Properties of the Neutron

A Neutron Primer:

Elementary Scattering Processes and the

Properties of the Neutron

Prof. Thomas Brückel

IFF - Institute for Scattering Methods

& RWTH Aachen - Experimental Physics IV c

Neutron Laboratory Course 2009

"Why

Scattering?"

→ Internal structure? (building blocks: atoms, colloidal particles, ...) ?

→ Microscopic dynamics? (atom movements, ...)

⇒ Macroscopic properties (thermal conductivity, elasticity, viscosity, susceptibility, ...)

Scattering

Scattering:

interaction sample ↔ radiation weak

⇒ non-invasive, non destructive probe

for structure & dynamics

Example: High Energy Physics

Wavelength

YBa

Cu

O

Er

Mn

Structures on Atomic Length Scale

Thermal motion

Eigenmode

(optical phonon)

Lattice Dynamics

ferromagnet

antiferromagnet

( ) ²

1 2 ..

Q iQr i

i iec

SQ fQe e ^{− ⋅ σ ⋅}

∈

= ⋅ ⋅

Spin Dynamics

"Elementary Scattering

Theory"

HADAS

Reflectometer

Scattering Experiment

• Fraunhofer approximation: Ø sample <<

distance source distance detector

λπ

=

=

=

= k k' k' 2

• elastic scattering (diffraction): k

• scattering vector Q=k−k'

θ

− +

=

= Q k2 k'2 2kk'cos2 Q

λπ θ

= 4 sin Q

(I.1) (I.2)

(I.3)

Elastic Scattering

wavelength λ

propagation direction

Schematics: Scattering Process:

• point-like scatterer without internal structure:

→ isotropic emitted wave

„s-wave scattering“

• scatterer with internal structure:

→ anisotropic amplitude distribution in emitted wave

„p, d, … -wave scattering“ (development into spherical harmonics)

sample

detector

Schematics of a Scattering Experiment

n': particles scattered per second into dΩ: solid angle seen by detector

under angle 2θ and into

dE^': energy interval between E^' and E^' + dE^'

j: incident beam flux (particles / (area · time))

double differential cross section:

differential cross section: total scattering cross section:

→ scattered intensity

(I.4)

(I.5) (I.6)

Cross Section

2

' '

d d

d d dE dE

σ ^∞ σ

−∞

Ω =

∫

0

d d d

π σ

σ = Ω

∫

2 '

' '

d n

d dE jd dE σ ₌

Ω Ω

• scattering amplitude:

( ) ^{e d r}

V A

A ∝

⋅ ∫ r ⋅

Fourier Transform!

• scattered intensity:

A2

~ I

→ phase problem

• phase difference:

( )

r

Qk' r k r

CD 2 AB

⋅

= ⋅ − ⋅

= λ

⋅ − π

= ΔΦ

→ Born approximation or "kinematic” scattering theory!

no refraction

v

no attenuation

single scattering event

Elastic Scattering in Born Approximation

C

A

D r B

k

k

λ

When can a real beam still be considered as „plane wave“,

i.e. when are interference patterns not significantly destroyed?

→ phase relationship between different components of the beam!

1. Temporal or longitudinal coherence due to wavelength spread:

λ+Δλ

( λ λ )

λ ⎟ + Δ

⎠

⎜ ⎞

⎝ ⎛ −

=

= 2

||

n n 1 l

λ λ

= Δ 2

2

l

Longitudinal coherence length:

Coherence of the Beam

2. Transversal coherence due to source extension:

θ λ

= Δ

⊥

2

Transversal coherence length: l d θ

( ) θ θ λ _{= ⋅ ≈ ⋅}

d d sin

First minimum for: 2

→ Phase relation destroyed for large source size or large divergence Δθ

Coherence of the Beam

Neglects:

