Neutron Scattering from Magnetic Materials part 9 docx

Spherical neutron polarimetry 237
Fig. 10. The h0l section of reciprocal space for the incommensurate structure of CuO. The fundamental (nuclear)
reflections are shown as open circles and the magnetic satellites as filled ones. The scattering vectors for the
002 +τ and 000 −τ reflections are shown as dashed lines.
4.4. Magnetic structures with zero propagation vector
It has already been pointed out in Section 2.3 that in structures with τ = 0, nuclear and
magnetic scattering occur in the same Bragg reflections and interference between them can
occur. In this case SNP can determine the ratio between magnetic and nuclear scattering,
allowing the magnitude of magnetic moments to be established from SNP alone without
recourse to supplementary integrated intensity measurements.
4.4.1. The magnetic structure of U
14
Au
51
. The intermetallic compound U
14
Au
51
crys-
tallises in the hexagonal Gd
14
Ag
51
structure with space group P 6/m [11]. The uranium
atoms occupy three crystallographically distinct sites 6(k), 6(j ) and 2(e) labelled U1,
U2 and U3, respectively. Susceptibility, specific heat and resistivity indicate a magnetic
phase transition at 22 K. This has been confirmed by neutron powder diffraction measure-
ments: an antiferromagnetic structure with zero propagation vector and magnetic space
group P 6

/m was proposed [12]. The ordered magnetic moments aligned parallel to c

were 0.5 and 1.6µ
B
on U1 and U2, respectively. No moment was assigned to the U3 atoms
which have a particularly small separation, it was argued that direct f -electron wave func-
tion overlap prevents a magnetic response. An SNP study of the magnetic structure was
undertaken because it proved impossible to reconcile the intensity of magnetic scattering
by single crystals of U
14
Au
51
with the proposed magnetic structure. The U
14
Au
51
crys-
tal was mounted with its [01.0] axis vertical. The polarisation matrices determined for the
20.0, 20.1 and 10.1 reflections are given in Table 1. They enable severe constraints to be
imposed on the possible magnetic structures.
1. For incident polarisation parallel to the scattering vector x the scattered beam is partly
depolarised, and reversed but not rotated. The depolarisation must be due to off-
diagonal terms (P
xy
, P
xz
) of opposite signs coming from 180
◦
domains. This means
that the J
ni
terms in (14) must be nonzero showing that the magnetic scattering is

238 P. J . B r o w n
Table 1
Polarisation matrices P
ij
measured for the 20.0, 20.1 and 10.1 reflections of U
14
Au
51
at 15 K
hkl
P
ij
=20.0 P
ij
=20.1 P
ij
=10.1
j
i
xyzxyzxyz
x −0.86 0 0 −0.28 0 0 −0.54 −0.08 −0.07
y 0 −0.83 0 0 −0.62 −0.38 −0.08 0.98 −0.12
z 0 0 0.98 0 0.37 0.73 0.01 −0.05 −0.55
in quadrature with the nuclear scattering. The reversal of direction shows that the
magnetic structure factor is greater than the nuclear one for both 20.0 and 20.1.
2. For the 20.0 reflection there are no significant off-diagonal terms and P
zz
is not sig-
nificantly different from 1 showing that for 20.0 M
⊥

must be parallel to z (crystal-
lographic [01.0]), and there are therefore no significant components of the magnetic
structure factor parallel to c.
3. For 20.1 there is some depolarisation for all three incident polarisation directions and
off-diagonal components P
yz
and P
zy
are observed. This is consistent with moments
in the a–b plane. The observation that the depolarisation for incident directions in
the y–z plane, that containing the magnetic interaction vector, is less than for the
x direction implies that all the depolarisation is due to the 180
◦
domains and that
there is none due to orientation domains. The magnetic structure therefore probably
has the full symmetry of the crystallographic space group.
There is just a single magnetic space group and basic model structure which is compati-
ble with all these constraints. The U1 and U2 sites lie on the mirror planes perpendicular to
the hexad and from (2) and (3) their moments lie in it. These mirror planes cannot therefore
invert the moments and must be combined with time reversal. To satisfy (1) which implies
that centrosymmetrically related atoms have opposite moments the hexad must operate
without time inversion. The magnetic space group is therefore P 6/m

and the magnetic
moments on the groups of 6 U1 (and U2) atoms related by the hexad have a star struc-
ture as illustrated in Figure 11. In this magnetic group any moment on the U3 atoms is
constrained to be parallel to c since these sites are on the hexagonal axes. The SNP mea-
surements show that the c component of moment is small or zero so it can be concluded
that the U3 moment is also small or zero. To describe the structure completely it is neces-
sary to determine the magnitude and the orientation within the a–b plane of the moments

on the U1 and U2 atoms.
The SNP data for the 10.1 reflection (Table 1) allow rough values of the moment direc-
tions within the a–b plane to be deduced. The incident polarisation parallel to y is hardly
changed on scattering so its magnetic interaction vector M
⊥
is nearly parallel to y.The
magnitude of M
⊥
can be obtained from
P
xx
=
1 −γ
2
1 +γ
2
=β, (27)
γ gives, as before (equation (10)), the ratio of magnetic to nuclear scattering. M
⊥
for
Spherical neutron polarimetry 239
Fig. 11. The hexagonal star arrangement of moments found in the U1 and U2 layers of U
14
Au
51
. The angle φ
used to fix the orientation of the moments is marked.
the 10.1 reflection is the sum of contributions from the U1 and U2 layers. The y and z
components of these were computed separately as a function of the angle φ in Figure 11
and are plotted in Figure 12. By moving one curve relative to the other it was found that

there is only a small range in which a pair of φ’s exist for which the M
⊥z
for U1 and U2
cancel whilst their M
⊥y
reinforce one another. It corresponds to φ
U1
∼ 140
◦
, φ
U2
∼ 90
◦
.
These initial values provided an adequate starting point for a least squares refinement of
the structure using both SNP and integrated intensity data.
4.4.2. Magnetoelectric crystals. The property of magnetoelectricity in centrosymmetric
crystals is restricted to those having antiferromagnetic structures with zero propagation
vector in which the centre of symmetry is combined with time-reversal. These are just
the requirements for J
ni
(equation (16)) to be finite giving rise to off-diagonal terms P
xz
,
P
zx
in the polarisation matrix (equation (14)). It is known that although the temperature
dependencies of magnetoelectric (ME) susceptibilities are unique to each material, their
magnitudes and even their signs are specimen dependent. This specimen dependence is
due to the existence of 180

