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University of LjubljanaFaculty of Mathematics and PhysicsDepartment of Physics

Seminar

Magnetic symmetry and

determination of magnetic structures

Gregor Posnjak1

Adviser: dr. Andrej Zorko

August 22, 2012

Abstract

In this seminar we take a look at symmetry operations in three dimensions(3D) and show how magnetic symmetries are derived from them. Then we

present the basics of group theory and demonstrate the strength of representa-tion analysis in narrowing down the spectrum of possible magnetic structuresin a magnetic material. In the end we apply these methods to the exampleof CuSe2O5: we briefly present its crystallographic structure, take a look atexperimental neutron scattering data on a single crystal sample, find the mag-netic propagation vector, determine the little group of the system, all possibleirreducible representations and finally determine the magnetic model that is inbest agreement with the experimental data.

1E-mail: [email protected]

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1 Introduction

1 Introduction

Determination of magnetic structures of materials is an important part of solidstate sciences as it sheds light on electronic interactions in a material. Usually

the strongest experimental tool for this is magnetic neutron diffraction as thermalneutrons typically have wavelengths comparable to the periodicity of the investigatedstructure and also similar energies as its phonons and magnons.

Determination of magnetic structures from experimental data is generally in-volved and there are no fail-safe methods that would be guaranteed to work everytime. Because of that incorrect magnetic structures are often reported in the liter-ature [1]. Typically the determination is a trail-and-error process where experienceand intuition are vital to successfully find the correct structure. There are of coursetheoretical methods for calculating magnetic structures by using the interactionsbetween atoms but typically these calculations are complex and it is difficult to pre-dict which interactions are important and which can be neglected [2]. Consequently

these methods have limited success rates.There is another set of methods based on symmetry properties of the crystallinestructure which requires no information about the interactions between atoms. Herethe interpretation of experimental data is aided by the analysis of what kind of con-straints the symmetries of the crystal structure impose on the magnetic structure.Magnetic symmetry analysis thus often helps to find models of magnetic struc-tures that would otherwise be overlooked. In this seminar we will make a shortintroduction to the use of symmetry relations and representation analysis in thedetermination of magnetic structures and apply this theory to an example.

There are two approaches to magnetic symmetry analysis. The first is the deriva-tion of possible magnetic space groups from the crystallographic space group of the

structure. The second is representation analysis of the little group of the magneticpropagation vector which gives all possible representations that agree with the pe-riodicity of the magnetic and crystallographic structures.

Because none of these approaches takes into account the precise nature of theinteractions between the magnetic atoms, the real magnetic structure can only beobtained by fitting the possible models of the structure we have found in the analysisto experimental data. To make the picture complete we will present in the lastchapter of this seminar how representation analysis is applied to neutron scatteringdata on a single crystal of CuSe2O5.

All formulations in this text are done under the assumption that there is onlya single magnetic propagation vector k and all magnetic atoms reside at a single

crystallographic site. The generalisations of equations for cases with multiple prop-agation vectors and magnetic sites can be found in the cited references.

2 Groups

In discussing symmetries one of the fundamental concepts are groups. A group Gis a set of elements {g1, g2, ...} with a multiplication operation which assigns toeach ordered pair gi, gj G another element ofG. This operation must satisfy thefollowing conditions [3]:

Closure: a product of two elements ofG is itself also an element ofG.

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2 Groups

12

3

21

3

m

Figure 1: An object exhibits a symmetry if applying the corresponding symmetry operationleaves it unchanged. In the picture we can see this by applying a reflection over thedashed line on the triangle. The corners of the triangle are numbered for clarity but areotherwise equivalent. We can see that the corners 1 and 2 swap places, but the triangleis left unchanged. We can find additional symmetry operations which leave the triangleinvariant - two more reflections and a three-fold rotation axis in the center.

Associativity: (gigj)gk = gi(gjgk). Existence of identity element e G: egi = gie = gi gi G.

Existence of inverse elements g1i G: g1i gi = gig

1i = e gi G.

When dealing with the crystallographic and magnetic structures we use point andspace groups to classify the possible structures. We will present them in the nextsection.

