Influence of temperature and strain amplitude on damping and modulus of electron-irradiated copper

INFLUENCE OF TEMPERATURE AND STRAIN AMPLITUDE ON DAMPING AND MODULWS OF ELECTRON-IRRADIATED COPPER* K. LOCKE,?

G. ROTH?

and G. SOKOLOWSKIi

The changes of damping and modulus in copper during 1.5 Me\*-electron irradiation were measured at different strain amplitudes, measuring and irradiation temperatures. It was found that the minimum in dislocation pinning rate at 160°K reported by Thompson and Buck occurs only as function of the Furthermore, it. was found that the depth measuring temperature and not of the irradiation t,emperature. of the minimum decreasss strongly with decreasing strain amplitude and that also the differences in pinning rates derived from damping and modulus measurements are caused mainly by amplitude dependent effects. It is concluded that the minimum does not, characterize the true pinning rate but is caused by a temperature and amplitude dependent damping effect. Hence also the interpretation that the minimum characterizes the thermal conversion of crowdions is no longer justified. INFLUENCE DE LA TEMPERATURE ET L’AMORTISSENENT ET LE MODULE

DE L’AMPLITUDE DE DEFORMATION DU CUIVRE IRRADIE AUS ELECTROES

SUR

Les variations de l’amortissement et du module dans le cuivre au tours d’irradiations aux 6lectrons de 1,5 MeV ont BM mesurbes pour diffkrentes amplitudes de deformation et differentes tempCratures d’irradiation et de mesures. Les auteurs ont trouv6 que le minimum de taux d’epinglage des dislocations B 160-K, mis en Bvidenee par Thompson et Buck, apparait seulement pour des variations de la temp& rature de mesure, mais pas pour des variations de la tempbrature d’irradiation. En outre, its ont trouv& que I’amplitude du minimum d&-oft fortement quand l’amplitude de la d&formation diminue, et Qgalement que les diff&enoes de taux d’bpinglage dCterminCes iL partir des mesures d’amortissement et de module rtisultent principalement d’effets d&pendant de l’amplitude. En conclusion, le minimum ne caractbrise pas le taux d’hpinglage r&el, mais il r&&e d’un effet d’amortissement d&pendant de la temperature et de l’amplitude. L’interprbtation selon laquelie le minimum caract&& la conversion thermique des orowdions n’est done plus justifibe. EINFLUD

DER

TEMPERATUR USD DEHNUNGSBMPLITUDE MODUL IN ELEKTRONENBESTRAHLTEM

AUF KUPFER

D;iMPFUXC

UND

Die iinderungen von Diimpfung und Modul von Kupfer wiihrend der Bestrahlung mit 1.5 MeV-Elektronen wurde bei verschiedenen Dehnungsamplituden, MeO-und Bestrahhmgstemperaturen gemessen. Es ergab sich, dafj das von Thompson und Buck gefundene ~~imimum der Verankerun~ra~ der Versetzungen bei 160’K nur als Funktion der ~Ie~~mperatur und nicht der Bestr~lun~~m~ratur auftritt. Weiterhin wurde gefunden, da8 die Tiefe des Minimums stark mit abnehmender Amplitude abnimmt und da13 aueh der Unterschied dcr aus Diimpfungs- und Modulmessungen gewonnenen Verankerungsraten haupts&chlich auf amplitudenabhiingige Effekte zuriickzufhiiren sind. Es wird daraus geschlossen, da13 es sich hier in Wirklichkeit nicht urn ein Minimum der Verankerungsrata handelt, sondern da13 dieses durch amplitudenabhhngige Effekte vorgetiiuscht wird. Damit entf&llt such die Deutung dieses Minimums durch die thermische Konversion von Crowdionen.

