ESR Intensity and Anisotropy of the Nanoscale Molecular
M a g n e t V15Manabu Machida*, Toshiaki Iitaka1* and Seiji Miyashita***Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan"*Computational Astrophysics Laboratory, RIKEN (The Institute of Physical and Chemical Research), 2-1
Hirosawa, Wako, Saitama 351-0198, Japan** Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo
113-0033, Japan
Abstract. The low temperature ESR intensity of the nanoscale molecular magnet V15 is studied theoretically. First wenumerically obtain the temperature dependence of the intensity. Then we analytically explore the effect of the DzyaloshinskyMoriya interaction on the intensity at zero temperature using the triangle model.
Keywords: ESR, molecular magnets, ultra-cold temperature
PACS: 75.10Jm,75.50.Xx,76.30.-v

INTRODUCTION

jThe nanoscale molecular magnet V15 is the complex offormula K6 [V^As 6 0 4 2 (H 2 0)] • 8H 2 0. In the molecule,fifteen vanadium ions of spin 1/2 are placed almost
forming a sphere: three ions in the middle are sandwiched by two hexagons formed by the upper and lower
twelve ions [1].
In this paper, we theoretically study the ESR of V15
at low temperatures. Using the triangle model, we study
the effect of the Dzyaloshinsky-Moriya (DM) interaction[2, 3, 4] on the intensity ratio (intensity divided by thatof a single isolated spin 1/2) at zero temperature. In
particular, we reveal how the intensity ratio deviatesfrom 1 due to the DM interaction when the ground statemagnetization is 1/2. The results of this analysis explain
the behavior of the numerically obtained ESR intensityof V15 quite well.
In experiments, the magnetization changes smoothly
at zero field from —1/2 to 1/2, and around 2.8T from
1/2 to 3/2, when the field is swept adiabatically at lowtemperatures. These facts imply that gaps open between
the ground state and the lowest excited state at OT and2.8T, and the origin of the gaps may be attributed to theDM interaction [5, 6, 7, 8, 9]. However, the details of theDM interaction in V15 are not yet fully understood.

ESR INTENSITY RATIO OF V15The V15 molecule is modeled by the following spin
Hamiltonian [10, 11].

^ = -^JijSrSj+^Dij.[SixSJ]-yZHs.Si.
(U)

(1)

FIGURE 1. Schematic picture of V15 showing the magneticinteractions J, J\, and J2.
For Jjj, we have three different values </, J\, and J2(\J\ > \J21 > \J\\), as shown in Fig. 1. Here we take
J= -800K, J2 = -350K, and Jx = -225K [12]. The
second term on the right-hand side in (1) describes theDM interaction. DM vectors are considered to exist onthe two hexagons along the bonds with J. We take thereference DM vector £>i 2 to be D\ 2 = D^ 2 = D^ 2 =40K. The other DM vectors are determined from the D3
symmetry of V15 [13]. We assume the static magnetic
field is applied parallel to the c-axis of the molecule (zaxis) and put H$ = (0,0,i7s)At low temperatures, the ESR absorption occurs only
from transitions among low-lying energy levels because
only these states are populated. Hence, we calculate theESR absorption of V15 by taking a small subspace ofthe total Hilbert space which concerns the lowest eightlevels [13]. Within this subspace, the ESR absorption isobtained by the explicit formulation [14] of the Kubo

(U)

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2\Hs—H$\, H^ = — I J, and ignore terms smaller than0(D). Note that the ground state magnetization changes
from 1/2 to 3/2 as the field//s increases across Hs=H$.
In the ultra-cold limit, Rtri(T) near H^ is obtained as [17]Rtn(T) :HS=2T( 56.0GHz)
i/s=4T(112.0GHz) —-o//s=6T(168.0GHz)
0

1

4

56

J__i_891+
3V3D(HS<HI)(HI<HS).

(4)This behavior of Rtri(T) gives a qualitative understanding of R(T) in Fig. 2 at zero temperature.
10ACKNOWLEDGMENTS
FIGURE 2.
R(T).

Temperature dependence of the intensity ratio
This work is supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology, and also by the NAREGI Nanoscience Project, Ministry of Education, Culture, Sports, Science and Technology, Japan. We also thank RIKEN (The Institute of Physical and Chemical Research) for computational resourcesof the RIKEN Super Combined Cluster (RSCC).

formula. That is, we obtain the imaginary part of thesusceptibility %"((D\T) by direct diagonalization of thesmall Hamiltonian. The intensity I(T) and the averageenergy absorption per unit time I(co; T) are given byI(T)ri(co;T)dco,I(co;T)(oHl ...

.

REFERENCESJo(2)where T is temperature, and co and / / R are the frequency
and amplitude of the radiation field / / R , respectively. We
note that /fR is perpendicular to the static field / / s • We
define the intensity ratio as
R(T)=I(T)/Il(T),(3)

where I\(T)is the intensity of an isolated spin 1/2:

h (T) = ^ ^tanh ( W . In Fig. 2, we plot R(T) for

Hs = 2T(56.0GHz), Hs = 4T( 112.0GHz), and Hs =6T( 168.0GHz). The behavior of R(T) is consistent withexperiment [15,16]. We note that, for H$ = 2T, R(T = 0)
deviates from 1.

TRIANGLE MODEL ANALYSIS FORWEAK DM INTERACTION
The triangle model is given by considering only three
spins in (1) with Jn = J23 = ^31 = J < 0. Assuming C3
symmetry, we determine the DM vectors as D\2 = Dx,Ifu = Dy, D*23 = (-Dx + y/3Dy)/2, lf23 = (~V3DX Dy)/29 D*3l = (-Dx - V3Dy)/2, L?3l =(V3Dx-Dy)/2,a n d K12 ^-23 E?3l = Dz. Let us further assume Dx
Dy=Dz=D>0.
If we put J= - 2 . 5 K mdD = 0.25K,
the energy levels of the triangle model reproduce thelowest eight levels of V15 quite well [11].
We analytically obtain the ESR intensity ratio ^tri(^)
of this triangle model. In the calculation, we assume D «C

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