1. Prolog Numerical Modeling in Magnetism
Macro-Magnetism: Solution of Maxwells Equations – Engineering of (electro)magnetic devices
Micromagnetism:
Domain Dynamics, Hysteresis
MFM image
Micromagnetic simulation.
Atomic Magnetism:
Instrinsic Magnetic Properties

2. Atomic Magnetism- Modeling Instrinsic Magnetic Properties
Band Models
Spin Polarized First Principle Methods:
restricted to simple Magnetic Structures, T=0, no dynamics, no rare earth elements ... there are attempts to overcome these restrictions
Localized Moment Models
Ising-, Heisenberg-, xy-, Standard Model of RE-Magnetism)
Exact Methods: e.g. branch and bound algorithm, transfer matrix algorithm
Monte Carlo Methods
Selfconsistent Mean Field Method

3. Atomic Magnetism- Modeling Instrinsic Magnetic Properties
M. Rotter, Institut für physikalische Chemie, Universität Wien
Atomic Magnetism- Modeling Instrinsic Magnetic Properties
Band Models
Spin Polarized First Principle Methods:
restricted to simple Magnetic Structures, T=0, no dynamics, no rare earth elements ... there are attempts to overcome these restrictions
Localized Moment Models
Ising-, Heisenberg-, xy-, Standard Model of RE-Magnetism)
Exact Methods: e.g. branch and bound algorithm, transfer matrix algorithm
Monte Carlo Methods
Selfconsistent Mean Field Method

4. The Standard Model of RE Magnetism - the Crystal Field Concept+
4f –charge density E Q
Hamiltonian
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

5. McPhase - the World of Rare Earth Magnetism
Example: NdCu2 a b c
Crystal Structure of RCu2
Imma (orthorhombic)
... 9 nonzero CF Parameters
you can use module pointc to calculate CF parameters by the pointcharge model +
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

6. NdCu2 – Crystal Field Excitations
orthorhombic, TN=6.5 K, Nd3+: J=9/2, Kramers-ion
McPhase can
solve CF Model
Calculate Intensities and Energies
Calculate and Plot Charge Density
...
Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

7. Make a Crystal Field Model using McPhase Module Cfield
Module simmannfit can do this again and again for you
to fit the result of the calculation to your spectrum by
variation of the CF-parameters
Make a Crystal Field Model using McPhase Module Cfield
CF Hamiltonian
Example files in directory /mcphas/examples/ndcu2b_new/cf
Edit file Bkq.parameter and enter CF parameters Blm
Start module cfield - type: cfield –r -B
View output file cfield.out: CF - energies, eigenstates, transition-matrixelements and corresponding neutron intensities
Use module convolute to convolute energy vs intensity results with spectrometer resolution function
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

8. Magnetism would be boring without a magnetic field
Use module cfield
to calculate magnetization
type: cfield –m
Hamiltonian
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

9. McPhase - the World of Rare Earth Magnetism
Specific Heat
Use module cpcalc to calculate specific heat
type: cpcalc
Tmin=5 Tmax=30 dT=1
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

10. Use modules chrgplot+javaview to plot 4f charge density
T=100 K
T=40 K
T=10 K
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

11. Use modules pointc+chrgplot+javaview
T=2K
H=0
Same CEF
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

12. McPhase - the World of Rare Earth Magnetism
Module mcphas
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

13. McPhase - the World of Rare Earth Magnetism
Input files for module mcphas:
mcphas.j (structure), mcphas.cf (single ion properties), mcphas.tst (table of initial values), mcphas.ini (H,T-range, ...)
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

14. Do you really want to see the MF equations ?
Cfield can calculate
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

15. Bulk Properties Calculated by module mcphas
Magnetization output file: mcphas.fum
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

16. McPhase - the World of Rare Earth Magnetism
NdCu2 Specific Heat
output file: mcphas.fum
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

17. Spontaneous Magnetostriction
Microscopic Source of Magneostriction:
Strain dependence of magnetic interactions
Crystal field
Exchange
T<TC(N)
L=0, L0
„exchange-striction“ T T
.... Symmetry
decreases
L0 +
T<TC(N)
T>TC(N) e- +
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

18. Forced Magnetostriction
Crystal Field
Exchange Striction
L0
L=0, L0
H <0 H + e- H
>0 +
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

19. Calculation of Magnetostriction
Crystal Field
Exchange
mit
Output file: mcphas.xyt
Output file: mcphas.jj* +
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

20. NdCu2 Magnetostriction
Exchange - Striction
Crystal Field
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

21. NdCu2 Magnetic Phase Diagram
F1   
F3  c
F1  b a
AF1 
lines=experiment
output file: mcphas.xyt
Use module phased or displaycontour for color plot of phasediagram
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

22. McPhase - the World of Rare Earth Magnetism
output file: mcphas.hkl
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

