Dislocation relaxation in silver, gold, palladium and platinum

SUPPLEMENTO AL VOLUME XVII1, SERIE X N. 1, 1960 DEL NUOVO CIMENTO 4 ~ T r i m e s t r e

Dislocation Relaxation in Silver, Gold, Palladium and Platinum.

P. G. BORDO~I

Is t i tuto di F i s ica-Matemat ica del l 'Universi th - P i sa

M. lquovo and L. VE~DINI

Is t i tu to Nazio~eale di Ultracustica (, O. M. Corbino ~) - Roma

( r icevuto il 15 N o v e m b r e 1960)

CONTENTS. - - 1. Introduction. - 2. Results - 3. Discussion. 1. Activation energy. 2. Relaxation spectra. 3. Frequency relaxation. 4. Influence of anneal and cold work. 5. ,~ubsidiary peaks. - 4. C o n c l u s i o n s .

1 . - I n t r o d u c t i o n .

The existence of a relaxation effect due to the dislocations in silver, gold,

palladium and plat inum has already been reported in a preliminary commu-

nication to the Third Internat ional Congress of Acoustic (Stuttgart , 1959) (~).

In the present paper a larger amount of experimental evidence concerning the same metals is analysed in order to obtain a more complete knowledge of the

effect, as it wns previously done with copper (2).

In addition to the fundamental parameters (activation energy and cha.r-

acteristie time) whose values are discussed in the light of the present theories

of dislocation motion, many other aspects of the effect have been considered,

such as the width of the relaxation spectrum, the relation between the total

(1) p . G. BORDONI, M. •UOVO a n d L. VERDINI: Relaxatio~t of dislocat~on~ in ]ace- centered cubic metals, in Proc. o] the Th ird In ternat ional Congress o] Acoust ics ( S t u t t g a r t , S e p t e m b e r 1959), in press.

(2) p . G. BOI~DONI, M. NUOVO and L. VERDL~I: Nuovo Cime~do, 14, 273 (1959).

56 P . G . BORDONI, M. NUOVO a n d L. VERDINI

number of dislocations and the characteristic time, the influence of anneal

and cold work upon the total number of active dislocations and upon the width

of the relaxation spectrum. Much at tent ion has also been given to the sub-

sidiary or ~ible t t and Wilks peaks, which seem to be a general if not very

evident feature of the relaxation effect.

I t may be hoped that the considerable amount of experimental evidence

which is now available shall provide a good basis for a more detailed model

of dislocation motion, and for a better theoretical t rea tment of the relaxation

effect.

2 . - R e s u l t s .

The measurements have been made on circular polycrystalline plates of

chemically pure silver, gold, palladium and plat inum whose main character-

istics are listed in Table I together with their thermal and mechanical his-

tory (a). The experimental technique is the same used in the previous in-

vestigation on copper (4).

The energy dissipation and the resonant frequency have been measured

as a function of the temperature T in the range (60 + 300) ~ for the two

lowest flexural symmetrical modes of the plates, having respectively one and

two nodal circles ~nd no nodal diameters. On the same specimens the dissi-

pation and the resonant frequency have also been measured for one or two

longitudinal modes. These measurements have proved more difficult th~n the

corresponding ones on copper, owing to the coarser grain structure of the

present specimens. I t has been found that the average grain size may be of the order of some millimeters when the material is machined from the cast.

When the wavelength is of the same order of magnitude, the amplitude vs.

frequency curve is no longer bell-shaped but exhibits side peaks and the meas- urement of dissipation is no longer possible. The above difficulty has been over-

come by obtaining a finer grain structure whenever possible by cold-rolling and

intermediate annealing (silver no. 3, Table I) ~nd in every ease by measuring

the dissipation for longitudinal waves with wavelengths larger than 4.5 mnl.

(3) Tile authors wish to thank the ~(Ditta E. l,ongobardi - Metalli Preziosi ~> for its co-operation in the preparation of tile samples and in their analysis. They also express their gT~titude to Prof. D. CTENTILON1 and to Mr. MACCARRONE of the Istituto di Scienza delle Costruzioni of the University of Rome for their kind help in the mechanical treatment of the specimens.

(4) See the paper quoted in footnote (2). Additional information on the experimental apparatus may be found in the following papers: P. G. BORDONI : NUOVO Cime.~tto, 4, 177 (1947); Ric. Scient., 18, 103 (1948); P. G. BORDONI and M. NL~ovo: ,4custica, 7, 1 (1957); Ric. Scieq~t., 27, 695 (1957); Ac,ust.ica, 8, 351 (1958).

c

D I S L O C A T I O N ll, I ~ L A X A T I O N I N S I L V E R , G O L D ~ P A L L A D I U M A N D P L A T I N U - ' ~ [ 5 7

I n p l a t i n u m an i n t e r m e d i a t e v i b r a t i o n m o d e has also been e m p l o y e d fo r

d i s s i p a t i o n m e a s u r e m e n t s .

TAJ3L~ I. - Characteristics o] the specimens.

3I:etM

Silver

P u r i t y Thermal and mechanical t rea tments I

__I . . . . .

Cross rolled after casting from 12 mm to 10 ram; machined.

[0.9980

Spee-,] imen

] (a)

2 (a)

(b)

3 (c) (d) (e)

Cross rolled after casting from 12 mm ~o 10 ram; machined.

(?old rolled from 10 mm to 7.5 ram; annealed 1 h at 900 ~ cold rolled from 7.5 to 5.0 inm; machined. Annealed 3 h at 460 ~ Annealed 4 h at 493 ~K. Permanent ly strained 2.7% a.t room temperature .

1)iam- Thick- eter ness

(mm) (ram)

36.0 10.00

29.0 5.00

I

36.0,) 4.55 :

i

, i ' ( a ) Machined from the cast. Gold ' 0.998 8

Pal ladium 0.999 1

i

Plat inum 0.9989

! ( b )

(e)

- i (a.)

(b) (c)

Annealed 7 h at 502 ~ 25.0 3.02 Permanent ly strained 3.8% at room temperature .

i~Iachined from the east and permanent ly strained 1% at room temperature. Annealed 6 h at 423 ~ Permanent ly s~rained 6.9% at room temperature .

i i

, 25.2 I 2.95 l

(a) Machined from the cast. (b) Annealed 4 h a.t 500~ (c) Annealed 11 h at 693 ~ (d) Annealed 1 h at 793 ~ (e) Permanent ly afrained 3.2% at, room

temperature .

I 36.0 : 3.03

(*) W i t h ~ r i n g o f 11 .0 m m e x t e r n a l d i a m ' ~ t e r a n d 0 . 6 m m t h i e k ~ m s g .

As i t is shown b y t h e lower d i a g r a m of F ig . 1, t h e d i s s i p a t i o n p e a k in s i lve r

is qu i te wel l m a r k e d , wh i l s t a m u c h lower a n d f l a t t e r p e a k was f o u n d in a

p r ev ious i n v e s t i g a t i o n on a n o t v e r y p u r e spec imen of t he s a m e m a t e -

~ 8 P. G, BOF~DONI, M. NUOVO and L. VERDINI

rial (+). As it was observed in lead, copper and aluminium, the t empera tu re T~

corresponding to the m a x i m u m dissipation increases with the v ibra t ion frequency ]m of the specimen (6). The same peak is also clearly shown by the dissipation

600. ! 0 "5

400 f -f R f q

~ I | 0

Ag

50 75 100 125 150 Temperature (~

(b

200

+5 ' I U

(2-1

./ /

100 %

Ag

0 50 75 100 125 150

Temperature (~

Fig. 1. - Dissipation peak and frequency relaxation in silver; curve A, ],+= 18.58 kHz; B, fm=50.55kHz; C, ]m=421.4kHz; D, ]+,+= 820.4 kHz.

