acns 2008 tutorial section sans and reflectometry for soft condensed matter research the basic...
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ACNS 2008 Tutorial Section
SANS and Reflectometry for Soft Condensed Matter Research
The Basic Theory for Small Angle Neutron Scattering
Wei-Ren Chen
Neutron Scattering Sciences Division Spallation Neutron Source
Oak Ridge National Laboratory
May 11th 2008
Outline
Two Aspects of Collision: Kinematics vs. Dynamics
Cross Section Calculation I: Method of Phase Shift
Cross Section Calculation II: Fermi Approximation
Expression of Scattering Cross Section
Coherent and Incoherent Scattering
Contrast variation
References
Kinematics Aspect of Collision
particle 1 (projectile)
particle 2 (target)
v1 v2
v1’
v2’
Conservation laws
• energy (1)• mass(1)
• momentum (3)• v1 and v2 are known (6)
1+1+3+6 = 11
12 variables: v1, v2, v1’ & v2’
Kinematics Aspect of Collision
What is the possibility that the projectile will scatter off the target at that specific angle?
Possible existence of NeutronJames ChadwickNature, 129, 312, 1932
Interaction: hidden in Cross Section
111
21 mm
origin
effective particle
closest approach
Is this reaction possible? Does it violate any conservation law?
Independent of the specific forces between the particles
scattering ≡ (initial constellation = final one), elastic scattering ≡ conservation of kinetic energy
A + B → A + B
Dynamics Aspect of Collision: Concept of Cross Section
d
azimuthal axis
polar axis
x
area A
density N
d
d
Intensity I
Beam size A (L2)Intensity of beam I (T-1)Thin sample thickness Δx (L)Number density of sample N (L-3)no. of reaction occurring per second (T-1)
Reaction probability ≡
To calculate one must be to be able to calculate reaction probability
: a proportionality constant of reaction probability with dimension of L2
A
xNA
I
A
xNA
I
Scattering Experiment
ikzrki ee in
r
ef
ikr
sc
vvJ in*inin
dRvdN 2sc
*sc
Given the interaction potential V(r), how can one calculate σ(θ)?
2
in
/ fJ
ddN
d
d
2
2
sc*scsc R
fvvJ
angular differential cross section
dd
d
Phase Shift Analysis
looking for far field solution (kr >> 1 , V(r) = 0) E > 0
Where is f(θ) in Schrödinger equation ? You put it in through boundary condition
ErV2
2
2
Schrödinger equation:
is introduced as one of the integration constants
LHS r
efe
r
efe
ikrikr
ikrikz cos RHS
expanded by partial wave
matching the coefficients ofexp(ikr) and exp(-ikr) from RHS and LHS
llk
2
02
sin124
2
02
cossin121
l
lli Pel
kl and
ll lkrAru 2/sin0
Reasoning of S-wave Scattering for Low Energy Scattering (kr0 << 1)
b
r0
v
z
Classically
Quantum Mechanically
sec1021 27 ergll
Only neutrons with l = 0 will be scattered
sec10
sec/101051067.130
61324
erg
cmcmgmbvL
u0
0
sin(kr)sin(kr+)
Definition of Scattering Length a
1 when sin1
cossin121
002
2
2
02
krk
Pelk l
lli l
and 1 when sin4
002
2 kr
k
0 → 0 as k → 0
k
fak
0
0lim
2a24 a
Example: Neutron-Proton Scattering
Lecture 2 Basic Theory - Neutron Scattering for Biomolecular ScienceRoger Pynn, UCSB, 2004
36 MeV-Vo
-EB 2.23 MeV
r0=2F
V(r)
r
Example: Neutron-Proton Scattering
From the capture of a low-energy neutron by hydrogen
Solving the Schrödinger equation with this binding energy, (E < 0)
V0 = -36 MeV and r0 = 2 F (F = 10-13 cm)
Matching the wave functions and their flux for the exterior and interior regions, (E > 0)
n + H1 → H2 + (2.23 MeV)
= 2.3 barns
~20 barns
2.3 barns
Example: Neutron-Proton Scattering
The “Barn Book”Brookhaven National Laboratory Report
BNL-325, 1955
Experimental Nuclear Reaction Data (EXFOR / CSISRS)National Nuclear Data Center
http://www.nndc.bnl.gov/
Example: Neutron-Proton Scattering
Eugene P. Wigner, Zeits. f. Physik 83 253 1933
spin dependence interaction
t
= 20 barns
S0
2T0
22
sin4
1sin
4
31
k
triplet state (bound state)I = 1, parallel, EB = -2.23 MeV
singlet state (virtual state)I = 0, antiparallel, E* = 70 keV
Fermi Approximation Step 1 – Born Approximation
Another way to solve the Schrödinger Equation
rVrirdf exp
4
2 32
''exp'exp'2
32
1 rkirVrkirdr
eer
ikrikz
Why we need Born Approximation?
