The multi-k structures cannot be distinguished from the single-k structure by neutron diffraction unless one applies magnetic fields and uniaxial stresses.. Irreducible matrix corepresen
Trang 184 T Chatterji
Fig 44 Crystal structures of Ruddlesden–Popper phases.
determined by Ruddlesden and Popper in 1958 [113] The Rietveld profile refinements onneutron and (both laboratory and synchrotron) X-ray powder diffraction data produced ahuge amount of literature on the structure analysis of Ruddlesden–Popper phases As al-ready mentioned in the previous section the structure is flexible with respect to the cationorder disorder between the 12-coordinated perovskite (P) and 9-coordinated rock-salt (R)sites and also with respect to the octahedral distortion To make the situation more compli-cated there exists the possibility of phase separation mentioned before The possibility ofelectronic phase separation also exists Despite these complications accurate structural pa-
rameters have been determined both as a function of hole doping x and also as a function of
temperature Also available are structural parameters as a function of both hydrostatic sure and applied magnetic field Of special interest are the Mn–O bond lengths There is
pres-only one Mn site (4e : 00z), but there are three O sites in the structure, O(1) (2a : 000), O(2) (4e : 00z) and O(3) (8g : 012z) As mentioned before there are two cation sites: La/Sr(P) (2b : 0012) and La/Sr(R) (4e : 00z) The essential result of the structural evolution at room temperature with x is that the JT distortion ∆JTwhich is about 1.04 for x = 0.3 decreases
continuously and becomes very small, ∆JT= 1.005 at x = 0.5 [114] This result remains
qualitatively the same at low temperature, but ∆JT is slightly smaller [114] The
tempera-ture variation of ∆JTfor La1.2Sr1.8Mn2O7(x = 0.4) shows a minimum close to TC[115].Investigation of the crystal structure of the same compound [115] as the temperature is
lowered through TCin a field of 0.6 T reveals a significant magnetostriction The torial Mn–O bond contracts and the apical Mn–O bond expands in a magnetic field For
Trang 2equa-Magnetic structures 85LaSr2Mn2O7 (x = 0.5) charge-orbital ordering [116] takes place at TCO≈ 225 K The
intensity of the superlattice reflections corresponding to the charge-orbital ordering
in-creases continuously below TCO≈ 225 K At about 165 K it shows a maximum and then
starts decreasing and becomes very small at about 100 K The intensity of the tice reflection starts increasing again below 50 K Ling et al [117] have investigated the
superlat-crystal and magnetic structure in the doping range 0.5 x 1.0 The crystal structure
be-comes orthorhombic (Immm) 0.8 < x < 1.0 but the transformation is never complete The
orthorhombic phase coexists with the tetragonal phase
The magnetic structure of La1−2xSr1+2xMn2O7has been investigated for 0.3 x 1.0
by various authors [117–121] Essentially there are three types of phases: ferromagnetic(FM) and antiferromagnetic (AFM) and canted (C) phases There are two types of fer-romagnetic phases FM-I and FM-II In FM-I the magnetic moments of the Mn ions are
in the a–b plane whereas in FM-II the moments are parallel to the c axis There are two
essential types of antiferromagnetic phases For LaSr2Mn2O7(x = 0.5) the
antiferromag-netic phase (AFM-I) is similar to the A-type AFM phase observed in perovskite Here theindividual ferromagnetic layers of the bilayers are antiferromagnetically stacked The bi-layers as units are also antiferromagnetically stacked In the other type of AFM-II phase,the two individual ferromagnetic layers of the bilayers are ferromagnetically stacked, butthe bilayers as units are stacked antiferromagnetically There exist a third type of phasecalled the canted phase This is similar to the AFM-I phase except that the individual lay-ers of the bilayers are canted by an angle which is different from 180 degrees The cantedphase can be thought of a combination of FM-I and AFM-I phases in which there exist
both ferromagnetic and antiferromagnetic components For x = 0.3–0.4 ferromagnetism
dominates at lower temperature whereas for x > 0.4 antiferromagnetism dominates The saturated magnetic moment is about 3µBat low temperatures Ling et al [117] have pub-lished a phase diagram for both crystal and magnetic phases in the concentration range
0.3 < x < 1.0 This phase diagram shows the type-C AF phase in the concentration range 0.74 < x < 0.9 and the G-type AF phase in the range 0.915 < x < 1.0 Close to x = 0.9
there is a small region where C- and G-type AF phases coexist There is no long-range
magnetic order in the range 0.64 < x < 0.74 Recently Okamoto et al [122] have
investi-gated the correlation between the orbital structure and the magnetic ordering temperatureand the type of magnetic structure of the bilayer manganites at low temperatures The two
e g orbitals 3d 3z2−r2 and 3d x2−y2 in a Mn3+ion are split in the crystal field of the layered
structure and only one of them is occupied by the e g electron The occupied orbital trols the anisotropy of the magnetic interaction and also its strength The bilayer manganite
con-La1−2xSr1+2xMn2O7becomes a metallic ferromagnet for a hole doping 0.32 x 0.42.
