Prolog
Numerical Modeling in Magnetism
Macro-Magnetism: Solution of Maxwells Equations –
Engineering of (electro)magnetic devices
MFM image
Micromagnetism:
Domain Dynamics, Hysteresis
Micromagnetic
simulation.
Atomic Magnetism:
Instrinsic Magnetic Properties
Atomic MagnetismModeling 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
Atomic MagnetismModeling 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
M. Rotter, Institut für physikalische Chemie, Universität Wien
The Standard Model of RE Magnetism the Crystal Field Concept
+
+
+
+
+
4f –charge density
+
+
E
+
+
+
Hamiltonian H cf 
m m
B
 l Ol (J i )
lm,i
Q
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Example: NdCu2
Crystal Structure of RCu2
+
+
+
+
c
+
... 9 nonzero CF Parameters
+
you can use module
pointc to calculate CF
parameters by the
pointcharge model
+
+
+
+
Imma (orthorhombic)
+
+
b
a
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
NdCu2 – Crystal Field Excitations
orthorhombic, TN=6.5 K, Nd3+: J=9/2, Kramers-ion
McPhase can
• solve CF Model
• Calculate Intensities and Energies
Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297
• Calculate and Plot Charge
Density
McPhase
• ... - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Make a Crystal Field Model
using McPhase Module Cfield
CF Hamiltonian H   B O (J )
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
m
l
cf
m
l
i
lm,i
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 Rio de Janeiro 2007
Magnetism would be boring
without a magnetic field
Hamiltonian
H   BlmOlm ( J i )   g J  B J i H
lm,i
i
Use module cfield
to calculate
magnetization
type: cfield –m
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Specific Heat
Use module cpcalc to
calculate specific heat
type: cpcalc 5 30 1
Tmin=5 Tmax=30 dT=1
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Use modules chrgplot+javaview to plot
4f charge density
 ˆ (r ) | R4 f (r ) |2
m
ec


O
 nm n n (J) T Z nm ()
n 0, 2, 4, 6
m 0 ,..., n
T=100
T=40
T=10 K
K
K
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Use modules pointc+chrgplot+javaview
T=2K
H=0
Same CEF
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
The magnetically ordered State
D1B
ILL, Grenoble
...investigated
by neutron scattering
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
TN= 42 K M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
GdCu2
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Module mcphas
.... do not fit moments – fit Hamiltonians !
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
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 Rio de Janeiro 2007
Do you really want to see the MF
equations ?
1
1
gJi JB JHi H
 J
ij J)Ji  J (ij )  J j 

HH 
B BOO( J( J) ) 
g

J
(
i i Ji B i 2 ij 2 iij
j
m m
ml ml
i
l
l
i
lm,i
lm,i
eff
i
Cfield can calculate
J i J j  J i  J j   J i  J j   J j  J i  
Mi  Mi Hieff  g Ji  B  (JJi 
T,HJeff )( J   J )
 
H
eff
i
i
i
i
J (ij )
 H
Jj 
j g Ji  B
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
j
j
I(κ) [counts]
GdCu2In
McPhase - the-1World of Rare Earth Magnetism
|κ|[Å
] 2007
nichtkollineare Struktur(mcphase)
Martin Rotter - McPhase Rio
de Janeiro
Single Crystal
Flate Cone Diffraction
E2 – HMI, Berlin
k
τ
O
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
3 crystals
(ca 120° rotated)
NdCu2
τ
5τ
3τ
l
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
AF2 Pattern
T=4.1K
H=0T
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
NdCu2
Magnetic
Structures
at T=0
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Pattern T=0
k
0.4
AF1
0.6
0.8
h
3τ
τ=(0.6 0 0)
F1
F2
1.0
τ=(0.6666 0 0)
2τ
τ=(0.625 0 0)
5τ
  
