Matteucci effect

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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

M.Rotter „Magnetostriction“ Course Lorena 2007

Isotropic Thermal Expansion

Thermal expansion Coefficients

Helmholtz free Energy

Compressibility

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Phase
Transitions

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

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

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

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

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

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


Anharmonicity of
lattice dynamics

+

Small
contribution of
band electrons

anharmonic

Potential

Harmonic
potential


with Debye function


Normal thermal Expansion

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Thermal expansion Coefficients

Magnetostriction Coefficients

Material

Crystal axis

Saturation

m
agnetostriction

l
||

(x 10
-
5
)


Fe

100

+(1.1
-
2.0)

Fe 111
-
(1.3
-
2.0)

Fe polycristal
-
0.8

Terfenol
-
D 111 200



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

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

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Cubic

(8 domains)

Assumption: in zero field all 8 domains are equally populated

18

M.Rotter „Magnetostriction“ Course Lorena 2007

field

magnetization

dL/L Measurement dir.

Zero field


Field || 111

... 8 domains

19

M.Rotter „Magnetostriction“ Course Lorena 2007

is zero

field

magnetization

dL/L Measurement dir.

20

M.Rotter „Magnetostriction“ Course Lorena 2007

field

magnetization

dL/L Measurement dir.

Zero field


Field || 011

... 8 domains


contributions cancel

21

M.Rotter „Magnetostriction“ Course Lorena 2007

field

magnetization

dL/L Measurement dir.

Zero field


Field || 0
-
11

... 8 domains


contributions cancel

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Atomic Theory of Magnetostriction



Band Models



Localized Magnetic Moments


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

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

M.Rotter „Magnetostriction“ Course Lorena 2007

k
y

k
x

2
-
D projection

of 3
-
D k
-
space

2
p
/L

k

dk



Each state can hold 2 electrons


of opposite spin (Pauli’s principle)




To hold
N

electrons

k
F
: Fermi wave vector

h
e
=N/V
: electron number density

Fermi Energy

Fermi Velocity:

Fermi Temp.

27

M.Rotter „Magnetostriction“ Course Lorena 2007

Fermi Parameters for some Metals


E
F

F:
Work Function

Energy

Vacuum

Level

Band Edge

free

electrons

electrons in
periodic potential

energy gap at
Brillouin zone
boundary

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,



F

Increasing

T

T


= 0 K

k T

B

Vacuum

Energy

μ

Enrico

Fermi

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

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Direct Exchange between delocalized Electrons

Spontaneously Split bands: e.g. Fe M=2.2
μ
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

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

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

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Gd metal

T
c
= 295 K ,

T
SR
= 232 K

M
||[001]
=7.55

B

LARGE VOLUME

MAGNETOSTRICTION !

...anisotropic MS

c/a(T)

not explained

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

T <T
N

kT >>

cf

kT <

cf

H

37

M.Rotter „Magnetostriction“ Course Lorena 2007

T >T
N

kT >>

cf

kT <

cf

38

M.Rotter „Magnetostriction“ Course Lorena 2007

T <T
N

T
N

NdCu
2

T
N

39

M.Rotter „Magnetostriction“ Course Lorena 2007

T <T
N

H

T <T
N

NdCu
2

40

M.Rotter „Magnetostriction“ Course Lorena 2007


2) Two ion effects




Exchange Striction








…spontaneous


magnetostriction




…forced


magnetostriction

T >T
N

T <T
N

T <T
N

H

41

M.Rotter „Magnetostriction“ Course Lorena 2007

T
N

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

M.Rotter „Magnetostriction“ Course Lorena 2007

Calculation of Magnetostriction

Crystal field

Exchange

+

with

43

M.Rotter „Magnetostriction“ Course Lorena 2007

NdCu
2
Magnetostriction

Crystal Field

Exchange
-

Striction

Calculation done by Mcphase

www.mcphase.de

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
/d
ε
)


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)/d
ε






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

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

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

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

M.Rotter „Magnetostriction“ Course Lorena 2007

The story of NdCu
2


Thermal expansion


cf influence


... Magnetoelastic parameters (A=dB
2
0
/d
ε
, B=dB
2
2
/d
ε
)





E. Gratz
et al.
, J. Phys.: Condens. Matter 5, 567
(
1993
)


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

M.Rotter „Magnetostriction“ Course Lorena 2007

NdCu
2

Magnetic Phase Diagram

F3


䅆ㄠ


ㄠ


a

b

c

F1






lines=experiment

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)

NdCu
2

F3



䅆ㄠ


F1


M. Rotter,
et al.
Applied Phys.
A 74 (
2002
) s751


How to start analysis


the story
of NdCu
2


Magnetostriction ... Confirmation of phasediagram model for H||a,b,c, and
determination of dJ(ij)/d
ε






M. Rotter,
et al.

J. of Appl. Physics
91 10(
2002
) 8885


55

M.Rotter „Magnetostriction“ Course Lorena 2007

56

M.Rotter „Magnetostriction“ Course Lorena 2007



M
c
P
h
a
s
e

-

t
h
e

W
o
r
l
d

o
f

R
a
r
e

E
a
r
t
h

M
a
g
n
e
t
i
s
m


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

M.Rotter „Magnetostriction“ Course Lorena 2007



and much more....

Magnetostriction


Crystal Field/Magnetic/Orbital Excitations


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,

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