2.1 Point and space groups

There exist different kinds of symmetry operations: translations, rotations, reflec-

tions, inversions, ... We say that an object has a particular symmetry if that sym-metry operation transforms the object into a new object which is identical to theoriginal one - it leaves it invariant (Figure 1). Classification of 3D structures onlyby the translational symmetries they exhibit yields the 14 Bravais lattices [4]. Theselattices are formed by taking all possible combinations of their primitive (basis) vec-tors. Every point in such a lattice has exactly the same surroundings as any other.On the other hand, if we consider only the symmetry operations that leave at leastone point fixed (rotations, reflections, inversions and improper rotations), we getpoint groups. There are many possible point groups, but in crystalline matter onlythose consistent with the translational symmetry of Bravais lattices are possible.There are 32 such point groups, which are called the crystallographic point groups

or the crystal classes [5].By combining the crystallographic point groups with the Bravais lattices we get

230 space groups. Space groups therefore exhibit translational and point symmetriesas well as two new types of symmetry - screw axes and glide planes, which arecombinations of rotations or reflections with translations. Lists of space groups andtheir symmetry operations can be found in literature [6, 7].

If we take a general position inside the zeroth unit cell of a crystal structure andapply one of the symmetry operations of its space group, we find that the position istransformed into another position inside the zeroth unit cell or into another positionin some other unit cell which can be translated back into the zeroth cell by some

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2 Groups

Table 1: Wyckoff symbols of general (8f) and special positions of the space group C2/c [8].

MultiplicityWyckoff Site Coordinates

letter symmetry Lattice: (0,0,0) + (1/2,1/2,0) +

8 f 1 (x,y,z) (-x,y,-z+1/2) (-x,-y,-z) (x,-y,z+1/2)4 e 2 (0,y,1/4) (0,-y,3/4)4 d -1 (1/4,1/4,1/2) (3/4,1/4,0)4 c -1 (1/4,1/4,0) (3/4,1/4,1/2)4 b -1 (0,1/2,0) (0,1/2,1/2)4 a -1 (0,0,0) (0,0,1/2)

linear combination of the Bravais lattices primitive vectors. All positions we get inthis way from a general position are related by the symmetry operations of the spacegroup and are therefore crystallographically equivalent. Their number is determined

by the number of symmetry operations in the space group.Not all positions inside the unit cell are transformed into a new position by thesymmetry operations of the space group - some are left unchanged by one or moreof the symmetry operations. These are those which lay on one of the symmetryelements of the space group - on a axis of rotation, a mirror plane or a center ofinversion. They are called the special positions. Lists of space groups with thegeneral and special positions are given in literature [8]. As an example lets take alook at the listing for the space group number 15 (C 2/c).

In the first column of Table 1 the multiplicity of a position is given - the number ofequivalent positions of particular symmetry in the zeroth cell. The general position,which is listed in the first row, is transformed into a new position by all the symmetry

operations of the space group and it therefore has the highest multiplicity (8 in thiscase). The special positions are left invariant by some of the symmetry operationsand therefore have lower multiplicity than the general position. In the last columnthe symmetry operations which transform the position into equivalent positions arelisted. In the case of the general position all the symmetry operations of the spacegroup are stated. The Wyckoff letter is a way of numbering the positions startingwith the one with the highest symmetry and lowest multiplicity, which is assignedthe letter a, all the way up to the general position.

Space groups tell us only what kind of arrangements of atoms or molecules arepossible in crystalline matter. Positions of atoms are represented with polar vectors.Magnetic moments, on the other hand, are axial vectors which transform differently

under symmetry operations. To visualise the difference we can think of the magneticmoments as arising from electric current loops. In Figure 2a we can see that mirrorplanes have exactly the opposite effect on axial vectors as they do on polar vectors.The same is true for inversion centres. Spatial arrangements of axial vectors aretherefore in general not left invariant by the symmetry operations of their spacegroup.

