by Roger Pynn LECTURES 1 and 2: Theory of Nuclear Scattering of Neutrons Overview Introduction and theory of neutron scattering • Advantages/disadvantages of neutrons for scattering measurements • Neutron properties • Comparison with other structural probes • Definition of scattering cross sections • Fermi pseudopotential • Refractive index for neutrons – two different calculations • Elastic scattering and definition of the structure factor, S(Q) • Coherent & incoherent scattering • Inelastic scattering • References
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Transcript
by
Roger Pynn
LECTURES 1 and 2: Theory of Nuclear Scattering of Neutrons
Overview
Introduction and theory of neutron scattering • Advantages/disadvantages of neutrons for scattering measurements
• Neutron properties
• Comparison with other structural probes
• Definition of scattering cross sections
• Fermi pseudopotential
• Refractive index for neutrons – two different calculations
• Elastic scattering and definition of the structure factor, S(Q)
• Coherent & incoherent scattering
• Inelastic scattering
• References
Why do Neutron Scattering?
• To determine the positions and motions of atoms in condensed matter– 1994 Nobel Prize to Shull and Brockhouse cited these areas
(see http://www.nobel.se/physics/educational/poster/1994/neutrons.html)
• Neutron advantages:– Wavelength comparable with interatomic spacings– Kinetic energy comparable with that of atoms in a solid– Penetrating => bulk properties are measured & sample can be contained– Weak interaction with matter aids interpretation of scattering data– Isotopic sensitivity allows contrast variation– Neutron magnetic moment couples to B => neutron “sees” unpaired electron spins
• Neutron Disadvantages– Neutron sources are weak => low signals, need for large samples etc– Some elements (e.g. Cd, B, Gd) absorb strongly– Kinematic restrictions (can’t access all energy & momentum transfers)
The 1994 Nobel Prize in Physics – Shull & Brockhouse
Neutrons show where the atoms are….
…and what the atoms do.
The Neutron has Both Particle-Like and Wave-Like Properties
λ (nm) = 395.6 / v (m/s)E (meV) = 0.02072 k2 (k in nm-1)
Comparison of Structural ProbesNote that scattering methods provide statistically averaged information on structure rather than real-space pictures of particular instances
Macromolecules, 34, 4669 (2001)
Thermal Neutrons, 8 keV X-Rays & Low Energy Electrons:- Absorption by Matter
Note for neutrons:
• H/D difference
• Cd, B, Sm
• no systematic A
dependence
Interaction Mechanisms
• Neutrons interact with atomic nuclei via very short range (~fm) forces.• Neutrons also interact with unpaired electrons via a magnetic dipole interaction.
Brightness & Fluxes for Neutron & X-Ray Sources
Brightness(s-1 m-2 ster-1)
dE/E(%)
Divergence(mrad2)
Flux(s-1 m-2)
Neutrons 1015 2 10 x 10 1011
RotatingAnode
1016 3 0.5 x 10 5 x 1010
BendingMagnet
1024 0.01 0.1 x 5 5 x 1017
Wiggler 1026 0.01 0.1 x 1 1019
Undulator(APS)
1033 0.01 0.01 x 0.1 1024
Cross Sections
dEd
dE & d into secondper scattered neutrons ofnumber
d
d into secondper scattered neutrons ofnumber
/ secondper scattered neutrons ofnumber total
secondper cmper neutronsincident ofnumber
2
2
ΩΦΩ
=Ω
ΩΦΩ
=Ω
Φ==Φ
dEd
d
d
d
σ
σσ
σ measured in barns:1 barn = 10-24 cm2
Attenuation = exp(-Nσt)N = # of atoms/unit volumet = thickness
ikzinc e=ψIncident plane wave:
Scattered (circular) wave:
ikzuRik
ikzikrs
eeuR
b
eer
b
rr
rr−
−−=
−=ψ
Scattering by a single nucleus
Squires Eq. 1.18 generalized for the origin at any location.
