Theory Isotropic Thermal Expansion Phase Transitions Lagrange Strain

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Theory • • • Isotropic Thermal Expansion Phase Transitions Lagrange Strain Tensor Anisotropic Thermal

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 1

Isotropic Thermal Expansion Thermal expansion Coefficients Helmholtz free Energy Compressibility M. Rotter „Magnetostriction“ Course

Isotropic Thermal Expansion Thermal expansion Coefficients Helmholtz free Energy Compressibility M. Rotter „Magnetostriction“ Course Lorena 2007 2

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

Approximation: compressibility is T independent (dominated by electrostatic part of binding energy) Subsystem r. . . phonons, electrons, magnetic moments M. Rotter „Magnetostriction“ Course Lorena 2007 3

Phase Transitions M. Rotter „Magnetostriction“ Course Lorena 2007 4

Phase Transitions M. Rotter „Magnetostriction“ Course Lorena 2007 4

Mechanics of Solids Kinematics i=1, 2, 3 Inf. Translation Inf. Rotation (antisymmetric matrix) Inf.

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 5

Lagrange Strain Tensor The strain tensor, ε, is a symmetric tensor used to quantify

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 6

T stress tensor is defined by: where the d. Fi are the components of

T stress tensor is defined by: where the d. Fi are the components of the resultant force vector acting on a small area d. A which can be represented by a vector d. Aj 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 7

Anisotropic Thermal Expansion Elastic Energy density. . strain can be written as Thermal expansion

Anisotropic Thermal Expansion Elastic Energy density. . strain can be written as Thermal expansion Coefficients Elastic Constants Elastic Compliances M. Rotter „Magnetostriction“ Course Lorena 2007 8

. . this can (as in the isotropic case) be written as sum of

. . 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 9

Grueneisens Approximation • Specific heat of subsystem r • Grueneisen Parameter of subsystem r.

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 10

Normal thermal Expansion Anharmonicity of lattice dynamics anharmonic Potential Harmonic potential + Small contribution

Normal thermal Expansion Anharmonicity of lattice dynamics anharmonic Potential Harmonic potential + Small contribution of band electrons with Debye function M. Rotter „Magnetostriction“ Course Lorena 2007 11

Magnetostriction is a property of magnetic materials that causes them to change their shape

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 12

Thermal expansion Coefficients Magnetostriction Coefficients Material Crystal axis Saturation magnetostriction l|| (x 10 -5)

Thermal expansion Coefficients Magnetostriction Coefficients Material Crystal axis Saturation magnetostriction l|| (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 13

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

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 M. Rotter „Magnetostriction“ Course Lorena 2007 14

Domain Effects T>TC T<TC M||111 rotation of the domains. migration of domain walls within

Domain Effects T>TC T<TC M||111 rotation of the domains. migration of domain walls within the material in response to external magnetic fields. M. Rotter „Magnetostriction“ Course Lorena 2007 15

In general the saturation magnetostriction will depend on the direction of the field and

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 And apply + consider symmetry Hexagonal M. Rotter „Magnetostriction“ Course Lorena 2007 16

Cubic (8 domains) Assumption: in zero field all 8 domains are equally populated M.

Cubic (8 domains) Assumption: in zero field all 8 domains are equally populated M. Rotter „Magnetostriction“ Course Lorena 2007 17

d. L/L Measurement dir. magnetization field Zero field. . . 8 domains Field ||

d. L/L Measurement dir. magnetization field Zero field. . . 8 domains Field || 111 M. Rotter „Magnetostriction“ Course Lorena 2007 18

d. L/L Measurement dir. magnetization field is zero M. Rotter „Magnetostriction“ Course Lorena 2007

d. L/L Measurement dir. magnetization field is zero M. Rotter „Magnetostriction“ Course Lorena 2007 19

d. L/L Measurement dir. magnetization field Zero field. . . 8 domains – contributions

d. L/L Measurement dir. magnetization field Zero field. . . 8 domains – contributions cancel Field || 011 M. Rotter „Magnetostriction“ Course Lorena 2007 20

d. L/L Measurement dir. magnetization field Zero field. . . 8 domains – contributions

d. L/L Measurement dir. magnetization field Zero field. . . 8 domains – contributions cancel Field || 0 -11 M. Rotter „Magnetostriction“ Course Lorena 2007 21