(5) I). G'. BORDONI: Journ. Acoust. Soc. Ant . , 26, 495 (1954); Ric. Scie~tt., 23, 1193 (1953).

(.i) As the resonant frequency changes wil~h temperature, ]m is the value measured at the temperature T,+.

DISLOCATION R E L A X A T I O N i N SILVER~ GOLD, P A L L A D I U M AND P L A T I N U M 5 9

measurements in gold (Fig. 2, lower diagram), whilst in palladium and platinum the peak is somewhat smaller and flatter bu t still very evident (Fig. 3, 4 lower diagrams). As in the ease of silver, the tempera ture Tm is an increasing function of the frequency 1,,.

a)

- 5 600" lu

40(

20(

fR

k i

100 200

Au

50 150 250 Temperature (*K)

b)

300.10 -5

200

100

01

A l l

\ ""---.o

50 100 150 200 250 Temperature (~

Fig. 2. - Dissipation peak and frequency relaxation in goht; curve A, ]~= 15.99 kHz; B, ]m--55.60kttz; C, ], ,=561.0kHz.

The upper diagrams of Fig. 1-4 give the relative difference ( / - - /~) / /~ between the resonant f requency ] measured at the tempera ture T, and the re-

laxed value ]R of the resonant f requency computed for the same tempera ture

(]0 P . G . ] ~ O R D O N I , M. N U O V O and ~.. V ] ] R D I N I

by means of a linear ext rapola t ion of tlle room tempera tu re slope (7). These

differences have been computed only for the flexural modes, as the exper imenta l

-5 600.10

a)

40C

20C

fR

l

Pd

0 100 200 300 400

Temperature (=K)

80.10 -s

if ' 6O

0 0

b) 20

/

.z

Pd

�9

|

lO0 200 300 zOO Temperature (~

Fig. 3. Dissipation peak and frequency relaxation in palladium ; curve A,/,,,= 24.92 kHz ; B, .f,~=88.35 kttz; C, ],~=387.9 kHz.

difficulties due to the grain s t ructure of the specimens make the f requency

measurements less accurate for longitudinal vibrations. In the upper d iagrams

of Fig. 1, 2 (silver and gold) an inflexion point is found at a t empera tu re ve ry

(7) See the paper quoted in footnote (2), p. 307.

DISLOCATION R]~LAXATION IN SI:LVEI~ GOLD, PALI~ADIUM AND PLAT1NU?~[ 61

near to t h a t of the dissipation peak for the same vibra t ion mode. This coinei- deuce shows t h a t the peaks are due to a relaxation ejJeet.

600 I0 -5

400

a) 20(]

f- fR q~ 7;

,%

Pt

0 0 100 200 300 400

Temperature (~

-5 150. I0

~oo Q-i ~'k Pi"

0 I00 200 300 400 Temperature (~

Fig'. 4. - Dissipation peak ~nd frequency relaxation in platinum ; curve A, f,, 9.6 kHz ; B, ]~=42.41 kHz; C, ],~=111.8 kHz.

The inflex~on is not evident in the upper diagrams of Fig. 3, 4 (pal ladium

and pla t inum) being masked by the pronounced curva ture of the elastic mo-

dulus v~. t empera tu re d iagram exhibi ted by the transi t ion Inetals in this tempe-

future range (8). This curva ture has an anomalous character as the concavi ty

(8) j . A. RAY.N'E: Phys. Rev., 118, 1545 (1960).

6 2 P. G. ~ O R D O N I , M. NUOVO and L. V E R D I N I

is tu rned towards the high values of the modulus, and it is not related to any re laxat ion effect but to the conduction electrons. However there can be no

doubt tha t a f requency re-

2.5. lO -3

2.0 - , \ \ ' , \ ' "

:; 1.s -,,

o 0.5 i.o 1.5. io -z -I

T m (~

Fig. 5. - Frequency dependence of the temperature T~ of maximum dissipation.

laxat ion takes place also for

pal ladium and p la t inum in the same t empera tu re interval in which the dissipation peak is observed.

The exper imenta l values of ]m and Tm are listed in

Table I I together with the

m a x i m u m energy dissipa-

t ion Q~I. When the loga-

r i t hm of ]m is p lo t ted

against ] '~ , as it is done in Fig. 5, the points for each meta l lie on a s t ra ight line; hence the re laxat ion effect is thermally activated, as it has already been found

in the case of lead, copper

and aluminium.

To establish an experimental relation between

the re laxat ion effect and

the mot ion of dislocations, the influence of anneal and cold work on the dis- s ipation peaks and on f requency relaxat ion has been sys temat ica l ly investigated.

I t has been found tha t a mode ra t e anneal decreases the height of the peak, as is shown for silver and p la t inum (Fig. 6, 9, lower diagrams) by the comparison of the curves A, obta ined when the mater ia l was in the original state (~), and the curves B, taken af ter an annealing t r e a t m e n t (10).

I f the t empera tu re of the t r ea tmen t is high enough, the peak m a y even-

tual ly be cancelled as it is shown by the curve C of Fig. 6 {lower diagram)

for silver and by the curves B of Fig. 7, 8 (lower diagrams) for gold and pal-

ladium. In the case of p la t inum the height of the peak is also dependent upon

the t empera tu re of anneal, bu t some trace of it is still present af ter the spec-

imen has been kep t for 1 h at 793 ~ (Fig. 9, lower diagram, curves C and D).

I t m a y be added tha t the effect of an h igh- tempera ture anneal is somewhat

(9) That is immediately after the mechanical treatments listed in Table I under (a) for gold, palladium and platinum, and after (a) and (b) for silver.

(lo) For the details of these treatments see Table I and the captions of the figures.

DISI, OCATION R.FLAXATION IN S I L V E R , GOLD, P A L L A D I U M AND P L A T I N U M

TABLE I I . - E.rperime,~tal datzt o~ the relaxation peal,:.

63

M e t a l

S i l ve r

~ p e e i m e n L a s t (*) t r ea , tm en t

a

(kHz) C K )

1 28.61 73.3 1 2 9 . 1 0 -5

2 a 79.99 79.3 150

18.58 75.8

50 .55

421.4

b

b

b

b

C

820.4

81.0

91 .0

93 .0

i 55 .28 72.5

228

254

d

(,

288 p

255

235

i I ]

I . . . . . . . . . . . i ! . . . .

~5.35 80.0 184

a i 15.99 I

a 55 .60 ,

( ;o ld a i 561 .0 284 . . . . . . . . i . . . .

b i I

c 15.36 118.0 150

117.7 202

127.7 221

153.0

30

59

a

a

24.92

88.35

387.9

180.0

125

9O

186.5

P a l l a d i u m

P l a t i n u m

a 200.0 75

b . . . .

e 21.69 187.0

173.2 9 .600 a

a 42:41 201 .0 116

a 111.8 213.0 138

b 9 .641 166.0 21 .5

c 9 .652 166.0 10.0

d 9 .653 166.0 9 .0

e 8 .778 180.0 137

( ') For t i le de t a i l s of t h e r m a l and m e c h a n i c a l t ren . tments , see T a b l e I .

