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COMPARISON BETWEEN EXPERIMENTS AND NUMERICAL CALCULATIONS ON S = 1/2 HEISENBERG-XY FERROMAGNETIC CHAINS

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COMPARISON BETWEEN EXPERIMENTS AND

NUMERICAL CALCULATIONS ON S = 1/2

HEISENBERG-XY FERROMAGNETIC CHAINS

K. Kopinga, J. Emmen, G. de Vries, L. Lemmens, G. Kamieniarz

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, SupplGment au no 12, Tome 49, d6cembre 1988

COMPARISON BETWEEN EXPERIMENTS AND NUMERICAL CALCULATIONS ON S =1/2 HEISENBERG-XY FERROMAGNETIC CHAINS

K. Kopinga, J. Emmen, G. C. de vriesl, L. F. ~ e m m e n s ~ and G. ~ a m i e n i a r z ~

Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Abstract.

-

Heat capacity measurements in fields 0 5 B 5 3 T were performed on [CGH11NH3] CUBIQ (CHAB), an excellent ferromagnetic S = 112 chain system with 5 % X Y anisotropy. A comparison of the data on AC = C (B) -C ( 0 )

with extrapolations of numerical results for finite chains reveals systematic deviations at low temperatures, which are attributed to the limited chain lengths (up to 11 spins) used in the computations.

One-dimensional S = 112 ferromagnetic systems in magnetic fields up t o 0.65 Tesla [3, 41. Attempts have been the subject of a large number of theo- have been made t o improve the description of the data retical investigations [I]. Especially the compound by changing the values of the anisotropy in the spin [C6H11NH3] CuBr3 or CHAB has been studied rather Hamiltonian of CHAB:

extensively, since the intrachain interaction contains about 5 % easy-plane anisotropy. This - in principle

- allows the presence of non-linear soliton-like exci- tations for external in-plane magnetic fields, since in that case the system can be mapped t o a sine-Gordon system [2]. In this mapping the spins are considered as classical vectors, their motion is confined t o the easy (XY) plane and the limit of zero lattice spacing is taken. Despite these approximations, the behav- ior of the excess heat capacity AC = C (8)

-

C (0)

of CHAB could be described fairly well by the sine- Gordon model [3]. A more detailed analysis, how- ever, revealed that this largely resulted from an ac- cidental cancelling of quantum effects by spin fluc- tuations out of the easy plane [4]. In view of this, there is a renewed interest in more direct calculations of the thermodynamic properties of easy-plane ferro- magnetic chain systems using the quantum mechani- cal spin Hamiltonian. Various theoretical approaches have been used. First, the thermodynamic properties of the infinite chain have been evaluated by extrap- olation of the numerical results for finite chains [5]. Alternatively, an appropriate version of the Trotter- Suzuki transformation has been used t o map the par- tition function of the one-dimensional quantum system onto the partition function of a two-dimensional lattice of Ising spins. The properties of this latter system are then evaluated using Monte Carlo or transfer matrix techniques [6]. The various predictions for AC show large differences, indicating that the error in the corre- sponding numerical results is still rather large. Apart from this, they show large deviations from the exper- imental data on CHAB, which have been collected

i

(1) where J x x / k ~ = 5 5 f 5 K, Jzz/Jxx = 0.95, and , ( I - J Y Y / J x x ) ~ 5 x In the calculations the small in-plane a4sotropy JYY - Jxx is usually neglected. In terms of Zeeman energy, however, this anisotropy cor- responds to 0.02 T , which hampers a detailed com- parison between theory and experiment, especially a t the lowest applied field, i.e., B = 0.125 T. Therefore we thought it worthwhile to measure AC of CHAB a t higher fields, where the in-plane anisotropy is rela- tively small. This study was also motivated by recent measurements [7] of the in-chain magnon dispersion re- lation of CHAB by inelastic neutron scattering, which yielded a value Jxx/kB=67 f. 1 K. This value is 20 %

higher than the value J x x / k ~ = 5 5 f 5 K deduced pre- viously from an analysis of the zero-field heat capacity [81

The measurements were performed on a single- crystal of CHAB with a mass of 527.5 mg for 1.2

<

T

<

20 K and B = 0, 1, 2 and 3 T along the crys- tallographic c-axis, which is located within the easy plane [9]. The zero-field heat capacity was found to be equal within experimental error (2 % ) t o the data obtained previously on a polycrystalline sample, ex- cept for the region between 2.5 K and 8 K, where the present results are up to 6 % lower. The data for B = 0 were analysed by simultaneous fits of the lattice heat capacity CL and a magnetic contribution CM. For

CL

a three-parameter expression appropriate for a chain-like structure was used [8]. Estimates for

etherla lands

Energy Research Foundation ECN, P.O. Box 1, 1755 ZG Petten, The Netherlands.

