1

Neutron scattering I

Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton

(Date: February 27, 2018)

The neutron interacts with the atomic nucleus, and not with the electrons as photon does.

This has important consequences. The response of neutrons from light atoms such as

hydrogen is much higher than for x-rays. Neutrons can easily distinguish atoms of

comparable atomic number. For the same wavelength as x-ray, the neutron energy is much

lower and comparable to the energy of elementary excitations in matter. Thus neutrons do

not only allow the determination of the static average structure of matters, but also the

investigation of the dynamic properties of atomic arrangements which are directly related to

the physical properties of materials. The neutron has a large penetration depth and the bulk

properties of matter can be studied. The neutron carries a magnetic moment which makes it

an excellent probe for the determination of the static and dynamical magnetic properties of

matters. The charge of neutron is zero. The neutron is a fermion with spin 1/2.

1. de Broglie relation: duality of wave and particle

The kinetic energy of slow neutrons with velocity v is given by

� = 12 ���

where m is the mass of neutron

� =1.674927471(21)×10−24 g

The de Broglie wavelength of the neutron is defined by

= � = ��, (de Broglie relation)

where h is the Planck constant. The wavevector k of the neutron has the magnitude

� = ��

.

The energy can be expressed by

2

� = ħ���� .

Here we note that the kinetic energy is related to the temperature T as

� = �� ���,

using the equipartition theorem since there are 3 degrees of freedom and �� ��� for each

freedom. The velocity is evaluated as

�� ����� = �� ���,

or

��� = ����� . (this velocity is called the root-mean square velocity)

However, we do not use this notation based on the equipartition relation. In the neutron

scattering, it is convenient to say that neutron with energy E corresponds to a temperature T;

� = �� ��� = ���.

The most probable velocity is evaluated as

� = ����� .

((Note-1)) Using the above relations with � = ���, we have the following wave-particle

relationships.

(a) Energy:

������ = 2.07212 [�"Å$�%]�

from the relation: � = ħ����

3

(b) Wavelength:

"Å% = ℎ√2�� = 9.04457,������

������ = [9.04457"Å% ]�

or

"Å% = 30.8107,��/�

(c) Wavevector:

�"Å$�% = ��

��.

(d) Frequency:

���01� = 0.241799 ������,

from the relation; � = 2�.

(e) Wavenumber:

��3�$�� = 33.3565���01�

from the relation �

= �5

(f) Velocity:

�6�� 78 9 = 0.629622 ��Å$��

= 3.95603��

4

from the relation � = ħ � = �.

(g) Temperature:

��/� = 11.6045 ������

from the relation � = ���

((Note-2))

Table from G. Shirane, S.M. Shapiro, and J.M. Tranquada, Neutron Scattering with a Triple

Axis Spectrometer (Cambridge, 2004).

((Note-3)) Example

For T = 300 K;

E = 25.852 meV. = 1.779 Å. v = 2.224 km/s

Some useful rules of thumb worth memorizing are

1Å = 81.80 meV, 1 meV = 8.07 cm-1 ≈ 0.2418 THz ≈ 11.61 K = 17.3 tesla.

2. Life time of neutron

A free neutron is unstable and undergoes radioactive decay. It is a beta-emitter, decaying

spontaneously into a proton, an electron, and an electron anti-neutrino. The life time of the

neutron is 886 1 s. The half-life ��/�, or the time taken for half of the neutrons to decay is

5

��/� = � ;<2 = 614 s.

At T = 300 K, the velocity of neutron is v = 2.224 km/s. It takes 45 ms to travel 100 m. So it

is safe to assume that a finite life time of the neutron is of no practical significance in a

scattering experiment.

3. Magnetic moment

The best available measurement for the value of the magnetic moment of the neutron is

�= = −1.91304272(45) �>,

where μN is the nuclear magneton,

�> = ?ħ�@5= 5.050783699(31)×10−24 emu (=erg/Oe)

4. Maxwell-Boltzmann distribution

We now consider the Boltzmann factor for the kinetic energy of free particles with a

mass m. We derive the form of Maxwell-Boltzmann distribution function. The probability

of finding the particles between v and dvv is

0

22

22

)2

exp(4

)2

exp(4)(

mvdvv

mvdvv

dvvf

�

�

.

Here we have

2/32/3

0

22 )

2()(

24)

2exp(4

�

�

�

�m

mmv

dvv

The Maxwell-Boltzmann distribution function is

6

)2

exp(4)2

(

)2

exp(4)2

()(

222/3

222/3

Tk

mvv

Tk

m

mvv

mvf

BB

��

�

�

where

1)(0

dvvf

The average velocity:

m

Tkmdvvvfv B

�

�8

)(2

2)( 2/1

0

The variance:

m

Tk

mdvvfvv B33

)(0

22

This can be rewritten as

Tkvm B2

3

2

1 2 (from the equi-partition theorem)

The root-mean square velocity is

m

Tkvv B

rns

32 .

What is the most probable velocity in which )(vf takes a maximum.

0dv

df

7

0)2

exp()()2

exp(22

22

Tk

mv

Tk

mvv

Tk

mvv

BBB

or

� = A2����

(most probable velocity)

or

12 ��� = ���

Which can be used conventionally for the neutron scattering.

Fig. Plot of the normalized max/)( fvf as a function of a normalized v/vmp.

