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Strain induced knight shift in rhodium metal
Solid State Communications,
Vol. 8, pp. 1151—1154, 1970.
Pergamon Press. Printed in Great Britain
STRAIN INDUCED KNIGHT SHIFT IN RHODIUM METAL P. Ségransan, J.C. Fanton and P. Averbuch, Laboratoire de Spectrométrie Physique,* Faculté des Sciences de Grenoble, Cédex n°53, 38 Grenoble-Gare, France. -
(Received 19 hay 1970 by E.F. Beriaut)
The second moment of N.M.R. lines of several cold worked samples are linearly related to the mean square distortion, measured by analysis of X-ray diffraction line shapes. This effect can be understood as coming partly from the variation of the susceptibility with shear stress and partly from dipole—dipole coupling between d electrons and the Rh nuclei.
GREGORY and Bommel12 have observed the absorption of ultrasonic waves by nuclear spins in tantalum metal. This absorption clearly arises from a quadrupolar coupling, given the huge quadrupole moment of the Ta’81 nucleus. In an attempt to study other coupling mechanisms between phonons and nuclear spins in transition metals, we have examined the effect of strains upon the N.M.R. of metallic rhodium, whose nuclear spin 1
coming from the same ingot, or to demagnetisation broadening; in either case, the following analysis is unaffected.
The experimental technique we employed on our powder samples was the same as that used for measuring quadrupolar coupling due to defromation in copper,3 i.e. mean square distorsions and second moments of N.M.R. lines compared for the same samples. As the nuclear spin is the only mechanism for line broadening is through Knight shift changes due to these strains.
center of the line. One obtains:
-
~.
The mean square strains are calculated from the Debye—Scherrer X-ray diffraction line shapes As shown by Warren and Averbach~the diffracted intensity can be Fourier-analysed and expressed as function of ~\0, the angle measured from the
IS,, (SO)
~,
=
1
+
2~A~(S, 1) cos(~n so’)
where S refers to the sample, I is the order of Bragg reflexion, and T, = r’tg 00 ~ I) is the product of an instrumental factor, of the number i4,, (S) of Bragg planes contributing to the process i.e. the coherence length connected with the stacking fault density, and of the coefficient A~(S,I) related to the deformation by: .
Five samples of powdered metallic rhodium were obtained by grinding in a ball mill. This produces cold work. Four of these samples were then annealed, with different degrees of annealing. The most annealed sample had an N.M.R. second moment of 0.2 G,2 equal to that of a reference specimen made of sponge metal. This is to be cornpared with the theoretical Van Vleck value of 0.0135G.2 The discrepancy can be attributed either to a small impurity content, common to all samples * Laboratojre associé au C.N.R.S. 1151
A~(S,I) = cos(277lZ~) = 1 2n2l2Z~(S)2 where Z,~is the distance of the nth neighbour from its nominal position, measured in atomic distances. The macroscopic mean square strain is given by: —
_=.-
—
=
lirn
72
—
‘\L) n-. 0 One can now eliminate the instrumental effect
1152
STRAIN INDUCED KNIGHT SHIFT IN RHODIUM METAL
as well as the .4~(S)by computing the ratio:
.4,(S,I)/A,(R,l) .4, (S,l ‘)/A,(R,l’)
__________________
=
1
—
2712(/2
j~2)
Z,, (S)2
where R designates the reference sample. In fact, as one is interested in the limit of small n, it is sufficient to take the second moments ot~I. The coherence effect of the sponge was such that it was more convenient to use the most annealed sample as reference. The results for the [1,0,0] and [1, 1, 11 directions obtained with a copper source, for 11,1,01 direction obtained with a molybdenum source, as well as the N.M.R second moments in a 12 kg field, measured at room ternperature with a Varian wide-line spectrometer, are summarized in Table 1.
5
r)
(100)
(~)2
4
2 0~
=
=
Vol.8, No.14
+
~
+
2 03
~
+
(1.1.0)
5ü2
(~L)2 (1
—
and one sees that the three mean square strains, measured on the same sample, are linearly dependent parameters. It is evident from Table 1 that the theoretical relation is not fulfilled and this gives a lower limit of 20 per cent to our error. As the Knight shift tensor, as function of
Table 1. Second moments of N.M.R. line at l2koe and square mean distorswn in 3 directions for ~ refere~icesample a’id four cold worked samples. __________________________________________ Samples R S 1 S2 S3S4 2 oe2 0.20 0.23 0.38 0.50 0.62
(!\H)
strain,in isa assumed to be this a linear function is of the cubic system, dependence characterized by three coefficients, giving for a powder: =
5 2 (~c,
2 0~
~
6
C2
2
3 03 ~
i~12~ 106 1_~t ic?
