03_dipolar


Tongco and Villagonzalo

8

Dipolar Interaction in a One-Dimensional Ising Ring

Gina Rose Tongco* and Cristine Villagonzalo
Structure and Dynamics Group,

National Institute of Physics, University of the Philippines,
Diliman, Quezon City 1101

E-mail: gtongco@nip.upd.edu.ph

ABSTRACT

Science Diliman (July-December 2004) 16:2, 8–9

As synthesis of low-dimensional magnetic systems become possible, the need for theoretical understanding
of their behavior arises. In this work, the authors explore a one-dimensional magnetic structure with the
spins having dipolar interaction.

The effects of a long-range interaction in a one-
dimensional (1D) magnetic ring is studied. This ring
consists of N equally spaced spins i. The spin interaction
is described by a modified Ising model and the system’s
total energy E is given by

(1)

where i = ±1 and r
ij
 is the distance between spins i and

j. The first sum is the exchange interaction restricted
to nearest-neighbor pairs in the original Ising model
(Yeomans, 1997). The second sum represents the
dipolar interaction over all pairs of moments within
the ring. The energy is measured in units of the
exchange constant J. In this study we set J = 1.0. Here
G is the strength of dipolar interaction in units of J.

We investigate the behavior of the ring as the dipolar
strength G is varied for N = 10 spins. We employ the
Metropolis algorithm (Gould & Tobochnik, 1996) to
obtain the equilibrium state of the system of spins. This
procedure realizes the importance sampling which
restricts the sampling space only to statistically feasible
configurations. Figure 1 shows the plot of the average
magnetization (〈M〉) as a function of temperature (T).

〈M〉 is obtained by summing all the spins and dividing
the result by the total number of spins.

At low temperatures, the behavior of the average
magnetization as a function of T is shown to depend
on the strength of the dipolar interaction (G).
Ferromagnetic states occur in the ground state (T = 0)
for a small value of G. On the other hand,

*Corresponding author

3

i j
i j

i j i j ij

s s
E J s s G

r≠ ≠
= − +∑ ∑

Fig. 1. Behavior of the magnetization with temperature for
different values of G. The temperature is given in energy units
(k

B
T), where k

B
 is the Boltzmann constant.

0 10 20 30 40
0

0.2

0.4

0.6

0.8

1

50

〈〈〈〈 〈M
〉〉〉〉 〉

k
B
T



Dipolar Interaction

9

antiferromagnetic states are favored when G is large
even at a finite temperature. The magnetization saturates
at high T. Also, an effective lowering of the total energy
is achieved when smaller values of G approach zero
and when large values of G are increased further.

Similar effects are observed for rings having a larger
number of spins. Increasing the number of spins in a
ring decreases the value of the transition point in G.

Our findings, which show the dependence of the ground
state with the dipolar strength, are in contrast to results
in higher dimensions, where dipolar interactions favor
ferromagnetic alignment of spins (Tang & Sun, 2002).
Indeed, the state of a magnetic system is dictated by its
dimension and spin interaction.

ACKNOWLEDGMENT

Ms. Tongco would like to acknowledge the DOST for
the Scholarship (2000–2005) support under RA 7687.

REFERENCES

Gould, H. & J. Tobochnik, 1996. Introduction to computer
simulation methods: Applications to physical
systems. Addison-Wesley, New York.

Tang, B. & G. Sun, 2002. Striped and rectangular phases in
multilayered Ising systems with dipolar interaction. Phys.
Rev. B. 65: 132418.

Yeomans, J.M., 1997. Statistical mechanics of phase
transitions. Oxford University Press, New York.