first elements of thermal neutron scattering theory (i) daniele colognesi istituto dei sistemi...
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First Elements of Thermal Neutron Scattering Theory (I)
Daniele Colognesi
Istituto dei Sistemi Complessi,
Consiglio Nazionale delle Ricerche,
Sesto Fiorentino (FI) - Italy
Talk outlines
0) Introduction.
1) Neutron scattering from nuclei.
2) Time-correlation functions.
3) Inelastic scattering from crystals.
4) Inelastic scattering from fluids (intro).4) Inelastic scattering from fluids (intro).
5) Vibrational spectroscopy from molecules.5) Vibrational spectroscopy from molecules.
6) Incoherent inelastic scattering from 6) Incoherent inelastic scattering from molecular crystals.molecular crystals.
7) Some applications to soft matter.7) Some applications to soft matter.
0) Introduction
Why neutron scattering (NS) from condensed matter?
Nowadays NS is relevant in physics, material science, chemistry, geology, biology, engineering etc., being highly complementary to X-ray scattering.
E. Fermi C. G. Shull B. N. Brockhouse
What is special with NS?1) Neutrons interact with nuclei and not with their electrons (neglecting magnetism). Ideal for light elements, isotopic studies, similar-Z elements, and lattice dynamics.
2) Neutrons have simultaneously the right and E, matching the typical distance and energy scales of condensed matter.
4) The neutron has a magnetic moment, ideal for studying static and dynamic magnetic properties (not discussed in what follows ).
3) Weakly interacting with matter due to its neutrality, then: (a) small disturbance of the sample, so linear response theory always applies; (b) large penetration depth for bulky samples; (c) ideal for extreme condition studies; (d) little radiation damage.
Basic neutron properties
Mass: 1.67492729(28) 10−27 KgMean lifetime: 885.7(8) s (if free)
Electric charge: 0 eElectric dipole moment: <2.910−26 e·cm
Magnetic moment: −1.9130427(5) μN
Spin: 1/2
Neutron wave-mechanical properties
Interested only in slow neutrons (E<1 KeV), where:
E=mv2/2 (m=1.675·10-27 Kg) and
=h/(mv)Using the wave-vector (k=2/), one has:
E(meV)=81.81 (Å)-2=2.072 k(Å-1)2
=5.227 v(Km/s)2=0.08617 T(K)
The slow neutron “zoology”
(a version of)
Name Energy range (meV)
Very cold / Ultra-cold <0.5
Cold 0.5 - 5
Thermal 5 - 100
Hot 100 - 103
Epithermal / Resonant >103
1) Neutron scattering from nuclei
The neutron-Nucleus interaction
1) Short ranged (i.e. 10-15 m).
2) Intense (if compared to e.m.).
3) Spin-dependent.
4) Complicated (even containing non-central terms).
Example: D=p+n, toy model (e.g. rectangular potential well)
width: r0=2·10-15 m
depth: Vr=30 MeV
binding energy: Eb=2.23 MeV
Coulomb equivalent (p+p) energy: EC=0.7 MeV
The slow neutron-Nucleus system
Good news: if nr0 (always true for slow neutrons) and dr0 (d: size of the nuclear delocalization) we do not need to know the detail of the n-N potential for describing the n-N system! Two quantities (r0 and the so-called scattering length, a) are enough.