23. McPhase - the World of Rare Earth Magnetism
Dispersive Magnetic Excitations
153
MF - Zeeman Ansatz
T=1.3 K
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

24. McPhase - the World of Rare Earth Magnetism
... Spinwaves (Magnons)
153
T=1.3 K
Bohn et. al.
PRB 22 (1980) 5447
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

25. McPhase - the World of Rare Earth Magnetism
Spinwaves (Magnons)
153 a
T=1.3 K
Bohn et. al.
PRB 22 (1980) 5447
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

26. McPhase - the World of Rare Earth Magnetism
Module Mcdisp – Calculate Magnetic Excitation Energies and the Neutron Scattering Cross Section
MF-RPA
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

27. McPhase - the World of Rare Earth Magnetism
Module Mcdisp – a novel fast algorithm for magnetic excitations – Rotter 2005
Transformation:
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

28. McPhase - the World of Rare Earth Magnetism
with definition:
(1)
all other components
of Ψ are zero
with definition:
Generalized eigenvalue problem (analogue to dynmical matrix
in the case of phonons!!)
Solution gives eigenvalues
and eigenvectors
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

29. McPhase - the World of Rare Earth Magnetism
may then be inverted to give the following expression for Ψ
back transformation...
+calculation of absorptive part...
using Diracs formula:
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

30. McPhase - the World of Rare Earth Magnetism
McDisp - fast algorithm - Cookbook 1) 2) 3)
...setup Matrix 4)
...solve generalized EV Problem 5)
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

31. NdCu2
F3 
F1 
AF1 

32. McPhase - the World of Rare Earth Magnetism
Diffuse Scattering
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

33. McPhase - the World of Rare Earth Magnetism
McPhase Modules
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

34. Symmetry - CF Local Point Symmetry limits the number of nonzero
Crystal Field Parameters
(mind: local symmetry at rare earth position may be lower than lattice symmetry, i.e. The lattice may be cubic, but the local symmetry tetragonal)
Point Group / Latt. Symmetry
Coordinate Orientation
Nonzero Blm
O cubic
xyz||abc
B40,B44,B60,B64
z||111
B40,B43,B60,B63,B66
D6h hexagonal
B20,B40,B60,B66
D4h tetragonal
B20,B40,B44,B60,B64
C3v (no lattice)
B20,B40,B43,B60,B63
C2h monoclinic
B20,B40,B60,B66,B66s
D3d (quasicubic in dhcp)
B20,B40,B43,B60,B63,B66
D2 orthorh.
B20,B22,B40,B42,B44,B60,B62,B64,B66

35. McPhase - the World of Rare Earth Magnetism
Example: 2nd order CF terms for point symmetry mm2=C2v
We choose here the basis of Racah instead of Stevens
operators for the Crystal field, because these transform
like the spherical harmonic functions
Group elements G
C2v 1E
1C2
1σy
1σx A1 1 B1 -1 A2 B2
Irr.
Repr.
These operators form a reducable representation T2(G) of the point group
Character table of mm2
Group Theory basics taken from: Elliott&Dawber Symmetry in Physics, McMillan Press, 1979
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

36. McPhase - the World of Rare Earth Magnetism
The representation T2(G) can be decomposed into irreducible
Representations (i.e. „the Olm can be linear combined to another
Basis so that in this basis the representation T2 bas block diagonal form with each block corresponding to a irreducible representation“)
The m‘s tell, how often a representation occurs. mA1 tells, how often the unit representation occurs in the decomposition, i.e. how many different independent basis vectors span this subspace, i.e. how many independent crystal field parameters will occur.
A little group theoretical trick for calculating m
Cp... Number of members of class p
g.... Number of group elements
χ.... Character of class
Class p a χp E 5 C2 π 1 σy σx
a... Angle of rotation
McPhase - the World of Rare Earth Magnetism
i.e. We expect 2 independent 2nd order CF parameters
Martin Rotter - McPhase Course TU Dresden 2005

37. McPhase - the World of Rare Earth Magnetism
The basis of the 2 A1 representation occuring in the decomposition of T2(G) can be found using the projection operator
In order to calculate it, we have to epxlicitely write down the reducable representation T2:
Jx‘=-Jx, Jy‘=-Jy
Jy‘=-Jy
Jx‘=-Jx
B20 and B22 are nonzero.
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

38. Symmetry – Bilinear Interaction
Isotropic interaction (J(ij) is a scalar)
Anisotropic Interaction (J(ij) is a tensor)
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

39. McPhase - the World of Rare Earth Magnetism
(quasi)hexagonal types of neighbors
neighbors related by symmetry must have related interaction constants J(ij)
CeCu2 Structure Cu Ce
M. Rotter et al., Eur. Phys. J. B 14, 29 (2000)
M. Rotter et al., JMMM. 214, 281 (2000)
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