λπ2
=kwhere
Scattering by a Single (fixed) Nucleus
units) (note 4 so bd
d b v
d
d
v v :is areasunit throughpassing neutronsincident ofnumber theSince
d b v /rb dS v dS v
:is scatteringafter secondper dSarea an throughpassing
neutrons ofnumber the),scatteringafter and before (same neutron theof velocity theis vIf
2total
22
2
incident
2222
scat
bπσσ
ψ
ψ
==ΩΦΩ
=Ω
==Φ
Ω==
• range of nuclear force (~ 1fm)is << neutron wavelength soscattering is “point-like”
• energy of neutron is too smallto change energy of nucleus &neutron cannot transfer KE to a fixed nucleus => scattering is elastic
• we consider only scattering farfrom nuclear resonances whereneutron absorption is negligible
Adding up Neutrons Scattered by Many Nuclei
0
)(
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kkQ
ebbebbd
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eebd
dS
vd
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d
ee
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i
ii
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jii
RR.kkij
jii
R.kki rkii
scat
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rrr
rr
rr
r
rrrrrrr
rrrrr
rrrrr
rr
−=
==Ω
=Ω>>
Ω=
Ω=
Ω∴
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
−−−−
−
−
∑∑
∑
∑
σ
ψσ
ψ
The Fermi Pseudo-Potential
)(2
)( so )(2
'')2(
21 So,
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so ''
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'k state in neutronsfor energy,unit per ,d in states momentum of # is where
)(12
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: RuleGolden sBy Fermi'
ns transitioall of iesprobabilitover is sum the where11
22).'(
2
2
2
23
23'
23
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222
3
'
2
).'(2'
2
'd in '
'
d in ''
rbm
rVrderVm
kVkdm
kY
d
m
k
Y
d
d
kmY
dm
kY
ddkk'Y
ddkk'dE
m
dkkdE
m
kE
Y
rderVY
kVkW
Wdd
d
rkki
k
k
k
Y
rkki
kkk
kk
kkk
vhvrv
h
rr
hhh
h
h
hh
v
rv
h
rr
h
vrv
r
r
vrv
rrr
rr
rrr
δπππ
πσ
πρ
πρ
π
ρ
ρπρπ
σ
=⎟⎠⎞
⎜⎝⎛=Ω
Ω=
Ω
===Φ
Ω==
Ω=Ω=
=⇒=
=
Ω
==
ΩΦ=
Ω
∫
∫∑
∑
−
−
Ω→
Ω→
Fermi pseudopotential
Use V(r) to Calculate the Refractive Index for Neutrons
materials.most from reflected externally are neutrons 1,ngenerally Since
2/1
:get we)A 10(~ small very is and ),definition(by index refractive Since
4or 2
222 givesenergy of onConservati
2 isenergy total themedium theInside
2
is vaccuumin neutron ofenergy total)(and kinetic The
tryreflectome & SANSfor used - (SLD) Density LengthScatteringnuclear thecalled is
1 where
2 :is medium theinside potential average theSo
nucleus. singlea for )(2
)( :by given is potential neutron-nucleus The
2
2-6-0
220
2222220
2
22
20
2
2
2
<=
==
=−+=+=
+
=
==
=
∑
πρλ
ρ
πρρπ
ρ
ρρπ
δπ
-n
nk/k
kkmm
kV
m
k
m
k
Vm
k
m
kE
bvolumem
V
rbm
rV
ii
hhhh
h
h
h
rhr
Now let’s do it the Hard Way – Calculate the Scattered Wavefunction….
( ) ⎥⎦⎤
⎢⎣
⎡
−+
+−≈
++−=
2
22
2222
2
11)(
)(
zd
yxzdr
yxzdr
for t & u << dViewgraph from M. R. Fitzsimmons
( )
( ) ( )
( )
ikds
tikd
zd
xik
zd
yikt
ikds
zd
yxik
ikd
ikzuRik
s
tei
k
zdi
zddze
dxedyezd
dze
dVeezd
bN
NdVeeuR
bd
ρλψ
πρ
ρψ
ψ
−=
−−
−=
−−=
−−≈
−−=
∫
∫∫∫ −−
−+
−
21
1
0
22
0
2
22
22
rr
rr A function with interesting consequences.
p.66 of Als-Nielsen & McMorrow
The scattered wave function at “d” from a plate of material illuminated by a plane wave.
Viewgraph from M. R. Fitzsimmons
Definition of the 1st Fresnel zone
The 1st Fresnel zone is the portion of the sample yielding phase from 0 to 180°, i.e., having the same sign of the imaginary component of exp(iku2/d).
m 10μλ ≈≤ du
d = 1 mλ = 1 nm
Viewgraph from M. R. Fitzsimmons
(1) The phase of the wave function in the forward direction ψt is related to the average (macroscopic) properties of the material, i.e., , and not its atomic structure, e.g., crystalline, liquid, etc.
(2)The phase can be retarded or advanced depending upon the sign of , since ρ = Nb.
