first elements of thermal neutron scattering theory (ii) daniele colognesi istituto dei sistemi...
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First Elements of Thermal Neutron Scattering Theory (II)
Daniele Colognesi
Istituto dei Sistemi Complessi,
Consiglio Nazionale delle Ricerche,
Sesto Fiorentino (FI) - Italy
Talk outlines
0) Introduction.0) Introduction.
1) Neutron scattering from nuclei. 1) Neutron scattering from nuclei.
2) Time-correlation functions.2) Time-correlation functions.
3) Inelastic scattering from crystals.3) Inelastic scattering from crystals.
4) Inelastic scattering from fluids (intro).
5) Vibrational spectroscopy from molecules.
6) Incoherent inelastic scattering from molecular crystals.
7) Some applications to soft matter.
4) Inelastic scattering
from fluids (intro)
Disordered systems (gasses, liquids, glasses, amorphous solids etc.): atomic order only at short range (if existing). For simplicity’s sake only monatomic fluid systems are considered here.
key quantities: density, , constant, and pair
correlation function, g(r)
ji
ji )(1
)( rrr N
rg
connected to the static structure factor, S(Q), via
a 3D spatial Fourier transform:
)exp()1)((1
)(exp1
)(ji
ji
rQr
rrQ
irgd
iN
QS
where both S(Q) and g(r) exhibit some special
values at their extremes:
1)(;0)0(
1)(;)0( BT
gg
STkS
Since S(Q)=I(Q,t=0), it is possible to generalize
g(r) by introducing the time-dependent pair
correlation function, G(r,t):
srrsrs
rQQ
dtN
tQIidtrG
ji,ji
3
)()0(1
),()exp(8
1),(
and the time-dependent self pair
correlation function, Gself(r,t):
srrsrs
rQQ
dtN
tQIidtrG
iii
self3self
)()0(1
),()exp(8
1),(
where the t=0 values of G(r,t) and Gself(r,t) are:
)()0,(
)()()0,(
self r
r
rG
rgrG
No elastic scattering,(), in fluids!
the elastic components in S(Q,) and Sself(Q,)
come from the asymptotic values of I(Q,t) and Iself(Q,t):
)exp(),(),(),(2
)(),(),()(),(
tfor0gapproachinasymptotic
inelasticelastic
rQr
itirGtrGrGdtd
QIQSQI
Due to the asymptotic loss of time correlation, and making use of =i(r-ri), one writes:
01
)()0(1
),(
;1
)()0(1
),(
iiiself
2
ji,ji
Ndt
NrG
VN
dtN
rG
srrsrs
srrsrs
so, finally:
0),(
)0:scatteringreal(no)(8),(
self
o3
QI
QI Q
Gas of non-interacting distinguishable particles: a useful “toy model”. No particle correlation: S(Q,)Sself(Q,). Starting from the definitions:
V
i
M
kE
EEnimpNQS
iiN1i
N
1i
2i
2
n
nm
2
nm,
N
1iin
1self
expn;
2with
exp),(
rk
rQ
one writes:
M
k
M
k
iV
dd
TMkd
VQS
TMk
k
22
'
'exp'2
e
8),(
2222
2/32B
2
3self
B
22
rkkQ
rkk
2
B2
2
B12
222
B
22
2/32B
self
2
2exp
2
1
22exp
2),(
MTkQ
MQ
TkMQ
MM
Q
TMk
k
TkM
dQS
Qkk
After some simple algebra:
Very important for epithermal neutron scattering!
recoil
Doppler broadening
Coherent inelastic scattering from liquids a.k.a. “Neutron Brillouin Scattering”: the acoustic phonons become pseudo-phonons (damped, dispersed). A new undispersed excitation appears too. Very complex, not discussed here.
SQD
QDQS
pQQc
QQS
Q
42Qs,
2
2Qs,
42Q
2Q
2Q
r
)1)((0
)(2
)())(()-exp(
:large) not too
for andlimit ichydrodynam 0 (in thetripletBrillouin
qτqQ
Liquid Al g(r)
Liquid Ni S(Q)
Incoherent inelastic scattering from liquids: the elastic component becomes quasi-elastic (diffusive motions), not discussed here in great detail.