◦
antiferromagnetic domains which have opposite ME effects.
The measured ME susceptibility χ
obs
is related to the intrinsic susceptibility χ
0
by
χ
obs
=ηχ
0
with η =
v
1
−v
2
v
1
+v
2
,
where v
1
and v
2
are the volumes of crystal belonging to each of the two domains.
SNP gives the possibility, for the first time, to obtain the intrinsic ME susceptibilities
since it allows the domain fraction η to be determined. For a centrosymmetric ME crystal
with domain fraction η and moments in the x–y plane the polarisation matrix (equations
(14) and (10)) can be simplified to

P =

β 0 ηξ
010
−ηξ 0 β

β is given by equation (27) and ξ = 2q
y
γ/(1 + γ
2
), (28)
240 P. J . B r o w n
Fig. 12. Curves showing the variation with φ of M
⊥y
(full) and M
⊥z
(dashed) components of the magnetic
interaction vector of the 10.1 reflection of U
14
Au
51
. (a) is for the U1 atoms and (b) for the U2 atoms. The origins
of the two figures are displaced so that on the vertical line marked ((a) φ =140
◦
,(b)φ = 90
◦
)thez components
of U1 and U2 cancel whilst their y components reinforce one another.
Fig. 13. The moment directions of Cr
3+

ions at the centres of the double octahedral coordination polyhedra
found in Cr
2
O
3
, after (a) cooling in parallel, (b) antiparallel electric and magnetic fields.
q
y
is +1ifM(Q) is parallel to y and −1 if it is antiparallel. Measurement of the polar-
isation matrix therefore allows both η and γ to be determined. The absolute directions
of rotation of the neutron spins when η = 0 determine the magnetic configuration of the
more populous domain. This in turn allows the effects of electric and magnetic fields on
the domain population to be studied. The results shed light on the fundamental mecha-
nisms leading to the ME effect. Perhaps the best known ME material is Cr
2
O
3
in which
the Cr
3+
ions are octahedrally coordinated by oxygen and the structure is made up of pairs
of octahedra, sharing a common face as illustrated in Figure 13, linked to other pairs by
sharing the free vertices. SNP has shown that electric and magnetic fields, applied parallel
to one another and to the c axis while cooling through the Néel transition, stabilise the
Spherical neutron polarimetry 241
domain in which the moments point towards the shared face of their coordinating oxygen
octahedra [13].
5. Determination of antiferromagnetic form factors
As has been shown in Chapter 4 the classical polarised neutron diffraction technique [14]
is widely used to study the magnetisation distribution around magnetic atoms and ions in

ferromagnetic and paramagnetic materials. It is very much more difficult to measure this
distribution in antiferromagnetic systems because in antiferromagnets the cross-section is
seldom polarisation dependent so the classical method is not applicable. As a consequence,
very few measurements of magnetisation distributions in antiferromagnetic materials have
been made since usually they require very precise integrated intensity measurements of
rather weak reflections. In the few cases where such measurements have been undertaken,
they have given very interesting results [15–17]. An antiferromagnetic magnetisation dis-
tribution is more sensitive than a ferromagnetic or paramagnetic one to the effects of cova-
lency because the overlap of positive and negative transferred spin on the ligand ions leads
to an actual loss of moment rather than just to a redistribution.
Until recently no precise measurements had been made for the class of antiferromagnetic
structures with zero propagation vector, in which magnetic atoms of opposite spin are re-
lated by a centre of symmetry. In such structures the magnetic and nuclear scattering are
superimposed, making separation of the nuclear and magnetic parts difficult. Additionally
the magnetic and nuclear structure factors are in phase quadrature so that is no interfer-
ence between them to give a polarisation dependent cross-section. However, it was shown
in the previous section that it is in exactly this case for which the magnetic and nuclear
scattering are in quadrature that the polarisation matrices depend sensitively on the ratio γ
between the magnetic and nuclear structure factors when there is an imbalance (η = 0) in
the populations of the two 180
◦
domains. The high precision with which the ratio γ can
be determined in favourable cases allows the magnetic structure factors to be determined
with good accuracy and so gives access to the antiferromagnetic form-factors.
The polarisation matrix of (28) allows two independent estimates of γ :
(a) P
xz
=−P
zx
=ηξ =

ηq
y
γ
1 +γ
2
,
(29)
(b) P
xx
=P
zz
=β =
1 −γ
2
1 +γ
2
,
the former only being useful if there is an imbalance in the 180
◦
domains. Assuming the
polarimeter (CRYOPAD) is free of aberrations the precision with which γ can be deter-
mined depends on the statistical error in the determination of the components of scattered
polarisation. It should be recalled that in this type of structure the cross-section is inde-
pendent of the polarisation direction. The counting rate summed over the two polarisation
states accepted by the detector is therefore constant, and independent of either incident
or scattered polarisation direction. The polarisation measured by the analyser is given by
P = (I
+
− I
−

)/(I
+
+ I
−
) where I
+
and I
−
are the counting rates in the two detector
242 P. J . B r o w n
channels. The variance in the measurement of a component of polarisation due to counting
statistics is
V
P
=
(1 −P
2
)
2
4

1
N
+
+
1
N
−

, (30)

where N
+
and N
−
are the counts recorded in each channel. The variance is minimised
by dividing the measuring time available in the ratio t
+
/t
−
=(1 −P)/(1 + P). With this
division, if the total number of neutrons counted is N equation (30) becomes
V
P
=
1 −P
2
N
. (31)
The variances in the values of γ derived from (29) are
V
γ
=
(1 +γ
2
)
4
16γ
2
V
P

from (a) and
(32)
V
γ
=
(1 +γ
2
)
4
4η
2
(1 −γ
2
)
2
V
P
from (b).
If η is small (nearly equal domains) or γ is close to unity, the best estimate of γ will
be obtained from (29)(a) whereas for very small or very large γ (29)(b) will give a better
value so long as η is nonzero. Figure 14 shows the regions of γ –η space in which one or
the other equation gives the better estimate of γ .
The first example of the use of this technique was to determine the Cr
2+
form-factor
in Cr
2
O
3
[18]. Samples were cooled in combined electric and magnetic fields to obtain dif-