Summary Cubic crystal, easy axis 111 Assumption: in zero field all 8 domains are

Summary 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 22

Atomic Theory of Magnetostriction • Band Models • Localized Magnetic Moments M. Rotter „Magnetostriction“

Atomic Theory of Magnetostriction • Band Models • Localized Magnetic Moments M. Rotter „Magnetostriction“ Course Lorena 2007 23

 Magnetism of Free Electrons Sommerfeld Model of Free Electrons Schrödinger equation Free electrons

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 M. Rotter „Magnetostriction“ Course Lorena 2007 24

Periodic Boundary Condition (1 d): Complex numbers Condition for phases Allowed k-vectors (3 dim)

Periodic Boundary Condition (1 d): Complex numbers Condition for phases Allowed k-vectors (3 dim) Possible wavefunctions (3 dim) M. Rotter „Magnetostriction“ Course Lorena 2007 25

2 -D projection of 3 -D k-space ky • Each state can hold 2

2 -D projection of 3 -D k-space ky • Each state can hold 2 electrons of opposite spin (Pauli’s principle) • To hold N electrons dk k 2 p/L kx k. F: Fermi wave vector he=N/V: electron number density Fermi Energy Fermi Velocity: Fermi Temp. M. Rotter „Magnetostriction“ Course Lorena 2007 26

Fermi Parameters for some Metals F: Work Function Vacuum Level Energy EF free electrons

Fermi Parameters for some Metals F: Work Function Vacuum Level Energy EF free electrons in periodic potential –energy gap at Brillouin zone boundary Band Edge M. Rotter „Magnetostriction“ Course Lorena 2007 27

Effect of Temperature Fermi-Dirac equilibrium distribution for the probability of electron occupation of energy

Effect of Temperature Fermi-Dirac equilibrium distribution for the probability of electron occupation of energy level E at temperature T Occupation Probability, f Enrico Fermi k. BT 1 T = 0 K Vacuum Energy Increasing T 0 Electron Energy, E M. Rotter „Magnetostriction“ Course Lorena 2007 μ Work Function, F 28

Number and Energy Densities Summation over k-states Integration over k-states Transformation from k to

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 Density of States Number of k-states available between energy E and E+d. E A tedious calculation gives: M. Rotter „Magnetostriction“ Course Lorena 2007 29

Free Electrons in a Magnetic Field Pauli Paramagnetism Spin - Magnetization for small fields

Free Electrons in a Magnetic Field Pauli Paramagnetism Spin - Magnetization for small fields B (T=0) Magnetic Spin - Susceptibility (Pauli Paramagnetism) 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 M. Rotter „Magnetostriction“ Course Lorena 2007 Nobel Price 1945 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). 30

Direct Exchange between delocalized Electrons Spontaneously Split bands: e. g. Fe M=2. 2μB/f. u.

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 31

total energy change: there is an energy gain by spontaneous magnetization, if Stoner Criterion

total energy change: there is an energy gain by spontaneous magnetization, if Stoner Criterion . . . 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 Edmund C. Stoner (1899 -1968) 32

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

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 33

Band Magnetostriction moving De(EF)δE/2 electrons from spin down to spin up band exchange energy

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 34

Gd metal Tc= 295 K , TSR= 232 K M||[001]=7. 55 m. B LARGE

Gd metal Tc= 295 K , TSR= 232 K M||[001]=7. 55 m. B LARGE VOLUME MAGNETOSTRICTION ! . . . anisotropic MS c/a(T) not explained M. Rotter „Magnetostriction“ Course Lorena 2007 35

Mechanisms of magnetostriction in the Standard model of Rare Earth Magnetism Ø microscopic origin

