Neutron Scattering

n = v - cap[i(pga . - wt)] (13 .22),
where a, designates thé lattice constant and b is numbering thé spins .
Like in thé case of thé phonon dispersion one may deduce thé magnon relation .
In
case of a simple chain one yields
r~,w(q) = VS(1- cos(ga)) (13 .23),
which may be approximated for small q by hcw(q) = 2JS(ga) 2 , i .e . a quadratic relation .
In contrant an acoustic phonon dispersion is always linear in that q-range . Equation
(13 .23) may be extended to thé there-dimensional case, in all cubic systems thé relation
hw(q) = 2JS(ga) 2 remains valid for small q .
Figure 15 shows thé magnon dispersion measured in Fe, from which one may obtain
thé exchange constant J ;
for comparison thé linear dispersion relation of thé LA branch
is added .
One may note, that in the case of Fe the magnon frequencies extend to higher
énergies than those of thé phonons .
Due to thé spécial form of thé résolution ellipsoid in
thé triple axis spectrometer, it is favorable to perform scann at constant energy revealing
both thé phonon and thé magnon, which exhibit approximately thé came intensities, see
right part of figure 15 .
Since magnons are bosons, their occupation is given by Bose-statistics like that of
thé phonons, équation (13 .11) .
Also for thé differential cross section one may deduce a
formula similar to équation (13 .10) valid for phonons
d2()-
( )}
c~QdE'
(k'/k) - S - covst . - cap(-2W(Q))F2(Q)(1 + Q2)
.
~Tq(n(w~) + 1/2 ± 1/2) - 4(w + w j (q» - a(Q + q -1) (13 .24) .
One recognizes thé ternis for momentum and energy conservation as well as thé Bosefactor,
which is increased by one in case of magnon création .
Furthermore thé intensity is
modulated by thé same Debye-Waller- factor as thé phonons . However, there is no terni
équivalent to thé dynamic structure factor, thé intensity is determined just by thé spin
S, thé form factor F(Q) and thé direction of thé momentum transfer .
13-22