Theory

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Theory

• Isotropic Thermal Expansion
• Phase Transitions
• Lagrange Strain Tensor
• Anisotropic Thermal Expansion
• Magnetostriction
• Matteucci effect
• Villari Effect
• Wiedemann Effect
• Saturation Magnetostriction
• (Phenomenological Description, Symmetry
  Considerations)
• Band Magnetostriction
• Local Moment Magnetostriction (Crystal Field
  Exchange Striction)

2
Isotropic Thermal Expansion
Thermal expansion Coefficients
Helmholtz free Energy
Compressibility
3
Approximation compressibility is T independent
(dominated by electrostatic part of binding
energy)
Subsystem r ..... phonons, electrons, magnetic
moments
4
Phase Transitions
5
Mechanics of Solids - Kinematics
i1,2,3
Inf. Translation
Inf. Rotation (antisymmetric matrix)
Inf. Strain (symmetric matrix)
Volume Strain
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Lagrange Strain Tensor

• The strain tensor, e, is a symmetric tensor used
  to quantify the strain of an object undergoing a
  small 3-dimensional deformation
• the diagonal coefficients eii are the relative
  change in length in the direction of the i
  direction (along the xi-axis) 
• the other terms eij 1/2 ?ij (i ? j) are the
  shear strains, i.e. half the variation of the
  right angle (assuming a small cube of matter
  before deformation).
• The deformation of an object is defined by a
  tensor field, i.e., this strain tensor is defined
  for every point of the object. In case of small
  deformations, the strain tensor is the Green
  tensor or Cauchy's infinitesimal strain tensor,
  defined by the equation

Where u represents the displacement field of the
object's configuration (i.e., the difference
between the object's configuration and its
natural state). This is the 'symmetric part' of
the Jacobian matrix. The 'antisymmetric part' is
called the small rotation tensor.
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T stress tensor      is defined by
where the dFi are the components of the resultant
force vector acting on a small area dA which can
be represented by a vector dAj perpendicular to
the area element, facing outwards and with length
equal to the area of the element. In elementary
mechanics, the subscripts are often denoted x,y,z
rather than 1,2,3.
Stress tensor is symmetric, otherwise the volume
element would rotate (to seet this look at zy and
yz component in figure)
Hookes Law
(Voigt) notation
1 11, 2 22 3 33 4 23 5 31 6 12
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Anisotropic Thermal Expansion
Elastic Energy density
.... strain can be written as
Thermal expansion Coefficients
Elastic Constants
Elastic Compliances
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.... this can (as in the isotropic case) be
written as sum of contributions of subsystems r
phonons, electrons, magnetic moments
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Grueneisens Approximation

• Specific heat of subsystem r
• Grueneisen Parameter of subsystem r ... Is in
  many simple model cases temperature independent

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Normal thermal Expansion
Anharmonicity of lattice dynamics
anharmonic Potential
Harmonic potential

Small contribution of band electrons
with Debye function
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Magnetostriction
Magnetostriction is a property of magnetic
materials that causes them to change their shape
when subjected to a magnetic field. The effect
was first identified in 1842 by James Joule when
observing a sample of nickel.
James Prescott Joule, (1818 1889)
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Thermal expansion Coefficients
Magnetostriction Coefficients
Material Crystal axis
Saturation magnetostrictionl (x 10-5)
Fe 100
(1.1-2.0) Fe 111
-(1.3-2.0) Fe
polycristal -0.8 Terfenol-D
111 200
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Villari Effect the change of the
susceptibility of a material when subjected to a
mechanical stress Matteucci effect
creation of a helical anisotropy of the
susceptibility of a magnetostrictive material
when subjected to a torque Wiedemann Effect
twisting of materials when an helical magnetic
field is applied to them
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Domain Effects
rotation of the domains.
migration of domain walls within the material in
response to external magnetic fields.
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In general the saturation magnetostriction will
depend on the direction of the field and the
direction of measurement ... Taylor expansion in
terms of cosines of magnetization direction (ax
ay az) and measurement direction (ßx ßy ßz)
(Cark Handbook of ferromagnetic materials,
Elsivier, 1980)
Write Energy in terms of strain and Magnetization
Zero in case of inversion symmetry
consider symmetry
And apply
Hexagonal
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Cubic
(8 domains)
Assumption in zero field all 8 domains are
equally populated
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dL/L Measurement dir.
magnetization
field
Zero field
... 8 domains
Field 111
19
dL/L Measurement dir.
magnetization
field
is zero
20
dL/L Measurement dir.
magnetization
field
Zero field
... 8 domains contributions cancel
Field 011
21
dL/L Measurement dir.
magnetization
field
Zero field
... 8 domains contributions cancel
Field 0-11
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Summary
Cubic crystal, easy axis 111
Assumption in zero field all 8 domains are
equally populated Magnetostriction due to domain
rotation is given by
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Atomic Theory of Magnetostriction

