Atomic Energy of Canada Limited
THE DEPENDENCE OF THE INTEGRATED INTENSITY
OF A SCATTERED NEUTRON GROUP
ON THE EXPERIMENTAL CONDITIONS
by
V.F. SEARS and G. DOLLING
Chalk River, Ontario
January 1972
AECL-4133
THE DEPENDENCE OF THE INTEGRATED INTENSITY OF A SCATTERED
NEUTRON GROUP ON THE EXPERIMENTAL CONDITIONS
V . F , SEARS & G. DOLLING
ABSTRACT
The accurate determination of phonon or magnon
dispersion relations by inelastic neutron scat-
tering with a triple-axis crystal spectrometer
is facilitated by choosing scanning modes which
give sharply focused scattered neutron groups.
Since the optimum scanning mode is in general
not a constant-Q mode, the integrated intensity
of a group should be corrected for its depen-
dence on the scanning mode before being used to
determine, for example, the polarization vector
of the phonon or magnon. The necessary correc-
tion factor is derived in this report and dis-
cussed for a number of cases of practical
interest.
Chalk River Nuclear LaboratoriesChalk River3 Ontario, Canada
Jar, iry, 19 72
AECL-4133
L'intensité intégrée d'un groupe neutronique diffusé
dépend des conditions expérimentales
par
V.F. Sears et G. Dolling
Résumé
La détermination précise des relations de dispersion
du phonon ou magnon, effectuée par diffusion neutronique
inëlastique au moyen d'un spectromètre à cristal à trois axes,
est facilitée si l'on choisit des modes de balayage donnant des
groupes neutroniques diffusés nettement focalisés. Etant donné
que le mode de balayage optimal n'est généralement pas un mode
constant-^, l'intensité intégrée d'un groupe devrait être
corrigée, pour le fait qu'elle dépend du mode de balayage,
avant d'être employée pour déterminer, par exemple, le vecteur
de polarisation du phonon ou du magnon. Le facteur de correction
nécessaire est donné dans ce rapport et commenté pour un certain
nombre de cas ayant un intérêt pratique.
L'Energie Atomique du Canada, Limitée
Laboratoires Nucléaires de Chalk River
Chalk River, Ontario
AECL-4133
1. INTRODUCTION
The double differential cross section for the
scattering of a neutron by a macroscopic system such as a
crystal, from an initial state with wave vector 5c . to a finalJ-
state with wave vector k\, can be expressed in the form1
d M ^ - = kT SCS.a), (1)
where S(Q,w) is the scattering function in which $ and CD
denote respectively the momentum and energy in units of Ti
which are transferred from the neutron to the crystal in the
scattering process:
Q = £ ± -fiki
2 -nkf2
25 25
where m is the neutron mass. To the extent that multiple
scattering is negligible, the scattering function defined by
(1) depends only on Q and ID and not on k. and k_ separately.
This permits a considerable amount of flexibility in choosing
the experimental conditions under which the scattering func-
tion is measured.
The scattering function for a non-magnetic
crystal is characteristic of the spatial arrangement and
vibrational motion of the nuclei in the crystal. If ui(§)
denotes a branch of the dispersion relation for the normal
- 2 -
modes of vibration of the nuclei then, in the neighbourhood
of the point co = a>($) , the scattering function takes the form
+ S'($,ai), (3)
in which the first term arises from the scattering of a neutron
by the creation of a phonon of wave vector Q and frequency
u)($) and the delta function expresses conservation of energy.
The factor A(Q) depends, among other things, on the polariza-
2 3tion vector of the phonon ' . The second term in (3) arises
from incoherent scattering and multi-phonon coherent scattering.
The scattering function also has a delta-func-
tion term at u = -w(Q) associated with phonon annihilation
proceases. This term need not be considered separately, how-
ever, because the results obtained below for phonon creation
can be extended to phonon annihilation simply by replacing
w(Q) by -u)(Q) . Additional delta-function terms appear in the
scattering functions of ordered magnetic crystals as a result
of magnon creation and annihilation processes.
To obtain the maximum amount of information
about interatomic forces and/or magnetic interactions in cry-
stals from inelastic neutron scattering experiments, it is
desirable to measure not only a)(§) but also A(§) . This report
is concerned essentially with how these quantities can best
be determined, taking advantage of the focusing properties of
the triple-axis spectrometer and the flexibility in the choice
of experimental conditions referred to above.