On the contrary, the inelastic component is not too dissimilar from the crystal case (pseudo-phononic excitations).
422
21)(
:bigtoonotandfor
QD
DQ
Qω
N
1nnn
INC
INC
2
)(iexp)0(iexp
exp2
'
4'
t
tidt
k
k
NdEd
d
RQRQ
Starting from the well-known:
it is possible to show (Rahman, 1962) that:
)(expexp2
'
4
)()(expexp2
'
4'
12INC
1LL
2LLINC
INC
2
tQtidt
k
k
tQtidt
k
k
dEd
d
where we made use of the Gaussian approximation in Q. The t-dependent factor has apparently a tough aspect:
d
titTk
fM
tB
0
liqn
1 sincos12
coth2
)(
but it is actually equal to Q-2[B(Q,0)-B(Q,t)].
Then fliq() has to be analogous to g() in solids… Surprising!
Let’s study it, starting from the velocity self-correlation function of an atom in a crystal: cvv(t).
Expanding in normal modes through the Bloch theorem, one gets (in the isotropic case):
0t
2
2
2
0n
z1,z1,
N
1nnnvv
),(3
sincos2
coth2
3
)()0(3)()0(1
)(
dt
tQBd
Q
dtitTk
gM
tvvtN
tc
B
vv
It applies to fliq() too. Using the fluctuation-dissipation theorem, linking Re[cvv(t)] with Im[cvv(t)], one writes:
dtttcTkM
f
0
vv
1Bd
liq cos)(Re3
)2(tanh4)(
However, there is a property distinguishing fliq() from g():
Tk
DMfttRDtcdt
B
dliq
0
12tvv
2)0()(lim33)(Re
where D is the self-diffusion coefficient, while g(0)=0.
-10 0 10 20 30 40 50 60
0.00
0.01
0.02
0.03
0 5 10 15 20 25 30
0.00
0.03
0.06
0.09
from CMD, T=14.7 via Gaussian approx.
exp., T=14.3 K
Sse
lf (Q
,E)
(meV
-1)
E (meV)
from CMD, T=14.7
E (meV)
f liq(E
) (m
eV-1)
Example: liquid para-hydrogen, measured on
TOSCA at T=14.3 K (Celli et al. 2002) and simulated through Centroid Monte Carlo Dynamics (Kinugawa, 1998).
5) Vibrational spectroscopy
from molecules
chemical-physical spectroscopy: studying the forces that:
- bind the atoms in a molecule [covalent bond: E400 KJ/mol].
- keep the functional groups close to one another
[hydrogen bond: E20 KJ/mol].
- place the molecules according to a certain order in a crystalline lattice [molec. crystals: E2 KJ/mol].
Wide range of energies! Here only intra-molecular modes (vibrational spectroscopy).
Cross-section summary
22
inc
2
coh2
tot
ˆˆ4
ˆ4;ˆ4
bb
bb
H case (ideal incoherent scatterer):
inc=80.27 b, coh=1.76 b
Proton selection rule
D case (quite different):
inc=2.05 b, coh=5.59 b
Then only incoherent scattering will be considered in the rest of this talk!
Comparing various spectroscopies
(neutron)10-28 m2/molec.
(Raman)10-32 m2/molec.
(IR)10-22 m2/molec.
Why neutron spectroscopy ?
1. In Raman polarizability generally grows along with Z: possible problems in detecting H.
2. In IR (sensitive to the electric dipole) the H-bond gives rise to a large signal, but it is distorted by the so-called electric anharmonicity (not vibrational).
3. Molecules with elevate symmetry: many modes are optically inactive (e.g. in C60 up to 70%!).
4. Direct relationship between neutron spectra
and vibrational eigenvectors.
Conclusions
Neutron spectroscopy is complementary to optical spectroscopies (Raman and IR) and is often essential for studying proton dynamics!