To find the operations that do leave them invariant, a new operation is neededwhich would invert only the axial vectors. Because electric currents are definedby I = de/dt, a time-inversion operation changes their direction and inverts theaxial vectors which represent them (Figure 2b). Positions of atoms are not changedby the time-inversion. To find new space groups which leave magnetic structures

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2 Groups

(a) (b)

Figure 2: On the left we can see how a mirror plane affects a polar (dipole moment) andan axial vector (electric current loop). In the configurations when the polar vector isinverted, the axial stays unchanged and vice-versa. Similar result can be found for aninversion center which always inverts a polar vector but never an axial. On the right wecan see the effect of the time-inversion operation on an electric current loop - the direction

of the current is reversed and the axial vector is inverted. [10]

invariant some of the symmetry operations of the space group must be combinedwith the time-inversion operation. A combination of symmetry operation and thetime-inversion operation is called an anti-symmetry operation or a primed operation.By finding all the possible ways of priming some or all the symmetry operations ofspace groups 1651 magnetic or Shubnikov groups are derived, which are listed in [9].

To find the possible magnetic groups of a material, we have to start with itsspace group and see which magnetic space groups can be derived from it. Mag-netic structures can, in principle, have lower symmetry than the underlying crystalstructure. Therefore it is often necessary to look for Shubnikov groups with lower

symmetry.One shortcoming of the magnetic space groups is that they can only describe

magnetic structures with periodicity which is in some way related to the periodicityof the crystallographic structure - the so-called commensurate structures [1]. Forincommensurate structures, having a periodicity unrelated to the crystallographicstructure, a generalisation in the form of magnetic superspace groups is needed [ 10].In the next chapter we will take a look at representation theory which is an evenmore powerful tool for finding the symmetry properties of magnetic structures butis also useful in other fields; e.g., in vibrational spectroscopy, particle physics andcrystal field calculations in solid state physics [3].

2.2 Representations

In this chapter we will present how representation theory can be used to reduce thenumber of possible magnetic structures and of parameters needed in the refinementof the structure. A very nice review of representation theory for the use in theanalysis of scattering data is presented in [10], a compact tutorial can be found in[1] but for a more rigorous mathematical derivation one should look at [3] or [11].

If we can find a set of linear operators (g) in a vector space L, which corre-spond to the elements g of a group G in the sense that [3]

(gi) (gj) = (gigj) gi, gj G, and that (E) = 1 , (1)

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2 Groups

then this set of operators is said to form a representation of the group G in thespace L. If the dimension of the space L is s, then the representation is said tobe s-dimensional [3]. From the relations (1) we can see that representations aresets of objects which have the same internal structure as the group regarding the

multiplication operation (see Table 2). In practice one usually defines an operatorby its matrix with respect to some chosen basis. Hence by choosing a basis e1, e2, ...in L we can form a matrix for each operator (gi) [3]. By applying an appropriaterotation to the matrices of all operators (by changing the basis), it is possible toform matrices with square blocks on the diagonal and zeros everywhere else. If U isthat rotation and:

(g) = U(g)U1 for g G, (2)

then we could for example get something like:

(g) =

A11 A12 0 0 0 0 0A21 A22 0 0 0 0 0

0 0 B11 0 0 0 00 0 0 B11 0 0 00 0 0 0 C11 C12 C130 0 0 0 C21 C22 C230 0 0 0 C31 C32 C33

= A(g) 2B(g) C(g) , (3)

where A(g), B(g) and C(g) are also representations of the group G [10]. We say wehave reduced the representation to a number of lower-dimensional representations.In terms of vector space the basis vectors belonging to these blocks form invariantorthogonal vector subspaces. If these smaller representations cannot be reduced toblock diagonal form, they are said to be irreducible.

A reducible representation can be decomposed into a direct sum of irreduciblerepresentations (IRREPs) in a unique way. We will note the different IRREPs with. Two representations are regarded equivalent if there exists a rotation that trans-forms one into the other. It is possible that a number of equivalent representationsappear in the reduction of a representation into its irreducible constituents. In sucha case they are regarded as the same IRREP and the appropriate is given aweight n equal to the number of its equivalent representations in the composition.The decomposition of a reducible representation can be written as:

=

n = n11 n22 ... nmm , (4)

where m is the number of distinct IRREPs ofG.The reductions of the representations of space groups to IRREP are given in [12]

and the mathematical relations for calculations with them in [10, 11]. These calcu-lations may become tedious, therefore it is much more convenient to use appropriatecomputer programs for such calculations (BasIreps in the FullProf suite [13]). Wecan see an example of the IRREPs calculated by BasIreps in the Table 2.