Green: most probable speed (vmp)

Blue: averaged speed (vavg)

Brown: root-mean squared speed (vrms)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

v

vmp0.0

0.2

0.4

0.6

0.8

1.0

f

fmax

8

)225.12

6()128.1(1

mp

rms

mp

avg

v

v

v

v

where

� = ����� , �B�C = �D���� , ��� = ����� .

((Example)) Neutron (obeying the Maxwell Boltzmann distribution)

(a) T = 300 K (thermal neutron)

� = �� ���=25.852 meV

= ��E@ = 1.77885 Å, � =2.2239 km/s.

(b) T = 20 K (liquid hydrogen or deuterium)

� = �� ���=1.72347 meV

= ��E@ = 6.8895 Å, � =574.211 m/s.

(c) T = 2 K (cold neutron)

� = �� ���=172.347 �eV

= ��E@ = 15.405 Å. � =256.8 m/s.

(d) T = 2200 K (hot neutron)

� = �� ���=189.582 meV

= ��E@ = 0.6569 Å. � =6.022 km/s.

9

For most reactors designed to produce high thermal neutron flux, the temperature of the

moderator is about 300 to 350 K, and the resulting velocity spectrum of the neutrons is

suitable for many experiments. However, in some experiments the velocities required for the

incident neutrons lie on the low-energy tail of the thermal spectrum, while in other

experiments neutrons on the high-energy side are required. It is therefore desirable to be able

to change the temperature of the velocity distribution. This is done by placing in the reactor

a small amount of moderating material at a different temperature.

________________________________________________________________________

Source Energy E(meV) Temperature T (K) Wavelength �Å)

Cold 0.1 – 10 1 – 120 4 – 30

Thermal 5 – 100 60 – 1000 1 - 4

Hot 100 – 500 1000 – 6000 0.4 – 1

________________________________________________________________________

Table: Approximate values for the range of energy, temperature, and wavelength for the

three types of source in a reactor.

10

5. Fundamental

Thermal neutron flux

11

The number of thermal neutrons which pass through a unit area (cm2) per second.

((Cross section))

12

We assume that the target is sufficiently small that the probability of scattering of a neutron

incident on the target is not large.

If the target does scatter some of the incident neutrons, then at a large distance from the target

we can detect neutrons that are scattered into a small element of solid angle d with energy

'.

''

2

0

dd

dd

dI s

The number of neutrons per second scattered into solid angle d between ' and ' + d'.

((Definition))

'

2

dd

d s

Partial differential cross section.

Integrating this over d', we put the number of neutrons per second scattered into solid angle

d (regardless of ') to be

d

ddId

dd

ddId

dd

ddI sss

00

2

00

2

0 ''

''

d

d s: differential cross section

If we now integrate over solid angle, we find that the number of neutrons scattered per second

is

ss I

d

ddI

00

. [number/sec]

since the unit of I0 is number/(cm2 sec) and the unit of s is cm2.

13

Finally we can write down the number of neutrons per second that have momentum changed

in the target as

tI 0

t is the total cross section,

sat

((Note))

[] = barn.

1 barn = 10-28 m2 = 10-24 cm2.

1 fermi = 10-15 m = 10-13 cm.

This is about the projected area of an atomic nucleus. If the target consists of many atoms,

we can either deal with the cross section for the whole target, or take it per atom or per

chemical formula unit.

((Note))

s : transmission

d

d s; neutron diffraction, time of flight method

d

d s : energy spectrum measurement

dd

d s

2

: inelastic scattering experiment

___________________________________________________________________

6. Differential cross section

14

Fig. Scattering of neutron by a target. ki = k'. kf = k'. The scattering vector Q = ki - kf.

,k : initial state of neutron

',' k : final state of neutron

where

k: neutron wave vector

: spin state

The target is initially in quantum state and after the scattering event is in quantum state

' .

m: mass of the neutron

V: interaction potential between the neutron and the target.

15

E: energy of the target in state .

The first order Born approximation is given by

)'(''')2

('

''

2

2

2

�

EEV

m

k

k

dd

d s kkℏ

from the quantum mechanics (see the Appendix)

7. Nuclear elastic scattering

For nuclear scattering, we describe the interaction potential by a quantity known as the

Fermi pseudopotential

Fig. The distance over which the nuclear force extends (10-12 cm). The wavelength

of the neutron is = 10-8 cm. The target is located at the origin.

pi

p f

pi

Dp

r

q

f

qê2

qê2b

b

z

h

x

y

O

16

)(2 2

Rr �

bm

Vℏ

,

where b is the scattering length. It is typically found to be of the order of 10 fm.

((Note))

The Fermi pseudopotential cannot be the correct interaction potential, since the

interaction is not actually infinite anywhere. However, the nuclear size and range of the

nuclear force are so small compared with the wavelength of a thermal neutron that the delta

function pseudopotential gives an excellent representation of the actual scattering.

Fig. Schematic diagram for the neutron scattering, fi kkQ . ki = k. kf = k'.

Q is the scattering wave vector.