Example: nadic anhydride (C9H8O3) on TOSCA
Molecular vibrations and normal modesPolyatomic Molecules: N atoms instantaneously in the positions {rα}, vibrating around their equilibrium positions {rα0}:
rα= rα0+uα
Normal modes
3 traslations
3 rotations (2 if linear)
3N-6 vibrations (3N-5 if linear)
Translations elimination (center-of-mass fixed):
αmαrα= αmαrα0 =R αmαuα=0
Rotations elimination (small oscillations):
αmαrαvα= J=0 αmα rα0tuα
αmα rα0uα=cost.0
The normal modes of a molecule can be classified according to the character of the atomic motions, starting from the symmetry of the equilibrium configuration of the molecule (group theory).
General Theory
of normal modes
with s d.o.f. qi:
ui=qi-qi0
approx.)(harmonic2
1
(rigorous)2
1
s
ji,jiij
s
ji,jiij
uuKU
uuMT
0s
jjijjij uKuM One gets s Lagrange
equations:
Oscillating test solutions:
Characteristic equation :
(in general one has s real and positive roots: 1,… s)
Eigenvectors aj
(s):
0ij2
ij MK
0(s)j
s
jij
2(s)ij aMK
)exp(jj tiau
General solution:General solution:
0
:bygiven is
mode normalth - thewhere
)(
)exp(
)(2)()(
)(
)(
α
)(j
1/2j
)()(
α
)(jj
Q
Qam
tiCau
Example: normal modes in H2O
a. Symmetric stretching
b. Bending
c. Anti-symmetric stretching
Normal mode quantization
α
f
1i
)(i
)(
α
f
1i
2)(i
2)(2)(
i
22
i
2)(i
α
2)(
αi,
2)(icl
α
α
)2/1(
2
1ˆ
2
1
2
1
n
H
QQH
)(0
2)(i
)(i)(
2)(i 22
1
quantized:amplitudesquareMean
QnQ
Diffusion from a harmonic oscillator
The mono-dimensional harmonic oscillator
is then the simplified prototype of the true
intra-molecular vibrations:
~1000 cm-1 <0<4400 cm-1 (H-H):
)()sinh(2
coth2
expexp),(
002
10
2
0nn
021
0
2
21
nEQ
I
QEEQS
Typical experiment : T=20 K (i.e. 14 cm-1<< 0) then:
,2!
1)(1
)sinh(2
1)sinh(,1)coth(70n
n02
10
2
021
021
021
x
nxI
Q
from which:
),(
!exp),( 0
0n
n
0
22
0
22 nEn
uQuQEQS
where u20 is the mean square displacement (at T=0).
.)cm()amu(
16759.0
2)nm(
10
2
0
2
u
Again on the harmonic oscillatorMass problem: what is μ in a molecule? It depends on all the atomic masses, but MH obviously plays a primary role! However, in general, μMH .
Elastic Line: there is no exchange of energy between oscillator and neutron, then n=0. It is intense, but it decreases rapidly with Q. Then it will be neglected:
Fundamental: for n=1 there is a peak centered
at 0, while in Q one gets a competition between the Debye-Waller factor and the term Q2u20 :
)(]exp[),( 00
22
0
221n EuQuQEQS
)(]exp[),(0
22el EuQEQS
The maximum of Sn=1(Q,E) appears at Q2=u20. So, the ideal measurement conditions for H are:
k1<<k0 k0Q for any value of E. Namely:
1)cm()amu(
16759.0
16759.0
)amu()cm(21
n1
0
2212
n21
m
ukEm
k
Overtones: excitations from the ground state (n=0) to states higher than the first (i.e. n=2,3…):
).(
!exp),( 0
n
0
22
0
221n nE
n
uQuQEQS
,2
,2
00
22
n221
uEm
Qk
considering that:
one obtains:).(
!
1exp
1),( 0
nn
1n
nEn
nEQS
The relative intensity of the overtones (with respect to n=1) quickly decreases along with μ. It is important to separate the high-frequency fundamental excitations from the overtones.
0 1000 2000 3000 4000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
n=3
n=2
n=1
S(Q
,E)
(arb
. un
its)
E (cm-1)
ZrH2, T=20 K
from J. Tomkinson (ISIS)
Example: fundamental and overtones in ZrH2, almost a harmonic oscillator (three-dimensional).
AnharmonicyIdeal vibrational model: set of decoupled harmonic oscillators (normal modes).
Anharmonicity: breaking of the harmonic approximation, implying inseparability and mixing of normal modes.