ferent domain ratios η as indicated in the previous section. The crystals were aligned with
Fig. 14. Plot of γ –η space. The shaded region is that in which equation (29)(a) gives a more precise estimation
of γ than (29)(b). The γ axis is plotted on a logarithmic scale.
Spherical neutron polarimetry 243
Fig. 15. The experimental values of the magnetic form factor measured at the h0. Bragg reflections of Cr
2
O
3
.
The smooth curve is the spin-only free-ion form factor for Cr
2+
normalised to the experimental value at the
lowest angle reflection (10.2).
a [
¯
12
¯
10] axis vertical so as to obtain reflections h0
¯
hl in the horizontal plane. The moment
direction is [0001] so that with this orientation M
⊥
(Q) is parallel to polarisation y.The
elements P
xz
, P
zx
of the polarisation matrices obtained with different domain ratios are
very different, but the magnetic structure factors deduced from them were found to agree
well. This confirms the supposition that extinction effects are not a major problem since

the measurements of polarisation are made with a constant cross-section. The points on
the Cr
3+
form factor obtained from the measured structure factors are plotted in Figure 15
where they are compared with the Cr
3+
free in form factor. It can be seen that for most
reflections an extremely good precision was obtained. Exceptions are the 20.
¯
2 and 10.
¯
10
reflections; for the former the nuclear structure factor is small so that γ  1 and in the
latter the geometric structure factor for the Cr atoms is small so the reflection is insensitive
to the Cr form factor.
This first pioneering experiment has shown that SNP can be used to make high preci-
sion measurements of antiferromagnetic magnetisation distributions. However is should
be emphasised that such measurements are only possible for a restricted class of antiferro-
magnets, those in which magnetic and nuclear scattering occur in quadrature in the same
reflections. Additionally high sensitivity can only be obtained if the population of the 180
◦
domains can be unbalanced. Nevertheless, this class of antiferromagnets includes the mag-
netoelectric classes, and for these electromagnetic annealing can unbalance the domains.
SNP therefore provides an important new tool for probing the magnetisation distributions
associated with magnetoelectricity.
244 P. J . B r o w n
References
[1] R.M. Moon, T. Riste and W.C. Koehler, Phys. Rev. 181 920 (1969).
[2] O. Schärpf, Physica B 182 376 (1992).
[3] F. Tasset, P.J. Brown, E. Lelièvre-Berna, T. Roberts, S. Pujol, J. Alibon and E. Bourgeat-Lami, Physica B

267–268 69 (1999).
[4] M. Blume, Phys. Rev. 130 1670 (1963).
[5] S.V. Maleev, V.G. Bar’yaktar and P.A. Suris, Sov. Phys. Solid State 4 2533 (1963).
[6] P.J. Brown, Physica B 297 198 (2001).
[7] J.B. Forsyth, P.J. Brown and B.M. Wanklyn, J. Phys. C 21 2917 (1988).
[8] D. Yablonski, Physica C 171 454 (1990).
[9] Yu.G. Raydugin, V.E. Naish and E.A. Turov, J. Magn. Magn. Mater. 102 331 (1991).
[10] P.J. Brown, T. Chattopadhyay, J.B. Forsyth, V. Nunez and F. Tasset, J. Phys.: Condens. Matter 3 4281
(1991).
[11] A. Dommann and F. Hullinger, J. Less-Common Met. 141 261 (1988).
[12] A. Dommann et al., J. Less-Common Met. 160 171 (1990).
[13] P.J. Brown, J.B. Forsyth and F. Tasset, J. Phys.: Condens. Matter 10 663 (1998).
[14] R. Nathans, C.G. Shull, G. Shirane and A. Andresen, J. Phys. Chem. Solids 10 138 (1959).
[15] H.A. Alperin, Phys. Rev. Lett. 6 520 (1961).
[16] J.W. Lynn, G. Shirane and M. Blume, Phys. Rev. Lett. 37 154 (1976).
[17] X.L. Wang, C. Stassis, D.C. Johnstone, T.C. Leung, J. Ye, B.N. Harmon, G.H. Lander, A.J. Shultz,
C K. Loong and J.M. Honig, J. Appl. Phys. 69 4860 (1991).
[18] P.J. Brown, J.B. Forsyth, E. Lelièvre-Berna and F. Tasset, J. Phys.: Condens. Matter 14 1957 (2002).
CHAPTER 6
Magnetic Excitations
Tapan Chatterji
Institut Laue–Langevin, B.P. 156X, 38042 Grenoble cedex, France
E-mail:
Contents
1. Introduction . . . . . . . . . . . 247
2. Experimental methods . . . . . 247
2.1.Triple-axisspectrometer 247
2.2.IntensityandresolutionfunctionofTAS 251
2.3.Sizeandshapeoftheresolutionfunction 254
2.4. TAS multiplexing . . . . . 256

2.5.Time-of-flightspectrometers 256
3.Spinwavesinlocalizedelectronsystems 259
3.1.SpinwavesinHeisenbergferromagnets 259
3.2.ThermalevolutionofspinwavesinHeisenbergferromagnets 263
3.3.SpinwavedampinginHeisenbergferromagnets 265
3.4.SpinwavesinHeisenbergantiferromagnets 268
3.5. Two-magnon interaction in Heisenberg antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . 274
3.6.SpinwavesinHeisenbergferrimagnets 275
4. Spin waves in itinerant magnetic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
4.1. Generalized susceptibility and neutron scattering cross-section . . . . . . . . . . . . . . . . . . . . 281
4.2. Spin dynamics of ferromagnetic Fe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
4.3. Spin dynamics of ferromagnetic Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
4.4. Spin dynamics of weak itinerant ferromagnet MnSi . . . . . . . . . . . . . . . . . . . . . . . . . . 297
5.SpinwavesinCMRmanganites 300
5.1. Spin waves A
1−x
B
x
MnO
3
,A= La, Pr, Nd; B =Ca,Sr,Ba 300
5.2. Thermal evolution of spin dynamics of A
1−x
B
x
MnO
3
313
5.3. Spin waves in bilayer manganite La
2−2x

Sr
1+2x
Mn
2
O
7
315
6.Concludingremarks 327
Acknowledgments 328
References 328
NEUTRON SCATTERING FROM MAGNETIC MATERIALS
Edited by Tapan Chatterji
© 2006 Elsevier B.V. All rights reserved
245
Magnetic excitations 247
1. Introduction
Spin systems coupled by exchange interactions have wavelike low-lying energy states.
The waves are called spin waves. The energy of a spin wave is quantized and a magnon
is the unit of energy of a spin wave. Neutron scattering is the unique probe for the exper-
imental investigation of spin waves and magnetic excitations in general. In Chapter 1 we
developed basic equations of inelastic magnetic neutron scattering and scattering from spin
waves. Spin waves have been studied in all types of ordered spin systems, ferromagnets,
antiferromagnets, ferrimagnets and other more complex magnetic structures described in
Chapter 2. We have derived in Chapter 1 the spin wave dispersion equations in simple spin
systems. The dispersion relation of ordered spin systems can be experimentally determined
by inelastic neutron scattering yielding eventually the sign and magnitudes of exchange in-
teractions. Before we explain how this may be achieved in particular cases we describe the
essential experimental techniques of inelastic neutron scattering.
2. Experimental methods