'kkQ

The matrix element is given by

RQrkrkRrrkk

iiibe

meedb

mV

2'

2 2)(

2'

ℏℏ �

�

17

where ħF is the momentum transfer in the scattering event. We assume that the scattering is

from an assembly of fixed nuclei at the position Rj that have scattering length bj. We will

further assume that this scattering length is independent of the neutron state .

j

jj

j

i

j

i

j

j Fbebeb jj RQRQ''

The differential cross-section for the elastic scattering from an assembly of fixed nuclei in

the system GG

can be given by

HH=

lj

ljlj

i

j

j FFbbeb j

,

*

2

' RQ

where we assume b to be real and

ji

j eFRQ '

It will normally be the case that the isotopic distribution is random, so that, if the nuclear

spins have random orientation, then the value of the scattering length bj at the j-th site will

not be correlated with l.

lj

lj

j

jj

lj

ljlj

j

jjj

lj

ljlj

FFbFFb

FFbbFFbFFbb

*2*2

**2

,

*

(1)

where < > indicates the mean value. If j and j’ refer to different sites (I ≠ I′� and there is no

correlation between the values of LM and LMNwe can write

⟨LMLMP⟩ = ⟨LM⟩ ⟨LMN⟩ = ⟨L⟩�

Note that

18

⟨L⟩ = ∑ SMLMM , ⟨L�⟩ = ∑ SMLM�M

with

∑ SMM = 1,

where SM denotes the probability that a nucleus has the scattering length LM. Thus Eq.(1) can

be rewritten as

HH= T LMLUVMVU∗

M,U= ⟨L⟩� T VMVU∗

M,U+ �⟨L�⟩ − ⟨L⟩�� T VMVM∗

M

The first term is a coherent term, while the second term is an incoherent term. The scattering

can be divided into two parts; coherent scattering and incoherent scattering. If the cross

sections for these parts are c and i, then for a single fixed atom,

2

4 bc � ,

)(422

bbi �

and

ics

The coherent scattering is the scattering the same system (same nuclei with the same

positions and motions) would give if all the scattering lengths were equal to <b>. The

incoherence scattering is the term we must add to this to obtain the scattering due to the actual

system. Physically the incoherent scattering arises from the random distribution of the

deviations of the scattering lengths from their mean value.

((From G.L. Squires, Thermal Neutron Scattering, Cambridge University Press, 19768)

Table Values of [\] and ^_[

Element or nuclide Z 5`� a=5 0� 1 1.8 80.2

19

0� 1 5.6 2.0

C 6 5.6 0.0

O 8 4.2 0.0

Mg 12 3.6 0.1

Al 13 1.5 0.0

V 23 0.02 5.0

Fe 26 11.5 0.4

Co 27 1.0 5.2

Ni 28 13.4 5.0

Cu 29 7.5 0.5

Zn 30 4.1 0.1

To simplify matters, we drop the operator formalism, since the atomic positions Rj are fixed.

Note that

VM = �abcd , VM∗ = �$abce

Thus we have

HH= fHH

g5`� + fHHga=5

with

fHHg5`� = ⟨L⟩� T VMVU∗

M,U= ⟨L⟩� T �ab�cd$ch�

M,U

fHHga=5 = �⟨L�⟩ − ⟨L⟩�� T VMVM∗

M= i�⟨L�⟩ − ⟨L⟩��

The incoherent elastic scattering is isotropic and yields a constant background. On the other

hand, the coherent elastic scattering provides the information about the mutual arrangement

of the atoms due to the phase factor. The coherent elastic scattering is rewritten as

fHHg5`� = ij⟨L⟩� T �abcd

M= ij⟨L⟩� �2���

�j T �b − k�l

20

leading to the Bragg reflection. Note that N0 is the number of unit cells and �j is the volume

of unit cell.

9. Structure factor for simple liquid

We discuss the formulation of m�b� for liquid, using the number density �>�n�

m�b� = 1i ⟨T �ab�cd$ch�M,U

⟩?=�?oU? = 1i ⟨pq Hn�abn�>�n�p�⟩

We use the relation

T �abcdM

= q Hn�abn T �n − cr�M= q Hn�abn�>�n�

where �>�n� is the nuclear number density

�>�n� = T �n − cr�M

The structure factor for thye simple liquid is given by

m�b� = 1i q Hn q HnP�ab"n$nN%⟨��n���n′�⟩

This can be rewritten as

m�b� = 1i q Hn q Hc�abc⟨��n���n − c�⟩

10. Structure factor for solid

For systems with more than one atom per unit cell, we define the position vector of the

atoms as

cs = cst + n�

21

with csj and n� being the position vectors of the j-th unit cell and of the s-th atom in the unit

cell, respectively.

fHHg5`� = ij⟨L⟩� T �ab�cd$cd,�

M,MPT L�L�P�ab�nu$nu,��,�P

Using the relation

T L�L�P�ab�nu$nu,��,�P

= |m�b�|�

where m�b� is the structure factor,

m�b� = T L��abnu�

The expression for the coherent elastic neutron cross section is

fHHg5`� = ij �2���

�j |m�b�|�exp [−2z�b�] T �b − k�l

Where exp [−2z�b�] is the Debye-Waller factor that accounts for thermal motion of atoms,

2z�b� = 13 {�⟨|�⟩

and ⟨|�⟩ is the measured squared displacements of atoms.

