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Use of neutron diffraction for the study of domains in nuclear magnetic ordering
M. Goldman
To cite this version:
M. Goldman. Use of neutron diffraction for the study of domains in nuclear magnetic ordering. Journal
de Physique, 1980, 41 (8), pp.885-896. �10.1051/jphys:01980004108088500�. �jpa-00208910�
Use of neutron diffraction for the study of domains
in nuclear magnetic ordering
M. Goldman
Service de Physique du Solide et de Résonance Magnétique, Centre d’Etudes Nucléaires de Saclay,
BP n° 2, 91190 Gif sur Yvette, France
(Reçe le 14 février 1980, accepté le 14 avril 1980)
Résumé.
2014On développe l’analyse théorique de l’influence des domaines dans les systèmes nucléaires ordonnés
sur la diffraction des neutrons. Dans une première partie, on montre que la variation avec l’orientation du cristal de la distribution bidimensionnelle des intensités de neutrons diffractés détermine la fonction de corrélation de la distribution de domaines. Dans une seconde partie on établit, dans un certain modèle, la relation entre la fonction de corrélation et la probabilité de distribution des dimensions de domaines dans diverses directions.
On présente une brève illustration expérimentale.
Abstract.
2014We develop a theoretical analysis of the influence of domains in ordered nuclear spin systems on the neutron diffraction pattern. In a first part, we show that the variation with crystal orientation of the 2-dimen- sional distribution of diffracted neutron intensities yields the correlation function of the domain distribution.
In a second part we establish, within a given model, the relationship between the correlation function and the
probability distribution of domain dimensions in various directions.
A short experimental illustration is given.
Classification Physics Abstracts 75.60G - 75.25
1. Introduction.
-Neutron diffraction has recently
been used for the detection and study of several nuclear magnetic ordered phases-ferromagnetic and antiferromagnetic in single crystals of lithium hydride [1, 2].
A striking différence was observed between the neutron diffraction lines corresponding to magnetic ordering and the crystalline neutron diffraction lines : the rocking curves of the former are much broader
than those of the latter. As an example, the rocking
curve of the 220 crystalline neutron diffraction line in our samples of LiH is Gaussian-like with a width at half intensity of 0.20. In the same samples the
110 diffraction line, corresponding to an antiferro- magnetic ordering of the 1H and ’Li spins, has a nearly Lorentzian rocking curve with a width at half intensity of 0.660 [1]. It was later observed, through
the use of a 2-dimensional neutron multidetector,
that the angular spread of the diffracted beam at a
given crystal orientation was much larger for the
diffraction lines corresponding to ordering than for
the crystalline diffraction lines.
The main factors contributing to the width of a
neutron diffraction line on a single crystal are well
known. They are :
1 ) The distribution of wavelengths of the incident
neutrons,
2) The angular width of the incident neutron beam, 3) The distortions within the crystal, generally
described by the mosaic spread : the crystal is sup-
posed to consist of perfect small blocks slightly
misoriented with respect to one another. The contri- bution to the width of the neutron line is larger, the larger the angular mosaic distribution and the smaller the individual blocks. This factor of broadening is
in general negligible compared with the first two.
The new broadening mechanism which is operative
on the diffraction lines corresponding to ordering
can only arise from a spatial distribution of the
spin-dependent nuclear scattering amplitudes within
the crystal. In order to account for the observed
broadening the scale of this variation must be rela-
tively small (numerical estimates will be given later)
and in fact much smaller than the size of the mosaic blocks. This scale is however still much larger than
the interatomic distance, otherwise it would merely
result in a scattered line so broad as to be lost in the continuous background. The variation of the
spin-dependent nuclear scattering amplitudes over quasi-macroscopic distances is attributed to the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004108088500
existence of magnetic domains in the ordered states.
These domains are unusually small compared with
the domains that are known to exist in other magnetic systems. A likely interpretation for this small domain size is that they are nucleated by the electronic para-
magnetic impurities present in the samples [3].