)wave()(1),(
0)(for),(),(),()(22
N0nN
nNnNnN0nNNN
2N
2
n
2n
2
sr
a
rEHUmm
r rrr
rrrrrrrrr
Localized isotropic impact model
The Schroedinger equation plus the boundary condition are exactly equivalent to:
potential]-pseudo[Fermi
)(2
)(2
)( :where
),(lim)(),(),(
Nnn
2
Nn
2
Nn
nN0
NnnNnN0
rrrrrr
rrrrrrrr
bm
aV
rr
VEHr
Tough equation… But it can be expanded
in power series of V: =0+1+2+… (if
na and da), where:
1k0
kk0
000
lim
0
rr
VEH
EH
r
The Fermi approximation is identical to the well-known first Born approximation:
00000
110 lim
VVrr
VEHrr
QM text-book solution:
where a spherical wave, modulated by the inelastic scattering amplitude f(k’,F;k,0) has been introduced:
state)ion(perturbat)()0,;,'(
exp
8
1
state)ed(unperturb)(exp8
1
Nucleus
NF
neutron
n
n
31
Nucleus
N0
neutron
n30
rkk
rrk
Ffr
rk'i
i
and the following energy conservation balance and useful definitions apply:
transfer)(momentum'
:defines oney analogousl
transfer)(energy2
'
22
'
2 0Fn
22
n
22
Fn
22
0n
22
kkQ
kkkk
EEmm
Em
Em
0F0*F
N0NnN*FnNn2
n
)exp()()('exp
)(),()('exp2
4
1)0,;,'(
RQRRRkkR
rrrrrkkrrkk
ibidb
Viddm
Ff
Slow neutron scattering from a nucleus
(k, E)
(k’, E’)
E0EF
Measurable quantity: number of scattered neutrons, n detected in the time interval t, in the solid angle between and +, and between and +, having an energy ranging between E’ and E’+ E’:
tEEIn ')',,(
Qk-k’ E-E’=EF-E0
Scattering problem: how is I(,,E’) related to the intrinsic target properties [i.e. to i(RN)]?
The concept of double differential scattering cross-section (d2/d/dE’) has to be introduced:
where Jin is the current density of incoming neutrons (i.e. neutrons per m2 per s), all exhibiting energy E. Analogously, for the outgoing neutrons, one could write: Jout(,,E’)= r
-2 I(,,E’) (spectral density current).
in
out
in J
EJr
J
EI
dEd
d )',,()',,(
'
2
E'E,φ,θ,
2
Going back to our QM text book, one finds the “recipe” for the neutron density current:
*
n
Imm
J
0F0
2
0F2
23
2
23
3
')(),'(),()(
)exp('
)0,;,'('
)(
EEEEΩJΩEJΩJdΩJ
ibr
k
mLFf
r
k
mLΩJ
kmL
J
outoutoutout
nnout
nin
RQkk
which, applied to 0 and 1 (box-normalized, L3), gives:
and finally, the neutron scattering fundamental equation:
0F
2
0F2
F02
')exp('
'EE,φ,θ,
EEEEibk
k'
dEd
d
RQ
for the transition from the nuclear ground state 0 to the excited state F, with the constraint: E-E’=EF-E0.
Summing over all the possible nuclear excited states F, one has to explicitly add the energy conservation:
0FF
2
0F2
2
)exp('
'EE,φ,θ,
EEibk
k'
dEd
d
RQ
Finally, if the target is not at T=0, one should also consider a statistical average (pI) over the initial nuclear states, I:
onlypropertyTarget
12
IIF
F
2
IFI2
2
),(
)exp('
'EE,φ,θ,
Q
RQ
SbE
E'
EEipbk
k'
dEd
d
where the inelastic structure factor or scattering law S(Q,) has been defined. Giving up to the neutron final energy (E’) selection, one writes the single differential s. cross-section:
I F
2
IFIF
I2
IF
2
0
)exp(
)for('
''EE,φ,θ,Eφ,θ,
RQiE
EEEpb
EEEdEd
ddE
d
d
Neutron scattering from an extended system
So, is everything so easy in NS? No, not quite…
Giving up to the selection of the scattered neutron direction () too, one writes the s. cross-section:
s.)c.(freefor4
s.)c.(bound0for4)( 2
2n
2
2
Eφ,θ,
Eb
m
Eb
d
ddE
potential) like-(comb)(2
)(2
states)body(many),...,,()(
jn
N
1jj
n
2
Nnn
2
N21FI,NFI,
Rrrr
RRRr
bm
bm
Neutrons and incoherence
“Real-life” neutron scattering (from a set of nuclei with the same Z)
INC
2
COH
22
'''
dEd
d
dEd
d
dEd
d
where: ),(4
'
'1COH
COH
2
QS
E
E
dEd
d
),(4
'
' self1INC
INC
2
QS
E
E
dEd
d
scattering laws defined for a many-body
system (set of nuclei with the same Z) as:
IIF
F
N
1j
2
IjFIself
IIF
F
2
I
N
1jjFI
)exp(),(