The Curie temperature TCis about 110 K for x = 0.32 It increases with increasing x and
becomes highest TC≈ 130 K for x = 0.35 On further increasing x, TCdecreases again
and becomes again TC≈ 110 K for about x = 0.42 The magnetic structure deviates from
a simple ferromagnetic for x < 0.32 and also for x > 0.42.
The bilayer manganite La1−2xSr1+2xMn2O7for x = 0.4 is a quasi-two-dimensional
fer-romagnet which shows colossal magnetoresistive properties and has therefore been tigated very quite intensively In Chapter 6 we will describe the spin dynamics of thisquasi-2D ferromagnet in detail The ferromagnetic La1.2Sr1.8Mn2O7 has Curie temper-
inves-ature TC≈ 128 K The ferromagnetic phase transition was determined by neutron
Trang 3dif-86 T Chatterji
Fig 45 (a) Temperature variation of the magnetic contribution of the intensity of the 110 reflection of
La1.2Sr1.8Mn2O 7 (b) Temperature variation of the total intensity of the 110 Bragg reflection close to TC The continuous curve is a power-law fit of the data to (13) (c) Temperature variation of the integrated intensity of the
rod scattering by doing a Q scan perpendicular to the rod through the reciprocal point Q = (0, 1, 1.833).
Trang 4Magnetic structures 87fraction Figure 45(a) shows the temperature variation of the magnetic contribution of theintensity of the 110 reflection The magnetic intensity decreases continuously with increas-
ing temperature and becomes zero at TC≈ 128 K Figure 45(b) shows the temperature
vari-ation of the total intensity of the 110 Bragg reflection close to TCand a fit to the equation
where In is the nuclear contribution to the intensity, I0 is the magnetic intensity at T = 0,
TC is the ferromagnetic transition or Curie temperature and β is the critical exponent The least squares fit gave TC= 128.7 ± 0.1 K and β = 0.35 ± 0.01 The critical exponent ob-
tained from the fit is closer to the three-dimensional (3D) Heisenberg value, β = 0.38, than
to the two-dimensional (2D) Ising value, β = 0.125 Although La 1.2Sr1.8Mn2O7behaves
like a quasi-2D ferromagnet, the ferromagnetic phase transition at TC≈ 128 K caused
by relatively weak inter-bilayer exchange interaction is ultimately of 3D-Heisenberg type
A second determination of TCfrom the temperature variation of the integrated intensity of
the rod scattering by doing a Q scan perpendicular to the rod through the reciprocal lattice
point Q= (0, 1, 1.833) (Figure 45(c)) also gave TC≈ 128 K
The spin dynamics of the quasi-2D bilayer manganite La1.2Sr1.8Mn2O7has been tigated in detail by inelastic and quasielastic neutron scattering, which will be described inChapters 6 and 7
appear simple, one must note the difficulty arising from multi-k ordering The multi-k structures cannot be distinguished from the single-k structure by neutron diffraction unless
one applies magnetic fields and uniaxial stresses We then introduced the more complexmagnetic structures found in rare-earth elements and other magnetic solids We described
in some details the incommensurate magnetic structures which appear below the magneticordering temperature and the interesting phase transitions they undergo at lower tempera-tures We also described and discussed magnetic phase transitions caused by the application
of magnetic field and pressure Qualitative and phenomenological arguments are given insome cases to rationalize such structures The magnetic structures of important electronicmaterials, namely, the high temperature superconducting cuprates and colossal magnetore-sistive manganites, have also been considered We have, however, left out many importantmagnetic structures that exist in heavy fermion and other actinide compounds There exist
Trang 588 T Chatterji
numerous review articles on heavy fermion and actinide compounds and interested ers are advised to consult those It is useful to keep in mind that the published magneticstructures that are usually determined from polycrystalline samples by unpolarized neu-tron diffraction, are merely models that lead to reasonably good agreement between theobserved and calculated magnetic intensities In many cases only a complete polarizationanalysis of the diffracted neutron intensity from monodomain single crystals can lead tounambiguous determination of magnetic structure The present chapter is meant to be anintroduction to prepare the reader for more specialized methods of solving magnetic struc-tures by group theoretical and polarized neutron diffraction described in Chapters 3–5