Another interpretation of the broadening could be
that the neutron diffraction would result from short- range correlations between the spin orientations that
are known to exist in magnetic systems even in the disordered phase in the immediate vicinity of the
transition temperature. Such an interpretation must
be discarded in the present case, for two reasons
examplified with the antiferromagnetic structure giving
rise to the 110 neutron reflection. First, the maximum proton sublattice polarization, determined from the
intensity of the diffraction line, is nearly 80 %, which
proves that the lowest temperature achieved is much below the transition temperature. Secondly, the
width of the rocking curve remains constant during
the warming up of the system until the diffraction line vanishes, whereas the pretransitional correlation
length decreases very rapidly when the temperature is increased above Tc, and would give rise to broader
and broader neutron diffraction rocking curves.
The purpose of this article is to give an analysis of
the influence of domains on the neutron diffraction
line, that enables one to obtain well-defined and detailed information on the distribution of sizes and shapes of the domains from the study of this
diffraction line. Since according to observation the size of the domains is much smaller than that of the mosaic blocks, it is a good approximation to assume
that the crystal is perfect, which brings about an
enormous simplification in the theory.
2. The basic équations.
-We consider a perfectly
collimated and perfectly monochromatic neutron beam hitting a perfect crystal. We first recall briefly
some results of the theory of thermal neutron scat-
tering by nuclei [4, 5].
An incident neutron wave of wave vector k :
scattered by a nucleus at the position R is transformed in to :
where k = k and a is the nuclear scattering ampli-
tude. The amplitude a is isotropic, that is independent
of the relative orientations of k and (r - R). This
property results from the fact that thermal neutron wavelengths, of the order of 10 - 8 cm, are much
larger than the range of the strong neutron-nucleus interaction responsible for the scattering, which is
of the order of 10-13 cm.
In the limit ru » R eq. (1) can be expanded
to the first-order in R )/[ r 1, which yields :
where
is a vector of modulus k pointing in the direction of r.
The neutron wave scattered by a lattice of nuclei of individual scattering amplitudes ai
=a(R;) is of
the form :
where we have assumed that the attenuation of the incident beam by the various scatterings is negligible.
The scattering cross-section in the direction r is
equal to ,2 y§*(r) y§(r) that is :
We consider for simplicity the case when all nuclei
are of the same species, are located on a Bravais
lattice with one nucleus per cell, and have a spin I;
The neutron-nucleus scattering amplitude is of the
form :
where 1 is the nuclear spin and s is the neutron spin.
In the case when the neutron beam is not polarized,
the average of the product ai aj over the state of the
neutron and the nuclear spins is :
Inserting eq. (7) into eq. (5) yields :
The spin-independent and the spin-dependent parts
of the scattering amplitude contribute independently
to the scattering cross-section. In the following, we
consider only the latter, which can be written :
We consider the case when the spin system is ordered. For a ferromagnet ordered along Oz, one
has :
where P is the polarization. For a two-sublattice
antiferromagnet :
depending on the sublattice to which the spin Ii belongs. P is the sublattice polarization. We add
and substract from eq. (9) the term :
and we obtain :
The first term on the right-hand side does not depend on k’, and contributes to the isotropic inco-
herent background. The second term corresponds to
the short-range correlation between the spin orien-
tations : at short distance [ R; - Rj 1, (Ii .Ij) is
not equal to Ii >. Ij >. However, when the pola-
rization P has a sizeable value, the term
is smaller than Ii >. Ij >. Furthermore, except in the immediate vicinity of the transition temperature, the short-range correlation length is small, of the order of a few atomic distances, and as will be seen
later its contribution to the neutron diffraction is a
broad line lost in the continuous background. In
any case, this term will be neglected. The third term
on the right-hand side of eq. (10) describes the cohe-
rent scattering due to the long-range order :
When the order is perfect ;
in a ferromagnet
in an antiferromagnet .
A Bragg scattering is observed when
is a constant independent of the spins i and j. This
is realized when the crystal is oriented with respect
to the incident beam in such a way that k’ - k can
be equal to a vector t of the magnetic reciprocal lattice, i.e. such that :
where the origin is properly chosen.
In practice, the incident beam has a well-defined orientation, i.e. k
=cst. As for the vector i, which
is linked to the crystal, its orientation can be changed by rotating the crystal. At the exact Bragg orientation the vectors k’ and i are called respectively ki and io
(Fig. 1), that is :
Fig. 1.
-Neutron diffraction in a perfectly ordered crystal. The
incident and diffracted neutron wave vectors k and ko make equal angles 0 with the diffracting planes.