0.0
1.2
5.9
6.8
0.0
1.2
3.6
5.3
0.0
0.8
4.5
5.0
9.9
Lj~110,
2
j.~L1 ~ L
io~
7.5
_____________________________________________ If the X-ray measurements had great accurac, it would be possible to generalize an analysis already made for isotropic systems.5 The six components of the strain tensor .4, are projected on to 0 group basis functions, giving .‘1’,~
2)H2
2
C5 05
0
ne cannot separate, from such experiments, the F co’ntribution, which should be obtained independently from hydrostatic pressure measurements. Nevertheless, as in isotropic systems, one has, for of a strain field due to a random distribution dislocations: 05 = o = 10 o~ we have neglected the O~2 contribution, thus making a systematic error not greater than our experimental error. In Fig. 1 is plotted as function of ±(2 2 2 relat ion ~ 03 + 3 ü~ ), giving the
th
belonging to the ni
row of I”
,
with
i
=
1, 3, 5
=
1.7 102
H 0
in the Bethe notation. As the random distribution of strains on a cold worked sample must have cubic symmetry, it follows from Schur’s lemma that: fl4)’7,1)
=
~h.
~t
2
and that the coefficients a, describe the sample. One obtains:
between shear strain and the extra line width. This result can be interpreted by saying that only d band of metallic is sensitive to shearthe strains which produce Rh a perturbing crystalline field on those electrons. In first order of perturbation, there is a change in level populations near the Fermi surface, and the magnetic field l-4~ gives a further re-population
Vol. 8, No. 14
STRAIN INDUCED KNIGHT SHIFT IN RHODIUM METAL
~
1153
&
x 10 0.0
0
5
10
15
20
FIG. 1. Plot of measured second moment of N.M.R. line in a l2koe field, as function of the mean square distortion ~ (2 c~ + 3 o~).
responsible for Pauli paramagnetism. We are interested in the effect which is linear both in H 0 and in the shear field; this effect is the sum of contribution of individual electrons, if one assumes integral J to be uniform within thethe d exchange band.6 An uncompensated d electron gives an extra contribution to the Knight shift, equal to _0.161.106 oe per d electron, calculated from the temperature dependence of the Knight shift,7 and confirmed by relaxation time measurements.8 However, such an electron produces a dipolar magnetic field, which is not balanced by the effect of other electrons. This can be estimated from the Van Vleck paramagnetic contribution to
the Knight shift, which is proportional to With the values of Seitchik ci a!. this is
K
3
r
0.29.106 oe per d electron. In our crude model, this contribution to bein multiplied by the 2 0)2 =is 4/5 order to get (1 — 3 cos contribution to second moment. So, we conclude that the dipolar interaction is responsible for 70 per cent of the observed broadening.
Acknowledgements
— We thank Mrs Bernasson and Descouts, from ‘Ecole de Physique de l’Université de Genéve’, who gave us the samples, Dr. de Bergevin, from Laboratoire d’Electrostatique et de Physique du Metal’ for helpful discussions about X-ray technics, and Mr d’Assenza, for technical assistance in X-ray experiments.
1154
STRAIN INDUCED KNIGHT SHIFT IN RHODIUM METAL
Vol. 8, No. 14
1.
REFERENCES GREGORY E.H. and BOMMEL H.E., Phys. Rev. Leit. 15, 404 (1965).
2.
GREGORY E.H., Ph. D. Thesis, University of California, Los Angeles (1965).
3.
AVERBUCH P., DE BERGEVIN F. and MULLER-WARMUTH W., C.r. hebd. sèanc. Acad. Sci., Paris 249, 2315 (1959).
4. 5.
WARREN B.E. and AVERBACH B.L., J. app!. Phys. 23, 497 (1952). AVERBUCH P., C.r. hebd. séanc. Acad. Sci., Paris 253, 2674 (1961).
6. 7.
SEGRANSAN P. and AVERBUCH P., to be published. SEITCHIK J.A., JACCARINO V. and WERNICK J.H., Phys. Rev. 138A, 148 (1965).
8.
NARATH A., FROMHOLD A.T. and JONES E.D., Phys. Rev. 144, 428 (1966).
Les seconds moments des raies de R.M.N. de plusieurs échantillons écrouis sont relies Iinéairement ~ Ia deformation quadratique moyenne, mesurée par analyse des formes de raies de diffraction des rayons X. La variation de susceptibilité avec le cisaillement, et le couplage dipolaire entre electrons d et noyaux, contribuent tous deux a l’effet observe.