In the case of point and space groups the objects in our representations arematrices which represent how the position vectors of atoms are transformed by thesymmetry operations. In the context of determining the magnetic structure thegoal of representation analysis is to find in how many different ways the symmetry

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3 Magnetic structure and the little groupGk

Table 2: The multiplication table of the elements of space group no. 15 on the left and itsirreducible representations in the right table. The notation is: 1 is the identity, -1 theinversion operation, 2 the screw axis and m the glide plane. In the table on the right the 4IRREPs of the same group are given. In each row of the table, a matrix representation of

the symmetry operation is stated for a given IRREP. To test if the multiplication relationsof the space group are preserved, we multiply the matrix representations of two elementsand check if the result is consistent with the multiplication table of the group on the left.For example we can check the relation for elements -1 and 2 in IRREP 3 - the operation-1 is in this case represented by the matrix 1 and 2 by the matrix -1. If we multiply thesematrices we get -1. In the table on the left we see the product of these two operationsis the operation m, represented by -1 in IRREP 3, which equals to the result of ourmultiplication. We can preform this test for all the pairs of operations in all IRREPS andsee, that the multiplication relations of the group still hold in its IRREPs.

1 2 -1 m

1 1 2 -1 m

2 2 1 m -1-1 -1 m 1 2m m -1 2 1

IRREPsSymmetry operators1 2 -1 m

1 1 1 1 12 1 1 -1 -13 1 -1 1 -14 1 -1 -1 1

restrictions of the crystal structure can be satisfied by the magnetic structure. Rep-resentation analysis is therefore a systematic way of finding all possible magneticstructures. In the next section we will show that it can also reduce the number ofrequired parameters of the model.

3 Magnetic structure and the little group Gk

The magnetic structure does not necessarily have the same periodicity as the crystalstructure - the magnetic unit cell can have some or all dimensions multiplied by somenumber compared to the crystallographic unit cell or it can even be incommensurate(Figure 3). Instead of describing the whole magnetic unit cell it is therefore moreconvenient to write down only the magnetic moments of all the magnetic atoms inthe zeroth crystallographic unit cell and the magnetic propagation vector k, whichtells us how the magnetic moments of atoms in equivalent crystallographic sites inother unit cells are related to the one in the zeroth cell. In the simplest case with asingle propagation vector this relation is [1]

mj(R) = jei2kR, (5)

where mj is the magnetic moment at the atomic site j in some unit cell, that isrelated to the zeroth by a translation R and j the magnetic moment in the sameatomic site in the zeroth unit cell.

The periodicity of the magnetic phase affects the positions of magnetic reflections- they are shifted from the positions of nuclear peaks by the vector k. Magnetic peaksare therefore present at moment transfer vectors Q,

Q = + k, (6)

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4 Effect of symmetry elements on atoms possessing magnetic moments

Figure 3: A few examples of magnetic unit cells that are larger than the crystallographicunit cell. In the first line we can see the positions of the atoms and the crystallographicunit cell, in the second the magnetic unit cell in the case of a ferromagnetic order and inthe third and fourth in the case of antiferromagnetic order. We can see that the size ofthe magnetic unit cell is doubled and quadrupled compared to the crystallographic cell. Inthe last line we can see an example of an incommensurate structure, which has periodicityunrelated to the crystallographic structure.

where is a vector of the crystallographic reciprocal space [1].This different periodicity means that the magnetic structure can have lower

symmetry than the crystallographic one. To find the magnetic symmetry we mustcalculate how the propagation vector k transforms under the symmetry operationsof the space group. Lets take an operation g which is an element of the crystalsspace group G. The operation can be written as g = {h|th}, where h and th are therotational and translational part of the symmetry operation, respectively. Those

operations g G for which it is true

hk = k + , (7)

where is some vector of the crystals reciprocal lattice, form the little group Gkwhich is a subgroup ofG [1]. The elements of the little group leave the magneticstructure invariant. We can use this fact to find all the possible structures bydetermining the irreducible representations of the little group and their base vectors.

