lezioni iins

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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
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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.

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4) Inelastic scattering from fluids (intro)

• Disordered systems (gasses, liquids, glasses,
  amorphous solids etc.) atomic order only at
  short range (if existing). For simplicitys sake
  only monatomic fluid systems are considered here.

key quantities density, ?, constant, and
pair correlation function, g(r)
connected to the static structure factor, S(Q),
via a 3D spatial Fourier transform
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where both S(Q) and g(r) exhibit some
special values at their extremes
Since S(Q)I(Q,t0), it is possible to generalize
g(r) by introducing the time-dependent pair
correlation function, G(r,t)
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and the time-dependent self pair correlation
function, Gself(r,t)
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where the t0 values of G(r,t) and Gself(r,t) are

• 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)

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Due to the asymptotic loss of time correlation,
and making use of ??i??(r-ri)?, one writes
so, finally
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• Gas of non-interacting distinguishable particles
  a useful toy model. No particle correlation
  S(Q,?)?Sself(Q,?). Starting from the definitions

one writes
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After some simple algebra
recoil
Doppler broadening
Very important for epithermal neutron scattering!
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• 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.

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Liquid Al g(r)
Liquid Ni S(Q)
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• 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).
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Starting from the well-known
it is possible to show (Rahman, 1962) that
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where we made use of the Gaussian approximation
in Q. The t-dependent factor has apparently a
tough aspect
but it is actually equal to Q-2B(Q,0)-B(Q,t). Th
en fliq(?) has to be analogous to g(?) in solids
Surprising! Lets study it, starting from the
velocity self-correlation function of an atom in
a crystal cvv(t).
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Expanding in normal modes through the Bloch
theorem, one gets (in the isotropic case)
It applies to fliq(?) too. Using the
fluctuation-dissipation theorem, linking
Recvv(t) with Imcvv(t), one writes
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However, there is a property distinguishing
fliq(?) from g(?)
where D is the self-diffusion coefficient, while
g(0)0.
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Example liquid para-hydrogen, measured on TOSCA
at T14.3 K (Celli et al. 2002) and simulated
through Centroid Monte Carlo Dynamics (Kinugawa,
1998).
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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).
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• Cross-section summary

H case (ideal incoherent scatterer) ?inc80.27
b, ?coh1.76 b ? Proton selection rule D case
(quite different) ?inc2.05 b, ?coh5.59 b
Then only incoherent scattering will be
considered in the rest of this talk!
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Comparing various spectroscopies ?(neutron)?10-28
m2/molec. ?(Raman)?10-32 m2/molec. ?(IR)?10-22
m2/molec.

• Why neutron spectroscopy ?
• In Raman polarizability generally grows along
  with Z possible problems in detecting H.
• 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).
• Molecules with elevate symmetry many modes are
  optically inactive (e.g. in C60 up to 70!).

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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
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• Molecular vibrations and normal modes

Polyatomic Molecules N atoms instantaneously in
the positions ra, vibrating around their
equilibrium positions ra0 ra ra0ua Normal
modes 3 traslations 3 rotations (2 if
linear) 3N-6 vibrations (3N-5 if
linear) Translations elimination (center-of-mass
fixed) ?amara ?amara0 R ? ?amaua0
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Rotations elimination (small oscillations) ?amara
?va J0 ??ama ra0??tua ? ?ama ra0?uacost.?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 uiqi-qi0
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One gets s Lagrange equations ? Oscillating test
solutions ? Characteristic equation (in
general one has s real and positive roots ?1,
?s) ? Eigenvectors aj(s)
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General solution
Example normal modes in H2O a. Symmetric
stretching b. Bending c. Anti-symmetric stretching
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Normal mode quantization
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• Diffusion from a harmonic oscillator
• The mono-dimensional harmonic oscillator
• is then the simplified prototype of the true
• intra-molecular vibrations
• 1000 cm-1 lt??0lt4400 cm-1 (H-H)

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Typical experiment T20 K (i.e. 14 cm-1ltlt ??0)
then
from which
where ?u2?0 is the mean square displacement (at
T0).
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• 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 n0. It is
intense, but it decreases rapidly with Q. Then it
will be neglected
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Fundamental for n1 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
The maximum of Sn1(Q,E) appears at Q2?u2?0. So,
the ideal measurement conditions for H
are k1ltltk0 ? k0?Q for any value of E. Namely
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Overtones excitations from the ground state
(n0) to states higher than the first (i.e.
n2,3)
considering that
one obtains
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The relative intensity of the overtones (with
respect to n1) quickly decreases along with µ.
It is important to separate the high-frequency
fundamental excitations from the overtones.
Example fundamental and overtones in ZrH2,
almost a harmonic oscillator (three-dimensional).
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• 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 ?gt0 (e.g. in the Morse potential).
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In practice, in real molecules one uses a
pseudo-harmonic approach in which the structure
factor for a single atomic species is
approximated by
where n labels the sum over the overtones and k
the multi-convolution in E over the normal modes,
from which
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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).
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Similarly to what seen for the internal modes, an
external structure factor for the molecular
lattice can be defined
making implicitly use of the decoupling
hypothesis between internal and external modes
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using the distributive property of the
convolution one gets
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
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• 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
Discrepancies between Biso and the inelastic mean
square displacements static disorder
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Example hexamethylenetetramine (C6H12N4) on TOSCA
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• 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)
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Sn1(Q,E) is maximum for f0 (Qx) and zero for
f90o (Q?x). Similar to E in IR. It is also
defined a displacement tensor Bij
In practice the powder spectrum will be a
spherical average containing various modes ?i
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One can prove that a good approximation of the
spherical mean is given, for the fundamental, by
where
This expression is formally identical to the
isotropic harmonic oscillator one all the
vibrations are visible, but wakened by a factor
1/3.
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• Example of the anisotropy importance in
  highly-oriented (gt90) polyethylene

• c ?

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• Example lattice modes in highly-oriented
  polyethylene simulated for TOSCA

Q?c (calc. by Lynch et al.) Qc
(calc. by Lynch et al.)
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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!

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• 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 ?ck, neutrons
  ??k2/(2mn)
• ii) interaction e.m. waves ?A?j??, neutrons
  (2??2/mn) b ?(r).

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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).
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• Why INS for soft matter?
• Limitations of IR and Raman selection rules
  (from ?fDi? and ?fPi?). 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

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i) Dispersion and acoustic modes collective
modes dispersion ??j(q), con 0ltqlt2?/a?20
nm-1. What q can be obtained through e.m.
waves? Green light (E2.41 eV) q0.0122
nm-1?0 X-rays are needed (Egt1 KeV)
IXS. Acoustic modes ?ac(q?0)csq?0.
Thermal neutrons (E25.85 meV) q35.2 nm-1.
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ii-iii) Selection rules and proton
visibility High symmetry many modes are
optically inactive (C60 70!). Neutrons
pseudo-selection rule for H (?H81.67 barn
gtgt?x?1-8 barn). Isotopic substitution H?D
(?D7.63 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 ?fDi?,
?fPi?). Example one-dimensional harmonic
oscillator (at T0)
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Example isotopic substitution in potassium
hydrogen phthalate. Two hydrogen-bond modes are
clearly pointed out.
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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).

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Acknowledgements
Many thanks to
Dr. R. Senesi (Univ. Roma II) for the kind
invitation to talk.
The audience for its attention and interest.