40. McPhase - the World of Rare Earth Magnetism
Anisotropic Interaction –Symmetry Considerations
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

41. McPhase - the World of Rare Earth Magnetism
ETC...
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

42. McPhase - the World of Rare Earth Magnetism
Example: bc mirror plane b 1 a
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

43. Symmetry – Quadrupolar Interaction
Derivation similar to CF operator using representation T(G)=T2(G)xT2 (G)
Isotropic Quadrupolar Interaction
dhcp –lattice: between hexagonal sites
dhcp –lattice: between quasicubic sites

44. Example for quadrupolar interactions: PrCu2H + + M + + + + +
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

45. McPhase - the World of Rare Earth Magnetism
PrCu2
Settai et. al. JPSJ 67 (1998) 636 +
GMS + + + + + +
Ferroquadrupolarer (Cij>0) Austausch (durch CF-Phonon WW)
Settai et. al. JPSJ 67 (1998) 636
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

46. McPhase - the World of Rare Earth Magnetism
PrCu2
Ferroquadrupolar (Cij>0) Interaction
Settai et. al. JPSJ 67 (1998) 636
The Model describes well:
the quadrupolar phasen diagram
the magnetisation
the magnetostriction
die temperature dependence of elastic constants
Whats about the Dynamics ?
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

47. Orbital Excitations (Orbitonen)+
4f – charge density E Q
Crystal field
+Antiferroquadrupolar (C<0) Interaction
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

48. McPhase - the World of Rare Earth Magnetism
PrCu2 + + + + + + +
Ferroquadrupolar (Cij>0) Interaction (via CF-Phonon coupling)
McPhase - the World of Rare Earth Magnetism
Settai et. al. JPSJ 67 (1998) 636
Martin Rotter - McPhase Course TU Dresden 2005

49. PrCu2 Orbital Modes T=5 K, H=0 T
MF-RPA Model
Experiment Г
2.5
Energy (meV) ? 1
00L 2
McPhase:
Rotter, JMMM (2003) 481
Kawarazaki et. al.,
J. Phys. Cond. Mat. 7 (1995) 4051
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

50. McPhase - the World of Rare Earth Magnetism
NdCu2
PrCu2
Magnetic Excitations
Rotter et. al., Europ. Phys. J. B 14 (2000) 29
NdCu2
Könnte nicht auch die Austauschwechselwirkung
zu der beobachteten Dispersion führen ?
Nur Quadrupolaustausch Г
[Interpretation von Kawarazaki et. al., J. Phys. Cond. Mat. 7 (1995) 4051]
2.5
Energy (meV) 1
00L 2
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

51. McPhase - the World of Rare Earth Magnetism
PrCu2
00L
Energy (meV)
2.5
+ magnetic Interactions 1 2
Quadruplar Interaction only
Nur Quadrupolaustausch Г
nur
magnetischer
Austausch
2.5
Energy (meV) 1
00L 2
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

52. PrCu2 Orbital modes in Magnetic field T=2 K, H||a
Rechnung
Messung
IN12(ILL) März 2004
(15 Tesla cryomagnet)
McPhase:
Rotter, JMMM (2003) 481
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

53. McPhase - the World of Rare Earth Magnetism
Quadrupolar Effects
Neutrons can be scattered by 4f - Orbitons
– Orbiton spectroscopy:
- Determination of multipolar Interactions
- Modeling of GMS
Crystal field + Ferroquadrupolar (Cij>0) Interactions
Settai et. al. JPSJ 67 (1998) 636
PrCu2
The model describes well:
macroscopic properties and quadrupolar Phase diagram
Magnitude of dispersion of orbital modes
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

54. How to start – the story of NdCu2
Suszeptibility: 1/χ(T) at high T
... Crystal Field Parameters B20, B22
Specific Heat Cp
...  first info about CF levels
Magnetisation || a,b,c on single crystals in the paramagnetic state,
... ground state matrix elements
Neutron TOF spectroscopy – CF levels
...  All Crystal Field Parameters Blm
Thermal expansion in paramagnetic state – CF influence
...  Magnetoelastic parameters (dBlm/dε)
Neutron diffraction: magnetic structure in fields || easy axis
...  phase diagram H||b - model
...  Jbb
Neutron spectroscopy on single crystals in H||b=3T
...  Anisotropy of Jij - determination of Jaa=Jcc
Magnetostriction
...  Confirmation of phase diagram models H||a,b,c, dJ(ij)/dε
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

55. McPhase - the World of Rare Earth Magnetism
The story of NdCu2
Inverse suszeptibility at high T
... B20=0.8 K, B22=1.1 K
Hashimoto, Journal of Science of the Hiroshima University A43, 157 (1979)
Θabc
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

