Q Data for magnetic structure factor calculations

Q cards have the format:

Q CCSL-word data

or

Q label data
where label is an atom-label or a scattering-factor-label.

DATA FOLLOWING ALLOWED CCSL WORDS:

In the first case the CCSL-word may be one of STYP PROP MSYM or NSYM , and in the second one of FORM MU SDIR CHI or PSI .

STYP
Data

A CCSL-word describing the magnetic structure type. The types defined are:

ANTI

ordinary comensurate antiferromagnetic structure

AMOD

incommensurate amplitude modulated antiferromagnetic structure

FERO

unmagnetised ferromagnet with equal population of all domains

FERA

aligned ferromagnet with spins in the direction given by the the $z$ diffractometer axis.

HELI

helical spin structure

PARA

aligned paramagnet: moments calculated from susceptibilities (used by program CHILSQ)

PROP
Data

The three components of the propagation vector in reciprocal space coordinates.

MSYM
Data

Pairs of integers defining the magnetic space-group. The first number of a pair is the identifying number of one of the generating elements of the group, and the second is $-1$ if the symmetry operation is combined with time-reversal, and 1 if not. The special operator number $-1$ is used to refer to a centre of symmetry.

There must be one such pair of integers for each of the generating elements of the magnetic group.

Note

In CCSL the space group symmetry operators are assigned identifying numbers which can be printed out using OPSYM. It is these numbers which are used to refer to individual operators when defining their magnetic symmetry.

NSYM
Data

Operators for symmetry elements not in the magnetic space group.

The magnetic space group M may or may not coincide with the nuclear group N but M must be a subgroup of N .

In the case that M $\ne$N then some NSYM cards are needed to describe how each symmetry element which is not in the magnetic group acts on the directions of the magnetic moments. There is one NSYM card for each element of the factor group F where M $\times$F = N .

The data given on the NSYM cards are the integer label which has been assigned to the operator, followed by nine real numbers. These specify the matrix describing the rotation with respect to the representative atom, of the moment on the atom generated from the representative atom, by the operator. The matrix is expressed in the CCSL orthogonal axes.

FORM
Data

A scattering-factor-label appearing on an F card, followed by FORM , followed by a list of atom-names.

Not all atoms appearing on A cards are necessarily magnetic. They are defined to be magnetic if their names appear to the right of FORM on a Q card.

The form factor will be used as the magnetic form factor applying to all the atoms whose atom-labels appear to the right of FORM on the card.

MU
Data

An atom-label corresponding to one on an A card, followed ny MU and, unless the structure type is HELI, a single number giving its magnetic moment in Bohr magnetons.

The CCSL-words MU and SDIR refer to individual magnetic atoms. There must be one of each of these cards for each of the A cards which refer to magnetic atoms.

If the structure type is HELI two numbers are required which are the major and minor axes of the elliptical envelope of the helix in Bohr magnetons.

Note

The moment values are referred to as MU and MU1 in the Least Squares programs.

SDIR
Data

An atom-label corresponding to one on an A card followed by SDIR and the spherical polar angles $\theta$ and $\phi$ of the moment direction of the atom with respect to CCSL orthogonal axes.

If the structure type is HELI a second pair of angles is required; the first pair give the orientation of the major axis of the elliptical envelope and the second pair the direction of the minor axis (the two directions must be perpendicular).

Details

The angles are given in degrees.

The vector describing a moment direction or one of the axes of the elliptical envelope thus has components:

$\mu \sin\theta \cos \phi$, $\mu \sin\theta \sin\phi$ and $\mu \cos\theta$ on the orthogonal CCSL axes

Note

These angles are referred to by the names THET, PHI, THE1, PHI1 respectively in the Least Squares Programs.

CHI
Data

If STYP = PARA: an atom-label corresponding to one on an A card followed by CHI and up to 6 numbers which are the coefficients X of the anisotropic magnetisation tensor for that atom in the order $X_{11}$, $X_{22}$, $X_{33}$, $X_{23}$, $X_{31}$ and $X_{12}$ in that order. If only one number is given the refinement will start from an isotropic magnetisation of that value.

Note

The coefficients are referred to as CH11, CH22 ...etc. by the least squares programs.

PSI
Data

An atom-label corresponding to one on an A card followed by up to four pairs of numbers. The first number of each pair is the integer label of an operator and the second is the phase shift in degrees to be applied to the sublattice generated by that operator.

Note

One or more PSI cards will be required for each magnetic atom when the structure type is AMOD or HELI and the magnetic symmetry is less than the nuclear symmetry, i.e. there are one or more NSYM cards.