read-Acknowledgment
I wish to thank P.J Brown for collaboration over the years
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Trang 9CHAPTER 3
Representation Analysis of Magnetic Structures
Rafik Ballou
Laboratoire Louis Néel, CNRS, B.P 166, 38042 Grenoble cedex 9, France
E-mail: ballou@grenoble.cnrs.fr
Bachir Ouladdiaf
Institut Laue–Langevin, B.P 156, 38042 Grenoble cedex 9, France
E-mail: ouladdiaf@ill.fr
Contents
Introduction 95
1 Crystallographic preliminaries 96
2 Mathematics of representations 101
2.1 Matrix representations 105
2.2 Irreducible matrix representations of space groups 113
2.3 Matrix corepresentations 116
2.4 Irreducible matrix corepresentations of magnetic space groups 121
3 Analysis of magnetic structures 124
3.1 Analysis without time inversion 127
3.2 Analysis with time inversion 130
4 Practical working scheme 133
5 Application 135
5.1 First example, k= 0 135
5.2 Second example, k = 0 146
References 150
NEUTRON SCATTERING FROM MAGNETIC MATERIALS
Edited by Tapan Chatterji
© 2006 Elsevier B.V All rights reserved
93
Trang 11Representation analysis of magnetic structures 95
Introduction
Neutron diffraction is the most powerful experimental technique for the determination ofthe magnetic structures of crystals: as from the directions of the magnetic scatterings withrespect to the incident neutron beam, the magnetic periodicity (defined by wave vectors k
in the reciprocal space) can be deduced, and from the intensities of these scatterings, therelative orientations of the magnetic moments can be computed Using the trial-and-errormethod to solve for the orientations, however, becomes tedious and inefficient if the num-ber of the magnetic moments in the primitive cell of the crystal structure is large
Complex magnetic structures can be solved by using methods of the mathematical theory
of representations of groups to select solely the magnetic configurations that are symmetrycompatible The general procedure consists, at first, of building up the matrix representa-tions (or, depending on the effects of the time-inversion symmetry, the matrix corepresen-
tations) associated with the action of the symmetry operators of the space group G k (or
of the magnetic space group M k) as defined below, on the k-component moments S u ( (u ∈ s) of each magnetic site s in the crystal Next, these matrix representations (or ma-
trix corepresentations) are reduced over the irreducible matrix representations of the space
group G k (or the irreducible matrix corepresentations of the magnetic space group M k).The symmetry allowed magnetic structures are then obtained in terms of the linear combi-nations of the basis vectors of the irreducible matrix representations (or of the irreduciblematrix corepresentations) appearing in the different decompositions The coefficients ofthese combinations are the unknown to be calculated from the magnetic contributions tothe neutron intensities to deduce the actual magnetic structure of the crystal A further sim-plification arises when the magnetic structure is stabilized through a second-order phasetransition from the paramagnetic state for it would then be fully described by a singleirreducible matrix representation (or a single irreducible matrix corepresentation) [1].The earliest uses of matrix representations of groups to analyze magnetic structures can