1. Attenuation of incident wave due to scattering 2. Multiple scattering events for

- attenuation of scattered wave

- local amplitude of total wave field

→ good approximation for

- weak scattering potential - small samples

First Born Approximation or Kinematical

Scattering Theory

- neutrons:

first Born approximation (+ multiple scattering corrections) except: strong Bragg peaks from perfect crystals

- x-rays:

cross section ~ 10 x larger than neutrons

→ kinematical theory breaks down rapidly

„Extinction correction“ for Bragg scattering - electrons:

strong Coulomb interaction

→ kinematical theory mostly inapplicable

In Praxi

"Scattering and Correlation

Functions"

( )r e iQ r V

r r d

Q ei r V r d Q

A

I − ⋅

⋅ ∫

∫ ^⎜_⎝^{⎛ ⎟}_⎠^⎞

⎟⎠

⎜ ⎞

⎝⎛ 2~ ' ' ' *

~ ^{3 3}

( ) ⋅^⎜⎝⎛ − ^⎟⎠⎞

=∫∫ ⎟

⎠⎞

⎜⎝

⎛ iQ r r

e r V r V r d r

d '

* ' ' ³

3

( )r eiQ R V

r R V r Rd

d ⋅

∫∫ +

= ^{3 3} ⎜⎝⎛ ⎟⎠⎞ *

R : r ' r − =

⇒ I Q d R P R e i Q ⋅ R

∫

⎟⎠

⎜ ⎞

⎝⎛

~

( )

⎟⎠

⎜ ⎞

⎝⎛

R = ∫ d r V r V r + R

P

*

"Pair Correlation Function"

Fourier-transform

Patterson-function

( )

S

V r e d r

A c = ⋅ ∫ ⋅

I ~ A

Born approximation:

Pair Correlation Function

( )

⎟⎠

⎜ ⎞

⎝⎛

R = ∫ d r V r V r + R

P

*

⇒ all difference vectors of original pattern!

(simplest example of correlation function!)

Patterson Function of a Triangle

Patterson function

original pattern

R

"Scattering

from a Crystal"

• 3d lattice of point-like scatterers:

( ) = ∑ ∑ ∑

⋅ − ⋅ + ⋅ + ⋅

=

−

=

−

=

⎟⎠

⎜ ⎞

⎝

⎛ ⎟

⎠⎞

⎜⎝

1 ⎛

0 1 0

1 0 N

n M m

P p

c p b m a

n r

r

V α δ

• scattering amplitude:

( )

(

(

) )

A ³

p , m , n

r Q i

0 = ∑ α

∫

∑

∑ α ∑

= ⁻

=

− ⋅

=

− ⋅

=

⋅ P 1

0 p

c Q 1 ip

M 0 m

b Q 1 im

N 0 n

a Q

in e e

e

geometrical series

a

b c

x y z

( )

a Q

in e

e ^⋅ = ^⋅

Scattering from a 3d-Lattice

• scattered intensity: Laue - function

"Laue function"

0 10 20

30 "Laue" function and

Intensity

Qa

N=5 N=10

0 π 2π

N²

2π/N

( ) ( )

c Q sin

c Q P sin b

Q sin

b Q M sin

a Q sin

a Q N α sin

Q A

~ Q I

21 221 2

12 212 2

21 221 2 2

2

⋅

⋅ ⋅

⋅

⋅ ⋅

⋅

⋅ ⋅

=

N-2 side maxima

Scattering from a 3d-Lattice

• Atomic model of a crystal:

V(r)= ∗ ⋅

basis lattice (δ-functions) cutoff for finite size

Scattering from a Real Crystal

• basis:

• scattering amplitude

:

( )

( ) [ ]

i

i

r r r

r u a v b w c S

V r

V ( ) ( ) ( ) ⋅

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ − + +

⎥ ∗

⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ∗ −

= ∑ ∑

∞

−

δ δ

basis lattice cut-off

• total potential:

( ) ( ( ) ) ( )

c e i

i

r r r

S

V Q

A

. .