- 3 -
It should be noted that the phonon or magnon
wave vector Q is arbitrary to the extent of the subtraction
of 2ir times any vector T of the reciprocal lattice of the
crystal. It is customary to define a reduced wave vector
q = Q-2TTT, where T is chosen so that q has the smallest pos-
sible magnitude. This convention makes no difference to the
arguments presented in this report, however5, and so for
simplicity we shall use Q throughout to refer to the phonon
(magnon) wave vector. The quantity A(Q) is in general not a
simple periodic function of the reciprocal lattice, and
measurements at several different Q, but the same q, are nor-
mally required to determine all the branches of the multi-
valued function OJ(Q) .
We begin in Section 2 with a brief review of
the theory of the triple-axis spectrometer in order to define
unambiguously what is meant by a "scanning mode" and by the
"intensity of a scattered neutron group". The integrated
intensity of a scattered neutron group is calculated in Section
3 for an arbitrary scanning mode assuming ideal instrumental
resolution. The effect of finite instrumental resolution ar.d
associated focusing effects " is discussed in Section 4.
Finally, a few concluding remarks are presented in Section 5.
2 . TRIPLE-AXIS SPECTROMETER2 2
The cross section d cr/dftdw,. is defined such that d a
equals the number of neutrons per unit time per unit incident
flux which are scattered into the solid angle dft with energy
in the interval (wf,wf+do)f) as illustrated in Fig. 1. To
measure S($,u)) we must therefore produce a collimated mono-
energetic beam of neutrons, allow it to fall on the crystal
and observe the scattered neutrons with an energy-sensitive
detector. The count rate (number of neutrons per unit time)
registered by the detector is given essentially by
R =
=
ddft
i
2c
dt
I
1 A
°f
fiAw
fiAwf F ( k
f F(ki)n
±)n
( k f
(k
)S
f )
(?, w ) ,
where F(k.) is the incident flux, AQ the solid angle sub-
tended by the detector at the crystal, Ao.y the band-width of
the detector and n(k.r) its efficiency.
9 10For the triple-axis spectrometer ' , which is
represented schematically in Fig. 2, the incident beam is
produced by a single-crystal monochromator and the energy-
sensitive detector consists of a single-crystal analyzer which
diffracts scattered neutrons with a preselected wave vector
k^ into a suitable proportional counter. The setting of the
spectrometer is characterized by the Bragg angles 0M and 6.
for the monochromator and analyzer, by the angle of scattering
- 5 -
<J>, and by the angle ty which defines the orientation of the
crystal sample relative to the incident beam. For given
sets of reflection planes the angles 0 and 0. determine k.
and k^ respectively while the angles \j) and $ determine the
directions of k. and k^ relative to the crystallographic
axes of the sample.
The incident beam is monitored by a low-sensi-
tivity fission counter. Since the monitor produces only a
small attenuation of the beam its count rate is given by
Rw=a(k.)F(k-)Ne, (5)M 1 1 '
where a(.k.) is the reaction cross section per atom in the
monitor, N the number of such atoms and e the fraction of
reactions which produce an output pulse from the monitor.
Hence the count rate per unit monitor count rate, usually
called simply the intensity, is given by
I = ~ -- C S($,u>), (6)RM
where
r - f f f (7)
^ " kToTkTTNe
The expression (6) ignores a number of effects which cannot
be eliminated entirely from the observed intensity, principally:
finite instrumental resolution, order contamination, background
radiation and various spurious scattering processes. The
- 6 -
effect of finite instrumental resolution will be considered
in Section 4 but the remaining effects will be neglected.
Consequently, the results obtained below apply to observed
intensities only after any necessary corrections for the
remaining effects have been made. In passing we may note
that it is almost always possible to avoid order contamination
effects by a suitable choice of experimental conditions but
that the precision of an intensity measurement is often limited
by the difficulty in subtracting the background radiation in
a reliable manner.
The intensity depends parametrically on the
vectors k. and kf and an inelastic neutron scattering experi-
ment consists in measuring I for a sequence of pairs of values
(k.,kf), i.e. for a sequence of settings (26M,iJ),<l>, 20.) . Such
a scan yields a set of numbers I(t) = I(k.(t),kf(t)) where
t = lj2,-«-,T say. The point number of the scan is denoted
by t in order to emphasize its obvious analogy with time.