be dated back to 1958 [2] A number of works then followed, attempting to further themethodology or extend its application, of which we mention a few [3–10] in our referencelist Owing to a general belief that the time inversion symmetry would be irrelevant [10],which is not always true [5], many fewer investigations were concerned with matrix corep-resentations Concrete instances of magnetic structures deduced from neutron patterns withthe help of matrix representations (or of matrix corepresentations) can also be found in theliterature, of which we mention a few [11–15] in our reference list, but these are moreexceptional than systematic A lot of magnetic structures are published without any indica-tion of a use of representation analysis Tables of the irreducible matrix representations forthe different space groups and irreducible matrix corepresentations for the different mag-netic groups for different types of wave vectors were reported [16–18] Various computerprograms calculating the irreducible matrix representations, as well as the associated basisvectors, were also developed in recent years, such as Karep [19], Mody [20], Sarah [21] andBasIreps [22] All these appear, however, rather esoteric without a minimum knowledge ofthe underlying methodology
The method of matrix representation analysis and of matrix corepresentation analysis
of magnetic structures, as well as the associated practical step-by-step procedures, will
Trang 1296 R Ballou and B Ouladdiaf
be presented in the following as pedagogically as possible, but not without rigor, ing the matter from several works on the topic [3–10,23,24] We start with a first section
borrow-on crystallographic preliminaries and recall in a secborrow-ond sectiborrow-on the mathematics of resentations at an elementary level We discuss the analysis of magnetic structures in athird section We consider, at first, the spatial symmetries and describe how to build up thetransformation-induced matrix representations for magnetic structures and how to reducethem We next include the time inversion in the treatment, indicating where matrix corep-resentations are required In a fourth section we write down the practical recipes to use inconcrete instances of representation analysis of magnetic structures We end with a fifthsection detailing two different examples A reader not interested in the abstract formalismcan skip, after the crystallographic preliminaries, directly to the practical recipes then tothe examples
rep-1 Crystallographic preliminaries
A crystal can be defined as a medium where to each point always corresponds a discreteinfinite set of equivalent points around which the atomic arrangements are either identical
or inverted Any recovery operation g, or isometry, that brings a point to its equivalent
is a discrete displacement (eventually combined with an inversion) so is most generallycomposed of a proper (eventually improper) rotation and a discrete translation A recovery
operation g is also an invariance symmetry of the crystal Using the Koster–Seitz notation [25,26], g will be written as {α|τα+ R n}, where α symbolizes the rotational part (proper
or improper) of the symmetry andτ α+ R nits translational part
The action of g = {α|τ α+ R n} on the space variable r is given as
x1
x2 x3
+
where (x1, x2, x3), (τ α1 , τ α2 , τ α3 ) and (n1, n2, n3) are the components of r, τα and R n
with respect to a basis{ai}i =1,3of primitive translations which always exist because the
medium is discrete; α is a 3 × 3 orthogonal matrix, the determinant of which is η(α) = +1
for a proper rotation and η(α) = −1 for an improper rotation We shall denote ε as the
unit 3× 3 matrix so that α = ε for no rotation, α = −ε for an inversion or α2= ε and
α = ε for a reflection τ α is either null or a translation vector with fractional components
which combines with α to produce a screw rotation if η(α)= +1 or a glide reflection if
η(α)= −1 R n is a lattice translation vector of which the components n1, n2and n3arealways integers Any point of the space can be reached with a vector
i =1,3 u i ai+ R n,
where u i 1 are real numbers The parallelepiped bounded by the aiis called a unit cell.The region of space bounded by the planes bisecting the origin to all the neighboring latticepoints is also a unit cell called the Wigner–Seitz cell
Trang 13Representation analysis of magnetic structures 97
The action of g = {α|τα+ R n} on a function ψ of the space variable r is given as
O(g)ψ ( r) = ψ(g−1r) = ψ(({α|τα+ R n})−1r) = φ(r), where O(g) is a function operator
which, acting on the function ψ , gives the new function φ = O(g)ψ.