1. INTRODUCTION

pinning points per length L,. If the dislocation resonance theory is valid, the values paI and pA at. small irradiation doses are generally due t,o the should be found to be equal, pinning of dislocations by the i~adiation-induced Utifizing damping measurements during y-irradiapoint defects. Therefore, modulus and internal t,ion of copper single crystals at different temperatures friction measurements can be used to study the Thompson and Buckc2) found in this way at 160°K behaviour of the pinning point defects. For kHz- a pronounced minimum of the pinning rate $ (i.e. the frequencies the dislocation resonance theory”) (which number of pinning points per dislocation segment Lo is valid at small strain amplitudes) predicts for the formed per unit time). This minimum was interpreted decrement A and the modulus defect AM/M of the by these authors@) and by Seeger@) as a direct proof dislocations of the thermal conversion of interstitials.(4~5) Very A = K, ..L4 = KA -L,4/( 1 + pa)4 (1) recently Simpson et al. verified this minimum on A~/M = K,*L2 = KM-L2/(1 -/-p&z (2) electron-i~adiated polyc~stalline copper by means of modulus measurements. These authors also Here Lo is the pre-irradiation Ioop-length of the dis- forwarded a theoretical treatment of this effect which location, K, and K, are constants (see Appendix 2) was also based on the thermal conversion of the interand PA and PM the numbers of irradiation induced stitial.(7) The experimental results of these two groups of authors are summarized in Fig. f (the modulus * Received June 12, 1972. t Institut fiir Allgemeine Metalikunde und ~~etallphysik results after having been transformed in the usual way der Technischen Hochschule Aaehen und Van de Graaffinto initial pinning rates). Labor Aachen der Kernforschungsanlage Jiilich. Changes in modulus and internal friction of metals

ACTA

METALLURGICA,

VOL.

21, MARCH

1973

237

238

ACTA

%fETALLURGICA, 100 150 2go 250

VOL.

21,

1973

100 150 200 250

-

Roth ,SOkdowski.LtickclB/ 0 sampI*

1

nSmpIc2 3 hicv ll*ctra .73 cv Ii EI.140”~

-

f?olh.%kdowskri,Li/ o!smpIc4 3Mev Bremsssrrahluq =25 *v h :1.6.n~+ EIn xc Simpson

-

Temperature

of Irradiation

[

*I al /6/

0 1 WV

electrons

W]

FIG. 1. Pinning rates i vs. temperature of irradiation Tirr as derived from damping and modulus measuremenIt of different authors.

In two foregoing papers, also the present authors have demonstrated the existence of such a minimum(s,9) as is shown also in Fig. 1. They argued, however, that the interpretation in terms of conversion of interstitials is not justified. They gave experimental evidence that the numbers for@ leading to such a minimum are not the true pinning rates but apparent numbers connected with internal friction effects not yet clearly understood. Their main reasons were the following (see Fig. 1) : (i) The (relative) depth of the minimum of @A found by the present authors after irradiation with 3 MeV-electrons and with 3 MeV-Bremsstrahlung was much smaller than that found by Thompson and Buck after irradiation with CoBo-y-rays of comparable dose. There was the suspicion that the depth of t,he minimum increases with decreasing mean energy {&) transferred to the primary knock-on, but a pronounced irreproducibility of this depth indicated an influence of parameters not yet recognized. (ii) The depth of the minimum obtained from modulus measurements (i.e. using pnl-values) was found to be considerably smaller than that obtained from damping measurements (i.e. using p,-values). Thompson and Buck had a similar result, but pointed out that their modulus might not be reliable. In their case the specimen had very little thermal contact t,o the surrounding so that by the y-heating the temperature of the specimen and, therefore, its modulus

might have changed in an uncontrolled way. In the experiments of the present authors this difficulty did not exist so that their pfil-values are as accurate as the p,-values. (iii) Preliminary results indicated that the minimum occurred not as function of the temperature of irradiat,ion Tirr (as necessary for the interpretation in terms of thermal conversion of interstitials) but as function of temperature of measurements T, (compare the open and filled triangles in Fig. 1). This could not be recognized by Thompson and Buck and by Simpson et al. since they took their measurements always at the temperature of irradiation. The purpose of the present paper is to contribute to further clarification of these points. Therefore, the previous pinning rate measurement for which irradiations with 3 MeV electrons have been used have been supplemented by measurements with 1.5 MeV and 0.52 RIeV electrons. In these measurements, the temperature of irradiat,ion and the temperature of measurement have been varied independent of each other and, in particular, the strain amplitude dependence has been investigated. 2. EXPERIMENTAL

PROCEDURE

Polycrystalline samples were prepared of 99.999 % ASARCO copper, the same material as it was used by Thompson and Buckt2) and in the former experiments of the present authors, (s*9)In order to achieve a homogeneous radiation damage samples with a thickness

LtfCKF,

et al.:

DAMPIKG

AND

MODULUS

~0.1 mm had to be used. They were punched out from 0.1 mm copper sheet in a form shown in Fig. 2(a). The essential part S of the sample (1 x 1 mm) is excited to torsional vibrations of about 5 kHz by lateral forces acting on the lever arm L. The strain amplitudes E have been varied from 1 x IO-’ to 1 x 10e6. In order not to obtain uncontrolled vibrations in the lever arm it was stiffened by spot-welding an additional piece of copper sheet to the lever arm and by bending it. parallel to its edge to a L-shape crosssection. The lower part (10 x 10 mm) of the sample is clamped to the sample holder (see Fig. 2b). The sample holder is attached to a liquid nitrogen cryostat, by which the temperature of the sample can be changed between 78 K and 800 K. The electrodes E, and E, (steel sheet, 0.1 mm thick) are attached to the sample holder with the insulat,ors 1; and I, (Mycroy, Molecular Dielectrics, Clifton, New Jersey). These electrodes which have a distance of about 0.1 mm to the lever arm, have the purpose to excite the oscillations electrostatically and to detect them using a frequency modulation of a 150 MHzoscillator. The AF-signal which is received after demodulation of this 150 MHz-signal and which is L

al

Fro. 2. Sample and sample holder. (a) Geometry of the sample. L is the lever arm for the torsional vibrations of the part S. The dimensions are given in mm. (b) Sample holder, to which the sample is clamped. The electrodes E, and .Ez we fixed on the insuiators 1, and I,.

OF

ELECTRON-IRRADI;ITr;I)

(‘11

239

proportionally to the sample vibration, is fed back to the exciting electrode, giving rise to a self-excitation of the sample. According to Thompson and Glassoo) this feedback can be used after correct phasing for measuring both damping and shear modulus: The amplification of the feedback circuit which is needed to produce a given amplitude of vibration can be used as a measure for the damping and the frequency of the self excited vibration as a measure for the shear modulus. In the present paper a slightly modified form of this method has been used for measuring the amplitude dependence of damping. Further information on sample and method of measurements are given in.“l). In contrast to the normally used flexural vibrations of thin cantilevered beam&+-i’) where the strain amplitude decreases along the sample in =-direction (see Fig. 2), the here used t.orsion-sample is deformed equally over its whole lengt,h*. Since each element, of the sample contributes to damping and modulus with the square of its amplitude, damping measurements of flexural vibrations are determined by the part of the sample nearest t,o the clamp. These parts, houever, are affected most by changes in clamping, e.g. during temperature cycles, so that it is diflicult to obtain good reproducibility. Such complications do not exist for torsional vibrations, and the good reproducibility of the present measurements is ascribed to this fact. All results presented here are obtained with one sample. Before irradiation, the sample, already mounted in the cryostat, was annealed at 800°K for $ hr. The irradiations were performed with elect,rons with an energy of either 0.52 or 1.5 MeV. After each irradiation the sample was recovered in &tu by annealing at 770°K for 5 min. This trestmcnt, led reproducibly to the condition of the sample before irradiat,ion. The irradiation dose cb is given in each figure. Equat~ions (1) and (2) were used to calculate the numbers p, and p, of irradiation-induced pinning points from the observed changes in damping and modulus. In order to apply these equations the background values for decrement and modulus, which characterize the dislocation-free crystal, are needed. They were determined by irradiation with 1 r; 10’6 electron per cm2 and subsequent annealing at 400°K for 5 min; this treatment led to complete dislocation * The stress state obtained by torsion of a thin rectangular beam is in first approximation a pure shear stress state with only the component cl,” (see Fig. 2a) different from zero. The magnitude of the stress is independent of z, nearly independent of z and increasing in y-direction nearly proportionally with the distance from the neutral plane.“8

_1CTA

"40

METALLURGIC-~,

VOL.

pinning. The p-values deduced either from modulus or from damping were normally found to be somewhat different so that it is necessary to distinguish between them. 3. EXPERIMENTAL

21,

1973

50 K

120

160 K

RESULTS

Figure 3 gives the results of isothermal irradiation experiments at 78 and 160°K with 0.52 MeV-electrons and a flux of 5.7 x 1Ol1 electrons/cm2 sec. The p,values obtained from measurements during irradiation

40 . *F 4 E 30.

240 K

z : 0”

20.

10

2

0

4 6 8 10 Strain Amplitude = 10’

12

FIG. 4. Decrement of the unirradiated sample vs. strain amplitude for T, = 78, 120, 160 and 240°K.

‘g h

015.