As in the case of the determination of the magnetic structure, the neutron scattering tech-
nique is unique for experimental investigation of spin waves and other excitations in mag-
netic crystals. The spin wave energies in magnetic solids are normally in the meV range
and therefore scattering of thermal neutrons is suitable for their investigation. Sometimes
the spin wave energy is of the order of 0.1 meV for which cold neutron scattering is more
appropriate. In some cases like transition metal ferromagnets like Fe, Ni, Co, etc., the spin
wave energies lie at 0–200 meV for which it is necessary to use scattering of hot neutrons
in addition to that of thermal neutrons.
2.1. Triple-axis spectrometer
Spin wave dispersion is usually determined with a neutron triple-axis spectrometer (TAS).
Magnetic and structural excitations have been investigated with triple-axis spectrometers
ever since Brockhouse [1] developed such a spectrometer at Chalk River in Canada. The
techniques of triple-axis spectrometers have been discussed in details by Shirane, Shapiro
and Tranquada [2] and Currat [3]. Here we will give an outline of this technique.
Figure 1 shows schematically typical triple-axis spectrometer. The three axes correspond
to the rotation axes of the monochromator, the sample table and the analyzer. The triple-
axis spectrometer is the instrument of choice whenever it is necessary to have precise
control on the positions in (Q,ω) space at which one wishes to measure the scattered
neutron intensity. The intensity at a single position in (Q,ω) space is measured in a step
by step manner where each spectrometer configuration corresponds to a well-defined value
of k
i
and k
f
, the incident and the scattered wave vector. Q and
¯
hω satisfy momentum and
energy conservation laws given by
k
f

−k
i
=Q, (1)
¯
h
2
k
2
i
2m
n
−
¯
h
2
k
2
f
2m
n
=
¯
hω, (2)
248 T. Chatterji
Fig. 1. A typical triple-axis spectrometer set-up at a reactor thermal beam-port (IN20, Institut Laue–Langevin,
Grenoble).
where m
n
is the mass of the neutron. The incident neutron wave vector k
i

is selected by
Bragg diffraction from a monochromator crystal. The monochromator Bragg angles are
labeled in Figure 1 by A1 = ω
m
and A2 = 2θ
m
. The orientation of the vector k
i
in the
reciprocal space of the sample crystal is controlled by orienting the sample with respect
to k
i
by the rotation of the sample table (A3 = ω
s
) and the double goniometer (tilt an-
gles) or Eulerian cradle. The modulus of scattered wave vector k
f
is selected by the Bragg
diffraction from the analyzer crystal (A5 = ω
a
and A6 = 2θ
a
) and its orientation in the
reciprocal space of the sample is determined by the scattering angle (A4 = 2θ
s
)atthe
sample position. Figure 2 shows the reciprocal space diagram corresponding to the spec-
trometer configuration of Figure 1. The magnitudes of the initial and final wave vector
are not equal since we are interested in measuring a finite energy transfer
¯

hω. The total
momentum transfer Q is decomposed into a reciprocal vector τ
hkl
and a wave vector q.
One can measure a collective excitation with a dispersion ω(q) with such a spectrometer
configuration. The measurement is done at the [hkl] Brillouin zone of the reciprocal space.
In principle the dispersion can be measured in any Brillouin zone but the intensity will
be different. In magnetic samples the intensity is severely reduced as we go further away
from the center of the reciprocal lattice due to the magnetic form factor. One can access
the same (Q,ω) point using infinite number of alternative combinations of k
i
and k
f
.This
has been illustrated by the dotted and dashed lines in Figure 2. However the intensity and
resolution characteristics are different for these alternative configurations and therefore a
proper choice of k
i
and k
f
is important for the measurement.
Two types of scan methods are normally employed: (1) constant-Q and (2) constant-E.
In the constant-Q method the spectrometer is set to a particular Q, which is kept fixed
Magnetic excitations 249
Fig. 2. The solid lines represent the reciprocal space representation of the inelastic measurement with a TAS
corresponding to Figure 1. The dotted and dashed lines represent alternative configurations leading to the
same (Q,ω) (from Currat [3]).
during the scan (hence called constant-Q) and the energy transfer is varied. Also these
scans are usually performed in two modes: (1) constant-k
i

or (2) constant-k
f
mode, the
later being more frequently used. Figure 3 illustrates these two ways of constant-Q scans,
keeping k
i
=|k
i
| constant in the first case and k
f
=|k
f
| in the second case.
The constant-Q scan is the most common mode of scan because the data collected in
this mode can be directly related to the dynamical susceptibility of the magnetic sample
investigated. The model calculations to which one likes to compare the experimental data
are usually given in terms of the dynamical susceptibility χ(Q,ω)at high symmetry points
in the reciprocal space. Also the integrated intensity of a constant-Q scan can give a direct
measure of S(Q,ω), multiplied by resolution volume associated with the analyzer arm of
the spectrometer. If the scattered neutron wave vector k
f
is held fixed (constant-k
f
mode),
so that the energy transfer is varied by varying k
i
, then the phase space volume remains
constant during the scan. In constant-E scans the spectrometer is set to detect a particular
energy transfer corresponding to the energy of the spin wave and Q is varied in a particular
direction in reciprocal space. The choice between these two scans is dictated by the slope

of the spin wave dispersion and the form of the resolution ellipsoid of the spectrometer,
which will be discussed in the next section.
Apart from the neutron source which is usually a reactor, the monochromator crystal
(or assembly of crystals) is the most important component of the triple-axis spectrometer
that determines the neutron intensity incident on the sample. The monochromator crys-
tal selects a specific neutron wavelength from the incident polychromatic neutron beam
from the reactor by Bragg diffraction from a given set of lattice planes of the crystal. The
choice of the monochromator crystal is mainly dictated by the maximum reflectivity and
less higher-order wavelength contamination. For maximum reflectivity it is desirable to
have crystals with small unit cell volumes, large neutron scattering lengths and low ab-
sorption coefficients. To keep the background low it is desirable to have monochromator
crystals with large Debye temperatures (rigid lattice) and small incoherent scattering cross-
sections. Phonons and incoherent scattering from the monochromator or analyzer crystals
250 T. Chatterji
Fig. 3. Upper panel: Illustration of a constant-k
i
constant-Q scan. Lower panel: Illustration of a constant-k
f
constant-Q scan (from Currat [3]).
give signals at unwanted and unintended wavelengths. Certain crystal lattices are suitable
for suppressing higher-order wavelength contamination. For example, the 111 reflection
from the diamond type structure (shared by silicon and germanium) will have no λ/2 con-
tamination because of the forbidden 222 reflection of this type of crystal structure. Another
important characteristic of the monochromator is the mosaic width. The horizontal mosaic
width should be consistent with horizontal collimation, typically from 20

