Setting up mcphas.j and other input files

In order to include the higher order interactions, the number of columns in the input file mcphas.j has to be increased. The additional columns then contain the higher order exchange constants.

Using the module so1ion, the input file mcphas.j may look as given for the example GdRu$_2$Si$_2$ in examples/gdru2si2:

# GdRu2Si2 anisotropic bilinear and biquadratic exchange 
#<!--mcphase.mcphas.j-->
#***************************************************************
# Lattice and Exchange Parameter file for
# mcphas version 3.0
# - program to calculate static magnetic properties
# reference: M. Rotter JMMM 272-276 (2004) 481
# mcdisp version 3.0
# - program to calculate the dispersion of magnetic excitations
# reference: M. Rotter et al. J. Appl. Phys. A74 (2002) 5751
#***************************************************************
#
# Lattice Constants (A)
#! a=4.165 b=4.165 c=9.654 alpha=  90 beta=  90 gamma=  90
#! r1a= 0.5 r2a=   0 r3a=   0
#! r1b= 0.5 r2b=   1 r3b=   0   primitive lattice vectors [a][b][c]
#! r1c= 0.5 r2c=   0 r3c=   1
#! nofatoms=1  nofcomponents=8  number of atoms in primitive unit cell/number of components of each spin
#****************************************************************************}
#! da=   0 [a] db=   0 [b] dc=   0 [c] nofneighbours=38 diagonalexchange=1 gJ=   2 sipffilename=Gd3p.sipf
# da[a]    db[b]      dc[c]       J11[meV]  J22[meV]  J33[meV]  J44[meV]  J55[meV]  J66[meV]  J77[meV]  Jbc[meV]  %%@
J88[meV]} 
-1.0 0.0 0.0 0.08390 0.08390 0.21890 -0.0024 -0.0096 -0.0008 -0.0096 -0.0024 
1.0 0.0 0.0 0.08390 0.08390 0.21890 -0.0024 -0.0096 -0.0008 -0.0096 -0.0024 
0.0 -1.0 0.0 0.08390 0.08390 0.21890 -0.0024 -0.0096 -0.0008 -0.0096 -0.0024 
0.0 1.0 0.0 0.08390 0.08390 0.21890 -0.0024 -0.0096 -0.0008 -0.0096 -0.0024 
-0.5 -0.5 -0.5 0.00380 0.00380 0.01000 0.0 0.0 0.0 0.0 0.0 
-0.5 -0.5 0.5 0.00380 0.00380 0.01000 0.0 0.0 0.0 0.0 0.0 
-0.5 0.5 -0.5 0.00380 0.00380 0.01000 0.0 0.0 0.0 0.0 0.0 
-0.5 0.5 0.5 0.00380 0.00380 0.01000 0.0 0.0 0.0 0.0 0.0 
0.5 -0.5 -0.5 0.00380 0.00380 0.01000 0.0 0.0 0.0 0.0 0.0 
0.5 -0.5 0.5 0.00380 0.00380 0.01000 0.0 0.0 0.0 0.0 0.0 
0.5 0.5 -0.5 0.00380 0.00380 0.01000 0.0 0.0 0.0 0.0 0.0 
0.5 0.5 0.5 0.00380 0.00380 0.01000 0.0 0.0 0.0 0.0 0.0 
-1.0 -1.0 0.0 -0.03630 -0.03630 -0.09445 0.0 0.0 0.0 0.0 0.0 
-1.0 1.0 0.0 -0.03630 -0.03630 -0.09445 0.0 0.0 0.0 0.0 0.0 
1.0 -1.0 0.0 -0.03630 -0.03630 -0.09445 0.0 0.0 0.0 0.0 0.0 
1.0 1.0 0.0 -0.03630 -0.03630 -0.09445 0.0 0.0 0.0 0.0 0.0 
1.5 -0.5 0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-1.5 -0.5 0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
1.5 -0.5 -0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
0.5 -1.5 0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
0.5 1.5 0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
0.5 1.5 -0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-0.5 1.5 0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-1.5 -0.5 -0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
1.5 0.5 0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-1.5 0.5 -0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-0.5 -1.5 -0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-0.5 -1.5 0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
1.5 0.5 -0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-0.5 1.5 -0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-1.5 0.5 0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
0.5 -1.5 -0.5 -0.00072 -0.00072 -0.00188 0.0 0.0 0.0 0.0 0.0 
-2.0 0.0 0.0 -0.01754 -0.01754 -0.04567 0.0 0.0 0.0 0.0 0.0 
2.0 0.0 0.0 -0.01754 -0.01754 -0.04567 0.0 0.0 0.0 0.0 0.0 
0.0 -2.0 0.0 -0.01754 -0.01754 -0.04567 0.0 0.0 0.0 0.0 0.0 
0.0 2.0 0.0 -0.01754 -0.01754 -0.04567 0.0 0.0 0.0 0.0 0.0 
0.0 0.0 1.0 0.24850 0.24850 0.13560 0.0 0.0 0.0 0.0 0.0 
0.0 0.0 -1.0 0.24850 0.24850 0.13560 0.0 0.0 0.0 0.0 0.0

here the meaning of the J11, J22, J33, J44, J55 ...is the exchange constant $J_{\alpha\alpha}$ between the operators $\hat I_1 \cdot \hat I_1$, $\hat I_2 \cdot \hat I_2$, $\hat I_3 \cdot \hat I_3$, $\hat I_4 \cdot \hat I_4$, ... etc. in equation (67). Note that in our notation the indices of the interaction operators $\hat \mathbf I$ are sometimes denoted by numbers, sometimes by characters: $\hat I_1,\hat I_2,\hat I_3, \hat I_4, \dots \equiv \hat I_a,\hat I_b,\hat I_c, \hat I_d, \dots$.