The action of g = {α|τα+ R m} on a vector is solely determined by its purely rotational
part {α|0} A translation transports a vector from one position to another but does not
change its orientation When the vector is polar V , its components (V1, V2, V3) with respect
to the{ai}i =1,3 basis are transformed by{α|0} similarly as the components of a position
vectorr Accordingly,
α |τ α+ R m V = α V =α11 α12 α13
α21 α22 α23 α31 α32 α33
V1
V2 V3
S1
S2 S3
where η(α) = 1 if α is a proper rotation and η(α) = −1 if α is an improper rotation.
Another way featuring the action of g on vectors [5–7,11], which we shall prefer to use for it gives immediate geometrical insights, is through the vectors S x = Sa1/|a1|, S y=
S a2/|a2| and S z = Sa3/|a3|, the components of which are, respectively, (S, 0, 0), (0, S, 0)
and (0, 0, S) with respect to the {ai}i =1,3basis Assuming these vectors are axial, we easilyshow that {α|0}S x = η(α)(α11Sx + α21Sy + α31Sz ), {α|0}S y = η(α)(α12Sx + α22Sy+
α32S z ) and {α|0}S z = η(α)(α13Sx + α23Sy + α33Sz ), which we can rewrite symbolically
where ¯α is the transpose of α Equation (1d) should be used cautiously as it is a condensed
mnemonic notation for a transformation of vectors and not a transformation of components
of vectors
Finally, we briefly recall that the action of g = {α|τα+ R n} on quantum states |ϕ and
O(g)†is its adjoint [24]
The canonical product of two symmetry operations is a symmetry operation defined as
Trang 1498 R Ballou and B Ouladdiaf
rotation, and associating an inverse{α−1|−α−1( τα+ R n ) } to each {α|τα+ R n}, this
prod-uct defines, on the set of the symmetries of any crystal, a group strprod-ucture G named space
group or Fedorov group
Any space group G contains an invariant (or normal) subgroup T of lattice translations
{ε| R n} T is a subgroup because
(i) the product of two translations is a translation,
(ii) T contains {ε|0} as its neutral element,
(iii) the inverse of any translation is a translation
T is an invariant subgroup because ∀{ε| R n} ∈ T , ∀{α|τα + R m} ∈ G, {α|τα +
R m}{ε| R n}{α−1|−α−1( τ α + R m ) } = {ε|α R n} ∈ T Any pair of points showing a same
atomic arrangement with the same orientation are indeed separated by a lattice translationvector and conversely Now, ifr1andr2 are two position vectors such thatr2− r1= R n
then{α|τα + R m}(r2− r1)= αr2− αr1= α R n , since α is linear Accordingly, α R n isnecessarily a lattice translation vector, since a same rotation{α|0} is applied to r1andr2:the changes in orientation of the atomic arrangement at the new positions are exactly the
same A particular outcome of this is that the elements of α, and therefore its trace Tr(α),
are integers Using an orthonormal basis (not necessarily crystallographic), we show, on
the other hand, that Tr(α) = ±(1 + 2 cos θ), where θ is the rotation angle Accordingly,
only 2nd-, 3rd-, 4th- and 6th-order axes can exist in a crystal A further outcome is that
α k = ε for a k-fold axis and a k-fold application of {α|τ α} in point space must lead to
a lattice translation, that is,{α|τ α}k = {α k |(α k−1+ α k−2+ · · · + ε)τ α } = {ε| R n}, which
constrains the allowed τ α Applying to a point all the translations{ε| R n} of T , we get a
periodic lattice L The set {{α|0} | α R n ∈ L} of all the rotations preserving a lattice L,
equipped with the product defined in (2), forms a group P Lcalled the holosymmetric pointgroup of the lattice Only seven such groups exist, leading to 14 different lattices called theBravais lattices
Any space group G can be expanded in terms of left cosets of T as
G = {ε|0}T + {α1|τ1}T + {α2|τ2}T + · · · + {αn G0 |τn G0 }T (3)
An expansion in terms of right cosets of T could be made as well Since T is invariant
(or normal) left and right cosets are identical The set of left cosets equipped with themultiplication law{αm|τ m}T · {αn|τn}T = [{αm|τm}{αn|τn}]T forms a group GT (read