Let us say that the magnetic atom is positioned at some Wyckoff site labelled jand the equivalent atoms of this site are j1, j2, ..., jpj , where pj is the multiplicityof the site. Each of these atoms has a magnetic moment, which can be describedby a vector, therefore we need to find nj = 3 pj components or a nj dimensional

vector to describe the magnetic structure. This vector corresponds to the magneticmoments of atoms in the zeroth cell, but we can find the moments of all atoms in acrystal by knowing the magnetic propagation vector k and using the relation (5).

4 Effect of symmetry elements on atoms possessingmagnetic moments

Symmetry operations effect the magnetic moments in two ways - they change theposition of the atom and reorient the magnetic moment. The combined effect isdescribed by the magnetic representation mag [1].

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4 Effect of symmetry elements on atoms possessing magnetic moments

rot

14

23

21

34

perm(g) =

0 0 0 11 0 0 00 1 0 0

0 0 1 0

, axial(g) =

0 1 01 0 0

0 0 1

Figure 4: The effect of a rotation of 90 about an axis in the centre of the square on thepositions of the atoms 1 trough 4 and the magnetic moment they carry. On the left wecan see, how the positions of atoms are moved and the magnetic moments reoriented andon the right we can see how these effects are described by the permutation representationperm(g) and the axial representation axial(g).

A symmetry operation moves the atom from the position i to position j of thesame crystallographic site. We can describe this by a permutation representation

perm, which has matrices of the order pj. If the symmetry operation results in anatomic position outside the zeroth cell, we have to include a phase factor exp(i),that relates the generated position to the one in the 0th cell. The phase is given by = 2k t, where t is the translation vector that relates the generated position tothe equivalent one in the 0th cell [1].

The second effect of the symmetry operation is that it transforms the momentumcomponents of the original atom as axial vectors. This can be described by the axialrepresentation axial, which is formed of the rotational parts h of the operationsg = {h|th} of the little group Gk, multiplied by -1 if g is an improper rotation[10].An illustration of both effects is given in Figure 4.

The magnetic representation mag is formed as the direct product of these two

representations mag = perm axial. (8)

Representation mag is nj dimensional and it affects the nj dimensional vector ofmagnetic moment components of the atoms in the zeroth crystallographic cell. magcan be decomposed to IRREPs of the Gk. We can calculate the number of possiblebasis vectors of an IRREP of Gk: NBV = n l, where NBV is the number ofbasis vectors of the representation , n the number of times is contained inthe decomposition of mag and l the dimension of [10]. These vectors formthe basis for the magnetic moments ordered according to a particular IRREP. Themagnetic moments can therefore be expressed as some linear combination of thesebasis vectors.

In most cases the magnetic structure is arranged according to a single IRREP[1], so we have effectively changed the number of parameters from nj to NBV.If NBV is smaller than nj, we have thus reduced the number of free parameters.When representation analysis fails to reduce the number of parameters, we mustfind another way to choose among all possible structures.

We will not go into the details of the procedure for calculating the basis vectorsof the possible IRREPs as they can be found in [10]. Instead we will just rely onthe program BasIreps for these calculations. We will add one more thing: in thecase of one dimensional real IRREPs, if a symmetry operation is represented by -1,that means that the operation is combined with time inversion in that IRREP [10].

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5 Example: application of symmetry analysis to neutron scattering data of CuSe2O5

Figure 5: Here we can see the crystal structure of CuSe2O5. The CuO4 plates are connectedalong thec direction by a bridge formed by the selenium and oxygen atoms. The unit cellof the compound is monoclinic with the angle 112 between the axesa and c. Strong

interaction between the copper atoms along the c direction results in an 1D magneticcharacter. The long chains also affect the macroscopic appearance - the single crystalsare long and thin and can crack along the c direction. [15]

This is an important fact we will use later to explain why only some of the IRREPsof the little group allow ordered magnetic moment at a particular site and othersdont.