( )( )
( )( ) ikdti
t
ikd
sinct
ikds
eed
eti
d
teid
ρλψ
ρλ
ψψψρλψ
−≈
−=
+=−=
1
t ρ
b
(scattering approach)
Viewgraph from M. R. Fitzsimmons
Let’s Calculate ψt(d) using a Macroscopic Approach
Relative to the front surface of Medium 1, the accumulated phase at d, φ(d) is:
( )tdktk −+′=+= 21 φφφ
Viewgraph from M. R. Fitzsimmons
( )( )
ikdktnit ee
kdktn
tdknkt
tdktk
)1(
)1(−=⇒
+−=−+=−+′=
ψ
φ
φ
From optics theory (Snell’s law), the perpendicular components of the wave vectors across an interface are related by the index of refraction n:
⊥⊥ =′ nkk
so
This is the result obtained from the macroscopic (optics)approach
Viewgraph from M. R. Fitzsimmons
(macroscopic approach)
Equating the Scattering & Macroscopic Approaches…ikdti
t ee ρλψ −=ikdktni
t ee )1( −=ψ( )
2
2
11
1
ρλπ
ρλ
−=⇒
=−
n
tktn
(scattering approach)
Index of refraction – same result as before
Phase advanced,
λ1<λ0
1+0.8x10-5-3.70.08Mn
Phase retarded,
λ1>λ0
1-2x10-510.30.09Ni
nb
(x10-5 Å)N (atoms/Å3)
Viewgraph from M. R. Fitzsimmons
Why do we Care about the Refractive Index?
• When the wavevector transfer Q is small, the phase factors in the cross section do not vary much from nucleus to nucleus & we can use a continuum approximation
• We can use all of the apparatus of optics to calculate effects such as:– External reflection from single surfaces (for example from guide surfaces)
– External reflection from multilayer stacks (including supermirrors)
– Focusing by (normally) concave lenses or Fresnel lenses
– The phase change of the neutron wave through a material for applications such as interferometry or phase radiography
– Fresnel edge enhancement in radiography
Coherent and Incoherent Scattering
The scattering length, bi , depends on the nuclear isotope, spin relative to the neutron & nuclear eigenstate. For a single nucleus:
)(
ji unless vanishes and 0but
)(
zero toaverages b where
22
,
).(2
2222
2
i
Nbbebd
d
bbbbb
bbb
bbbbbbbb
bbb
ji
RRQi
ii
ji
jijiji
ii
ji −+=Ω
∴
−=−=
==
+++=
+=
∑ −−rrrσ
δ
δδδ
δδδδ
δδ
Coherent Scattering(scattering depends on the direction & magnitude of Q)
Incoherent Scattering(scattering is uniform in all directions)
Note: N = number of atoms in scattering system
Nuclear Spin Incoherent Scattering
[ ] [ ]222 )())(1(12
1b and )1(
12
1 Thus
2)(2I)/(4I is state of occurence offrequency The
2)2)/(4I(2I is state of occurence offrequency The
y.probabilit same the
has state spin each d,unpolarize are spinsnuclear theand neutrons theIf
21)2/1(2 is )2/1( spin withstates ofnumber The
221)2/1(2 is )2/1( spin withstates ofnumber The
).2/1(or )2/1( be can system
neutron-nucleus theof spin The . spin withisotope singlea Consider
−+−+
−−
++
+++
=+++
=
+=
++=
=+−−+=+++
−+
bIbII
IbbII
b
fb
fb
a priori
III
III
II
I
Values of σcoh and σinc
0.024.936Ar0.01.5Al
0.57.5Cu0.04.2O
5.21.0Co0.05.6C
0.411.5Fe2.05.62H
5.00.02V80.21.81H
σincσcohNuclideσincσcohNuclide
• Difference between H and D used in experiments with soft matter (contrast variation)• Al used for windows• V used for sample containers in diffraction experiments and as calibration for energy
resolution• Fe and Co have nuclear cross sections similar to the values of their magnetic cross sections• Find scattering cross sections at the NIST web site at:
http://webster.ncnr.nist.gov/resources/n-lengths/
Coherent Elastic Scattering measures the Structure Factor S(Q) I.e. correlations of atomic positions
only. R of functiona is )( where
}.)(.{1)( ie
)()(1
)'()(.'1
or
)(.1
)( so
densitynumber nuclear theis where)(.)(.Now
1
)( whereatomssimilar ofassembly anfor )(.
00
.
.)'.(
2.