2τ 
2τ


F3
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
NdCu2 Magnetic Phase Diagram
F1   
F3 
c
F1 
a
b
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 Rio de Janeiro 2007
output file: mcphas.hkl
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Bulk Properties Calculated by
module mcphas
Magnetization output file: mcphas.fum
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
NdCu2 Specific Heat
output file: mcphas.fum
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Spontaneous Magnetostriction
Microscopic Source of Magneostriction:
Strain dependence of magnetic interactions
Crystal field
T .... Symmetry
decreases
+
Exchange
L0
T<TC(N)
L=0, L0
T<TC(N)
e-
+
„exchange-striction“
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Forced Magnetostriction
Crystal Field
+
Exchange Striction
H <0
+
L0
H
H
>0
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
L=0, L0
Calculation of Magnetostriction
Crystal Field
Exchange
1
H ex    J (ij ,  )J i J j
2 ij
H cf   Blm ( )Olm (J i )
i ,lm
H  Eel  H cf (  0)  H ex (  0)   
mit Eel 
Z  Tr{e
 H / k BT
1
2

 
c

  

}
  ( H cf  H ex )
 
F
0


F  kBT ln Z
Output file: mcphas.xyt
lm 
    l  Olm (J ) T ,H
B m
+
Output file: mcphas.jj*
k
 
i ik
T ,H

J (i, i  k )
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
 ...
J J

NdCu2 Magnetostriction
Crystal Field
H  H cf (  0)  
H cf

 H ex (  0)
Exchange - Striction

H ex

McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Dispersive Magnetic Excitations
153
1
H    J (ij )Si  S j
2 ij
MF - Zeeman Ansatz
T=1.3 K
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Martin Rotter - McPhase Rio de Janeiro 2007
... Spinwaves (Magnons)
153
1
H    J (ij )Si  S j
2 ij
T=1.3 K
Bohn et. al.
PRB 22 (1980) 5447
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Spinwaves (Magnons)
1
H    J (ij )Si  S j
2 ij
153
a
T=1.3 K
Bohn et. al.
PRB 22 (1980) 5447
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Module Mcdisp – Calculate Magnetic
Excitation Energies and the Neutron
Scattering Cross Section
d 
k '  e 

 N 
2 
ddE '
k  mc 
2
S  mag
2
inel
(κ ,  ) 
2
1
2N b
(   κˆ  κˆ  )S 


mag
(κ ,  )
 iκ ( B d  B d ' ) Wd Wd ' 
1
1
{
gF
(

)}
{
gF
(

)}
e
e
S dd ' (κ ,  )
2
d 2
d'
dd '


1 
 dd ' ' ' ( z ) 
 dd ' ( z )   d'd ( z*)
2i


 (κ,  )   0 ( ) 1   0 ( ) J (κ )
 0 ( )  


1
S  2
1
1  e  / kT
MF-RPA
 i | J   J  H ,T | j  j | J    J   H ,T | i 
ij
 j   i  
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
 ''
(ni  n j )
Module Mcdisp – a novel fast
algorithm for magnetic
excitations
M. Rotter, Comp. Mat. Sci. 38 (2006) 400
 0 ( ) 
s
M 
 s  

s
M 
 s  

s
s*
U 
 s  1U 
 s  

s*
s
U 
 s  1U 
  s  
1   ( ) J (κ) (κ, )   ( )
0
0
Transformation: s ''''s ' ' (κ,  )  

s'
U s* ''  s ''s ' (κ,  )U 
'
 s '  s ''
*
ss ''
s* ss ''
s ''
L
(
κ
)

U
J
(
κ
)
U
  
''
 ''

McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
with definition:
 ss'
 ss''
 1 if  s  0
  ss''  
 1 if  s  0
*
ss ''

   ss'' ( s   )   s L11 (κ )  s '' 11s ''s ' (κ,  )


s '' 
(1)
all other components
of Ψ are zero
with definition: Ass''   ss''  s   s L (κ )  s ''
ss ''
11
*
Generalized eigenvalue problem (analogue to dynmical matrix
in the case of phonons!!)