E=4.10S’ T,,, =78K

0.1 b)

dependence of the amplitude dependent damping of the unirradiated sample has been investigated. Figure 4 shows that the amplitude dependence is the more pronounced the lower the measuring temperature T,. Examples for the amplitude dependence of the decrement of the irradiated sample give Figs. 5 and 6. Here the curves for different electron doses (E = 1.5 MeV) are plott,ed for Tirr = T, = 78 and 160°K. Except for the small increase at low doses and large amplitudes at 160°K (which is close to the experimental

1

L, = T,,, q 78 K E = 1.5 MeV 0

5

10 irradiation

T, = T,,,) are plotted vs. the irradiation time t for the strain amplitudes E = 1 x lo-’ (Fig. 3a) and 4 x lo-’ (Fig. 3b). One recognizes that p, is larger for 78°K than for 160”K, but. this difference is much larger for t,he larger strain amplitude. The initial slope of these curves is generally used as pinning rate $. Thus one obtains at small strain amplitudes a pinning rate rather independent of temperature and at large strain amplitudes a decrease in # by going from 78 to 160°K. Because of this temperature dependence of Ij at large strain amDlitudes also the temperature (i.e.

I

.

40

.

Time [min]

FIQ. 3. Pinning point numbers pa derived from damping measurements vs. irradiation time for T, = Tirr = 78°K and T, = Ti,,= 160°K. (a) for the strain amplitudr R = 1 x IO-‘, (b) fore = 4 x IO-:.

”

50

*

“p Q & 30. E F x a 20. 160 K lo-

I,

0

.

2

.,

4 Shin

,

,

6 8 Amplitude ~10’

,

,

10

,

,

12

Flc. 5. Decrement vs. strain amplitude for T, = Ti,, = 78°K and different electron doses 0 (in cl/cm*).

LUCKE

et al.:

DAMPING

AXD

MODULUS

OF

order

are determined

$510” 40.9 0 Q w-

5.10” 25.1d3 1 ,lO” 15.10U

.

3,10y

-

of equations

(1)

of the influence of strain amplitude

on

the effect of a dose of 5 x 1013 el/cm2 is considered. For

78°K

T, = Tirr =

(filled squares)

an increase of p, with E and for (open circles) a decrease.

g 20. Y n

these two

curves

minimum,

one recognizes

much

gives

strain amplitude. are introduced

0

*

2

.

4

s

.

.

.

.

’

*

for

1

L

6 8 10 12 Strain Amplitude =lO’ FIG. 6. Decrement vs. strain amplitude for T, = Tirr = 160°K and different electron doses @ (in el/cme).

inaccuracy) rement.

In order to check measurements

the irradiation generally decreases the dec-

are, as in the unirradiated

from

At large doses, however, these differences

tend to disappear

and the curves for both tempera-

tures look rather similar. are obtained

case, rather different

Corresponding

for the modulus

Since equations amplitude

independent

and

Pm

as

functions

however, consideration

often

of

strain

p-values

0

’

2

.

.

4

into Fig. 7 (filled circles). case of a fixed measuring (6

from

both PA and

0)

78 to 160°K

crease at any point. and

amplitude. are

.

6

P,

and also

One recognizes temperature increase,

In

calculated

.

’

8

In

.

Strain

of

introduced that in the

of

when

This

(W -+ a).

temperature

of the strain amplitude.

o- .

were performed

valid for the

modulus change, it has no physical meaning to present literature,

at

160’K

Tirr =

for the various strain amplitudes,

part of damping

(open ellipses).

T, = Tirr were considered. the effect of T, or Tirr singly, also T, = 78°K after irradiation at

sets of curves

change.

(1) and (2) are only

is obtained

So far only cases with

At small doses the curves for 78 and 160°K

each other.

the

the results of a second run

160°K are plotted

T, = Tirr =

Also

int,o Fig. 7.

Finally, in order to demonstrate

the good reproducibility,

.

160°K

of the discussed

Also here a variation with strain amplitude

01 .

without

T,,= Tirr =

that this depth varies very

values forpl,

but less pronounced.

PA

one obtains

Since the distance between the depth

with the applied

corresponding

lo-

strain

pA(&) and P,,(F)

the apparent pinning point numbers gives Fig. 5 where

1 10’5

&

with these values.

such quantities

by formal application

and (2). An example

“41

Cu

an easy comparison

to permit

also in the following

4 [el/cm2]

T,,,=T,u=l60 K E =t5MeV

ELECTROS-IRRADIATED

T, = 78°K Tirr is raised

increase

differs

but there is no de-

In the case of fixed irradiation 160°K

Tirr =

(0,

0)

however,

both

p, and P_,~decrease, when the measuring temperature is raised from 78 to 160°K (0 ---f 0).

