FINITE TEMPERATURE EXCHANGE CORRELATION : APPLICATION TO NMR LINEWIDTHS IN MAGNETIC INSULATORS

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FINITE TEMPERATURE EXCHANGE CORRELATION : APPLICATION TO NMR LINEWIDTHS IN MAGNETIC INSULATORS

D. Hone, B. Silbernagel

To cite this version:

D. Hone, B. Silbernagel. FINITE TEMPERATURE EXCHANGE CORRELATION : APPLICATION TO NMR LINEWIDTHS IN MAGNETIC INSULATORS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-761-C1-762. �10.1051/jphyscol:19711265�. �jpa-00214096�

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RESONANCE MAGN~TIQUE ET D YNA MIQUE DES SPINS

FINITE TEMPERATURE EXCHANGE CORRELATION : APPLICATION TO NMR LINEWIDTHS IN MAGNETIC INSULATORS

c)

D. W. HONE and B. G. SILBERNAGEL

Department of Physics University of California, Santa Barbara, California, 93106

Rksumk. - On calcule les variations avec la temperature des largeurs de raie des resonances nucleaires paramagne- tiques resserrees par l'echange, AH(T), de la resonance de F19 dans MnF2, KMnF3, et RbMnFs. Au premier degre de (TNIT), l'on obtient, par des developpements valables pour des periodes courtes, les fonctions correlatives qui d6ter- minent A H ( T ) . L'on calcule aussi le coefficient de dispersion qui caractkrise leur comportement apres longtemps.

Ensuite l'on offre des courtes interpolations entre ces limites. Les predictions qui en resultent pour le coefficient de ( T N I T ) dans la formule pour AH s'accordent avec l'experience, ayant un kcart dans le pire cas de 20 pour cent seulement. Ceci constitue une nette amelioration sur les predictions d'autres theories frequemment employbs.

Abstract. - The temperature dependent exchange narrowed paramagnetic linewidths AH(T) of the F19 NMR in MnF2, KMnF3, and RbMnF3 are calculated. To first order in (TNIT), the auto- and pair correlation functions which determine AH(T) are obtained through short time expansions, calculation of the diffusion coefficient characterizing long time behavior, and short interpolations between these limiting regions. The resulting predictions for the term in AH linear in ( T N / T ) agree with experiment to within 20 % in the worst case. This is a substantial improvement over the predictions of other commonly used theories.

Recent interest in exchange induced spin correlations in the paramagnetic state of magnetic materials has been stimulated by the possibility of directly compar- ing cr first principles ^Dcalculations with experimental data. For example, high temperature NMR linewidth studies in the magnetic insulators MnF,, KMnF,, and RbMnF, provide a very stringent test of the theory of exchange narrowing, since in these parti- cularly simple and well-studied systems the precision of the linewidth calculations (- + ⁷^{X )}and of the

lower temperatures. Experimental values of the coefficient a are presented in Table I.

Experimental and theoretical values for the coef- ficient of the linear term in the linewidth. The present

calculati~n agrees with the experimental values to within the uncertainties quoted and is in signzJicantly better agreement than the Mori-Kawasaki theory.

experimental measurements ^'(- f 5-73 are both Experimental Present Mori- extremely high [I]. We have extended these studies to Values Calculation Kawasaki

finite temperatures, where experimental linewidth data - - -

( A H ( ~ " ) ) can be compared t o calculations made t o MnF2 0.44 Oa05 0.41 ⁺0.04 first order in /?X. This provides an opportunity to study KMnF3 0.60 + Oa05 0.49 + 0.05

0.95 1.02 the temperature dependence of such microscopic RbMnF3 0.55 +0.05 0.49 + 0.05 1.02 urouerties as the time evolution of spin correlation

and such properties as Since the region of linear temperature dependence is diffusion in paramagnets.

so extensive, it is useful to extend the infinite tempera- The ''' NMR linewidth in MnF2, ^KMnF3, and ture theory of exchange narrowed linewidths to first RbMnF, has been chosen for this test because the order in p.p. The linewidth has the form : source of linewidth is almost exclusivelv the fluctuating

transferred hyperfine fields of neighboring magnetG _{S ( S}+ ¹⁾

spins. The problem is thus reduced to understanding A H ( T ) = 7 3 x the dynamics of the auto-correlation and pair correla-

tions between these nearest neighbors. The simplicity

of the magnetic structures and of the exchange interac- ^x( ~ ~ I * $ ~ ( t ) d t + ~ ~ I o $ ~ ( t ) d t ] o (1) tions in these systems has facilitated a precise deter-

mination of th&ir magnetic parameters. It also makes where A is the hyperfine coupling constant7 and NA possible detailed calculations of the correlation func- and NP are the number auto- and pair- tions which can then be with experiment. correlation contributions. $ ~ ( t ) and $ ~ ( t ) are the The linewidth measurements [2], on large single temperature dependent auto- and air-correlation

crystals of the three substances, were made between functions : 84 OK and 573 OK, or from T -- TN ^{to T}-- 7 TN. In all

three cases, AH(T) was found to be approximately linear in 1/T: AH(T) = AH(oo) [l - a(B)/T)], where l3 = zJS(S + 1)/3 k,, over a large temperature range (T > 2.5 TN). Significant deviations occur only at

(*) Supported in part by the National Science Foundation.

where < > denotes the thermal average and the time evolution of the spin operators is governed by an isotropic Heisenberg exchange hamiltonian.