5 Example: application of symmetry analysis to neutronscattering data of CuSe2O5

Now we will present the procedure which has led to the successful determinationof the magnetic structure of CuSe2O5 by applying the theory we have presented tosingle-crystal neutron scattering data. After finding the little group of the magneticpropagation vector k we will preform its representation analysis and try to find thebest fit of the reduced parameters of the magnetic structure to the single-crystaldata.

The compound CuSe2O5 crystallises in the monoclinic space group C2/c. Thematerial is quasi-1D with zig-zag chains of CuO4 plates bridged by Se2O5E2 (E is alone pair of electrons - they are not shared in a bond) dimers along the c direction[14] as we can see in Figure 5. The Cu2+ ions carry S = 1/2 spins which order

antiferromagnetically below TN = 17 K [15].The experimental data we will use for the refinement of the magnetic structure

was acquired with the instrument TriCS at the SINQ facility at the Paul ScherrerInstitute (PSI) in Switzerland [16]. The sample used was a CuSe2O5 single crystalwith a mass m = 11.1 mg. Monochromatic neutrons with = 2.46 A were used tofind the integrated intensities of magnetic peaks. The experimental set-up is shownin Figure 6. The type of scan used was an omega scan - at a position of a given peakthe scattering angle 2 was held constant and the sample was rotated about its axis.The magnetic peaks were found by comparing scattering patterns at T = 6 K and T= 20 K. No structural deformation is associated with the magnetic phase transition

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Figure 6: The experimental set-up of the TriCS instrument at PSI. In the presented casethe positions of the incoming neutron beam and the detector were fixed for a given peak,thus defining the scattering angle 2. To collect the integrated intensity of a peak, thesample was rotated about its axis - the so-called omega scan. [16]

at T = 17 K, therefore all differences in the scattering patterns were attributed tomagnetic scattering.

The propagation vector k would usually be found by a neutron scattering exper-iment on a powder sample. In this case the whole scattering pattern is collected anda computer program is used to find the propagation vector with which the magneticpeaks can be indexed. However, if the quality of data is not sufficiently good, thisprocedure may fail [2]. This was the case with CuSe2O5 powder sample and thecorrect k had to be determined from the single crystal data. By finding scatteringpeaks shifted according to (6), the magnetic propagation vector k = (1,0,0) was

determined. Then calculations were made with probable models of the magneticstructure to find which magnetic peaks could be the strongest and intensities of 20magnetic reflections were collected. Usually as many as possible peaks are measuredto improve the statistics but in this case that was not feasible because of long dataacquisition times.

5.1 Representation analysis

Now that we know the propagation vector k, we must calculate its little group Gkand find the irreducible representations of this group. For this we used the programBasIreps in the FullProf suite [13]. As input we specify the space group of the crystalstructure of the material and the propagation vector of the magnetic structure andBasIreps returns the irreducible representations of the magnetic structure. TheIRREPs for space group C2/c and k = (1, 0, 0) are listed in Table 3.

If we specify the position of the magnetic atom (copper in this case) before theprocedure, the program also calculates the magnetic irreducible representations andtheir respective base vectors. In our case only two IRREPs can represent magneticmoments at the site of copper, 1 and 3. This is because the Wyckoff positionof the copper atom (4a) has site symmetry -1 which means it is left invariant bya space inversion. Space inversions also leave axial vectors invariant, which meansthe magnetic moment of the copper atom is not changed by the space inversion.Therefore only those representations in which the space inversion {-1|000} is not

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Table 3: Irreducible representations of Gk, where p = 1/2. The numbers below the symbolsfor the symmetry operations are the matrices representing each operation in a IRREP.We can see all representations are one dimensional in this case.

