THE SNOEK-KÖSTER RELAXATION IN BODY-CENTRED CUBIC METALS

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THE SNOEK-KÖSTER RELAXATION IN BODY-CENTRED CUBIC METALS

M. Weller

To cite this version:

M. Weller. THE SNOEK-KÖSTER RELAXATION IN BODY-CENTRED CUBIC METALS. Journal

de Physique Colloques, 1983, 44 (C9), pp.C9-63-C9-82. �10.1051/jphyscol:1983906�. �jpa-00223328�

JOURNAL DE PHYSIQUE

Colloque C9, supplément au n°12, Tome kk, décembre 1983 page C9-63

THE SNOEK-KOSTER RELAXATION IN BODY-CENTRED CUBIC METALS M. Weller

Max-Planok-Institut ftir Metallforsohung, Institut filr Werkstoffwissensshaften, D-7000 Stuttgart 1, F.R.G.

Résumé - Dans les expériences de frottement intérieur sur le fer<x et les métaux du groupe transitoire V contenant des atonies lourds

(0,N,C) en solution solide interstitielle (ISA) des pics de Snoek-Kôster (SK) apparaissent,après êcrouissage, à des tempéra- tures beaucoup plus élevées que les pics de Snoek des ISA. La position et la hauteur du pic dépendent du degré d'êcrouissage et du teneur en ISA. Des expériences sur des monocristaux de Nb-O, Ta-O et Fe-N montrent clairement que deux processus de relaxation SK-1 (pic d'êcrouissage "classique") et SK-2, existent. On peut comprendre la plupart des résultats expérimentaux avec un modèle de dislocation fondé sur un "draggihg" des ISA mobiles par des boucles de dislocations agitées par. la contrainte. Dans une théorie plus raffinée, la relaxation de Snoek-Kôster dans les métaux c e . est attribuée à la formation thermiquement activée des paires de décrochements sur les dislocations vis a/2 < 111 > e n présence des ISA. Les conceptions élémentaires et les paramètres atomiques de cette théorie sont en bon accord avec les faits expérimentaux; particulièrement elle est en mesure d'expliquer pourquoi le pic SK-2, à température plus élevée, peut posséder une énergie d'activation plus faible que le pic SK-1,qui apparait à plus basse température.

Abstract - In internal friction experiments on<X-Fe and group-V transition metals containing heavy interstitial solute atoms

(ISA) (0,N,C) Snoek-Koster (SK) peaks appear after cold work at temperatures substantially higher than the ISA Snoek peaks. Peak position and height depend on degree of cold work and ISA con- tent. Experiments on monocrystals of Nb-O, Ta-0 and Fe-N clearly indicate that there exist two relaxation processes SK-1 (the

"classical" cold-work peak) and SK-2. Most experimental results may be understood with a dislocation model based on dragging of mobile ISA by dislocation segments moving under stress. In a more refined theory the SK relaxation in bcc metals is attributed to the thermally activated formation of kink pairs in a/2 <111>

screw dislocations in presence of ISA. Basic assumptions and atomistic parameters within this kink theory are in good agree- ment with experimental facts; it is especially able to explain that the high temperature SK-2 peak may have a lower activation enthalpy than the low temperature SK-1 peak.

1. Introduction

Dislocations in metals give rise to different types of anelastic rela- xation. The intrinsic properties of dislocations which correlate with

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983906

C9-64 JOURNAL DE PHYSIQUE

the crystal structure, and which lead e.g. to the well known Bordoni relaxation, were treated in recent reviews /1,2,3/. Additional relaxa- tion phenomena occur due to the interaction of dislocations with defects of atomic size, i.e. point defects. These interactions may be classified into two categories:

(i) Interactions with intrinsic point defects produced e.g. by cold work (at low temperatures) or irradiation. To these belong the relaxa- tions of the Hasiguti type (see e. g. /4,5,6/)

.

(ii) Interactions with foreign atoms. In bcc metals heavy interstitial solute atoms (ISA) such as C,N,O exert a strong influence on disloca- tions due to their local, anisotropic strain field. This leads to re- laxation phenomena which have long been known as "cold-work" peaks.

Recently the nomenclature Snoek-Kbster (SK) relaxation /4/ has become more usual (to differentiate it e.g. from other "cold-work" effects).

A common feature of all SK relaxations is that mobile ISA are involved.

They require a dislocation "dragging" mechanism as was first recog- nized by Schock /7/ (see below). The SK relaxations of heavy ISA in bcc metals occur at substantially higher temperatures than the corres- ponding ISA Snoek peak temperatures (see e.g. /3,4,8/). By this fact the SK relaxations may be distinguished from intrinsic dislocation relaxations which occur at substantially lower temperatures.

In this survey hydrogen SK relaxations are excluded. For group V metals (high solubility for H and other ISA) the oc -peaks were considered erroneously for some time as being an intrinsic dislocation relaxation.

Recently it was recognized that they are of SK-H type and masked the real intrinsic dislocation relaxations (see e.g. reviews /1,2,3/).

The Snoek-Koster relaxation was first reported by Snoek /9/ in inter- nal friction (IF) experiments on iron which was loaded with nitrogen after cold rolling. Fig. 1 shows Snoeks first measurement. An IF peak appeared a t 1 450 K (f = 0.2 H z ) , i.e. substantially higher than the corresponding N Snoek peak (%285 K). Later experiments of K$ /lo/

showed the same phenomenon. KBster, Bangert and Hahn /11/ were the first to study systematically the influence of cold work, carbon and nitrogen contents,and annealing treatments. Fig. 2 has been taken from their work and shows the influence of different degrees of deformation.

K6stey et al. concluded that the phenomenon is due to an interaction of carbon and nitrogen (i.e. ISA) with dislocations. For these

historical reasons the nomenclature ~ ~ ~ & y g ~ s ~ ~ ~ > e l a x a t i o n was suggested / 4 / . The SK relaxation should not be restricted to iron but should also occur in other bcc metals, especially group V metals which exhibit a relatively high solubility for 0 and N. This was first shown by experiments on Ta and Nb /12-14/.

In the following experimental part (2) we try to summarize the experi- mental situation for iron and group V metals. Theoretical aspects are treated in ( 3 ) . Earlier surveys were given in books /4,5,8/.

2. Experimental Situation

First IF experiments on the SK relaxation in a - F e /9,10/ were carried out after doping with nitrogen, i.e. they showed the nitrogen SK rela- xation. In the investigations of West / 1 5 / and K6ster et al. /11/ low alloy steels containing C and N were used. By this it is not possible to distinguish from these experiments if N and C (or only N) caused the relaxation phenomena. Kamber, Keefer and Wert /16/ first showed

F i g . 1: Snoek's f i r s t I F curve on deformed Fe w i t h 0 . 2 5 a t . % N a f t e r cold r o l l i n g /9/ ( f = 0 . 1 9 HZ).

F i g . 2: I n f l u e n c e o f d i f f e r e n t amounts of p l a s t i c deformation on t h e I F of i r o n ( f x 1 HZ) /11/.

t h a t f o r i r o n doped w i t h C t h e c o r r e s p o n d i n g carbon S K r e l a x a t i o n e x i s t s a t a b o u t t h e same t e m p e r a t u r e a s f o r N . The peak h e i g h t f o r t h e c a r b o n peak was, however, l e s s pronounced and developed o n l y a f t e r h e a t i n g t o t e m p e r a t u r e s above t h e peak p o s i t i o n i n t e m p e r a t u r e . T h i s was l a t e r confirmed by Barrand and Leak / 1 7 / , P e t a r r a and B e s h e r s /18/

and F e r r o n e t a l . /19/.

t L -

-

In _P

-

-

C . 2 -

0

- 2

- L -

The A r r h e n i u s diagram o f F i g . 3 ( 1 n Z v s . 1/T) g i v e s a s u r v e y on t h e e x p e r i m e n t s c a r r i e d o u t on p l a s t i c a l l y deformed i r o n f o r t h e tempera- t u r e r a n g e from 4 1 5 K t o 1000 K .

*)

Measurements on Fe doped w i t h N a r e r e p r e s e n t e d by open symbols, on Fe-C by f i l l e d symbols. For samples

1 0 0 0 1 ~

I

^K-I

I

F i g . 3: Survey of t h e r e l a x a t i o n p r o c e s s e s i n deformed Fe-N and Fe-C between 415 K and 1000 K ( A r r h e n i u s diagram o f r e l a x a t i o n t i m e s ) .

