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# Physics with Neutrons II SS 2011 - E21 - TUM

Physics with Neutrons II SS 2011 - E21 - TUM

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Physics with Neutrons II

SS 2011

31/05/2011

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

VL19, Physik mit Neutronen II, 30 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. Basics

6.2. Master equation for spin-only systems

6.3. Master equation with L ≠ 0

6.4. Elastic magnetic scattering

6.4.1 Paramagnetic scattering

6.4.2 Magnetic form factor

6.4.3 Ferro- and Antiferromagnetism

Helimagnetism

6.3. Inelastic Magnetic Scattering

Magnons

6.4. Flux Line Lattices in Superconductors

2

1

Repetiton: Neutron Scattering - Susceptibility

cross section:

(T >> T C )

⎛ dσ

⎜ ⎟

⎝ dΩ

pm

=

2 N

0

)

Q

n

3

2

2 −2W

( )

( γ r ) S(

S + 1) F ( Q e

(ideal paramagnet)

susceptibility:

χ

g µ µ S(

S + 1)

3k

( T −T

)

2 2

0 B

= = cS(

S + 1)

B

C

TC

T −T

C

fluctuation-dissipation theorem:

⎛ dσ

⎜ ⎟

⎝ dΩ

pm

n + 1 Im( χ)

correlation function

physical quantity

3

Magnetic Form Factor of Fe, Co, Ni

Fe: Z = 26

Co: Z = 27

Ni: Z = 28

distribution function for

the unpaired 3d electrons

in metallic iron, cobalt,

and nickel, produced by

Fourier transform of the

neutron form-factor data.

4

2

Induced Magnetic Form Factor of Uranium

5f form factor of U 3+ ion

UPt 3

B = 5 T

T = 4.2 K

C. Stassis et al., Phys. Rev. B 34, 4382 (1986).

5

z-axis:

quantisation axis

for M

Ferromagnetic Order

B ext = 0

Heisenberg exchange interaction: V jj’ = -2J S j ⋅S j’

6

3

T-Dependence of (111) Bragg Peak in EuO/EuS

J. Als-Nielsen et al., Phys. Rev. B 14, 4908 (1976).

7

T-Dependence of Magnetisation in EuO/EuS

Bloch’s T 5/2 law

I ( T ) ⎛ M ( T ) ⎞

= ⎜ ⎟

I(0)

⎝ M (0) ⎠

2

critical exponent:

exp: 0.36 ± 0.01

theory: 0.3647

bulk magnetiz.:

0.368 ± 0.005

J. Als-Nielsen et al., Phys. Rev. B 14, 4908 (1976).

8

4

Quantum Phase Transitions in Ferromagnets

apply pressure: suppress T C

UGe 2 : different fm phases

C.Pfleiderer et al.

Questions:

• magnetic structure

• magnetic moments

• transition temperatures

9

T-Dependence of Susceptibility in EuO

Q

(111)

2

κ1

S(

Q)

∝ χ(

Q = 0)

2

Q + κ

2

1

HWHM

Wave number Q (Å -1 )

J. Als-Nielsen et al., Phys. Rev. B 14, 4908 (1976).

10

5