The $A(t) and $,(t) functions are determined unambiguously a t short times by expansion techniques : $i(t) = ai(T) + bi(T) t 2 + ci(T) t4, where

51

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711265

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C 1 - 762 D. W. HONE AND B. G . SILBERNAGEL i = A, P. A major assumption of the theory is that the

long time, long wavelength, spin diffusion description is adequate even at 2 or 3 times the microscopic corre- lation time ^t,and at near neighbors of the reference spin. With an appropriate choice of initial conditions, this description appears to be quite accurate, as we will discuss below.

In this diffusion picture, the spatial and temporal Fourier transform of $(t), P(k, o), has the form Ak2/[02 + (/lk2)'] at small k and o. The temperature dependent diffusion constant, A(Q, in this case can be determined directly in terms of a,, b,, and ci using moment techniques [3]. In accordance with more sophisticated theories [4], we have chosen a gaussian envelope for P(k, u ) rather than the sharp u cutoff of reference 3. The resulting diffusion coefficient has the form A(T) = A ( a ) [l - 6(8/Q], where 6 = 0.125 for MnF, and 6 = 0.138 for KMnF,, RbMnF,, and the value of A(a) is in very close agreement with previously obtained values [5]. The initial conditions for the long time diffusion solution are fixed by taking values of ll/,(t) and $,(t) from the short time expansion at the maximum time for which the expansions are valid, i. e. t

-

^t,⁼^{(h/J) (2}^zS(S⁺1)/3)-s. The form for the long time solution is thus

where $,(t) = $(O, t), +,(t) = $ ( A , t), A being the near neighbor distance. This choice for the long time form of $, and $, is found to match the short time expansion excellently. This is indicated in figure 1, where $,(t) is plotted for BIT = 0,0.2,0.4 respectively.

The region of interpolation is found to be extremely small. This plus the fact that the hydrodynamic solution is known to be correct at very long times, suggests that our procedure is justified. The corresponding figure for $,(t) has not been included since the temperature dependence of the autocorrelation function is found to be considerably weaker.

Assuming 8/T < 1, the functions $, and $, are integrated explicitly in the short (0 < ^t< ^7,)^and

long (t 2 2 ^7,)time regions, while the area of the intermediate region is estimated using a trapezoidal approximation. The resulting values for the coefficient a are presented in Table I, and are found to be in

FIG. I. - A plot of the pair correlation vs. reduced time

( t l ~ e ) for KMnF3 for temperatures corresponding to B/T = 0.0,

0.2, and 0.4. The short time behavior comes directly from a series expansion, while the long time behavior is characteri.zed by the diffusion solution, Eq. 3. Of special interest is the extremely small region of interpolation (indicated by a dashed line)

required.

substantial agreement with experiment. The uncertainty of the values is associated with the estimates involved in the diffusion tails. As predicted the bulk of the temperature dependence arises from the pair correlation terms. In KMnF, for example, pair correlations contribute only 20 % of the infinite temperature linewidth, while they are responsible for 60 % of the temperature dependence.

We have described a simple model which avoids the correlative approximations of Green's function and related theories. Within the limited temperature region of its validity, it then serves as a useful tool for the evaluation of these more general but approximate theories. For example, the results obtained here can be compared with the predictions of the Mori-Kawasaki theory for the values of A(a), 6, and a discussed above [4, 71. Although the values of A(o3) agree to within 1 %, the Mori-Kawasaki theory predicts a considerably stronger temperature dependence of A(T) : 6 = 1, compared with the 6 -- 0.13 obtained here. Further, the prediction for a is larger than experiment by a factor of two (see Table I). A detailed study of this and other theories will appear elsewhere.

References

[I] GULLEY (J.), HONE (D.), SCALAPINO (D.), SILBER- [5] REITER (G.), to be published.

NAGEL (B.), Phys. Rev. B, 1970, 1 , 1020. [6] SILBERNAGEL (B.), JACCARINO (V.), PINCUS (P.), WER- [2] GULLEY (J.), SILBERNAGEL (B.), JACCARINO (V.), J. _NICK (J.), Phys. Rev. Letters, 1968, 20, 1091.

Appl. Physics, 1969, 40, 1318.

[3] DE GENNES (P.), J. Phys. Chem. Solids, 1958, 4 , 223. [71 (T.)y Phys. (Kyoto)7 [4] MORI (H.), KAWASAKI (K.), Progr. Theoret. Phys. ^28, 371.

(Kyoto), 1962, 27, 529.