Example - how to evaluate the crystal field of NdCu$_2$ using so1ion

All the files described in the following procedure are available in the directory examples/ndcu2b_new/cf.

A simple input file is made up in the following:

• The first line in the input file tells so1ion that it should use simple input format.
• Comment lines start with #.
• The type of ion has to be given plus several lines containing the crystal field parameters.

It follows an example, we take the values for NdCu$_2$ from literature [27]. Note that if the crystal field parameters are not known, there are different possibilities to obtain values, such as ab initio calculation, point charge calculations (use module pointc) and fitting to experimental data (see section 17).

#!MODULE=so1ion
#<!--mcphas.cf-->
# comment followed by
# the ion type and the crystal field parameters [meV]
 IONTYPE=Nd3+
 GJ=0.727273
# - note you can also do any pure spin problem by entering e.g. IONTYPE=S=2.5 
 B20=  0.116765                                           
 B22  =  0.134172                                           
 B40  =  0.0019225                                          
 B42  =  0.0008704                                          
 B44  =  0.0016916                                          
 B60  =  0.0000476                                          
 B62  =  0.0000116                                          
 B64  =  0.0000421                                          
 B66  =  0.0003662       
# instead of the Stevens parameters Blm 
# second order crystal field parameters Dx^2 Dy^2 and Dz^2 can be entered in meV
# - this coresponds to Hamiltonian H=+Dx2 Jx^2+Dy2 Jy^2+Dz2 Jz^2
Dx2=0.1
Dy2=0
Dz2=0.4
# the temperature in Kelvin
TEMP= 10
# if you want you can apply a magnetic field in Tesla
Bx=0
By=0
Bz=0

Note: you can also create an input file for so1ion with more comments for the crystal field parameters, the type of ion etc. This is done by running so1ion with the option -c, i.e.: so1ion -c, a help screen appears ...

1. The calculation of the single ion properties is performed using so1ion with the option -r, i.e. type the command so1ion -r filename -B where filename refers to the name of the single ion property file. Files so1ion.out and levels.cef (short summary, usable as input for mcdiff) are generated, which contain the results of the calculation, i.e. the diagonalisation of the Hamilton operator and the neutron scattering cross section. The program so1ion outputs a variety of results, such as eigenvectors and energies of the crystal field states. In addition it provides the total neutron powder cross section for each crystal field transitions (in barn/ion) at a given temperature according to the formula

   
   

   $$\sigma(i\rightarrow k)=4\pi \left(\frac{\hbar \gamma e^2}{mc... ...pha=x,y,z} \vert\langle i\vert J^{\alpha}\vert k\rangle\vert^2$$ (7)

   
   

   Note: in this calculation energy and Q dependence of the double differential scattering cross section are not considered and integration over all energies and scattering angles has been performed. In order to get a more realistic scattering intensity, the form factor (giving a $Q$ dependence), the factor $k'/k$ and the Debye Waller factor $exp(-W(Q))$ should be considered.

   Here comes the output file so1ion.out:

   #{------------------------------------------------------------- 
   #                  C F I E L D / S O 1 I O N    5.60           |
   #                    A crystal field program                   |
   #               __________________________________              
   #              |         Peter  Hoffmann          |             
   #              |    Forschungszentrum Juelich     |             
   #              |Institut fuer Festkoerperforschung|             
   #               __________________________________              
   #    O U T P U T         Tue Aug 30 12:44:30 2011
   #-------------------------------------------------------------- 
   #!Temperature of the sample       T=  10.00 Kelvin                
   #!Ion                    IONTYPE= Nd3+                          
   #!Lande factor of the ion   gJ= 0.727273                       
   #                                                              
   # Total angular momentum J of the                               
   #!Spin - orbit - level     J=  4.5                            
   #!Electrons in 4f shell   Ne=  3                              
   #-------------------------------------------------------------- 
   # Parameter           :  Akq   in  meV   /a0**k                  
   # (compare Hutchings Solid State Physics 16 (1964) 227, p 255 eq 5.5)                             
   #!A20  =        -16.306372                                     
   #!A22  =        -18.737280                                     
   #                                                              
   #!A40  =         -2.269445                                     
   #!A42  =         -1.027477                                     
   #!A44  =         -1.996875                                     
   #                                                              
   #!A60  =         -0.083369                                     
   #!A62  =         -0.020317                                     
   #!A64  =         -0.073736                                     
   #!A66  =         -0.641377                                     
   #-------------------------------------------------------------- 
   #-------------------------------------------------------------- 
   # Parameter           :  Bkq   in  meV                           
   #                        Bkq are the Stevens Parameters  - see Hutchings Solid State Physics 16 (1964) 227
   #!B20  =          0.116765                                     
   #!B22  =          0.134172                                     
   #                                                              
   #!B40  =          0.001922                                     
   #!B42  =          0.000870                                     
   #!B44  =          0.001692                                     
   #                                                              
   #!B60  =          0.000048                                     
   #!B62  =          0.000012                                     
   #!B64  =          0.000042                                     
   #!B66  =          0.000366                                     
   #-------------------------------------------------------------- 
   #-------------------------------------------------------------- 
   # Parameter           :  Dkq   in  meV                           
   #!D20 =         -36.330596                                     
   #!D22 =         -17.043002                                     
   #                                                              
   #!D40 =         -52.832675                                     
   #!D42 =          -3.782031                                     
   #!D44 =          -5.556290                                     
   #                                                              
   #!D60 =         -20.048458                                     
   #!D62 =          -0.476801                                     
   #!D64 =          -1.579686                                     
   #!D66 =         -10.148138                                     
   #-------------------------------------------------------------- 
   #-------------------------------------------------------------- 
   # Parameter           :  Lkq   in  meV                           
   #                        Lkq are the Wybourne Parameters - see A. Kassmann J. Chem. Phys. 53 (1970) 4118
   #!L20  =        -36.330596                                     
   #!L22  =        -17.043002                                     
   #                                                              
   #!L40  =        -52.832675                                     
   #!L42  =         -3.782031                                     
   #!L44  =         -5.556290                                     
   #                                                              
   #!L60  =        -20.048458                                     
   #!L62  =         -0.476801                                     
   #!L64  =         -1.579686                                     
   #!L66  =        -10.148138                                     
   #-------------------------------------------------------------- 
   #-------------------------------------------------------------- 
   # Parameter           :  Vkq   in  meV                           
   # (NOT the same Vlm as in Hutchings p255 or Elliot and Stevens)
   #!V20 =           0.116765                                     
   #!V22 =           0.067086                                     
   #                                                              
   #!V40 =           0.001922                                     
   #!V42 =           0.000435                                     
   #!V44 =           0.000846                                     
   #                                                              
   #!V60 =           0.000048                                     
   #!V62 =           0.000002                                     
   #!V64 =           0.000021                                     
   #!V66 =           0.000183                                     
   #-------------------------------------------------------------- 
   #-------------------------------------------------------------- 
   # Parameter           :  Wkq   in  meV   /a0**k                  
   #!W20 =         -16.306372                                     
   #!W22 =          -9.368640                                     
   #                                                              
   #!W40 =          -2.269445                                     