)exp(),(
EEipN
S
EEipN
S
RQQ
RQQ
S(Q,) is obvious, but where does Sself(Q,) come from? From the spins of neutron (sn,mn) and nucleus (IN,MN), so far neglected! b depends on IN+sn
- e.g. full quantum state for a neutron-nucleus pair:|k’,sn,mn; F, IN, MN
How does it work? Assuming randomly distributed neutron and nuclear spins, one can have a simple idea of the phenomenon:
2
spinj'spinjspinj'j
2
spinjjspinj'j
positionspin
N
1j'j'j'
positionspin
N
1jjj
2
positionspin
N
1jjj
:for
:for
)'exp()exp(
)exp(
bbbbbj'j
bbbbbj'j
ibib
ib
RQRQ
RQ
The incoherence origin (rigorous theory):
22
INC
2
COH2
TOT
ˆˆ4
ˆ4;ˆ4
bb
bb
since b is actually not simply a number, but is the scattering length operator acting on |iN, mN (nucleus) and on |sn, mn (neutron) spin states:
Nˆˆ
2ˆ iσ
BAb
This implies the existence of b+ and b- (if iN>0) for any isotope (N, Z), respectively for iN½:
bibii
A NNN
)1(12
1
bbi
B12
2
N
)1(4
,;,ˆ,;,)12(2
1
;,;,ˆ,;,)12(2
1
NN
22
m,MnnNN
2nnNN
N
m,MnnNNnnNN
N
nN
nN
iiB
AmsMibmsMii
AmsMibmsMii
After some algebra, only for unpolarized neutrons and nuclei, one can write:
j jN,
2jjN,
2jjN,
j2
j jN,
jjN,jjN,j
12
)())(1(ˆ
12
)1(ˆ
i
bibicb
i
bibicb
Important case: hydrogen
(protium H, iN=1/2): b+=10.85 fm, b-=-47.50 fm
TOT=82.03 b, COH=1.7583 b
(deuterium D, iN=1): b+=9.53 fm, b-=0.98 fm
TOT=7.64 b, COH=5.592 b
Deuterium
Hydrogen
TOT:
With various isotopes (cj ) one gets:
High-Q spatial incoherence
DIS
2
INC
2
INC
TOT2
INC
2
INC
COH
COH
2
DIS
2
'''
'''
dEd
d
dEd
d
dEd
d
dEd
d
dEd
d
dEd
d
(rearranging…)
0),(),(),(
0for''
distself
INC
2
INC
TOT2
QQQ SSS
dEd
d
dEd
d
Incoherent approximation
When does it apply in a crystal?When does it apply in a crystal?
2
2/12
2d
uQ
Practical example : D2SO4 (T=10 K)
d(DO)=0.091 nmu21/2(D)=0.0158 nm2u21/2/d2=11.9 nm-1
|Qinc|100 nm-1
2) Time-correlation functions
S(Q,) and Sself(Q,) are probe independent, i.e. they are intrinsic sample properties. But what do they mean?
Fourier-transforming the two spectral functions, one defines I(Q,t) and Iself(Q,t), the so-called intermediate scattering function and self intermediate scattering function:
),(exp),(
),(exp),(
selfself
StidtI
StidtI
After some algebra (e.g. the Heisenberg representation), one writes I(Q,t) and Iself(Q,t) as time-correlation functions (with a clearer physical meaning):
),(),(),(
)(iexp)0(iexp1
),(
selfdist
jjjself
tItItI
tN
tI
QQQ
RQRQQ
kj,
kj )(iexp)0(iexp1
),( tN
tI RQRQQ
So far we have dealt only with a pure monatomic system (set of nuclei with the same Z).
s
tIbbsctdtk
k
dEd
d
tIbbzcsctdtk
k
dEd
d
),(][iexp2
1'
'
),(][][iexp2
1'
'
(s)self
2s
2s
INC
2
s z
z)(s,zs
COH
2
Q
Q
Sum over ”s” distinct species (concentration c[s]):
sN
1j
sj
sj
s
(s)self )(iexp)0(iexp
1),( t
NtI RQRQQ
But what about “real-life” samples (e.g. chemical compounds)?
where I(s)self(Q,t) is the so-called self intermediate
scattering function for the sth species:
),(exp2
1),( (s)
self(s)self tItidtS QQ
and where S(s)self(Q,) is the so-called self
inelastic structure factor for the sth species. Properties similar to those of Sself(Q,):
The coherent part is slightly more complex
I(s,z)(Q,t) is the so-called total intermediate scattering function for the sth species (if sz), or the cross intermediate scattering function for the s,zth pair of species (if sz):
s zN
j
zk
N
k
sj
zs
z)(s, )(iexp)0(iexp),( tNN
NtI RQRQQ
and S(s,z) (Q,) is the so-called total inelastic
structure factor for the sth species (if sz), or the cross inelastic structure factor for the s,zth pair of species (if sz):
),(exp2
1),( z)(s,z)(s, tItidtS QQ
The total contains a “distinct” plus a “self” terms, while the cross only a “distinct” term.
