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.
1) Neutron scattering from nuclei.
2) Time-correlation functions.
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)
1
g (r ) 
 (r  ri  rj )

 N i j
connected to the static structure factor, S(Q), via
a 3D spatial Fourier transform:
1
S (Q) 
N

i j

exp i Q  (ri  rj )

 1    dr ( g (r )  1) exp(i Q  r )
where both S(Q) and g(r) exhibit some special
values at their extremes:
S (0)  T kBT ; S ()  1
g (0)  0; g ()  1
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):
G (r , t ) 
1
8
3
 dQ exp(i Q  r) I (Q, t )
   s  r (0) s  r  r (t ) ds
1

N
i
j
i, j
and the time-dependent self pair
correlation function, Gself(r,t):
Gself (r , t ) 
1

N
1
8
3
 dQ exp(i Q  r) I
self
   s  r (0) s  r  r (t ) ds
i
i
i
(Q, t )
where the t=0 values of G(r,t) and Gself(r,t) are:
G(r ,0)   (r)  g (r )
Gself (r ,0)   (r)
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):
I (Q, ) ( )  S (Q,  )  I (Q, ) ( )

 
 


elastic
inelastic

dtdr 



(r , )  G (r , t )  G (r , ) exp(it  iQ  r )
G

 
2  asymptotic
approaching 0 for t  


Due to the asymptotic loss of time correlation,
and making use of =i(r-ri), one writes:


i, j
1 2
 s  ri (0)  s  r  rj (t ) ds   V   ;
N
1
Gself (r , ) 
N
1
 s  ri (0)  s  r  ri (t ) ds    0
N
1
G ( r , ) 
N


i
so, finally:
I (Q, )  8 3   (Q) (norealscattering:   0o )
I self (Q, )  0
Gas of non-interacting distinguishable
particles: a useful “toy model”. No
particle correlation: S(Q,)Sself(Q,).
Starting from the definitions:
2
N
Sself (Q,  )  N 1  pn  m expiQ  ri  n    Em  En 
i 1
m,n
 2 ki2
with E n  
;
i 1 2 M
N
n 
N
i 1
expik i  ri 
V
one writes:
 2k 2

2 Mk BT
V
e
Sself (Q,  )  3  d k
8
2MkBT 2


 2 k '2  2 k 2 

    

2M
2M 


3/ 2
dr
 d k ' V expiQ  k 'k  r
After some simple algebra:
Sself (Q,  )  
d k
2 M k T 
2 3 / 2
B
  2k 2  
 2Q 2  2k  Q 
    

exp 

2M
M 
 2MkBT  
2 2
 
Q 

 
  
2M  
1
 

exp
2
2
1
2
Q
k BT 
2 Q M k BT


M


recoil
Doppler broadening
Very important for epithermal neutron scattering!
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.
Brillouin triplet(in theQ  0 hydrodynamic limit and for 
not toolarge) :
QQ 2
S (Q)
exp(Q  q - τ ) (  r (q)) 
 p 
2 4
2
2  (  cQQ)  Q Q
0
S (Q)(  1)
Ds, QQ 2

  Ds, Q Q
2
2
4
 S 
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.
for ω and Q not toobig :
DQ2
 ( ) 
  2  D 2Q 4
1
On the contrary, the inelastic component is not
too dissimilar from the crystal case (pseudophononic excitations).
Starting from the well-known:

 d 2 
 INC k ' dt

 
exp i t 

 ddE'  INC 4 N k  2
N
  exp  iQ  R n (0)exp iQ  R n (t )
n 1
it is possible to show (Rahman, 1962) that:

 d 2 
 INC k '  dt
L
2L

 


exp

i

t
exp
(

)
Q
 L (t )


L 1
 ddE'  INC 4 k  2 


 INC k ' dt
2




exp

i

t
exp

Q
 1 (t )

4 k  2

where we made use of the Gaussian
approximation in Q. The t-dependent factor
has apparently a tough aspect:

 1 (t ) 
2M n


0

 d
  
1  cost   i sin t 
f liq  coth
 2 k BT 

 
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 selfcorrelation function of an atom in a crystal: cvv(t).
Expanding in normal modes through the Bloch
theorem, one gets (in the isotropic case):
1
cvv (t ) 
N
N

n 1
v n (0)  v n (t ) 3 v1, z (0)v1, z (t )


  
3
 cos t   i sin  t  d 

g   coth

2M n 0
 2 k BT 



3  d 2 B(Q, t ) 
 2 

Q  dt 2  t 0
It applies to fliq() too. Using the fluctuationdissipation theorem, linking Re[cvv(t)] with
Im[cvv(t)], one writes:

4M d tanh  (2kBT )
f liq ( ) 
3 

1 
 Rec
vv
(t )cos t dt
0
However, there is a property distinguishing
fliq() from g():

 dt Rec
0
vv
(t )  3D  3 lim t 
2M d D
R (t ) t  f liq (0) 
 kBT
2
1
where D is the self-diffusion coefficient,
while g(0)=0.
from CMD, T=14.7
-1
fliq(E) (meV )
0.09
0.06
0.03
0.00
0
5
10
15
20
25
30
E (meV)
-1
Sself (Q,E) (meV )
0.03
0.02
0.01
from CMD, T=14.7 via
Gaussian approx.
exp., T=14.3 K
0.00
-10
0
10
20
30
E (meV)
40
50
60
Example: liquid parahydrogen, 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: E400
KJ/mol].
- keep the functional groups close to one another
[hydrogen bond: E20 KJ/mol].
- place the molecules according to a certain order in a
crystalline lattice [molec. crystals: E2 KJ/mol].
Wide range of energies! Here only intra-molecular
modes (vibrational spectroscopy).
Cross-section summary
2
 tot  4 bˆ ;  coh  4 bˆ 
2
 inc
2
2

 4 bˆ  bˆ 


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 modes
Polyatomic 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):
αmαrαvα= J=0 αmα rα0tuα 
αmα rα0uα=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
1 s
T   M ijui u j (rigorous)
of normal modes
2 i, j
with s d.o.f. qi:
1 s
U   K ijui u j (harmonicapprox.)
ui=qi-qi0
2 i, j
One gets s Lagrange
equations:

Oscillating test solutions:

Characteristic equation :
(in general one has s real and
positive roots: 1,… s)

Eigenvectors aj(s):
s
 M u  K u
ij j
ij j
0
j
u j  aj exp(it )
K ij   2 M ij  0
 K
s
j

(s) 2
(s)


M
a
ij
ij
j 0
General solution:

u j   a (j )  C ( ) exp(i ( )t )

α
  (m j ) 1/2 a (j )Q ( )
α
where theQ ( ) - th normalmode
is given by :
( )
( ) 2 ( )


Q  Q  0
Example: normal modes in
H2O
a. Symmetric stretching
b. Bending
c. Anti-symmetric stretching
Normal mode quantization
1
1
( ) 2

H cl   Qi
   ( ) 2  Qi( ) 2
2 i,α
2 α
i

fα
2

1

2
( ) 2
( ) 2 
ˆ

H      

Qi
( ) 2
2 α i 1 
Qi

fα
   ( )  ( ni( )  1 / 2)
α
i 1
Mean square amplitude: quantized
( ) 2
i
Q
  ( ) 1 
 ( )  ni    Qi( ) 2
 
2
0


2
( )
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):
2



Q
1
1
S (Q, E )  exp 2 E  exp
cot h 2 0 
 20



Q 2
  ( E  n0 )
  I n 
1
2

sinh(
n 0
0
2  0 ) 


Typical experiment : T=20 K (i.e. 14 cm-1<< 0) then:
1
2
 0  70  cot h(12  0 )  1, sinh(12  0 )  1
n
Q 2
1  x
 1  I n ( x)    ,
1
2 0 sinh(2  0 )
n!  2 
from which:

S (Q, E )  exp  Q u
2

Q


2
0
n 0
2
u
2
n!
0

n
 ( E  n0 ),
where u20 is the mean square displacement (at T=0).

0.16759
u (nm ) 

.

1
0
2 0  (amu) (cm )
2
2
Again on the harmonic oscillator
Mass problem: what is μ in a molecule? It
depends on all the atomic masses, but MH
obviously plays a primary role! However, in
general, μMH .
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:
S el (Q, E )  exp[ Q 2 u 2 ]  ( E )
0
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 Q2u20 :

S n 1 (Q, E )  exp[ Q 2 u 2 ] Q 2 u 2
0
0
 (E   )
0
The maximum of Sn=1(Q,E) appears at Q2=u20.
So, the ideal measurement conditions for H are:
k1<<k0  k0Q for any value of E. Namely:
2mn
2
k1  2 E  k12 u 2

0

 (cm1 )mn (amu)
0.16759
0.16759
1
1
 (amu) (cm )
Overtones: excitations from the ground state (n=0)
to states higher than the first (i.e. n=2,3…):

S n 1 (Q, E )  exp  Q u
2
2
0
 Q
2
u
2
0

n!
n
 ( E  n0 ).
considering that:
2mn
k  Q  2 E,

2
1
2
one obtains:
u
2
0


20
,
n
1
 1  n n
S n 1 (Q, E )   exp 
 ( E  n0 ).
   n!

