Physics with Neutrons II
SS 2011
16/05/2011
Peter Böni
Physik-Department E21
Technische Universität München
D-85747 Garching
E-mail:Peter.boeni@frm2.tum.de
Web: www.ph.tum.de
https://campus.tum.de/tumonline/lv.detailclvnr=950011566
VL17, Physik mit Neutronen II, 16 May 2011
Physics with Neutrons II
5. Neutron Scattering from Disordered Systems
5.1. Introduction
5.2. Pair correlation functions and structure factor
Determination of interactions potentials
5.3. Incoherent single particle dynamics and diffusion
5.4. Coherent dynamics and collective modes
The scattering law in the hydrodynamic limit
6. Magnetic Neutron Scattering: Structures, Magnons
6.1. Magnetic Cross Section
Master Formula
Spin and Orbital Coupling
6.2. Elastic Magnetic Scattering
Paramagnetism and magn. form factors
Ferro- and Antiferromagnetism
Helimagnetism
6.3. Inelastic Magnetic Scattering
Magnons
6.4. Flux Line Lattices in Superconductors
2
1
Repetition
J and µ are usually not parallel to each other
magnetic moment:
µ
B B
J B
= − g µ S + µ L = −g
µ J
Landé-Splitting Factor g J :
g J
J ( J + 1) + S(
S + 1) − L(
L + 1)
= 1+
2J
( J + 1)
3
Magnetic Moments of Various Ions
saturation moment µ s
Gd 3+ (S = 7/2)
Fe 3+ (S = 5/2)
Cr 3+ (S = 3/2)
slope
B
M = C
T
M
B
C =
= χ =
2
p µ
0µ
3k
B
C
T
2
B
Brillouin function:
0 1 2 3 4
B/T (T/K)
M = ng
p: effective magnetic moment (µ eff )
or: paramagnetic moment
⎡2J
+ 1 ⎛ 2J
+ 1⎞
1 ⎛ a ⎞⎤
⎛ gµ
µ
B
JH
⎟ ⎞
J ⎢ coth⎜a
⎟ − coth⎜
⎟⎥
= ngµ
JB
⎜
B j
⎣ 2J
⎝ 2J
⎠ 2J
⎝ 2J
⎠⎦
⎝ k
BT
⎠
0
µ
B
4
2
Ions of Fe-Group
calculated calculated measured
5
Ions of the Rare-Earth Group
Eu: (J = 1) level only 500 K above J = 0 excited states important!
6
3
Magnetic Interaction Vector 1
Q: scattering vector
S: spin of unpaired electron
S Q
Q
θ
S
Ssinθ
θ
Scosθ
Q (S Q)
M = Q×
( M × Q)
⊥
only components perpendicular to Q
contribute to scattering cross section
7
Magnetic Interaction Vector 2
x
S X
S
S Y
S Z
Q
z
y
S z does not contribute to scattering
8
4
Magnetic Form Factor of Mn 2+ Ion
Variation of the magnetic amplitude form factor F for the Mn 2+ ion.
nucleus
F(Q cryst )
electrons
unpaired electrons
Q cryst
Θ
= sin λ 2
1 - 1
(Å
)
Note:
Q = 4πQ cryst
9
5
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