11. Inelastic neutron scattering: van Hove expression for the nuclear scattering

In inelastic neutron scattering experiments the energy transfer ħ� may be positive and

negative, corresponding to neutron energy-loss and energy-gain processes, respectively. The

partial differential cross section is given by

),('

'2

2

�

QSNb

k

k

dd

d s

,

22

where

''

2

)('1

),(

�� ℏEEePN

Sj

i jRQQ

and

'� ℏ ,

Using the relation for the Dirac delta function,

tEEi

dteEE)(

'

'

2

1)(

�

��

ℏℏ

ℏℏ

we get

�

�

�

��

��

�

�

' ,

'

*)(

'

2)(

'

'

'

''2

1

''2

1

'2

1),(

lj

tEi

itEi

iti

j

i

l

itEE

i

j

itEEi

eeeeedPN

eePedN

ePdteN

S

jl

jl

j

ℏℏ

ℏℏ

ℏℏ

ℏ

ℏ

ℏ

RQRQ

RQRQ

RQQ

Here we note that

Ht

iiHt

itE

iitE

i

eeeeee jj ℏℏℏℏ RQRQ

'''

We use the Heisenberg representation

)(''

tiHti

iHti

jj eeeeRQRQ ℏℏ

Then we have

23

� �

��

' ,

)(''

2

1),(

lj

tiiti jl eeedPN

SRQRQ

Qℏ

Noting that

1'''

and

OPO

(O is any operator).

The final expression can be derived by

lj

tiiti jl eedteN

S,

)(ˆˆ

2

1),(

RQRQQ

�

��

ℏ (van Hove)

This expresses the cross section as the temporal Fourier transform of a correlation function

between the position vector of the l-th atoms at t = 0 and the j-th atom at time t.

j

j trt ))(ˆ(),(ˆ Rr � .

We introduce the Fourier tranform as

j

ti

j

j

i

i

je

ted

tedt

)(ˆ

))(ˆ(

),(ˆ)(ˆ

RQ

rQ

rQ

Q

Rrr

rr

��

Then we have

24

)(ˆ)0(ˆ2

1),( tdte

NS

ti

QQQ

���

� �

ℏ

which shows that S(Q, �) can be expressed in terms of a density-density correlation function.

)0(ˆ)0(ˆ1

)(ˆ)0(ˆ)(1

)(ˆ)0(ˆ2

1),(

QQ

QQ

QQQ

��

��

����

�� �

N

ttdtN

teddtN

dS ti

ℏ

ℏ

ℏ

________________________________________________________________________

x

y

z

q

dq

f df

r

drr cosq

r sinq

r sinq df

rdq

er

ef

eq

dS

25

REFERENCES

((Nuclear scattering))

G.E. Bacon, X-ray and Neutron Diffraction (Pergamon, 1966).

W. Marshall and S.W. Lovesey, Theory of Thermal Neutron Scattering: The Use of Neutrons

for the Investigation of Condensed Matter (Oxford, 1971).

G.E. Bacon, Neutron Diffraction (Oxford, 1962).

G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering (Cambridge, 1978).

B. Dorner, Coherent Inelastic Neutron Scattering in Lattice Dynamics (Springer, 1982).

S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter. Volume 1: Nuclear

Scattering. Volume 2: Polarization Effects and Magnetic Scattering (Oxford, 1984).

K. Sköld and D.L. Price, Neutron Scattering (Academic Press, 1986).

M. Bée, Quasi-elastic Neutron Scattering: Principles and Applications in Solid State

Chemistry, Biology and Materials Science (Adam Hilger, 1988).

A. Authier, S. Lagomarsino, and B.K. Tanner, X-ray and Neutron Dynamical Diffraction

Theory and Applications (Plenum, 1996).

R. Hempelmann, Quasi-elastic Neutron Scattering and Solid State Diffusion (Oxford, 2000).

V.M. Nield and D.A. Keen, Diffuse Neutron Scattering from Crystalline Materials (Oxford,

2001).

M.T. Dove, Structure and Dynamics: An Atomic View of Materials (Oxford, 2002).

M.T. Dove, Introduction to Lattice Dynamics (Cambridge, 2003).

G. Shirane, S.M. Shapiro, and J.M. Tranquada, Neutron Scattering with a Triple-Axis

Spectrometer: Basic Techniques (Cambridge, 2004).

E.H. Kisi and C.J. Howard, Applications of Neutron Powder Diffraction (Oxford, 2008).

A. Furrer, J. Mesot, T. Strässle, Neutron Scattering in Condensed Matter Physics (World

Scientific, 2009).

B.T.M. Willis and C.G. Carlile, Experimental Neutron Scattering (Oxford, 2009).

J.M. Carpenter and C.-K. Loong, Elements of Slow-Neutron Scattering: Basics, Techniques,

and Applications (Cambridge, 2015).

R. Pynn, Introduction to Neutron Scattering

https://neutrons.ornl.gov/sites/default/files/intro_to_neutron_scattering.pdf

_________________________________________________________________________

((Magnetic neutron scattering))

P.G. de Gennes, Theory of Neutron Scattering by Magnetic Crystals, p.115-147, in

Magnetism volume III, edited by G.T. Rado and H. Suhl (Academic Press, 1963).

W. Marshall and R.D. Lowde, Magnetic Correlations and Neutron Scattering, p.705, Reports

on Progress in Physics, Vol.31, No.2, p.705 (1968).

Y.A. Izyumov, Magnetic Neutron Scattering (Springer, 1970).

26

M.F. Collins, Magnetic Critical Scattering (Oxford, 1989).