N ...
ensemble,
).(2
rrrrr
rrr
rrrrrrrrrr
rrr
rrrrr
rr
rr
rrrrr
rr
rrrrrr
rrr
∑
∫∫ ∫∫∫
∫
∫∑∫∑
∑
≠
−
−−−
−
−−−
−−
+−=
−+=
−==
=
=−=
==Ω
ii
RQi
NNRQi
NNrrQi
NrQi
NrQi
ii
rQi
i
RQi
ji
RRQi
RRR)Rg(
eRgRdQS
RrrerdRdN
rrerdrdN
)QS(
rerdN
QS
rerdRrerde
eN
QSQSNbd
d
i
ji
δ
ρ
ρρρρ
ρ
ρρδ
σ
g(R) is known as the static pair correlation function. It gives the probability that there is anatom, i, at distance R from the origin of a coordinate system, given that there is also a (different) atom at the origin of the coordinate system at the same instant in time.
S(Q) and g(r) for Simple Liquids
• Note that S(Q) and g(r)/ρ both tend to unity at large values of their arguments
• The peaks in g(r) represent atoms in “coordination shells”
• g(r) is expected to be zero for r < particle diameter – ripples are truncation errors from Fourier transform of S(Q)
Neutrons can also gain or lose energy in the scattering process: this is called inelastic scattering
General Expression for d2σ/dΩdE
• Squires (eqn 2.59) derives the following expression:
• Note that, because of the operators and the average over the states of the system, this expression is not easy to evaluate in the general case
• Note also that the exponential operators do not commute – each contains H and therefore p, and p and R do not commute.
λλ
λ
πσ
λλ
ω
Ap
Heeee
tR
dteeebbk
k
dEd
d
iHtRQiiHttRQi
i
ii
titRQiRQiii
ii
ii
∑
∑ ∫
=
=
=Ω
−−−
∞
∞−
−−
A operator,any for i.e. --scatterer
theof , states, possible over the average a thermal denotes and
scatterer theof an Hamiltoni theis where
i.e.operator rga Heisenbe is )( where
'2
1'
'
/./)(.
',
)(.)0(.'
2'
hrr
hrr
rrrr
rh
Correlation Functions
• Note again that the operators do not commute. If we ignore this fact, we can do the integration and obtain
∑
∑∫
∫
∑∫
+−=
−+−=
=⎟⎟⎠
⎞⎜⎜⎝
⎛Ω
=
=
−
−−
','
','
22
).(
',
)('.)0('.'.3
)}0()({1
),(
')}('{)}0('{1
),(
thatshows 4.17) to4.14 (eqn Squires
atom of typeoneonly is thereprovided ),('
'
find wen the.),(2
1),( and
'1
)2(
1),( :define weSuppose '
jjjjclassical
jjjj
coh
coh
trQi
jj
tRQiRQirQi
RtRrN
trG
rdtRrrRrN
trG
QNSk
kb
dEd
d
dtrdetrGQS
QdeeeN
trG jj
rrrr
rrrrrrr
r
rr
h
r
rr
rr
rrrrrr
δ
δδ
ωσπ
ω
π
ω
Correlation Functions (cont’d)
• We expressed the coherent scattering cross section in terms of G(r,t)
• If we use the classical variant given above, there is a clear physical meaning – G(r,t) is the probability that if particle j’ is at the origin at time zero, particle j will be at position r at time t.
• We can do the same thing with the incoherent scattering and express it in terms of a self-correlation function whose classical version is
• This says that the incoherent scattering is related to the probability that if a particle is at the origin at time zero, the same particle will be at position r at time t.
In term of the pair correlation functions, one finds
dtrdetrGQSdtrdetrGQS
QNSk
kb
dEd
d
QNSk
kb
dEd
d
trQiss
trQi
sinc
inc
coh
coh
rr
h
rrr
h
r
r
r
rrrr
∫∫∫∫ −− ==
=⎟⎟⎠
⎞⎜⎜⎝
⎛Ω
=⎟⎟⎠
⎞⎜⎜⎝
⎛Ω
).().(
22
22
),(2
1),( and ),(
2
1),(
where
),('
.
),('
.
ωω
πω
πω
ωσ
ωσ
• Inelastic coherent scattering measures correlated motions of differentatoms
• Inelastic incoherent scattering measures self-correlations e.g. diffusion
(h/2π)Q & (h/2π)ω are the momentum & energy transferred to the neutron during thescattering process
Elastic Scattering as the tö¶ Limit of G(r,t)
• Elastic scattering occurs at ω = 0. Since it involves a δ(ω), only the part of G(r,t) which is constant contributes
• G(r,t) decays as t increases, so the constant part is G(r,¶)
• Since we only need the part of the correlation that is time-independent, we can write (noting that the correlation between the positions of j and j’are independent of t as t-> ¶)
any timeat operator density particle theis )'( where
')'()'(1
'}'{}'{1
),(
')}('{)}0('{1
),(
','
','
r
rdrrrN
rdRrrRrN
rG
rdtRrrRrN
trG
jjj
j
jjjj
v
vrrv
vrrrrrr
rrrrrrr
ρ
ρρ
δδ
δδ
+=
−+−=∞
−+−=
∫
∑∫
∑∫
G(r,¶) is called the Patterson function
Note – no trulyelastic scatteringfor a liquid
The Static Approximation
• In diffraction measurements, we measure scattered neutron intensity in a particular direction, independent of the change in neutron energy – i.e. we integrate the cross section over E = hω/2π. This is the Static Approximation.
