Relative importance of the T-5 and T-7 terms in spin-lattice relaxation time

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Relative importance of the T-5 and T-7 terms in spin-lattice relaxation time

C. Blanchard, B. Gaillard, A. Deville

To cite this version:

C. Blanchard, B. Gaillard, A. Deville. Relative importance of the T-5 and T-7 terms in spin-lattice relaxation time. Journal de Physique, 1979, 40 (12), pp.1179-1184.

�10.1051/jphys:0197900400120117900�. �jpa-00209205�

Relative importance of the T-5 and T-7

terms

in spin-lattice

relaxation time >>

C. Blanchard, ^B. Gaillard and A. Deville

Université de Provence, Département d’Electronique (*).

Centre de St-Jérôme, ^rue Henri-Poincaré, 13397 Marseille Cedex 4, ^France

(Reçu ^{le 2} juillet 1979, accepté ^{le 30 aout} 1979 )

Résumé. ²⁰¹⁴ R. Orbach et M. Blume ont montré la possibilité d’un processus de relaxation de type ^{Raman où} les niveaux intermédiaires appartiennent ^à l’état fondamental. Ceci conduit à une loi de variation en T5 I4(03B8D/T)

pour la probabilité ^de transition spin-réseau ^entre deux états conjugués de Kramers. Nous avons fait un calcul

complet pour Sm3+ en site cubique, ^{et avons} montré que ^cette dépendance ^en température ^reste vraie pour des transitions entre des états non conjugués de Kramers.

En étudiant le cas d’ions 3d5, ^{nous avons} montré que le processus de relaxation ^en T20147 de type ^{Raman est} plus

vraisemblable _que celui en T20145. Nous avons alors fait une nouvelle interprétation ^de quelques temps ^{de relaxa-} tion spin-réseau ^{et avons trouvé} qu’ils ^{devraient suivre une loi en T7} I6(03B8D/T) où 03B8D ^est donné par les phonons acoustiques transverses.

Abstract. ²⁰¹⁴ R. Orbach and M. Blume showed the possibility ^{of a} Raman relaxation process, where the interme- diate levels belong ^{to the} ground ^state. This leads to a T5 I4(03B8D/T) variation law for the spin-lattice ^transition probability between Kramers conjugate ^{states. We made a} complete calculation for Sm3+ in a cubic environment and showed that this temperature dependence still holds for transitions between non Kramers conjugate ^states.

Studying ^{the case of 3d5} ions, ^we showed that the T-7 Raman relaxation process is ^more likely ^{than the T-5}

one. We then made a new interpretation ^{of some} spin-lattice relaxation times and found that they ^{should follow}

a T7 I6(03B8D/T ) ^{variation law,} 03B8D being given by ^the transverse acoustic phonons.

1. Introduction. ^- R. Orbach and M. Blume

reported

[1] ^the

possibility

^{of a} T -5 temperature

dependence

^{of the}

spin-lattice

relaxation law Tl(T) ⁱⁿ

the low temperature part ^{of the Raman}

region.

They suggested ^a

simple

criterion for this law to dominate the classical T-’ one. However we

point

^{out that} they

omitted some terms in their numerical

application

for Sm3+ in a cubic environment. In section 2, ^we

make the complete calculation of the transitions

probabilities

^{and show that even for} non-Kramers

conjugate

states, ^{it is still}

possible

^to get ^a T-5 tem-

perature

depending

^{term. We} compare in section 3 the relative

importance

^{of the T -5 and T-7 terms}

in

spin-lattice

relaxation time. We show that for a

sufficiently high

temperature, ^{the T-’ term should} overwhelm the T-’ one. We make in section 4 a

(*) E.R.A. No 375.

new

interpretation

^{of the} temperature dependence ^of

the

spin-lattice

relaxation times for some 3dn ions.

We show that insteàd

of having

^{a T-’ variation} they

have ^a normal T-’ one.

