1. 1 M.Rotter „Magnetostriction“ Course Lorena 2007 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. 2 M.Rotter „Magnetostriction“ Course Lorena 2007 Isotropic Thermal Expansion Thermal expansion Coefficients Helmholtz free Energy Compressibility

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

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

5. 5 M.Rotter „Magnetostriction“ Course Lorena 2007 Inf. Rotation (antisymmetric matrix) Inf. Strain (symmetric matrix) Inf. Translation i=1,2,3 Mechanics of Solids - Kinematics Volume Strain

6. 6 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 ε ii are the relative change in length in the direction of the i direction (along the x i -axis) ; the other terms ε ij = 1/2 γ 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.

7. 7 M.Rotter „Magnetostriction“ Course Lorena 2007 T stress tensor is defined by: where the dF i are the components of the resultant force vector acting on a small area dA which can be represented by a vector dA j 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. Hookes Law Stress tensor is symmetric, otherwise the volume element would rotate (to seet this look at zy and yz component in figure) (Voigt) notation 1 = 11, 2 = 22 3 = 33 4 = 23 5 = 31 6 = 12

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

9. 9 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

10. 10 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

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

12. 12 M.Rotter „Magnetostriction“ Course Lorena 2007 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)

13. 13 M.Rotter „Magnetostriction“ Course Lorena 2007 Thermal expansion Coefficients Magnetostriction Coefficients Material Crystal axis Saturation magnetostriction  (x 10 -5 ) Fe 100 +(1.1-2.0) Fe 111 -(1.3-2.0) Fe polycristal -0.8 Terfenol-D 111 200

14. 14 M.Rotter „Magnetostriction“ Course Lorena 2007 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

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

16. 16 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 (α x α y α z ) and measurement direction (β x β y β z ) (Cark Handbook of ferromagnetic materials, Elsivier, 1980) Write Energy in terms of strain and Magnetization And apply + consider symmetry Hexagonal Zero in case of inversion symmetry

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

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

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

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

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

22. 22 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 Summary

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

24. 24 M.Rotter „Magnetostriction“ Course Lorena 2007 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

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

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

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

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

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

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

31. 31 M.Rotter „Magnetostriction“ Course Lorena 2007 Direct Exchange between delocalized Electrons Spontaneously Split bands: e.g. Fe M=2.2μ B /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 χ P are large anough. Quantitative estimation: what is the condition that the system as a whole can save energy by becoming ferromagnetic ? kinetic energy change: moving D e (E F )δE/2 electrons from spin down to spin up band exchange energy change:

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

33. 33 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

34. 34 M.Rotter „Magnetostriction“ Course Lorena 2007 Band Magnetostriction moving D e (E F )δE/2 electrons from spin down to spin up band kinetic energy change: exchange energy change:

35. 35 M.Rotter „Magnetostriction“ Course Lorena 2007 Gd metal T c = 295 K, T SR = 232 K M ||[001] =7.55   LARGE VOLUME MAGNETOSTRICTION !...anisotropic MS c/a(T) not explained

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

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

38. 38 M.Rotter „Magnetostriction“ Course Lorena 2007 T <T N TNTN NdCu 2 TNTN

39. 39 M.Rotter „Magnetostriction“ Course Lorena 2007 T <T N H NdCu 2

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

41. 41 M.Rotter „Magnetostriction“ Course Lorena 2007 TNTN T=4.2K M. Rotter, J. Magn. Mag. Mat. 236 (2001) 267-271 Spontaneous Magnetostriction Forced Magnetostriction GdCu 2 (Gd 3+ shows no CEF effect... only exchange striction)

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

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

44. 44 M.Rotter „Magnetostriction“ Course Lorena 2007 How to start – the story of NdCu 2 Suszeptibility: 1/χ(T) at high T...  Crystal Field Parameters B 2 0, B 2 2 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 B l m Thermal expansion in paramagnetic state – CF influence...  Magnetoelastic parameters (dB l m /dε) Neutron diffraction: magnetic structure in fields || easy axis...  phase diagram H||b - model...  J bb Neutron spectroscopy on single crystals in H||b=3T...  Anisotropy of Jij - determination of J aa =J cc Magnetostriction...  Confirmation of phase diagram models H||a,b,c, dJ(ij)/dε

45. 45 M.Rotter „Magnetostriction“ Course Lorena 2007 The story of NdCu 2 Inverse suszeptibility at high T... B 2 0 =0.8 K, B 2 2 =1.1 K Hashimoto, Journal of Science of the Hiroshima University A43, 157 (1979) Θ abc

46. 46 M.Rotter „Magnetostriction“ Course Lorena 2007 The story of NdCu 2 Specific haet Cp and entropy – first info about levels Rln2 Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297

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

48. 48 M.Rotter „Magnetostriction“ Course Lorena 2007 How to start analysis – the story of NdCu 2 Neutron TOF spectroscopy – CF levels... B l m Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297 B 2 0 =1.35 K B 2 2 =1.56 K B 4 0 =0.0223 K B 4 2 =0.0101 K B 4 4 =0.0196 K B 6 0 =4.89x10 -4 K B 6 2 =1.35x10 -4 K B 6 4 =4.89x10 -4 K B 6 6 =4.25 x10 -3 K

49. 49 M.Rotter „Magnetostriction“ Course Lorena 2007 The story of NdCu 2 Thermal expansion – cf influence... Magnetoelastic parameters (A=dB 2 0 /dε, B=dB 2 2 /dε) E. Gratz et al., J. Phys.: Condens. Matter 5, 567 ( 1993 )

50. Neutron diffraction+ magnetization: magstruc, phasediag H||b-> model... J bb The story of NdCu 2 n(k)=sum of J bb (ij) with ij being of bc plane k M. Loewenhaupt et al., Z. Phys. B: Condens. Matter 101, 499 ( 1996 )      

51. 51 M.Rotter „Magnetostriction“ Course Lorena 2007 NdCu 2 Magnetic Phase Diagram F3  AF1  F1  a b c lines=experiment

52. 52 M.Rotter „Magnetostriction“ Course Lorena 2007 The story of NdCu 2 Neutron spectroscopy on single crystals in H||b=3T... Anisotropy of J(ij) - determination of J aa =J cc F3  M. Rotter et al., Eur. Phys. J. B 14, 29 ( 2000 ) Jaa=Jcc(R)

53. NdCu 2 F3  AF1  F1  M. Rotter, et al. Applied Phys. A 74 (2002) s751

54. How to start analysis – the story of NdCu 2 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

55. 55 M.Rotter „Magnetostriction“ Course Lorena 2007

56. 56 M.Rotter „Magnetostriction“ Course Lorena 2007 McPhase - the World of Rare Earth MagnetismMcPhase - 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. 57 M.Rotter „Magnetostriction“ Course Lorena 2007 and much more.... Magnetostriction Crystal Field/Magnetic/Orbital Excitations

58. 58 M.Rotter „Magnetostriction“ Course Lorena 2007 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,M. Rotter 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 NdCu 2 Appl. Phys. A74 (2002) S751 M. Rotter, M. Doerr, M. Loewenhaupt, P. Svoboda, Modeling Magnetostriction in RCu 2 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 Epilog