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### Magnetism of Free Electrons

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)

M.Rotter „Magnetostriction“ Course Lorena 2007

Isotropic Thermal Expansion

Thermal expansion Coefficients

Helmholtz free Energy

Compressibility

M.Rotter „Magnetostriction“ Course Lorena 2007

Approximation: compressibility is T independent (dominated by electrostatic part of binding energy)

Subsystem r ..... phonons, electrons, magnetic moments

M.Rotter „Magnetostriction“ Course Lorena 2007

Phase Transitions

M.Rotter „Magnetostriction“ Course Lorena 2007

Mechanics of Solids - Kinematics

i=1,2,3

Inf. Translation

Inf. Rotation (antisymmetric matrix)

Inf. Strain

(symmetric matrix)

Volume Strain

M.Rotter „Magnetostriction“ Course Lorena 2007

Lagrange Strain Tensor

- The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation:
- the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ;
- the other terms εij = 1/2 γij (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.

M.Rotter „Magnetostriction“ Course Lorena 2007

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Anisotropic Thermal Expansion

Elastic Energy density

.... strain can be written as

Thermal expansion Coefficients

Elastic Constants

Elastic Compliances

M.Rotter „Magnetostriction“ Course Lorena 2007

.... this can (as in the isotropic case) be written as sum of contributions of subsystems r = phonons, electrons, magnetic moments

M.Rotter „Magnetostriction“ Course Lorena 2007

Grueneisens Approximation

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Normal thermal Expansion

Anharmonicity of lattice dynamics

anharmonicPotential

Harmonic potential

+

Small contribution of band electrons

with Debye function

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)

M.Rotter „Magnetostriction“ Course Lorena 2007

Thermal expansion Coefficients

Magnetostriction Coefficients

MaterialCrystal 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

M.Rotter „Magnetostriction“ Course Lorena 2007

Villari Effect the change of the susceptibility of a material when subjected to a mechanical stress

Matteucci effectcreation 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

M.Rotter „Magnetostriction“ Course Lorena 2007

Domain Effects

T<TCM||111

T>TC

rotation of the domains.

migration of domain walls within the material in response to external magnetic fields.

M.Rotter „Magnetostriction“ Course Lorena 2007

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 (αx αy αz) and measurement direction (βx βy βz)

(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

M.Rotter „Magnetostriction“ Course Lorena 2007

(8 domains)

Assumption: in zero field all 8 domains are equally populated

M.Rotter „Magnetostriction“ Course Lorena 2007

magnetization

field

Zero field

... 8 domains

Field || 111

M.Rotter „Magnetostriction“ Course Lorena 2007

magnetization

field

Zero field

... 8 domains – contributions cancel

Field || 011

M.Rotter „Magnetostriction“ Course Lorena 2007

magnetization

field

Zero field

... 8 domains – contributions cancel

Field || 0-11

M.Rotter „Magnetostriction“ Course Lorena 2007

Cubic crystal, easy axis 111

Assumption: in zero field all 8 domains are equally populated

Magnetostriction due to domain rotation is given by

M.Rotter „Magnetostriction“ Course Lorena 2007

Atomic Theory of Magnetostriction

- Band Models
- Localized Magnetic Moments

M.Rotter „Magnetostriction“ Course Lorena 2007

Schrödinger equation

Free electrons (positive energy)

Schrödinger equation of

free electrons

Solution

Characteristic equation

Momentum

Wavevector k

Sommerfeld Model of Free Electrons

M.Rotter „Magnetostriction“ Course Lorena 2007

Periodic Boundary Condition (1d):

Complex numbers

Condition for phases

Allowed k-vectors (3 dim)

Possible wavefunctions (3 dim)

M.Rotter „Magnetostriction“ Course Lorena 2007

of 3-D k-space

- Each state can hold 2 electrons
- of opposite spin (Pauli’s principle)
- To hold N electrons

ky

dk

2p/L

k

kx

kF: Fermi wave vector

he=N/V: electron number density

Fermi Energy

Fermi Velocity:

Fermi Temp.