56. McPhase - the World of Rare Earth Magnetism
The story of NdCu2
Specific haet Cp and entropy – first info about levels
Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297
Rln2
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

57. How to start analysis – the story of NdCu2
Magnetization: Kramers ground state doublet |+-> matrix elements
P. Svoboda et al. JMMM 104 (1992) 1329
Module cfield can also calculate magnetization using a full set of CF parameters
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

58. How to start analysis – the story of NdCu2
Neutron TOF spectroscopy – CF levels
... Blm
Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297
B20=1.35 K
B22=1.56 K
B40= K
B42= K
B44= K
B60=4.89x10-4 K
B62=1.35x10-4 K
B64=4.89x10-4 K
B66=4.25 x10-3 K
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

59. McPhase - the World of Rare Earth Magnetism
The story of NdCu2
Thermal expansion – cf influence
... Magnetoelastic parameters (A=dB20/dε, B=dB22/dε)
E. Gratz et al., J. Phys.: Condens. Matter 5, 567 (1993)
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

60. The story of NdCu2 Neutron diffraction+ magnetization:
magstruc, phasediag H||b-> model
... Jbb
M. Loewenhaupt et al., Z. Phys. B:
Condens. Matter 101, 499 (1996)
n(k)=sum of Jbb(ij) with ij being of bc plane k
 


 

61. NdCu2 Magnetic Phase Diagram
F1   
F3  c
F1  b a
AF1 
lines=experiment
output file: mcphas.xyt
Use module phased or displaycontour for color plot of phasediagram
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

62. McPhase - the World of Rare Earth Magnetism
Jaa=Jcc(R)
The story of NdCu2
Neutron spectroscopy on single crystals in H||b=3T
... Anisotropy of J(ij) - determination of Jaa=Jcc
F3 
M. Rotter et al., Eur. Phys. J. B 14, 29 (2000)
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

63. NdCu2 F3  F1  AF1 
M. Rotter, et al. Applied Phys. A 74 (2002) s751

64. How to start analysis – the story of NdCu2
Magnetostriction ... Confirmation of phasediagram model for H||a,b,c, and determination of dJ(ij)/dε
M. Rotter, et al. J. of Appl. Physics 91 10(2002) 8885

65. McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

66. McPhase - the World of Rare Earth Magnetism
Nonsense !
„The Standard Model of Rare Earth Magnetism has been well established and can describe the magnetic properties of Rare earth compounds. There is no need for a program like McPhase.“
In very few RE systems a large number of results of the SM have been compared to experimental data: e.g. the full magneto-striction tensor has been analysed only in 1 case (NdCu2)
Quadrupolar Excitations have not been compared to the SM
There is a number of wrong predictions of the SM: e.g.
-magnetoelastic paradoxon in L=0 AF-systems
-extra magnetic modes or no modes (CeCu2, CeNi9Ge4, Nd2CuO4),
-wrong saturation moments, e.g. in Eu-Skutterudite
- ...
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

67. The magnetoelastic Paradoxon for L=0 demonstrated at GdNi2B2C
Orthorhombic Distortion ?
Exchange-Striction
Standard Model of RE Mag
... McPhase Simulation
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

68. McPhase - the World of Rare Earth Magnetism
McPhase is a program package for the calculation of
magnetic properties of rare earth based systems.
          Magnetization                       Magnetic Phasediagrams
    Magnetic Structures            Elastic/Inelastic/Diffuse                                               Neutron Scattering                                              Cross Section
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

69. McPhase - the World of Rare Earth Magnetism
Crystal Field/Magnetic/Orbital Excitations
Magnetostriction 
and much more....
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005

70. Epilog McPhase - the World of Rare Earth Magnetism
McPhase runs on Linux and Windows and is available as freeware.
McPhase is being developed by
  M. Rotter, Institut für Physikalische Chemie, Universität Wien, Austria   M. Doerr, R. Schedler, Institut für Festkörperphysik,
Technische Universität Dresden, Germany   P. Fabi né Hoffmann, Forschungszentrum Jülich, Germany   S. Rotter, Wien, Austria
  M.Banks, Max Planck Institute Stuttgart, Germany
Important Publications referencing McPhase:
M. Rotter, S. Kramp, M. Loewenhaupt, E. Gratz, W. Schmidt, N. M. Pyka, B. Hennion, R. v.d.Kamp Magnetic Excitations in the antiferromagnetic phase of NdCu2 Appl. Phys. A74 (2002) S751    
M. Rotter, M. Doerr, M. Loewenhaupt, P. Svoboda, Modeling Magnetostriction in RCu2 Compounds using McPhase J. of Applied Physics 91 (2002) 8885
M. Rotter Using McPhase to calculate Magnetic Phase Diagrams of Rare Earth Compounds J. Magn. Magn. Mat (2004) 481
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Course TU Dresden 2005