to 40

, while
the vertical mosaic width should be as narrow as possible. An ideal monochromator is

pyrolytic (or oriented) graphite (PG) which has highly preferred orientation of the (00l)
planes, but all other (hkl) planes are oriented at random giving rise to powder peaks (De-
bye rings). PG(002) is also often used as an analyzer crystal. PG is also used as a filter for
higher-order wavelength contamination. Other typical monochromator crystals used are
Be(002), Cu(111), Cu(200), Cu(220), Ge(111), Si(111) and Zn(002). For a more complete
list the readers can consult Table 3.1 of Shirane et al. [2]. High intensity at the detector
can be achieved by focusing the beam by curved monochromator and analyzer crystals.
The vertical focusing of the monochromator crystal is commonly used since good Q res-
olution within the scattering plane is desired, while poor resolution in the vertical plane is
tolerated. However, the analyzer crystal is often horizontally curved. The focusing of the
Magnetic excitations 251
monochromatic beam is more frequently achieved by an assembly of small single crystal
pieces oriented in such a way as to focus the monochromatic beam on to the sample. The
nature and characteristics of the crystal used as analyzer are similar to those of the mono-
chromator. The analyzer d-spacing must be adapted to the scattered neutron energies to be
analyzed and the energy resolution required.
The detector is generally a simple
3
He-gas proportional counter. A counting efficiency of
about 80–95% in the relevant neutron energy range is achieved by choosing the thickness
of the counter and the gas pressure. Unlike the highly collimated X-ray beam from the syn-
chrotron sources the neutron beam from the reactor emerge in all directions. Although the
Bragg diffraction from the monochromator and the analyzer crystals puts some constraint
on the angular divergence of the beam, it is necessary to have additional adjustable control
of the beam divergence. The horizontal beam divergence in the scattering is typically con-
trolled by the Soller collimators. The horizontal collimators are normally placed before the
monochromator, before the sample, before the analyzer and also before the detector. How-
ever care should be taken regarding the compatibility of the collimation with the focusing.
A variety of other devices can be inserted in the neutron beam, viz. adjustable diaphragms,
low efficiency counters to monitor the intensity of the beam incident on the sample (M1) or

the analyzer (M2), filters (oriented PG, polycrystalline Be or BeO, resonance filters, etc.)
to eliminate higher-order wavelength contaminations. Cooled polycrystalline Be or BeO
are generally used for the cold triple-axis spectrometers to eliminate all neutrons above the
energy corresponding to the Bragg cut-off. The elimination of neutrons with unwanted en-
ergies by the filters are achieved by the Bragg diffraction process. This implies that, unless
carefully shielded, the filters may cause increase in the background signal.
2.2. Intensity and resolution function of TAS
The determination of the resolution function of a TAS spectrometer is quite complicated.
Here we give the definition of the resolution function and some simple relations between
the intensity measured during a scan and the norm of the resolution function following
Currat [3] and Dorner [4] closely. The intensity or the neutron counts recorded by the de-
tector for a given spectrometer configuration corresponding to the nominal values (Q
0
,ω
0
)
is given by
I(Q
0
,ω
0
) =N

J(k
i
, k
f
)dk
i
dk

f
(3)
with
J(k
i
, k
f
) =A(k
i
)p
i
(k
i
)S(Q,ω)p
f
(k
f
), (4)
where A(k
i
) gives the spectrum of the source, p
i
(k
i
) and p
f
(k
f
) refer to the transmission
of the monochromator (analyzer) crystal for each incident (scattered) neutron wave vector

and N is the number of scattering particles in the irradiated sample volume. The variables
(k
i
, k
f
) and (Q,ω) are related through the energy and momentum conservation relations
252 T. Chatterji
given by equation 1. The integral on the right-hand side of (3) can be performed first over
the variables k
i
and k
f
at fixed Q and ω and subsequently over the variables Q and ω. Thus
I(Q
0
,ω
0
) = N

S(Q,ω)dQdω

A(k
i
)p
i
(k
i
)p
f
(k

f
)δ(Q −k
i
+k
f
)
×δ

ω −
¯
h
2
(k
2
i
−k
2
f
)
2m
n

dk
i
dk
f
(5)
or
I(Q
0

,ω
0
) =N

A(k
i
)


R(Q −Q
0
,ω−ω
0
)S(Q
0
,ω
0
)dQ dω, (6)
where the resolution function R(Q −Q
0
,ω−ω
0
) is given by
R(Q −Q
0
,ω−ω
0
)
=


p
i
(k
i
)p
f
(k
f
)δ(Q −k
i
+k
f
)δ

ω −
¯
h
2
(k
2
i
−k
2
f
)
2m
n

dk
i

dk
i
. (7)
The norm of the resolution function is given by

R(Q −Q
0
,ω−ω
0
)dQ dω =

p
i
(k
i
)p
f
(k
f
)dk
i
dk
f
=V
I
·V
F
, (8)
where
V

I
=

p
i
(k
i
)dk
i
,
V
F
=

p
f
(k
f
)dk
f
.
(9)
We note that the resolution function given by (7) is the convolution of the two distribution
functions p
i
(k
i
) and p
f
(k

f
) in the reciprocal space and the norm of the resolution function
given by (8) is the product of two integrated distributions V
I
and V
F
. Equations (3)–(9) are
quite general and can be equally applied to TAS and TOF spectrometers.
The integrals V
I
and V
F
can be determined as a function of the characteristics of the
monochromator and analyzer crystals and the angular divergence of the neutron beam. For
a nonfocusing flat mosaic monochromator crystal one gets [4,5]
V
I
= P
m
(k
I
)k
3
I
cotθ
m
(2π)
3/2
×
β