Coherent sum rules
factorstructureStatic)(),()0th
QQ SSd
EnergyRecoil2
),()1 R
22st
EM
SdQ
Q
I
I
N
jk,jkII )exp()exp(
1)( RQRQQ iip
NS
where:
0
n
nnn ),()(),(:from
t
tIt
iSd QQ
ionNormalizat1),()0 selfth
QSd
EnergyRecoil2
),()1 R
22
selfst
EM
SdQ
Q
2Nk
22self
2R
nd
2 :energy Kinetic
),()2
v
QvQ
ME
SEd
Incoherent sum rules
0
selfn
nn
selfn ),()(),(:from
t
tIt
iSd QQ
U
UQQM
SEd
potentialtheofLaplacian
2),()3
ji,jjii2
4
self3
Rrd
Q
UUM
SEd
potentialtheofgradientSquare
),()4
2
2
4
44self
4R
th
Q
QvQ
Detailed balance
),(exp),(
),(exp),(
selfself
SωS
SωS
Tk
Tk
B
B
from the microscopic reversibility principle:
q.e.d.exp
exp
exp
nm
2
nm,1
n
mn
mn
2
mn,1m
nm
2
nm,1n
EEnimp
pp
EEminp
EEnimp
RQ
RQ
RQ
Analogous proof for the scattering law
The zoo of excitations…
…and their (|Q|-E) relationships
3) Inelastic scattering
from crystals
Scattering law from a many-body system: analytically solved only in few cases (e.g. ideal gas, Brownian motion, and regular crystalline structures, with a purely harmonic dynamics.
n'n,n'nn'n
m
2m
)(iexp)0(iexp
exp2)(
1),(
tbb
tidt
b
RQRQ
Q
Generalized scattering law (sometimes used for mixed systems)
nnnnINC,
mmINC,
self
)(iexp)0(iexp
exp2
1),(
t
tidt
RQRQ
Q
Generalized self scattering law (sometimes used for mixed systems)
In a harmonic crystal the time correlation functions are exactly solvable in terms of phonons due to the Bloch theorem for a 1D harmonic oscillator (X is its adimensional coordinate):
221expexp XX
Three-dimensional crystalline lattice (N cells and r atoms in the elementary cell: “l” and “d” indexes):
)()( dl,mequilibriu
l.d tt udlR
Harmonicity (expansion of ul,d(t) in normal modes; e.g. phonon “s”; quantized):
tiia
tiiaNM
t
ssds,
3Nr
1sssds,
2/1s
ddl,
exp
exp2
)(
lqe
lqeu
with es,d polarization versor, a†s (as) creation
(annihilation) operator of the sth phonon with s frequency and 2|q|-1 wavelength.
Collective index “s”: {qx,qy,qz,j} with q1BZ (first Brillouin zone: N points). “j” labels the phonon branches (3 acoustic e 3r-3 optic).
Polarizations: 2 transverse and 1 longitudinal.
Total: 3Nr d.o.f.