...

) ( )

( ∗ ℑ

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝ ℑ ⎛

⎪⎭ ⋅

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ℑ ⋅ ℑ −

= ∑ ∑

∞

−

∈

δ δ

structure factor

(Fourier transform of scattering potential in unit cell)

Scattering from a Real Crystal

V

(r)= ∗δ(r-r

) + ∗δ(r-r

)

• Lattice (translational invariance) :

( )

( ) ( ) ( ) ( )

. . , ,

1 2 2 2

h k l e c

r ua vb wc a Q h b Q k c Q l

δ V δ π δ π δ π

∞

−∞

⎛ ⎞

ℑ⎜ − + + ⎟ = ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅

⎝

∑

∑

Structure factor (continuous function) is only observable for discrete positions in reciprocal Q space:

*

*

*

* h a k b l c G

Q = = + + vector of reciprocal lattice

Bragg Condition

G*

k

Ewald sphere

* 2 ; * 0; * 0; ...

a a⋅ = π a b⋅ = a c⋅ =

* 2 ( )

* 2 ( )

* 2 ( )

a b c

a b c b c a

a b c c a b

a b c π

π π

= ⋅ ×

⋅ ×

= ⋅ ×

⋅ ×

= ⋅ ×

⋅ ×

(elastic scattering!) _'

' 2 Q k k k k π λ

= −

= =

• Lattice (translational invariance) :

( )

( ) ( ) ( ) ( )

. . , ,

1 2 2 2

h k l e c

r ua vb wc a Q h b Q k c Q l

δ V δ π δ π δ π

∞

−∞

⎛ ⎞

ℑ⎜ − + + ⎟ = ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅

⎝

∑

∑

Structure factor (continuous function) is only observable for discrete positions in reciprocal Q space:

*

*

*

* h a k b l c G

Q = = + + vector of reciprocal lattice

λ π π λ

π λ

θ π

2 2 2

2 2 2

sin = Q 2 = G = d

Braggs law: 2d sin θ = λ

⇒ translational invariance ⇒

Bragg Condition

Q=G*

Θ 2θ k

k´

Ewald sphere

* 2 ; * 0; * 0; ...

a a⋅ = π a b⋅ = a c⋅ =

* 2 ( )

* 2 ( )

* 2 ( )

a b c

a b c b c a

a b c c a b

a b c π

π π

= ⋅ ×

⋅ ×

= ⋅ ×

⋅ ×

= ⋅ ×

⋅ ×

Braggs law: 2d sin θ = λ 2dsinθ θ

θ

lattice planes

Bragg Condition

• single atom: ^ℑ ( ^V

( ) ^r )

averaged scattering potential = atomic scattering potential ∗ thermal displacement

( ^V

( ) ^r )

ℑ

( ^V

^r ) ^{= ℑ} ( ^V

( ) ^r ) ^{⋅ ℑ} ( ^T

( ) ^r )

ℑ ( )

( )

( ) ⎟

⎠

⎜ ⎞

⎝ ⎛−

= ℑ

⎟ ⇒

⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

⋅

=

2 exp 1

2

2 exp ) 1

( x T r Q

r

T σ

σ πσ

• temperature factor:

1d:

3d: ( ( ) ) ( ) ^⎟

⎠

⎜ ⎞

⎝

⎛ − ⋅

=

ℑ

2

exp 1 Q σ

r T

only movement along Q visible

Q

σ

Single Atom Scattering

∫

∫ ⋅

≡

⎟⎠

⎜ ⎞

⎝⎛

⎟⎠

⎜ ⎞

⎝⎛

0 0

3 3

j j

V V

'r V r' d

'r Q e i 'r V r' Q d

f

E.g.: homogeneous sphere:

( )

⎩

⎪⎨

⎧

≤

= >

R r

R r r

V 1

0

( ) ⋅ = ∫ ⋅

∫ R

Qr Qr r

r dr Q e i r V r

d 0

2 sin 4 π

spherical co-ordinates

( ) ( )

QR cos QR

QR 3 sin

Q

f = ⋅ − ⋅

⇒

Form-Factor

·

2R

QR

form factor f(Q) homogeneous

spherical particle

QR

form factor f(Q) homogeneous

spherical particle

≡ Fourier-transform of scattering potential within the elementary cell:

( ) ( )

^{( )}

. .