0
β
1

4sin
2
θ
m
η
2
m
+β
2
0
+β
2
1
η
m
α
0
α
1

4η
2
m
+α
2
0

+α
2
1
(10)
Magnetic excitations 253
with
k
I
=V
−1
I

k
i
p
i
(k
i
)dk
i
, (11)
where α
0
,α
1
,β
0
and β
1
are horizontal and vertical collimations before and after the mono-

chromator, η
m
and η

m
are the horizontal and vertical mosaic widths of the monochromator
crystal and P
m
(k
I
) is the peak reflectivity of the monochromator crystal. The analogous
expression for V
F
can be similarly obtained. In the Gaussian approximation the four-
dimensional resolution function R(Q −Q
0
,ω−ω
0
) can be given by
R(Q −Q
0
,ω−ω
0
)
=R
0
(Q
0
,ω
0

) exp

−
1
2
4

k=1
4

=1
M
kl
(Q
0
,ω
0
)X
k
X


, (12)
where the matrix M
kl
defines the resolution ellipsoid and the four coordinates X
k
are linear
combinations of Q −Q
0

and ω −ω
0
.
We shall show below that the norm of the resolution function can give the integrated in-
tensity of the excitation mode under reasonable approximations. During a constant-Q scan
across a dispersion curve, shown schematically in Figure 4 the nominal value of the energy
transfer ω
0
varies at each point and therefore both R
0
(Q
0
,ω
0
) and M
kl
(Q
0
,ω
0
) vary as
well. Thus the resolution ellipsoid actually changes at each point of the scan. However in
order to measure the integrated intensity of the mode we can neglect these changes to a
very good approximation and assume that the resolution ellipsoid is translated along the
ω axis without deformation during a constant-Q scan. Thus the integrated intensity of the
Fig. 4. Schematic illustration of a constant-Q scan across a dispersion surface in a focused condition (from Currat
[3]).
254 T. Chatterji
mode can be determined as


I(Q
0
,ω
0
)dω
0
∼NA

k
∗
I


R

Q − Q
0
,ω−ω
∗
0

dQdω ∼ NA

k
∗
I

V
∗
I

V
∗
F
, (13)
where “
∗
” refers to the point of maximum overlap between the resolution ellipsoid and the
dispersion surface.
We give another example in which the integrated intensity is related directly to the norm
of the resolution function. This occurs in the case of a slowly varying scattering function on
the scale of the resolution ellipsoid, viz. diffuse scattering in both Q and ω. The integrated
intensity of the diffuse scattering can be given by
I(Q
0
,ω
0
) = NA(k
I
)

R(Q −Q
0
,ω−ω
0
)S(Q,ω)dQ dω
≈ NA(k
I
)

R(Q −Q

0
,ω−ω
0
)S(Q
0
,ω
0
)dQ dω
= NA(k
I
)V
I
V
F
S(Q
0
,ω
0
). (14)
The intensity measured in a constant-Q scan should be corrected for the variation of the
norm of the resolution function. However some simplifications arise for constant-Q scans
in constant-k
f
and constant-k
i
modes. For a constant-Q scan obtained in the constant-k
f
mode, V
F
is constant during the scan and therefore the intensity data have to be corrected

for the variation of A(k
I
)V
I
. Since A(k
I
)V
I
k
I
measures the flux of the neutron beam at
the sample [5], it is sufficient therefore to normalize the intensity by using a monitor with
a count rate is proportional to 1/k
I
before the sample. For a constant-Q scan obtained in
a constant-k
i
mode, A(k
I
)V
I
remains constant but V
F
varies. Therefore one must make
corrections for the variation of V
F
which is proportional to k
3
f
cotθ

a
as can be seen from
the analogue of (10) for V
F
. The peak reflectivity of the analyzer crystal P
a
(k
f
) may vary
rapidly due to the parasitic multiple Bragg reflections.
2.3. Size and shape of the resolution function
We have already noted that the resolution function given by (7) is the convolution of the
two distributions p
i
(k
i
) and p
f
(k
f
). The size and shape of the resolution function are com-
pletely given by the size and shape of these two distributions and by the value of the scat-
tering angle 2θ
s
(A4) which controls the way in which the two distributions are combined.
The size and shape of distribution p
i
(k
i
) depend on the Bragg angle of the monochroma-

tor θ
m
, the horizontal (α
0
,α
1
) and vertical (β
0
,β
1
) beam collimations before and after the
monochromator and the horizontal and the vertical mosaic widths (η
m
,η

m
) of the mono-
chromator crystal. The size and shape of the distribution p
f
(k
f
) depend on a similar set
of independent parameters (α
2
,α
3
,β
2
,β
3

,η
a
,η

a
). It is convenient to choose a coordinate
Magnetic excitations 255
system defined relative to Q
0
with Q

along the direction of Q
0
, Q
⊥
perpendicular and in
the scattering plane and Q
z
perpendicular to the scattering plane in the vertical direction.
The calculation of the resolution function is quite involved. However, if one uses Gaussian
approximation, i.e., if one assumes that collimator transmission functions and the mosaic
distributions of the monochromator and the analyzer crystals are Gaussian, it is possible
to derive an analytic formula for the resolution function expressed as a four-dimensional
Gaussian distribution given by (12) as already noted. We do not go into the detailed calcu-
lations of the resolution matrix. Interested reader may consult the original paper by Cooper
and Nathans [5] taking care of a few mistakes pointed out by Dorner [4] or the Appendix 4
of the book by Shirane, Shapiro and Tranquada [2]. In general the resolution matrix M
kl
is not diagonal, and hence the principal axes of the resolution ellipsoid do not coincide
with the axes defined by Q

0
,ω
0
. If the incident beam divergence is small then the resolu-
tion in the vertical direction (δQ
z
) is uncoupled to the other three coordinates. Hence the
matrix M separates into a 3 × 3 matrix coupling δω, δQ

and δQ
⊥
and a 1 × 1matrix
for δQ
z
.
While measuring a dispersive excitation by a constant-Q scan, it is of practical impor-
tance to optimize the orientation of the resolution ellipsoid such that the two longer axes of
the resolution ellipsoid are parallel to the dispersion surface. In this “focused” condition,
the measured width of the excitation is small. At the opposite extreme, with the longest
axis of the resolution ellipsoid orthogonal to the dispersion surface, the peak may be so
wide as to be undetectable. One does not usually attempt to achieve a perfect focusing but
simply choose between more focused and less focused measurement condition. For gen-
eral rules for making this choice the reader is referred to the book by Shirane, Shapiro and
Tranquada [2].
In order to gain intensity most of the modern triple-axis spectrometers use large curved
monochromator and/or curved analyzer crystals in open geometry, i.e., without Sollar
collimators. Many of the comments and conclusions given above are not strictly valid
for such spectrometers. The calculations of the resolution matrix become more complex.
Popovici [6] and Popovici et al. [7] reformulated the procedure of calculations of the res-
olution matrix of the triple axis spectrometers to make allowance for the spatial config-

