Time Correlations and MagneticResonance Linewidth in Finite Heisenberg LinearChainsFernando Carboni and Peter M. Richards Citation: Journal of Applied Physics 39, 967 (1968); doi: 10.1063/1.1656346 View online: http://dx.doi.org/10.1063/1.1656346 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/39/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A method for an accurate T 1 relaxation-time measurement compensating B 1 field inhomogeneity inmagnetic-resonance imaging J. Appl. Phys. 97, 10E107 (2005); 10.1063/1.1857393 Uniformity of Limits in the Calculation of the Magnetization of the Heisenberg Linear Chain J. Appl. Phys. 42, 1416 (1971); 10.1063/1.1660271 TimeDependent Spin Correlations in Heisenberg Linear Chains J. Appl. Phys. 42, 1380 (1971); 10.1063/1.1660258 Magnetically Coupled Impurities in a LinearChain Heisenberg Ferromagnet J. Appl. Phys. 40, 1120 (1969); 10.1063/1.1657554 Recording MagneticResonance Spectrometer Rev. Sci. Instrum. 27, 596 (1956); 10.1063/1.1715645

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JOURNAL OF APPLIED PHYSICS VOLUME 39, NUMBER 2 1 FEBRUARY 1968

Magnetic Scattering and Finite Systems M. F. COLLINS AND E. O. WOLLAN, Chairmen

Time Correlations and Magnetic-Resonance Linewidth in Finite Heisenberg Linear Chains*

FERNANDO CARBONIt AND PETER M. RICHARDS

Department of Physics, University of Kansas, Lawrence, Kansas

We have performed exact calculations of two-spin and four-spin time-correlation functions, such as (S,'Ct) s,r) and (Si'Ct) S;+Ct) Sk'S,-), for finite linear chains with periodic boundary conditions containing as many as ten spin-! particles coupled by a nearest-neighbor Heisenberg exchange interaction. Previous workers have computed thermodynamic properties of finite linear chains, but this to our knowledge is the first calculation of time-dependent properties, which requires the eigenfunctions as well as eigenvalues. Determination of the correlation functions enables us to compute the variation of magnetic-resonance linewidth with frequency in a linear-chain salt such as CuCNHa).SO,·H20 by applying the methods of Kubo and Tomita. Values of linewidth computed with our exact correlation functions give very much better agreement with Rogers' experimental values for Cu (NH,) 4S0,· H20 than the usual Gaussian approxi­mation. We also show results for the simple two-spin correlation function, such as measured by inelastic neutron scattering. Its frequency transform is non-Gaussian, showing a steep rise near zero frequency.

INTRODUCTION

Expressions for the frequency dependence of mag­netic resonance linewidths1 and neutron-scattering cross sections2 in materials with strong exchange inter­actions involve Fourier transforms of two-spin and four­spin time-correlation functions of the general form

and (Sia(t) S/(t) Ska'(O) Sl'(O»,

where S;a(t) is the ath component of the spin at lattice site i at time t and the triangular brackets stand for thermal average. Time dependence is given explicitly by

where JCex is the Heisenberg exchange Hamiltonian. Physically, these expressions state that the dipolar, hyperfine, etc., interactions are modulated by exchange­induced spin flips; and the relation between this flipping frequency and the measuring frequency is a pertinent factor. At high temperatures it is expected that the time-correlation functions will be appropriate to random fluctuations rather than well-defined collective modes of oscillation. An exact quantum-mechanical description would require finding the eigenfunctions I )J.) and eigen­values Ep. = hwp. and combining them according to

(A (t)B(O) )= L,Ap.vBvp. exp( -iwp.vt)Pp., (2) p.,

where Ap.. and Bvp. are matrix elements of any operators

* Supported by the United States Atomic Energy Commission. t Present address: Department of Physics, University of Costa

Rica, San Jose, Costa Rica. 1 R. Kubo and K. Tomita, J. Phys. Soc. Japan 9, 888 (1954). 2 P. G. de Gennes, in Magnetism (Academic Press Inc., New

York, 1963), Vol. III, p. 115.