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.
1.2
n=1
ZrH2, T=20 K
from J. Tomkinson (ISIS)
1.0
S(Q,E) (arb. units)
Example:
fundamental and
overtones in ZrH2,
almost a harmonic
oscillator (threedimensional).
0.8
0.6
n=2
0.4
n=3
0.2
0.0
0
1000
2000
3000
-1
E (cm )
4000
Anharmonicy
Ideal 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).
 0,(0n)  ne 1 (n 1)   n0,(01)
In practice, in real molecules one uses a pseudoharmonic approach in which the structure factor
for a single atomic species is approximated by:

  Qi u 2



(tot)
ij

2
S (Q, E )  exp  Qi u
Qj   
ij
n!
 ij
 k n 0
(k)
ij
  ( E  0(k)n )
where n labels the sum over the overtones and k
the multi-convolution in E over the normal modes,
from which:
u
2 (tot)
ij
 u
k
2 (k)
ij

Qj 

n
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 intermolecular 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).
Similarly to what seen for the internal modes, an
external structure factor for the molecular lattice can
be defined:

S ext (Q, E )  exp  Qi U 2
 ij
  ( E  npk ),

  Qi U 2


(tot)
ij
Qj    
ij
n!
 k n 0
making implicitly use of the decoupling hypothesis
between internal and external modes:
S (Q, E)  Sext (Q, E)  Sint (Q, E)
(k)
ij

Qj 

n
using the distributive property of the convolution
one gets:

S (Q, E )  exp  Qi U 2
 ij


2 (k')
  Qi U
Q j 

ij
ij

 
k, k'
n '!
n, n'
n'
(tot)
ij


Q j  exp  Qi u 2

 ij

  Qi u 2

 ij
n!
(tot)
ij

Qj 

n
(k)
ij

Q j 
  ( E  n' p  n ),
k'
k
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:
S (Q, E )  S orig (Q, E )  S branc (Q, E )

S orig (Q, E )  exp  Qi U 2
 ij
(tot)
ij

Q j  Sint (Q, E )

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:
1
Tr u 2
3
(tot)
ij
1
 Tr U 2
3
(tot)
ij
of Bragg peaks): Tr U
Biso

8 2
2 (tot)
ij
(cont rolling t heint ensit y
 Tr u
2 (tot)
ij
Discrepancies between Biso and the inelastic mean
square displacements: static disorder
Example:
hexamethylenetetramine
(C6H12N4) on TOSCA
C6H12N4, T=20 K
from J. Tomkinson (ISIS)
S(Q,E) (arb. units)
5
<--- Lattice modes
4
Pure vibration/libration
|
|
Phonon wing
|
|
|
|
3
2
1
0
0
200
400
600
-1
E (cm )
800
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):

Sn 1 (Q, E )  exp  Q  u
where : Q  u
2
2
 Qu
 Qu cos
2
2
 E  x 
 QT u x Q  QT B x Q
Sn=1(Q,E) is maximum for φ=0 (Q||x) and zero for φ=90o
(Qx). Similar to E in IR. It is also defined a displacement
tensor Bij:
Bx  u  uT
In practice the powder spectrum will be a spherical average
containing various modes i:

S n 1 (Q, E )  exp  Q T AQ
 Q
i
where : A   B i   u i  u iT
i
i
T
B i Q  E  i 
θ,
One can prove that a good approximation of the
spherical mean is given, for the fundamental, by:

 

Q2
S n 1 (Q, E )   exp   iQ
Tr Bi  E  i 
3
i
where:
2

Tr Bi A
1
 i  Tr A   2
5 
Tr Bi
 



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 –––––––
Example: lattice modes in highly-oriented
polyethylene simulated for TOSCA
Qc (calc. by Lynch et al.)
Q||c (calc. by Lynch et al.)
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: 02/t; |k|2/|r| and 2/t.
•Differences:
i) probe [e.m. waves: =c|k|, neutrons: =|k|2/(2mn)];
ii) interaction [e.m. waves: Aj, neutrons: (22/mn) b (r)].
E = Ei – Ef
Q = ki – kf
Main spectroscopic techniques
for soft matter
i) Nuclear Magnetic Resonance (NMR).
ii) Infrared absorption and Raman scattering (IR and Raman).
iii) Dielectric Spectroscopy
iv) Visible and ultraviolet optical spectroscopy
v) Inelastic neutron scattering (INS).
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/a20
nm-1.
What |q| can be obtained through e.m. waves?
Green light (E=2.41 eV): |q|=0.0122 nm-10…
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 visibility
High symmetry: many modes are optically inactive (C60: 70%!).
Neutrons: pseudo-selection rule for H (H=81.67 barn >>x1-8
barn). Isotopic substitution: HD (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 Interpretation
Direct 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):

S (Q, E )  exp  Q u
2

Q


2
0
n 0
2
u
2
n!
0

n
 ( E  n0 ),
+
-
+
-
K [C6H4(COOD)COO]
K [C6H4(COOH)COO]
5
da D. Colognesi, TOSCA
(OH)
S(Q,E) (a.u.)
¦
4
(OH)
¦
3
2
1
0
25
50
75
100
125
150
175
200
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.