uration of the experimental set-up and for the curvature of the monochromator and the
analyzer crystals. The concept of two independent distributions p
i
(k
i
) and p
f
(k
f
) is no
longer valid in this case and calculations become too complex to gain any qualitative es-
timate of the size and shape of the resolution ellipsoid. The situation can be only handled
by a computer program which can readily display graphically the resolution ellipsoid for
a particular set of spectrometer parameters and also capable of simulating scan profiles
from the approximate knowledge of the dispersion of the excitations to be investigated.
Fortunately such a program (RESTRAX, Saroun and Kulda [8]) for the calculation of TAS
resolution and also for the simulation of scan profiles is already available. This program is
based on the commonly used formalism of Cooper and Nathans (program RESCAL) com-
bined with the more recent TRAX code written by Popovici et al. [7] by using the transfer
matrix formalism [6,9]. The Monte Carlo (MC) ray-tracing simulation procedure has also
been implemented in this program. The program package including user and installation
guide is available with an anonymous ftp at the server ftp.ill.fr.
256 T. Chatterji
2.4. TAS multiplexing
The triple-axis spectrometry is notoriously a time consuming slow technique. So there have
been efforts to increase the data acquisition rate by multiplexing the secondary spectrome-
ter. This is done by operating several analyzer–detector arms in parallel. The price to pay is
of course to give up part of the selectivity of the conventional triple-axis spectrometer, i.e.,
most of the collected data will not correspond to a constant-Q scan at preselected high-
symmetry points or a high-symmetry direction in the reciprocal space. However, some-

times it is worthwhile to have a coverage of a considerable part of the (Q,ω) space and
this can be achieved. There exist several ways of multiplexing the TAS spectrometer. The
first method is to use a multiblade analyzer in combination with a two-dimensional position
sensitive detector (PSD). The RITA (Re-Invented Triple-Axis) spectrometer previously at
the RISO reactor and now installed at Paul Scherrer Institute in Switzerland, as well as
SPINS spectrometer at NIST in USA belong to this class of instruments. The (Q,ω) space
covered by such instruments is more or less continuous but limited to the neighborhood
of a preselected point (Q
0
,ω
0
). This is also called a local reciprocal space imaging. The
main difficulty of this method is that it can lead to a high background level and conse-
quently a low signal-to-noise ratio. Therefore it is important to enclose the multianalyzer
and PSD in a common evacuated protection and also to design the analyzer mount in such
a way as to minimize the parasitic scattering processes. A comprehensive review of this
technique is given by Lefmann et al. [10]. The other method is to have a set of independent
analyzer–detector arms, tuned to transmit the same final neutron energy covering an angu-
lar range as wide as possible (60–90
◦
). This is the principle of the Multi-arm Analyzer–
Detector (MAD) spectrometer described by Demmel et al. [11]. It is possible to perform a
constant-Q scan with one of the analyzer–detector arms only, while the other arms describe
mixed trajectories in the (Q
0
,ω
0
) space. The MAD spectrometer can be quite effectively
used for mapping magnetic excitations in a low-dimensional system. Figure 5 shows the
mapping of an one-dimensional dispersion with a multiple analyzer detector (MAD).

An alternative way of multiplexing a TAS secondary spectrometer is illustrated in Fig-
ure 6. This is the so-called flat-cone geometry, which has been successfully implemented at
the Hahn-Meitner-Institut in Berlin and a similar set-up is under construction at the Institut
Laue–Langevin, Grenoble.
2.5. Time-of-flight spectrometers
The main disadvantage of a triple-axis spectrometer is that it can only examine one posi-
tion at a time in the (Q,E) space. However as has already been discussed above, attempts
have been made to explore a part of the reciprocal space by TAS “multiplexing”. Time-of-
flight (TOF) spectrometers are capable of collecting energy spectra simultaneously for a
wide range of wave vectors by employing detector arrays. They are very useful for mea-
suring the dynamical response of the sample when it is wide in energy or Q or in cases
where the dispersion is very small (viz. crystal-field excitations). This technique is mainly
employed to study polycrystalline and amorphous samples. However, time-of-flight spec-
trometers viz. HET and MARI at the spallation (pulsed) neutron source ISIS have been
Magnetic excitations 257
Fig. 5. Mapping of an one-dimensional dispersion with a multiple analyzer-detector (MAD) (from Demmel
et al. [11]).
recently proved to be quite useful to investigate spin wave dispersions in magnetic crys-
tals, especially in low-dimensional magnetic samples. They have the added advantage of
studying spin wave excitations at very high energy transfers not possible at the reactor
neutron sources. Some investigations of the spin wave dispersion have also been recently
performed on the time-of-flight spectrometers IN6 and IN4 situated at the reactor of the
Institut Laue–Langevin in Grenoble.
Neutron time-of-flight spectrometers [13] yield information about the change in neutron
energy caused by the elementary excitations of the sample by measuring the time that
a neutron takes to reach the sample from a known starting point and also that to cover
the distance from the sample to the detector after the scattering process. The total time is
compared with the known flight time of the neutrons that are scattered from the sample
elastically, i.e., without change in energy. For this purpose one needs a sharp neutron pulse
with known start time and location in the primary spectrometer. Additionally one needs to

know (also fix) either the incoming neutron energy (or velocity) or the outgoing neutron
energy (velocity) after the scattering process. Accordingly the spectrometer is called a
direct-TOF (d-TOF) or an inverted-TOF (i-TOF) spectrometer. For a d-TOF spectrometer
a monochromatic beam hits the sample and the scattered polychromatic beam is counted
as a function of the flight time from the chopper to the detector, the flight time (before
the scattering process) from the chopper to the sample being fixed to a preselected known
258 T. Chatterji
Fig. 6. The flat-cone set-up for excitation mapping on a triple-axis spectrometer. Only seven analyzer channels
are drawn for clarity. Each channel contains two pairs of crystal analyzers and detectors in vertical scattering
geometry (from Kulda [12]).
value. For an i-TOF spectrometer a polychromatic beam hits the sample, but only a fixed
neutron velocity (energy) is accepted after the scattering process. In this case the flight
time from the sample to the detector is known. So from the total flight time from the
chopper to the detector it is possible to determine the initial energy and the energy transfer
caused by the scattering process. TOF instruments for which the monochromatic incident
beam is produced by choppers, i.e., by TOF and also for which the neutrons are detected
according to their flight time on the secondary part of the spectrometer are called TOF–
TOF instruments. If instead, monochromator or analyzer crystals (X) are used for energy
determination, the spectrometers are called X-TOF or TOF-X spectrometers.
Magnetic excitations 259
3. Spin waves in localized electron systems
3.1. Spin waves in Heisenberg ferromagnets
Spin dynamics of localized electron systems can be well described by the Heisenberg
Hamiltonian
H =−

l,l

J


l −l


S
l
·S
l

−gµ
B
H

l
S
z
l
, (15)
where it is assumed that a magnetic field H is applied in the z direction. J(l − l

