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.
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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