Since 1 k’ 1 = 1 k 1
=k, the diffraction planes, per-
pendicular to to, make an equal angle 0 with k and k’O,
with :
which is the well-known Bragg condition.
We consider now the case when the nuclear magnetic ordering is not perfect. This may arise from the existence of domains : Pi exp( - iT. Ri) changes from
+ P to - P in adjacent domains. We may also have
Pi * P in the domain walls and in perturbed regions
around defects such as the paramagnetic impurities.
The coherent scattering cross-section (eq. (11)) can
be written in the form :
where r is a vector of the crystalline lattice.
We define a correlation function cp(r) through :
When the ordering is perfect, qJ(r)
=1. When it is not, (p(r) decreases from 1 at r
=0 to zero when
r 1 -+ oo. Its correlation length in a direction r, loosely defined as the distance over which it decreases
by a substantial amount (say lie) is comparable
with the average domain size in this direction. We will assume that this size is much larger than the
interatomic distance.
Let A(q) be the space Fourier transform of qJ(r) :
Since the average domain size is large, the distri- bution of A(q) has a width Aq « T.
According to eqs. (16) and (17), eq. (15) becomes :
Neutrons will be diffracted in the direction k’
when :
with an intensity proportional to A(q). In order for
conditions (19) to be fulfilled, the vectors k’ and t
are no longer restricted to ki and io as when the
ordering is perfect, but can vary by an amount of
the order of Aq. As a consequence of the existence of domains, the rocking curve is broadened, as well
as the angular spread of the diffracted beam.
Remark. - The short-range order contribution to the neutron diffraction (second term on the right-
hand side of eq. (10)) exhibits the same characteristics.
However, since this short-range order extends over
a few interatomic distances only, the spread of its
Fourier transform is much larger than that of the domain correlation function. The corresponding rocking curve and neutron angular spread are very
large and hardly distinguishable from the continuous
background which, as noted earlier, is a justification
for neglecting this contribution.
3. The neutron difiraction pattern.
-We consider the angular distribution of diffracted neutron inten- sities for a given orientation of the crystal with respect to the incident beam. This distribution can
be experimentally determined by the use of a 2-
dimensional neutron counter. The experimental geo- metry is shown in figure 2. The neutron detection
Fig. 2.
-Geometry of neutron diffraction on a nuclear magneti- cally ordered crystal with domains. The neutrons diffracted with
wave vector k’ are detected in point M.
takes place in a plane perpendicular to kô at a dis-
tance D from the sample.
We choose a set of axes Oxyz, with Ox parallel
to k’ 0 and Oy in the plane of k, k’ 0 and io. The crystal
can be rotated around the axis Oz. A set 0’ YZ is defined in the detection plane, with 0’ Yeoy and
0’ Z/Oz. The origin is the point at which the vec-
tor k’ 0 hits the plane. Neutrons diffracted with a wave vector k’ #- k’ 0 hit the detection plane at a point M of coordinates Y and Z. The detector mea- sures simultaneously the diffracted neutron inten- sities (i.e. the number of neutrons per unit time) at
the various points (Y, Z).
The vectors k’ 0 and io, corresponding to the Bragg
reflection for a perfect ordering, have the following
components :
Let us rotate the crystal, so that the angle between
k and the diffracting planes becomes equal to e + E,
and consider the neutrons diffracted with a given
wave vector k’. The vector T defining the ordering is
linked to the crystal, and it varies when the latter is
rotated.
Let :
The condition (19) is then of the form :
or else, taking eq. (13) into account :
bk’ and ôr are comparable with q and since the spread Aq is small, we can use first-order expansions in àk’
and bi (or e).
Since k’ 1 = k and 1 r 1
=i, one has bk’ 1 k’ 0
and ôT 1 io that is :
Eq. (22) becomes :
The coordinates of the point M in the detection plane hit by the vector k’, at the distance D from
the sample, are :
The diffracted neutron intensity 3(e, Y, Z) measured
at this point is proportional to :
The components of q correspond to a frame which is not fixed with respect to the crystal, whereas they
describe a domain structure linked to the crystal.
