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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),(
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).