0 4 :P. G. ]~ORDONI, ~I. 57UOVO anr L. V]~RDINI

m a s k e d in si lver b y the large increase of the d i ss ipa t ion measu red ,~t room

s which has a l r eady been observed in copper af ter a t h e r m a l

57s

55.(~

" I "

"~ 53.C

51.C

49 .C 0

-5 300.10

50 100

X~x~x ~

150

Temperature (~

Ag

- x

250 200 300

200

100 b)

,I

Ag

50 100 150 200 250 300

Temperature (~

Fig. 6. - Influence of anneal on the height of the dissipation peak and on the frequency relaxation in silver. Curve A, after the treatments (a) and (b) of Table I ; B, after

3 h st 460 ~ C, after 4 h at 493 ~

D I S L O C A T I O N R E L A X A T I O N IN .~ILVER, G O L D , P A L L A D I U M A N D P L A T I N U M 6 5

17.4

16.!

T

16.,

a)

15,$

15.4

Au

50 100 150 200 Temperai:,re (~

250 300

300.10 -s

200

100

b~

0

ffi Au

I I

* - T X ~ [

0 50 100 150 200 250 300 T e m p e r a t u r e ( ~

Fig. 7. - In f luence of a n n e a l on t h e h e i g h t of t h e d i s s ipa t ion peak and on t h e fre- q u e n c y r e l a x a t i o n i n gold. Curve A, a f t e r t h e t r e a t m e n t (a) of Tab le I ; B, a f t e r

7 h a t 502 ~

'g 5 - Supplemento al Nuovo Cimento.

66 P . G . BORDONI, M. NUOVO and L. VERI) INI

25.4

25.2

N " r - ..x

25.1 U r r

r 01

a)

24.6

% Pd

50 100 150 200 250 300 Temperature (~

b)

30-10 -5

Q-1

2(

10

0 0

/ Pd

/ X

i 50 100 150 200 250 300

Temperature (~

Fig. 8. - In f luence of a n n e a l on t h e h e i g h t of t h e d i s s ipa t ion peak and on t h e fre- q u e n c y r e l a x a t i o n in pa l l ad ium. Curve A, a f te r t h e t r e a t m e n t (a) of T a b l e I ; B, a f t e r

6 h a t 423 ~

D I S L O C A T I O N R ~ L A X A T I O N I N S I L V E R , G O L D , P A L L A D I U M A N D P L A T I N U M 6 7

t r e a t m e n t (~1) ~nd which is no t re lu ted to the r e l axu t ion effect. ~ o such

increuse is no t iced in the o ther metuls , which seem therefore to be b e t t e r

9.8

9.7

A N

9.6

9.5

a)

9.4

100.~0 -s

Pt

50 100 150 200 250 300

Temperature (~ 350

b)

75

0-1 P~

50

2~

/ t

F

0 50 100 150 200 250 300 350

Temperature (~

Fig. 9. - Influence of ~nneal on tile height of the dissipation peak and on the frequency relaxation in platinum. Curve A, after the t reatment (a) on Table I ; B, after

4 h at 500~ C, after l l h at 693~ D, after 1 h at 793~

(11) See the paper quoted in footnote (2), p. 292, or D. H. NIBLETT and J. WII, KS: Phil. Mag., 1, 415 (1956); 2, 1427 (1957); Adv. in, Phys'., 9, 1 (1960).

6~ P . G . BORDONI~ M. NUOVO a n d L. VERDINI

suited for an investigation on the influence of anneal u p o n the dissipation peak.

All the annealing t rea tments were made in an evacuated container and they lasted long enough to allow an equilibrium state to be reached by the material . One of the resonant frequencies of the specimen was measured during the t rea tment , and the anneal was not stopped until the f requency was ap- proximate ly constant with time, as it is shown for silver and gold, in Fig. 10.

1.018

Fig. 10.

1.012 [En' of anneal

1 .oo6 , . /

,j f / T o 1 3 4

Tim e ( Hours )

Increase of frequency ](t) with time t at the constant temperature of 493 ~ for silver and of 502 ~ for gold.

The upper diagrams of Fig. 6-9 show the influence of anneal upon the f requency of the lowest flexural mode of the specimens. For silver and gold (Fig. 6, 7, upper diagrams) this influence is similar to tha t already observed in copper (12). The frequency vs. t empera ture curve is shifted after anneal

towards higher values of the frequency. I ts inflexion decreases with the height of the dissipation peak, as it is required by the relaxation theory, whilst its

slope is little or not changed at temperatures higher than T,.. After an high- tempera ture anneal, when the dissipation peak has been cancelled, the frequency curve no longer exhibits any trace of inflexion.

(12) See the paper quoted in footnote (~), p. 291.

DISLOCATION RELAXATION IN SILVER, GOLD, PALLADIU}[ AND PLATINUM 69

In palladium and plat inum (Fig. 8, 9, upper diagrams) the effect of anneal on the resonant frequency is less obvious, as the frequency vs. temperature

58O

~56.0

r

u. 54.o a)

52.0

Ag

300.t~ 5

50 I00 150

Temperature (~

200 250 300

200

100

b)

o

if ' Ag

0 50 100 150 200 250 300 Temperature (~

Fig. 11. Influence of cold work on the height of the dissipation peak and on tile frequency relaxation in silver. Curve A, after an anneal of 4 h at 493 ~ B, after

2.7% permanent axial strain.

curve, even after a complete anneal, is a curved line turning its concavity

towards the high frequencies, in agreement with the previous remark on the

elastic modulus of the transition elements. However a slight reduction of the

7 0 P . G . B O R D O N I , M . N U O V O a n d L . V E R D I N I

1"

O-

C

c u r v a t u r e is observed af ter each annea l i ng t r e a t m e n t a n d this is wha t one

would expect f rom the r e l axa t ion theory .

17.5

17.0

16.5

16.0

a)

15.5

Au

0 50 100 150 200

Temperature ( ~ 250 300

b)

15o.lo -s

100

5O

0

~ w r , .

\ \

Au

L 50 100 150 200 250

Temperature (~

Fig. 12. - Influence of cold work on the height of the dissipation peak and on the frequency relaxation in gold. Curve A, after an anneal of 7 h at 502 ~ B, after


300

D I S L O C A T I O N R]gl ' .AXATION IN S]I , vi.~lg, G O L D , P A L L A D I U M A N D P L A T I N U M

25.4

71

25.2

" i -

v

~25o

24.1

24.1

2oo. lo -5

Pd

50 100 150 200 250 Temperature ( ~

300

150

0-1

100

50

b)

0 0

./ ~x

, / /

. /

x x.,,,If, x ' x ~ /

Pd

..7 \

x , , , , . x

50 I00 150 200 2 50 300 Temperature (~

Fig. 13. - Influence of cold work on tile height of the dissipation peak and on the frequency relaxation in palladium. Curve A, aider an anneal of 6 h a~ 423 ~ B, after

6.9% permanen~ axial s~rain.

7 2 P . G . BORDONI , M. NUOVO and L. V E R D I N I

The influence of cold work is the opposite of tha t of anneal. I f the peak has been cancelled by a high t empera tu re t rea tment , as it is the case with

9.80

a)

9.70

:K

g.6o

o. ~. 9~o

9.40"

9.30 50 100 150

Temperature ( ~

PI"

C

200 250 300

200.16 s

t ~

-! 0

lOC - -

b) 5C

Pt

/ / "

~/

8 / / " - " ' " . . <

A /

,,.--- _.,_..-,-.-,-- ~ O0 50 100 150 200 300

Temperature (~

Fig. 14. - Influence of cold work o n the height of the dissipation peak and on the frequency relaxation in platinum. Curve A, after an anneal of 1 h at 793 ~ B, after


silver af ter the t r ea tmen t (d) of Table I, a pe rmanen t strain of 2.7%~, due to an axial compression of the plate at room tempera tu re reintroduces the dis-

DISLOCATION RELAXATION IN SILVER, GOLD, PALLADIUM AND PLATINUM ~3

sipation maximum (Fig. 11, lower diagram, curve B). At the same time an inflexion appears in the f requency vs. t empera ture curve (Fig. 11, upper diagram, curve B). The same influence is observed in gold, when it is permanent ly strained 3.8% after a complete anneal (Fig. 12) and in palladium after a permanent strain of 6.9% {Fig. 13). In the last case the peak is even higher than in the original state of the material corresponding to a permanent strain of only 1% (Table I).