• Band Models
• Localized Magnetic Moments

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Magnetism of Free Electrons

• Sommerfeld Model of Free Electrons

Schrödinger equation Free electrons (positive
energy) Schrödinger equation of free
electrons Solution Characteristic
equation Momentum Wavevector k
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Periodic Boundary Condition (1d)
Complex numbers
Condition for phases
Allowed k-vectors (3 dim)
Possible wavefunctions (3 dim)
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2-D projection of 3-D k-space

• Each state can hold 2 electrons
• of opposite spin (Paulis principle)
• To hold N electrons

ky
dk
2p/L
k
kx
kF Fermi wave vector heN/V electron number
density
Fermi Energy
Fermi Velocity
Fermi Temp.
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Fermi Parameters for some Metals
Vacuum Level
free electrons
F Work Function
EF
electrons in periodic potential energy gap at
Brillouin zone boundary
Energy
Band Edge
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Effect of Temperature
Fermi-Dirac equilibrium distribution for the
probability of electron occupation of energy
level E at temperature T
Enrico Fermi
f
1
T
0 K
Vacuum
Occupation Probability,
Energy
Increasing
T
0
µ
F
Work Function,

Electron Energy,
E
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Number and Energy Densities
Summation over k-states
Integration over k-states
Transformation from k to E variable
Integration of E-levels for number and
energy densities
Number of k-states available between energy E and
EdE
Density of States
A tedious calculation gives
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Free Electrons in a Magnetic Field
Pauli Paramagnetism
Spin - Magnetization for small fields B (T0)
Magnetic Spin - Susceptibility
(Pauli Paramagnetism)
Pauli paramagnetism is a weak effect compared to
paramagnetism in insulators (in insulators one
electron at each ion contributes, in metals only
the electrons at the Fermi level contribute).
The small size of the paramagnetic susceptibility
of most metals was a puzzle until Pauli pointed
out that is was a consequence of the fact that
electrons obey Fermi Dirac rather than classical
statistics.
W. Pauli Nobel Price 1945
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Direct Exchange between delocalized Electrons
Spontaneously Split bands e.g. Fe M2.2µB/f.u.
is non integer .... this is strong evidence for
band ferromagnetism
Mean field Model all spins feel the same
exchange field ?M produced by all their
neighbors, this exchange field can magnetize the
electron gas spontaneously via the Pauli
Paramagnetism, if ? and ?P are large
anough. Quantitative estimation what is the
condition that the system as a whole can save
energy by becoming ferromagnetic ?
moving De(EF)dE/2 electrons from spin down to
spin up band
kinetic energy change
exchange energy change
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total energy change
there is an energy gain by spontaneous
magnetization, if
Stoner Criterion
Edmund C. Stoner (1899-1968)
... Coulomb Effects must be strong and density of
states at the Fermi energy must be large in order
to get sponatneous ferrmagnetism in metals.
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Spontaneous Ferromagnetism splits the spin up and
spin down bands by ? If the Stoner criterion is
not fulfilled, the susceptibility of the electron
gas may still be enhanced by the exchange
interactions
energy change in magnetic field
this is minimized when
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Band Magnetostriction
moving De(EF)dE/2 electrons from spin down to
spin up band
exchange energy change
kinetic energy change
35
Gd metal Tc 295 K , TSR 232 K M0017.55mB
LARGE VOLUME MAGNETOSTRICTION !
...anisotropic MS c/a(T) not explained
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Mechanisms of magnetostriction in the Standard
model of Rare Earth Magnetism

• microscopic origin of magnetostriction
• strain dependence of magnetic interactions

1) Single ion effects ? Crystal Field
Striction spontaneous magnetostriction
forced magnetostriction
T gtTN
kT gtgt?cf
kT lt?cf
T ltTN
T ltTN
H
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T gtTN
kT gtgt?cf
kT lt?cf
38
T ltTN
NdCu2
TN
TN
39
T ltTN
NdCu2
T ltTN
H
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2) Two ion effects ? Exchange Striction
spontaneous magnetostriction forced m
agnetostriction
T gtTN
T ltTN
T ltTN
H
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GdCu2 (Gd3 shows no CEF effect... only exchange
striction)
Forced Magnetostriction
Spontaneous Magnetostriction
T4.2K
TN
M. Rotter, J. Magn. Mag. Mat. 236 (2001) 267-271
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Calculation of Magnetostriction
Crystal field
Exchange
with

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NdCu2 Magnetostriction
Calculation done by Mcphase www.mcphase.de
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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
48
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)
50
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
52
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)
53
NdCu2
M. Rotter, et al. Applied Phys. A 74 (2002) s751
54
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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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
57

Crystal Field/Magnetic/Orbital Excitations
Magnetostriction 
and much more....
58
Epilog

• McPhase runs on Linux and Windows and is
  available as freeware.
• www.mcphase.de
• 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