Influence of moving dislocations on the nuclear spin-lattice relaxation time in the rotating frame

Solid State Communications, Vol. 17, pp. 85—87, 1975

Pergamon Press.

Printed in Great Britain

INFLUENCE OF MOVING DISLOCATIONS ON THE NUCLEAR SPIN-LATTICE RELAXATION TIME IN THE ROTATING FRAME G. Hut and A.W. Sleeswyk Laboratorium voor Fysische Metaalkunde, Materials Science Centre, Universiteitscomplex Paddepoel, Groningen The Netherlands and H.J. Hackelöer, H. Selback and 0. Kanert Institut für Physik der Universität Dortmund, 46 Dortmund-Hombruch, W. Germany (Received 23 January 1975 by M. Cardona)

Dislocation motion at various velocities in Na~Qsingle crystals was studied using the spin-locking technique. The resulting spin—lattice relaxation time in the rotating frame, T1~,is strongly dependent on the plastic deformation rate è, but not on the plastic strain e. The experimental results are in accord with a theoretical expression for T1,, based on the relaxation model of Rowland and Fradin for atomic diffusion.

THE PRESENT communication is a sequel to the results of a NMR study on dislocation motion in alkali 1 In halide crystals using the Jeener pulse sequence. the new experiments the spin locking technique is utifised to obtain the spin—lattice relaxation time in the rotating frame, T 1~,which is known to be influenced by slow atomic movements. Na23Cl single crystals, ~ 11.5mm, 25 mm long, with the (100> crystal axis parallel to the specimen axis were umaxially compressed at room temperature with different strain rates e. The r.f. coil, connected

The total relaxation rate T~is composed of a static andthe a dynamic part due mentspart during deformation, or: to the atomic move/ 1 \ / 1 / 1 ~ = + (1) ‘~

~

~T1PJ

~Ti~s

.

\T1PJD

Adapting the Ailion3 expression by Rowland 2 and for derived the analogous case ofand selfFradin the term (l/T1p)D can be written as: diffusion ~,

2

,,

(—~—-) \TipJD

=

Rb°~= GD

2

2

2

1 + with a Bruker SXP 4-60 spectrometer, was mounted coaxially with the specimen and perpendicularto the external magnetic fieldH0.

2

7 H 1 — + GQ T

(WD>

+ (WQ)

~2> 2 2 2 2 7H1+(WD)+(WQ>

—

T

(2)

with GD and GQ geometry factors determined by the distribution of the nuclear dipoles and electric field gradients, respectively, and the mean dipolar and energies generally 2 },quadrupolar r the average waitingdetermined time between atomic by tr {R jumps, which is much longer than the actual jumping time TS and H1 the strength of the locking field. This equation holds under the conditions mentioned before :1

Figure 1 shows two typical decay curves obtained after the locking pulse from which it is obvious that Ti,, is reduced during plastic deformation, just as is in ~he case of the zero-field Similar to the fmdings for Trelaxation time T10 10, T1~exhibits a strong dependence of this reduction on the deformation rate ê and not on the compressional strain e.

(‘~.‘L> (4>

.~

85

—

86

MOVING DISLOCATIONS IN THE ROTATING FRAME

is the Larmor frequency (a) T~~ (woY1, where c~(~ of the spins considered. frequency. (b) r ~

(o.D)~, where ~

An expression for GD in equation (2) can be derived from the change in the dipolar Hamiltonian due to dislocation jumps, assuming that its spin dependent part is a constant during jumps (condition a). The resulting expression is:

~

I

k~ti ~ 1

I

GD

=

Pmflp flo

—

A~ A?j n

=

Pmflp .

gD,

_____

T~.20C ~/2

•

is the dipolar line width

A “jump” of a dislocation — the terminology is borrowed from the similar self-diffusion model — is the sudden advance of a stretch of dislocation line from a temporary stationary position to a second position of this type. The progress of the dislocation through the crystal is a sequence of such “jumps”.

r

FR)

with Pm mobile dislocation density per cm ber of involved nuclei per cm 3, A ~ dislocation geometricallength, part ofn0: the number of spins per cm dipolar spin Hamiltonian of spin i due to spin k before a dislocation jump, A~:the same after a jump, g~: geometry factor for 1 cm dislocation length, nc: number of spins contributing to ~

(Tip~&B3eC) tim.

FIG. 1. Showing the spin locking sequence two (see decay insert),curves with F(t) zero after and finite plastic deformation rates è. T 1~is calculated from the expression: S(r) = S(0) exp (— r/T1~),with r the duration of the locking pulse and S the maximum of F(t).

4 T~.20C Na~Ct

10 1

0

~

~

{

.~-

I =

Il — I.