In practice overtones are not simple multiples of the fundamental frequency any more, i.e. there is an anharmonicity constant, . One often has that >0 (e.g. in the Morse potential).
)10(,0e)0(,0 )1(1 nnnn
In practice, in real molecules one uses a pseudo-harmonic approach in which the structure factor for a single atomic species is approximated by:
)(
!exp),(
(k)n0
0n
n
ijj
(k)
ij
2i
kij
j
(tot)
ij
2
E
n
QuQ
QuQES iQ
where n labels the sum over the overtones and k the multi-convolution in E over the normal modes, from which:
k
(k)
ij
2(tot)
ij
2 uu
6) Incoherent inelastic scattering
from molecular crystals
External molecular modes
So far only isolated molecules have been dealt with, having a fixed center-of-mass (no recoil). In reality, at low temperature, one observes molecular crystals kept together by inter-molecular interations: weak (van der Waals), medium (H bond), or strong (covalent).
External modes (pk, lattice vibrations and undistorted librations): in general (but not always…) softer than the internal ones (e.g. lattice v. ~150 cm-1).
),(
!exp),(
k
0n
n
ijj
(k)
ij
2i
kij
j
(tot)
ij
2iext
pnE
n
QUQ
QUQES
Q
Similarly to what seen for the internal modes, an Similarly to what seen for the internal modes, an external structure external structure factorfactor for the molecular lattice can for the molecular lattice can be defined:be defined:
making implicitly use of the decoupling hypothesis between internal and external modes:
),(),(),( intext ESESES QQQ
using the distributive property of the convolution
one gets:
),'(!!'
expexp),(
kk'
n
ijj
(k)
ij
2i
n'n,
n'
ijj
)(k'
ij
2i
k'k,
ijj
(tot)
ij
2i
ijj
(tot)
ij
2i
npnEn
QuQ
n
QUQ
QuQQUQES
Q
then for each internal mode k there is also a shifted replica of all the external spectrum {pk’} (phononic branch), but with a strong intensity reduction due to the external Debye-Waller factor:
),(exp),(
),(),(),(
intij
j
(tot)
ij
2iorig
brancorig
ESQUQES
ESESES
QQQ
At low Q, Sorig(Q,E) is intense and Sbran(Q,E) has a shape similar to that of Sext(Q,E) (but translated).At high Q, Sbran(Q,E) is dominated by the multiphonon terms (difficult to be simulated).
Comparison to the mean square displacements worked out by diffraction:
(tot)
ij
2(tot)
ij
2
2iso
(tot)
ij
2(tot)
ij
2
:peaks)Braggof
intensitytheng(controlli83
1
3
1
uTrUTr
BUTruTr
Discrepancies between Biso and the inelastic mean square displacements: static disorder
0 200 400 600 8000
1
2
3
4
5 C
6H
12N
4, T=20 K
from J. Tomkinson (ISIS)
Pure vibration/libration||
Phonon wing||||
<--- Lattice modes
S(Q
,E)
(arb
. uni
ts)
E (cm-1)
Example: hexamethylenetetramine (C6H12N4) on TOSCA
Anisotropy and spherical mean
We have seen that, owing to the presence of various normal modes, scattering depends on the orientation of Q with respect to the molecule (anisotropy).
Toy-model: 1-D harmonic oscillators with frequency
x , all oriented along the x axis(e.g. parallel diatomic molecules and one lattice site only):
QBQQuQuQ
uQuQQ
x
T
x
T22
x
22
1n
cos:where
exp),(
Qu
EES
Sn=1(Q,E) is maximum for φ=0 (Q||x) and zero for φ=90o (Qx). Similar to E in IR. It is also defined a displacement tensor Bij:
T
xuuB
In practice the powder spectrum will be a spherical average containing various modes i:
i
Tii
ii
θ,
ii
i
TT1n
:where
exp),(
uuBA
QBQQAQ
EEQS
One can prove that a good approximation of the spherical mean is given, for the fundamental, by:
ii
2
i
2i1n 3
exp),( ETrQ
QEQS B
where:
i
ii 2
5
1
B
ABA
Tr
TrTr
This expression is formally identical to the isotropic harmonic oscillator one: all the vibrations are visible, but wakened by a factor 1/3.