Q iQ ri

i i e c

S Q f Q e

e

∈

= ∑ ⋅ ⋅

Structure Factor

( ) ( ( ) ) ( )

c e i

i

r r r

S

V Q

A

. .

...

) ( )

( ∗ ℑ

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝ ℑ ⎛

⎪⎭ ⋅

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ℑ ⋅ ℑ −

= ∑ ∑

∞

−

∈

δ δ

structure factor

(Fourier transform of scattering potential in unit cell)

( ) Cw Cw Bv

Bv Au

ABC Au S

π π π

π π

π ) sin( ) sin( )

3

= ⋅ sin( ⋅ ⋅

ℑ

A B

C

⇒ sharp Bragg condition smeared out by finite crystal size or finite size of coherent regions

Finite crystal size

( ) ( ( ) ) ( )

c e i

i

r r r

S

V Q

A

. .

...

) ( )

( ∗ ℑ

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝ ℑ ⎛

⎪⎭ ⋅

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ℑ ⋅ ℑ −

= ∑ ∑

∞

−

∈

δ

δ

• position of Bragg peaks in reciprocal lattice ⇒ unit cell metric

• extinction rules ⇒ symmetry elements with translational component (centering, screw axis,…) in the unit cell

• width of Bragg peaks determined by - instrumental resolution

- size of coherently scattering regions - mosaic structure of imperfect crystals

• intensity of Bragg peaks - atom species

- temperature movement - position within unit cell

( ) ( )

^{( )}

. .

Q iQ ri

i i e c

S Q f Q e

e

∈

= ∑ ⋅ ⋅

Bragg Scattering from Real Crystal

Questions?

"Properties

of the Neutron"

Hadrons Mesons (2 Quarks)

Baryons (3 Quarks) Proton

Neutron Nucleons Neutron:

decay of free neutron:

bound state of 3 Quarks:

ν + +

⎯

⎯ →

⎯ p e

n

on interracti weak

889

mass ≈ 1 u (1.675 · 10

g)

charge 0

spin 1/2

⇒ magnetic dipolar moment

N

n

µ

µ = − γ ⋅ ^; γ ^{= 1 . 913}

p

N

m

µ e

2

= h d

gluon

10

m (1 fm)

“nuclear magneton”

Neutron

five orders of magnitude smaller than an atom !

d

u

• matter wave (de Broglie):

momentum energy

; p / p m v= ⋅ =hk =h λ

2 2 2

2 2

1

2 2k 2h ^{B eq}

E mv k T

m mλ

= = h = ≡

in practical units:

[ ] [ ] [

]

[ ]

E

s / m v400 nm

λ2

=

= λ

Example: “thermal neutrons”

T_eq = 300 K E = 25 meV ≈ ΔE elementary excitations.

λ = 0.18 nm ≈ d _atoms v = 2 200 m/s

Charge: (-0.4±1.1)·10^-21 e ; Magnetic monopole moment: (0.85 ± 2.2)·10^-20 e/2α Electric dipole moment: (-0.1 ± 0.36)·10^-25 e·cm

Neutron

"Neutron Matter

Interaction: Nuclear

Interaction"

First Born Approximation: 2 2

|

|

|'

2 ⎟ | < >

⎠

⎜ ⎞

⎝

= ⎛

Ω m k V k

d d

π h σ

• strong interaction n ↔ nucleus

• magnetic dipole-interaction with B-field of unpaired e^- major

• magnetic dipole-interaction with - E-field of nuclei and e^-

- magnetic dipole moment of nucleus

observable (relativistic correction)

?

r Q i

r k i r

k i

3 3 '

) (

) (

⋅

−

−

∫

∫

⋅

=

Neutron-Matter-Interaction

… bound nucleus → no recoil !

