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 = 1 2 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
57
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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
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-