If in the plane Oxy we define two new axes, Ox’ll T and 0y’ we have :
Since however e is small we have approximately :
The 2-dimensional distribution of diffracted neutron intensities as a function of the orientation e of the
crystal yields a map of the Fourier transform A(q)
of the domain correlation function qJ(r), from which
the latter can be determined.
It may be more convenient in practice to make a
less exhaustive analysis, corresponding to partial
summations of neutron intensities. We describe two
examples of such a procedure.
3.1 THE ROCKING CURVE.
-The rocking curve
is the variation, as a function of the crystal orienta- tion e, of the total diffracted intensity :
There is no need of a 2-dimensional neutron detector in that case. The rocking curve is what is normally
obtained with an ordinary neutron counter : the aperture of the counter is large enough to collect all
diffracted neutrons, which are counted whatever their point of impact.
We express A(q) as a function of cp(r) by the inverse Fourier transform of eq. (17). The function g(r) is
defined only for discrete values of r, equal to inter-
atomic distances. Since however the decay of cp(r)
takes place on distances much larger than the inter- atomic spacing, it is a good approximation to treat r
as a continuous variable.
We have :
which yields for eq. (27) :
The intégration over qy and qZ yields a product of
b-functions :
and we obtain finally :
The fundamental result expressed by eq. (30) is
that the rocking curve is the Fourier transform,
with wave vector ei cos 0, of the domain correlation function in the direction of the diffracted beam.
This results is valid only insofar as no other factors
contribute to the broadening of the rocking curve.
In practice, the observed rocking curve is a convo-
lution of the form (30) and of the rocking curve
that would be observed were the ordering perfect.
The latter is essentially determined by the imper-
fection of the incident neutron beam : angular spread and distribution of wavelengths.
When the ordering is ferromagnetic, one observes, superimposed on the ordered neutron curve, a rocking
curve originating from the spin-independent scat- tering amplitude. This crystalline curve is the only
one observed above the transition. It has to be substracted from the curve observed below the tran- sition in order to obtain the purely ferromagnetic
curve. The shape of the crystalline curve is the normal
shape resulting from the beam imperfection, and it
can be used for a deconvolution of the ferromagnetic
curve in order to obtain the shape 3(e) of eq. (30).
When the ordering is antiferromagnetic, there is
no crystalline line : the diffraction arises entirely
from the antiferromagnetic order. The normal rocking
curve to be used for a deconvolution of the experi-
mental curve can be deduced from that observed on a crystalline reflection at a different Bragg angle,
with some uncertainty due to the change of diffrac- tion angle. If the observed width is substantially larger that the normal width, this uncertainty does not seriously affect the accuracy of 3(e).
3.2 PARTIAL SUMMATION OVER THE 2-DIMENSIONAL
NEUTRON PATTERN.
-We have seen in the preceding
section that when the crystal misorientation is e,
neutron diffraction originates from the Fourier com- ponents with vectors q in a plane defined by
This plane is called A in figure 3.
If, in the 2-dimensional neutron pattern observed
for this crystal orientation we sum the neutron
intensities over Y at constant ordinate Z, the resulting integral :
is, according to eqs. (25) and (26) proportional to :
that is to the Fourier components of cp(r) with the
vectors q represented by the straight line D of plane A (Fig. 3).
Fig. 3.
-Exploitation of the pattern of neutron diffraction by a
nuclear magnetically ordered system. For a given misorientation e
of the crystal, the integrated neutron intensity on a line Z
=cst
of the detection plane corresponds to vectors q on a line D of plane A.
When the crystal is rotated, the plane A moves perpendicularly
to Ox, and the line D generates the plane B, with qz
=cst.
When the crystal is rotated, the plane A moves perpendicularly to Ox and the line D generates the plane B. The integral of the intensity 3(e, Z) for afl
orientations of the crystal :
is proportional to the integral of A (q) over the plane B.
According to the calculation developed in the preceding section, 3(Z) is proportional to the Fourier transform, with wave vector qz
=kZ jD, of the domain correlation function (p(z) in the direction z perpen- dicular to the plane B, that is to the plane of k and T.
This procedure can be generalized : an adequate
summation can yield the neutron intensity propor- tional to the integral of A(q) over any given plane, corresponding to a constant component ql normal
to the plane. This intensity is proportional to the
Fourier transform, with wave vector ql, of the domain correlation function in the direction normal to the
plane.