1000 900 800 700 600 500 L50 T I K I

1 1 1 1 1 1 I 1 I 1 I

-

-

(21 ( 1 )

^-

-

191eV 3 3 10-~'s IN]+ ^{+ -}

- 184eV 7 -

A

-

?7HZ&@ 5 -

@@ @

5eV 10-,~s(Nj-

-

1 1 1 1 1 1 1 1 1 1 1 1

10 12 1 L 16 18 2 0 2 2 2L

0 Snoek (19Lll

-&KC (19L81

A Kamber.Keefer,Wert 11961 1 n Barmnd. Leak 1196LI

0 ~etarra.~eshers+ 119671 V @lno.Sugeno (19671

D

Buchmann ~enned~~'ll9751

@ ~1etze[O.O~5N.25%1 119831

@ T1etze10.06N. 10%1 (1983)

@ west (19L1 I

Koster. Bangert. Hahn (195Ll

@ Bungordt, Pre~sendanz 119581 A Kamber. Keefer.Wert (1961 1

@ Lebedev. Postn~kov 119631

d Bernshste~n et al 119631 r Barrand Leak 1196Ll Petarra. Beshers 119671 r w Ferron et a1 (19731

+

Magalas 1 0 1 ~ . lo%]* 11981 1

C9-66 J O U R N A L DE PHYSIQUE

F i g . 4 : I F o f Fe w i t h O . l a t . % C a f t e r 10% deformation a t RT and 3.5% a t 77K.

1 : r u n up, 2: r u n down (f-2 H Z ) /21/.

- N-Snoek

15 -

10-

5 ^-

200 300 LOO 500 600 700 800 900

T I K I

F i

.

5: I F of Fe w i t h 0.045% N ( c u r v e s 1 and 5 ) and w i t h 0.06% N ( c u r v e s 2-41 be:ore ( 1 ) and a f t e r d i f f e r e n t d e g r e e s o f p l a s t i c deformation ( 2 a , b : 5 % , 3a,b 7 . 5 % , 4a,b 10%; 5: 25% - a t RT) ( f r 2 HZ) /23/.

which c o n t a i n e d C a n d N we used h a t c h e d symbols. Experiments which comprehended d i f f e r e n t d e g r e e s o f d e f o r m a t i o n a r e r e p r e s e n t e d by h o r i - z o n t a l dashed l i n e s w i t h b a r s i n d i c a t i n g t h e t e m p e r a t u r e r a n g e of t h e peaks'. Measurements on s i n g l e c r y s t a l s a r e c h a r a c t e r i z e d by e n c i r l e d symbols. +)

P u b l i s h e d v a l u e s f o r t h e r e l a x a t i o n p a r a m e t e r s , H and

Z

( a c c o r d i n g t o Z = Zoexp(H/kT) a r e r e p r e s e n t e d i n F i g . 3 by i n c l i n e d : f u l l l i n e s

( t h e f r e q u e n c y r a n g e covered i s i n d i c a t e d by b a r s , i f n o t known by a r r o w s ) . We r e s t r i c t t h e d i s c u s s i o n t o e x p e r i m e n t s which used samples which c o n t a i n e d o n l y C o r N (doped s a m p l e s ) . For

Fe-N

P e t a r r a and B e s h e r s /18/ p u b l i s h e d H = 1.65 eV To = 10-I8s and Buchmann and Kennedy /20/ 1 , 9 1 eV, Z = 3 . 3 - I O - ~ I S . For

Fe-C

t h e r e was p u b l i s h e d t o o u r knowledge o n l y oXe v a l u e by Magalas e t a l . /21/: H = 1.84 eV, Z = 7 - 1 0 - I 9 s ( t h e a u t h o r s s p e c i f i e d To = 0.7-10-19s; t h i s i s p r o b a b l y a t y p i n g e r r o r s i n c e t h i s A r r h e n i u s l i n e does n o t f i t t h e i r experimen- 0

t a l p o i n t ) . There e x i s t o n l y a few e x p e r i m e n t s on m o n o c r y s t a l s from I n o and Sugeno /22/ and r e c e n t l y from T i e t z e e t a l . / 2 3 / , b o t h Fe-N.

A s can be s e e n from F i g . 3 t h e low f r e q u e n c y I F e x p e r i m e n t s

-

o n l y t h e e x p e r i m e n t s of West / 1 5 / r e p r e s e n t mechanical a f t e r - e f f e c t measure- ments

-

on deformed Fe-C and Fe-N show d e f o r m a t i o n induced peaks

(known a s S K p e a k s ) i n t h e t e m p e r a t u r e from a b o u t 450 K t o 620 K . A p p a r e n t l y t h e SK-peaks f o r C and N a r e s i t u a t e d i n a b o u t t h e same

t e m p e r a t u r e r a n g e . D i f f e r e n c e s between measurements of d i f f e r e n t a u t h o r s a r e p r o b a b l y due t o ( i ) d i f f e r e n t C and N c o n t e n t s ( t o t a l and i n i n t e r s t i t i a l s o l u t i o n ) , ( i i ) d i f f e r e n t d e g r e e s o f p l a s t i c deforma- t i o n and ( i i i ) d i f f e r e n t ( o v e r a l l ) p u r i t y o f t h e i r o n samples. Most

( p r o b a b l y a l l ) i r o n samples used up t o t h e end o f t h e s e v e n t i e s con- t a i n e d a r a t h e r h i g h amount of m e t a l l i c i m p u r i t i e s (up t o s e v e r a l hundred a t - p p m ) , samples from low a l l o y s t e e l s even more.

A new g e n e r a t i o n o f e x p e r i m e n t s which used h i g h p u r i t y i r o n s t a r t e d o n l y r e c e n t l y . A s we s h a l l s e e below a s i m i l a r s i t u a t i o n e x i s t s f o r +) 2-values c a l c u l a t e d from t h e peaks i n I F c u r v e s p u b l i s h e d i n l i t e r a -

t u r e a c c o r d i n g t o Z = 1/27Tf (f = measuring f r e q u e n c y ) . The d o t t e d h o r i z o n t a l l i n e i n F i g . 3 i s f o r f = 1 Hz

( r =

0.16 s ) .

group-V transition metals. Magalas et al. /21/ studied the SK relaxa- tion in high purity Fe doped with

0.1

at.% carbon. Fig. 4 shows mea- surements on a sample deformed 10% at room temperature and additional- ly 3.5% at 77 K. A stable, well pronounced peak appears at 545

K.

Recently Weller, Tietze, Diehl and Seeger /23/ published measurements on monocrystals of high purity Fe which were doped with nitrogen.

Fig. 5 shows measurements on two monocrystals which contained 0.045 and 0.06 at% N and which were subjected to different amounts of plas- tic deformation. As can be seen from Fig. 5 two deformation induced peaks appear in the temperature range from 550 K to 800 K, designated as peak

( 1 )

and (2) in Fig. 5. With increasing degree of deformation both peaks increase and are shifted to lower temperatures. Peak (1) is less stable against annealing than peak

(2) as

is shown in Fig. 5 by subsequent runs for the same degree of deformation.

The data from /23/ ark included in the Arrhenius diagram In Fig. 3.

Deformation induced peaks (i-e. peaks which increase with degree of deformation) above 700 K were already reported earlier for poly- crystalline iron /24,25,26/. However, they were interpreted as being due to a grainboundary relaxation mechanism as suggested by KC /lo/.

As far as we know, peak (2) in /23/ is reported for the first time for deformed Fe-monocrystals. In this light its earlier interpreta- tion as a grain-boundary sliding mechanism must be doubted.

In Fig. 3 not all measurements obtained on deformed ferritic steels could be included. Additional studies exist on plain carbon or alloyed steels with

C,

in which martensitic structures exist after quenching.

These steels show IF peaks between ab ut 470 K and 570

K

- even with- out cold work (see e.g. /27,28,29/) .+?

2.2 Group-V transition metals

The experimental situation for Nb and Ta was recently discussed in detail /30/. So we confine co summarize the essential points.

2.2.1 Niobium

The Arrhenius diagram of Fig.

6

shows published measurements on de- formed niobium together with recent experiments obtained for Nb-0 /30/.