   #!W42 =          -0.513739                                     
   #!W44 =          -0.998438                                     
   #                                                              
   #!W60 =          -0.083369                                     
   #!W62 =          -0.003386                                     
   #!W64 =          -0.036868                                     
   #!W66 =          -0.320689                                     
   #-------------------------------------------------------------- 
   #-------------------------------------------------------------- 
   #                Anisotropy parameters in meV.                  
   #-------------------------------------------------------------- 
   #         H= + Dx2 Jx ^ 2+ Dy2 Jy ^ 2+ Dz2 Jz ^ 2               
   #! Dx2 =          0.10                                     
   #! Dy2 =          0.00                                     
   #! Dz2 =          0.40                                     
   #                                                              
   #-------------------------------------------------------------- 
   #-------------------------------------------------------------- 
   # Energy Eigenvalues are in  meV   .          	            
   #--------------------------------------------------------------
   #!Nr of different energy levels   noflevels= 5                 
   #!Energy shift   Eshift=                       -2.6744 
   #                                                              
   # Because of the calibration freedom the smallest energy       
   # eigenvalue is shifted to zero. You can get the energy        
   # eigen-value of the applied eigen-value problem               
   # by shifting the energy by the added energy value above       
   #!*E( 1)=        0.0000        Degeneracy =   2 -fold            
   #!*E( 2)=        2.4850        Degeneracy =   2 -fold            
   #!*E( 3)=        4.5048        Degeneracy =   2 -fold            
   #!*E( 4)=        7.8548        Degeneracy =   2 -fold            
   #!*E( 5)=       19.1522        Degeneracy =   2 -fold            
   # Them with  *  marked Energy eigenvalues have a non-          
   # vanishing Matrix element of the Ground state E( 1).          
   #-------------------------------------------------------------- 
   #
   #-------------------------------------------------------------- 
   # The orthonormal characteristic Eigenvectors  |i,r>  with     
   #                                                              
   #             H |i,r>  = ( E  + E      ) |i,r>  , r=1 ... n    
   #                           i    shift                     i   
   #                                                              
   #                            and                               
   #                                                              
   #                    <i,r|j,s> = D   D                         
   #                                 ij  rs                       
   #                                                              
   #              D  = Kronecker- Delta function                  
   #               ij                                             
   #                                                              
   # the |i,r> are also orthonormal .                             
   # We consider in this program  Crystal Field spliting          
   # at the lowest Spin-orbit-level (Ground state)                
   #                                                              
   # 2S+1                                                         
   #     L  of the calculated Ion. The |i,r> are in               
   #      J                                                       
   #                                                              
   # a more developed  Eigenfunction   			    
   #                                                              
   # system  [ | J  M   > ]               ,where                  
   #                 J     M = -J,...,J                           
   #                        J                                     
   #                               ,                              
   #              <  J  M   |  J  M  >  =  D    ,    .            
   #                     J         J        M  M                  
   #                                         J  J                 
   #--------------------------------------------------------------
   #                                                              
   # I 1, 1> =     0.0414               I 4.5  -4.5>              
   #          -    0.3955               I 4.5  -2.5>              
   #          +    0.7083               I 4.5  -0.5>              
   #          -    0.4563               I 4.5   1.5>              
   #          +    0.3633               I 4.5   3.5>              
   #                                                              
   # I 1, 2> =     0.3633               I 4.5  -3.5>              
   #          -    0.4563               I 4.5  -1.5>              
   #          +    0.7083               I 4.5   0.5>              
   #          -    0.3955               I 4.5   2.5>              
   #          +    0.0414               I 4.5   4.5>              
   #                                                              
   # I 2, 1> =    -0.0402               I 4.5  -4.5>              
   #          +    0.1455               I 4.5  -2.5>              
   #          +    0.5713               I 4.5  -0.5>              
   #          +    0.1221               I 4.5   1.5>              
   #          -    0.7975               I 4.5   3.5>              
   #                                                              
   # I 2, 2> =     0.7975               I 4.5  -3.5>              
   #          -    0.1221               I 4.5  -1.5>              
   #          -    0.5713               I 4.5   0.5>              
   #          -    0.1455               I 4.5   2.5>              
   #          +    0.0402               I 4.5   4.5>              
   #                                                              
   # I 3, 1> =    -0.0255               I 4.5  -3.5>              
   #          -    0.7164               I 4.5  -1.5>              
   #          -    0.0601               I 4.5   0.5>              
   #          +    0.6945               I 4.5   2.5>              
   #          -    0.0098               I 4.5   4.5>              
   #                                                              
   # I 3, 