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 s(q) Coherent and Incoherent Scattering Contrast variation References
Kinematics Aspect of Collision
12 variables: v1, v2, v1’ & v2’
v1’ v2 particle 2 (target)
particle 1 (projectile) v1
Conservation laws • energy (1) • mass(1) • momentum (3) • v1 and v2 are known (6)
v2’
1+1+3+6 = 11
Kinematics Aspect of Collision
Possible existence of Neutron James Chadwick Nature, 129, 312, 1932
closest approach
effective particle
1 1 m1 m2
1
Is this reaction possible? Does it violate any conservation law?
Independent of the specific forces between the particles
origin What is the possibility that the projectile will scatter off the target at that specific angle?
Interaction: hidden in Cross Section
scattering ≡ (initial constellation = final one), elastic scattering ≡ conservation of kinetic energy
A+ B→A+ B
Dynamics Aspect of Collision: Concept of Cross Section
d d dq area A q polar axis
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 Q (T-1) Reaction probability ≡
Intensity I
density N x
Q NAx I A
azimuthal axis
Q NAx s I A
s : a proportionality constant of reaction probability with dimension of L2
To calculate s one must be to be able to calculate reaction probability
Scattering Experiment
* J in vin in v
e ikr sc f (q ) r
J sc v sc v
* sc
f (q )
2
R2 * dN (vsc sc )R 2 d
ds dN / d 2 s (q ) f (q ) d J in
angular differential cross section
in e i k r e ikz
ds s d d
Given the interaction potential V(r), how can one calculate σ(θ)?
Phase Shift Analysis
Schrödinger equation: Where is f(θ) in Schrödinger equation ?
2 2 V (r ) E 2
You put it in through boundary condition
looking for far field solution (kr >> 1 , V(r) = 0) E > 0
LHS
e
ikz
e ikr e ikr ikr cosq f (q ) e f (q ) r r
RHS
u 0 (r ) Al sinkr (l / 2) d l
d0 is introduced as one of
the integration constants
expanded by partial wave matching the coefficients of exp(ikr) and exp(-ikr) from RHS and LHS
1 s (q ) 2 k
(2l 1)e
l 0
id l
sin d l Pl (cosq )
2
and
4 s 2 k
(2l 1)sin 2 d l
0
Reasoning of S-wave Scattering for Low Energy Scattering (kr0 << 1)
v b z r0
Classically
L mbv (1.67 1024 g ) (5 1013 cm) (106 cm / sec) 1030 erg sec
Quantum Mechanically
l (l 1) 2 1027 erg sec
Only neutrons with l = 0 will be scattered
Definition of Scattering Length a
1 s (q ) 2 k
u0
(2l 1)e d
i l 0
l
sin d l Pl (cosq )
2
and
s
1 sin 2 d 0 when kr0 1 k2
4 sin 2 d 0 when kr0 1 k2
d0 → 0 as k → 0
a lim f (q )
k 0
d0
k
d0
s (q ) a 2
sin(kr) sin(kr+d0)
s 4a 2
Accurate Measurements of the Scattering Length
http://physics.nist.gov/MajResFac/InterFer/text.html
Physical Significance of Sign of Scattering Length
u 0 A sin (kr d 0 ) A sin (kr ka) Ak (r a )
uo r0
a<0
0
a>0
r
Example: Neutron-Proton Scattering
Lecture 2 Basic Theory - Neutron Scattering for Biomolecular Science Roger Pynn, UCSB, 2004
Example: Neutron-Proton Scattering
From the capture of a low-energy neutron by hydrogen
V(r) r0=2F -EB 2.23 MeV r
n + H1 → H2 + g (2.23 MeV)
Solving the Schrödinger equation with this binding energy, (E < 0)
V0 = -36 MeV and r0 = 2 F (F = 10-13 cm)
-Vo 36 MeV
Matching the wave functions and their flux for the exterior and interior regions, (E > 0)
s = 2.3 barns
Example: Neutron-Proton Scattering
The “Barn Book” Brookhaven National Laboratory Report BNL-325, 1955
~20 barns
2.3 barns
Experimental Nuclear Reaction Data (EXFOR / CSISRS) National Nuclear Data Center http://www.nndc.bnl.gov/
Example: Neutron-Proton Scattering
spin dependence interaction
Eugene P. Wigner, Zeits. f. Physik 83 253 1933
triplet state (bound state) I = 1, parallel, EB = -2.23 MeV
t singlet state (virtual state) I = 0, antiparallel, E* = 70 keV
s (q )
1 3 2 1 sin d 0 T sin 2 d 0S k2 4 4
s = 20 barns
Fermi Approximation Step 1 – Born Approximation
Why we need Born Approximation? The many-body problem of thermal neutron scattering What is Born Approximation? Another way to solve the Schrödinger Equation
eikr (1) (r ) eikz d 3r ' exp ( i k r ')V (r ) exp (i k ' r ') 2r 2
Compare with
eikr e f (q ) r
ikz
2 f (q ) d 3r exp( i r )V (r ) 4 2
Born approximation eliminates the need of solving Schrödinger equation
Can Born Approximation be Applied to Neutron Scattering?