T. Chatterji (editor), Neutron Scattering from Magnetic Materials (Elsevier, 2006).

APPENDIX: Born approximation

1. Green's function in scattering theory

We start with the original Schrödinger equation.

)()()()(2

22

rrrr � kEV

ℏ,

or

)()(2

)()2

(22

2 rrr �

�

VEkℏℏ

.

under the potential energy )(rV . We assume that

22

2kEE k �

ℏ ,

We put

)()(2

)(2

rrr �

Vfℏ

,

Using the operator

22kL r .

we have the differential equation

)()()()( 22rrrr fkL .

Suppose that there exists a Green's function )(rG such that

)'()',()( )(22rrrr

r Gk ,

with

27

'4

)'exp()',()(

rr

rrrr

�ik

G (Green function)

We will discuss about the derivation of this Green function later. Then )(r is

formally given by

)'()'(

'4

)'exp('

2)()'()',(')()( )(

2

)()( rrrr

rrrrrrrrrr

��

Vik

dfGdℏ

, where )(r is a solution of the homogeneous equation satisfying

0)()( 22 rk ,

or

)exp()2(

1)(

2/3rkkrr i

� , (plane wave, k is continuous)

with

kk

Note that

)(

)'()'('

)'()',()(')()()()( )(222222

r

rrrr

rrrrrr

f

fd

fGkdkk

2. Born approximation

We start with

)'()'(

'4

)'exp('

2)()( )(

2

)(rr

rr

rrrrr

��

Vik

dℏ

,

)exp()2(

1)(

2/3rkkr ir

� , (plane wave).

28

Fig. Vectors r and r’ in calculation of scattering amplitude in the first Born

approximation

29

Fig. rr eu

Here we consider the case of )()(r

rr errr ''

rkek '

'')'(' rkerrr iikrrikik

eeee r for large r.

r

1

'

1

rr

Then we have

O r'

r

r'ÿur

»r-r'»

30

)'()'('4

12)exp(

)2(

1)( )(''

22/3

)(rrrrk

rk �

��

Vedr

eir

iikr

ℏ

or

)],'([)2(

1)(

2/3

)(kk

rkf

r

eer

ikri

�

The first term denotes the original plane wave in the propagation direction k. The second term denotes the outgoing spherical wave with amplitude, ),'( kkf ,

)'()'('2

)2(4

1),'( )(''

2

2/3 rrrkk rk �

��

Vedf i

ℏ.

The first Born approximation:

')'(

2

'''

2

)'('2

)'('2

4

1),'(

rkk

rkrk

rr

rrkk

i

ii

eVd

eVedf

ℏ

ℏ

��

��

when )()(r

is approximated by

rk ier2/3

)(

)2(

1)(

� .

Note that ),'( kkf is the Fourier transform of the potential energy with the wave

vector Q; the scattering vector;

kkQ ' .

Formally ),'( kkf can be rewritten as

kk

kkkk

V

Vf

ˆ'4

ˆ')2(2

),'(

2

2

3

2

ℏ

ℏ

��

���

31

where

)()2(

1

)('ˆ'

)'(3

3

3

rr

krrrkrkk

rkkVed

VdV

i

�

with

)exp()2(

12/3

rkkr i�

((Forward scattering))

Suppose that k' = k (Q = 0). Then we have

)'('2

)'('2

4

1)0(

2

''

2

rr

rr rkrk

Vd

eVedf ii

ℏ

ℏ

��

��

Suppose that the attractive potential is a type of square-well

0

)( 0VrV

Rr

Rr

Then we have

3

203

20

3

2)0( R

VR

Vf

ℏℏ

��

The total cross section (which is isotropic) is obtained as

23

202

4)0(4

RV

fℏ

��� .

The units of )0(f is cm, and the units of is cm2

.

3. Differential cross section

32

We define the differential cross section d

d as the number of particles per unit time

scattered into an element of solid angle d divided by the incident flux of particles. The probability flux associated with a wave function

ikzi

k ee2/32/3 )2(

1

)2(

1)(

�� rkkrr ,

is obtained as

33

**

)2()2(

1)]()()()([

2 ���

�vk

zziJN kkkkzz

ℏℏ

rrrr

Fig.

z

Detectord

0

V(r)

incidentplane wave

sphericalwave

eikz fk(,)

re

ikz

z

unit area

ℏk

� vrel

relativevelocity

33

volume = 1�kℏ

12ikz

e means that there is one particle per unit volume. Jz is the probability flow

(probability per unit area per unit time) of the incident beam crossing a unit surface perpendicular to OZ The probability flux associated with the scattered wave function

)()2(

12/3

�

fr

eikr

r (spherical wave)

is

2

2

3

** )(

)2(

1)(

2 r

fk

rriJ rrrrr

��

�

ℏℏ

drdA 2

dfv

drr

fvdAJN r

2

3

2

2

2

3)(

)2(

)(

)2(

�

�

where Jr is the probability flow (probability per unit area per unit time)

The differential cross section

dfN

Nd

d

d

z

2)(

or

r

d

dS

Detector

dS r2d

34

2)(

f

.

First-order Born amplitude:

)'('2

)'('2

'2

)2(4

1),'(

3

2

)'(3

2

2

3

rr

rr

kkkk

rQ

rkk

Ved

Ved

Vf

i

i

ℏ

ℏ

ℏ

����

��

�

,

which is the Fourier transform of the potential with respect to Q, where

kkQ ' : scattering wave vector.