M.Rotter „Magnetostriction“ Course Lorena 2007

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

M.Rotter „Magnetostriction“ Course Lorena 2007

B

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

M.Rotter „Magnetostriction“ Course Lorena 2007

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 E+dE

Density of States

A tedious calculation gives:

M.Rotter „Magnetostriction“ Course Lorena 2007

Free Electrons in a Magnetic Field

Pauli Paramagnetism

Spin - Magnetization for small fields B (T=0)

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Direct Exchange between delocalized Electrons

Spontaneously Split bands: e.g. Fe M=2.2μB/f.u. is non integer

.... this is strong evidence for band ferromagnetism

Mean field Model: all spins feel the same exchange field λM produced by all their neighbors, this exchange field can magnetize the electron gas spontaneously via the Pauli Paramagnetism, if λ and χP are large anough.

Quantitative estimation: what is the condition that the system as a whole can save energy by becoming ferromagnetic ?

moving De(EF)δE/2 electrons from spin down to spin up band

kinetic energy change:

exchange energy change:

M.Rotter „Magnetostriction“ Course Lorena 2007

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.

M.Rotter „Magnetostriction“ Course Lorena 2007

Spontaneous Ferromagnetism splits the spin up and spin down bands by Δ

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Band Magnetostriction

moving De(EF)δE/2 electrons from spin down to spin up band

exchange energy change:

kinetic energy change:

M.Rotter „Magnetostriction“ Course Lorena 2007

Tc= 295 K ,

TSR= 232 K

M||[001]=7.55mB

LARGE VOLUME

MAGNETOSTRICTION !

...anisotropic MS

c/a(T) not explained

M.Rotter „Magnetostriction“ Course Lorena 2007

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 >TN

kT >>cf

kT <cf

T <TN

T <TN

H

M.Rotter „Magnetostriction“ Course Lorena 2007

Exchange Striction

…spontaneous

magnetostriction

…forced

magnetostriction

T >TN

T <TN

T <TN

H

M.Rotter „Magnetostriction“ Course Lorena 2007

GdCu2 (Gd3+ shows no CEF effect... only exchange striction)

Forced Magnetostriction

Spontaneous Magnetostriction

T=4.2K

TN

M. Rotter, J. Magn. Mag. Mat. 236 (2001) 267-271

M.Rotter „Magnetostriction“ Course Lorena 2007

Calculation of Magnetostriction

Crystal field

Exchange

with

+

M.Rotter „Magnetostriction“ Course Lorena 2007

NdCu2 Magnetostriction

Exchange - Striction

Crystal Field

Calculation done by Mcphase

www.mcphase.de

M.Rotter „Magnetostriction“ Course Lorena 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

... Confirmation of phase diagram models H||a,b,c, dJ(ij)/dε

M.Rotter „Magnetostriction“ Course Lorena 2007

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

M.Rotter „Magnetostriction“ Course Lorena 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

M.Rotter „Magnetostriction“ Course Lorena 2007

How to start analysis – the story of NdCu2

- Magnetization: Kramers ground state doublet |+-> matrix elements

P. Svoboda et al. JMMM 104 (1992) 1329

M.Rotter „Magnetostriction“ Course Lorena 2007

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=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

M.Rotter „Magnetostriction“ Course Lorena 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)

M.Rotter „Magnetostriction“ Course Lorena 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

NdCu2 Magnetic Phase Diagram

F1

F3

c

F1

b

a

AF1

lines=experiment

M.Rotter „Magnetostriction“ Course Lorena 2007

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)

M.Rotter „Magnetostriction“ Course Lorena 2007

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-theWorldofRareEarthMagnetism

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Crystal Field/Magnetic/Orbital Excitations

Magnetostriction

and much more....

M.Rotter „Magnetostriction“ Course Lorena 2007

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 NdCu2Appl. 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

M.Rotter „Magnetostriction“ Course Lorena 2007

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