2> =     0.0098               I 4.5  -4.5>              
   #          -    0.6945               I 4.5  -2.5>              
   #          +    0.0601               I 4.5  -0.5>              
   #          +    0.7164               I 4.5   1.5>              
   #          +    0.0255               I 4.5   3.5>              
   #                                                              
   # I 4, 1> =     0.4804               I 4.5  -3.5>              
   #          +    0.5046               I 4.5  -1.5>              
   #          +    0.4065               I 4.5   0.5>              
   #          +    0.5712               I 4.5   2.5>              
   #          -    0.1522               I 4.5   4.5>              
   #                                                              
   # I 4, 2> =    -0.1522               I 4.5  -4.5>              
   #          +    0.5712               I 4.5  -2.5>              
   #          +    0.4065               I 4.5  -0.5>              
   #          +    0.5046               I 4.5   1.5>              
   #          +    0.4804               I 4.5   3.5>              
   #                                                              
   # I 5, 1> =     0.9866               I 4.5  -4.5>              
   #          +    0.1175               I 4.5  -2.5>              
   #          +    0.0557               I 4.5  -0.5>              
   #          +    0.0948               I 4.5   1.5>              
   #          +    0.0261               I 4.5   3.5>              
   #                                                              
   # I 5, 2> =     0.0261               I 4.5  -3.5>              
   #          +    0.0948               I 4.5  -1.5>              
   #          +    0.0557               I 4.5   0.5>              
   #          +    0.1175               I 4.5   2.5>              
   #          +    0.9866               I 4.5   4.5>              
   #                                                              
   #-------------------------------------------------------------- 
   #
   #!magnetic moment(mb/f.u.): mx= 0.000 my= 0.000 mz= 0.000
   #
   #-------------------------------------------------------------- 
   #                                                              
   #                  M A T R I X  E L E M E N T                  
   #                  S I N G L E  C R Y S T A L                    
   #                                                              
   #--------------------------------------------------------------
   #    Only the marix elements that are non-zero are written     
   #                                                              
   #--------------------------------------------------------------
   #                              2 |            2 |            2 
   #    a  <-->  b      |<a|J |b>|  |  |<a|J |b>|  |  |<a|J |b>|  
   #                         x      |       y      |       z      
   #--------------------------------+--------------+--------------
   #    1  <-->  1       0.002008   |  36.711411   |   0.031042   
   #--------------------------------+--------------+--------------
   #    2  <-->  1       1.949420   |   0.270090   |   2.638097   
   #--------------------------------+--------------+--------------
   #    2  <-->  2       1.517583   |   2.490016   |   8.199948   
   #--------------------------------+--------------+--------------
   #    3  <-->  1       1.415357   |   0.424874   |   2.727140   
   #--------------------------------+--------------+--------------
   #    3  <-->  2      10.148894   |   8.149608   |   0.176745   
   #--------------------------------+--------------+--------------
   #    3  <-->  3       9.078618   |  11.619418   |   0.380198   
   #--------------------------------+--------------+--------------
   #    4  <-->  1       0.009374   |   1.424032   |   1.021019   
   #--------------------------------+--------------+--------------
   #    4  <-->  2       2.500806   |   2.020701   |   5.118137   
   #--------------------------------+--------------+--------------
   #    4  <-->  3       0.239888   |   0.004479   |   4.938522   
   #--------------------------------+--------------+--------------
   #    4  <-->  4      26.164401   |   0.690989   |   0.070056   
   #--------------------------------+--------------+--------------
   #    5  <-->  1       0.710568   |   0.137243   |   0.028325   
   #--------------------------------+--------------+--------------
   #    5  <-->  2       2.282293   |   2.029398   |   0.008264   
   #--------------------------------+--------------+--------------
   #    5  <-->  3       0.057679   |   0.000101   |   0.138477   
   #--------------------------------+--------------+--------------
   #    5  <-->  4       3.524062   |   1.023958   |   0.749577   
   #--------------------------------+--------------+--------------
   #    5  <-->  5       0.060712   |   0.019195   |  38.730148   
   #-------------------------------------------------------------- 
   #
   #
   #-------------------------------------------------------------- 
   #                                                              
   #                  M A T R I X  E L E M E N T S                
   #                    P O L Y C R Y S T A L                     
   #                                                              
   #--------------------------------------------------------------
   #                                                              
   #                             ----                2            
   # Matrix element       :      >     |<i,r|J |k,s>|             
   #                             ----         T                   
   #                              r,s                             
   #                                                              
   #                                                              
   # for the transition :      E  ->  E                           
   #                              i      k                        
   #                                                              
   #--------------------------------------------------------------
   #                                                              
   #                         SUm rule :                           
   #                                                              
   #             ----                2       2                    
   #             >     |<i,r|J |k,s>|  = n  --- J(J+1)            
   #             ----         T           i  3                    
   #             k,r,s                                            
   #-------------------------------------------------------------- 