By default in module so1ion the operator sequence is $\hat I_1,\hat I_2,\hat I_3, \hat I_4, \dots =$ $\hat O_{11}$,$\hat O_{11}^s$, $\hat O_{10}$, $\hat O_{22}^s$, $\hat O_{21}^s$,$\hat O_{20}$, $\hat O_{21}$,$\hat O_{22}$,$\hat O_{33}^s$, ...,$\hat O_{66}$,$\hat J_x^2$,$\hat J_y^2$,$\hat J_z^2$, $\hat J_x^4$,$\hat J_y^4$,$\hat J_z^4$. according to equation (67) and table 2. The exchange constants for the products of these Stevens operators have to be given according to the following scheme:

   11 22 33 12 21 13 31 23 32 (3x3 matrix)
   11 22 33 44 12 21 13 31 14 41 23 32 24 42 34 43 (4x4 matrix)
   11 22 33 44 55 12 21 13 31 14 41 15 51 23 32 24 42 25 52 34 43 35 53 45 54 (5x5 matrix)
   etc ...

The other input and output files have the same format as usual, but with the modification, that there are now $n$ components of the 'spins' in mcphas.sps instead of 3 ($<\hat J_a>$,$<\hat J_b>$,$<\hat J_c>$). The expectation values of the operators $<\hat I_a>$,$<\hat I_b>$,$<\hat I_c>$,$<\hat I_d>$,$<\hat I_e>$, etc. correspond to the interaction operator sequence of the single ion module, which for so1ion is by default $<\hat J_a>$,$<\hat J_b>$,$<\hat J_c>$, $<\hat O_{22}^s>$, $<\hat O_{21}^s>$,$<\hat O_{20}>$,$<\hat O_{21}>$,$<\hat O_{22}>$, $<\hat O_{33}^s>$, ... respectively. Table 2 contains a list of first and second order Stevens parameters. For a full list please refer to appendix G.

Note that the $\hat I_{\alpha}$ may be completely redefined by perl parsing the single ion property file - see section 13 for details. This may be useful to shorten notation, if by symmetry the coupling (67) may be rewritten in products of linear combinations of operators such as needed for pseudo spin models of multipolar order [31], e.g $(5\hat O_{44}(\hat \mathbf J^n)+\hat O_{40}(\hat \mathbf J^n)) \cdot (5 \hat O_{44}(\hat \mathbf J^{n'})+\hat O_{40}(\hat \mathbf J^{n'}))$

Table 2: Stevens operator equivalents $T=10$ and corresponding notation used in modules cfield and so1ion ( $a$. For a full list please refer to appendix G.

Stevens Operator	Notation used in module so1ion	cfield
$\hat O_{00}=1$ ( $O_{00}({\rm so1ion})=0$)
$\hat O_{11}=\frac{1}{2}[\hat J_++\hat J_-]=\hat J_x$	$\hat I_a$	$\hat I_c$
$\hat O_{11}(s)=\frac{-i}{2}[\hat J_+-\hat J_-]=\hat J_y$	$\hat I_b$	$\hat I_a$
$\hat O_{10}=\hat J_z$	$\hat I_c$	$\hat I_b$
$\hat O_{22}(s)=\frac{-i}{2}[\hat J_+^2-\hat J_-^2]=\hat J_x\hat J_y+\hat J_y\hat J_x$($=2\hat P_{xy}$)	$\hat I_d$	$\hat I_d$
$\hat O_{21}(s)=\frac{-i}{4}[(\hat J_+-\hat J_-)\hat J_z+\hat J_z(\hat J_+-\hat J_-)]=\frac{1}{2}[\hat J_y\hat J_z+\hat J_z\hat J_y]$($=\hat P_{yz}$)	$\hat I_e$	$\hat I_e$
$\hat O_{20}=[3\hat J_z^2-J(J+1)]$ ($O_2^0(s)=0$)	$\hat I_f$	$\hat I_f$
$\hat O_{21}=\frac{1}{4}[(\hat J_++\hat J_-)\hat J_z+\hat J_z(\hat J_++\hat J_-)]=\frac{1}{2}[\hat J_x\hat J_z+\hat J_z\hat J_x]$($=\hat P_{xz}$)	$\hat I_g$	$\hat I_g$
$\hat O_{22}=\frac{1}{2}[\hat J_+^2+\hat J_-^2]=[\hat J_x^2-\hat J_y^2]$	$\hat I_h$	$\hat I_h$