This decrease

is much larger for p, than for p,.

It is most pronoun-

ced for large strain amplitudes

and nearly vanishes

at small amplitudes.

’

10 Amplitude

’

2

.

g

4

*

6

*

8

’

10

I 10’

Fm. ‘7. The pinning point numbers pa from damping measurements (left part) and JJ~ from modulus measurements (right part) vs. strain amplitude for different combinations of T, and Tirr after a dose of Cp = 6 x 10’3 el/cmz.

ACTA

242

METALLURGICA,

E = 1.5MeV T,,,=T,=78 K

/To=

200

cl15c

alot

annealed

2

4 Strain

for 5 minutes at each T,

6

21,

1973 4. DISCUSSION

$I= 5.10’3 $*

PM

VOL.

8

10

Amplitude I 10’

8. pa VS. strain amplitude for a sample stepwise annealed after irradiation (5 min. per step). Electron dose CD= 5 x 1Or3 el/cms, T, = Z’I,, = 78°K.

FIG.

The effect of an isochronal annealing subsequent to the irradiation is shown in Fig. 8 for T, = Tirr = 78’K. With increasing annealing temperature TA the amount of additional pinning increases*. The particularly strong increase near TA = 200°K indicates the effect of a new recovery stage lying near this temperature”‘). The curves for TA = 160°K are also plotted in Fig. 7 (filled triangles). These curves agree rather well with the curves obtained directly by irradiation at 160°K and measuring at T, = 78°K (filled circles). In order to demonstrate the effect of the irradiation dose the curves p, and p,f vs. dose are plotted for T, = 78 and 160°K after Tirr = 160°K for a medium strain amplitude E = 4 x lo-’ (Fig. 9). The curves show that for high doses all the p-values converge to a single one. For low doses, however, just where the initial pinning rate is determined, the differences between values obtained at different temperatures of measurement or from either damping or modulus are rather large. At very small strain amplitudes (c.f. Fig. 3, upper part) these differences are considerably smaller. * This behaviour differs from that observed by Keefer’lg) who found that after Tfrr = 78°K an snneal at !!‘A = 150°K leads to an increase of px for T, = 4.2”K but to a decrease for T, = 78°K. Since Keefer worked at very large amplitudes (> 1O-B), it is suspected that this again is an effect of strain amplitude.

The most important result of the present paper is the reconfirmation of the earlier preliminary observation of the authors(g) that the so-called pinning rate minimum at 160°K is only connected with the temperature T, of measurement and not with the temperature Tirr of irradiation. As to be seen in Fig. 7. even an increase of both p and p,, is found if Tirr is raised from i8 to 160°K but T, is kept constant, (78°K). Such an increase ofp, with Tirr is the behaviour generally expect,ed because of the increasing mobility of the migrating point. defects. Also the effect of the annealing at 160°K after irradiation at 78°K by which nearly the same p-values are reached as by irradiat,ion directly at 160°K (Fig. 7) indicates, that there is no anomaly in point defect production or migration near 160’K. Only if at constant Tirr(160°K) T, is changed from 78 to 160°K a decrease in p is found (full and empty circles in Fig. 7). Moreover, this T,-dependence of p, i.e. the depth of the minimum, decreases strongly with decreasing st,rain amplitude E and reaches very small values at the smallest used amplitude of 1 k IO-‘. It cannot be decided, whether at still smaller amplitudes (in the true amplitude independent region) the minimum would disappear completely; but it can be concluded that in the present measurements the minimum is caused, at least to a large extent, by a strain amplitude dependent damping mechanism. Such mechanism must also be assumed for the work o_

iz..

I

10’3

10” Electrons

lo5 per cm’

FIG. 9. The pinning point numbers PA and pa vs. electron dose for T, = 78°K and T, = 160’K. The strain amplitude is E = 4 x IO-’ and TI,, = 160“K.