   #----------------------------- 
   # \ E |   |   |   |   |   |Zei|
   #E \ k|E  |E  |E  |E  |E  |len|
   # i \ | 1 | 2 | 3 | 4 | 5 |sum|
   #----|---|---|---|---|---|---|
   #| E  | 24|  3|  3|  1|   | 33|
   #|  1 |.50|.24|.04|.64|.58|   |
   #|----|---|---|---|---|---|---|
   #| E  |  3|  8| 12|  6|  2| 33|
   #|  2 |.24|.14|.32|.43|.88|   |
   #|----|---|---|---|---|---|---|
   #| E  |  3| 12| 14|  3|   | 33|
   #|  3 |.04|.32|.05|.46|.13|   |
   #|----|---|---|---|---|---|---|
   #| E  |  1|  6|  3| 17|  3| 33|
   #|  4 |.64|.43|.46|.95|.53|   |
   #|----|---|---|---|---|---|---|
   #| E  |   |  2|   |  3| 25| 33|
   #|  5 |.58|.88|.13|.53|.87|   |
   # ---------------------------- 
   #
   #
   #
   #-------------------------------------------------------------- 
   #               Transition intensities in barn.                
   #                                                              
   #                                                              
   #                                                              
   #                                                              
   # =                    - E /T     -----                        
   # |                   e   i       \                   2        
   # |     = const --------------     >    |<i,r|J |k,s>|         
   # |             ----    - E /T    /            T               
   # =             >    n e   i      -----                        
   #  E -> E       ----  i            r,s                         
   #   i    k       i                                             
   #                                                              
   #                            with                              
   #                                                              
   #                                                              
   #                                -----                         
   #                      2     2   \                    2        
   #        |<i,r|J |k,s>|   = ---   >     |<i,r|J |k,s>|         
   #               T            3   /             u               
   #                                -----                         
   #                             u = x,y,z                        
   #                                                              
   #                                                              
   #                             und                              
   #                                                              
   #                                   1          2               
   #                  const  = 4*pi*( --- r   g  )                
   #                                   2   0   J                  
   #                                                              
   #                                      -12                     
   #                  r      = - 0.54 * 10    cm                  
   #                   0                                          
   #                                                              
   #--------------------------------------------------------------
   #                                                              
   #                       1.Sum rule :                           
   #                                                 - E /T       
   #                                            n   e   i         
   #  ----  =            2                       i                
   #  >     |         = --- * const * J(J+1) * ----------------   
   #  ----  =            3                     ----     - E /T    
   #   k     E -> E                            >    n  e   i      
   #          i    k                           ----  i            
   #                                            i                 
   #--------------------------------------------------------------
   #                                                              
   #                       2. sum rule :                          
   #                                                              
   #                                                              
   #            ----  =            2                              
   #            >     |         = --- * const * J(J+1)            
   #            ----  =            3                              
   #             k,i   E -> E                                     
   #                    i    k                                    
   #-------------------------------------------------------------- 
   #-------------------------------------------------------------- 
   #!Temperature of the sample               T=   10.00 Kelvin         
   #-------------------------------------------------------------- 
   #!parition function            Z  =   2.12                 
   #-------------------------------------------------------------- 
   #!Total_magnetic_scattering_intensity =   7.94 barn            
   #-------------------------------------------------------------- 
   #----------------------------- 
   # \ E |   |   |   |   |   |Zei|
   #E \ k|E  |E  |E  |E  |E  |len|
   # i \ | 1 | 2 | 3 | 4 | 5 |sum|
   #----|---|---|---|---|---|---|
   #| E  |  5|   |   |   |   |  7|
   #|  1 |.55|.73|.69|.37|.13|.48|
   #|----|---|---|---|---|---|---|
   #| E  |   |   |   |   |   |   |
   #|  2 |.04|.10|.16|.08|.04|.42|
   #|----|---|---|---|---|---|---|
   #| E  |   |   |   |   |   |   |
   #|  3 |   |.01|.02|   |   |.04|
   #|----|---|---|---|---|---|---|
   #| E  |   |   |   |   |   |   |
   #|  4 |   |   |   |   |   |   |
   #|----|---|---|---|---|---|---|
   #| E  |   |   |   |   |   |   |
   #|  5 |   |   |   |   |   |   |
   # ---------------------------- 
   #
   #-----------------------------------------------------------
   #!Total_quasielastic_intensity =             5.68 barn           
   #-----------------------------------------------------------
   #                  Neutron-Energy-loss                      
   #!middle_position_of_the_energy        =        2.41 meV    
   #!relative_error_in_the_middl_Position  =        5.64 %      
   #!Intensity_of_the_middle_position   =        0.89 barn   
   #                  Neutron-Energy-Gain                 
   #!middle position of the energy        =       -2.36 meV    
   #!relative_error_in_the_middl_Position  =        7.74 %      
   #!Intensity_of_the_middle_position   =        0.06 barn   
   #-----------------------------------------------------------
   