As in the preceding section, the experimental
neutron pattern is a convolution of the theoretical pattern described above with the normal neutron
pattern that would be observed in the absence of domains. This normal pattern is determined not
only by the imperfections of the incident neutron beam, but also by the size of the sample. This may
impose practical limitations to the determination of qJ(r) : if in a given direction of the YZ plane, the
width of the neutron pattern is not significantly larger than that of the normal pattern the determi- nation of the contribution of the domains to the neutron intensity integrated along lines normal to that direction will suffer from a large inaccuracy
and will be of no practical use.
4. Analysis of the domain corrélation function. - In the preceding section, we have established an
exact relationship between the observed neutron
diffraction pattern and the correlation function qJ(r)
associated with the imperfect ordering, attributed to the existence of domains.
To go further, that is to establish a neat relationship
between the function (p(r) and the distribution of domain dimensions, it is necessary to use an admit-
tedly idealized model for the domain distribution.
The model to be used is based on plausible arguments and the validity of the analysis will of course depend
on the adequacy of this model.
We consider the one-dimensional problem of relating the function (p(r) in a given direction, to the
distribution of domain dimensions in this direction.
We make the following assumptions.
1) The thickness of the domain walls is negligible compared to the domain size. In the case of a nuclear
ferromagnet at least, this assumption is supported
both by experiment and by the theory which predicts
a wall thickness comparable with the interatomic
spacing [3]. A wall thickness of this size has a wide Fourier spectrum and affects the neutron diffraction line far in the wings where it is lost in the continuous
background. For practical purposes it makes no dif- ference if we assume sharp transitions between the domains. We neglect the possibility of more extended perturbations of the ordering around the para-
magnetic impurities.
2) The dimension 1 of an individual domain in the direction r is a continuous variable. This approxi-
mation which simplifies the analysis, rests on the
fact that 1 is much larger than the interatomic spacing.
3) There is no correlation between the dimension 1 of a given domain and those of the neighbouring
domains. Insofar as the domain structure is deter- mined by randomly distributed paramagnetic impu- rities, this is a plausible assumption.
The domain structure is then entirely determined
by the probability S(1) of the occurrence of domains of dimension 1, with 1 > 0 and :
The probability S(1) is defined as the statistical
weight of the domains with dimension 1.
The aim of the analysis is to extract the proba- bility S(1) from the knowledge of the correlation function cp(r) (or its Fourier transform). By a gene- ralization of eq. (12) to the case when there are
domains, we write :
where, according to assumption 1), g(Ri) = ± 1, depending on the domain to which the spin i belongs.
The function cp(r) is then, according to eq. (16), of
the form :
We consider an ensemble of independent infinite
lines in the direction r, with domain walls distributed
according to the law S(1) for domain dimensions.
The correlation function ç(r) in the direction r is equal to :
the average of g(0) g(r) over this ensemble. Statistical
weights of various quantities over this ensemble will be called probabilities.
Since g(r) changes from 8 to - e from one domain
to the next (e
=± 1), the value of g(0) g(r) on a given line depends on the number n of domain walls in the interval r. We have :
When r is small, the probability of having more
than one wall in r is negligible. The probability of having one wall is equal to r [/la where 10 is the
average domain dimension, defined by :
The probability of having no wall in r is equal to
so that the function cp(r) in the limit of small r is equal to :
LE JOURNAL
DEPHYSIQUE.
- T.41, N° 8, AOUT I9ôO
For larger values of r, we call Tn(r) the probability
of finding n walls in the interval 0, r, with :
The correlation function cp(r), average of g(O) g(r) is
then :
The problem is then to calculate the various Tn(r)
in terms of (T(/). We first introduce the probability X(r) of finding no wall within a distance r from an
existing wall :
and we proceed to calculate Tn(r).
Let us consider in the interval 0, r, a distribution of n walls at the points ri, r1 +r2, ..., r1 +r2+"’+rm with the last one r. We list the following condi-
tional probabilities.
The a priori probability of finding a wall between ri and ri + drl is, as noted earlier, equal to 1. 1 drl.
Once this wall is known to be present, the probability of finding no wall between 0 and ri is equal to X(rl).
