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Theory

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

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Theory

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  1. 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

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

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

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

  5. 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

  6. 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

  7. 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

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

  9. .... 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

  10. 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

  11. Normal thermal Expansion Anharmonicity of lattice dynamics anharmonicPotential Harmonic potential + Small contribution of band electrons with Debye function

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

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

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

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

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

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

  22. 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

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

  24. Magnetism of Free Electrons 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

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

  26. 2-D projection 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

  27. 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

  28. k T 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

  29. 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 E+dE Density of States A tedious calculation gives: M.Rotter „Magnetostriction“ Course Lorena 2007

  30. 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

  31. 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

  32. 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. M.Rotter „Magnetostriction“ Course Lorena 2007

  33. 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

  34. 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

  35. Gd metal 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

  36. 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

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

  38. T <TN NdCu2 TN TN M.Rotter „Magnetostriction“ Course Lorena 2007

  39. T <TN NdCu2 T <TN H M.Rotter „Magnetostriction“ Course Lorena 2007

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

  41. 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

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

  43. NdCu2 Magnetostriction Exchange - Striction Crystal Field Calculation done by Mcphase www.mcphase.de M.Rotter „Magnetostriction“ Course Lorena 2007

  44. 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

  45. 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

  46. 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

  47. 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

  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 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

  49. 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

  50. 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      

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