The many-body problem of thermal neutron scattering
What is Born Approximation?
Born approximation eliminates the need of solving Schrödinger equation
Compare with r
efe
ikrikz
Can Born Approximation be Applied to Neutron Scattering?
12
200
rV
7.310
104106.11036106.154
2612624
2
200
rV
If we use the potential parameters for n-p scattering
No with real potential, too large for Born Approximation to be applicable
Fermi Approximation Step 2 – Fermi Pseudopotential
Real potential
constant
7.3
10~
300
2
200
40
rV
rV
kr
02*
0
06*
0
10~
10~
rr
VV
Fictitious potential
300
320
~20
rVrrVdm
fakr
300
3*0
*0
22
200
2*0
1103
10~
rVrV
rV
kr
With this fictitious potential, Born Approximation is valid
Requirement
constant
1
1
300
2
200
0
rV
rV
kr
V(r)
r0
-V0
V(r)
r0
-V0*
*
actural neotron-nucleusinteraction potential
Fermi pseupotential
Fermi Approximation Step 2 – Fermi Pseudopotential
actual neutron-nucleusinteraction potential Fermi pseudopotential
300
320
~20
rVrrVdm
fakr
V0* ~ 10-6V0
r0* ~ 102r0
Enrico Fermi, Ricerca Scientifica 7 13 1936
N
iii rrb
mrV
1
2* 2
Why delta function?What is b ?
cmr
cmr
cm
11*0
130
8
10~
10~
10
Chemical Binding Effect
~ 2
high energy (~10 eV)
5.0
21
1
1
11
barns 80420
111
Tmmn
low energy (0.025 eV)
1~
1~)water(18
1
1
11
barns 20
A Typical Reactor-based SANS Diffractometer
Lecture 5 Small Angle Scattering - Neutron Scattering for Biomolecular ScienceRoger Pynn, UCSB, 2004
angular differential cross section
d
d
Expression of (): Coherent & Incoherent Contribution
N
iii rrb
mrV
1
2* 2 rVrirdf
exp
4
2 32
N
i
N
jjiji rrkibbf
d
d
1 1
2exp
2inc
222coh
2
222
bbbbb
bbbbbbb jijii
kSNbNb
rkibbbNN
ii
2coh
2inc
2
1
222
exp
Example: Neutron-Proton Scattering
t
kSNbNb
rkibbbNN
ii
2coh
2inc
2
1
222
exp
12 :number quantum magnatic , 2
1or
2
1 sIsIs
F
bbb 8.3
4
27.2324.53
4
3
222
2 6494
3 F
bbb
barns 8.18.3 cohcoh Fbb
barns 2.805.25 inc
22inc Fbbb
For D
F
bbb 7.6
4
10.0295.04
6
24
22 5.60 Fb
barns 6.5coh
barns 0.2inc
For H 0 1 ss
2
1
2
3 ss
F = 10-13 cm
Basis of Contrast Variation
t
For H For D For O
Fb 8.5 Fb 8.3 H Fb 7.6
D
For H2O
Fb 8.1 OH2
For D2O
Fb 2.19 OD2
can be adjusted to take on any value between these two extremessolvent b
Scattering Length Density Calculatorhttp://www.ncnr.nist.gov/resources/sldcalc.html
Lecture 1 Overview of Neutron Scattering & Applications to BMSE –
Neutron Scattering for Biomolecular ScienceRoger Pynn, UCSB, 2004
F = 10-13 cm
References and Further Reading
Roger Pynn - An Introduction to Neutron Scattering (http://www.mrl.ucsb.edu/~pynn/)
- Neutron Physics and Scattering (http://www.iub.edu/~neutron/)
Sidney Yip et al. - Molecular Hydrodynamics
Sow-Hsin Chen et al. - Interaction of Photons and Neutrons With Matter
Peter A. Egelstaff - An Introduction to the Liquid State
M. S. Nelkin et al. - Slow Neutron Scattering and Thermalization
Anthony Foderaro - The Element of Neutron Interaction Theory
Paul Roman - Advanced Quantum Theory
Jean-Pierre Hansen et al. - The Theory of Simple Liquids
Stephen W. Lovesey - Condensed Matter Physics: Dynamic Correlations
Peter Lindner and Thomas Zemb – Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter
Ferenc Mezei in Liquids, Crystallisaton et Transition Vitreuse, Les Houches 1989 Session LI
Léon Van Hove Physical Review 95 249 1954