2sin2

kQ Q for the elastic scattering.

The Ewald sphere is given by this figure. Note that the scattering angle is here. In the case of x-ray and neutron diffraction, we use the scattering angle 2, instead of .

35

Fig. Ewald sphere for the present system (elastic scattering). ki = k. kf = k’. Q =q

= k – k’ (scattering wave vector). 2

sin2

kQ .

((Ewald sphere)) x-ray and neutron scattering

ffê2

2q

OO1

Q

ki ki

k f

Ewald sphere

36

Fig. Ewald sphere used for the x-ray and neutron scattering experiments. ki = k. kf

= k’. Q =q = k – k’ (scattering wave vector). Note that in the conventional x-ray and neutron scattering experiments, we use the angle 2, instead of for both the x-ray and neutron scattering,

4. Spherical symmetric potential

When the potential energy V(r) is dependent only on r, it has a spherical symmetry. For simplicity we assume that ’ is an angle between Q and r’.

'cos'' Qr rQ .

We can perform the angular integration over ’.

0 0

2'cos'

2

0 0

2'cos'

2

2

)1(

)'('sin'''

)'('sin'2''2

4

1

)'('2

4

1)(

�

�

�

��

�

��

rVreddr

rVreddr

Vedf

iQr

iQr

i

ℏ

ℏ

ℏrr

rQ

Note that

)'sin('

2'sin'

0

'cos' QrQr

ed iQr �

37

Then

0

2

0

2

2

)1(

)'sin()'(''21

)'sin('

2)'('2'

2

4

1)(

QrrVrdrQ

QrQr

rVrdrf

ℏ

ℏ

�

��

�

(spherical symmetry)

Then the differential cross section is given by

2

0

2

22

2)1( )'sin()'(''21

)(

QrrVrdrQ

fd

d

ℏ

�

We find that )()1( f is a function of Q.

x

y

z

Q

r '

'

38

)2

sin(2

kQ ,

where is an angle between k’ and k (Ewald’s sphere). 5 Lippmann-Schwinger equation

The Lippmann–Schwinger equation is equivalent to the Schrödinger equation plus the typical boundary conditions for scattering problems. In order to embed the boundary conditions, the Lippmann–Schwinger equation must be written as an integral equation. For scattering problems, the Lippmann–Schwinger equation is often more convenient than the original Schrödinger equation (http://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation) ________________________________________________________________________

The Hamiltonian H of the system is given by

VHH ˆˆˆ0 ,

where H0 is the Hamiltonian of free particle. Let be the eigenket of H0 with the

energy eigenvalue E,

EH 0ˆ .

The basic Schrödinger equation is

EVH )ˆˆ( 0 . (1)

Both 0H and VH ˆˆ0 exhibit continuous energy spectra. We look for a solution to

Eq.(1) such that as 0V , , where is the solution to the free particle

Schrödinger equation with the same energy eigenvalue E.

)ˆ(ˆ0HEV .

Since EH 0ˆ or )ˆ( 0HE = 0, this can be rewritten as

)ˆ()ˆ(ˆ00 HEHEV ,

39

which leads to

VHE ˆ))(ˆ( 0 ,

or

VHE ˆ)ˆ( 10

.

The presence of is reasonable because must reduce to as V vanishes.

Lippmann-Schwinger equation:

)(1

0)( ˆ)ˆ( ViHEkk

by making Ek (= )2/22 �kℏ slightly complex number (>0, ≈0). This can be

rewritten as

)(10

)( ˆ'')ˆ(' ViHEd k rrrrkrr

where

rkkr ie2/3)2(

1

�,

and

''ˆ'0 kk kEH ,

with

22

' '2

kEk �ℏ

.

')ˆ()',(2 1

0)(

02rrrr

�iHEG k

ℏ

Note that

40

)('

2

)()(

02

)(10

)(

')'('4

'2

')'()',('2

')'(')ˆ('

�

�

�

rrrr

rkr

rrrrrkr

rrrrkrr

rr

Ve

d

VGd

ViHEdr

ik

k

ℏ

ℏ

The Green's function is defined by

'10

)(

0 '4

1')ˆ(

2)',(

2

rr

rrrrrr

ik

k eiHEG�

�

ℏ

((Proof))

'""')'2

('"'2

')ˆ(2

122

10

2

2

rkkkkkrkk

rr

��

�

iEdd

iHEI

k

k

ℏℏ

ℏ

or

�

��

��

��

iE

ed

iEd

iEddI

k

i

k

k

22

)'('

3

122

122

'2

)2(

'

2

'')'2

(''2

'")"'()'2

('"'2

2

2

2

k

k

rkkkrk

rkkkkkrkk

rrk

ℏ

ℏ

ℏℏ

ℏℏ

where

22

2k

�ℏ

kE .

Then we have

41

)(')2(

'

)'(2

)2(

'

2

22

)'('

3

222

)'('

3

2

�

�

��

ikk

ed

i

edI

i

i

rrk

rrk

k

kk

k

ℏ

ℏ

So we have

'4)(''

)2(

1)'(

'

22

)'('

3

)(0

rrkrr

rrrrk

��

iki e

ikk

edG

where >0 is infinitesimally small value (see the Appendix III for the derivation) In summary, we get

)(10

)( ˆ'')ˆ(' ViHEdr k rrrkrr

)()(

02

)( ˆ')',('2

�

VGd rrrrkrrℏ

or

)()(

02

)( ')'()',('2

�

rrrrrkrr VGdℏ

.