   #-------------------------------------------------------------- 
   # Transition Energy (meV   ) vs Intensity (barn)                  
   #-------------------------------------------------------------- }
     0.000  5.554803
     2.485  0.734343
     4.505  0.690467
     7.855  0.371045
    19.152  0.132449
    -2.485  0.041064
     0.000  0.103197
     2.020  0.156181
     5.370  0.081489
    16.667  0.036519
    -4.505  0.003705
    -2.020  0.014986
     0.000  0.017097
     3.350  0.004204
    14.647  0.000159
    -7.855  0.000041
    -5.370  0.000160
    -3.350  0.000086
     0.000  0.000448
    11.297  0.000088
   -19.152  0.000000
   -16.667  0.000000
   -14.647  0.000000
   -11.297  0.000000
     0.000  0.000000
   

2. at the end of the file so1ion.out the neutron scattering cross section of the different transitions is given as a list of energy vs intensity. In order to calculate a spectrum these results have to be convoluted with the resolution function of a neutron spectrometer. This can be done by the programs convolut (available under windows only) or convolute. For example, the command convolut so1ion.out 12 stp=0.05 mode=gauss fwhm=0.5 convolutes the calculated neutron transitions with a Gaussian resolution function of 0.5 meV full width half maximum. The step size of the output spectrum is 0.05 meV. The output spectrum is contained in file so1ion.cvt.
3. Use program cuthead (available under windows only) to shorten the file header and display to view the spectrum: cuthead 10 so1ion.cvt and display 1 2 so1ion.cvt. In order to create an image file for printing the viewed spectrum press the save button, it creates a file display.jpg which is shown in figure 5.