Dispersion curves: s=j(q)
Coherent scattering
Plugging the equations for Rl,d(t) and ul,d(t) (i.e. phonon quantization) into the coherent d. d. cross-section, one gets:
)(exp)0(expexp2
''exp'
'
d',l'd0,
d d',l'd'd
COH
2
tiitidt
ibbNk
k
dEd
d
uQuQ
lddQ
and using the Bloch theorem together with the commutation rules: eAeB=eA+Be[A,B]/2, one writes:
)()0(exp
)()0(2
1exp
)(exp)0(exp
d',l'd0,
2d'0,
2d0,
d',l'd0,
t
t
tii
uQuQ
uQuQ
uQuQ
),(expexp2
)/20,()/20,(exp
''exp'
'
d',l'd,d'd
d d',l'd'd
COH
2
tBtidt
BB
ibbNk
k
dEd
d
QQQ
lddQ
and then:
)(2122
)0,( ds
ss
2
sd,
dd0, Q
eQQ Wn
NMB
where the static term is the Debye-Waller factor, whose exponent is:
with ns number of thermally activated phonons; while the dynamic term contains:
)]''(exp[)]''(exp[
1))((
2),(
sssss
ss
s
*s,d'sd,
d'd
d',l'd,
lddqlddq
eQeQQ
itiniti
nMMN
tB
Coherent elastic scattering
Phonon expansion
Expanding exp[Bd,d’,l’(Q,t)] in power series, one gets a sum of terms with n phonons (created or annihilated):
...),(!
1
...),(2
1),(1),(exp
nl',d'd,
2l',d'd,l',d'd,l',d'd,
tBn
tBtBtB
Q
QQQ
one gets for the first term, 1, an elastic contribution:
)/20,()/20,(exp
''exp'
d'd
d d',l'd'd
1
elastCOH,
2
lddQ
BB
ibbNdEd
d
Integrating over E’ and making use of the reciprocal lattice () sum rule: l exp(iQl) = 83vcell
-1 (Q-), one obtains the well-known Bragg law:
τ
Q
τQ
QQddQ
)peaksBragg()(
)()(exp'exp8
factorstructure cell-unitnuclear:)(
d'dd'd,
d'dcell
3
elastCOH,
2n
F
WWibbNvd
d
Neutron powder diffraction pattern from Li2NH (plus Al container) at room temperature. Abscissa: d=2Q -1
One-phonon coherent contribution
If ER,d(Q)</s-1 [the lightest Md] then:
),(1),(exp l',d'd,l',d'd, tBtB QQ
and one obtains the single phonon (created or annihilated) d. d. coherent cross section:
),(exp2
)()(exp
''exp'
'
d',l'd,d'd
d d',l'd'd
1COH,
2
tBtidt
WW
ibbNk
k
dEd
d
QQQ
lddQ
Plugging the equation for Bd,d’,l’(Q,t) into the one-phonon coherent d. d. cross-section and performing the Fourier transforms and the reciprocal lattice sums, one gets:
)0creation ),( phonon,1(1
)(exp2
8'
'
jsss
2
dsd,d
ds
1s
cell
3
1COH,
2
qτqQ
eQQdQ
τ
--n
WiM
b
Nvk
k
dEd
d d
)0onannihilati),(phonon,1(
)(exp2
8'
'
jsss
2
dsd,d
ds
1s
cell
3
1COH,
2
qτqQ
eQQdQ
τ
-n
WiM
b
Nvk
k
dEd
d d
DispersionCurvepractical example: lithium hydride LiD (cubic, Fm3m, i.e. NaCl type) with r=2 j=1,2,3
Red: Acoustic
Blue: Optic
Full: Transverse
Dash: Longitudinal
First Brillouin zonein a face-centered cubic lattice
(f.c.c.)
face-centered cubic lattice
1st Brillouin zone f.c.c.
One-phonon incoherent contribution
),(exp)0,(exp
)(iexp)0(iexp
dd
d(d)INC
dl,dl,dl,
(d)INC
tBBN
t
RQRQ
Bloch theorem:
where the static term is the Debye-Waller factor, whose exponent is:
s
ss
2
sd,
dd 12
2)0,( n
NMB
eQ
Q
tin
tinNM
tB
ss
sss
s
2
sd,
dd
exp
exp12
),(
eQ
Q
with ns number of thermally activated phonons; while the dynamic term contains:
Single phonon and density of states
Expanding exp[Bd(Q,t)] in power series, one gets a sum of terms with n phonons (created or annihilated):
...),(!