The quantity C is independent of the crystal
sample and is therefore a calibration constant for the spec-
trometer. For thermal neutrons e is independent of k. whereas
aCk^) is closely proportional to k7 . Hence, C is independent
of k^ and depends only on k^. For many commonly used modes
of operation 20A, and hence kf, is held fixed throughout an
experimental scan so that C is constant and the intensity is
simply proportional to S($,a.O . The physical content of the
scattering function lies in its variation with § and w, rather
- 7 -
than in its absolute value, so that the value of C is not
usually required. One simply omits the factor C in (6) and
calls I the scattering function in arbitrary units. Modes
of operation in which 20 varies during the scan are compli-
cated somewhat by the need to make allowance for the variation
of n(kf).
Suppose that xyz refers to Cartesian axes
fixed in the crystal sample and that the sample is mounted with
the xy-plane in the plane of the spectrometer so that
kiz = kfz = ^z = °' T h e f o u r a n S l e s 2eM)4
l5*52eA then determine
uniquely the four remaining variables k. ,k. ,kf sk.p . With
26. held fixed the number of independent variables is reduced
to three which can be taken to be Q , Q and OJ. In this case
I(t) = S($(t) ,io(t)) so that from (3)
I(t) = I (t) + I'(t), (8)
where
I (t) = A($(t))6{dj(t)-(o($(t))}s
(9)
The quantity I (t) is called the intensity of the scattered
neutron group for the scanning mode Q(t),u(t).
The observed intensity of a scattered neutron
group is of course not a delta function but has a finite
width. The broadening of the group is due to three distinct
effects:
- 8 -
(i) finite instrumental resolution,
(ii) finite mosaic spread in the sample,
(iii) finite phonon (or magnon) lifetimes.
For a good "single" crystal at low temperatures (ii) and (iii)
are usually negligible compared with (i).
I'(t) may not be a completely smooth function
of t. The better the experimental resolution, however, the
less chance there is for irregularities in I'(t) to occur in
the region of the I (t) peak. It is customary to draw a
smooth line under the peak in order to estimate the peak area.
In what follows we shall assume that this can be done with
reasonable precision, so that we may confine our attention
to I (t).
- 9 -
3. INTEGRATED INTENSITY OF A SCATTERED NEUTRON GROUP
For ideal instrumental resolution the integra-
ted intensity of the scattered neutron group is given by
/"Ta = / I (t)dt
Jo g
= I A($(t))6{ui(t)-co(Q(t))}dt (10)
•Jo
A($(tQ))
where the dot denotes differentiation with respect to t, and
tn is defined by the relation
o)(tQ) = w($(to))s
i.e. tQ is the value of t at which the group occurs.
Introducing the group velocity,
we have
o -
v/here
J =
and
- 10 -
Aw = i(t0) = ai(to + l)-(i)(to), (15)
AQ = $(tQ) = $(to+l)-$(to).
The expression (14) for the Jacobian |J| applies to an arbi-
trary scanning mode. In special cases it can be simplified
further.
I constant-Q scan
If ($ is held constant so that A$ = 0 then J - Aw. For such
a scan it is usual to plot I as a function of u), rather
than t, so that the relevant integrated intensity is
5 = /Igdw = et|Au| = A($). (16)
This direct correspondence between A(§) and the measured
integrated intensity is one of the many excellent reasons
for the popularity of the constant-^ scan.
II constant-w scan
Here w is held constant so that Aw = 0 and J = -A§*v($).
If the direction of ($ is also held constant then I can beO
plotted as a function of Q in which case the relevant inte-
grated intensity is
S = /lgdQ = a|AQ| = ff^ , (17)J S |e-v(Q)|
where e is a unit vector in the direction of §.
- 11 -
III constant-^ scan
Consider the case, which occurs typically in experiments using
time-of-flight spectrometers with fixed detectors, in which £.
and <|> are held constant and i<r.(t) = kf(t)e where e is a unit
vector independent of t. In this case |J| = |Aoif | |J| where |J|
is the Jacobian' of Waller £ Froman ,
With I plotted as a function of u . the corresponding inte
grated intensity is given by
a = I I duf = a|Au
IV general linear scan
For a general linear scan, in which AQ = aAioe where a is a
constant and e a unit vector, we have J = Au>(l-ae«v(($)) so
that, with I plotted as a function of oi,
a-f I dw = a|Aui| = J ^ > m (20)S | l ( $ ) |
Successive steps in such a general linear scan
are illustrated in Fig. 3(a) for the case in which kf remains
constant. The corresponding points in (Q,co)-space for two
See Section S3 however.
- 12 -
such scans are shown as dots labelled IVa and IVb in Fig. 3(b).
The dotted lines I and II represent constant-Q and constant-o)
scans respectively while the full line represents a typical
dispersion curve. The constant-<f> scan (case III), which is
not shown in Fig. 3(b), would give dots lying on a parabola.