   Figure 5: Calculated crystal field neutron spectrum of NdCu$_2$ at a temperature of 10 K. Horizontal axis is energy transfer in meV and vertical axis is neutron intensity in barns per meV and Nd atom. [plot created by program display]

   
4. In order to calculate the magnetisation for our problem (in the paramagnetic state), the direction of the magnetic field has to be given in bkq.parmeter and program so1ion has to be started with the option -m, i.e. so1ion -m -B 0 30 10. The numbers denote the field range (0-30 Tesla) and the temperature (10 K). The results are written to moment.rtplot, fig. 6 shows the result.

   Figure 6: Calculated magnetic moment for field applied along $x$ direction ($c$ axis, note the notation of the crystal field $xyz$ axes with respect to the crystal lattice is $xyz\vert\vert cab$ in our example) of NdCu$_2$ at a temperature of 10 K. [plot created by program gnuplot]

   

5. In a similar way the temperature dependence of the susceptibility can be calculated by so1ion -s.
6. The crystal field contribution to the specific heat may be calculated from the output file so1ion.out using the program cpso1ion, e.g. cpso1ion 10 100 1 [options] calculates the specific heat in the temperature interval 10-100 K with a step width of 1 K. Alternatively a comparison to experimental data can be made by cpso1ion 1 2 cpexp.dat, where the temperatures are given in column 1 and the experimental specific heat in column 2 of file cpexp.dat. The calculated specific heat is compared to the experimental data and a standard deviation sta is calculated and output is written to stdout. Other quantities can be calculated using the options: -s (calculate entropy (J/molK) instead of cp), -f (calculate free energy (J/mol) instead of cp),-u (calculate magnetic energy (J/mol) instead of cp), -z (calculate partition sum instead of cp). Fig. 7 shows an example.

   Figure 7: Calculated specific heat of NdCu$_2$ in zero magnetic field as calculated by cpso1ion (crystal field contribution) in comparison with experimental data [27]. The dashed line shows the results of a calculation, which in addition to the crystal field takes into account the two ion interaction and using so1ion as a module in mcphas - see below and chapter 7. [plot created by program gnuplot]

   

7. The spin-disorder resistivity due to scattering of conduction $s$-electrons with the localised $f$-electrons via the exchange interaction can be calculated in the first Born approximation as [29],

   
   

   $$\rho_{s-f}(T) = \frac{3\pi N m}{\hbar e^2 E_F} G^2(g-1)^2 \s... ...' \vert {\mathbf s \cdot J} \vert m_s, i \rangle^2 p_i f_{ii'}$$ (8)

   
   

   where $G$ is the exchange constant, $p_i$ is the Bose factor $e^{-\beta E_i}/Z$ and $f_{ii'}$ is the Fermi function $2/(1+e^{-\beta(E_i-E_{i'})})$. The program rhoso1ion can be used to calculate this resistivity for magnetic ions in a crystal field. The wavefunctions $\vert i\rangle$ are taken from the file levels.cef output by so1ion. The matrix elements of ${\mathbf s \cdot J}$ are calculated according to the formulae of Dekker [30]. rhoso1ion calculates only the sum in the above equation, however. The constant coefficient $\rho_0=(3\pi N m/\hbar e^2 E_F) G^2(g-1)^2$ is set to unity, or may be specified using the option -rho0 or -r. The syntax is otherwise the same as cpso1ion. For example, to calculate the resistivity from 10 to 100 K in 1 K steps with $\rho_0 = 0.2$ $\Omega$.cm, the command rhoso1ion 10 100 1 -r 0.2 can be used. Alternatively, for comparison with data, rhoso1ion 1 2 rhoexp.dat -r 0.2 can be used. Note that as the temperature dependence of the resistivity in this case is mainly a function of the Bose and Fermi functions, at high temperatures where all $2J+1$ crystal field levels are thermally occupied, the resistivity will saturate to a value $\rho_0 J(J+1)$.

Exercises:

• Use so1ion to calculate the energies, eigenvectors and transition matrix elements (for inelastic neutron scattering) for Nd$^{3+}$ in an orthorhombic crystal field. Use the parameters given in section 6.3.