The probability of finding the second wall within
dr2 of the distance r2 from the first wall is equal to
the probability of a domain of dimension between r2
and r2 + dr2, that is W(r2) dr2.
The probability of finding the third wall between r3 and r3 + dr3 from the second wall is (r3) dr3.
According to the model, it is independent of the
presence of a wall at rl. The same reasoning applies
to the following walls.
Finally, the probability of finding no wall in the
interval r - r,
...- rn after the last wall is equal
to X(r - rl ... rn).
The probability Tn(r) is obtained by integrating
over ri,
...,rn the product of the probabilities listed
above :
This equation is valid only for n > 1. The case n = 0 will be treated later.
Let us now introduce the Laplace transforms of
the various probabilities :
We will use the following well-known properties
of Laplace transforms :
1) The Laplace transform of r" is equal to :
In particular, the Laplace transform of 1 is equal
to 1 /z.
2) Let A(z) and B(z) be the Laplace transforms of
a(r) and b(r).
The Laplace transform of
is equal to :
In particular, when b(r)
=1, the Laplace transform
According to these properties, and to eq. (38), the Laplace transform K(z) of X(r) is equal to :
Let us consider the Laplace transform of eq. (39).
The Laplace transform of its right-hand side is
obtained by an iteration of the property (43) and
we obtain :
which is valid only for n > 1. The Laplace transform 00 of To(r) is obtained by equating the Laplace trans-
forms of both sides of eq. (36) :
or else :
The Laplace transform of the correlation func- tion g(r) is, according to eqs. (37), (45) and (46’) :
Physically, the function 0 (z) does not diverge,
which implies that H (z) 1. We have then :
What is actually known from experiment is the
Fourier transform of the correlation function p(r) :
Since p(r) = ç( - r) and ç(0)
=1, we have :
According to eq. (50) :
where lP(iq) must be understood as :
Some information on S(1) is readily obtained
from eqs. (35) and (48). The short-distance behaviour of qJ(r) (eq. (35)) corresponds, according to eq. (42)
to an expansion of 0 (z) as a function of 1 /z of the
form :
or else, for z = iq and according to eq. (52) :
In the wings (large q), the function I(q) behaves
like a Lorentzian curve. The amplitude of "l(q) in
the wings yields the value of the average domain dimension 10. Alternatively, one may calculate (p(r)
as the Fourier transform of I(q) and obtain 10 from
its short-distance behaviour (eq. (35)). In practice
the experimental shape I(q), as deduced from the neutron diffraction pattern, extends only to finite
values of q. This truncation leads to a small rounding
off of its Fourier transform near the origin.
Except for very special shapes of I(q), this should
not hamper too severely the determination of 10.
The expansion of II (z) for small z is, according to
eq. (40), of the form :
where a2 is the second moment of S(1). By inserting
eq. (55) into eq. (48) we obtain for 0 (z) :
that is :
The average dimension 10 being known from the wings of I(q), the amplitude at the origin of the (normalized) experimental curve I(q) yields the r.m.s.
width (U2 _ lô )’2 of the domain size distribution around lo.
A complete information on S(1) requires more
calculations. Eq. (48) yields :
Re [il(iq)] is the cosine Fourier transform of s(1),
and its Fourier transform yields S(1).
In eq. (58), Re [O(iq)] is equal to nl(q). As for
Im [O(iq)], it can be obtained either by a Kramers- Kronig transformation of Re [O(iq)] [6], or by a
sine Fourier transformation of cp(r).
To sum up, in order to determine the probability distribution S(1) of domain dimensions in a given
direction n, one must :
-
measure the integrated scattered neutron inten- sity in planes of the reciprocal space normal to n as a function of the component of q parallel to n,
-
perform a deconvolution with respect to the normal curve, and normalize to unit area the resulting
curve in order to obtain the function I(q)’,
-
determine the average domain dimension 10,
either by fitting the wings of I(q) to a Lorentzian,
or measuring the initial linear decrease of cp(r) as
obtained by a Fourier transformation of 1 ( q),
-
determine Im [P(iq)], either by a Kramers- Kronig transformation of nI(q) or a sine Fourier
transformation of cp(r),
-
use 10, Re [0 (iq)] = nI(q) and Im [P(iq)] in
eq. (58) to compute the cosine Fourier transform
of e(l),
-
