Bilbao Crystallographic Server NEUTRON Help |

NEUTRON is a computer program for
calculating the phonon extinction rules for inelastic neutron
scattering experiments. Given the space group and the phonon symmetry
specified by the wave vector **q**, the program examines the
inelastic neutron scattering activity of the corresponding phonons
for all possible types of scattering vectors **Q**=**H**-**q**,
where **H** is a reciprocal lattice vector.

The applied algorithm is based on a
recently proven theorem that demonstrates the existence of
symmetry-based selection rules for the phonon activity in inelastic
neutron scattering experiments: the resulting phonon absences depend
only on the mode symmetry and the scattering vector **Q,** and not
on the specific atomic positions of the crystal structures
(Perez-Mato *et al.*, 1998). Using the distribution of the
reciprocal lattice vectors (Brillouin zones) into types with respect
to the symmetry group of the scattering vector, it is possible to
show that Brillouin zones belonging to the same type are
characterized by the same set of selection rules. The derived
extinction rules prove to be very useful for the identification of
the symmetries of the measured phonons, and their systematic use
helps the optimization of inelastic neutron scattering experiments
(Aroyo *et al.*, 2002a). These selection rules also apply to
phonon *X*-ray inelastic scattering, and their use has been
successfully extended to thermal diffuse scattering studies (Aroyo *et
al*., 2002b).

For more details related to the program
NEUTRON, its algorithm and the procedure for the calculation of the
phonon selection rules, the user is referred to Kirov *et al*.,
2003.

**REFERENCES**

- Perez-Mato, J.M., Aroyo, M.I.,
Hlinka, J., Quilichini, M. & Currat, R. (1998).
*Phys. Rev. Lett.***81**2462-2465. - Aroyo, M.I., Boysen, H. &
Perez-Mato, J.M. (2002a).
*Physica***316-317,**154-157. - Aroyo, M.I., Boysen, H. &
Perez-Mato, J.M. (2002b).
*Applied Physics*A**74**, S1043-S1045. - Kirov, A.K., Aroyo, M.I. &
Perez-Mato, J.M.(3003).
*J.**Applied Crystallogr*. (to appear).

**Space-group specification:***Option 1:*Enter the sequential numbers of the supergroup G and subgroup H as given in*International Tables for Crystallography volA*(referred to as ITA) or choose them from the table of space groups. Please note the default choices of the space-group settings which are used in this program.*Option 2:*It is also possible to carry out the calculations for a non-default setting of the space group. In this case the user is expected to provide the transformation between the user-chosen coordinate system of the space group and that of the default ITA-setting. The transformation consists of a rotational (3x3) matrix and an origin shift (3x1) column.

**q-vector data:**NEUTRON expects**q**-vector coefficients in the form of 3 decimal numbers and a label of the wave-vector. The**q**-vectors are neither restricted to the representation domains nor to the Brillouin zones. The program accepts**q**-vector coefficients determined with respect to different coordinate systems of the reciprocal space:*Option 1*:**q**-vector coefficients referred to the primitive basis of the reciprocal space as found, for example, in Cracknell , Davies, Miller and Love (1979)*Kronecker Product Tables, vol.1. General Introduction and Tables of Irreducible representations of Space Groups*(IFI/Plenum: New York). The wave vector database KVEC on the Bilbao Crystallographic server provides a complete list of the primitive**q**-vector coordinates of all special**q**-vector points, lines and planes for all space groups.*Option 2*: Another possibility is to refer the wave vectors with respect to bases of the reciprocal space which are dual to the conventional ITA coordinate systems, the so-called*conventional dual***q**-vector**q**-vector coefficients are used for the calculations. Please note that primitive coefficients coincide with the conventional dual ones only for space groups with primitive lattices. The following table gives the relation between the conventional dual**q**-vector**Table 1:**Conventional**q**-vector coefficients (q_{1}, q_{2},q_{3}) expressed by the primitive**q**-vector coefficients (q_{p1},q_{p2},q_{p3})