2. Transitions probabilities ^{for Sm3 + in a cubic}

environment. ^- We take the same notations as

R. Orbach and M. Blume. The orbit-lattice interaction for SM31 is written [2] :

We consider an effective

spin

J, ^the transition proba- bility

WM, ’ WM J _ K

^{between two levels M. and}

mi -

^{K is :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197900400120117900

1180

Because of time

conjugation

2.1 T-5 RELAXATION PROCESS. ^- If the interme- diate state

Te belongs

^{to the}

ground spin multiplet

then, ^{for a} temperature ^{such that the}

prevailing

pho-

nons have an energy hw »

1 EMJ - Ere

^1, the expres- sion in curly ^brackets

of eq.

(1) reduces to :

There is no a priori ^{reason for this term to reduce to}

zero when

M.

^and

Mi -

^{K are} non-Kramers conju- gate ^states.

The calculation of the T- 5 contribution is easier if we use, instead of the C(r, 1, m) operators, ^a

spin-

Hamiltonian

acting

^{on the four}

F.

^levels.

VOL is

^{then :}

The

expressions

^{of the symmetry adapted operators}

X(Fi,,m)

^are

^given

ⁱⁿ

^appendix.

^The

A(I’i,)

coefficients

are :

:’

Using

this formalism it is _easy to

verify

^{that the}

W3/2-1/2, W 3/2- -1/2

transitions and those between the time conjugate states, neglected ^{in ref.} [1] ] ^are

non zero. However, ^{one cannot use this}

equivalent

Hamiltonian for the other Raman processes.

The three différent transition

probabilities

^{are :}

OD

^{is the} Debye temperature,

In(eo/T )

^{is the} Debye

integral

of order n and we set

VL

⁼

VT

^{= V.}

When hco «

1 EMJ - E,r

^{1 =}

^de,

^{we have to consi-}

der, ^{in our case, both the T-7} and the Orbach reso-

nant process.

2.2 T - 7 RELAXATION PROCESS. ^- The non zero transition

probabilities

^{are :}

(1) ^{Our numerical} coefficient is different from that obtained by

Orbach et al. because ^we take into account the different polari-

zations of the propagating ^mode [3].

2.4 NUMERICAL ESTIMATION FOR THE DIFFERENT PROCESSES. ^- Taking ^the numerical values

given

ⁱⁿ

ref. [ 1 ] : V ^{= 5 000} m/s, p ⁼ 2

g/cm’, d e

^{= 50}

cm -1, V(r ig,l)

^{= 500 cm -1 1 and}

^supposing

^that

T « 0,

^we

obtain

y) ^{Orbach resonant} process :

For T 6.5 K, ^{the T - 5} process is more efficient than the T-’ one. In fact, ^both processes should be hidden either by ^{the resonant Orbach} one, which is 103 times faster than the

previous

^{ones at T = 6.5} K, ^{or the} direct one at very low temperature.

To sum up, ^we will be unable to see a T- 5 relaxa- tion law for Sm" in a cubic environment because the gap between the

ground

and first excited states is too small.

Orbach and Blume’s choice of Sm3 + has not been

fortunate, but their conclusions still remain valid. It is therefore

possible

^{to have a} transition

probability

between non-Kramers states with a T5 dependence

without

considering

^the

fully symmetrical

^coordinate

Fl.

introduced by ^{M. B. Walker} [4].

For rare earth ions with a half-filled shell, ^{the first} excited state is

sufficiently

^far away from the ground

state to allow the observation of a T-’ relaxation law. For instance Eu" and Gd 31 in

CaF2

[5, 6]

exhibit such a dependence, ^{whereas Sm3+ in L.E.S.}

shows a T - 9 variation in the Raman region.

3. Relative importance ^{of the T - 5 and T -’} pro-

cesses for a 3d" ion in an orbital singlet ^{state. - If}

we suppose that the T-’ and T -’ _processes are

simultaneously operating,

^the

spin-lattice

^relaxation

rate obeys ^the

following

expression :

If

T.

^{is the} temperature ^{for which the two contribu-} tions are

equal

^{we then have :}

The ratio of the two contributions at any temperature ^T is

designated 3(xo’

^x), where x is equal ^to

eD/T

^and

xo ^to

OD/TO

We have drawn in

figure

^{1 a set of curves} 3(xo, x) ^for

différent values of xo which correspond ^{to the most} frequent situations.