● phenomenological approach:

- range of strong interaction: ≈ 10^-14 m (∅ nucleus) - wavelength of thermal neutrons: ≈ 10^-10 m (1 Å)

⇒ isotropic s-wave scattering (L = 0), elastic

⇒ Fermi-Pseudo-Potential:

) 2 (

) (

2

R r

m b r

V = π _h δ −

|

| b d

d =

Ω σ σ = 4 π | b |

Nuclear Interaction

● nucleon - nucleus - interaction : complex ! Hadron - physics ( COSY )

● First Born approximation (applicable?):

Scattering Cross Sections

σcoh [barn]

0.66

H

24

C

416

Mn

450

Fe

522

Ni

1408

Pd

2986

Ho

5631

U

x 10^-1 σcoh[barn]

1 2

58 60 62

x-ray neutrons

1 6 25 26 28

46 67

92

element

Z

1 barn = 10^-28m²

outside:

inside:

2 mE k 2

= h

( )

2 V0 E m q 2

h

= +

eikr fr r k ei ) r

( = +

Ψ scattering amplitude

0 ) R ( = Ψ

hard sphere:

⇒ b: = - f = R

Potential Scattering

Bound state (standing wave)

Compound nucleus ) 2

1 N 2 ( R

q⋅ ≈ + ⋅π

2 2i R 1 E E

const R

2 =4π + − + Γ σ

Breit-Wigner Formula

→ strong energy dependence

→ imaginary part (neutron capture)

→ negative scattering length

Compare anomalous x-ray scattering:

n+K→C*→K+n

simplified potential well picture (shell model etc.) :

Resonance Scattering

due to nuclear reactions. Off resonance:

a

1 v

~

~ λ σ

Examples:

σ_a (25 meV) [barn]

5333 n + ³He → ⁴He^* → p + ³T 940 n + ⁶Li → ⁷Li* → ³T + ⁴He

3837 n + ¹⁰B → ¹¹B* → ⁴He + ⁷Li + γ 681 n + ²³⁵U → fission

neutron detection

(n, γ)-resonances:

nucleide σ_γ[barn] E_resonance[meV]

113Cd 20600 178

151Eu 9200 321

155Gd 60900 26.8

157Gd 254000 31.4

λ = 1.6 Å (compare photoel.

abs. of x-rays!)

Fermi Golden rule:

f 2

O ρ i σ ∝

Absorption Processes

Neutron Cross Sections Vol. II D.I. Garber, R.R. Kinsey

Jan. 1976; BNL Report

48

Cd

Data Bases

λ = 0,2 Å σ ~ 8 barn λ = 0,64 Å

σ ~ 20 kbarn

Arrangement of atomswith scattering length b_ion fixed positons R_i

• scattering potential:

• scattering amplitude:

• scattering cross section:

↑ _{averageovertherandom}distribution↑

(

)

Q i

i j bibj e ⁻

= ∑ ∑ ^*

( )

⎪⎩

⎪⎨

⎧

=

− +

=

≠

= =

j i b

b b

b

j i b

b b b

b_{i j 2 2 2}

2

(

)

Coherent and Incoherent Scattering

( ) ²

(

)

n i

V r b r R

m

π δ

= ^h ∑ −

( )

i

A Q = ∑ b e

( ) ( )

i j

d Q A Q b e b e

d

σ

= = ⋅

Ω ∑ ∑

( )

( )

2 2

2

" "

" "

iQ Ri

i

d Q b e coherent

d

N b b incoherent

σ = ⋅

Ω

+ −

∑

scattering from the regular mean lattice

Interference

+

scattering from randomly distributed defects isotropic scattering +

k

k‘

N x

crystal with two isotopes: , N atoms.

