i m e n t a l e x p e n d i t u r e ) y i e l d s no d o u b l i n g o f t h e i n f o r m a t i o n g a i n e d . However, we l i k e t o mention s e v e r a l advantageous applications o f t h e p r e s e n t t e c h n i q u e : i ) a t low f r e q u e n c i e s ( f < < f M A X ) t h e modulus de- f e c t a t t a i n s i t s maximum v a l u e MDLOW which can b e measured e a s i e s t whereas t h e decrement d e c r e a s e s t o v e r y low v a l u e s ( s i n c e a + a g ) . T h i s

2

s i t u a t i o n i s e x p e c t e d f o r impure samples ( s h o r t L, f<<fMAx-1/L

,

equ.(4)). The c o n t r a r y s i t u a t i o n i s met w i t h p u r e samples ( l a r g e L, f 2 f M A X ) which e x h i b i t r e l a t i v e l y s m a l l MD a t h i g h f a s compared w i t h 6 . I n t h i s c a s e a t t e n u a t i o n measurements a r e more s u i t a b l e t h a n MD-measurements. ii) An

even more i m p o r t a n t a d v a n t a g e of MD-measurements o c c u r s a t low tempe- r a t u r e s : I n c o n t r a s t t o a t t e n u a t i o n t h e sound v e l o c i t y i s n o t i n f l u e n - c e d by t h e phonon-electron i n t e r a c t i o n t o any p r e s e n t l y m e a s u r a b l e de- g r e e / 1 2 / . Thus measurements of t h e d i s l o c a t i o n modulus d e f e c t a r e im- p e r a t i v e a t t e m p e r a t u r e s where t h e a t t e n u a t i o n i s dominated by phonon- e l e c t r o n a b s o r p t i o n s /13/ ( e . g . w i t h Cu, R R R = I O O O , 50 M H Z a t T < 8 0 ~ ) . iii) F o r a simple Debeye relaxation(or w ~ < < l f o r an e x p o n e n t i a l d i s t r i b u - t i o n i n .r) G/MD=m. f

-

-c (m=numerical c o n s t a n t

relaxation

t i m e /l ,IO//. In the-

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C5-386 JOURNAL DE PHYSIQUE

yield T as function of e.g. temperature, deformation, pinning

...

with-

out knowledge of the relaxation strqngth of the process (however, such

measurements are misleading if more than one anelastic process contri-

butes to the measured 6 and MD resp.).

5. Conclusions.- 1. MHz-sound velocity measurements v(f) of the modulus

defect caused by dislocations are valuable in addition to measurements

of attenuation if they are done with an accuracy

<

5 x 1 0 - ~ over aboutone

decade in frequency. The v-measurements are especially valuable at low frequencies and low temperatures where they provide information about dislocation effects not easy t.o obtain from at.tenuation. 2. The present investigation shows that the frequency dependence of dislocation decrement as well as the frequency dependence of dislocation modulus defect is in agreement with dislocation resonance theory. The observed loop length distribution is given by the exponential distribution function. The same distribution holds for changes of the mean loop length

(produced by irradiation pinning). 3. Simultaneous measurements of dislocation attenuation and modulus defect provide a more stringent quantitative evaluation of dislocation pinning experiments (as compared with the less difficult attenuation measurement). This advantage is very important for quan<itative measurements of dislocation parameters

A

and L as function of temperature and/or plastic deformation /14/.

Acknowledgement.- Work supported by the Deutsche Forschungsgemeinschaft. References

/l/ D. Lenz, K. Liicke: Proc.5th ICIFUA, Vol.II,p.48, Springer Berlin 1975

/ 2 / J.S. Koehler: "Imperfections in Nearly Perfect Crystals", J.Wiley New York 1952

/3/ A.V. Granato, K. Liicke: J.Appl.Phys.Z;r, 583 (1956) /4/ P. Winterhager, K. Lucke: J.Appl.Phys.fi, 4855 (1973)

/5/ E.P. Papadakis: J. Acoust. Soc. Am.

9,

1045 (1967)

/6/ B. Chick, G. Anderson, R. Truell: J. Acoust. Soc. Am. ~ , 1 8 6 (1960)

/7/ P. Winterhager: Thesis T.H. Aachen (1968)

/8/ E.P. Papadakis: J. Acoust. Soc. Am.'%, 863 (1966)

/9/ H.J.Mc. Skimin: J. Acoust. Soc. Am.

33,

12 (1961)

/10/ A.S. Nowick, B.S. Berry: "Anelastic Relaxation in Crystalline Solids", Academic Press, New York 1972

/11/ R.M. Stern, A.\. Granato: Acta Met.

10,

358 (1962)

/12/ M.S. Steinberg: Phys. Rev.

111,

425 (1958)

/13/ A.B. Pippard: Phil. Mag.

46,

104 (1955)