IRREPsSymmetry operators

{1|000} {2 0y0|00p} {-1|000} {m x0z|00p}

1 1 1 1 12 1 1 -1 -13 1 -1 1 -14 1 -1 -1 1

combined with the time inversion operation (the matrix for the inversion must be 1and not -1) leave the magnetic moment of the copper invariant. The other twoIRREPs (with the {-1|000} matrix in Table 3 equal to -1) invert the magneticmoment of the copper, which means it must be zero - they are paramagnetic.

Because we have two magnetic atoms, the basis vectors have nj = 3 pj = 6components each. The first three components of each BsV correspond to the threemagnetic moment components of the Cu atom at the site with symmetry x,y,z andthe last three to those of the atom at the site -x,y,-z+1/2.

By making a linear combination of the basis vectors we can form the vectorswhich represent the magnetic moments of both atoms. In the case of 1 this vectoris (u,v,w,u,v,w) and for 3 (u,v,w,u,v, w). In we need only 3 parametersfor each representation to write down the two magnetic moments:

1 3Crys. pos. x,y,z x,y,z

Mag. symm. (u,v,w) (u,v,w)Crys. pos. -x,y,-z+1/2 -x,y,-z+1/2

Mag. symm. (-u,v,-w) (u,-v,w)

The transformation properties of the moments are implicit in this notation. Wecan see that in both possible representations of the magnetic structure the ampli-tudes of the components of the magnetic moment do not change when transformingfrom one atom to the other, only their direction does. With this symmetry-relatedtransformation of the magnetic moment and the propagation vector k, we have thefull information about the magnetic structure. The transformation tells us whathappens to the moment when we move from the atom at position (0,0,0) to the one

at (0,0,1/2) - the atoms are related by a screw axis operation, but the magneticmoment changes the direction ofa and c components in the case of 1 or just the bcomponent in the case of 3.

The atoms at positions (1/2,1/2,0) and (1/2,1/2,1/2) are related to those at(0,0,0) and (0,0,1/2) respectively, by a lattice translation t = (1/2, 1/2, 0), so theirmoments change according to (5), which means they are multiplied by a factorexp(i2k t) = exp(i) = 1. The moments of these two atoms therefore havethe same symmetry as those of the first two, but they are oriented antiferromag-netically to them. This phasing effect of the k vector is important, because in thecase k = (0, 0, 0) the symmetry transformations alone would leave a non-zero netmagnetic moment and the material would be ferromagnetic.

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0 50 100 150

0

50

100

150

Iobs

Icalc

1

(a)

0 50 100 150

0

50

100

150

Iobs

Icalc

3

(b)

Figure 7: The agreement between experimental (Iobs) and calculated (Icalc) integrated in-tensities of 20 magnetic peaks for both possible models. We can see a significantly betteragreement for the representation 3 with

2 = 4.28 andR = 21.6 contrasted by2 = 21.1andR = 43.3 for the 1 model.

With representation analysis we have reduced the number of parameters neededto describe the magnetic structure from 6 (3 for each of the two atoms) to 3 (u,v and w). All that is left to do is to find the true magnetic structure of CuSe2O5by comparing both possible solutions to experimental neutron diffraction data anddetermining which provides the best fit.

5.2 Refinement and comparison of models

For the refinement process we used the program FullProf [13]. The input dataof the refinement process are the cell parameters of the crystal structure togetherwith the positions of magnetic atoms, the magnetic symmetry information of theexpected irreducible representation that BasIreps generated and the experimentalintegrated intensities of the peaks we have measured. The program then minimisesthe difference between the experimental and calculated intensities by finding theoptimal parameters of the magnetic moments. We have chosen the parameters tobe the components of the magnetic moment along the crystallographic axes of thecrystal in units of the Bohr magneton (B).