Additionally the Snoek relaxations for 0 , C and N in Nb (acc. to /30/) are represented (long, thin lines)++) . Measurements from /30/ obtained for a deformed Nb-0 (100 at.ppm) monocrystal are shown as thick lines, experiments on two 0-doped polycrystals (Nb-A, Nb-B) with intermediate lines (for details see /30/). Stress relaxation experiments by Oytana and Varchon /31/ are presented by dashed lines (the thin line corres- ponds to an extrapolation to zero annealing time, the thicker one to annealing to 418

K

for 70 h). The (dotted) horizontal line represents

+)~pparently dislocations originating from martensitic structures give rise to relaxations which may be of SK-type. Due to the rela- tively complicated structure of these samples it may be understood that abnormally low values for the activation enthalpy are some- times reported (e.g. H

=

0.5 eV, To

= I O - ~ S

/28/).

++)For Nb and Ta it has turned out as useful to include the Snoek rela- xation data /30/. In literature sometimes SK relaxations were con- fused with Snoek relaxations (especially when the shift of the peak temperature with measuring frequency were not considered). Earlier data on the carbon Snoek relaxation in Nb have to be revised

/ 3 0 ,

46/.

J O U R N A L DE PHYSIQUE

1000 800 700 600 500 LOO 350 T I K I

r~ r

I I I ^I I ^I

-

^{15 - 0} Gebhardt u. Rothenbacher (1963)

/ 1

^--

U) Boone and Wert (1963)

-

⁰ de Lamotte and Wert (196L) N Snoek C Snoek I-

-

'

-

C

V Igata et al. (1977)

a

Zolotukhin et al. I1975

5-

1.0 1.5 2.0 2.5 3.0

F i g . 6 : A r r h e n i u s d i a g r a m ( 1 n Z v s . 1 / T ) f o r r e l a x a t i o n p r o c e s s e s i n Nb between 300 K and 1000 K .

Fig. I F o f a Nb m o n o c r y s t a l w i t h 100 at.ppm 0 a s a f u n c t i o n of s u c c e s s i v e p l a . s t i c d e f o r m a t i o n /30/. The Q-I s c a l e above 600 K h a s b e e n i n c r e a s e d by a f a c t o r o f 2 . 5 .

-

1 - b e f o r e p l a s t i c d e f o r m a t i o n ( f = 1.20 Hz)

--2-- a f t e r 0 . 6 % t o r s i o n ( f = 1.21 Hz)

-.-3-.-after a d d i t i o n a l 1.5%

t e n s i o n ( f = 1.21 Hz) ---4--- a f t e r a d d i t i o n a l 0 . 3 %

t o r s i o n ( f = 2.8 H Z ) .

.. . ..

⁵

.. .. .

a f t e r t o t a l p l a s t i c d e f o r - mation o f 0 . 6 % i n t o r s i o n

+ 1.5 i n t e n s i o n + ( 0 . 3 % + 0 . 6 % ) i n t o r s i o n (f =

4.9 I i z ) .

a measuring frequency of 1 Hz. In Fig. 6 all individual measurements which are represented by filled symbols by Boone and Wert /13/ van Ooiien and van der Goot /32,33/ and Schlat /34/ are attributed to the S~-;ela tion of nitrogen. An Arrhenius fit yields H = 1.7 eV,

'Go

=

3.2- 10-"s (dotted line)

.

All data points from literature which we correlate with the oxygen SK relaxation in Nb are represented as open symbols. These may be sub- divided into two groups. One group of measurements for f z l Hz is situated at lower temperatures at about 480 K - 600 K: de Lamotte and Wert /14/ Gebhardt and Rothenbacher /35/ and Igata et al. /36/. On the other hand van Ooijen et al. /32,33/ claimed that the 0-SK relaxation should occur at higher temperatures ( ~ 7 0 0 K). Schlat /34/ observed two peaks at 600 K and 800 K (f = 8 Hz). Zolotukhin et al. /37/ investiga- ted the IF of niobium (and Ta) vacuum condensates (which usually exhibit a relatively high dislocation density) in the kHz-range. A peak a t z 7 6 0 K which appears in the first IF heating curve correlates with the low temperature group of measurements for the oxygen SK rela- xation in Fig. 6 for f -1 Hz.

The controversy in literature whether the oxygen SK relaxation in Nb may be located at lower (c500 K) or higher ( ~ 7 0 0 K) temperature was recently clarified by experiments on ultrapure Nb which was carefully doped with only oxygen /30/ (in Fig. 6 thin lines for samples Nb-A and Nb-B, thick lines for a Nb monocrystal; for details see /30/).

These experiments showed that there exist

two

deformation induced peaks in Nb (designated ( 1 ) and (2) in Fig. 6). Fig. 7 shows these IF measurements on a Nb-0 monocrystal from ultrapure Nb doped with

100 at.ppm 0 , which was subjected to different degrees of plastic deformation /30/. Two peaks designated as ( 1 ) (atz545 K for f =

1.2 Hz) and (2) (at z= 730 K for f = 1.2 Hz) can be seen. The heights of both peaks increase with degree of deformation. (The shift in peak temperature is caused by different measuring frequencies between 1.2 and 4.9 Hz)-For peak (1) and (2) the following relaxation parameters were determined: (1): H = 2.00 eV,

To

= 4.2-10-~0s; (2):

H

= 1.68 eV,

To

= 3.4.10-~ 3s. It is remarkable that the activation enthalpy of the low temperature peak was found to be higher than that of the high temperature peak.

2.2.2 Tantalum

Fig.

8

shows the Arrhenius diagram for measurements obtained on de- formed tantalum above 400 K. Again the Snoek relaxations for 0 , N and C in Ta are included (long, thin lines, acc. to /30/). All data for temperatures above about 550 K which are represented by open symbols we assign to the oxygen SK relaxation. To these belong the data from Schock and Mondino /12/ and de Lamotte and Wert /14/ as well as one point from van Ooijen et al. /32,33/ which was assigned in our opinion erroneously to the nitrogen Snoek relaxation (this point lies suffi- ciently far away from the N-Snoek line). Probably the IF peak observed by Delobelle et al. /39/ which was attributed to the carbon Snoek re- laxation is in reality an oxygen SK relaxation. The data marked in Fig. 8 by full symbols /32,33/ belong in our opinion to the

nitrogen

SK-relaxation.

Experiments on ultrapure tantalum which was only doped with oxygen were recently performed in Stuttgart /30,40/. Measurements on poly-

crystalline Ta doped with about 700 at.ppm 0 and after 30% deformation (sample Ta-A) and after 5% deformation (Ta-B) are shown in Fig.

8

by thick full lines /30/. These experiments show that phe assignment of the open symbol points to the oxygen SK-relaxation is justified. Mono- crystals of high purity Ta which were doped with different amounts of oxygen were studied by Rodrian and Schultz /40/. Their data on "pure"

JOURNAL DE PHYSIQUE

T [ K l

1000 900 800 700 600 500 400

1 5 ' I ' I ' I

^{I I}

-

_In 0 Schoeck and Mondino (1963)

1

C 0 de Larnotte and Wert (1964) C Snoek N Snoek

//

-

C

("c.w.p.N,O")

- ^{lo -}

_A Zo\otukhin ei at. (1975) V Delobelle et al. (1978)

*

^{Oytana 11977)}

0 -

1.0 1.5 2.0 2.5

10001T [ K - ' I

F i g . 8: Arrhenius diagram [ l n Z v s . 1/T) f o r r e l a x a t i o n p r o c e s s e s i n T a between 400 K and 1000 K .

100 200 300 LOO 500 600 T(K)

c1

xlo'

10

X I O ~ + 0-Snoek 0 - SKR

2 -

1 -

0

300 400 500

-

600 ^T(K1 700

. . i . 9

.:.

Tantalum .C :

Ill11 e: 1 to

- 0-doped (25alpprn)

I..?..

f. 1 Hz

. .

. .

1 2 3 4 5 6 7 - / : :

I I I.

1 1

I

1

..

_... j::

....,:.. . .