The orbit lattice interaction

VoL

^has no non zero

matrix elements between the ground

spin-multiplets

1

Mj )

^{for an iron} group ion whose ground ^{state is}

an orbital

singlet.

Therefore, ^{we have to take into}

account the

mixing

^of the

1 Mj >

^{states with the}

excited | | ri>

^states through

spin-orbit coupling.

^The

modified ground ^state wave-functions

| Mj

^{y, are in}

the first order perturbation theory :

is the

spin-orbit coupling

coefficient.

1182

Fig. ^1. - log-log plot of the ratio i(x,,, ^{x) of the T-’} process ^over the T-5 one versus reduced temperature ^{x =} 8DIT. xo ^corres-

ponds ^{to the} temperature ^{where the two} processes ^are equal.

From eq. (1) ^{it is} easy ^{to see} that :

d is the _energy

difference

^{between the} ground ^state

and the first excited state

coupled

by

spin-orbit

interaction. We also have

so that :

Taking ,

^{= 400 cm -1 1 and d = 20 000 cm -1 1 as}

average values, A’/B’ N ^{T 2. The T - 5 and T -’} pro-

cesses will be equal ^for

To

^{= 1 K ; at this} temperature

both processes ^are hidden by ^{the direct} process. When

the temperature ^increases the Raman T-’ prevails

over the T-’ one. For

example taking

xo ^{= 200 or} more, which corresponds ^{to the most common matri-}

ces, ^at 5 K the T-’ _process is already ^{20 times faster}

than the T- 5 one. This shows that we will be unable to see a T -5 relaxation _process for such ions.

4. New interprétation of the relaxation law for

some 3d 5 ions. ^- The spin-lattice relaxation rates of Mn2+ in

SrF2, BaF2

[7] ^and

CaF2

[8] ^{have been}

previously interpreted,

^{in the Raman} region, ^{as follow-}

ing

^{a T-5 law. In the}

preceeding

^{section we showed}

that such a law was

highly

unlikely. ^{We have thus}

fitted the

experimental

values of

T1-

^{1 with a}

T’

16«(}oIT)

relaxation law, presented ⁱⁿ

figures

2, ³

Fig. ^{2. -} log-log plot ^{of the} spin-lattice relaxation time Tl ^versos

temperature ^for Mn" in SrF2 (from ^ref. [6]). ^{The best fit corres-} ponds ^{to :}

Fig. ^{3. -} log-log plot ^of T11 ^{versus T for Mn2+ in} BaF2 (from

ref. [6]). The best fit corresponds ^{to :}

and 4. We deduced a

0§

^{value from}

experimental

^data

on the

density

^{of the} transverse phonon ^modes [9, 10, 11]. This value

corresponds

^to the maximum in the density ^{of the} transverse

phonon

modes, ^which

have a smaller

velocity

^{than the}

longitudinal

ones, and are thus the most efficient in the relaxation _pro- cess, ^as shown by eq. (1). This choice is

justified

^since

this density shows, first, ^a

nearly parabolic

variation, and then, ^a sharp decrease after its maximum. This

0p

^{value, which is} temperature

independent,

^differs

from that obtained from

specific

^heat measurements, where the whole phonon spectrum ^{has to be taken} into account

[12].

^{We must note} that in the above

experimental

temperature range, T’

16«(JDIT)

^behaves

Fig. ^{4. -} log-log plot ^of Tl! ^{ver.çus T for Mn2 + in} CaF2 (from

ref. [7]). The best fit corresponds ^{to :}

approximately

^{as T5. We must} then, ^to

interpret

^the

experimental

^{results, consider the relative}

importance

of the différent processes. Moreover, ^{we must not} only rely ^{on the} exponent value of the best

fitting

curve for

Ti ’,

^{but also take into account the value}

of (Jo.

Marshall et al. [13]

interpreted,

^{in the} temperature

range 2-200 K, ^the

experimental

^{values of}

T1- 1

^for

Fe3 + in CaCo3 using ^the

following

expression :

This value of463 K does not

correspond

^to the upper limit of the transverse ,acoustical phonon ^branch

given

^{in ref.} [14].