Mechanisms of magnetostriction in the Standard model of Rare Earth Magnetism Ø microscopic origin of magnetostriction = strain dependence of magnetic interactions 1) Single ion effects Crystal Field Striction …spontaneous magnetostriction …forced magnetostriction M. Rotter „Magnetostriction“ Course Lorena 2007 T >TN k. T >> cf k. T < cf T <TN H 36

T >TN M. Rotter „Magnetostriction“ Course Lorena 2007 k. T >> cf k. T

T >TN M. Rotter „Magnetostriction“ Course Lorena 2007 k. T >> cf k. T < cf 37

T <TN Nd. Cu 2 TN TN M. Rotter „Magnetostriction“ Course Lorena 2007 38

T <TN Nd. Cu 2 TN TN M. Rotter „Magnetostriction“ Course Lorena 2007 38

T <TN Nd. Cu 2 M. Rotter „Magnetostriction“ Course Lorena 2007 T <TN H

T <TN Nd. Cu 2 M. Rotter „Magnetostriction“ Course Lorena 2007 T <TN H 39

2) Two ion effects Exchange Striction T >TN …spontaneous magnetostriction T <TN …forced magnetostriction

2) Two ion effects Exchange Striction T >TN …spontaneous magnetostriction T <TN …forced magnetostriction T <TN M. Rotter „Magnetostriction“ Course Lorena 2007 H 40

Gd. Cu 2 (Gd 3+ shows no CEF effect. . . only exchange striction)

Gd. Cu 2 (Gd 3+ shows no CEF effect. . . only exchange striction) Spontaneous Magnetostriction TN Forced Magnetostriction T=4. 2 K M. Rotter, J. Magn. Mag. Mat. 236 (2001) 267 -271 M. Rotter „Magnetostriction“ Course Lorena 2007 41

Calculation of Magnetostriction Crystal field Exchange with + M. Rotter „Magnetostriction“ Course Lorena 2007

Calculation of Magnetostriction Crystal field Exchange with + M. Rotter „Magnetostriction“ Course Lorena 2007 42

Nd. Cu 2 Magnetostriction Crystal Field Exchange - Striction Calculation done by Mcphase www.

Nd. Cu 2 Magnetostriction Crystal Field Exchange - Striction Calculation done by Mcphase www. mcphase. de M. Rotter „Magnetostriction“ Course Lorena 2007 43

How to start – the story of Nd. Cu 2 • Suszeptibility: 1/χ(T) at

How to start – the story of Nd. Cu 2 • Suszeptibility: 1/χ(T) at high T . . . Crystal Field Parameters B 20, B 22 • 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 (d. Blm/dε) • Neutron diffraction: magnetic structure in fields || easy axis . . . phase diagram H||b - model . . . Jbb • Neutron spectroscopy on single crystals in H||b=3 T . . . Anisotropy of Jij - determination of Jaa=Jcc • Magnetostriction 44 . . . Confirmation of phase diagram models H||a, b, c, d. J(ij)/dε M. Rotter „Magnetostriction“ Course Lorena 2007

The story of Nd. Cu 2 • Inverse suszeptibility at high T . .

The story of Nd. Cu 2 • Inverse suszeptibility at high T . . . B 20=0. 8 K, B 22=1. 1 K Hashimoto, Journal of Science of the Hiroshima University A 43, 157 (1979) Θabc M. Rotter „Magnetostriction“ Course Lorena 2007 45

The story of Nd. Cu 2 Specific haet Cp and entropy – first info

The story of Nd. Cu 2 Specific haet Cp and entropy – first info about levels Gratz et. al. , J. Phys. : Cond. Mat. 3 (1991) 9297 Rln 2 M. Rotter „Magnetostriction“ Course Lorena 2007 46

How to start analysis – the story of Nd. Cu 2 • Magnetization: Kramers

How to start analysis – the story of Nd. Cu 2 • Magnetization: Kramers ground state doublet |+-> matrix elements P. Svoboda et al. JMMM 104 (1992) 1329 M. Rotter „Magnetostriction“ Course Lorena 2007 47