_{. .}

-. :TI1

Fig. 9a: IT ( f 2 1 . 3 H z ) of Ta w i t h F i g . 9b: I F ( f r 0 . 8 Hz) f o r Ta w i t h 25 at.ppm 0 a f t e r p l a s t i c deformation 167 a t .ppm 0 a f t e r p l a s t i c deforma- and subsequent a n n e a l i n g t r e a t m e n t s t i o n ( c u r v e 1 ) and a f t e r a n n e a l i n g ( f o r d e t a i l s see / 4 0 / ) . t o 730 K ( c u r v e 2 ) / 4 0 / . The O-

Snoek peak of undeformed T a i s shown i n a d d i t i o n .

li-SKR 1

Ta ( - 5 at.ppm 0) may be correlated with stress relaxation experiments on Ta-0 from Oytana /41/. Fig. 9 shows IF curves from Rodrian and Schultz /40/ for a Ta-monocrystal doped with 25 at.ppm 0 (Fig. 9a) and a monocrystal containing 167 at-ppm 0 (Fig. 9b) after plastic deforma- tion. The oxygen SK peaks are located at 600 K (9a) or 660 K (9b).

(The Snoek peak which is increased by deformation is designed by these authors as " y " )

.

The SK peak in Fig. 9b for Ta-0 (167 at.ppm) which correlates well with polycrystalline samples Ta-A and Ta-B (see Fig.8) is superimposed to a rapidly increasing background. In our opinion this belongs to a second deformation induced peak in Ta-0 which is situated at higher temperatures. This peak has not been measured up to now on monocrystals.

That is we expect that the same situation exists for Ta-0 as for Nb-0, i.e. two deformation induced SK-peaks. For the low temperature peak

( I ) , studied up to now, good coincidence exists between experiments

on ultrapure poly- and monocrystalline Ta-0 /30,40/ and earlier experi- ments as can be seen from Fig.

8.

2.2.3 Vanadium

To our knowledge only unpublished IF measurements of Mondino et al.

/42/ exist on cold-worked vanadium. Their samples contained oxygen in excess to nitrogen. The IF curves at the first heating up after cold- work showed a peak at about 660 K (f=l

Hz)

which was decreased drasti- cally by subsequent annealing to about 720 K.

Presumably this IF peak at 660 K can be assigned as oxygen SK-I(?) re- laxation in deformed vanadium.

Experiments on V suffer from the difficulties (i) in producing V-samp- les with low content of metallic impurities, and (ii) in doping with only one species of ISA, 0 or N. The latter one is necessary to dis- tinguish clearly between 0 and N SK relaxations since the Snoek peak temperatures for 0 and N in V ( f

= I

Hz) are only about 85 K apart from each other /5/ (compared with 140 K for Nb and 196 K for Ta, see e-g.

/30/),i.e. the migration enthalpies of 0 and N in V are close together.

2.3 Group-VI transition metals

As far as we know there exist no experiments on SK relaxations in de- formed Cr, Mo or W (group-VI transition metals). The reason may be that these metals exhibit, in contrast to group-V metals an extremely low solubility for ISA (O,N,C). Even the location of the Snoek peaks in these metals is not without doubt as recent controversies on Mo /43,44/ show. An additional handicap for the investigation of possible SK relaxations in these metals is their tendency to brittleness after doping with ISA. For. chromium, which becomes extremely brittle by small amounts of oxygen, even the intrinsic dislocation relaxations are unknown; in addition there is strong interference in IF experiments

with effects resulting from its antiferromagnetic nature (see e.g.

/ 4 5 / ) .

3. Theoretical Aspects

Theoretical models that account for the SK-relaxation in bcc metals must involve an interaction between dislocations and ISA. In the first model proposed by Koster et al. /11/ diffusion

(by

reorientation) of ISA in the core of the dislocations, i.e. a Snoek like mechanism, was suggested. Other models proposed by Mura et al. /47/ and Boone and Wert /13/ assumed a cyclical rearrangement of ISA around precipitates.

C9-72 JOURNAL DE PHYSIQUE

Other models were suggested for the SK relaxation in iron. Rosenfield /48,49/ interpreted the SK-relaxation in iron as an oxygen Snoek rela- xation. Okazaki /50/ proposed that the SK relaxation in iron may be caused by the presence of an fcc structure (induced by deformation) in the bcc lattice. These latter two models can be ruled out clearly from the fact that the SK relaxation occurs in other metals, too. For the other earlier models mentioned above /I 1,13,47/ the anelastic strain (relaxation strength) should ensue from the reorientation of the ISA. That is the SK peak height should be in the same range of order as that of the Snoek peak. This is in contradiction to the experi- mental finding that the relaxation strength per solute atom at the dislocations contributing to the SK-relaxation may be about one order of magnitude higher than that for the Snoek peak.

Schack /7/ first proposed that the SK relaxation should be based on the motion of dislocations, i.e. the species responsible for energy dissipation (anelastic strain) should not be the ISA's but the moving dislocations. According to this model (see below) the SK relaxation should be caused by stress induced motion of dislocation segments (strings), dragging the ISA which become mobile at temperatures higher than the Snoek peak. That is the strength of the SK relaxation should be deter- mined by dislocation movement and can thus considerably exceed that of the Snoek relaxation.Sch6ckB dislocation-dragging-model has so far found wide acceptance since it could account for most experimental facts.

Recently Seeger /51/ developed a more refined model by taking into account the structure of the dislocations in bcc metals. He proposed that the SK relaxation should be caused by formation of kink pairs in screw dislocations in the presence of ISA. Hirth /52/ proposed a similar model for hydrogen in &-iron. A recent view-point-set (pub- lished in Scripta Metallurgica) /53-57/ enlightens the positions of the different authors envolved in the development of theoretical models for the SK relaxation in connection with historical surveys.

3.1 String model of the SK relaxation

The dislocations are surrounded by an atmosphere of ISA as shown in Fig. 10a and thus locked at low enough temperatures (i.e. as long as the ISA are immobile). At high enough temperatures (substantially higher than that of the Snoek peak the dislocations may bow out bet-

ween immobile locking points

a = 0 under influence of the applied

stress by dragging of the now

. . . .-.

X

= - - - ^a

mobile ISA (Fig. lob). In the

• e e • • dislocation string model this dragging may be introduced by a viscous damping term for the dislocation movement in which

d > O inters the concentration of the

ISA at the dislocation and their diffusion coefficients.

Anelastic relaxation, i.e.

b

mechanical loss, results from this hindered dislocation move- ment. We discuss this model briefly:

F i g . 10: Models of t h e SK r e l a x a t i o n .

- -

.

^{. e} a ) p o s i t i o n o f d i s l o c a t i o n w i t h o u t

s t r e s s . b ) s t r i n g model, c ) k i n k p a i r

c 1

f o r m a t i o n i n a / 2 ( 1 11) s c r e w d i s l o c a - t i o n ( k i n k model)

..

The l o s s a n g l e , Q - I , f o r a s i m p l e Debye e q u a t i o n i s Q-I = O U Z . / ( I + ( U I Z ) ~ )

w i t h w = 2 T f ( f = measuring f r e q u e n c y ; An I F peak o c c u r s a t ( W Z ) = 1 w i t h t h e h e i g h t

Q , '

= A / 2 , ( l a )

from which t h e r e l a x a t i o n s t r e n g t h d can be d e t e r m i n e d .

The t e m p e r a t u r e dependence o f t h e r e l a x a t i o n t i m e Z i s u s u a l l y d e t e r - mined by an A r r h e n i u s e q u a t i o n

(HSK = a c t i v a t i o n e n t h a l p y f o r t h e SK r e l a x a t i o n ,

-

Z = p r e - e x p o n e n t i a l f a c t o r ) . The I F peak t e m p e r a t u r e Tm r e s u l t s from (27 w i t h 2 = I / & = c o n s t .

SchBck /7,55/ d e r i v e d t h e f o l l o w i n g e q u a t i o n s f o r t h e r e l a x a t i o n s t r e n g t h A and t h e r e l a x a t i o n t i m e Z f o r t h e d i s l o c a t i o n s t r i n g model o f t h e SK r e l a x a t i o n :

n

^{= p n ~ 2 ( 3 )}

(where i s t h e mean l e n g t h o f t h e d i s l o c a t i o n segments i n v o l v e d and A i s t h e d i s l o c a t i o n d e n s i t y , i . e . t h e i r t o t a l l e n g t h p e r u n i t volume;

B i s

a n u m e r i c a l f a c t o r i n t h e r a n g e o f t o 10-1)and

(OC i s a n u m e r i c a l p a r a m e t e r o f v a l u e a b o u t u n i t y ) . I n ( 4 ) b d e n o t e s t h e Burgers v e c t o r , G t h e s h e a r modulus, cd t h e ISA c o n c e n t r a t i o n a t t h e d i s l o c a t i o n s and D ' t h e i r d i f f u s i o n c o e f f i c i e n t ( n e a r t h e d i s l o c a - t i o n , ""')

.