Example of the anisotropy importance in highly-oriented (>90%) polyethylene
–––––––––– c –––––––
Qc (calc. by Lynch et al.) Q||c (calc. by Lynch et al.)
Example: lattice modes in highly-oriented polyethylene simulated for TOSCA
7) Some applications
to soft matter
What is soft matter?
Soft matter: it is often macroscopically and mechanically soft, either as a melt or in solution. On a short scale: there is a mesoscopic order together with weak intermolecular force constants [v/(3kBT)1]. It is in
between solids and liquids (both for its structure and for its dynamics). It is not yet rigorously defined. Main classes (after Hamley, 1999): polymers, colloids, amphiphiles and liquid crystals. Good picture, but there is still some overlap!
What is spectroscopy?
•A microscopic dynamical technique: spectral analysis (k,) of a probe, before and after its interaction with a sample.
•Absorption (0) or scattering (k, ).
•Basic idea: 02/t; |k|2/|r| and 2/t.
•Differences: i) probe [e.m. waves: =c|k|, neutrons: =|k|2/(2mn)]; ii) interaction [e.m. waves: Aj, neutrons: (22/mn) b (r)].
Main spectroscopic techniquesfor soft matter
i) Nuclear Magnetic Resonance (NMR).ii) Infrared absorption and Raman scattering (IR and Raman).iii) Dielectric Spectroscopy iv) Visible and ultraviolet optical spectroscopyv) Inelastic neutron scattering (INS).
E = Ei – Ef Q = ki – kf
Why INS for soft matter?
• Limitations of IR and Raman: selection rules (from f|D|i and f|P|i). Group theory.
• General problems with optical techniques: i) dispersion and acoustic modes; ii) selection rules; iii) proton visibility; iv) spectral interpretation.
INS is always complementary and often essential
i) Dispersion and acoustic modes
collective modes dispersion: =j(q), con 0<|q|<2/a20
nm-1.What |q| can be obtained through e.m. waves? Green light (E=2.41 eV): |q|=0.0122 nm-10… X-rays are needed (E>1 KeV): IXS.
Acoustic modes: ac(|q|0)=cs|q|0.
Thermal neutrons: (E=25.85 meV): |q|=35.2 nm-1.
ii-iii) Selection rules and proton visibilityHigh symmetry: many modes are optically inactive (C60: 70%!).
Neutrons: pseudo-selection rule for H (H=81.67 barn >>x1-8
barn). Isotopic substitution: HD (D=7.63 barn). Proton visibility in
Raman: Tr (P) grows along with Z. Proton visibility in IR: strong signal for H-bonds (e.g. O-H), but there is also the electric anharmonicity (distortions).
iv) Spectral InterpretationDirect interpretation of the spectral line intensities: vibrational eigenvectors (IR and Raman: f|D|i, f|P|i).Example: one-dimensional harmonic oscillator (at T=0):
),(
!exp),( 0
0n
n
0
22
0
22 nEn
uQuQEQS
25 50 75 100 125 150 175 2000
1
2
3
4
5
(OH)¦
(OH)¦
K+[C6H
4(COOD)COO]-
K+[C6H
4(COOH)COO]-
da D. Colognesi, TOSCA
S(Q
,E)
(a.u
.)
E (meV)
Example: isotopic substitution in potassium hydrogen phthalate. Two hydrogen-bond modes are clearly pointed out.
Would you like to know more?
(from easy to difficult)
“Introduction to the Theory of Thermal Neutron Scattering” by G. L. Squires (1978).
“Vibrational Spectroscopy with Neutrons” by P. C. H. Mitchell et al. (2005).
“Molecular Spectroscopy with Neutrons” by H. Boutin and S. Yip (1968).
“Neutron Scattering in Condensed Matter Physics” by A. Furrer, J. Mesot and T. Straessle (2009).
“Slow Neutrons” by V. F. Turchin (1965).
“Theory of Neutron Scattering from Condensed Matter I” by S. W. Lovesey (1984).
Acknowledgements
Many thanks to:
Dr. R. Senesi (Univ. Roma II) for the kind invitation to talk.
The audience for its attention and interest.