More conveniently the Lippmann-Schwinger equation can be rewritten as

)(10

)( ˆ)ˆ( ViHEkk

with

10

2)(

0 )ˆ(2

ˆ �

iHEG k

ℏ,

and

ViHEVG kˆ)ˆ(ˆˆ2 1

0

)(

02

�

ℏ

42

When two operators A and B are not commutable, we have very useful formula as follows,

AAB

BBAB

ABA ˆ1

)ˆˆ(ˆ1

ˆ1

)ˆˆ(ˆ1

ˆ1

ˆ1

,

We assume that

)ˆ(ˆ0 iHEA k , )ˆ(ˆ iHEB k

VHHAB ˆˆˆˆˆ0

Then

10

1110 )ˆ(ˆ)ˆ()ˆ()ˆ( iHEViHEiHEiHE kkkk ,

or

110

10

1 )ˆ(ˆ)ˆ()ˆ()ˆ( iHEViHEiHEiHE kkkk .

Here we newly define the two operators by

10 )ˆ( iHEk , 1)ˆ( iHEk

Note that 10 )ˆ( iHEk denotes an outgoing spherical wave and 1

0 )ˆ( iHEk

denotes an incoming spherical wave. Then we have

])ˆ(ˆ1[)ˆ(

)ˆ(ˆ)ˆ()ˆ()ˆ(1

01

10

1110

iHEViHE

iHEViHEiHEiHE

kk

kkkk

])ˆ(ˆ1[)ˆ(

)ˆ(ˆ)ˆ()ˆ()ˆ(11

0

110

10

1

iHEViHE

iHEViHEiHEiHE

kk

kkkk

Then )( can be rewritten as

43

k

kk

k

k

k

]ˆ)ˆ(1[

ˆ)ˆ(

ˆ])ˆ((ˆ)ˆ(

ˆ])ˆ(ˆ1[)ˆ(

ˆ)ˆ(

1

1

)(10

)(1

)(10

1

)(10

)(

ViHE

ViHE

ViHEViHE

ViHEViHE

ViHE

k

k

kk

kk

k

or

kk ViHEk

ˆˆ

1)(

.

7 The higher order Born Approximation

From the iteration, )( can be expressed as

...ˆ)ˆ(ˆ)ˆ(ˆ)ˆ(

)ˆ)ˆ((ˆ)ˆ(

ˆ)ˆ(

10

10

10

)(10

10

)(10

)(

kkk

kk

k

ViHEViHEViHE

ViHEViHE

ViHE

kkk

kk

k

The Lippmann-Schwinger equation is given by

kkk TiHEViHE kkˆ)ˆ(ˆ)ˆ( 1

0)(1

0)(

,

where the transition operator T is defined as

kTV ˆˆ )(

or

kkk TiHEVVVT kˆ)ˆ(ˆˆˆˆ 1

0)(

This is supposed to hold for any k taken to be any plane-wave state.

TiHEVVT kˆ)ˆ(ˆˆˆ 1

0 .

44

The scattering amplitude ),'( kkf can now be written as

kkkkk TVf ˆ')2(4

12ˆ')2(4

12),'( 3

2

)(3

2�

��

��

�ℏℏ

.

Using the iteration, we have

...ˆ)ˆ(ˆ)ˆ(ˆˆ)ˆ(ˆˆ

ˆ)ˆ(ˆˆˆ

10

10

10

10

ViHEViHEVViHEVV

TiHEVVT

kkk

k

Correspondingly we can expand ),'( kkf as follows:

.......),'(),'(),'(),'( )3()2()1( kkkkkkkk ffff

with

kkkk Vf ˆ')2(4

12),'( 3

2

)1( ��

�ℏ

,

kkkk ViHEVf kˆ)ˆ(ˆ')2(

4

12),'( 1

03

2

)2( ��

�ℏ

,

kkkk ViHEViHEVf kkˆ)ˆ(ˆ)ˆ(ˆ')2(

4

12),'( 1

01

03

2

)3( ��

�ℏ

.

45

fk

V

fk '

fk

V

V

G0

fk '

fk

V

V

G0

G0

V

fk '

46

Fig. Feynman diagram. First order, 2nd order, and 3rd order Born approximations.

kk is the initial state of the incoming particle and '' kk is the final

state of the incoming particle. V is the interaction. 8 Optical Theorem

The scattering amplitude and the total cross section are related by the identity

tot

kf

�

4)]0(Im[ ,

where

),()0( kkff : scattering in the forward direction.

dd

dtot

.

This formula is known as the optical theorem, and holds for collisions in general.

47

Fig. Optical theorem. The intensity of the incident wave is �/kℏ . The intensity

of the forward wave is )]0(Im[)/4()/( fk ��� ℏℏ . The waves with the total

intensity totkf ��� )/()]0(Im[)/4( ℏℏ is scattered for all the directions, as

the scattering spherical waves. ((Proof))

]ˆIm[2

)2(4

1

]ˆIm[2

)2(4

1)],(Im[

)(

2

3

2

3

�

��

��

�

V

Tf

k

kkkk

ℏ

ℏ

)(1

0)( ˆ)ˆ( ViHEkk

,

or

)(10

)( ˆ)ˆ( ViHEkk,

Ok

ktot

k ktot

48

or

10

)()( )ˆ(ˆ iHEV kk.