I n Schacks model t h e main t e m p e r a t u r e dependence i n ( 4 ) r e s u l t s from t h e d i f f u s i o n c o e f f i c i e n t of t h e ISA

where

is is

t h e m i g r a t i o n e n t h a l p x o f t h e ISA n e a r t h e d i s l o c a t i o n . I n t h e i n d i s t u r b e d l a t t i c e HM'= H ( a c t i v a t i o n e n t h a l p y f o r t h e Snoek r e l a x a t i o n )

.

3.2 Kink model o f t h e S K - r e l a x a t i o n

The k i n k model i n t r o d u c e d by S e e g e r / 5 1 / , t a k e s i n t o t h e a c c o u n t t h e s p e c i f i c s t r u c t u r e of d i s l o c a t i o n s i n b c c m e t a l s , whereas t h e s t r i n g model from 3.1 i s a more phenomenological one. Screw d i s l o c a t i o n s w i t h b = a / 2 < I 11

>

( a = l a t t i c e p a r a m e t e r ) a r e h i g h l y immobile ( - i n con- t r a s t t o edge d i s l o c a t i o n s ) . I n I F ( a n e l a s t i c ) e x p e r i m e n t s t h e bowing o u t of d i s l o c a t i o n s under s t r e s s i s d e t e r m i n e d by n u c l a t i o n o f kink p a i r s and t h e i r l a t e r a l movement t o t h e immobile a n c h o r i n g p o i n t s . I n p u r e m e t a l s , and f o r screw d i s l o c a t i o n s , t h i s mechanism i s r e s p o n s i b l e a c c o r d i n g t o S e e g e r and S e s t s k / 5 8 / , f o r t h e y-peak ( s e e e . g . / 1 , 2 , 3 , 5 9 / ) . I n p r e s e n c e o f ISA, a s shown s c h e m a t i c a l l y i n F i g . IOc, k i n k movemeqt i s s t r o n g l y impeded by d r a g g i n g o f t h e ISA. T h i s b a s i c model was proposed by S e e g e r /51/ f o r t h e S K - r e l a x a t i o n i n bcc m e t a l s . S e e g e r / 5 1 / showed t h a t w i t h i n t h i s k i n k t r e a t m e n t f o r moving screw d i s l o c a t i o n s e s s ' e n t i a l l y t h e same e q u a t i o n s f o r

A

and 2 - a s f o r Schijcks s i m p l e r s t r i n g model ( e q s . 3 and 4 ) f o l l o w . An e s s e n t i a l r e s u l t of t h e

C9-74 JOURNAL DE PHYSIQUE

" k i n k model" e n t e r s i n t o t h e e x p r e s s i o n f o r t h e a c t i v a t i o n e n t h a l p y /51/:

H~~ = 2 H~

+

^{H ~ '}

,

( 6 )

where 2 Hk i s t h e a c t i v a t i o n e n t h a l p y f o r f o r m a t i o n o f a k i n k p a i r . R e c e n t l y S e e g e r /60/ p r e s e n t e d a more r e f i n e d t r e a t m e n t o f t h e k i n k model f o r t h e SK r e l a x a t i o n . He showed t h a t t h e r e may b e i n t e r f e r e n c e of k i n k p a i r s g e n e r a t e d u n d e r ( o s c i l l a t i n g ) s t r e s s w i t h k i n k s a l r e a d y p r e s e n t i n t h e r m a l e q u i l i b r i u m . The ( l i n e a r ) d e n s i t y o f k i n k s o f o n e s i g n i n t h e r m a l e q u i l i b r i u m o n d i s l o c a t i o n l i n e s i s

yEq.

Then t h e mean d i s t a n c e between k i n k s p r e s e n t i n t h e r m a l e q u i l i b r i u m ( o n e s i g n ) ,

xk, i s ( / 5 6 / ) : e q

Xk =

/ y k

^{( 7 )}

The r e l a x a t i o n t i m e f o r k i n k p a i r f o r m a t i o n u n d e r s t r e s s i n p r e s e n c e o f a n i n t e r a c t i o n w i t h k i n k s i n t h e r m a l e q u i l i b r i u m h a s b e e n c a l c u l a - t e d by S e e g e r / 6 0 , 3 0 /

H e r e t h e a d d i t i o n a l p a r a m e t e r

Yd

^{i s} t h e l i n e t e n s i o n o f . t h e d i s l o c a - t i o n s . A d e t a i l e d d i s c u s s i o n o f eq. ( 8 ) ( s e e / 3 0 , 6 0 / ) shows t h a t , a s a c o n s e q u e n c e , d i f f e r e n t t e m p e r a t u r e d e p e n d e n c i e s o f C ( i . e . a c t i v a - t i o n e n t h a l p i e s ) may r e s u l t . F o r t h i s t h e mean d i s t a n c e o f k i n k s , x k ( e q . 7 ) , h a s t o b e compared w i t h t h e d i s l o c a t i o n l o o p l e n g t h L b e t - ween immobile a n c h o r i n g p o i n t s . F o r low d e n s i t y o f t h e r m a l l y p r o d u c e d k i n k s ( l o w t e m p e r a t u r e s )

,

i . e . xk ("1

/

Eq)

>)

L t h e r e l a x a t i o n t i m e i s

(Dk = k i n k d i f f u s i v i t y ) , a n d f o r h i g h e r d e n s i t y o f k i n k s xk

<<

L ( h i g h t e m p e r a t u r e s )

L* ( k ~ ) 3/2exp ( H ~ / ~ T )

/

L I ~ . ( 9 b ) A c o m p a r i s o n o f e q s . ( 9 a ) a n d ( 9 b ) shows t h a t d e p e n d e n t on t h e c r i t e - r i o n xk))L o r xk(<L (low o r h i g h t e m p e r a t u r e s ) t h e r e l a x a t i o n t i m e e x h i b i t s different d e p e n d e n c i e s on l o o p l e n g t h ( L o r L2) a n d (more d r a s t i c a l l y ) i s d e t e r m i n e d by d i f f e r e n t a c t i v a t i o n e n t h a l p i e s . ( 2 Hk o r H k )

.

T h i s may b e u n d e r s t o o d i n a s i m p l i f i e d model: The r e l a x a t i o n t i m e f o r bowing o u t o f d i s l o c a t i o n s between immobile a n c h o r i n g p o i n t s may b e e x p r e s s e d a s

( A = a r e a s w e p t o u t , vd

;

mean d i s l o c a t i o n v e l o c i t y ) . S i n c e A C C L ~ ( s e e e . g . / 4 , 6 1 / ) 2 may b e w r l t t e n a s

The d i f f e r e n c e b e t w e e n ( 9 a ) a n d ( 9 b ) may b e s e e n w i t h ( 1 1 ) a s f o l l o w s : ( a ) low t e m p e r a t u r e s : t h e d e n s i t y o f t h e r m a l l y p r o d u c e d k i n k s ( k i n k p a i r s ) i s low, i . e . t h e i r a v e r a g e d i s t a n c e e x c e e d s t h a t o f t h e l o o p l e n g t h (xk)>L). T h e r e i s no i n t e r f e r e n c e o f s t r e s s g e n e r a t e d k i n k s w i t h t h e r m a l l y p r o d u c e d o n e s . Thus t h e d i s l o c a t i o n v e l o c i t y i s con- t r o l l e d by t h e maximum s i d e w a r d d i s t a n c e a newly g e n e r a t e d k i n k p a i r c a n e x t e n d . T h l s i s L, t h e (mean) a i s t a n c e between t h e a n c n o r l n g p o i n t s , w h e r e t h e k i n k s p i l e up ( s e e F l g . ~ O C ) . T h i s g i v e s vdWL a n d w i t h e q . ( 1 1 ) t o Z W L . The a c t i v a t i o n e n t h a l p y i s t h a t f o r k l n k p a i r

n u c l e a t i o n , 2 Hk.