As it has already been said, even an annealing t rea tment at 793 ~ does not remove completely the peak in plat inum, bu t reduces it to about one tenth of its original value. This small peak is increased again to nearly twice its original value by a pe rmanent strain of 3.2~ (Fig. 14). Hence the influence of cold work is the same as in the other metals.

All the previous measurements have been made on polycrystall ine specimens. However the dissipation peak cannot be due to the grain structure, owing to its low tempera ture and to the fact tha t in copper the same peak has been observed by various experimenters in single crystals and in the polycrystalline material (13).

The opposite influences of anneal and cold work point out tha t the relaxation effect is due to some geometrical imperfections of the crystal lattice. As it has been suggested by S~EGER (~4) the comparat ively high value of the ~nnealing tempera ture required to cancel the peak excludes tha t the relaxation effect can be due to the reorientation of divacancies or similar lattice defects created by the plastic deformations as these defects anneal out rapidly at room temperature, and indicates t h a t the peaks are produced by the motion of dislocation segments near their equilibrium positions.

3 . - D i s c u s s i o n .

3"1. Activation energy. - The straight lines of Fig. 5 show tha t the vibrat ion frequency ] . is related to the tempera ture T . at which the dissipation peak is observed by an Arrhenius equation

I ] (1 ) L,, = 1,,~,0 e x p - - k T . , '

where W is obtained from the slope of the lines, and characterizes the activation energy associated with the displacement o] the peak; J,~,o is obtained from

(13) See t ). G. BORDONI: Journ. Aaoust. Soc. Am., 26, 495 (1954); Ric. Scient., 23, 1193 (1953); H. L. CASW~LL: Journ. Appl. Phys., 29, 1210 (1958); V. K. PAR~: Thesis {'ornell University, AFOSR-TR 58-92.

(~) A. SE~(~ER: Phil. Mag., 1, 651 (1956}.

74 P . G . B O R D O N I , M. N U O V O ~ n d L. V E R D I N I

t h e in te rcep t of the stra.ight lines wi th the f r equency axis and gives the l imit ing

va lue of the f r equency at which the peak would be observed for an infinite

value of the t em pe ra t u r e T~.

F r o m a pure ly formal s t and poin t a new parameter~ the charac ter i s t ic

t ime ~, m a y be subs t i tu ted to ]a b y means of the re la t ion

The t e m p e r a t u r e dependence of ? is ob ta ined f rom (1) and (2)

(3) f - - G e x p ~ ,

where 7o is the value given b y (2) when ],~,o is subs t i tu ted to J ....

The values of W and ~, ob ta ined f rom the exper imenta l d a t a b y means

of (2) and (3) are listed in T a b l e I I I . F o r compar i son purposes the values

ob ta ined for the same pa rame te r s in previous inves t igat ions on c o p p e r (~5)

TABLE I I I . - Fundamental paramelers o] the relaxation effect.

Metal

Silver (**)

Last (*) treatment

(b)

(eV)

0.124 (•

Gold (a) 0.158 (~o.oo2)

PMladium (a) 0.260 (~=0.013)

Platinum (a) 0.192 (g-o.oo6)

Copper (***) 0.122 (•

(s -1)

40- ] 011

0.7

12.0

0.0 5 3.8

Aluminium (***) --- 0 . i l 0.13

Debye's frequency VD

45" 10 ll

35

57

"Too t VV

b41t

6.0" 10 -=1 0.04 I

8.6 2.2 !

10.3 0.3

47 3.2 i 23.0

67 4.2 0.6

82 5.0 24.0

(*) F o r t h e d e t a i l s of t h e r n m l a n d m e e h a n i c M t r e a t m e n t s , see T a b l e 1. (**) M e a s u r e m e n t s my, de on t h e s p e c i m e n no. 3. (%*) See t h e p a p e r q u o t e d in f o o t n o t e if), 11. 296. (***) Scc t h e p ~ p e r q u o t e d in f o o t n o t e (16).

('~) See tile paper quoted in footnote (2), p. 297.

DISLOCATION RELAXATION IN SILVER, GOLD, PALLADIUM AND PI~ATINUM 75

and aluminium (~) are also listed at the end of the Table. I t is worth of notice

that the activation energies W for the different metals are rather near euch

other, the largest value being tha t meusured in palladium (0.260 eV), and the smallest tha t found in aluminium (1~) (0.11 eV). This similarity shows ttmt W

is ussociated with some intrinsic proper ty of the dislocations and not with their density or with the impuri ty content.

The inverse ~o ~ of the limiting vMue of the churacteristic time is rather

close to the Debye 's frequency VD computed by means of the simple equation

/COl) ( 4 ) ~ . - ,

h

and given in the fourth column of Table I I I (~8).

The experimental values of W and ?o ~ may be employed to test the vMidity

of the present theories of dislocation motion. As it hns been shown in a recent

analysis (~9) the fundamenta l hypotheses adopted by WEE~TMA~ (2o), SEEGEU (21)

and by DO.~'TH (~2) are ruther different, but the final formulae for the numerical

eomputut ion of the potential barrier H which opposes the motion of dislo-

cations and of the frequency v associated with this motion (~3) are surprisingly similar.

In Weer tman ' s theory the value H,v of the potential barrier is given by

/ (70 1

where b is the length of Burger 's vector; # is the torsional nlodulus; %0 is the

Peierls' stress for the limiting case of a vanishing thermal agitation.

In Seeger~s theory the potential barrier H s depends upon the stress pro-

(~G) N. G. EL~SPRucg and R. TRUELL: Phys. Rev., 109, 652 (1957). (1:) An even smaller value of W seems to be characteristic of lead. See W. I'. MASON :

Physical Acoustics and the Properties o] Solids (New York, 1958), p. 267. However the experimental evidence for this metal is still too limited to allow any eonclusions to be drawn safely on the value of W.

(is) A more refined computation of the Debye's frequencies associated with the different types of waves is obviously not necessary as the comparison between v0 1 and v o has only a qualitative meuning.

(19) p. G. BORDONI: Dislocation Relaxation at High Frequencies, in Suppl. Nuovo Cimento, vol. XVII, pag. 43 (1960).

(:o) j . WEERTMA~: Journ. Appl. Phys., 26, 202 (1955). (.2~) See the p~per quoted in footnote (in). (22) H. DO_~TU: Zeits. ]. Phys., 149, 111 (1957). See also A. SEE(~E.~, H. DONTJL

and 1 a. PFA~'F: Discussions Faraday Society, 23, 19 (1957). (23) In this case the word (( ]requency )> has only a statistical meaning.