G modulo T ) of order n G0 called the factor group A similar canonical law can be defined
for the right cosets leading to an isomorphous factor group G T is also isomorphous with
the isogonal point group G0, of order n G0, made of the set of the elements{αi|0} equipped
with the product defined in (2) On the other hand, the set of the elements{αi|τ i} equipped
with the same product does not form a group, since {αn|τ n}{αm|τm} = {ε| R nm = τn+
α n τ m − τ nm}{αnm = αn α m|τ nm} and the lattice translation vector R nmcan be finite Given
a space group, the isogonal point group is always included in the holosymmetric point
group: G0⊆ PL With the constraint on the order of the axes associated with translationsymmetry, 32 point groups can be formed, all described in the International Tables [27]
T and G T fully feature the space group G, so that the catalogue of all the space
groups is finite Assuming that two space groups are the same if an affine transformationexists that sends one group to the other, or equivalently if the two groups are isomorphic as
Trang 15Representation analysis of magnetic structures 99abstract groups, we get a total of 219 different space groups On the other hand, assumingthat two space groups are the same if an orientation preserving transformation exists thatsends one group to the other, we get a total of 230 different space groups, which are thosedescribed in the International Tables [27].
As from the linearly independent primitive translations ai (i = 1, 3), which generates
the periodic lattice L, a dual periodic lattice L∗ can be deduced by defining dual basisvectorsa∗
i (i = 1, 3), such that ak · a∗
l = 2πδkl (k, l = 1, 3) This is solved as
a i∗= 2πai ( · (aj aj × ak × ak )
{ai}i =1,3is a covariant basis and{a∗
i}i=1,3a contravariant basis The lattice formed by thevectors K n = n1a∗
1+ n2a∗
2+ n3a∗
3, where n1, n2 and n3are positive or negative integers,
is called the reciprocal lattice The space generated from the vectors
i =1,3 u i a∗
i + K n,
where u i 1 are real numbers, is the reciprocal space The Wigner–Seitz cell built in this
space is called the (first) Brillouin zone
A basis change generally induces a change in the symmetry operations In particular,symmetry in real (or direct) and reciprocal spaces must not be confused [28] A basischange consists in transforming the basis{ai}i =1,3to another basis{bi}i =1,3as
where the M lk and N lk coefficients defines two inverse matrices: N = M−1 Any triplet
of coordinates X with respect to {ai}i =1,3 will become the triplet Y = ( M)−1· X with
respect to{bi}i =1,3, and conversely, X= M · Y where M is the transpose of the matrix M Consider now a symmetry operation g α = {α|τ α } that transforms to gβ = {β|τβ} under the {ai}i =1,3to{bi}i =1,3 basis change g α brings a point X to the point X α and g β a point Y
to the point Y β according to (1a) as
X is related to Y and X α is related to Y β through the base transformation as X= M · Y
and X α= M · Yβ Using (6a), we deduce that Y β = ( M)−1α M Y + ( M)−1τα, which byidentification with (6b), gives
β= M −1
α M and τβ= M −1
When the basis change corresponds to the passage from the direct to the reciprocal
space, M in (5) is equal to Q−1= Q∗, where Q and Q∗are transposes of themselves, since
symmetric, and are the metrics of the direct and reciprocal spaces, that is, Q kl = ak · al,
...= 146References 150
NEUTRON SCATTERING FROM MAGNETIC MATERIALS< /small>
Edited by Tapan... determination ofthe magnetic structures of crystals: as from the directions of the magnetic scatterings withrespect to the incident neutron beam, the magnetic periodicity (defined by wave vectors k... unknown to be calculated from the magnetic contributions tothe neutron intensities to deduce the actual magnetic structure of the crystal A further sim-plification arises when the magnetic structure