By considering typical values for the quantities
which determine the Jacobian, it is readily seen that the inte-
grated intensities a for scans I, II and IVa may differ typi-
cally by amounts up to 50%. The Jacobian vanishes for scan
IVb giving an infinite integrated intensity. This signals
the fact that effects of finite instrumental resolution cannot
be ignored, even in first approximation, for such a tangential
scan.
- 13 -
4. INSTRUMENTAL RESOLUTION & FOCUSING EFFECTS
As a result of finite collimation and finite
mosaic spread in the monochromator and analyzing crystals
the incident beam is, in reality, neither perfectly collimated
nor monoenergetic and the values of A°, and Au)f are not
infinitesimal as was tacitly assumed in (4). Consequently,
if the spectrometer is nominally set for momentum and energy
transfers Q and a), the observed value of I actually arises
from a finite distribution of momentum and energy transfers
centered about these nominal values. In general, therefore,
1 = fd^'du'RC^uj^1 jU^SC^'.u'), (21)= f
where R(^,CJ; Q1 ,03' ) is the resolution function. Cooper £
Nathans have derived an approximate analytic expression for
the resolution function assuming that the transmission func-
tion for a Soller slit collimator and the mosaic spreads of
the monochromator and analyzer are gaussian distributions.
With these assumptions they find, not surprisingly, that the
resolution function is also gaussian:
-3s I M ± j ($,aj)XiXj J.i,j=l >
(22)
where
- 14 -
Xl = V - Qx'X2 = V " V
(23)
A convenient way of visualizing the resolution
function is by considering the locus of points (Q',u)') for
which the resolution function equals one half its maximum
value. According to (22) the locus is an ellipsoid centered
at the point (§,<D) . The important point about the resolution
ellipsoid is not that it is ellipsoidal, which in any case is
7 12only an approximate result, but that it is highly non-spherical'*
As a result the shape of the scattered neutron group,
= rdI (t) = rd^fda)IR(^(t),w(t);$',a,')A($')6{u'-a3(^1)}, (24)
is very sensitive to both the orientation of the resolution
ellipsoid relative to the dispersion surface u = w($) and the
direction in which the resolution ellipsoid is scanned through
the dispersion surface. Depending on these conditions the
group may be sharp (focused) or broad (defocused).
To the extent that one can neglect the variation
in R Q and M^., i.e. the change in the size, shape and orienta-
tion of the resolution ellipsoid, as it is scanned through the
dispersion surface, it follows from (22) that
- $ ' ,01-0)'). (25)
- 15 -
Hence, (24) becomes a convolution which can be expressed
equivalently as
IgCt) = /d5tda)'R($l,a)')A($(t)-$')6-{a)(t)-aJ
I-a3(5(t)-$')}. (26)
The integrated intensity of the scattered neutron group can
then be calculated as before:
f= JT y t ) d t
,a3l) I dt A($(t) -$' )6{u(t) -OJ1 -to($(t)-$') } (27)
where t, = t,(^',cu') is defined by the relation
1 ) - ^1 ) , (28)
so that
tn = t + x 2 + » C29)
1 ° Q(t ) . $ i
in which t is defined by (11) as before. The integrated
intensity can be expressed as
a =
- 16 -
A = A ( Q ( t 1 ) - O ' ) , (31)
J = d)(t1)-i(t1)-v(Q(t1)-$l).
It is always possible, and usually most con-
venient in practise, to choose constant step lengths so that•
do(t) = Aw and (J(t) = A§ are independent of t. In this case
the Jacobian becomes
J = Aw-A(!)*v(^(t, )-($' ) . (32)
If, in addition, the dispersion surface is planar in the
sense that v((^(t, )-($') does not vary appreciably over the
range of the resolution function then (32) reduces to (14)
and
ia = -=— <A>. (33)
Finally, if A($(t.. )-$') is also essentially constant over
the range of the resolution function then (33) reduces to the
expression (13) for ideal resolution.
It will be noted that for a constant-Q scan
in which A$ = 0 the expression (33) is valid even if the
dispersion surface is non-planar. In this case
- 17 -
a = ot | Aa) | = <A(^-^')>
= <A($)-$'«^*A(^) + h $'§' : ^*v>A($) - .•••> (34)
where we have used the fact that <1> = 1 and <Q'> = 0
because the resolution function is even. The first correction
to S due to finite instrumental resolution does not depend
on the detailed form of t]
its second moment <Q'Q'>.
