Download Lecture 8

Introduction
Saturday, November 03, 2007
10:11 AM
Magnetism
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•
•
•
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How does magnetism come about?
What is magnetic susceptibility measuring?
How does the neutron interact with electrons?
Is the magnetic scattering length a constant?
Diffraction from magnetic crystals.
LectureNotes4_nov_2007 Page 1
Introduction
Saturday, November 03, 2007
10:36 PM
In a metal, degeneracy b/n electrons of opposite spin sharing the same
orbital state is resolved by applying a magnetic field.
This causes a redistribution of elections b/n the two spin orientations.
The difference in the population of up and down spins give rise to a
moment
This gives rise to Pauli paramagnetism. It does not depend on
temperature and measures the density of states at the Fermi level.
LectureNotes4_nov_2007 Page 2
Curie-Weiss law
Saturday, November 03, 2007
10:35 AM
Suppose each atom behaves like a small magnet such as in d- or felectrons
when a magnetic field is applied, magnetic moments aligned in the
direction other field will have energy
The magnetic susceptibility is given by
This is the Curie law for paramagnetic susceptibility. It works for
solids with magnetic atoms that are well separated from one another.
In the case of interactions b/n spins, then
which is known as the Curie-Weiss law.
Spontaneous magnetization in a ferromagnet.
LectureNotes4_nov_2007 Page 3
Saturday, November 03, 2007
10:37 AM
Magnetic scattering
Q: In the interaction of a neutron with an atom, how does magnetic
scattering come about?
Neutron has a magnetic dipole moment.
Electron has a magnetic dipole moment.
LectureNotes4_nov_2007 Page 4
Interaction potential
Saturday, November 03, 2007
10:39 AM
First consider the neutron interaction with a single bound atom. The
interaction potential is given by
B(r) is the magnetic-flux density (or magnetic induction) arising from
the atom.
The scattering length
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Evaluation of the potential
Saturday, November 03, 2007
12:09 PM
over all electrons in atom
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Saturday, November 03, 2007
6:22 PM
form factor
Q: How does the magnetic form factor vary with the sin θ?
The form factor and the electronic distribution are related as Fourier
transforms
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Saturday, November 03, 2007
10:41 AM
Supplement
B(q) consists of 2 terms.
What are they?
Orbital and spin angular momentum
The total magnetic field due to an electron of momentum p is
Let's compare with nuclear scattering
For nuclear scattering, the matrix element is
For magnetic scattering, the matrix element is
The magnetic interactions have long-range and the interactions are
not very simple.
For atoms that possess orbital and spin angular momentum
interactions b/n the two occur.
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Unpolarized neutrons
Saturday, November 03, 2007
10:41 AM
Consider the case of unpolarized neutrons
Q: Does magnetic scattering vanish?
Unpolarized beams consists of particles with spin in an incoherent
superposition The interference effects involve the coherent wave
function of a single neutron in whatever spin state it happens to be in.
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Diffraction patterns
Saturday, November 03, 2007
10:42 AM
It is also the case that the nuclear-magnetic interference terms average
to zero. The structure factor is given by the sum of 2 terms
representing the nuclear and magnetic intensities
From a diffraction pattern, need to identify Fmagn for a magnetic
material and then use it to determine the magnetic structure
Example: Ferromagnetic material
The magnetic peaks will occur at the same angular position as the nuclear
peaks
Example: Antiferromagnetic material
The magnetic unit cell is twice as long as the chemical unit cell in one
direction. Extra reflections are observed.
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Orbital and spin angular momentum
Saturday, November 03, 2007
5:16 PM
In general, magnetic ions in a crystal have both spin and orbital
angular momentum
Both contributions need to be taken into account.
The magnetic field due to the orbital motion of the electron
is given by
As we saw in the previous lecture
Spin part
Now, Orbital part
The interaction potential in this case is given by
Need to evaluate the matrix element
The combined effect for the magnetization
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Saturday, November 03, 2007
6:22 PM
We defined the spin magnetization operator as
The corresponding orbital term is given by (see Appendix H1 of Squires
for details)
Thus, the operator for the total magnetization is given by
Neutron magnetic scattering is due to the interaction of the magnetic
dipole moment of the neutron w/ the magnetic field produced by the
unpaired electrons
The field is determined by the magnetic moments due to
spin and orbital motion
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Rare earth
Sunday, November 04, 2007
6:54 PM
Rare earth ions have spin and orbital angular momentum
The calculation of the matrix elements of M is not so simple
LS coupling is considered and the angular momentum state is specified
by quantum numbers L, S and J
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Paramagnetic scattering
Sunday, November 04, 2007
7:11 PM
Let us first consider the case of paramagnetic systems
- the spins of the ions are randomly oriented; no internal field.
- no external field is applied
- spin flips don't change the energy of the system
Spin matrix element
For large spin values, paramagnetic scattering goes up.
There is no coherent scattering b/c paramagnetic ions are randomly oriented
in space.