V0 r02
2
1
If we use the potential parameters for n-p scattering
V0 r02
2
1.6 10 24 36 10 6 1.6 10 12 4 10 26 3.7 54 10
No with real potential, too large for Born Approximation to be applicable
Fermi Approximation Step 2 – Fermi Pseudopotential
a f (q ) kr 0
0
m d 3rV (r ) ~ V0 r03 2 2
Fictitious potential
V0* ~ 106 V0 r0* ~ 102 r0
Requirement
Real potential
kr0 1
kr0 ~ 10 4
kr0* ~ 102
V0 r02
2 V0 r03 constant
1
V0 r02
2 V0 r03 constant
3 .7
V0 r02
2
* *3 0 0
3 10 2 1
V r V0 r03
With this fictitious potential, Born Approximation is valid
Fermi Approximation Step 2 – Fermi Pseudopotential
V(r) V(r)
0
a f (q ) kr 0
m d 3rV (r ) ~ V0 r03 2 2
2 2 V (r ) m
*
b d (r r )
N i 1 i i
V0* ~ 10-6V0 r0* ~ 102r0
r0 * -V0
Why delta function? What is b ?
r0 *
actual neutron-nucleus interaction potential
actural neotron-nucleus interaction potential -V0
Fermi pseudopotential
Fermi 8 cm 10pseupotential
r0 ~ 10 13 cm r0* ~ 10 11 cm
Enrico Fermi, Ricerca Scientifica 7 13 1936
Neutron Scattering Data for Elements and Isotopes
Neutron Diffraction George E. Bacon
Chemical Binding Effect
s ~ 2
1 1 mn mT
1
low energy (0.025 eV)
1 1 ~1 1 18( water) ~1
1
high energy (~10 eV)
20 4 80 barns
1 1 2 1 1 0.5
1
20 barns
Lecture 2 Basic Theory - Neutron Scattering for Biomolecular Science Roger Pynn, UCSB, 2004
A Typical Reactor-based SANS Diffractometer
angular differential cross section
ds s (q ) d
Lecture 5 Small Angle Scattering - Neutron Scattering for Biomolecular Science Roger Pynn, UCSB, 2004
Expression of s(q): Coherent & Incoherent Contribution
2 2 V (r ) m
*
b d (r r )
N i 1 i i
f (q )
2 d 3r exp( i r )V (r ) 4 2
ds 2 s (q ) f (q ) d
bi
2
b b
i 1 j 1 i
N
N
j
exp i k ri rj
2
(
)
b2
2 bcoh
bi b j bi b j b
b
2
(b
2
2
b
2
2
) b
2 inc
s (q ) N b 2 b
(
) b
exp( i k r )
N i 1 i
2
2 2 Nbinc Nbcoh S (k )
Example: Neutron-Proton Scattering
s (q ) N b 2 b
(
2
)
b
2
exp( i k r )
N i 1 i
2
2 2 Nbinc Nbcoh S (k )
F=
10-13
cm
s I
1 1 or s I , magnatic quantum number : 2s 1 2 2 t
b
(3b b ) 3 (5.4 2) ( 23.7 2) 3.8F
For H s
1 s 0
bcoh b 3.8F s coh 1.8 barns
binc
b
2
(3b
4
2
b 4
2
) 649 F
4
2
(b
2
b
2
) 25 .5F s
inc
80 .2 barns
b
(4b 2b ) 4 (0.95) 2 (0.10) 6.7 F
6 4
3 1 s For D s 2 2
s coh 5.6 barns
b2 60.5F 2
s inc 2.0 barns
Contrast Variation
10-12
F = 10-13 cm
Neutron Diffraction George E. Bacon
Basis of Contrast Variation
For H For D For O
b
H
3.8F
For H2O
b
D
6.7F
b 5.8F
For D2O
b
H 2O
1.8F
t
b
D2O
19.2F
b
solvent
can be adjusted to take on any value between these two extremes
Scattering Length Density Calculator http://www.ncnr.nist.gov/resources/sldcalc.html
F = 10-13 cm
Lecture 1 Overview of Neutron Scattering & Applications to BMSE – Neutron Scattering for Biomolecular Science Roger Pynn, UCSB, 2004
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