( b ) h i g h t e m p e r a t u r e s : f o r t h e o p p o s i t e c a s e , xk(<L, t h e r e e x i s t on t h e a v e r a g e one o r more k i n k p a i r s i n t h e r m a l e q u i l i b r i u m p e r d i s l o c a - t i o n l o o p l e n g t h L. T h i s l e a d s t o an i n f e r e n c e o f t h e k i n k p a i r s pro- duced under ( o s c i l l a t i n g ) s t r e s s w i t h k i n k s p r e s e n t i n t h e r m a l e q u i l i - brium and t h u s t o r e c o m b i n a t i o n p r o c e s s e s : The mean d i s t a n c e t r a v e l l e d by k i n k s , i s reduced t o xk ( a n d becomes t e m p e r a t u r e d e p e n d e n t ) . The d i s l o c a t i o n v e l o c i t y i s vdo(xk. The r e l a x a t i o n t i m e 'i;,w i t h e q . ( l l ) , i s ZcC L ~ / x ~ . , The a c t i v a t i o n e n t h a l p y i s reduced t o Hk ( s i n c e t h e d e n s i t y of k l n k s i n k s i n c r e a s e s w i t h T) f o r xk<< L.

The t r a n s i t i o n from c a s e ( a ) (xk>>L) t o c a s e ( b ) ( x <<L) o c c u r s a c c o r - d i n g t o S e e g e r e t a 1 /30/ f o r xk = L/2 and f o r a va9ue o f

The p r e s e n c e o f mobile ISA i n t h e neighbourhood o f a d i s l o c a t i o n l e a d s t o a n i n t e r f e r e n c e by t h e s t r a i n f i e l d s u r r o u n d i n g t h e k i n k . T h i s r e s u l t s i n an i n c r e a s e o f t h e k i n k v i s c o s i t y o r i n a d r a g g i n g f o r c e on t h e moving k i n k , which may b e e x p r e s s e d a c c o r d i n g t o S e e g e r /51,60/

i n t e r m s o f t h e k i n k d i f f u s i v i t x Dk which e n t e r s i n t o e q s . ( 8 ) and ( 9 ) a s

1 -HS

D oc-exp (F) . +)

Cd ( 1 3 )

With r e s p e c t t o cd i n e q . (13) one has t o d i s t i n g u i s h between low and h i g h ISA c o n c e n t r a t i o n s ( f o r d e t a i l s s e e / 3 0 / , / 5 6 / , /6O/).Low cd may l e a d t o a t e m p e r a t u r e dependent d i s t r i b u t i o n o f ISA between d i s l o c a - t i o n s i t e s ( c d ) and b u l k s i t e s ( c b ) a c c o r d i n g t o

cd = c b exp ( H ~ / ~ T ) ( 1 4 )

By t h i s an a d d i t i o n a l t e r m c o n t a i n i n g a b i n d i n g e n e r g y HB e n t e r s i n t o ( 1 4 ) and t h u s a s an a d d i t i o n a l t e m p e r a t u r e dependence i n t o Dk ( e q . 13)

.

F o r h i g h ISA c o n c e n t r a t i o n t h e d i s l o c a t i o n s i t e s may b e s a t u r a t e d and no t e m p e r a t u r e dependent t e r m a p p e a r s i n (1 3 )

.

From t h e p r e c e d i n g d e s c r i p t i o n i t f o l l o w s t h a t f o u r l i m i t i n g c a s e s may be d e r i v e d a c c o r d i n g t o S e e g e r /56/ f o r t h e e f f e c t i v e a c t i v a t i o n e n t h a l p i e s Heff = d l n '77 / d ( l / k T ) o f t h e SK r e l a x a t i o n . The d i f f e r e n t c a s e s a r e o b t a l n e d from e q u a t i o n s ( 9 a ) f o r low t e m p e r a t u r e s o r ( 9 b ) f o r h i g h t e m p e r a t u r e s t o g e t h e r w i t h eq. ( 1 3 ) ( i f t h e r e e n t e r s a b i n d i n g e n t h a l p y o r n o t ) . The r e s u l t s a r e l i s t e d i n T a b l e 1. The kT-terms a r e a consequence o f t h e p r e - e x p o n e n t i a l t e r m s i n e q s . ( 9 a ) and ( 9 b ) . The d i f f e r e n c e i n H e f f between t h e c a s e of low and h i g h t e m p e r a t u r e s i s

HA$&

^-

HA$^

^{= Hk}

- 3

k T r H k . T h i s w i l l be t e s t e d i n s e c t . 4 w i t h r e s p e c t t o e x p e r i m e n t a l r e s u l t s .

I

c d s m a l l c d high

(low t e m p e r a t u r e s ) 1

xk

<<

L

;

H , + H ~ + H ~ - 3/2 kT H ~ + H ' - J / z k ~ ( h i g h t e m p e r a t u r e s )

;

Table 1 : E f f e c t i v e a c t i v a t i o n e n t h a l p i e s H e f f f o r t h e SK r e l a x a t i o n a c c . t o

Seeger

/56/.

+'

^{1 n eq. (} 13) a t e r m exp (-HM/kT') w i t h = k i n k m i g r a t i o n e n t h a l p y can b e o m i t t e d s i n c h s 2 f o r kcc m e t a l s i s r n t h e r a n g e of I O - ~ ~ V ( s e e e . g . / 3 0 / )

,

i - e . Hk

<<

kT f o r u s u a l t e m p e r a t u r e s .

C9-76 J O U R N A L D E PHYSIQUE

The temperature dependence of the relaxation time for the low and high temperature case according to eqs. (9a), (9b), and (13) (assuming HB = 0 , i.e. cd high enough) is shown schematically in Fig. 1 1 in an Arrhenius diagram for two dislocation loop lengths L 1 and L 2 > L 1 (this will be discussed below). The break for the transition from low

temperatures (right in 1/T sca1e)to high temperatures (left) is con- trolled by eq. (12) (see 4.2.2).

4. Discussion of Experimental Results on the SK relaxation

In the preceding survey on theoretical models for the SK-relaxation it was shown that the strength of the relaxation as given by the two models is the same, since both models involve the concept of bowing out of dislocations fixed at immobile nodes (see e.g. / 6 1 / . Thus in both models the specific nature of the SK-relaxation, the interaction with foreign atoms

-

and the statements from different models

-

enters into the relations for the relaxation time.

4.1 Relaxation strength

4.1.1 Temperature dependence.- According to the dislocation models for the SK-relaxation, the relaxation strength

A

should be independent of temperature (eq. 3 ) . For a deformed Nb-0 (100 at.ppm) monocrystal the temperature dependence of the high temperature relaxation process (2) has been studied in / 3 0 / by a combination of IF and mechanical after- effect experiments. The results for three different degrees of plastic deformation are shown in Fig. 12. As can be seen from Fig. 12

a

^is

indeed independent of temperature. As far as we know this has been verified for the first time for a deformation induced peak. This gives strong support for the dislocation model and clearly rules out other models /11, 47-50/, in which a Snoek type reorientation

-

for which A X l / T is expected - was assumed as basic mechanism of the SK rela- xation.

4.1.2 Dependence on amount of plastic deformation.- According to eq.

(3) it is expected that Q ~ ( & L ~ , i.e. the product of the dislocation densityn and the square. of the (mean) loop length ( L ~ ) enters into the expression for

A .

One expects that

A

i-ncreases with increasing amount of plastic deformation due to an increase of the dislocation density (unless L decreases very drastically with increasing deforma- tion). This is seen in most experiments dealing with the influence of degree of deformation on the SK relaxation (and considered as a criterion that indeed a SK-process occurs). Detailed investigations of the influence of degree of deformation were performed e.g. on Fe(/ll/, / 1 8 / , /21/-/23/,/37/), Ta (/12, 40/) and Nb (/14,30/). Examples are reproduced in Fig.2 and Fig. 5 for Fe and Fig. 7 for Nb. These clearly show that the deformation induced peaks increase with amount of plas- tic deformation (in Fig. 5 and 7 for both peaks (1) and (2)).