'~nstitute for Applied Mathematics, University of Antwerp, B 2020 Antwerp, Belgium.

'~nstitute for Theoretical Physics, University of Leuven, B 3030 Leuven, Belgiu111 (on leave from Instytut Fizyki, Uniwersytet im. A. Mickiewicza, 60-769 Poznan, Poland).

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C8 - 1452 JOURNAL DE PHYSIQUE

CM were obtained from extrapolations of numerical re- sults for finite chains up to N = 11 spins. The details of the numerical procedure have been reported before 151. The accuracy was estimated from a comparison of the results of extrapolations for N = 10 and N = 11, respectively. For T

>

7 K, both extrapolations coin- cide within 1 %

,

but a t lower T large deviations occur. Therefore the fit was limited to T

>

7 K. CM was com- puted for a fixed value ( J x x

-

Jz") /k~=2.75 K [Q], and for various values of J x x / k ~ . The best description of the data was obtained for Jxx/Ic~=63 f 3 K, which is in fair agreement with the value J x x / k ~ = 6 7 f 1 K obtained from the neutron scattering experiments. For this reason we use a value of 63 K in the analysis of our data on AC. First, however, we wish to note that this choice of J a X / k ~ instead of the value J x x / k ~ = 55 K does not affect the basic conclusions drawn in the interpretation of the magnetic properties of CHAB reported up till now [3, 4, 101. In the present detailed comparison of AC with numerical data, however, the various parameters should be as accurate as possible. An analysis of A C has the advantage that errors in the determination of the empty sample holder and CL are cancelled out. Therefore, it may serve as a direct check on the accuracy of theoretical pregctions. The experimental data on AC are plotted in figure 1 for B = 1, 2 and 3 T. The results of the corresponding numerical calculations are also included. The dashed and dotted curves represent extrapolations based on

chains up t o N = 10 and N = 11, respectively.

Full

curves reflect the regions where both extrapolations coincide within 1 %

.

The overall agreement of the calculations with the data is ratha: good. At low T

deviations occur, which are comparable to the differ- ence between the results for N = 10 and N = 11.

The discrepancies that were fou~ld in previous de- scriptions [5, 61 of AC of CHAB by the results of nu- merical calculations based on the quantum mechan- ical Hamiltonian (1) are probably due to the small in-plane anisotropy, which is relatively important at low fields. Apart from this, these calculations were based on an exchange parameter J x ' " / k ~ = 5 5 K, which is about 15 % smaller than the value 63 f 3 K result- ing from the present analysis of C (0). We wish to em- phasize that especially below 7 K the convergence of the extrapolated numerical results is still rather poor, which may affect the reliability of procedures in which exchange parameters are estimated from simultaneous fits of CL and CM to experimental heat capacity data.

Acknowledgments

One of the authors (G. K.) would like to thank the Institute of Physics of the Lodz University for a partial support via project CPB 01.08 and the Department of

Physical Sciences in Naples for the! use of their com- puter facilities.

[l] See, for instance, Johnson, J. D. and Bonner, J.

1.5 I I I C., Phys. Rev. B 22 (1980) :!51, and references

therein.

[C6H11 NH3I CuBr3 [2] Mikeska, H. J., J. Phys. C 111 (1978) L29; 13

(1980) 2913.

1.0- B I I C [3] Kopinga, K., Tinus, A. M. C. and de Jonge, W.

J. M., Phys. Rev. B 29 (1984) 2868.

[4] Tinus, A. M. C., Kopinga, K. and de Jonge, W.

-

Y J. M., Phys. Rev. B 32 (1985) 3154.

[5] Kamieniarz, G. and Vanderzartde, C., Phys. Rev.

B

35 (1987) 3341.

[6] Wysin, G . M. and Bishop, A. R.., Phys. Rev. B 34

(1986) 3377.

[7] de Vries, G. C., Frikkee, E., Kakurai, K., Steiner, M., Dorner, B., Kopinga, I<. and de Jonge, W. J. M., contributed paper, 1:CNS88.

o B - 1 T

B - 2 T [8] Kopinga, K., Tinus, A. M. C, and de Jonge, W.

+ B = 3 T J. M., Phys. Rev.

B

25 (1982:) 4685.

-

[9] Phaff, A. C., Swiiste, C. H. W., de Jonge, W. J.

I I I

0 5 10 15 20 M., Hoogerbeets, R. and van Duyneveldt, A. J.,

T I K I . .... J . ph?/~. C 17 (1984) 2583.

Fig. 1. - Excess heat capndty AC = C(B) - ~ ( 0 ) of [lo] ~ o p i n g a , K.,

tin us,'^.

M. C., de Jonge, W. J.

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