A and Band Pp. is the Boltzman factor for state I !J.). This becomes a virtually impossible task for a Heisen­berg model at high temperatures. Thus previous workers have invoked assumptions based on the notion of ran­dom fluctuations. Generally, at some stage of the cal­culation either a Gaussian decay is postulated or transi­tion probabilities are used. Prior to our work there have existed to the best of our knowledge no "first-principle" calculations, by which we mean using Eq. (2) directly without any stochastic models.

FIG. 1. Histogram of (fCw», Fourier trans­form of (Sl'Ct) Sl'(O) ) at infinite temperature, as extrapolated to N = 00.

1.2

We have calculated time-correlation functions for finite closed (periodic boundary conditions) chains of up to N = 10 particles of spin-! coupled by a nearest­neighbor exchange interaction J. Bonner and Fisher and others have computed thermodynamic properties of finite chains, which require eigenvalues only, and made plausible extrapolations to N = 00. In particular, it was shown in Ref. 3 that one could extrapolate to within 0.1% of the exact antiferromagnetic ground­state energy for N = 00. These results give encourage­ment to our own extrapolations.

3 J. C. Bonner and M. E. Fisher, Phys. Rev. 135, A640 (1964). 967

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968 F. CARBONI AND P. M. RICHARDS

1.5

(\10 114

X 1.3

I~ 1.2

g 1.1

to

7 •

012345678 {10/NY'

FIG. 2. Values of (j(O.lJ /h) > for closed chains of N =6,7,8,9,10 spins plotted vs (lO/N) 4.

RESULTS

Our computed results give the frequency transform (J(w) > of (2) defined as

(J(w»= (1j~w) LAl'yByl'PI' 1"

X[ -H~w)+w~wp.P<w+H~w)J, (3)

so that (j(w) > represents the average value of the Fourier transform within an interval of width ~w centered at w. For infinite temperature a regular varia­tion with N from 6 to 10 is obtained by taking ~w= 0.4Jjh for w<0.6Jjh and ~w=0.8Jjh for higher fre­quencies.

Figure 1 shows (j(w) ) associated viith (SlzCt) SlzCO) > at infinite temperature extrapolated to N = 00. It is certainly non-Gaussian, showing a steep rise near zero frequency. Other correlation functions such as (Sl"Ct) S2"CO) ) and the four-spin functions give similar behavior near zero frequency. Our extrapolation pro­cedure is shown in Fig. 2 for the interval -0.2Jjh~ w~ +0.2J jh. This is the most critical interval in terms of the general feature of a steep rise near zero fre-

quency. Although the value of (fCw» does decrease with N for both odd and even N, a plot of (fCw» vs 1/N4 shows slight upward curvature for even N as N-+oo; a plot vs 1jNZ (not shown) gives strong up­ward curvature. The picture for odd N is not so clear since calculations show that N = 5 is too small to be meaningful and N = 11 requires a prohibitive amount of computer time. However, the relatively small difference between N = 9 and 10 should be noted. The complete histogram of Fig. 1 was obtained by making plots similar to Fig. 2 for each of the frequency inter­vals involved. At higher frequencies the extrapolation procedure is considerably more certain than for the interval shown, since the variation with N is less.

COMPARISON WITH EXPERIMENT

The dependence of exchange-narrowed magnetic resonance linewidth on frequency has been computed from the two-spin and four-spin correlation functions. Dipole-dipole interaction is assumed to be the major source of broadening. Results are in excelJent agree­ment of Rogers' measurements on the linear-chain salt Cu(NHa)4S04·H20, whereas the Gaussian approxima­tion gives very poor agreement. This comparison is treated in a separate publication.4

It would be most desirable to have experimental evidence, such as might be obtained from neutron scattering, on the simple two-spin correlation functions since this could be a more stringent and direct test of our calculations.

Calculations were made with facilities (IBM 7040 and GE 625) of The University of Kansas Computation Center, whose staff is gratefully acknowledged.

4 R. N. Rogers, F. Carboni, and P. M. Richards (to be pub­lished in Phys. Rev. Letters).

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