13on the detailed form of the resolution function but only on
- 18 -
5. CONCLUDING REMARKSIf the scattered neutron group is well-focused,
so that the results of Section 3 are applicable, the frequency
w(§) can be obtained from the position of the group and A(Q)
from its integrated intensity. It is clear from Fig. 3(b)
that the peak height of a well-focused group is independent
of the scanning mode so that the width is proportional to the
integrated intensity and, hence, to |j|~ . Fig. 3(b) also
illustrates the fact that a (but not a) is inversely propor-
tional to the step length Aw (or AQ). The step length should
normally be chosen comparable with the band-width in eq. (4):
a finer step length would be redundant while a coarser one
would result in a loss of information.
Generally speaking, the constant-Q scan is
superior to all others because A(Q) then equals the integrated
intensity and the accuracy is not limited by the accuracy
with which v($) can be determined. However, if the constant-^
scan gives a badly defocused group, as often happens when
v(Q) is large, very long counting times followed by extensive
numerical analysis will be necessary to obtain w(^) and A(($)
from the observed intensity. In many cases there will be
other scanning modes for which the group is better focused and
the determination of u)($) and A(§) is more straightforward and
reliable. Graphical methods ' can be employed, if necessary,
to determine the optimum scanning mode.
- 19 -
We conclude with a brief remark concerning
the well-known Jacobian of Waller £ Froman . The expression
given by these authors, and quoted in many subsequent papers,
differs from (18) in that the minus sign is replaced by a
plus sign (for phonon creation processes). We believe that
the minus sign as given in (18) is in fact correct. We have
1M- 15since discovered that other authors * have also obtained
the minus sign without commenting upon the discrepancy with
Waller & Froman.
- 20 -
REFERENCES
1. L. Van Hove, Phys. Rev. 9_5_(1954) 249 .
2. W.M. Lomer S G.G. Low, In Thermal Neutron Scattering,
edited by P.A. Egelstaff (Academic Press, New
York, 1965), p.l.
3. G. Dolling S A.D.B. Woods, Ref. 2, p.193.
4. M.F. Collins, Brit. J. Appl. Phys. 14(1963)805.
5. G. Peckham, AERE Report No. R4380 (1964).
6. R. Stedman £ G. Nilsson, Phys. Rev. 145(1966)492.
7. M.J. Cooper & R. Nathans, Acta Cryst. 23(1967)357.
8. H. Bjerrum Miller S M. Nielsen, in Instrumentation for
Neutron Inelastic Scattering Research. (Interna-
tional Atomic Energy Agency, Vienna, 1970), p. 49.
9. B.N. Brockhouse, in Inelastic Scattering of Neutrons in
Solids & Liquids (International Atomic Energy
Agency, Vienna, 1961), p.113.
10. P.K. Iyengar, Ref. 2, p.97.
11. I. Waller S P.O. Froman, Ark. Fys. 4_ (1952)183.
12. B. Dorner S H.H. Stiller, Ref. 8, p.19.
13. A similar remark has been made by R. Stedman, Ref. 8, p.74,
14. A. Sjolander, in Phonons & Phonon Interactions, edited
by T.A. Bak (Benjamin, New York, 1964), p. 76.
15. B.N. Brockhouse, Ref. 14, p.221.
H-1
Fig. l(a),(b). Two equivalent representations of a particular point of an experi-
mental scan. x and y are axes defined within the specimen, whose orientation with
respect to the incident neutron beam (k.) is specified by I|J. The angle of scatter-
ing is' cj>.
NEUTRON
SOURCE
MONOCHROMATOR
SAMPLEDETECTOR
I
Fig. 2. Schematic diagram of a triple axis crystal spectrometer. C denotes
collimators which define the direction of the neutron beams at various points,
y is an axis defined within the specimen, as in Fig. 1.
(b)
Fig. 3(a). Representation of 5 successive->•
points on a general linear scan in which co and Q•*• ->
change by constant steps AUJ,AQ. e is a unit vec-
tor parallel to AQ. x and y axes are as defined
in Fig. 1.
ETa
Fig,. 3(b). The dotted straight lines
represent various possibilities for the
linear scan illustrated in Fig. 3(a).
I and II are constant-Q and constant
energy scans, respectively. The solid
line is a typical dispersion curve for
excitations in a crystal.
Additional copies of this documentmay be obtained from ,.
Scientific Document Distribution OfficeAtomic Energy of Canada Limited
Chalk River, Ontario, Canada
Price - $1.00 per copy