The only dependence to Q comes from the form factor
Magnetic scattering is diffuse
Q: what happens in the presence of a magnetic field?
An the presence of an H field and at low enough temperatures, a
paramagnet will be polarized
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Part of the scattering that is proportional to the component of spin pointing
in the same direction as the field will contribute to Bragg peaks; the rest
will be part of the diffuse scattering
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FM scattering
Saturday, November 03, 2007
11:21 PM
Consider scattering from magnetic crystals where spins are oriented.
Time independent spin correlations
elastic scattering
G
G
dσ
2 g
= (γ r0 ) F (κ ) e − 2W (κ ) ∑ (δ αβ − κˆα κˆ β )∑ eiκ ⋅(rl −rl′ ) Sαl
dΩ
2
ll ′
αβ
2
S lβ′
1. Ferromagnetic crystals
Even with no field, spins are aligned in one direction within domains
a. Since dσ depends on
, it becomes very temperature dependent.
b. form factor falls off rapidly with |τ|
Q: what happens with external field?
If a magnetic field is applied in the direction τ, the spin directions within
domains will align so that η is along -τ. Then τ.η = -1.
In this case, magnetic scattering will disappear
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Antiferromagnet
Sunday, November 04, 2007
4:39 PM
Continuation
ms
μB
zˆ
b
b a
m
G G ms zˆ G −2W (κG )
F (κ ) =
F (κ )e
8sinπ h sinπ k sinπ l
am
2μB
2
3
⎛
(2π ) ‘δ (κG −τG )
ms ⎞ −2W (κG ) G 2
dσ
⎟⎟ e
= N⎜⎜γ r0
F(κ ) (1−κ z2 )
∑
m
G
τ
dΩ
2
μ
v
B⎠
⎝
G
No magnetic diffraction for κ
S⊥
Remember:
S
G
κ
m
S
b*m
b*
a*
a*m
Reciprocal Space
gS ≡
Rea l spa ce
Simple cubic antiferromagnet
G G
G
gd
G − 2Wd (κG )
iκ ⋅d
F (κ ) = ∑
Fd (κ )e
Sd e
d 2
What happens at TN, the Neel transition temperature?
LectureNotes4_nov_2007 Page 18
Sunday, November 04, 2007
11:20 AM
Helimagnets
Other types of magnetic ordering exist besides FM and AFM
Not so simple Heli-magnet :
MnO2
c*
c
a
b
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a*
Spin waves
Sunday, March 12, 2006
12:23 PM
Spin waves are quantized with energy relative to the ground state,
At finite temperatures, spins are not all aligned. The spin deviations are
represented by traveling sinusoidal waves
SPIN waves
G
G
G
S
S ⊥ (κ , ω ) = {δ (ε (κ ) − =ω )(n(=ω ) + 1) + δ (ε (κ ) + =ω )n(=ω )}
2
Magnon
Magnoncreation
creation
Magnon
Magnondestruction
destruction
Dispersion relation
Gadolinium
G
G
ε (κ ) = 2S ( J (0 ) − J (κ ))
Magnon occupation prob.
n( E ) =
Magnon frequencies
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1
⎞ −1
exp⎛⎜ E
⎟
k
T
⎝ B ⎠
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Spin waves in AFM
Sunday, March 12, 2006
1:22 PM
Spin waves in an antiferromagnet
(
G
iκ ⋅d
)
G
S J 1 − ∑d e
S ⊥ (κ ,ω ) =
G
2
ε (κ )
G
G
× {δ (ε (κ ) − =ω )(n(=ω ) + 1) + δ (ε (κ ) + =ω )n(=ω )}
1
z
Dispersion relation
G
ε (κ ) = 2S
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G2
2
J (0 ) − J (κ )
susceptibility
αβ
S
Sunday, November 04, 2007
9:51 PM
(κG, ω ) and the magnetic susceptibility
G
Sαβ (κ , ω ) = ∫ dt e − iωt
1
N
G
iκ (r -r )
α
β
∑ e l l′ Sl (0 )Sl ′ (t )
ll ′
Compare to the generalized susceptibility
2
G
(
gμ B )
iκ (r -r )
− iωt
χ q (ω ) =
dt
e
e
∑
∫
αβ
l
N
[S
l′
ll ′
α
l
(t ), Slβ′ (0)]
They are related by the fluctuation dissipation theorem
S
αβ
(q, ω ) =
{
}
Im χ qαβ (ω )
π (gμ B )2
1
1 − e − β =ω
αβ
χ
We convert inelastic scattering data to q (ω ) to
• Compare with bulk susceptibility data
• Isolate non-trivial temperature dependence
• Compare with theories
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Saturday, November 03, 2007
3:23 PM
Summary 1
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Saturday, November 03, 2007
3:24 PM
Summary 2
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Summary 3
Saturday, November 03, 2007
3:25 PM
LectureNotes4_nov_2007 Page 26