) is the
exchange interaction parameter between the spins located at the site l and l

, g is the gyro-
magnetic ratio and µ
B
is the Bohr magneton. Introducing the spin angular momentum op-
erators S
±
=S
x

±S
y
and making a linear approximation (valid for small spin deviations)
one can derive the expression for the spin wave energy as a function of the momentum
transfer q
¯
hω =gµ
B
H +2S

J (0) −J (q)

, (16)
where
J (q) =

l
J(l) exp(iq ·l). (17)
This is the equation for the spin wave dispersion. For nearest-neighbor exchange interac-
tion J
J (0) = rJ,
J (q) = rJγ
q
(18)
with
γ
q
=
1
r


ρ
exp(iq ·ρ), (19)
where r is the number of nearest neighbors and ρ is the vector connecting nearest neighbor
atoms. If q is small
γ
q
1 −
1
6
q
2
ρ
2
(20)
so that
¯
hω =gµ
B
H +Dq
2
, (21)
260 T. Chatterji
where
D = 2JSa
2
(22)
for a cubic system with a lattice constant a. So we expect the spin wave dispersion of a
Heisenberg ferromagnet to be quadratic in q for small q. This is in contrast to the phonon
dispersion which is linear in q at small q. The Heisenberg antiferromagnets have also a

linear dispersion at small q.
We will now borrow the expression for the neutron scattering cross-section for an as-
sembly of unpaired electrons derived in Chapter 1. This is given by
d
2
σ
dΩ dE

=

γe
2
mc
2

2

1
2
gF(Q)

2

k

k

exp

−2W(Q)


×

α·β
(δ
αβ
−Q
α
Q
β
)
1
2π
¯
h

∞
−∞
dt exp(iωt)

S
α
Q
(0)S
β
−Q
(t)

. (23)
We recall that γ is the nuclear magneton, e is the electronic charge, m is the mass of

electron and c is the velocity of light, k and k

are the incoming and outgoing wave vector
of the neutron, Q = k

−k is the scattering vector and F(Q) is the magnetic form factor,
which is the Fourier transform of the spin density of the magnetic electrons. S
α
Q
(0)S
β
−Q
(t)
is a spin correlation function between the spin with Cartesian coordinate α at time 0 and the
spin with Cartesian coordinate β at time t. The brackets denote the thermal average. The
term exp[−2W(Q)] is the Debye–Waller factor. The spin raising and lowering operators
are given by
S
+
Q
=S
x
Q
+iS
y
Q
, (24)
S
−
Q

=S
x
−Q
−iS
y
−Q
. (25)
The neutron scattering cross-section can be rewritten in terms of spin raising and lowering
operators as
d
2
σ
dΩ dE

=

γe
2
mc
2

2

1
2
gF(Q)

2

k


k

exp

−2W(Q)

×


1 −Q
2
z

1
2π
¯
h

∞
−∞
dt exp(−iωt)

S
z
Q
(0)S
z
−Q
(t)


(26)
+

1 +Q
2
z

1
2π
¯
h

∞
−∞
dt exp(−iωt)
×
1
4

S
+
Q
(0)S
−
−Q
(t) +S
−
−Q
(0)S

+
Q
(t)


. (27)
Here the neutron scattering cross-section has been broken into two parts: (1) the part mul-
tiplied by (1 −Q
2
z
) is the longitudinal part of the cross-section, while the term multiplied
Magnetic excitations 261
by (1 + Q
2
z
) is the transverse part. In the linear spin wave approximation S
z
(t) = S
z
(0),
therefore the longitudinal part leads to elastic scattering and does not concern us here. So
the transverse neutron scattering cross-section from spin waves from a crystal lattice at low
temperature can be given by
d
2
σ
dΩ dE

=


γe
2
mc
2

2

1
2
gF(Q)

2

k

k

exp

−2W(Q)

×
1
4

1 +Q
2
z

1

2π
¯
h

∞
−∞
dt exp(−iωt)
×

ll

exp

iQ ·

l −l


S
+
l
(0)S
−
−l

(t) +S
−
−l
(0)S
+

l

(t)

. (28)
The neutron scattering intensity for spin wave creation is related to that of spin wave anni-
hilation by the relation known as the principle of detailed balance given by

d
2
σ
dΩ dE


Q,ω
=exp

¯
hω
k
B
T

d
2
σ
dΩ dE


−Q,ω

. (29)
Here k
B
is the Boltzmann constant and T is the temperature. Now we are only left with the
evaluation of the spin correlation function for the lattice for single spin wave creation or
annihilation. By doing this the scattering cross-section can be written as

d
2
σ
dΩ dE


±
=

γe
2
mc
2

2

1
2
gF(Q)

2

k


k

exp

−2W(Q)

1 +Q
2
z

1
2
S
×
2π
3
v
0

q,τ

n
q
+
1
2
±
1
2


δ(
¯
hω
q
∓
¯
hω)δ(Q ∓q − τ ), (30)
where v
0
is the unit cell volume, τ is the reciprocal lattice vector and n
q
is given by
n
q
=
1
exp(
¯
hω/k
B
T)−1
. (31)
Two of the best examples of ferromagnetic insulators for which the Heisenberg Hamil-
tonian is appropriate is EuO and EuS. Both these compounds have been investigated exten-
sively by neutron scattering. However, in order to apply the spin wave theory to the neutron
scattering results on EuO and EuS, one has to generalize equation (16) to include dipolar
effects. This has been worked out by Holstein and Primakoff [14]. The spin wave energies
including dipolar effects are given by
¯

hω =

gµ
B
B +2S

J (0) −J (q)

×

gµ
B
B +2S

J (0) −J (q)

+gµ
B
M sin
2
θ
q

1/2
, (32)