TimeDependent Spin Correlations in Heisenberg Linear ChainsF. B. McLean and M. Blume Citation: Journal of Applied Physics 42, 1380 (1971); doi: 10.1063/1.1660258 View online: http://dx.doi.org/10.1063/1.1660258 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/42/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Timedependent correlation functions in the hightemperature limit for the XYchain and the Isingchain in atransverse magnetic field J. Appl. Phys. 50, 1771 (1979); 10.1063/1.327215 SpinPeierls transitions in Heisenberg antiferromagnetic linear chain systems AIP Conf. Proc. 29, 504 (1976); 10.1063/1.30415 ESR of TMPD–TCNQ: Spin Excitations of the Heisenberg Regular Linear Chain J. Chem. Phys. 52, 4011 (1970); 10.1063/1.1673601 Time Correlations and MagneticResonance Linewidth in Finite Heisenberg Linear Chains J. Appl. Phys. 39, 967 (1968); 10.1063/1.1656346 TimeDependent SpinCorrelation Function in Ferromagnet and Antiferromagnet J. Appl. Phys. 39, 1353 (1968); 10.1063/1.1656298

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1380 TUCCIARONE, CORLISS, AND HASTINGS

J /kBT. We make the three following observations: (1) the strong convergence of XT(i, i, 1), experimentally observed, as T is increased, even in the temperature range 2TN-3.55TN; (2) the exact agreement between the calculated and observed temperature dependence of XT(i, i, 1) in the temperature range 2TN-3.55TN; (3) the small effect of short range order on X3.SSTN(1,1, 1). As a consequence of the above, we can put the measured value of X, at T= 3.55TN and q = (i, 1, 1), equal to its calculated value (0.97) and thus set all our measured values of XT(q) on an absolute scale. Finally, we would like to point out the generally good agreement between measured and calculated susceptibilities. It is ind~ed remarkable that a nearest-neighbor isotropic Heisenberg Hamiltonian can account so well for the behavior of this system.

Figure 3 shows the characteristic frequencies WT(q) defined by the equation

The rapid drop of WT(l, 1, 1) immediately above TN is paralleled by the evident gradual transition of the shape function from the characteristic three-peaked structureS to a single peak. Note that at room tempera­ture the width at q= (I, I, I) is still larger than the one at (i, i, i), so that the widths at 3.55TN have still

not acquired the expected infinite temperature order. The shape function F(q, w) is obtained on an absolute basis from the curves in Fig. 1 by dividing the ordinate by the total area. This area is equal to 37800 in the case T=3.55TN and q= (t, t, i) and the other areas are obtained from this value by scaling with h(q) lJ(q) 12, where If(q) 12 is equal to 0.957, 0.817, 0.735 for the three values of q that have been measured.

It is hoped that comparison, on an absolute basis, of theoretical calculations of the spectral density for the Heisenberg paramagnet with the experimentally deter­mined curves can provide not only a general test for validity, but also an insight into the physical signifi­cance of the inevitable approximations.

ACKNOWLEDGMENT

We should like to thank M. F. Collins for making the results of his calculations available to us in advance of publication.

* Research performed under the auspices of the U.S. Atomic Energy Commission.

1 See, for example, W. Marshall, in Critical Phenomena, edited by M. S. Green and J. V. Sengers (National Bureau of Standards, U.S. Government Printing Office, Washington. D.C. 20025, 1966).

2 H.-Y. Lau, L. M. Corliss, A. Delapalme, J. M. Hastings, R. Nathans, and A. Tucciarone (unpublished).

3 G. S. Rushbrooke and P. J. Wood, Mol. Phys. 6, 409 (1963). 4 M. F. Collins, Phys. Rev. B 2,4552 (1970). 5 H.-Y. Lau, L. M. Corliss, A. Delapalme, J. M. Hastings, R.

Nathans, and A. Tucciarone, Phys. Rev. Lett. 23,1225 (1969).

JOURNAL OF APPLIED PHYSICS VOLUME 42, NUMBER 4 IS MARCH 1971

Time-Dependent Spin Correlations in Heisenberg Linear Chains*

F. B. McLEAN AND M. BLUME

Brookhaven National Laboratory, Upton, New York 11973

We have used the self-consistent theory of Blume and Hubbard l to calculate the Fourier trans­form Seq, w) of the time-dependent spin pair correlation function for the Heisenberg linear chain at finite temperatures. The spins are treated as classical unit vectors2 and the time-independent cor­relation functions for the linear chain with nearest-neighbor interactions are used as initial condi­tions for the approximate integrodifferential equations. Numerical solution of these equations yields Seq, w) for both ferromagnetic and antiferromagnetic interactions. At low temperatures in the paramagnetic regime, "spin-wave" type peaks are found in the scattering funCtion. The shape and position of these peaks as a function of temperature agrees qualitatively with the experimental results of Birgeneau et al.

* Work performed under auspices of the U.S. Atomic Energy Commission. 1 M. Blume and J. Hubbard, Phys. ReV. B 1,3815 (1970); J. Hubbard, J. Phys. C (to be published). 2 Quantum corrections have been shown by E. D. Siggia (private communication) to be of order

(1/ S)2.

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