At   t
 
Solution gives eigenvalues r and eigenvectors  (t1 , t2 ,...)
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
(1) may then be inverted to give the following expression for Ψ
11ss' (κ,  )   sr ( r   ) 1 rr ' sr* '
rr '
back transformation...
 (κ ,  )   s '  s
ss '
*
U 
s
1
ss '
11
(κ ,  )U s '*1
r


1 
 ss' ( z )   s's ( z*)
+calculation of absorptive part...
2i
1
1
lim

P
 i ( r   )
using Diracs formula:
 0     i





r
r
 ss' ' ' ( z ) 
 ' ' (κ ,  )   s '  s
ss '
*
s
*
s '*
U


(


(
κ
)



)

U
 1 sr
r
s 'r  1
r
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
McDisp - fast algorithm - Cookbook
s
1) s  ij : M  i | J   J  H ,T | j  j | J    J   H ,T | i  (ni  n j )
s
s
2) ...diagona lize M ...   s ,U
3) ...setup Matrix Ass''   ss''  s   s
4) ...solve generalized EV Problem
ss '
 ' '
(κ ,  )   s '  s
5)
S  2
S  mag
inel
(κ ,  ) 
1
2N b
*
1
U 
s
1
U


s*
1
J  (κ )U
ss ''
s ''
1
 s ''
*


At   t ...   r ,
*
s '*

(


(
κ
)



)

U
sr
r
s 'r  1
r
1  e  / kT
 ''
 iκ ( B d  B d ' ) Wd Wd ' 
1
1
{
gF
(

)}
{
gF
(

)}
e
e
S dd ' (κ ,  )
2
d 2
d'
dd '
d 2
k '  e 2 

 N 
2 
ddE '
k  mc 
2

ˆ
ˆ
(


κ
κ
)
S
    mag (κ,  )

McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
F3 
NdCu2
F1 
AF1 
Diffuse Scattering
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
McPhase Modules
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
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. Coordinate Orientation
Nonzero Blm
Symmetry
O cubic
xyz||abc
B40,B44,B60,B64
O cubic
z||111
B40,B43,B60,B63,B66
D6h hexagonal
xyz||abc
B20,B40,B60,B66
D4h tetragonal
xyz||abc
B20,B40,B44,B60,B64
C3v (no lattice)
B20,B40,B43,B60,B63
C2h monoclinic
B20,B40,B60,B66,B66s
D3d (quasicubic in
dhcp)
xyz||abc
B20,B40,B43,B60,B63,B66
D2 orthorh.
xyz||abc
B20,B22,B40,B42,B44,B60,B62,B64,B66
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

3 2
~ 2
O2 
O2  i 2 Pxy
8

3
~
O21   2 Pxz  i 2 Pyz 
8
~
O20  O20
Irr.
Repr.
These operators form a reducable
representation T2(G) of the point group
~m
~ m'
2
O2 (J ' )   Tmm' (G )O2 (J )
Group elements G
C2v
1E
1C2
1σy
1σx
A1
1
1
1
1
B1
1
-1
-1
1
A2
1
1
-1
-1
B2
1
-1
1
-1
Character table of mm2
m'
Group Theory basics taken from: Elliott&Dawber Symmetry in Physics, McMillan Press, 1979
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
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“)
T 2 (G)  mA1 A1  mB1B1  mA2 A2  mB 2 B2
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
m A1 
1
c p  pA1*  p

g p
l
 p   Tmm
(G) 
m
sin(( l  1 / 2)a)
sin( a / 2)
a... Angle of rotation
1
m A1  (5  1  1  1)  2
4
Cp... Number of members of class p
g.... Number of group elements
χ.... Character of class
Class p
a
χp
E
0
5
C2
π
1
σy
π
1
σ
π
1
x
the World2ndoforder
RareCFEarth
Magnetism
i.e.McPhase
We expect 2-independent
parameters
Martin Rotter - McPhase Rio de Janeiro 2007
The basis of the 2 A1 representation occuring in the decomposition of
T2(G) can be found using the projection operator
P
A1
1
   A1 (G )T 2 (G )
g G
In order to calculate it, we have to epxlicitely write down the reducable
representation T2:
2
Tmm
' ( E )   mm'

