Journal of Magnetism and Magnetic Materials 226}230 (2001) 417}419

g-factor and linewidth of the quasi-one-dimensional Heisenbergantiferromagnet copper benzoate

B. Pilawa*, E. Herrling, I. Odenwald

Physikalisches Institut, Universita( t Karlsruhe (TH), Engesserstr. 7, D-76128 Karlsruhe, Germany

Abstract

The principal values and axes of the anisotropic exchange tensor in the antiferromagnetic spin chain compoundCu-benzoate are determined by electron spin resonance (ESR) at 9.5GHz from an analysis of the temperature-dependentshift of the g-factor and the anisotropy of linewidth. � 2001 Elsevier Science B.V. All rights reserved.

Keywords: Electron paramagnetic resonance; Heisenberg chain; Spin correlation function

The anisotropy of the g-factor of exchange-coupledspin systems is mainly a "nger-print of the ions or rad-icals under investigation. On the other hand, the temper-ature dependence of the g-factor re#ects the in#uence ofstatic internal magnetic "elds which are usually causedby demagnetization [1]. In quasi-one-dimensional spinsystems the g-shift is mainly governed by the nearest-neighbor interaction [2]. The ESR-signal of the uniformantiferromagnetic spin-chain compound Cu-benzoate isstudied in order to demonstrate that reliable informationabout the anisotropic nearest-neighbor interactions canbe obtained from the analysis of the temperature-depen-dent g-shift and the anisotropy of the linewidth. The Cu��ions of Cu-benzoate are exchange-coupled via carboxylgroups along the c-direction (J

��/k

�"!17.3K). The

spin chains are well separated by benzoate-groups. Thecrystallographic unit cell of Cu-benzoate is monoclinic�+903 (space-group I2/c, lattice parameters a"6.98As ,b"34.12As , c"6.30As , ��903 [3]). The principal axesa�, b�, c� of the g-tensor are rotated within the ac-plane( (c�, c)+383 and b��b [4]). Fig. 1 shows the temperaturedependence of the g-factor of Cu-benzoate at��"9.5GHz within the ac-plane. Depending on the

orientation of the magnetic "eld the g-factor shiftsto higher or lower values when the temperature is

*Corresponding author. Tel.:#49-721-608-3452; fax:#49-721-608-6103.E-mail address: bernd@piobelix.physik.uni-karlsruhe.de

(B. Pilawa).

decreased. The center of the resonance curve is given bythe "rst moment of the ESR-line [5]:

��h"!

�[S�, [H,S�]]��S��

. (1)

S�"�is��denotes the x-component of the total spin and

H the Hamiltonian of the spin system

H"!J��

��

s�s���

#g���B���

s��

# �

�����������������

A����s�� s��� # �

�������������

B����(i, j)s�� s��� (2)

with the isotropic exchange-, the Zeeman-, the pseudo-dipolar exchange-A���� (PD) and the dipolar interactionB����(i, j). The coordinates (�"x, y, z) specify the laborat-ory frame of reference with the static magnetic "eldB�applied along z. Due to the large inter-chain distances

the dipolar interaction is mainly determined by theintra-chain interaction and points along the c-axis. ThePD-interaction is de"ned by the symmetric traceless ten-sorA���� which can be characterized by the orientation ofthe principal axes and an anisotropy parameter�"(A

��!A

��)/A

��. The principal axes of the PD inter-

action are expected to coincide approximately withthose of the g-tensor. The evaluation of the commutatorsof Eq. (1) leads to

h��"g

���B�(¹)#h���, (3)

0304-8853/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 9 7 - 5

Fig. 1. Temperature dependence of the g-shift at 9.5GHz whenthe magnetic "eld is applied in the ac-plane (symbols). Solidlines: "t of Eq. (3).

