### Section 2: Spin Waves

On the previous page in this tutorial, it was seen
how the magnetization vector of a saturated
ferrite sample underwent precession when
it was misaligned from the external field direction. Since the magnetization
vector is the volume average of the dipole magnetic moment of the sample,
it follows that the individual moments which are responsible for the
magnetic character of the sample each undergo precession.

The precession of the net magnetization vector implies that the individual
moments, or "spins" are each precessing with the same frequency and the
same phase. This situation is commonly referred to as the
*uniform mode.*

A spin wave is similar in that the individual moments throughout a material
precess with the same frequency about the external field direction. The
distinction which sets spin waves apart, however, is that the phase of the
precession varies from spin to spin.

##### How to use this applet:

The Java applet below provides a simple graphical depiction of spin waves
on a two-dimensional lattice. The applet shows a 6-by-6 array of
precessing spins. Each of these precessing moments obey the same equations
of motion as the magnetization vector shown in the previous applet.
For simplicity, damping has been eliminated in this model.

As with any other wave phenomenon, the direction and wavelength of the
spin waves are determined by the wavevector **k**. In the applet below,
use the mouse to click and drag the position of the **k**-vector, in
order to change the propagation direction and the wavelength of the spin
waves. As a special exercise, make the **k**-vector very small and
observe the result. You can adjust the frequency of the precession by
changing the external field strength H. Pressing Finish will stop the
applet execution.

Source code for this applet

When the wavevector magnitude becomes small, the wavelength becomes
large, and the phase difference becomes negligible. This gets us back
to the concept of the uniform mode: The uniform mode is a spin wave
with wavevector zero.

This applet only demonstrates the conceptual nature of spin waves
and is far from being physically rigorous. A well known property of
spin waves is that the frequency depends on the wavevector magnitude and
direction. This relationship, quantified by the spin wave *dispersion
relations*, will be explored later in the tutorial.

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*Last updated: March 1998*

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