Advanced Solid State Physics

Figure 78: Magnon dispersion relation (1 dimensional, for nearest neighbor interaction). The slope for
long wavelengths is proportional tok^2. [from Kittel]

. And the density of states in three dimensionsD(ω)is

D(ω)dω =

1

(2π)^3

4 πk^2

dk

dω

dω (162)

=

1

(2π)^3

4 πk^2 (2

√

ω·

√

3 JSa^2

~

)dω (163)

=

1

4 π^2

(

~

2 JSa^2

)

32 √

ω (164)

where we used the dispersion relationω∝k^2 obtained obove. Plugging these into the integral 160 we
find for the total amount of magnons

Nexc= 0. 0587

kBT

2 JSa^2

3

2

(165)

Therefore the relative change in magnetizationM∆M(0)is proportional toT

(^32)

11.3.4 Plasmons

Suppose you have some metal where you have a uniform distribution of the positive ions and a uniform
distribution of the negative electrons as seen in fig. 79. We assume that the electrons are much lighter
than the ions so that the ions don’t move and are fixed. But the electrons can move. So if you pull
them off to a side they are not uniformly distributed anymore but the ions still are. This causes a force
on the electrons and if you let them go they will start to oscillate back and forth (the oscillation will
of course decay after a certain time). These oscillations are called plasma oscillations, because they