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
ln V T , P
Thermal expansion Coefficients
T P
ln V T , P PT , V PT , V
T
P T T V T V
Helmholtz free Energy dF SdT PdV
1
1 V T , P 1 F T ,V
2
Compressibility T
V P T V V
2
T
M.Rotter „Magnetostriction“ Course Lorena 2007 2
2 F T ,V S T ,V
T T
TV V T
Approximation: compressibility is T independent
(dominated by electrostatic part of binding energy)
S r T , V
(T ) r T
r r V T
Subsystem r ..... phonons, electrons,
magnetic moments
M.Rotter „Magnetostriction“ Course Lorena 2007 3
Phase
Transitions
M.Rotter „Magnetostriction“ Course Lorena 2007 4
x' x u ( x)
Mechanics of Solids - i=1,2,3
Kinematics
Inf. Translation
Inf. Rotation Inf. Strain
(antisymmetric matrix) (symmetric matrix)
0 3 2
dx ' dx 3 0 1 dx
1 0
2
Volume Strain
M.Rotter „Magnetostriction“ Course Lorena 2007 5
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 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)
6
Hookes Law c
1
(Voigt) notation 1 = 11, 2 = 22 3 = 33 4 = 23 5 = 31 6 = 12
M.Rotter „Magnetostriction“ Course Lorena 2007 7
Anisotropic Thermal Expansion
dF SdT V d
1 6
Elastic Energy density d c
2 1
1 F (T , )
.... strain can be written as
V
T ,
Thermal expansion Coefficients
T
Elastic Constants c
6
c s
1
Elastic Compliances s
M.Rotter „Magnetostriction“ Course Lorena 2007 8
T , 6
T , 6
s 2 F 6
s S
T 1 T 1 V T 1 V
.... this can (as in the isotropic case) be written as
sum of contributions of subsystems r = phonons,
electrons, magnetic moments
r T , 6
s S r
r
r r T r 1 V
s Fr
6
r
r r 1 V T , 0
M.Rotter „Magnetostriction“ Course Lorena 2007 9
Grueneisens Approximation
S r
cV
r r
• Specific heat of subsystem r
• Grueneisen Parameter of
subsystem r ... Is in many
simple model cases
temperature independent
s r r
6
(T ) r c V (T )
r r 1 V
M.Rotter „Magnetostriction“ Course Lorena 2007 10
Normal thermal Expansion
Anharmonicity of C
u
Y
.0
0
10 2
lattice dynamics
.9
9
09
a
anharmonic .8
9
09
Potential
.7
9
09
c
Giterpametr,normiertauf30K
.6
9
09
b
.5
9
09
Harmonic
potential 0 0 5 0 5 0
0 0 0 0 0
0 5 1 1 2 2 3
[
K
T]
l
el phon K1T 2 K 2TD( D / T )
+ Small l
contribution of z 3
3 x dx
z3 e x 1
band electrons with Debye function D( z )
0
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 12
T , H
Thermal expansion Coefficients T, H
T
T , H
Magnetostriction Coefficients T, H
H
H
l|| T , H || e
l T , H || e
l T , H e
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 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
rotation of the domains.
T>TC TTN
kT >>cf kT TN
kT >>cf kT TN
…spontaneous
magnetostriction T matrix elements
M g B tanh( g B ( H lM ) /( 2kT ))
P. Svoboda et al. JMMM 104 (1992) 1329
a, b, c
g a / g J J y 2.1
g b / g J J z 2.8
g c / g J J x 1.5
M.Rotter „Magnetostriction“ Course Lorena 2007 47
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 48
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 49
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
f(B) [arb.units] T=0K
BcAF1F3 B
Bc1 AF1
Bc2
Bc3
F1
F2
F3
NdCu2 Magnetic Phase Diagram
F1
F3
c
F1
b
a
AF1
lines=experiment
M.Rotter „Magnetostriction“ Course Lorena 2007 51
The story of NdCu2
Jaa=Jcc(R)
• 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 52
F3 NdCu2
F1
AF1
M. Rotter, et al. Applied Phys. A 74 (2002) s751
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
M.Rotter „Magnetostriction“ Course Lorena 2007 55
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
M.Rotter „Magnetostriction“ Course Lorena 2007 56
Crystal Field/Magnetic/Orbital Excitations
Magnetostriction
and much more....
M.Rotter „Magnetostriction“ Course Lorena 2007 57
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
M.Rotter „Magnetostriction“ Course Lorena 2007 58