Fig. 2. Angular variation of the anisotropic nearest neighborintra-chain interaction

��(dots) and the g-factor g

�(circles)

obtained with Eq. (3) from the "t of the temperature-dependentg-shift shown in Fig. 1. Solid line: "t of the anisotropic nearestneighbor intra-chain interaction. Broken line: "t of the angularvariation of the g-factor.

Fig. 3. Dots: Angular variation of the linewidth in three ortho-gonal planes at room temperature. Solid lines: calculation basedon Eq. (6).

which describes the temperature dependence of theg-factor according to the resonance condition��h"g (¹)�

�B�(¹). The isotropic exchange does not

contribute to Eq. (3) and the main contribution comesfrom the Zeeman interaction. The combined contribu-tion of the dipolar and PD interaction is abbreviated byh���. The most important contribution to h��� is due tothe nearest-neighbor intra-chain interaction and thecommutators of Eq. (1) give

h���

"3 ������

(A��

#B��(i, j))

��s��s��!s�

�s���/�s�

��. (4)

The neighbored spins are strongly correlated via theintra-chain exchange interaction. For the analysis of theexperiment, the correlation function �s�

�s��!s�

�s���/�s�

�� is

numerically calculated with a periodic boundary condi-tion of 15 and 16 spin, respectively. A reliable predictionis possible for temperatures above T+0.5J

��/k

�. The

dipolar interaction also leads to contributions withjOi$1. In this case, the correlation function can beapproximated by �s�

�s��!s�

�s���/�s�

��+�s�

�� for

T�0.5J��/k

� and h��� becomes

h���"h���

#3�������

�������

B��(i, j)�s�

��

#

3

2(g

���)�n (N!�

)�s�

��. (5)

N denotes the demagnetization factor and n the densityof spins (n"2.667�10�� spins/cm for Cu-benzoate).The sum �

�������is approximated by an exact summa-

tion within a sphere � �������������

B��(i, j)"!0.0216 cm��

and the demagnetization due to the shape of the crystalswhich accounts for the rest of the sample. N is negligiblysmall (N+0), when the magnetic "eld is applied withinthe ac-plane. The experimental data of g(T) within theac-plane are "tted with N"0, the g-factor of the Zeemaninteraction g

�and the nearest neighbor intra-chain

interaction ��

"������

(A��

#B��(i, j)) (solid lines in

Fig. 1). The angular variation of ��and g

�are shown in

418 B. Pilawa et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 417}419

Fig. 2. B��(i, j) is "xed by the crystallographic data

and the g-tensor so that the "t of the angular variationof

��"xes the anisotropic exchange with

A�

"!0.0506$0.0004 cm�� and �"0. This leads toa ratio of A

�/B"!0.914 with respect to the dipolar

interaction estimated by B"(g��)�/c and g"2,

c"3.15As . These results are used to calculate thelinewidth according to [6]

�B���

J(���#10

��#

��) (6)

with �"

��,

��"!�

(

��Gi

��) and

��"

!�(

��!

��G2i

��). � accounts for the enhancement

of the secular contribution due to spin di!usion. Thecomparison between �B

���at room temperature and

Eq. (6) in Fig. 3 yields �"2.3 and con"rms that theparameters of the anisotropic interaction determinedfrom the temperature-dependent g-shift describe excel-

lently the anisotropy of the linewidth. This analysis of theanisotropic interaction in Cu-benzoate provides thestarting point for an investigation of dynamic spin cor-relation which determine the temperature dependence ofthe ESR-linewidth.

References

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[2] K. Nagata, Y. Tazuke, J. Phys. Soc. Japan 32 (1972) 337.[3] H. Koizumi, K. Osaki, T. WatanabeH , J. Phys. Soc. Japan 18

(1963) 117.[4] M. Date, H. Yamazaki, M. Motokawa, S. Tazawa, Suppl.

Progr. Theor. Phys. 46 (1970) 194.[5] K. Kambe, T. Usui, Progr. Theor. Phys. 8 (1952) 302.[6] R. Kubo, K. Tomita, J. Phys. Soc. Japan 9 (1954) 888.

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