4.2 Relaxation time

4.2.1 Pre-exponential factorc,.

-

The pre-exponential factor for

-

both, the string model (eq. 4) and the kink model (eqs. 9a, 9b) de- pends on the state of the sample. The main parameters which enter into eqs. (4) or (9) are the loop length L (in the string model L ~ , in the kink model L or L ~ ) and the concentration of ISA at the dislocationsr cd. According to eq. (2a) the IF peak temperature Tm directly reflects variations in

'iT,

(Tm O( In T o )

.

-A*

- - --+

⁽⁰¹

T 1x1

I F i g . 12: Temperature dependence of t h e

I > r e l a x a t i o n s t r e n g t h A f o r Nb-0 and t h r e e

1 1

- - -

¹ d e g r e e s of p l a s t i c d e f o r m a t i o n ( a , b ,c ; kTDIWl kTplSKII kT f o r d e t a i l s s e e / 3 0 / ) .

F i g . 1 1 : S c h e m a t i c A r r h e n i u s diagram ( l n r v s . l / k T ) f o r t h e k i n k model and t w o d i s c r e t e d i s l o c a t i o n segment l e n g t h s il and L2 /23/.

( a ) I n f i u e n c e o f p l a s t i c d e f o r m a t i o n . - With i n c r e a s i n g p l a s t i c deforma- t i o n t h e l o o p l e n g t h L , i . e . t h e d i s t a n c e between n o d e s i n t h e d i s l o c a - t i o n network s h o u l d d e c r e a s e . Thus, a l s o Tm s h o u l d d e c r e a s e w i t h p l a s - t i c d e f o r m a t i o n . T h i s was found i n many p u b l i c a t i o n s on t h e SK r e l a x a - t i o n . F i g . 5 , e . g . shows, f o r Fe-N,a s l i g h t d e c r e a s e o f Tm w i t h i n c r e a - s i n g amount o f d e f o r m a t i o n f o r b o t h p e a k s ( 1 ) and ( 2 ) . A s i m i l a r be- h a v i o u r i s s e e n i n F i g . 7 f o r Nb-0. T h i s r e s u l t i s i n good q u a l i t a t i v e a g r e e m e n t w i t h t h e d i s l o c a t i o n m o d e l s .

( b ) I n f l u e n c e o f a n n e a l i n g . - A n n e a l i n g t r e a t m e n t s a f t e r p l a s t i c d e f o r - m a t i o n a r e e x p e c t e d t o a g a i n i n c r e a s e t h e l o o p l e n g t h s . T h i s i s e . g . d e m o n s t r a t e d i n F i g . 5 f o r t h e example o f Fe-N.

( c ) Peak b r o a d e n i n g . - A f t e r p l a s t i c d e f o r m a t i o n t h e d i s l o c a t i o n n e t -

-

work u s u a l l y c o n s i s t s of-a d i s t r i b u t i o n w i t h d i f f e r e n t l o o p l e n g t h s L p e a k i n g a t a mean v a l u e L . A d i s t r i b u t i o n i n L l e a d s t o a d i s t r i b u t i o n i n Z , i . e . a b r o a d e n i n g o f t h e SK p e a k s . 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 L v a l u e s was a l r e a d y i n t r o d u c e d by Schock / 7 / . I n most e x p e r i m e n t s on t h e SK r e l a x a t i o n it i s r e p o r t e d t h a t t h e SK p e a k s a r e e s s e n t i a l l y b r o a d e r t h a n a s i m p l e Debye peak. Only a few e x p e r i m e n t s a r e known i n which a q u a n t i t a t i v e e v a l u a t i o n o f t h e peak s h a p e was t r i e d . U s u a l l y a log-normal d i s t r i b u t i o n which may b e c h a r a c t e r i z e d ( a c c o r d i n g t o Nowick and B e r r y / 4 / ) by a d i s t r i b u t i o n p a r a m e t e r , ,fi

,

was a p p l i e d . Buchmann a n d Kennedy / 2 0 / o b t a i n e d f l z 3 f o r Fe-N, Maqalas e t . a l . /21/

b

r 3 . 5 f o r Fe-C. I n d e t a i l e d s t u d i e s o n Nb-0 /30/,by u s i n g I F and mechanical after- e f f e c t e x p e r i m e n t s , a t e m p e r a t u r e i n d e p e n d e n t

/3 =

^{3 . 5} was o b t a i n e d . T h i s s t r o n g l y i n d i c a t e s , t h a t , a s a n a l y s e d i n / 3 0 / , t h e b r o a d e n i n g o f t h e I F p e a k s i n d e e d r e s u l t s from a l o o p l e n g t h d i s t r i b u t i o n and n o t , a s may a l s o b e d i s c u s s e d , f r o m a d i s t r i b u t i o n i n a c t i v a t i o n e n t h a l p i e s .

4.2.2 A c t i v a t i o n e n t h a l p y H~~

.-

The a c t i v a t i o n e n t h a l p y o f t h e SK-rela- x a t i o n i s t h e p a r a m e t e r which most s p e c i f i c a l l y r e f l e c t s t h e p h y s i c a l n a t u r e o f t h e r e l a x a t i o n p r o c e s s i n t h e d i f f e r e n t m o d e l s ( s t r i n q o r k i n k ) . The e x p e r i m e g f i a l f a c t t h a t H~~ i s c o n s i d e r a b l y h i g h e r t h a n IiS - t h e d i f f e r e n c e H

-

^{'H i s} i n t h e r a n g e o f 0 . 5 eV

-

i s i n t e r p r e t e d i n b o t h models i n a d i f f e r e n t manner:

C9-78 JOURNAL DE PHYSIQUE

( i ) W i t h i n t h e f r a m e o f t h e s t r i n g model two e x p l a n a t i o n s w e r e s u g g e s - t e d . Schbck and Mondino / 1 2 / a r g u e d t h a t f o r d i f f u s i o n n e a r t h e d i s l o - c a t i o n c o r e i n t e r s t i t i a l l o c a t i o n s w i t h d i f f e r e n t t e t r a g o n a l i t i e s may l e a d t o a n e x t r a e n e r g y b a r r i e r . As p o i n t e d o u t by S e e g e r / 4 9 / t h i s e x p l a n a t i o n c a n n o t h o l d f o r a / 2 (111, s c r e w d i s l o c a t i o n s i n b c c m e t a l s s i n c e t h e s t r e s s f i e l d of t h e d i s l o c a t i o n s h a s t r i g o n a l symmetry s o t h a t t h e v a r l o u s s i t e s o f < 1 0 0 ) o r i e n t e d ISA d i p o l e s a r e e n e r g e t i c a l l y e q u i v a l e n t . A n o t h e r o b j e c t i o n a g a i n s t I S A w i t h t e t r a g o n a l s t r a i n f i e l d s i s d e r i v e d f r o m t h e e x p e r i m e n t a l f a c t t h a t pronounced S K p e a k s o f h y d r o gen

-

t h e t e t r a g o n a l i t y o f w h i c h i s e x t r e m e l y s m a l l

-

a r e o b s e r v e d i n

M-Fe and g r o u p V t r a n s i t i o n m e t a l s . The same d i f f i c u l t y a r i s e s i n e x p l a i n i n g r e c e n t e x p e r i m e n t s i n which a S K peak was r e p o r t e d f o r f c c N i c o n t a i n i n g c a r b o n / 6 2 / which o c c u p i e s i s c t r o p i c i n t e r s t i t i a l s i t e s .

( i i ) W i t h i n t h e k i n k model a s o u t l i n e d i n s e c t i o n 3.2 a n d summarized i n Tab.1 t h e d i f f e r e n c e H~~ - H~ may v a r y w i t h i n w i d e l i m i t s . The m i n i - mum v a l u e i s Hk, t h e maximum v a l u e 2Hk

+ ^{H ~ .}

An e x t r a t e r m o f a b i n d i n g e n e r g y H~ was a l r e a d y p r o p o s e d by De B a t i s t / 5 / who s u g g e s t e d t h a t t h e ISA a r e partitioned between l a t t i c e ( b u l k ) s i t e s a n d d i s l o c a t i o n s i t e s a c c o r d i n g t o e q . ( 1 4 )

.