Then

)(10

)()()( )ˆ(ˆIm[(ˆIm[ iHEVV kk.

Now we use the well-known relation ( i representation, see the Appendix IV)

)]ˆ(ˆ

1[

ˆ1

)(ˆ

0

0

0

0

HEiHE

P

iHEiEG

k

k

k

k

�

Then

]ˆ)ˆ(ˆIm

ˆˆ

1ˆ[ImIm[(ˆIm[

)(0

)(

)(

0

)()()()(

�

VHEiV

VHE

PVVk

The first two terms of this equation vanish because of the Hermitian operators of V and

VHE

PV ˆˆ

1ˆ

0

.

Therefore,

kkk THETVHEVV kˆ)ˆ(ˆˆ)ˆ(ˆˆIm[ 0

)(0

)()( ��

or

49

)2

'(ˆ''

)2

'(ˆ''ˆ'ˆIm[

222

22)(

��

��

kETd

kETTdV

k

k

ℏ

ℏ

kkk

kkkkkk

)]'([))]')('(2

[)]'(2

[

)2

'

2()

2

'(

2222

2

222222

kkk

kkkkkk

kkkE

�

�

�

��

�

ℏℏℏ

ℏℏℏ

or

)'()2

'(

2

22

kkk

kE

��

ℏ

ℏ

2

2

22

2

)(

ˆ''

)'(ˆ''''ˆIm[

kk

kkk

Tdk

kkTdkdkk

V

ℏ

ℏ

��

�

�

Thus

]ˆ''16

4

ˆ'')(2

)2(4

1

]ˆIm[2

)2(4

1)]0(Im[

2

4

24

2

22

3

)(

2

3

kk

kk

k

Tdk

Tdk

Vf

ℏ

ℏℏ

ℏ

���

����

�

�

��

Since

2

4

24

2

4

26

2

2

ˆ'16

ˆ'4

)2(16

1

),'('

kk

kk

kk

T

T

fd

d

ℏ

ℏ

��

��

�

,

50

2

4

24

ˆ''16

'' kk Tdd

ddtot

ℏ

��

Then we obtain

tot

kf

�

4)]0(Im[ . (optical theorem)

9. Summary

Comparison between the partial wave approximation (low-energy scattering) and high-energy scattering (Born approximation). (i) Although the partial wave expansion is “straightforward”, when the energy of

incident particles is high (or the potential weak), many partial waves contribute. In this case, it is convenient to switch to a different formalism, the Born approximation.

(ii) At low energies, the partial wave expansion is dominated by small orbital angular

momentum. APPENDIX-II

Inelastic neutron scattering data

We use the conversion relation of the units for photon with the dispersion relation

� = ℎ� = 3ℎ

1 meV = 0.241799 THz

1 THz = 4.13567 meV

100 THz = 0.413567 eV

1 cm-1 = 0.123984 meV

1 meV = 8.06556 cm-1

3 THz = 100.069 cm-1.

(a) Rh

51

Fig. Phonon dispersion curve of Rh. A. Eichler, K.P. Bohnen, W. Reichardt, and J.

Hafner, Phonon dispersion relation in rhodium: Ab initio calculations and neutron-

scattering investigations, Phys. Rev. B57, 324 (1998).

(b) Neon

52

Fig. Acoustic mode dispersion curves for neon (ccp structure) measured by inelastic

neutron scattering. (Data taken from Endoh et al. Phys. Rev. B11, 1681, 1975). M.T.

Dove, Structure and Dynamics: An Atomic View of Materials (Oxford, 2002).

(c) Lead

Fig. Acoustic mode dispersion curves for lead (fcc) measured by inelastic scattering (Data

taken from Brockhouse et al. Phys. Rev. 128, 1099, 1962). M.T. Dove, Structure and

Dynamics: An Atomic View of Materials (Oxford, 2002).

(d) Potassium (fcc)

53

Fig. Acoustic mode dispersion curves for potassium (fcc) measured by inelastic neutron

scattering. (Data taken from Cowley et al., Phys. Rev. 150, 487, 1966). M.T. Dove,

Structure and Dynamics: An Atomic View of Materials (Oxford, 2002).

(e) NaCl

54

Fig. Dispersion curves for NaCl (inelastic neutron scattering). (Data taken from Raunio et

al. Phys. Rev. 178, 1496, 1969). M.T. Dove, Structure and Dynamics: An Atomic

View of Materials (Oxford, 2002).

(f) Qaurtz

55

Fig. Dispersion curves for quartz. (Data are taken from Strauch and Dorner, J. Phys. Cond.

Matter 5, 6149, 1993). M.T. Dove, Structure and Dynamics: An Atomic View of

Materials (Oxford, 2002).

56

Fig. Phonon dispersion curves of quartz obtained by inelastic x-ray scattering. (Data

from Ch. Halcoussis, J. Phys,: Cond. Matter 13, 7627, 2001).

57

Fig. Dispersion curves for potassium bromide at 90 K. (Woods et al. Phys. Rev. 131,

1025 (1963). C. Kittel, Introduction to Solid State Physics, 4-th edition (John Wiley,

1971).