Schematic Illustration

example:

Nickel: ₂₈Ni

isotope abundance nuclear spin scattering length [fm]

58Ni ≈ 68 % 0 14.4 ←

60Ni ≈ 26 % 0 2.8 ←

(⁶¹Ni ≈ 1 % ³/₂ 7.6)

62Ni ≈ 4 % 0 -8.7 ←

(⁶⁴Ni ≈ 1 % 0 -0.4)

( )

[ ]

b ≈ 0.68⋅14.4+0.26⋅2.8+0.04⋅ −8.7 ≈10.2

4 b

π

σ =

⇒

( ) ( )

[ ^{( )} ]

isotope

incoherent = 4π 0.68⋅ 14.4−10.2 ² +0.26⋅ 2.8−10.2 ²+0.04⋅ −8.7−10.2 ^{2 2} ≈5.1 σ

barn barn

incohisot coherent

1 . 5

; 1

. 13

. .

≈

=

σ σ

Isotope-Incoherence

example: Hydrogen: ₁H

⎩⎨

= ⎧

±

=

+ 0 singlet

triplet 1

2 I 1 J : n H spin Total

⇒ + =

+ =

− =

−

− =

⎪⎭

⎪⎬

⎫

4

; 3 85 . 10

4

; 1 5 . 47

c fm b

c fm b

( ) ( )

b 10.85 3.74

4 5 3 . 4 47

1 − + ⋅ = −

= ⎥⎦⎤

⎢⎣⎡

barn 76

. 2 1

b

coherent =4π =

σ

⇒

( ) (

)

4 2 3 74 . 3 5 . 4 47

4 1

incoherent^{nuclear spin} ⎥⎦⎤

⎢⎣⎡ − + + +

π

= σ

barn 2

.

=80

barn 2

. 80

; barn 76

. 1

spin nuclear

. incoh coherent

≈ σ

≈ σ

glue!

Diffusion of hydrogen

Nuclear Spin Incoherence

2D Deuterium 1

with nuclear spin 1:

barn 1

. 2 barn

6 .

5 _inc.

.

coh ≈ σ ≈

σ

contrast variation by isotope substitution:

fm b

fm

b_H = −3.74 ; _D = 6.67

• Example: vanadium ₂₃V

barn 08

. 5

; barn 02

.

0 ^nuclear_incoh. ^spin

.

coh ≈ σ ≈

σ

Isotope Natural abundance Nuclear spin Scattering length [fm]

( ⁵⁰V 0.25 % 6 7.6 )

51V 99.75 % 7/2 -0.40 ⇐

calibration standard;

sample container

Deuterium

Example:

Core-Shell micelle in solution

"Neutron Matter

Interaction: Magnetic

Interaction"

magnetic moment of the neutron:

⋅σ γμ

−

=

μ_{n N}

magnetic field of the electron:

L

S B

B

B = +

dipolar field of the spin moment: ; 2 S

R R

B x _{e B}

e 3

S ⎟⎟⎠ μ = − μ ⋅

⎜⎜ ⎞

⎝

×⎛ μ

∇

=

field due to the movement of the electron (Biot-Savart): _L ^e ₃ R

R v

c

B = −e × n

B

m

= − μ ⋅ V

Zeeman energy:

Magnetic Interaction Potential

e

v

μ

R

B μ

n

σz Vm k

σz‘ k‘

z 2 m

z 2

n2 k' ' k

2 m d

d ⎟⎟⎠ σ σ

⎜⎜ ⎞

⎝

⎛

= π Ω

σ V

h

( )

( )

B

0 2 ' M Q

2 r 1

d

d σ ⋅ σ

− μ γ

Ω = σ

σ ⊥

cm 10

539 . 0

r₀ = ⋅ ⁻¹² γ

→"equivalent scattering length" for 1 µ_B (S=

2

1): 2.696 fm ≈ b_co

( )