Vol. 8, pp. 1151—1154, 1970.
Pergamon Press. Printed in Great Britain
STRAIN INDUCED KNIGHT SHIFT IN RHODIUM METAL P. Ségransan, J.C. Fanton and P. Averbuch, Laboratoire de Spectrométrie Physique,* Faculté des Sciences de Grenoble, Cédex n°53, 38 Grenoble-Gare, France. -
(Received 19 hay 1970 by E.F. Beriaut)
The second moment of N.M.R. lines of several cold worked samples are linearly related to the mean square distortion, measured by analysis of X-ray diffraction line shapes. This effect can be understood as coming partly from the variation of the susceptibility with shear stress and partly from dipole—dipole coupling between d electrons and the Rh nuclei.
GREGORY and Bommel12 have observed the absorption of ultrasonic waves by nuclear spins in tantalum metal. This absorption clearly arises from a quadrupolar coupling, given the huge quadrupole moment of the Ta’81 nucleus. In an attempt to study other coupling mechanisms between phonons and nuclear spins in transition metals, we have examined the effect of strains upon the N.M.R. of metallic rhodium, whose nuclear spin 1
coming from the same ingot, or to demagnetisation broadening; in either case, the following analysis is unaffected.
The experimental technique we employed on our powder samples was the same as that used for measuring quadrupolar coupling due to defromation in copper,3 i.e. mean square distorsions and second moments of N.M.R. lines compared for the same samples. As the nuclear spin is the only mechanism for line broadening is through Knight shift changes due to these strains.
center of the line. One obtains:
-
~.
The mean square strains are calculated from the Debye—Scherrer X-ray diffraction line shapes As shown by Warren and Averbach~the diffracted intensity can be Fourier-analysed and expressed as function of ~\0, the angle measured from the
IS,, (SO)
~,
=
1
+
2~A~(S, 1) cos(~n so’)
where S refers to the sample, I is the order of Bragg reflexion, and T, = r’tg 00 ~ I) is the product of an instrumental factor, of the number i4,, (S) of Bragg planes contributing to the process i.e. the coherence length connected with the stacking fault density, and of the coefficient A~(S,I) related to the deformation by: .
Five samples of powdered metallic rhodium were obtained by grinding in a ball mill. This produces cold work. Four of these samples were then annealed, with different degrees of annealing. The most annealed sample had an N.M.R. second moment of 0.2 G,2 equal to that of a reference specimen made of sponge metal. This is to be cornpared with the theoretical Van Vleck value of 0.0135G.2 The discrepancy can be attributed either to a small impurity content, common to all samples * Laboratojre associé au C.N.R.S. 1151
A~(S,I) = cos(277lZ~) = 1 2n2l2Z~(S)2 where Z,~is the distance of the nth neighbour from its nominal position, measured in atomic distances. The macroscopic mean square strain is given by: —
_=.-
—
=
lirn
72
—
‘\L) n-. 0 One can now eliminate the instrumental effect
1152
STRAIN INDUCED KNIGHT SHIFT IN RHODIUM METAL
as well as the .4~(S)by computing the ratio:
.4,(S,I)/A,(R,l) .4, (S,l ‘)/A,(R,l’)
__________________
=
1
—
2712(/2
j~2)
Z,, (S)2
where R designates the reference sample. In fact, as one is interested in the limit of small n, it is sufficient to take the second moments ot~I. The coherence effect of the sponge was such that it was more convenient to use the most annealed sample as reference. The results for the [1,0,0] and [1, 1, 11 directions obtained with a copper source, for 11,1,01 direction obtained with a molybdenum source, as well as the N.M.R second moments in a 12 kg field, measured at room ternperature with a Varian wide-line spectrometer, are summarized in Table 1.
5
r)
(100)
(~)2
4
2 0~
=
=
Vol.8, No.14
+
~
+
2 03
~
+
(1.1.0)
5ü2
(~L)2 (1
—
and one sees that the three mean square strains, measured on the same sample, are linearly dependent parameters. It is evident from Table 1 that the theoretical relation is not fulfilled and this gives a lower limit of 20 per cent to our error. As the Knight shift tensor, as function of
Table 1. Second moments of N.M.R. line at l2koe and square mean distorswn in 3 directions for ~ refere~icesample a’id four cold worked samples. __________________________________________ Samples R S 1 S2 S3S4 2 oe2 0.20 0.23 0.38 0.50 0.62
(!\H)
strain,in isa assumed to be this a linear function is of the cubic system, dependence characterized by three coefficients, giving for a powder: =
5 2 (~c,
2 0~
~
6
C2
2
3 03 ~
i~12~ 106 1_~t ic?