76 P . G . B O R D O N I , M. N U O V O and L. V E R D I N I

duced by the vibrat ions. However , when the strain is f rom 10 -8 to 10 -6 as in the present measurements , a sat isfactory average of H~ is given by the very

simple relation

(6) H s = 1.25 Hw �9

In the more recent theory due to SEEGER, DoNTH and PFAFF (2e), the relation between v and the t empera tu re is not exact ly of the exponential type. As the

t empe ra tu r e in terval in which the peaks are observed is ra ther limited, the

t empe ra tu r e dependence of the dislocation frequency v can be represented with a sat isfactory approx imat ion by an Arrhenius equation, with an ac t iva t ion

energy H s , r given b y

(6') Hs~i, ~ 1.4Hw �9

The l imit ing value v0 assumed by the f requency associated with the dislocation mot ion for very high values of T has the following expression ac-

cording to SEEGEI~

C~ (~o~�89

(7) = J '

where et is the velocity of t ransversa l waves. In the S.D.P. theory the coeffi-

cients (Vo)s.~. of the exponential depends upon the t empera tu re according to

the equat ion

(7') (Vo)s.~ :- :~ ~/~ c~kT b~/~

where e is the base of logar i thms and k is the Bol tzmann ' s constant . Howeve r

it mus t be remembered tha t the pa rame te r given by the exper iment is not the f requency ]mo but its logari thm. Hence in the narrow interval of tempe-

ra tures in which the peaks are observed the t empera tu re dependence of (Vo)s[,. can be neglected, taking for T an in termedia te value between those listed in

Table I I for Tm (54). When this is done the approx imate values of (Vo)st)r are

near to those given by (7). A first check of the theory m a y be obtained assuming with SEEGEI~ t h a t

the value H s of the potent ia l barr ier equals the act ivat ion energy, tha t is

(8) H~ : W,

(~4) The relative error introduced in this way in the computation of In (Vo)s.~. does not exceed ~-6~ . For the derivation of (7') see the paper quoted in footrmte (19).

D I S L O C A T I O N R E L A X A T I O N I N S ILV~I~ , CrOLD, P A L L A D I U M A N [ ) P L A T I N U M ~'7

and employing the exper imenta l da ta for W given in Table I I I to compute

the rat io of the Peierls ' stress a ~ to the torsional modulus # by means of (5)

and (6).

The values obta ined are listed in the sixth column of Table H I and are of the order of a few units in 10 -4, t ha t is somewhat larger than those obta ined

in static experiments . This has been explained b y SEEGE~ (~5) observing t ha t

a great port ion of dislocations in a crystal does not lie along close-packed di- rections. Therefore their effective Peierls ' stress is several orders of magni tude

o and, even a t low temperatures , the crysta l deforms at stresses smaller than a~, o Only if all dislocations involved in an exper iment lie along smaller than a , .

a crystal lographic direction it is possible to observe the Peierls ~ stress in static experiments .

Another significant test of the theory can be made b y el iminating (a~/#)

be tween (5), (6) and (7) and observing tha t the ratio (C~Hs)/[2z(vo)sbd/~ ] is an invar ian t for all metals . I f we make the reasonable assumpt ion t ha t the

unknown rat io between the theoretical value of (vo)s ~nd the exper imenta l value of ~o 1 has the same value for all metals having the same s t ructure (26),

the above expression mus t also be invar ian t when the exper imenta l values

of W and ~ are subs t i tu ted to Hs and 2~r(vo)s. The last column of Table I H shows tha t this is not the case. As the hypothesis (8) cannot be reasonably

questioned, it seems tha t the lack of agreement between theory and exper-

iment is essentially due to the evaluat ion of (vo) s which is the less sat isfactory pa r t of the present theories of dislocation motion.

3"2. R e l a x a t i o n spectra. - I t has been shown elsewhere (27) t ha t all the

exper imenta l points obta ined in energy dissipation measurements made a t different frequencies mus t fall on a single line when the rat io Q 1/Q~I is p lo t ted against W k - I ( T ~ 1 - - T-~), provided the re laxat ion effect is due to the sum of m a n y e lementary effects having the same act ivat ion energy W.

Fig. 15-18 show tha t this is the case for the four meta ls ; i t is also evident t ha t the exper imenta l peaks are much broader than the theoretical curve for a single re laxat ion t ime (dotted line), and tha t the low- tempera ture branch is in every case higher than the h igh- tempera ture one. These results are quali- t a t ive ly the same which have been obtained for copper, and they show tha t

the dissipation peak is associated with a spec t rum of re laxat ion times. The

(25) See the paper quoted in footnote (14) The value of 0 �9 a~/t~ would essentially be t)m same if Hw or HSD e were taken instead of Hs.

(2G) SEEGER takes /V0)s- ~ol/(2n). (27) See the paper quoted in footnote ('~), p. 300, or P. G. BORDONI: Theory o] rela-

xation e]/ects with a continuous spectrum, in Proe. of the Third International Congress of Acoustics (Stuttgart, September 1959), in press�9

7~ P . G . BORDONI, M . NUOVO and L. V]~RDINI

ac t iva t ion energy W associated with each t ime ~ has essential ly the same

va lue W which character izes the f r equency dependence of the t e m p e r a t u r e

of the diss ipat ion peak.

1.0

Q-1

0.5

o c

0

/

o. yi .

I I t I

/ / ~

/ / \

/ \ / \

Ag

"10 -5 0 +5 +1

~, k-l tr;nl-r - ')

Fig. 15. - Values of the ratio Q-1/Q~nl as a function of Wk-I(T;~ 1 - - T -1) for silver, after the treatments (a) and (b). Measurements at: (o) 19 kHz; (x) 51 kItz; (O) 421 kHz; (A) 820 kHz. Heavy line, sech [0.27 Wk-I(Tj~ ~ - T-l)]; dotted line, Zener's dissipation

curve for a single relaxation time.

The symmet r i cM p a r t of the dissipat ion curve, p lo t t ed as a func t ion of

W k - ~ ( T ~ 1 - T - l ) , is due to a symmet r i ca l spec t rum of re laxa t ion times, whose

1.0

0-.~ 1

051

x

�9 x

0

0

/ ! \ / I \ I \

\ 0

Au

-10 -5 +5 *10

k-' (r~,'-r-')

Fig. 16. - Values of the ratio Q-1/Q~I as a function of Wk-1(TL 1 - - T -1) for gold, after the treatment (a). Measurements at: (e) 16 kHz; (.:) 56 kHz; (@) 561 kHz. Heavy line, sech [0.35 Wk-I(T;, , ~-- T-l)]; dotted line, Zener's dissipation curve for a single

relaxation time.

center coincides wi th the charac ter i s t ic t ime u which is d i rec t ly re la ted t h r o u g h

eq. (1) and (2) to the f r equency dependence of the diss ipat ion peak.

D I S L O C A T I O N R]gT ,AXATION IN SILVJ~I~,, GOI, D, P A L L A D I U M A N D P L A T I N U M 7 9

T h e w i d t h of t he s p e c t r u m can be e v a l u a t e d w i th good a p p r o x i m a t i o n b y

a p p l y i n g to t h e h i g h t e m p e r a t u r e b r u n c h of t h e e x p e r i m e n t a l curves , wh ich

1.0

Q-.

0.5

0 -10

/ , !