Lattice types

q

_{1}q

_{2}q

_{3}aP, mP, oP, tP, cP, rP, hP

q

_{p1}q

_{p2}q

_{p3}mA, oA

q

_{p1}q

_{p2}-q_{p3}q

_{p2}+q_{p3}mC, oC

q

_{p1}+q_{p2}-q

_{p1}+q_{p2}q

_{p3}oF, cF

-q

_{p1}+q_{p2}+q_{p3}q

_{p1}-q_{p2}+q_{p3}q

_{p1}+q_{p2}-q_{p3}oI, cI, tI

q

_{p2}+q_{p3}q

_{p1}+q_{p3}q

_{p1}+q_{p2}hR

q

_{p1}-q_{p2}q

_{p2}-q_{p3}q

_{p1}+q_{p2}+q_{p3}*Option 3*: There is also a possibility to input the wave-vector coordinates referred to a coordinate system which is dual to the non-default setting of the space group defined by the user (see*Option 2*in**Space-group specification**)

The **OUTPUT** of the program
consists of three main blocks: a Space-group data block, a **q**-vector
data block including the little group irreducible representations
(irreps), and a data-block with **Q**-vector types and extinction
rules.

**Space-group data block:**The listed data starts with the ITA number of the space group and its lattice type. Then follows the set of non-translation generators used by the program for the construction of the matrix-column pairs of the coset representatives of the group with respect to its translation subgroup. Both the generators and the coset representatives are listed in a (3x4) matrix form.

**q-vector data block:**The wave-vector block starts with the information on the

**q**-vector coordinates given by the user. For the calculations the program uses conventional dual wave-vector coordinates,*i.e*. those referred to a coordinate system of the reciprocal space which is dual to the default conventional settings of the space group. The translational coset representatives of the little group of the wave vector are listed as matrix-column pairs in a (3x4) matrix form. The matrices of the little group representations are listed for all translational coset representatives of the little group in a consecutive order. The matrix coefficients (in general complex) are given in the polar form (modulus and phase angle [deg]. The labels of the representations consist of the wave-vector letter(s) and a consecutive number determined by the order of the irrep generation.**Q-vector types and extinction rules:**A line with the conventional dual coordinates of the general form of the scattering vector

**Q**=**H**-**q**= (h-q_{1}, k-q_{2}, l-q_{3}) indicates the beginning of the block. The extinction rules are listed in subblocks corresponding to the different types of**Q**-vectors. These types are distinguished by the different sets of restrictions on the values of the components h,k,l of the lattice vector**H**(h,k,l). The strict point group of a**Q**-vector type consists of all those elements of the little co-group which leave the**Q**-vector invariant. The phonon extinction rules are checked for pnonon symmetries,*i.e.*for all irreps of the little group. A symmetry type (irrep) of phonons can be active in inelastic neutron scattering experiments if it is listed as 'Allowed'.

** Example 1: **Phonon extinction rules for

**INPUT:**Specification of the space-group and the wave-vector data:This is the look and feel of the input form of the program NEUTRON. The user must enter the ITA number of Pnma which is 62, and the

**q**-vector coordinates (0,0,0), and its label: GM.**OUTPUT:**To see the results for the extinctions the space group Pnma,**q**=(0,0,0), click here.The space-group data-block of the OUTPUT contains the list of the non-translational generators as chosen in ITA. For example, in the case of

*Pnma*this set is <(2z|1/2,0,1/2),(2y|0,1/2,0),(-1|0,0,0)> (shown in the form of (3x4) matrices). The sets of 8 translational coset representatives for the group corresponds to the 'general position' lists of ITA. The wave-vector data-block starts with the**q**-vector coordinates followed by a list of the translational coset reprsentatives of the little group. As the wave vector we are considering is GM, its little group coincides with the space group Pnma and the 8 little group irreps for both cases are simply related to the irreps of the point group*mmm*.The results listed in the data-block on the

**Q**-vector types start with the general form of the scattering vector**Q**=**H**-**q**, which in this case coincides with the reciprocal lattice vector**H**. There are 7 special (nontrivial)**Q**-vector types, represented by the corresponding Brillouin zone vectors**H**. The strict point groups refer to the representives of the scattering-vector types which are listed. For example, the strict point group of**H**(0,0,l) is*2mm*= {1, 2_{z}, m_{y}, m_{x }}. The extinctions rules are given in blocks for each scattering vector type. Due to the non-symmorphic character of*Pnma*, one gets a different set of selection rules depending on the parity of the values of the h, k,l, components of**H**. For example, in the case of the Brillouin zone**H**(0,0,l), the phonons of symmetry GM5 and GM8 can be INS active if l=2n, while for l=2n+1, the active irreps are GM1 and GM4.

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