F o r a q u a n t i t a t i v e d i s c u s s i o n o f t h e e x p e r i m e n t a l a c t i v a t i o n e n t h a l p i e s p u b l i s h e d i n l i t e r a t u r e w i t h i n t h e t h e o r e t i c a l framework o f t h e k i n k model o u t l i n e d a b o v e , i t i s e v i d e n t t h a t t h e i n t e r n a l s t a t e o f t h e s a m p l e s (Cdr d e g r e e and mode o f d e f o r m a t i o n ) must b e known i n d e t a i l . Only f o r s p e c i a l c a s e s a s l i s t e d i n T a b l e 1 , may t h e o r e t i c a l e q u a t i o n s

d i r e c t l y be applied - a s was shown f o r Nb-0 / 3 0 / .

A d d i t i o n a l i n f o r m a t i o n may b e o b t a i n e d : ( i ) From I F e x p e r i m e n t s on deformed p u r e b c c m e t a l s , i . e . d a t a o n t h e y - p e a k ( C X -Fe / 6 3 - 6 5 / , Nb / 6 6 / , Ta / 4 0 / ) , f r o m which v a l u e s f o r t h e k i n k p a i r f o r m a t i o n e n t h a l p i e s may b e d e r i v e d .

( i i ) From measurements o f t h e t e m p e r a t u r e d e p e n d e n c e o f t h e c r i t i c a l r e s o l v e d s h e a r s t r e s s i n b c c m e t a l s f o r w h i c h , a s shown by Seeger/67/thc t h e o r y o f k i n k p a i r f o r m a t i o n i s a l s o a p p l i c a b l e . F o r a q u a n t i t a t i v e d i s c u s s i o n o f t h e e x p e r i m e n t a l a c t i v a t i o n e n t h a l p i e s of t h e SK r e l a x a - t i o n i n Nb-0 and Ta-0, t o g e t h e r w i t h ( i ) and ( i i ) , s e e / 3 0 / , and f o r Fe s e e / 5 1 / .

4.3 Two S n o e k - K o s t e r P e a k s

From e x p e r i m e n t a l r e s u l t s on deformed F e , N b and Ta a s r e p r e s e n t e d i n t h e A r r h e n i u s d i a g r a m s o f F i g s . 3 , 6 and 8, i t c a n b e c o n c l u d e d t h a t t h e r e e x i s t two d e f o r m a t i o n - i n d u c e d p e a k s d e s i g n a t e d as p e a k s ( 1 ) and ( 2 ) . The o c c u r e n c e o f two SK p e a k s may b e e x p l a i n e d by t h e d i s l o c a t i o n model by-assuming t h a t

two

d r a s t i c a l l y d i f f e r e 2 t d i s t r i b u t i o n s o f l o o p l e n g t h s L e x i s t , which peak a t two v a l u e s L1 and L2 ( s e e a l s o d i s -

c u s s i o n i n / 2 3 / ) and t h a t d i f f e r a t l e a s t o n e o r d e r o f m a g n i t u d e from e a c h o t h e r .

T h i s b e h a v i o u r i s shown s c h e m a t i c a l l y i n F i g . 1 1 i n a n A r r h e n i u s t y p e d i a g r a m ( I n Z v s . l / k T / 2 3 / ) f o r two d i s c r e t e l o o p l e n g t h s L and L2

>

L, At t h e t r a n s i t i o n from t h e low t e r n p e r a t u r q c a s e ( X > > L , ? W ~ , e q . 9 a ) t o t h e h i q h t e m p e r a t u r e c a s e ( x k < < L, TCX L

,

e q . 9 b f , a t T ( e q . 121, t h e s l o p e o f t h e l i n e s , H'~, i s r e d u c e d by H k . F i g . 11 shows t h a t i n I F - e x p e r i m e n t s which a r e c a r r i e d o u t a t c o n s t a n t 2 = 1 / 2 7 l f ( d a s h e d l i n e ) two p e a k s may r e s u l t u n d e r t h e s e c o n d i t i o n s . W i t h i n t h e I n Z - 1 / T r a n g e , marked by d o t t e d l i n e s , t h e low t e m p e r a t u r e peak i s c h a r a c - t e r i z e d by a h i g h e r s l o p e ( a c t i v a t i o n e n t h a l p y ) t h a n t h e h i g h tempera- t u r e p e a k . A p p a r e n t l y t h i s i s t h e c a s e f o r Nb-0 / 3 0 / .

The o r i g i n of two L d i s t r i b u t i o n s i s n o t y e t c l e a r . I n / 3 0 / i t w a s s u g g e s t e d t h a t s m a l l p r e c i p i t a t e s may s u b d i v i d e l o n a d i s l o c a t i o n s i n t o

smaller segments which would cause the low temperature peak (1).

An alternative interpretation was offered in /23/. In deformed metals there may exist wide regions with long dislocation segments and narrow regions with dengely packed, much shorter segments. Thus two groups of lengths, L and L2, may result and thus lead to two, well separated SK peaks. another specific prediction from the kink model is that for two SK peaks the activation enthalpy for lower temperatures may exceed that for higher temperatures (compare Table 1); the difference should be equal to H

-

1/2 kT. This prediction was verified for Nb-0 /30/

(see sect. 2.5.

I

)

.

According to /301 the activation enthalpy for the low temperature peak (xk >) L) is 'H (1 ) = (2 .OO f 0.18) eV, and for the high tem erature peak (x c<

L)

e f f ~ ~ $ (2) = (1 - 6 8 0.015) eV and thus

Hk

= ( 1 )

- (2f +

i / Z kTe=f0.32 eV

+

0.03 eV = 0.35 eV is obtained. This Hk va ue 1s (within experimental error) in good agreement with de Lima's result /66/ derived from the )Y-peak in pure Nb (Hk = 0.33 eV).

4.4 Miscellaneous and open questions

(i) Different behaviour of Fe-C and Fe-N.

-

Originally it was claimed

...

/18/ that C does not contribute to the SK-relaxation in Fe, but then Kamber et.al. /16/ demonstrated that the SK-relaxation of C in H - F e exists but only after annealing to higher temperatures than the SK peak temperature (see 2.1). An explanation for this apparently different behaviour was given by Mondino and Seeger / 6 8 / . Cold-working generates both dislocations and vacancies. At heating up in IF experiments, complexes of C or N with vacancies are formed (migration of C or N at T s 3 S O K). IF and positron annihilation experiments on neutron irra- diated iron doped with C or N demonstrated that the dissocation of these complexes

-

which is a pre-requisite for appearance of the SK- peak

-

occurs for Fe-N at about 500 K and Fe-C at 600 K /69-7?/. This explains why the SK-N(1) peak appears in the first heating up curve, whereas the SK-C(l) peak appears only after heating to higher tempera- tures ( 2 6 0 0 K) in a second run. The influence of vacancy-ISA-inter- action on the development of the SK relaxation should not be confined to Fe but should also dominate in other bcc metals if the dissociation of vacancy -1SA complexes occurs at higher temperatures than the SK peak temperature. Considering group-V transition metals - non metal systems, Nb-N may behave similarly as Fe-C, Ta-0 and Nb-0 like Fe-N /68/.

Another explanation which additionally may play a r61e was advanced by Fast /8/. Carbon in oc-Fe has a lower solubility than nitrogen and may precipitate as carbide near dislocations and/or grain boundaries /72/.

Heating to higher temperatures dissolves the precipitates and developes the SK peak. This mechanism may dominate for low degrees of cold-work

(low vacancy concentration)

.

(ii) "Undeformed" metals. - Dahlstrom et .al. /73/ showed that an SK

...

peak appeared in quench-aged Nb-N without external deformation (as in / 1 3 / ) . Transmission electron microscopy studies showed that disloca- tion loops and tangles of significant density were created around coarsened (quench-aged) precipitates, which apparently are responsible for the SK-relaxation. Similar conditions may predominate in Fe-(C+N) alloys /74/ which showed SK(1)-peaks after rapid quenching.

The SK(2) peak was observed in / 2 3 / in undeformed Fe-N monocrystals (see Fig. 5, curve 1) and in deformed high purity iron. By this an intrinsic relaxation mechanism for this high temperature SK (2) -peak may not be ruled out completely. Other relaxation mechanisms occuring at high temperatures ( ~ 0 . 5 Tme - ) may be discussed within this context

(for a recent review see / 7 5 ) ) . To these belong mechanisms such as climb of dislocations by movement of jogs and/or participation of vacancies in polygonised dislocation arrangements /76-78/. These structures may also be present in monocrystals. By using magnetic after-effect measurements