! / I /

/ /

\ \ \ \ \

Pd

\

- 5 0 +5 *10

k "1 ( r~ - r "~ )

Fig. 17. - VMues of the rat io Q ~/~)~1 as a function of Wk-~(T~J - - T -z) for palladium, after the t r ea tment (a). Measurements ~t: (O) 25 kHz; (• 88 kHz; (o) 388 kI-[z. Heavy line, seeh [0.22 W/c 1(T~1-- T-~)]; dot ted line, Zener~s dissipation curve for a

single relaxat ion time.

is less a f fec ted b y t h e a s y m m e t r y of t h e s p e c t r u m , t h e F u o s s - K i r k w o o d e q u a -

t i on (2s)

(9) (2 - ' = sech [ y W k ~(T~ ~ - T 1)] ,

1.0

Q-1

0.5

0 - 10

0 o

~o ["

I I

I / /

\ \ \ \

o

,-1r - r -b

Pt

o4o

+' +10

Fig. 18. - Values of the ratio Q t/Q~I as a function of ITk-I(T~ 1 - 2 ' - l) for plat inum, a.flier the t r ea tment (a). Measurements s t : (e) 10 kHz; (x) 42 kHz; (�9 112 kHz. t i eavy line, sectl [0.35ITk-~(T~ 1 - T-~)); dot:ted line, Zener's dissipation curve for a

single relaxation time.

(is) R. M:. Fuoss and J. G. KIRXWOOD: Jo~rn. Chem. Pl~ys., 63, 385 (1941). For additional information on this approximation of the spectrum see also the paper quoted in footnote (~), p. :305.

8 0 P. G, BORDONI , M. NUOVO ~tnd L. V E R D I N I

where X is a pa rame te r which characterizes the spec t rum width; its value is un i ty for a single re laxat ion t ime, whilst i t vanishes for an infinite spect rum

width. According to the theory developed by Fuoss and Kirkwood, the den-

s i ty (~(z) of the re laxat ion spec t rum is a bell-shaped curve, having its m a x i m u m

TABLE I V . - Relaxatio~ spectra.

M e t a l " . . . . . . . . . . . . . L a s t t r e a t m e n t (*) 7 ' I n z . /T .

Silver ('*) (a) 0.28 5.30

i (b) 0.27 5.C0

Silver {c) 0.30 4.87

(e) 0.18 (***) 9.05

i (a) 0.35 3.91 [

Gold (c) 5.60 i

Palladium

Platinum

(a)

(e)

0.27 ('.*) i . . . .

0.22

0.24 ('.')

7.38

6.55

(a) 0.35

(b) 0.35

(c) 0.35 3.91

(d) 0.35 3.91

(e) 0.24 (',') 6.55

Copper - - 0.39 (.'*) 2.89

3.9~ I i

3,91

(*) F o r t h e de t a i l s of t h e r m a l a n d mechunic 'M t r e a t m e n t s , see T a b l e I.

(**) M e a s u r e m e n t s m~de on spce in lens no. 1 a n d 2. (***) C o m p u t e d w i th t he va.lue of 15" g i v e n b y Tab le I l I a n d m o a s u r e d be fore cold work.

(***) See t h e p a p e r q u o t e d in f o o t n o t e (~), p. ?,09.

a t 2; the t imes T2 and T1 for which the height of the spec t rum is l /~/ff of the m a x i m u m height m a y be t aken as upper and lower limits of the spec t rum;

the logar i thm of their ratio gives a physically significant measure of the

spec t rum width. The heavy lines of Fig, 15-18, which have been computed by means of (9),

D I S L O C A T I O N R]~LAXATION IN SILVER, GOL[)~ P A L L A D I U M ANY) P L A T I N U M 81

fit well the exper imenta l points for T ~ T.~, showing tha t the Fuoss-Kirk-

wood theory gives a good approx imat ion of the symmetr ica l pa r t of the spec-

t rum. The values of y and of ]n (~2/T~) have been computed for all the measurements made, and the values obta ined are listed in Table IV.

The widths of the spectra are somewhat larger than the value previously found in copper which is given for comparison purposes at the end of the same

Table. I t m a y be noticed t ha t anneal decreases the width, whilst the opposite effect is produced by cold work.

3"3. Frequency relaxat ion. - As it has been said, the exper imenta l v~.lues

of ( f - - fR) / ] , are avai lable only for the lowest v ibra t ion modes (Fig. 1-4, upper

diagrams).

For silver and gold the agreement between theory and exper iment is satis- factory. According to the Fuoss -Ki rkwood ' s computa t ion , the m a x i m u m

dissipation Q~i is related to the total re laxat ion of the resonant f requency by the following equat ion

where S is the re laxat ion strength, and ( ] v - - ] , ) is the value taken by the

difference ( I - -JR) for T<<T.~, tha t is when ? is much longer than the vibra- t ion period (.~9).

The rat io ( ] v - - ] S ] ~ can be obtained f rom the upper diagrams of Fig. 1

~md 2, ex t rapola ted towards the low temperatures . The values given b y this ex t rapola t ion are in sat isfactory agreement with those computed b y means

of equat ion (10) taking for Q~I and y the values given respect ively b y Tables I [ and IV. I t mus t however be noticed tha t the inflexion point of the curve which gives ( ] - - ]~) / /~ as a funct ion of T, is not found at the t empera tu re T~ as theory requires, bu t at a somewhat lower tempera ture .

The above comparison between the m a x i m u m dissipation and the relaxat ion s t rength cannot be made for pal ladium and pla t inum, as the computa t ion of ]~ cannot be made b y a linear ex t rapola t ion owing to the anomalous behaviour of the modulus vs. t empera tu re curve which has a l ready been pointed out.

3"4. I n f l u e n c e of anneal and cold work. - The influence of anneal on the re laxat ion effect, t ha t is the reduct ion of the dissipation peak and of the re laxat ion observed in the resonant f requency (Fig. 6-9) is readily explained

(~.o) The difference ( f v - JR) must be computed at temperatures lower than T,~, but not too near to the absolute zero, otherwise the extrapolation which gives ]n as a linear function of T is not valid.

6 - Supplemento al Nuovo Cimento.

8 2 P . G . B O R D O N I , M. N U O V O and L . V E R D I N I

b y a dec rease of t he n u m b e r of d i s loca t ions which are ac t i ve in the r e l a x a t i o n

process . The resu l t s o b t a i n e d t h r o w also some l igh t on a n o t h e r effect due to

annea l , t h a t is t h e t e m p e r a t u r e - i n d e p e n d e n t inc rease of t he f r e q u e n c y of t h e

spec imen , w h i c h was f irst o b s e r v e d in copper (~0). This i nc rease is shown b y

t h e u p p e r d i a g r a m s of F ig . 6-9. I n s i lver , as in copper , a c o n t e m p o r a r y in-

c rease of t h e r o o m t e m p e r a t u r e d i s s i p a t i o n is o b s e r v e d (Fig'. 6, l ower d i a g r a m ) .

N o change in d i s s i p a t i o n is f o u n d in go ld (Fig. 7, lower d i a g r a m ) whi l s t in

p a l l a d i u m a n d p l a t i n u m a n n e a l g ives r ise to a decrease of t he r o o m t e m p e r a -

t u r e d i s s i p a t i o n (Fig. 8, 9, lower d i ag rams) .

The r e l a t i v e va lue s of t he p e r m a n e n t changes o b s e r v e d in f r e q u e n c y a n d

d i s s i p a t i o n a f t e r annea l a re l i s t ed in T a b l e V, t o g e t h e r wi th those o b t a i n e d for

TABLE V. - Permanent changes in resonant ]requeney and energy dissipation at roo*~* temperature alter anneal.

l~etal

Silver

Gold

Pal ladium ___

Pla t inum

Speci- nl e n

Last treatment, (*)

Changes in

Dissipation ( A Q - 1 ) Frequency (A///)

(e) + 9000" 10 -5 + 80" i0 -5

I +13o (d) ] +11 6O0 !