2
mm'
(C2 )  (1)  mm'
Jx‘=-Jx, Jy‘=-Jy
2
mm'
( y )  (1)  m,  m '
Jy‘=-Jy
T
T
m
m
2
Tmm
' ( x )   m ,  m '
A1
mm '
P

Jx‘=-Jx
3 2
~
O2 2 
O2  i 2 Pxy
8
3
~
O21   2 Pxz  i 2 Pyz 
8
~0
O2  O20

1
  mm'  (1) m  mm'  (1) m  m ,  m '   m,  m '
4

1 ~
3
~
~
P A1O2 2  O22  O22  O22
2
8
~
~
~
P A1O21  0, P A1O20  O20

B20 and B22 are nonzero.
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007

Symmetry – Bilinear Interaction
Isotropic interaction (J(ij) is a scalar)
1
H ex    J i J (ij )J j
2 ij
Anisotropic Interaction (J(ij) is a tensor)
1
H ex    J i J (ij )J j
2 ij
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
neighbors related by
symmetry must have
related interaction
constants J(ij)
(quasi)hexagonal types of neighbors
CeCu2 Structure
c
a
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 Rio de Janeiro 2007
Anisotropic Interaction –
Symmetry Considerations
1
H ex    J i J (ij )J j
2 ij
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
ETC...
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Example: bc mirror plane
1
1
H   J 0 J J1   J '0 J J '1
2
2
1 0 0 


J '   0  1 0 J  S J
 0 0  1


 J aa

 J  SJS  0
 0

0
J bb
J cb
0 

J bc 
J cc 
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
b
1
0
a
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
Example for quadrupolar
interactions: PrCu2
+
+
M
++
++
++
++
+
+
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
H
PrCu2
www.mcphase.de
O22
T
0
O22
T
0
+
+
1
2 T
O
0
++
++
++
++
Settai et. al. JPSJ 67 (1998) 636
+
+
Ferroquadrupolarer (Cij>0) Austausch (durch CF-Phonon WW)
H Q   Cij  O (J i )  O (J j )
2
2
2
2
Settai et. al. JPSJ 67 (1998) 636
McPhase -ijthe World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
PrCu2
Ferroquadrupolar (Cij>0) Interaction
H Q   Cij  O22 (J i )  O22 (J j )
ij
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 Rio de Janeiro 2007
Orbital Excitations (Orbitonen)
+
+
+
+
+
4f – charge density
+
+
+
E
+
+
Crystal field H cf 
m m
B
 l Ol (J i )
lm,i
+Antiferroquadrupolar (C<0) Interaction
H Q   C  O22 (J i )  O22 (J j )
Q
ij
McPhase - the World of Rare Earth
Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
PrCu2
+
+
++
++
++
++
+
+
Ferroquadrupolar (Cij>0) Interaction (via CF-Phonon coupling)
H Q   Cij  O (J i )  O (J j )
2
2
2
2
Settai et. al. JPSJ 67 (1998) 636
McPhase -ijthe World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
PrCu2
Orbital Modes T=5 K, H=0 T
MF-RPA Model
Experiment
Г
Energy (meV)
2.5
?
0
1
00L
2
McPhase: www.mcphase.de
Rotter, JMMM 272-276 (2003) 481
Kawarazaki et. al.,
J. Phys. Cond. Mat. 7 (1995) 4051
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
NdCu2
PrCu2
Nur Quadrupolaustausch Г
[Interpretation von Kawarazaki
et. al., J. Phys. Cond. Mat. 7
(1995) 4051]
Energy (meV)
2.5
0
NdCu2
Könnte nicht auch die
Austauschwechselwirkung
zu der beobachteten
Dispersion führen ?
Magnetic Excitations
Rotter et. al., Europ. Phys. J. B 14 (2000) 29
1 00L
2
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
PrCu2
H   C  O
2
2
Q
1
 H ex    J (ij )S i  S j
2 ij
(J i )  O (J j )
2
2
ij
2.5
2.5
Energy (meV)
+ magnetic Interactions
Energy (meV)
Quadruplar
Nur Quadrupolaustausch
Interaction only Г
0
0
1 00L
2
nur
magnetischer
Austausch
1
00L
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
2
PrCu2 Orbital modes in Magnetic field
T=2 K, H||a
Rechnung
Messung
IN12(ILL) März 2004
(15 Tesla cryomagnet)
McPhase - the World of Rare Earth Magnetism
McPhase: www.mcphase.de
Rotter, JMMM 272-276 (2003) 481
Martin Rotter - McPhase Rio de Janeiro 2007
Quadrupolar Effects
Neutrons can be scattered by 4f - Orbitons
– Orbiton spectroscopy:
- Determination of multipolar Interactions
- Modeling of GMS
(Cij>0) Interactions
PrCu2 Crystal field + Ferroquadrupolar
2
2
H Q   Cij  O2 (J i )  O2 (J j ) Settai et. al. JPSJ 67 (1998) 636
ij
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 Rio de Janeiro 2007
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
McPhase - the World of Rare Earth Magnetism
...  Confirmation of phase diagram models H||a,b,c, dJ(ij)/dε
Martin Rotter - McPhase Rio de Janeiro 2007
The story of NdCu2

