Prolog Numerical Modeling in Magnetism

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PrologNumerical Modeling in Magnetism
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Atomic Magnetism- Modeling Instrinsic Magnetic
Properties

• Band Models
• Spin Polarized First Principle Methods
• restricted to simple Magnetic Structures,
  T0, 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

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Atomic Magnetism- Modeling Instrinsic Magnetic
Properties

• Band Models
• Spin Polarized First Principle Methods
• restricted to simple Magnetic Structures,
  T0, 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

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The Standard Model of RE Magnetism - the Crystal
Field Concept
4f charge density
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Example NdCu2
Crystal Structure of RCu2 Imma (orthorhombic) ...
9 nonzero CF Parameters
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NdCu2 Crystal Field Excitations
orthorhombic, TN6.5 K, Nd3 J9/2, Kramers-ion
Gratz et. al., J. Phys. Cond. Mat. 3 (1991) 9297
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Make a Crystal Field Modelusing 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

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Magnetism would be boring without a magnetic field
Hamiltonian
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Specific Heat
Use module cpcalc to calculate specific
heat type cpcalc 5 30 1
Tmin5 Tmax30 dT1
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Use modules chrgplotjavaview to plot 4f charge
density
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Use modules pointcchrgplotjavaview
T2K H0 Same CEF
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Module mcphas
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Input files for module mcphas mcphas.j
(structure), mcphas.cf (single ion properties),
mcphas.tst (table of initial values), mcphas.ini
(H,T-range, ...)
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Do you really want to see the MF equations ?
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Bulk Properties Calculated by module mcphas
Magnetization output file mcphas.fum
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NdCu2 Specific Heat
output file mcphas.fum
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Spontaneous Magnetostriction
Microscopic Source of Magneostriction Strain
dependence of magnetic interactions
Crystal field
T?
L?0
.... Symmetry decreases
T?

TltTC(N)
TgtTC(N)
e-

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Forced Magnetostriction
Crystal Field
Exchange Striction

L?0
L0, L?0
H ?lt0
H

e-
H ?gt0

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Calculation of Magnetostriction
Crystal Field
Exchange
mit
Output file mcphas.jj
Output file mcphas.xyt

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NdCu2 Magnetostriction
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NdCu2 Magnetic Phase Diagram
F1 ? ? ?
F3 ??
c
F1 ???
b
a
AF1 ??????????
linesexperiment
output file mcphas.xyt Use module phased or
displaycontour for color plot of phasediagram
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output file mcphas.hkl
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Dispersive Magnetic Excitations
153
MF - Zeeman Ansatz
T1.3 K
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... Spinwaves (Magnons)
153
T1.3 K
Bohn et. al. PRB 22 (1980) 5447
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Spinwaves (Magnons)
153
a
T1.3 K
Bohn et. al. PRB 22 (1980) 5447
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Module Mcdisp Calculate Magnetic Excitation
Energies and the Neutron Scattering Cross Section
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Module Mcdisp a novel fast algorithm for
magnetic excitations Rotter 2005
Transformation
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with definition
(1)
all other components of ? are zero
with definition
Generalized eigenvalue problem (analogue to
dynmical matrix in the case of phonons!!)
Solution gives eigenvalues
and eigenvectors
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1. may then be inverted to give the following
   expression for ?

back transformation...
calculation of absorptive part...
using Diracs formula
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McDisp - fast algorithm - Cookbook
1)
2)
...setup Matrix
3)
...solve generalized EV Problem
4)
5)
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NdCu2
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Diffuse Scattering
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McPhase Modules
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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 xyzabc B40,B44,B60,B64
O cubic z111 B40,B43,B60,B63,B66
D6h hexagonal xyzabc B20,B40,B60,B66
D4h tetragonal xyzabc B20,B40,B44,B60,B64
C3v (no lattice) B20,B40,B43,B60,B63
C2h monoclinic B20,B40,B60,B66,B66s
D3d (quasicubic in dhcp) xyzabc B20,B40,B43,B60,B63,B66
D2 orthorh. xyzabc B20,B22,B40,B42,B44,B60,B62,B64,B66
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Example 2nd order CF terms for point symmetry
mm2C2v
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 1sy 1sx
A1 1 1 1 1
B1 1 -1 -1 1
A2 1 1 -1 -1
B2 1 -1 1 -1
Irr. Repr.
These operators form a reducable representation
T2(G) of the point group
Character table of mm2
Group Theory basics taken from ElliottDawber
Symmetry in Physics, McMillan Press, 1979
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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 ms 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 ?.... Character of class
Class p a ?p
E 0 5
C2 p 1
sy p 1
sx p 1
a... Angle of rotation
i.e. We expect 2 independent 2nd order CF
parameters
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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.
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Symmetry Bilinear Interaction
Isotropic interaction (J(ij) is a scalar)
Anisotropic Interaction (J(ij) is a tensor)
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(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)
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Anisotropic Interaction Symmetry Considerations
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ETC...
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Example bc mirror plane
b
1
a
0
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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