(b) d- 6 800

(b) + 460 - - 25

(e) + 630 - - 27

3 1 h at 515 ~ + 980 + 29

Copper (*') 6 h at 516 ~ + 2370 +130

i + 2 920

2

l ( b ) " 1 h at 873 ~ + 120

(% For the details of thermal and mechanical treatments, see Table I. (**) See the paper quoted in footnote (2), p. 291.

c o p p e r in t he p rev ious work . I n e v e r y ease t h e changes in f r e q u e n c y a re a t

l e a s t one o r d e r of m a g n i t u d e l a rge r t h a n those m e a s u r e d for Q-1. M o r e o v e r

(30) See the paper quoted in footnote (3), p. 291.

DISLOCATION RELAXATION IN SILVER, GOLD, PALLADIU.~I AND PLATINUM 83

the former changes are always positive, whilst the dissipation m a y either in-

crease or decrease af ter anneal. Hence no direct relation seems to exist between the influence of anneal upon the room tempera tu re f requency and upon the dissipation at the same tempera ture .

The annealing tempera tures a t which the pe rmaneu t increase of f requency begins to be observed show tha t this effect in some cases m a y be related to

the recrystall isation of the specimen. On the other hand the changes in dissipation, which have also been observed in copper single crystals, m a y be

associated with some changes in the mobi l i ty of the dislocations.

For p la t inum the values of Q~I have been plot ted in Fig. 19 in a logari thmic

100

50

20

1C

-5 lO

r-

-1 Or.

5

/ ,P

2

1 0

J

, , f / #,

J

/

j '

/ / /

] 0.1 0.2 0.3 0.4.10 -2

Ta I (~ -1)

Fig. 19. - Exponential relation between the height of tile dissipation peak and tile inverse of the annealing temperature for pla~ilmm.

scale against the inverse of the annealing t empera tu re Ta. As the exper imenta l

points fall on a s t ra ight line, the number of residual act ive dislocations is re-

la ted to the annealing t empera tu re by an Arrhenius equation, with an act iv-

ation energy of abou t 0.09 eV. This energy is near ly one half of the acti-

vat ion energy associated with the re laxat ion effect (Table I I I ) . The reason why at room tempera tu re dislocations move near their equil ibrium position

wi thout annealing', is tha t the t ime required to f o r m a kink is of the order

~)4 P . G . BORDONI, M. NUOVO a n d L. ViERDI~I

of ~o, tha t is ~ 10 -~z s, and the probabi l i ty of an energy f luctuation higher than W is large for such a short time. To make the dislocations inact ive with respect to the re laxat ion effect, the energy fluctuations required mus t last a much longer time, and the corresponding probabi l i ty is small a t room temperature , notwi ths tanding the fact t ha t the energy barr ier seems to be lower than W.

The influence of cold work on the re laxat ion effect (Fig. 11-14) is explained

as easily as t ha t of anneal, b y an increase of the number of act ive dislocations.

A tempera tu re - independen t decrease of v ibra t ion frequency after cold work is

ve ry evident in gold (Fig. 12, upper diagram) and in p la t inum (Fig. 14, upper

diagram). This decrease is due to an effective change of the elastic coefficient, as the values of f requency given in Fig. 11-14, (upper diagrams, curves B) have been corrected for the thickness reduct ion of the plate and for the increase in diameter , due to the pe rmanen t strain.

For a given vibra t ion f requency a slight dependence of the t empe ra tu r e T,,

upon the annealing t empera tu re T~ has already been observed in previous

exper imenta l work (31). As T~ decreases together with Q ~ when T~ increases, i t was concluded tha t the dislocations more easily removed in copper are

those associated with the largest values of the re laxat ion time, as the ave-

rage value u is decreased by anneal. The same dependence is observed in the four metals , being very evident in silver (Fig. 6, lower diagram) and in

p l a t inum (Fig. 9, lower diagram). I t m a y be added tha t cold-work has ap- pa ren t ly the opposite effect, as T~ for a given f requency ineregses slightly with the amoun t of cold-work and therefore with the height of the dissipation peak (Fig. 11 and 14, lower diagram).

The mos t significant pa rame te r to which the observed changes of T~ m a y be related is the to ta l number of dislocations which are act ive in the relaxat ion effect. In the Fuoss-Kirkwood approx imat ion this number is propor- t ional to the rat io Q~/y, which m a y be computed f rom the da ta of Tables I I

and IV. The diagrams of Fig. 20 show tha t the t empera tu re of the peak in-

creases regularly with the number of dislocations; in the narrow range of values

considered this dependence m a y be satisfactorily approx ima ted b y a s t raight

line. The changes of Y corresponding to a given change of the to ta l numbe r of act ive dislocations m a y be easily computed f rom the diagrams of Fig. 20

and f rom the values of the ac t ivat ion energy given in Table I I I . Fo r silver

when Qj~/~ goes f rom 1500.10 -5 to 750.10 -s the value of Y is decreased b y a fac tor of abou t 5. In p la t inum, when Q~/y goes f rom 500-10 -5 to 250-10 -~ the characteris t ic t ime ~ is reduced b y a fac tor o f gbout 2 (3~).

(31) See the papers quoted in footnotes (2), (5) and (11). (32) In the paper quoted in footnote (2), p. 312, a factor of 1.5 was computed for

che change of Y in copper. This value is correct, but one must read: (, The change in the value of (T,~ 1 - T -1) is of the order of 0.3" 10 -3 ~ -1 ~>.

DISLOCATION RELAXATION IN SILVER, GOLD, PALLADIUM AND PLATINUM ~5

The above results show tha t anneal and cold work are effective in

changing both the number of active dislocations and their average relaxation

time. In fact when the number is reduced by a factor of 2, the characteristic time may be reduced by a larger factor.

a)

b)

90

Trot~ 80

Ag

S " 70

0 500 1000 1500 2000.10 -s

190

180

TmCKI

170

1600

7 J

J

J J

f -

soo 6oo.16 S 100 200 300 400 -1 -)

Om'Y ]:lg. 20. - Dependence of the temperature of the peak upon the total number of active dislocations. Silver, experimental data reduced to the vibration frequency of 50 kHz;

platinum, vibration frequency of 9.6 kHz.

The experimentM evidence which is presently available is not sufficient to

decide if the changes of 7, which are observed at low temperatures, are due

to corresponding changes of the limiting value u of the characteristic time or,

as it is more probable, to a slight variation of the activation energy. To give

~6 1 ~. G. BORDONI, M. NUOVO a n d L. VERDINI

a sat isfactory answer to this question the ac t ivat ion energy mus t be measured

af ter different anneals with an accuracy of the order of a few units percent.

3"5. Subsidiary peaks. - The dissipation curves for pal ladium (Fig. 3, 8 and 13, lower diagrams) and those for p la t inum (Fig. 4, 9 and 14, lower dia-

grams) clearly show a subsidiary peak below the t empera tu re of the main max-

imum, of the same type as tha t found by NIBLE~T and WXLKS in high pur i ty

(99.999%) polycrystal l ine copper, and successively invest igated in the same mater ia l by PAI~]~ and b y others (33).

In silver and gold the existence of a N.W. peak is less obvious (Fig. 1, 2, 6, 7, 11, 12, lower diagrams). This is pa r t ly due to the fact tha t the exper-

imenta l values are p lo t ted against tempera ture . As it has been shown (~4)

the dissipation curves due to a symmetr ica l re laxat ion spect rum are not symmetr ica l with respect to T but to T -1. When the values of the dissipation are p lo t ted against the (( symmetrical ~> variable Wk-I(T~, ~ - - T-l), a flat (( b u m p )) is

observed in the low t empera tu re branch of the curves for silver (Fig. 15) and gold (Fig. 16).