the integral over ω picks out the t = 0 value of G to give
∑∫
∫−−
−
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛Ω
',
)(.)0(..3
).(22
'1
)2(
1),( where
.),(2
1'
'
:had Earlier we
jj
tRQiRQirQi
trQicoh
coh
jj eeeN
trG
dtrdetrGNk
kb
dEd
d
rrrrrr
rr
r
rr
h
π
πσ ω
)(2
1)( Because ω
πδ ω hh det ti∫
∞
∞−
=
∫
∫=
=⎟⎠⎞
⎜⎝⎛Ω
−
rderGNb
ddtrdetrGNbd
d
rQicoh
trQicoh
static
coh
rr
rr
rr
rr
.2
).(2
)0,(
..),( ωσ ω
Comparison of Elastic Scattering and the Static Approximation
• These are not the same, except in an (unreal) system with no motion
• The elastic scattering cross section gives the true elastic scattering that results when the positions of different atoms are correlated for all times, as they are in a crystalline solid, even when phonons arepresent
• The static approximation, as its name suggests, gives the scattering for a system that is frozen in time
),(
)0,(
.2
.2
∫
∫
∞=⎟⎠⎞
⎜⎝⎛Ω
=⎟⎠⎞
⎜⎝⎛Ω
rderGNbd
d
rderGNbd
d
rQicoh
elastic
coh
rQicoh
static
coh
rr
rr
rr
rr
σ
σ
The Intermediate Scattering Function
• Another function that is often useful is the Intermediate Scattering Function defined as
This is the quantity measured with Neutron Spin Echo (NSE)
• It is not possible to derive exact expressions for I, G or S except for simple models. It is therefore useful to know the various analytical properties of these functions to ensure that models preserve them. Squires shows:
• There are also various sum & moment rules on these quantities that are sometimes useful (see Squires for details)
∫= rdetrGtQI rQi rrr rr.),(),(
),(),( );,(*),(
/
)/,( ),( );,(*),(
/ ωωωω ω −−==
+==+−−=−=
QSeQSQSQS
T)ki,-trG(-,t)rG(,-t);rG*(-,t)rG(
TkitQItQItQItQI
Tk
B
B
Brrrr
hrrrr
hrrrr
h
0 .001 0 .01 0 .1 1 10 100
Q (Å-1
)
En
erg
y T
ran
sfe
r (m
eV
)
Ne utro ns in Co nde ns e d Matte r Re s e arc h
S pa lla tio n - Ch opp er ILL - w ithou t s p in -ec ho ILL - w ith s p in -e ch o
Crystal Fields
Spin Waves
Neutron Induced
ExcitationsHydrogen Modes
Molecu lar Vibrations
Lattice Vibrations
and Anharmonicity
Aggregate Motion
Polymers and
Biological Systems
Critical Scattering
Diffusive Modes
Slower Motions
Resolved
Molecular Reorientation
Tunneling Spectroscopy
Surface Effects?
[Larger Objects Resolved]
Proteins in Solution Viruses
Micelles PolymersMetallurgical
SystemsColloids
Membranes Proteins
C rystal and Magnetic
Structures
Amorphous Systems
Accurate Liquid Structures
P recision Crystallography Anharmonicity
1
100
10000
0.01
10-4
10-6
Momentum DistributionsElastic Scattering
Electron- phonon
Interactions
Itinerant Magnets
Molecular Motions
Coherent Modes in Glasses
and Liquids
SPSS - Chopper Spectrometer
Neutron scattering experiments measure the number of neutrons scattered at different values of the wavevector and energy transfered to the neutron, denoted Q and E. The phenomena probed depend on the values of Q and E accessed.
References
• Introduction to the Theory of Thermal Neutron Scatteringby G. L. SquiresReprint edition (February 1997)Dover Publications ISBN 048669447
• Neutron Scattering: A Primerby Roger PynnLos Alamos Science (1990)(see www.mrl.ucsb.edu/~pynn)
• Elements of Modern X-Ray Physicsby Jens Als-Nielsen and Des McMorrowJohn Wiley and Sons