~

~_-

~

~

INa~I

I

_______

T~.20C

L1’18 R1g) 10 6 ii

14i1

3 -2

1

uiuu~ 0

1

2

3

4

iii

5 -

~

Bj

I

=

g~, (4)

sec_I

FIG. 2. Showin~the linear relationship between (R~~’and H 1 and è’ for two ê-values.

equation (2) has a negligible effect on the relaxation process, which is confirmed by the experimental data)4

GQ

,~~H1~190

~ R

3. Pm iO~cm~, l0~’cm n~ iO~cm_’ Because andn0 ZD S 1, it is evident that the dipolar part of

flp Pie ~ ~
~

~

3~C

I

~

f

The value of GD can be estimated by substituting some realistic values of the parameters e.g.:

The quadrupolegeometry factorGQ inequation (2) can be derived analogously, yielding:

fF(t)

Sbk ~ ~15G

\(~~&5sec)

(3) 2, n~:num-

k=i i=i

Vol. 17, No.1

6 7 H 2 1

8

02

FIG. 3. Comnarison between Jeener (RD) and spin locking (Rn!) experiment, showing the proportionality between RD andR~,and è.

~

with p: total dislocation density, Bik the electric field gradient produced by atom k at nucleus i before a jump, Bg’ie the same after a jump, gQ: geometry-factor for 1 cm dislocation length.

Comparing GQ to GD and taking into account the g~is of the order of ~ it is easy to see that GQ ~ GD. Therefore equation (2) can be rewritten as:

Vol. 17, No. 1

R1~~ = (-J—-~ \Tip/D

MOVING DISLOCATION IN THE ROTATING FRAME

(WL>+ ~2H~ + (c4>

=

________________

~

P Pm

(5)1

—

—.

Became r ~ ?~ the Orowan equation for dislocation movement can be written as

e

1 =

nct)b~P~

(6)

(where b: burgers vector, nb: mean distance covered by a jumping dislocation, çb: geometry factor (~ 0.5)). Together with the result of Kanert and Mehring):5 =

Ap

(7)

with A = cci~stant(A ~ 36 cm2 sec2 for Na23C1), the foregoing equations result in: 2H2 + ~A

+

2

•

1

gQ

e.

(8)

~ = Measurements verifying this expression are shown in Fig. 2. From this picture it can be deduced that (R~0))i is approximately linearly dependent on H~as required by equation (8). The scatter in the experimental data might be due to an orientation effect. The value for the proportionality constant R~°~/è ~ 390 for H 1 -~0 is obtained from these spin locking experiments, which is in good agreement with the result of the Jeener

87

yie1ds~ H~+H~~ 2G~.SubstitutingHD ~ 0.9 experiments, viz. Rj~/è of the experimental curves of400.’ Fig. 2 The withintersection the H~ axisG6 one gets forHQ : HQ ~ 1.1 G. The latter is roughly comparable to the result obtained by Kanert et al.:7 HQ ~ 1.7 G. The values indicate that (~4) is sufficiently small to allow a strong cross relaxation between the dipolar and quadrupolar systems, i.e. to enable the total spin system to come to thermal equilibrium between dislocation jumps. The derivation of equation (2) is valid only in the case of the assumption of such rapid energy transfer. So far only two è-values were used. However, in Fig. 3 Rj~°~ andfor RDa dependence on éhasand been checked in more detail number of è-values a few values ofH 1, one of which is too large to be adequately future are in theFig. accuracy of the results such as presented presented 2. Points to be investigated in the in Fig. 2, and the detailed calculation of gQ• Acknowledgements The authors wish to thank Prof. M. Mehring for his stimulating discussions of this work. l’his study was performed as part of a project on the investigation of dislocation dynamicsonjointly sponsored by the Foundation for the research Matter, F.O.M. —

and the Metaalinstituut T.N.O. at Apeldoorn. Financial support of the “Herbert-Quandt—Stiftung” is gratefully acknowledged.

2.

REFERENCES HUT G., SLEESWYK A.W., HACKELOER H.J., SELBACH H. and KANERT 0. Solid State Commun~15, 1115 (1974). ROWLAND TJ. and FRADIN F.Y., Phys. Rev. 182, 760(1969).

3.

AJLION D.C.,Advances inMagneticResonance, Vol. 5. P. 177ff. Academic Press, New York (1971).

4.

6.

HUT G., HACKELOER H.J., SELBACH H., KANERT 0. and SLEESWYK A.W.,Proceedings of the 18th Colloque Ampere, Nottingham (1974) (to be published). KANERT 0. and MEHRING M., NMR Basic Principles and Progress, Vol. 3, p. 62ff. Springer Verlag, Berlin (1971). HEBEL L.C., Solid State Phys. 15, p. 457ff. Academic Press, New York (1963).

7.

KANERT 0., KOTZUR D. and MEHRING M. Phys. Status Solidi 36, 291 (1969).

1.

5.