( )

M_⊥ = × ×

( )

( )

M ^{iQ r 3}

( )

( )

( )

M = _S + _L

( )

( )

(

)

i i i

B B

S r 2 S r 2 r r S

M

1. Born approximation

Magnetic Scattering Cross Section

Q k‘

M

k

M_⊥

( ) ^{Q Qˆ M Qˆ}

M

= × ×

Illustration: scattering from the dipolar field

Only the component of the magnetisation perpendicular to the scattering vector gives rise to magnetic scattering!

M || Q M

Q

Planes with equal phase factor

M⊥Q M

Q

Directional Dependence

Ri rik

tik Sik

Si

Atom i

Separation of intra-atomic quantities for localised moments:

( )

(

)

=

ik ik ik

S B i ik

ik R t ; M r 2 r r s

r

( )

( )

M _S ^{iQ r 3}

∑ ∑ ⋅

∑ =

= ^{⋅ ⋅ ⋅}

i iQR k iQ t ik ik ik

r Q

i s e e s

e ^{i i ik}

Expectation value of the operator for the thermodynamic state of the sample:

Pure Spin Scattering

pure organic FM p-O₂N·C₆F₄·CNSSN:

( )

( )

i

d r f Q S e

d

σ = γ _⊥

Ω

∑

( ) ^{( )}

m s

Atom

f Q =

∫

( )

( )

M Q = − μ ⋅ f Q ⋅

∑

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

normalized form factor

sin(Θ/λ)

nuclear scattering

orbital

x-ray

spin

Chromium

0.0 0.2 0.4 0.6 0.8 1.0

sin(Θ/λ)

Q || b Q || a

a b

form factor

in general anisotropic:

Form Factor: Spin, Orbit, Anisotropy

M(r)

λ

F

Orbital and Spin Formfactor

+

v e- B

B

re r

+

s e-

B

re r

Bohr orbit

magnetic field distribution due to

Orbital angular momentum Spin angular momentum

Summary

• Scattering amplitude:

(1^st Born approximation)

• Intensity:

• Pair correlation function:

(Patterson function)

( )

A

A∝ ₀⋅

∫

( ) phase problem2

d A Q

I d σ Ω

⇒ I Q d R P R eiQ ⋅R

∫ ^⎜_⎝^{⎛ ⎟}_⎠^⎞

⎟⎠

⎜ ⎞

⎝⎛ ~ 3

( ) ⎜⎝⎛ ⎟⎠⎞

⎟⎠

⎜ ⎞

⎝⎛R = ∫ d r V r V r + R

P ³ *

Fourier transform!

( ) ( )

c Q sin

c Q P sin b

Q sin

b Q M sin a

Q sin

a Q N α sin

Q A

~ Q I

12 221 2 12

221 2 21

221 2 2

2

⋅

⋅ ⋅

⋅

⋅ ⋅

⋅

⋅ ⋅

• Laue function:

(3d lattice)

• Bragg condition:

• Structure factor:

• Fermi pseudo potential:

• Absorption:

• Coherent & incoh.:

• magnetic cross section:

*

*

*

* ha kb lc G

Q = = + + 2dsinθ = λ ( ) ( ) ^{12 ( )2}

. .

Q iQ ri

i i e c

S Q f Q e^{− ⋅σ} e ^⋅

∈

= ∑ ⋅ ⋅

) 2 (

) (

2

R r m b

r

V = π_h δ −

a 1v

~

~λ σ

( ) ^{2 iQ Ri 2 ( )2}

i

d Q b e N b b

d

σ = ⋅ + −

Ω ∑

( ) _z ( ) _z ²

B

0 2 ' M Q

2 r 1

d

d σ ⋅ σ

− μ γ

Ω = σ

σ ⊥

scattering length b