LUCKE

et al.:

DAMPING

AND

MODULUS

OF ELECTRON-IRRADIATED

damping

@:

haviour

5 010’~ cl/cm*

o Ti,=l60 K T,,, -160K

P

difference

the ratio phf/p, as obtained

Fig. 7 is plotted that

This

Ti,,. In

One recognizes

temperature

For small amplitudes

one

of the irradiat,ion

case of measuring

160”K, however, p,/p,

78°K

T, =

has ~,,~/p~ < 1 and independent

T, =

be-

in Fig. 10.

from the data of

vs. strain amplitude.

for the measuring

temperature

in the

and p, is also demonstrated

of p,

There,

10 -

measurements.

243

Cu

temperature

> 1 is generally

the two curves

found.

approach

each

other at a value only little below one. of p,/pA

In Fig. 11 the dose dependence

oL___l 0

according

of 4 x 10-7). Again, for T, = 78°K values p,,Jp, < 1 and for T, = 160°K values p,/p, > 1 are found.

10

5 Slrain

Amplilude

a 10’

> 10-s. amplitude

and

Buckc2)

did not report

used in that particular

The meaning

can best

at 78’K than the curves at larger however

(cf. Fig.

5), this

be recog-

T. After

irradiation,

dependence

st,rongly reduced so that then the 78”K-curve a steepness means

damping

that

due to irradiation

ent pinning

rate

Thus the increase

“minimum”

at

very strong

strain

T, =

sample

with

78”K,

which is not yet known.

of

the appar-

increasing

strain

reflects

mainly the

dependence

of the un-

the

physical

In contrast,

9 by going from the ‘Lminimum” temperatures

reduction

reason

of

the increase

in

at 160°K

is caused by a true increase

in pinning.

temperature. 7 also shows that the quantities

much less with strain agreement

amplitude

with the earlier

amplitude then

again

a strong

dependence

independent

the dislocation

irradiation

in both cases only the

damping resonance

or large

significant

T,

remains. theory

Since only

is valid, rates,

doses from

meaning

at the present Several p,

Whether

the

values at small ampli-

the value

of one have

or are only accidental

still not small enough

only

so that

close to one

are obtained.

of the p,/p,

small deviations tudes

of

which

has the same effect

only in these t,wo limit cases pi,/p,-values and independent

can

Since the strong

and pA give the true pinning

then p,

a value

This behaviour

from one are due to the ampli-

damping,

as going to small amplitudes:

amplitudes

cannot

than p,.

findings

P,,~ change

be decided

other

proposals

to interpret

appears satisfactory

differences

in the present case : (i) It has been

.

10

firr.160 c m’78

K K

I

aI aQ 1

This is in

* Similarly, the decrease of p with E for T, = 2’1~~= 16O’K (Fig. 7) indicates only that the curve obtained after irradiation approaches at large strain amplitudes the curve before irradiation.

0.1

10”

in

but none of them

of the authors’g)

that the minimum derived from modulus measurements is less pronounced than that derived from

a

or due to

state of investigat,ions.

and p, are given in literature,

to higher

It is due to the increase of the mobile fraction of irradiation induced point defects with increasing Figure

approach

in $J by going from the

160 to 78’K

amplitude at

the

e.g. to

(cf. Fig. 6).

and, therefore,

~5 increases

amplitude*.

irradiated

16O’K

T, = Tirr = for T, = 78°K

is

exhibits

similar to that of the other curves,

that of the curve for This

the

runs much steeper

amplitude

of pJpa

reduces this amplitude

It is to be seen (cf. Fig. 4) that

the A vs. E-curve before irradiation

below one.

deviations

set of measurements.

of the minimum

nized from Figs. 4-6.

curves

equal to or somewhat

tude dependent

et al.@), since they used strain amplitudes

Thompson

For high doses both

be reduced to that shown in Fig. 10.

FIG. 10. The ratios par/p&vs. strain amplitude as derived from Fig. 7 (T, = Firr = 160°K and T, = 78”K, !Z’i,, = 160’K) and Fig. 5 (T, = Tirr = 78’K). @ = 5 x 1Ol3cl/cm*.

of Simpson

is plotted

to the data of Fig. 9 (i.e. for an amplitude

10’4

10’5

Eleclrons per cm2 Fro. 11. The ratios px/p* 8~ derived from Fig. 9 vs. electron dose. Strain amphtude E = 4 x IO-‘, Ti,, = 160PK, T, = 78’K and 160°K.

244

_-ICTA METALLURGICA,

VOL.