Considering

^the phonon

density

curve of CaCo3, ^{the best fit is} given by :

The second and third terms

correspond

^{to a Raman}

process due to acoustic and

optical

phonons respec-

1184

Fig. ^{5. -} log-log plot ^of Tl! ^{versus T for Fe3+ in} CaC03 (from

ref. [12]). ^{The solid curve} corresponds ^{to :}

The dashed ^curve from 40 K onwards arises from the first two terms in Tî ’. ^The deviation between the two curves corresponds ^{to the}

last term in Tï ^{1 which is due to} optical phonons.

tively. The results are presented ⁱⁿ

figure

^{5. In} fact,

we do not have well separated

optical

phonon ^bran- ches, ^{but the value of 430 K} corresponds ^{to a} peak amplitude ^{in the} phonon

density

^{curve. A similar}

behaviour has been found for Fe" and Mn" in ZnS [15, 16].

5. Conclusion. ^- In this _paper we showed that the transition

probabilities

given by ^R. Orbach and M. Blume for Sm3+ in a cubic environment

leading

to Tl ^oc

T -’,

^were

incomplete.

^In fact, ^{we have a} T - 5

dependence

^even for transition

probabilities

between non-Kramers states and there is no need to invoke a

r 19

vibrational mode to obtain non zero

transition

probabilities

^{as M. B. Walker did. An order}

of

magnitude

calculation indicates that for a 3dn ion

having

^a

singlet

^orbital ground ^{state, it is} very

unlikely

to get ^a T- 5

spin-lattice

relaxation rate.

We

reinterpreted

^some

published

results for such ions and showed that the variation of

Tl

^{versus T,}

in the Raman

region,

^{was more}

likely

^{due to a T-’}

process.

6. Acknowledgments. ^{- We are}

greatly

^indebted

to Pr. K. W. H. Stevens for his fruitful remarks

concerning

^{the use of}

spin-Hamiltonian

^formalism.

Appendix

References

[1] ORBACH, R., BLUME, M., Phys. Rev. Lett. 8 (1962) ^478.

[2] BLUME, M., ORBACH, R., Phys. ^{Rev. 127} (1962) ^1587.

[3] ^VAN VLECK, ^J. H., Phys. ^{Rev. 57} (1940) ^426.

[4] WALKER, M. B., Can. J. Phys. ⁴⁶ (1968) ^1347.

[5] HUANG, ^C. Y., Phys. ^{Rev. 139} (1965) ^{A 241.}

[6] BIERIG, ^R. W., WEBER, ^M. J., WARSHAW, ^S. I., Phys. ^Rev.

134 (1964) ^{A 1504.}

[7] HORAK, ^J. B., NOLLE, A. W., Phys. ^{Rev. 153} (1967) ^372.

[8] LAY, ^F. M., NOLLE, ^A. W., Phys. ^{Rev. 161} (1967) ^266.

[9] HARIDASAN, T. M., KRISHNAMURTHY, N., KRISHNAN, ^R. S., J. Indian Inst. Sc. 51 (1970) ^347.

[10] HURRELL, ^J. P., MINKIEWICZ, ^V. J., Solid State Commun. 8

(1970) 463.

[11] CHOPRA, ^K. K., DAYAL, B., Pramaña 8 (1977) ^408.

[12] ZIMAN, ^J. M., ^Electrons and phonons (Oxford, ^{at the Clarendon} Press), 1960.

[13] MARSHALL, ^S. A., NISTOR, ^S. V., SERWAY, ^R. A., Phys. ^Rev.

B 6 (1972) ^1686.

[14] PLIHAL, M., Phys. ^{Status Solidi} (b) ⁵⁶ (1973) ^495.

[15] DEVILLE, A., GAILLARD, B., BLANCHARD, C., LANDI. A.

J. Physique ⁴⁰ (1979) ^1173.

[16] ROGER, G., MORE, C., BLANC, ^C. (to ^be published).