Environmental Engineering Reference
In-Depth Information
q
=
Q
, the coupling between the modes at
Q
and
−
Q
leads to an
ampli-
tude mode
and a
phason mode
, corresponding respectively to an in-phase
and a 90
◦
out-of-phase modulation of the lengths of the moments. The
energies of the two longitudinal modes at
q
=
Q
are approximately given
by
√
2
E
amplitude
σ
∆
(7
.
4
.
9)
E
phason
8
β
∆
p
01
1
/
2
σ
∆
,
where
p
01
is the average value of
p
01
(
j
). The scattering intensity, pro-
portional to 1
/
[
hω
], of the lowest-lying phason mode is
much larger than that of the amplitude mode. The low-intensity ampli-
tude mode is indicated by the dashed line at
q
-vectors close to
Q
in Fig.
7.15, and it was not clearly resolved in the experiments. The phason
mode has a dispersion relation, indicated by the solid lines in the fig-
ure, which increases linearly from the magnetic Bragg peak at
Q
, except
for the presence of the small gap
E
phason
at
q
=
Q
. In the incommens-
urable case, the free energy is invariant to a change of the phase constant
ϕ
in (7.4.7), so that the longitudinal component of the zero-frequency
susceptibility diverges at the wave-vector
Q
. The corresponding gener-
ator of an infinitesimal phase shift is 1
{
1
−
exp(
−
βhω
)
}
|
j
. fthis
generator commuted with the Hamiltonian, a specific choice of
ϕ
would
break a continuous symmetry of the system, implying the presence of a
well-defined linearly-dispersive Goldstone mode, as discussed in Section
6.1. However, as may be verified straightforwardly, it does not in fact
commute with the Hamiltonian. On the contrary, within the RPA the
longitudinal response contains an elastic contribution, due to the final
term in (7
.
4
.
8
a
), and hence the scattering function contains a diffusive
peak at zero frequency. It is the intensity of this peak which is found
to diverge in the limit
q
→
Q
.As
q
departs from
Q
, the diffusive re-
sponse at zero frequency rapidly weakens, and the phason mode begins
to resemble a Goldstone mode. The presence of the inelastic phason
mode at the wave-vector
Q
can be explained as a consequence of the
modulation of the population difference
n
01
(
j
), which is proportional to
p
01
. This mode corresponds to an oscillation of the phase-constant
ϕ
in (7.4.7), except that the adiabatic condition, which applies within the
RPA as soon as the oscillation frequency is non-zero, constrains
n
01
(
j
)
to remain constant, without participating in the oscillations. This con-
dition, in turn, gives rise to the restoring force which determines the
frequency of the oscillations. However, if the oscillations are so slow (i.e.
essentially zero in the present approximation) that
n
01
(
j
) can maintain
its thermal-equilibrium value, there are no restoring forces. In the zero-
temperature limit,
n
1
vanishes exponentially, in which case
n
01
(
j
)=1,
and the diffusive elastic response disappears together with
E
phason
.The
− iδϕ
j
|
1
><
1
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