21,

1973

0

l CO+

E not reported Thompson,BucklZi

= ’ MeV- t Bremstr 4 OS NV-e4 a5 t.ieV-ev 1.5 MN-eT 1.5 MN-e0 3.0 r&V-e-

20

40 Mean

FIU. 12. The pinning rate ratios 1; (lGO’K)/$

Transferred

not reported Roth et o1./9/ ~=l+lO-’ c:L~lo” present authors E=l .lO-’ ~=4.10-’ ~=1.5~1O~~Rolhet al.181

60 Energy [eV]

(78°K) measured by different, authors vs. kansferred energy?(Q) during irradiation.

statedf3) that such a difference occurs if the pinning is due to point defects migrating in only one dimension, i.e. in t’he case of crowdions. As will be shown in Appendix 1, it is felt that this is not correct and that also in this case p’a and p, should come out to be equal. (ii) As pointed out by several authorsf20-22) equations (1) and (2) used for calculating pa and p, are only first approximations neglecting the higher modes of dislocation vibration. It is shown in Appendix 2 that a calculation considering also higher order terms leads to values pM # p,, but that in the kHz-region this effect is very small and far below the accuracy of measurement. (iii) The existence of two dislocation components has been proposed as a reason for a disagreement of the p-values.c2) But also this interpretation leads to difficulties as will be shown at another place.(23) Finally, the question whether or not the mean energy (Q) transferred to the primary knock-on has an influence on the depth of the observed minimum shall be checked. In Fig. 12 the ratios of the $avalues found at 160 and 78°K (T, = T,,,) for all existing measurements are plotted vs. (Q). It can clearly be seen that there is no systematic energydependence, but again a great influence of the strain amplitude used in the experiments. For the lowest amplitudes (open symbols) a ratio near one is obtained. Summarizing the results of this investigation it can be stated that the so-called pinning rate minimum at 160°K has been found here too, but only as function of the temperature of measurement T, and not, as implicitly assumed by Thompson and Buck(2) and Seegerc3), as function of the temperature of irradiation Tirr.Furthermore, its depth has been found to strongly decrease with decreasing strain amplitude. This means this minimum indicates the occurrence of a temperature-denendent and stronnlv strain amnlitude-de-

pendent damping mechanism, but is not related to a temperature-dependence of the arrival rate of point. defect,s at the dislocations. Thus, in particular, no argument in favour of the assumption of thermal conversion of the crowdion can be derived from this minimum. Quite in the contrary, the very normal behaviour of the pinning rate near Tirr= 160°K (at constant temperature of measurement) favours the conclusion that no new process starts to occur in this temperature range. This conclusion is in agreement with the conclusion of Schilling et al.(s) who found that the model of the thermal conversion of the crowdion is in contradiction with the results of their resistivity measurements. REFERENCES 1. A. GRANATO and K. LOCKE. Phusicat Acoustica. Vol. 4. Part A, edited by W. P. MASON. “Academic Pres$ (1966): 2. D. 0. THOMPSON and 0. BUCK, Phqs. Status Solidi 37,

53 (1970). 3. A. SEEGER, Phys. Status Solidi 88, 235 (1970). 4. W. FRANK and A. SEEGER, Radiation Effect8 1,117 (1969). 5. W. SCHILLINQ, K. SCHR~DER and H. WOLLENBERCER, Phys. Status Solidi 38, 245 (1970). 6. H. M. SIMPSON, A. SOSIN and S. L. SEIFFERT, J. appl. Phys. 42, 3977 (1971). 7. H. M. SIMPSON and A. SOSIN, Radiation Effects 2, 299 (1970). 8. G. ROTH, G. SOKOLOWSKI and K. LOCKE, Phys. Stab Solidi 40, K77 (1970). 9. G. ROTH, G. SOKOLOWSKI and K. L~~cKE, J. Phys. 32, C2-145 (1971). 10. D. 0. THOMPSON and F. M. GLASS, Rev. Scient. In&rum. 29, 1034 (1958). 11. G. ROTH and K. F. RITTINIXIAUS, 2. ange2u. Phys. 32,331 (1972). 12. A. SOSIN, Acta Met. 10, 390 (1962). 13. D. KBNIG, J. V~LKL and W. SCHILLINO, Phya. St&w, Solidi 7, 591 (1964). 14. R. KAMEL and K. Z. BOTROS: Phys. Statue Solidi 12, 399 (1965). 15. J. A. DI CARLO and J. R. TOWNSEND, Acta Met. 14, 1716 (1966). 16. G. ROTH and V. NAUNDORF, Radiation Effect8 2, 187 (1970). 17. V. NAUNDORF, G. ROTH and K. LOCKE, Crystal Lattice Defecta 2, 205 (1971). 18. S. TIMOSHENKO and J. N. GOODIER, Theory of Ek&city. McGraw-Hill (1951).