0.0
1.2
5.9
6.8
0.0
1.2
3.6
5.3
0.0
0.8
4.5
5.0
9.9
Lj~110,
2
j.~L1 ~ L
io~
7.5
_____________________________________________ If the X-ray measurements had great accurac, it would be possible to generalize an analysis already made for isotropic systems.5 The six components of the strain tensor .4, are projected on to 0 group basis functions, giving .‘1’,~
2)H2
2
C5 05
0
ne cannot separate, from such experiments, the F co’ntribution, which should be obtained independently from hydrostatic pressure measurements. Nevertheless, as in isotropic systems, one has, for of a strain field due to a random distribution dislocations: 05 = o = 10 o~ we have neglected the O~2 contribution, thus making a systematic error not greater than our experimental error. In Fig. 1
th
belonging to the ni
row of I”
,
with
i
=
1, 3, 5
=
1.7 102
H 0
in the Bethe notation. As the random distribution of strains on a cold worked sample must have cubic symmetry, it follows from Schur’s lemma that: fl4)’7,1)
=
~h.
~t
2
and that the coefficients a, describe the sample. One obtains:
between shear strain and the extra line width. This result can be interpreted by saying that only d band of metallic is sensitive to shearthe strains which produce Rh a perturbing crystalline field on those electrons. In first order of perturbation, there is a change in level populations near the Fermi surface, and the magnetic field l-4~ gives a further re-population
Vol. 8, No. 14
STRAIN INDUCED KNIGHT SHIFT IN RHODIUM METAL
~
1153
&
x 10 0.0
0
5
10
15
20
FIG. 1. Plot of measured second moment of N.M.R. line in a l2koe field, as function of the mean square distortion ~ (2 c~ + 3 o~).
responsible for Pauli paramagnetism. We are interested in the effect which is linear both in H 0 and in the shear field; this effect is the sum of contribution of individual electrons, if one assumes integral J to be uniform within thethe d exchange band.6 An uncompensated d electron gives an extra contribution to the Knight shift, equal to _0.161.106 oe per d electron, calculated from the temperature dependence of the Knight shift,7 and confirmed by relaxation time measurements.8 However, such an electron produces a dipolar magnetic field, which is not balanced by the effect of other electrons. This can be estimated from the Van Vleck paramagnetic contribution to
the Knight shift, which is proportional to With the values of Seitchik ci a!. this is
K
3
r
0.29.106 oe per d electron. In our crude model, this contribution to bein multiplied by the 2 0)2 =is 4/5 order to get (1 — 3 cos contribution to second moment. So, we conclude that the dipolar interaction is responsible for 70 per cent of the observed broadening.
Acknowledgements
— We thank Mrs Bernasson and Descouts, from ‘Ecole de Physique de l’Université de Genéve’, who gave us the samples, Dr. de Bergevin, from Laboratoire d’Electrostatique et de Physique du Metal’ for helpful discussions about X-ray technics, and Mr d’Assenza, for technical assistance in X-ray experiments.
1154
STRAIN INDUCED KNIGHT SHIFT IN RHODIUM METAL
Vol. 8, No. 14
1.
REFERENCES GREGORY E.H. and BOMMEL H.E., Phys. Rev. Leit. 15, 404 (1965).
2.
GREGORY E.H., Ph. D. Thesis, University of California, Los Angeles (1965).
3.
AVERBUCH P., DE BERGEVIN F. and MULLER-WARMUTH W., C.r. hebd. sèanc. Acad. Sci., Paris 249, 2315 (1959).
4. 5.
WARREN B.E. and AVERBACH B.L., J. app!. Phys. 23, 497 (1952). AVERBUCH P., C.r. hebd. séanc. Acad. Sci., Paris 253, 2674 (1961).
6. 7.
SEGRANSAN P. and AVERBUCH P., to be published. SEITCHIK J.A., JACCARINO V. and WERNICK J.H., Phys. Rev. 138A, 148 (1965).
8.
NARATH A., FROMHOLD A.T. and JONES E.D., Phys. Rev. 144, 428 (1966).
Les seconds moments des raies de R.M.N. de plusieurs échantillons écrouis sont relies Iinéairement ~ Ia deformation quadratique moyenne, mesurée par analyse des formes de raies de diffraction des rayons X. La variation de susceptibilité avec le cisaillement, et le couplage dipolaire entre electrons d et noyaux, contribuent tous deux a l’effet observe.