In Figure 7 we can see the agreement between the experimental integrated in-tensities of the measured peaks and their calculated values for the best parameters

found with the refinement in FullProf for both irreducible representations. Figure7a shows that there are many peaks with higher measured intensities than predictedby the 1 model (those below the diagonal) but also some that the model predictsto have higher intensities than were measured. The values of 2 and R for thisrefinement are 21.1 and 43.3 respectively. Both of these factors are measurementsof goodnes-of-fit of the model to the data. They are not absolute but can be used tocompare how well different models fit the same data set, the one with the lower val-ues being the best [17]. Figure 7b shows that the agreement between the calculatedand measured intensities is significantly better for the model 3, with

2 = 4.28and R = 21.6, but there are still some peaks that significantly deviate from themodel. During the cooling of the sample the nuclear peaks widened and split. This

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6 Conclusion

a

b

c

Figure 8: The magnetic structure of CuSe2O5 for the refined magnetic parameters of therepresentation 3. We can see that the magnetic moments are oriented predominantlyalong the b axis with c components being 0.

was probably the result of different thermal expansion coefficients of the crystal andthe glue that was used to secure the crystal to the mount [18]. These imperfections

probably contributed to deviations between the calculated and measured intensities.The magnetic moment components along the crystallographic axes in units of

B found in the refinement of the model 3 are ma = 0.13(7), mb = 0.50(1),mc = 0.00(8), yielding the magnitude |m| = 0.52(2). The magnitude of the or-dered magnetic moment is lower than expected (1 B). This is most likely causedby quantum fluctuations which can be present in low dimensional systems even atthe lowest temperatures [18] and are most pronounced in the extreme quantum limitof S = 1/2. The most probable magnetic structure of CuSe2O5 is depicted in Figure8.

6 ConclusionThis seminar is a brief introduction to a broad and complex field of magnetic struc-ture determination with the help of symmetry analysis. I tried to make the text asconcise as possible to a first time reader, as not many things are said about symme-try analysis and magnetic structures in the undergraduate course of physics. For amore in-depth look at the subject one should consult the references.

We have shown that representation analysis provides us with a systematic wayto derive the possible magnetic structures form the symmetry information of thecrystallographic structure and even help us reduce the number of parameters neededto describe the structure. The development of computer programs such as FullProfhas simplified and sped-up many tasks in representation analysis and experimentaldata analysis but the understanding of the theoretical fundamentals is still necessaryto critically asses when a wrong solution is derived. The reader should rememberthat the theory we have presented is just a tool to help us look for the magneticstructure in the right place. In the end true magnetic structure of a material canonly be found in agreement with experimental data.

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References

References

[1] A. S. Wills, J. Phys. IV France 11, PR9-133 (2001).

[2] D. E. Cox, IEEE Trans. Magn. 8, 161 (1972).

[3] J. P. Elliot, P. G. Dawber, Symmetry in Physics: Principles and Simple Appli-cations; Volume 1 (MacMillan, London, 1986).

[4] N. W. Ashcroft, N. D. Mermin, Solid State Physics (Brooks/Cole, Belmont,1976).

[5] http://en.wikipedia.org/wiki/Crystallographic point group (24. 5. 2012).

[6] International Tables for Crystallography Vol. A, edited by Th. Hahn (Wiley,2006).

[7] A Hypertext Book of Crystallographic Space Group Diagrams and Tables(Birkbeck College, University of London, 1999)http://img.chem.ucl.ac.uk/sgp/mainmenu.htm.

[8] Bilbao Crystallographic Serverhttp://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list.

[9] D. B. Litvin, Magnetic space groups (2011)http://www.bk.psu.edu/faculty/litvin/Download/THE Magnetic Group Tables J11.pdf(18.8.2012).

[10] J. Rodrguez-Carvajal, F. Bouree, EPJ Web of Conferences 22, 00010 (2012).

[11] Z. L. Davies, Application of Representation Theory to Magnetic and StructuralPhase Transitions (Phd dissertation, University College London, 2009).

[12] O. V. Kovalev, Irreducibe Representations of the Space Groups (Gordon andBreach, New York, 1961).

[13] FullProf, http://www.ill.eu/sites/fullprof/.

[14] R. Becker, H. Berger, Acta Cryst. E 62, i256 (2006).

[15] M. Herak et al., Physical Review B 84, 184436 (2011).

[16] http://www.psi.ch/.[17] B. H. Toby, Powder Diffraction 21, 67 (2006).

[18] M. Herak et al., One-dimensional and long-range order in quasi-one-dimensional CuSe2O5 (unpublished experimental report).

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