Lt’CKE

et al.:

DAMPIR’G

AND

NODULL-S

19. D. IV. REEFER. Actn Met. 17. 611 (1969). 20. A. GRANATO, Thesis, Brown bniversity iIS%). 21. D. HOL.MES, Oak Ridge Report, personal communication. 22. W. HORNCSG, person&l co&n&ation. 23. G. ROTH, G. SOKOLOWSKI and Ii. LOCKE, To be published. 24. A. GRANATO and K. LCXE, J. a&. Phys. 27,583 (1956).

Seegert3) calculated the fraction

annihilated

By expanding

,u~(Z) of crowdions

instead

during their motion.

to Ref.

A = li, f AM/M

for pro-

of pinning

time tirr is given b!

s

if tirr is large compared

to the time needed

crowdion

to

equations

(1) and (2) these pinning

following

reach

relative

the

dislocation.

. L2 . [l + 0.9868 . g . L? T 0.9856

= K,,

u’} * L4]

g = LOO.Alr2.C a2 = Co2. K,

by the

According

to

K,

points cause the

changes of damping

0.9986u~) . L4

and modulus:

. C2

~7~4

= 1.0014.8.b2.A.G.

B.w/T~.C’

= 1.0147 * 8 . b” . A . G/n4 . C

A the mass per unit length, B the damping C the line tension, A the density

AtlAo= (1 + P(tin-))-4 =

By inserting the values A = lo-l4 g/cm,

than that with a2 so that

least two orders of magnitude the equations

P(tirr))-2

AM/M in Ref. (3) equations

AtlAo=

s 0

(AM/M),

reduce to

inating

[1 +

from

different

ever, has no physical

P,,

p,/p,

the pinning points orig1 would

and modulus.

meaning.

In reality,

contribute

With

This, how-

4

damping

and modulus are only able to see the total number of pinning points so that no difference pa can be derived in this way.

between p,

=

-

1

[(AM/Wo/(A~/W,l”*

- 1

one obtains for p < 1

IV1+ Y . tirr - .P~m(~)l~ distances

0.9856 - cl2 * P]

1

p, sz (A0/A,)“4

K&l4 and

to decrement

- L2 * [l -

p = L,/L -

P(Z) . dl

to these equations

independently

Y - tirr *

= K,

0.9986 - u2 - LJ]

Ritjh the notations

P(Z) . dl

lm,

(AM/M), = s o According

are

to

Imax

B = 10-a

nizes that the terms with g and g2 are smaller by at

A = K, . L4. [l -

used corresponding

and

dyn sec/cm2, C = low5 dyn, L, = 10W4cm one recog(3)

Inst’ead of these expressions,

constant,

of dislocations,

G the shear modulus.

[l + y.tirrl”“‘ ,uu,(l) .p(i).d;l-4

WfIW, = t1 + (AM/M),

and using in-

Here is

max pm(Z) . P(2) . dl

0

(2.9958 . g’ -

. (g2 -

1

p(ti,,) = y . tirr

of

* L4 . [l T l.9975gLz

of being otherwise

(3) the number

points p(t) after an irradiation

distribution

loops into a power series

obtains

ducing such defect in a distance between 1 and 1 + dl. according

for a delta function

the length of the dislocation

y is the production

rate of crowdions ‘and P(Z) dl the probability Then

the exact expressions of the Granato-

Liicke theory’%)

at a distance 1 from the dislocation

and reach the dislocation

of higher mode3

st,ead of only the first term the first three terms, one

by crowdions

which are produced

2. Consideration

245

Cu

with respect to the angular frequency

APPENDIX

1. Pinning

OF ELECTROS-IRRADIATED

and

x

for

= 1 + 0.9726 * a2 * Lo4(1 + p)

the values lo-’ o/2n

for

given

0/2x

= 10’ Hz.

region the correction of measurement. of observable

above

= 104 Hz This

one obtains

a2L,4 =

and a2L,4 = 4 x 10-I

means

that

in the kHz

term is far below the accuracy

Only for the MHz-region

magnitude

can be expected.

deviations