1
...),(2
1),(1),(exp
nd
2ddd
tBn
tBtBtB
Q
QQQ
if ER,d(Q)</s-1 then:
),(1),(exp dd tBtB QQ
and one obtains the single phonon (creation or annihilation) d. d. incoherent cross section:
)0,(exp1
2
1
4
'
'
dssss
s s
2
sd,
dd
(d)INC
1INC,
2
Q
eQ
Bnn
NMk
k
dEd
d
Density of (phonon) states: density probability for a phonon of any kind with frequency between and +d:
3rN
ss )(
3
1)(
rNg
)(2exp]})(exp[1{
)()(
24
'
'
d1
2
d
d
2
d
(d)INC
1INC,
2
Qe
WTk
gr
M
Q
k
k
dEd
d
B
The single-phonon incoherent d. d. cross section (creation or annihilation) becomes more simply:
where:
(creat.)0
(annih.)0for
1)(
)()(exp1
11B
forn
nTk
weak point (“”), i.e. the meaning of the averaged eigenvector:
Δω
2sd,
2
d
s)(3
1)( ee
rNg
The separation from Q is rigorous only in cubic lattices:
2
d
2
s
2sd, )(
3eeQ
Q
otherwise one has the isotropic approximation.
In addition, using this approximation, one proves that:
2d
2
dd 3)(2)0,( uQQ
QWB
link between the exponent of the Debye-Waller factor
and the mean square displacement of the d species.
It is often used the density of states projected on d:
dTk
G
M
grG
B0
d
d
2d
2
dd
2coth
)(
2
3
)()()(
u
e
2d
2
1B
d
d
2
d
(d)INC
1INC,
2
3exp
]})(exp[1{
)(
24
'
'
uQ
Tk
G
M
Q
k
k
dEd
d
from which:
70 80 90 100 110 120 130 140 150
0.00
0.01
0.02
0.03
0.04
0.05
LiH, Optic
ZH()
(m
eV
-1)
(meV)
DDM13, (Dyck & Jex, 1981) SM-IV, (Verble, Warren & Yarnell, 1968) TOSCA-II (Colognesi et al., 2002)
Density of states projected on Hpractical example: lithium hydride LiH
—————— transverse—————— longitudinal
Multiphonon incoherent contributions
1INC,
2
INC
2
MultINC,
2
'''
dEd
d
dEd
d
dEd
d
Coherent multiphonon terms are too complex and not very useful (e.g. for powders: Bredov approximation). Here only incoherent terms. Definition:
ddd
(d)INC
INC
2
),(exp)0,(exp4
exp2
'
'
tBB
tidt
k
k
dEd
d
remembering that:
...),(!
1
...),(2
1),(1),(exp
nd
2ddd
tBn
tBtBtB
Q
QQQ
and that:
one gets for the first term, 1, an elastic contribution:
dd
(d)INC
dd
(d)INC
ElasINC,
2
)(2exp4
')(2exp
4
exp2
'
'
QQ Wk
kW
tidt
k
k
dEd
d
not to be confused with the incoherent s. d. cross-section:
''0 INC
2
INC
dEdEd
d
d
d
For the second term one gets, B(Q,t), the single phonon contribution (1, created or annihilated) already known:
)(2exp]})(exp[1{
)(
24
'
'
d1B
d
d
2
d
)(
1INC,
2
QWTk
G
M
Q
k
k
dEd
d dINC
While for the (n+1)th term, one gets Bn(Q,t), a contribution with n phonons (created and/or annihilated). Using the convolution theorem:
n
d
timesn
dddnd
0
),(~
),(~
...),(~
),(~
),(exp2
QB
QBQBQBtQBtidt
where::
)(2)]}/(exp[1{
)(
2),(
~d
d
2
B
d
d
2
d
fM
Q
Tk
G
M
QQB
we obtain:
)(2exp!
)(
24
'
'
d
nd
n
d
2
d
(d)INC
nINC,
2
QWn
f
M
Q
k
k
dEd
d
Self-convolution shifts and broadens fd(), but blurs its details too…
Sjölander approximation: [fd()]n is replaced by an appropriate Gaussian (same mean and variance ):
2ddk2
dd
d
1d2
dd
d
2dd
d
d
2
;2
3
;3
2
:)(for
mEM
v
AM
m
MA
f
u
u
u
d
2d
n
dd
nd 2
exp1
2
1)(
nv
nm
mnvf
Then we have:
Properties
of fd()
0 20 40 60 80 100 120 140 160 1800.0
0.5
1.0
1.5
2.0
2.5
3.0
self(
Q,
) (a
.u.)
E (meV)
Not always appropriate…
-D-glucose at T=19 K, example from TOSCA