Physics with Neutrons II SS 2011 - E21 - TUM

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Physics with Neutrons II SS 2011 - E21 - TUM

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


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


)

Note:

Q = 4πQ cryst

9

5

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