Phase shifts must be defined for operators not in the magnetic group which relate different atomic positions of an equivalent set. At present there is space for only four phase shifts per atom which means that the programs can not deal with structures in which magnetic atoms occur on sites whose multiplicity due to symmetry operators not in the magnetic group is greater than four.

EXAMPLES:

The following is an example of part of the Crystal Data describing the magnetic structure of Mn₃ Sn:

A Mn 0.8415 0.68291 0.25000 0.00000
S x-y, x, 1/2+z
S x, y, 1/2-z
S y, x, 1/2+z
F Mn 1 -0.37300
F MnM 2 0.4220 17.6840 0.5948 6.0050 0.0043 -0.6090 -0.0219
Q PROP 0 0 0
Q STYP ANTI
Q MnM FORM Mn
Q MSYM -1 1 4 -1 8 -1
Q NSYM 2 -.5 .866 0 -.866 -.5 0 0 0 1
Q NSYM 3 -.5 -.866 0 .866 -.5 0 0 0 1
Q Mn MU 3.0
Q Mn SDIR 90.0 60.0

The first two Q cards indicate an antiferromagnetic structure with zero propagation vector (nuclear and magnetic cells are the same).
The S cards define space group $P6_3/mmc$ and the output from OPSYM(1) is:

           General equivalent positions are:
         0                   0                   0       +-
  1      x                   y                   z
  2      x-y                 x                  1/2+z
  3     -y                   x-y                 z
  4     -x                  -y                  1/2+z
  5     -x+y                -x                   z
  6      y                  -x+y                1/2+z
  7      y                   x                  1/2+z
  8     -x+y                 y                   z
  9      x                   x-y                 z
 10     -x                  -x+y                1/2+z
 11      x-y                -y                  1/2+z
 12     -y                  -x                   z

This information is needed to understand the Q MSYM and Q NSYM cards. The interpretation of the card Q MSYM -1 1 4 -1 8 -1 is as follows:

Operator $-$1:

(the centre of symmetry) is not combined with time reversal so that the spins on atoms related by the centre of symmetry are parallel.

Operator 4:

$-x,-y,1/2+z$ (a screw diad parallel to $z$) is combined with time-reversal and atoms related by this operator have parallel $x$ and $y$ components and anti-parallel $z$ components.

Operator 8:

$-x+y,y,z$ (a mirror plane bisecting the angle between $x$ and $y$) is also time-reversing and, since spins are axial vectors, the components parallel to the plane, of spins which it relates, are parallel to one-another whereas those perpendicular to the plane are anti-parallel.

The NSYM cards

Q NSYM 2 -.5 .866 0 -.866 -.5 0 0 0 1
Q NSYM 3 -.5 -.866 0 .866 -.5 0 0 0 1
indicate that the operators 2 and 3 which describe the screw hexad (6₃ ) and triad operations respectively are not in the magnetic space group. The matrix on the NSYM 2 card implies that atoms related by the screw hexad have spin directions rotated with respect to one-another by 60^o about $z$ but with the rotation direction opposite to that of the symmetry axis. The matrix acompanying element 3 implies that the spin rotation associated with the triad axis is also opposite to that of its symmetry operation.

The MU , SDIR and FORM cards indicate that the Mn atom is magnetic with a spin of 3.0 $\mu_B$. The representative Mn atom (that whose position is given on the A card) has its spin direction in the $x-y$ plane ( $\theta=90^\circ$) at $60^\circ$ to orthogonal $x$ ($10\overline10$). The magnetic form factor for Mn is MnM.

A second example desribes the magnetic structure of the helical phase of monoclinic CuO:
S 1/2+x, 1/2+y, z
S 1/2+x, 1/2-y, 1/2+z
S -x, -y, -z
A Cu 0.25000 0.25000 0.00000 0.00000
Q CuM FORM Cu
Q PROP .507 0 -.482
Q STYP HELI
Q Cu MU 0.5013 0.4860
Q Cu SDIR 90.0000 90.0000 28.2215 0.0000
Q NSYM 2 1 0 0 0 1 0 0 0 1
Q NSYM -1 1 0 0 0 1 0 0 0 1
Q Cu PSI 2 4.5

Note the irrational values on the Q PROP card for this incommensurate structure. In this example neither the twofold axis (operator 2) nor the centre of symmetry is in the magnetic group, but in both cases the atoms related by these operators have parallel spins. The major and minor axes of the elliptical envelope of the spin helix are 0.5013 and 0.4860 $\mu_B$, oriented parallel to b and at $28.2^\circ$ to c in the (010) plane, respectively. The phase of the spiral based on the Cu atom at ( ${3\over4} {3\over4} {1\over 2}$) leads that on the representative Cu atom ( ${1\over4} {1\over4} 0$) by 4.5^o.