Inverse suszeptibility at
high T
1
(2 J  1)( 2 J  3) 0
k a  J ( J  1) J (q  0) 
( B2  B22 )
3
10
1
(2 J  1)( 2 J  3) 0
kb  J ( J  1) J (q  0) 
B2
3
5
1
(2 J  1)( 2 J  3) 0
k c  J ( J  1) J (q  0) 
( B2  B22 )
3
10
... 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 Rio de Janeiro 2007
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 Rio de Janeiro 2007
How to start analysis – the story of
NdCu2

Magnetization: Kramers ground state doublet |+-> matrix elements
M   g  B tanh( g  B ( H  M ) /( 2kT ))
  a, b, c
P. Svoboda et al. JMMM 104 (1992) 1329
g a / g J   J y   2.1
g b / g J   J z   2.8
g c / g J   J x   1.5
Module cfield can also calculate
magnetization using a full set of
CF parameters
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
How to start analysis – the story of
NdCu2

Neutron TOF spectroscopy – CF levels
Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297
... Blm
B20=1.35 K
B22=1.56 K
B40=0.0223 K
B42=0.0101 K
B44=0.0196 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 Rio de Janeiro 2007
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 Rio de Janeiro 2007
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
f(B) [arb.units] T=0K
BcAF1F3

B
 AF1
Bc1
Bc2

Bc3

F1

F2

F3
NdCu2 Magnetic Phase Diagram
F1   
F3 
c
F1 
a
b
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 Rio de Janeiro 2007
The story of NdCu
Jaa=Jcc(R)
2

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 Rio de Janeiro 2007
F3 
NdCu2
F1 
AF1
M. Rotter, et al. Applied Phys. A 74 (2002) s751

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
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
„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.“
Nonsense !
• 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 Rio de Janeiro 2007
The magnetoelastic
Paradoxon
for L=0
demonstrated
at GdNi2B2C
Orthorhombic Distortion
?
Exchange-Striction
H  H ex  Eel 
  A( aa   bb )( J i J i ( 010)  J i J i (100) ) 
i
 B( aa   bb )( J i J i ( 010)  J i J i (100) )

Standard Model of RE Mag
 aa   bb ~ J i J i ( 010) T ,H   J i J i (100) T ,H ... McPhase Simulation
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
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 Rio de Janeiro 2007
Crystal Field/Magnetic/Orbital Excitations
Magnetostriction
and much more....
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
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
McPhase - the World of Rare Earth Magnetism
Martin Rotter - McPhase Rio de Janeiro 2007
Workshop
Magnetostrictive Materials and
Magnetic Refrigeration (MMMR)
13.-15. August 2007,
Vienna, Austria
http://www.univie.ac.at/MMMR/