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Example for quadrupolar interactions PrCu2






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PrCu2
GMS






Settai et. al. JPSJ 67 (1998) 636
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PrCu2
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 ?
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Orbital Excitations (Orbitonen)
4f charge density
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PrCu2






Settai et. al. JPSJ 67 (1998) 636
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PrCu2 Orbital Modes T5 K, H0 T
Experiment
MF-RPA Model
?
2.5
Energy (meV)
?
0
00L
1
2
McPhase www.mcphase.de Rotter, JMMM
272-276 (2003) 481
Kawarazaki et. al., J. Phys. Cond. Mat. 7 (1995)
4051
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PrCu2
Könnte nicht auch die Austauschwechselwirkung zu
der beobachteten Dispersion führen ?
Nur Quadrupolaustausch
?
Interpretation von Kawarazaki et. al., J. Phys.
Cond. Mat. 7 (1995) 4051
2.5
Energy (meV)
0
00L
1
2
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PrCu2
Nur Quadrupolaustausch
Quadruplar Interaction only
?
2.5
Energy (meV)
0
00L
1
2
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PrCu2 Orbital modes in Magnetic field
T2 K, Ha
Rechnung
Messung
IN12(ILL) März 2004 (15 Tesla cryomagnet)
McPhase www.mcphase.de Rotter, JMMM
272-276 (2003) 481
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Quadrupolar Effects
Neutrons can be scattered by 4f - Orbitons
Orbiton spectroscopy
- Determination of multipolar Interactions -
Modeling of GMS

• The model describes well
• macroscopic properties and quadrupolar Phase
  diagram
• Magnitude of dispersion of orbital modes

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How to start the story of NdCu2

• Suszeptibility 1/?(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/de)
• Neutron diffraction magnetic structure in fields
  easy axis
• ... ? phase diagram Hb - model
• ... ? Jbb
• Neutron spectroscopy on single crystals in
  Hb3T
• ... ? Anisotropy of Jij - determination
  of JaaJcc
• Magnetostriction
• ... ? Confirmation of phase diagram
  models Ha,b,c, dJ(ij)/de

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The story of NdCu2

• Inverse suszeptibility at high T
• ... B200.8 K, B221.1 K
• Hashimoto, Journal of Science of the Hiroshima
  University A43, 157 (1979)

Tabc
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The story of NdCu2

• Specific haet Cp and entropy first info about
  levels

Gratz et. al., J. Phys. Cond. Mat. 3 (1991) 9297
Rln2
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How to start analysis the story of NdCu2

• Magnetization Kramers ground state doublet -gt
  matrix elements

P. Svoboda et al. JMMM 104 (1992) 1329
Module cfield can also calculate magnetization
using a full set of CF parameters
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How to start analysis the story of NdCu2

• Neutron TOF spectroscopy CF levels
• ... Blm

Gratz et. al., J. Phys. Cond. Mat. 3 (1991) 9297
B201.35 K B221.56 K B400.0223 K B420.0101
K B440.0196 K B604.89x10-4 K B621.35x10-4
K B644.89x10-4 K B664.25 x10-3 K
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The story of NdCu2

• Thermal expansion cf influence
• ... Magnetoelastic parameters (AdB20/de,
  BdB22/de)

E. Gratz et al., J. Phys. Condens. Matter 5, 567
(1993)
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The story of NdCu2

• Neutron diffraction magnetization
• magstruc, phasediag Hb-gt 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
??????????
??
???
???
??????????
??
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NdCu2 Magnetic Phase Diagram
F1 ? ? ?
F3 ??
c
F1 ???
b
a
AF1 ??????????
linesexperiment
output file mcphas.xyt Use module phased or
displaycontour for color plot of phasediagram
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The story of NdCu2

• Neutron spectroscopy on single crystals in
  Hb3T
• ... Anisotropy of J(ij) - determination
  of JaaJcc

F3 ??
M. Rotter et al., Eur. Phys. J. B 14, 29 (2000)
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NdCu2
M. Rotter, et al. Applied Phys. A 74 (2002) s751
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How to start analysis the story of NdCu2

• Magnetostriction ... Confirmation of phasediagram
  model for Ha,b,c, and determination of
  dJ(ij)/de

M. Rotter, et al. J. of Appl. Physics 91 10(2002)
8885
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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 L0 AF-systems
• -extra magnetic modes or no modes (CeCu2,
  CeNi9Ge4, Nd2CuO4),
• -wrong saturation moments, e.g. in
  Eu-Skutterudite
• - ...

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The magnetoelastic Paradoxonfor
L0demonstrated at GdNi2B2C
Orthorhombic Distortion
?
Exchange-Striction
Standard Model of RE Mag ... McPhase Simulation
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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
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Crystal Field/Magnetic/Orbital Excitations
Magnetostriction 
and much more....
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Epilog

• 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. 272-276 (2004) 481