How to start analysis – the story of Nd. Cu 2 • Neutron TOF

How to start analysis – the story of Nd. Cu 2 • Neutron TOF spectroscopy – CF levels Gratz et. al. , J. Phys. : Cond. Mat. 3 (1991) 9297 . . . Blm B 20=1. 35 K B 22=1. 56 K B 40=0. 0223 K B 42=0. 0101 K B 44=0. 0196 K B 60=4. 89 x 10 -4 K B 62=1. 35 x 10 -4 K B 64=4. 89 x 10 -4 K B 66=4. 25 x 10 -3 K M. Rotter „Magnetostriction“ Course Lorena 2007 48

The story of Nd. Cu 2 • Thermal expansion – cf influence . .

The story of Nd. Cu 2 • Thermal expansion – cf influence . . . Magnetoelastic parameters (A=d. B 20/dε, B=d. B 22/dε) E. Gratz et al. , J. Phys. : Condens. Matter 5, 567 (1993) M. Rotter „Magnetostriction“ Course Lorena 2007 49

The story of Nd. Cu 2 • Neutron diffraction+ magnetization: magstruc, phasediag H||b-> model

The story of Nd. Cu 2 • 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 M. Rotter „Magnetostriction“ Course Lorena 2007 50

Nd. Cu 2 Magnetic Phase Diagram F 1 F 3 c F 1 a

Nd. Cu 2 Magnetic Phase Diagram F 1 F 3 c F 1 a b AF 1 lines=experiment M. Rotter „Magnetostriction“ Course Lorena 2007 51

The story of Nd. Cu Jaa=Jcc(R) 2 • Neutron spectroscopy on single crystals in

The story of Nd. Cu Jaa=Jcc(R) 2 • Neutron spectroscopy on single crystals in H||b=3 T . . . Anisotropy of J(ij) - determination of Jaa=Jcc F 3 M. Rotter et al. , Eur. Phys. J. B 14, 29 (2000) M. Rotter „Magnetostriction“ Course Lorena 2007 52

F 3 Nd. Cu 2 F 1 AF 1 M. Rotter „Magnetostriction“ Course Lorena

F 3 Nd. Cu 2 F 1 AF 1 M. Rotter „Magnetostriction“ Course Lorena 2007 M. Rotter, et al. Applied Phys. A 74 (2002) s 751 53

How to start analysis – the story of Nd. Cu 2 • Magnetostriction. .

How to start analysis – the story of Nd. Cu 2 • Magnetostriction. . . Confirmation of phasediagram model for H||a, b, c, and determination of d. J(ij)/dε M. Rotter Course Lorena 20078885 M. Rotter, et al. J. of„Magnetostriction“ Appl. Physics 91 10(2002) 54

M. Rotter „Magnetostriction“ Course Lorena 2007 55

M. Rotter „Magnetostriction“ Course Lorena 2007 55

Mc. Phase - the World of Rare Earth Magnetism Mc. Phase is a program

Mc. Phase - the World of Rare Earth Magnetism Mc. Phase 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 56

Crystal Field/Magnetic/Orbital Excitations Magnetostriction and much more. . M. Rotter „Magnetostriction“ Course Lorena 2007

Crystal Field/Magnetic/Orbital Excitations Magnetostriction and much more. . M. Rotter „Magnetostriction“ Course Lorena 2007 57

Epilog Mc. Phase runs on Linux and Windows and is available as freeware. www.

Epilog Mc. Phase runs on Linux and Windows and is available as freeware. www. mcphase. de Mc. Phase 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 Mc. Phase: • 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 Nd. Cu 2 Appl. Phys. A 74 (2002) S 751 • M. Rotter, M. Doerr, M. Loewenhaupt, P. Svoboda, Modeling Magnetostriction in RCu 2 Compounds using Mc. Phase J. of Applied Physics 91 (2002) 8885 • M. Rotter Using Mc. Phase to calculate Magnetic Phase Diagrams of Rare Earth Compounds J. Magn. Mat. 272 -276 (2004) 481 M. Rotter „Magnetostriction“ Course Lorena 2007 58