The N.W. peak is clearly related to the dislocation motion, like the main one, as it is reduced or cancelled b y anneal (Fig. 6-9, lower diagram) and

increased or reintroduced b y cold work (Fig. 11-14).

To separate the N.W. peak f rom the main max imum, the exper imenta l

values of dissipation have been plo t ted against T -~ as it is shown in Fig. 21

(heavy line). A Fuoss-Kirkwood 's curve (eq. (9)) has been drawn to fit the exper imenta l points, and the values of T,~, W, u and y given in the Tables I I , I I I and IV have been employed to this purpose (Fig. 21, dot ted line A). The difference between the exper imenta l points of the low- tempera ture branch and

the A curve has been computed, and two m a x i m a have been found for T -~ = 0.99.10 -2 ~ -1 and T -~ = ~.37.10 -~ ~ -~. A second Fuoss-Kirkwood 's

curve has been drawn choosing the t empera tu re of the first subsidiary max- i m u m T:~ and the new value of the product y 'W 'k -~ by a trial and error method,

to fit the computed differences in the best possible way (Fig. 21, dot ted line B).

The procedure has been repeated for the second subsidiary max imum, and a

th i rd curve of the type (9) has been drawn through the new differences (Fig. 21,

do t ted line C). In this way the exper imenta l dissipation curve has been ap-

p rox imated by the sum of three <( symmetrical ~) curves, corresponding to symmetr ical re laxat ion spectra.

The same procedure has been followed whenever possible for the other

metals , and for the different v ibrat ion frequencies. The results obtained, t ha t is the tempera tures T'~ and T:~ of the two subsidiary peaks, their heights

(33) See the papers quoted in footnote (11) and (13). (34) See the papers quoted in footnote (2), p. 304 and in footnote (~7).

D I S L O C A T I O N R N ] C . A X A T I O N I N S I L V E R , G O L D , P A L L ~ k D I U M A N D P L A T I N U M ~ 7

120.11j s

110

10o I

90

8 0 - -

70-

0"160-

50

4 0 - -

3O

20

tO

0 '

A~

I _

P ~

>Cf ! / | - ' - . I

, -. / ' l > . | /

0 O1 0.2 0.3 0.4 Q5 0.6 07 08 0.9 1.0 1.1 1.2 1-3 1.4 1.5 1.6"10 -2 T-1 (OK-l)

F i g . 21. - S e p a r a t i o n of t h e N . W . p e a k s f r o m t h e m a i n d i s s i p a t i o n m a x i m u m . H e a v y l i ne , e x p e r i m e n t a l d a t a ; d o t t e d l i n e s A, B, C, c o m p u t e d P u o s s K i r k w o o d ' s c u r v e s .

TABLF V I . - Experimenta~ data ]or the subsidiary peaks.

L a s t F r e - M a i n p e a k F i r s t s u b s i d i a r y p e a k S e c o n d s u b s i d i a r y p e a k

t r e a t " u e n c v i~ . . . . . . i ~_ . . . . i' _ . . . . . . i - - - . . . . . . T --~ ~ ~ Ie ta l .~ t t . ' ~ ~ - - ! 'l ~ 1 ' , I~Yt 11 , ! I U - - u L " i m e n t a ~ ' ~ ' ? ~ " T / ? i r - - [ y ~/~ k - - 7 ?y~ . . . . . ] ! V t l ][;I I T m ' . . . . . . 1 ~] W k - l i

. . . . . . (*) : (kHz)_- , (~ ' ''~ , (OK) ' (oK) (W,,~ ]' ( ~ [(OK ) (t~,,) i ( ~ i

g i l v e r (**) (b) !421 .4 91 .0 288.10-~! 430 61 120-10 .5 460 ! - - - - r : i , i . . . . . . . . . . . . ; ] . ] : . . . . . . . . . . . . . . . . . ] . . . . . . . . . . . J . . . . . . . . .

G o h t . . . . ' i127 7 221 i ; l (a) 5 5 . 6 0 i 635 i < ' i 4 > 1 1 0 - - i - - - - - - - i i , ,! i

(a) 24 .92 [180.0 30 650 107 20 i, 650 ! < 7 1 ! > 1 3 " 1 0 -5 - - i l ! ' i

P a l l a d i u m

P l a t i n u m

186.5 I I ' (a) 88 .35 59 650 1 1 1 . 5 36 i 650 < 8 0 [ > 20 L __

(a) 387 .9 200 .0 75 : 650 - - - - - ~ " i i

(a) 9.60 2 _ ; 0 80-101 , . 630

(a) 4~.41 201.0116 0 111 65 6ao I . . . . . . . . . .

(*) F o r t h e d e t a i l s o f t h e r m a l r m e c h a n i c a l t r e a t m e n t s , s e e T a M e I .

(**) . M e a s a r e m e n t ~ m a d e o n s p e c i m e n n o . 3.

8 8 P . G . B O R D O N I , M. N U O V O and ]~. V E R D I N I

! 1 (@,,)- and (O:'~)=~ and the values of the product y'~7'k-~ for the first subsidiary peak are listed in Table VI.

The exper imenta l da ta are too l imited for a sys temat ic evaluat ion of the

~ctivation energy W' associated with the first N.W. peak. However , the energy W ' = 0.]6 eV which m a y be computed for pal ladium f rom the values of frequency and t empera tu re given in Table VI, and the energy W ' = 0.11 eV

which is obta ined for plat inum, are bo th consistent with the hypothesis t ha t

a t high tempera tures the corresponding values of the characterist ic t ime are of

the same order as the period of the Debye ' s frequencies. As it was found for

copper, these energies are considerably smaller than the energies of the main

peak. In silver and gold the Iq.W. peaks fall below the exper imenta l r~nge

of t empera tu res and no informat ion can be derived f rom Table V I on their ac t iva t ion energies.

4 . - C o n c l u s i o n s .

The fundamenta l pa ramete rs W and Yo of the relaxat ion effect due to dislocations in silver, gold, pal ladium and p la t inum have values near to those

found for the same paramete rs in other f.e.e, metals. This seems to point

out t ha t the mot ion of dislocations has an << intrinsic ~> character , essentially in-

dependent of the impur i ty content, in agreement with the present theoret ical

t r ea tmen t s of the same problem . F r o m a quant i t a t ive s tandpoint , the agreement betwen exper iment and the-

ory is sa t is factory as far as the ac t ivat ion energy is concerned. The evaluat ion of the characterist ic t ime mus t be improved, as the present values hardly give the r ight order of magni tude. In this respect the theory mus t also be extended to account for the wide spec t rum of characterist ic t imes which is always associated with the effect, and for the relation between the to ta l numbe r

of dislocations and the value of T at low temperatures . The a s y m m e t r y of the dissipation vs. t empera tu re curves is a general pro-

per ry of the effect, and is possibly due to subsidiary peaks on the low temper-

a ture side of the main dissipation max imum. These peaks are of the same type

as tha t found by NIBLETT and WIL~S in copper. The dislocations responsible

for this a s y m m e t r y seem to have a lower ac t ivat ion energy than those asso-

ciated with the main peak. The influence of anneal and cold work on the re laxat ion effect is now satis-

factor i ly understood. The new results on silver, gold, pal ladium and p la t inum have removed an apparen t contradict ion with theory, showing tha t the

t empera tu re - independen t increase of f requency which is produced by ~nneal,

and the opposite effect due to cold work are not related to the change observed

in dissipation bu t are p robab ly due to recrystal l ization effects.

Documents

Dislocation relaxation in silver, gold, palladium and platinum