First Elements of Thermal Neutron Scattering Theory (II)

1. First Elements of Thermal Neutron Scattering Theory (II) Daniele Colognesi Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Sesto Fiorentino (FI) - Italy

2. 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.

3. 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) connected to the static structure factor, S(Q), via a 3D spatial Fourier transform:

4. where both S(Q) and g(r) exhibit some special values at their extremes: Since S(Q)=I(Q,t=0), it is possibleto generalize g(r) by introducing the time-dependentpair correlation function, G(r,t):

5. and the time-dependentselfpair correlation function, Gself(r,t):

6. where the t=0 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):

7. Due to the asymptotic loss of time correlation, and making use of =i(r-ri),one writes: so, finally:

8. Gas of non-interacting distinguishable particles: a useful “toy model”. No particle correlation: S(Q,)Sself(Q,). Starting from the definitions: one writes:

9. After some simple algebra: recoil Doppler broadening Very important for epithermal neutron scattering!

10. 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.

11. Liquid Al g(r) Liquid Ni S(Q)

12. Incoherent inelastic scattering from liquids: the elastic componentbecomesquasi-elastic(diffusive motions), not discussed here in great detail. On the contrary, the inelastic componentis not too dissimilar from the crystal case (pseudo-phononic excitations).

13. Starting from the well-known: it is possible to show(Rahman, 1962)that:

14. where we made use of the Gaussianapproximation in Q. The t-dependent factor has apparently a tough aspect: but it is actually equal toQ-2[B(Q,0)-B(Q,t)]. Thenfliq()has to be analogous to g() in solids… Surprising! Let’s study it, starting from the velocityself-correlation function of an atom in a crystal: cvv(t).

15. Expanding in normal modes through the Bloch theorem, one gets (in the isotropic case): It applies tofliq() too. Using the fluctuation-dissipationtheorem, linkingRe[cvv(t)]withIm[cvv(t)], one writes:

16. However, there is a property distinguishing fliq() from g(): where D is the self-diffusioncoefficient, whileg(0)=0.

17. Example:liquidpara-hydrogen, measured on TOSCAat T=14.3K (Celli et al. 2002) and simulated throughCentroid Monte Carlo Dynamics(Kinugawa, 1998).

18. 5) Vibrational spectroscopy from molecules • chemical-physical spectroscopy:studying the forces that: • bind the atoms in a molecule [covalent bond: E400KJ/mol]. • keep the functionalgroups close to one another • [hydrogen bond:E20KJ/mol]. • place the molecules according to a certain order in a crystallinelattice[molec. crystals:E2KJ/mol]. Wide range of energies! Here only intra-molecularmodes (vibrational spectroscopy).

19. Cross-section summary Hcase (ideal incoherent scatterer): inc=80.27 b, coh=1.76 b  Proton selection rule Dcase (quite different): inc=2.05 b, coh=5.59 b Then only incoherent scattering will be considered in the rest of this talk!

20. 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 electricdipole) the H-bond gives rise to a large signal, but it is distorted by the so-called electricanharmonicity (not vibrational). Molecules with elevate symmetry: many modes are optically inactive (e.g. in C60 up to 70%!).

21. 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

22. 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

23. 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 of normal modes with s d.o.f. qi: ui=qi-qi0

24. One gets s Lagrange equations:  Oscillating test solutions:  Characteristic equation : (in general one hassreal and positive roots: 1,… s)  Eigenvectors aj(s):

25. General solution: Example: normal modes in H2O a. Symmetric stretching b. Bending c. Anti-symmetric stretching

26. Normal mode quantization

27. Diffusion from a harmonic oscillator The mono-dimensional harmonicoscillator is then the simplified prototype of the true intra-molecular vibrations: ~1000 cm-1 <0<4400 cm-1 (H-H):

28. Typical experiment :T=20 K (i.e. 14 cm-1<< 0) then: from which: whereu20is the mean square displacement (atT=0).

29. Again on the harmonic oscillator Mass problem: what isμin a molecule? It depends on all the atomic masses, but MHobviouslyplays a primary role!However, in general,μMH . Elastic Line: there is no exchange of energy between oscillator and neutron, thenn=0. It is intense, but it decreases rapidly withQ. Then it will be neglected:

30. Fundamental: forn=1there is a peak centered at0, while inQone gets a competition between the Debye-Waller factor and the termQ2u20: The maximum ofSn=1(Q,E)appears atQ2=u20. So, the ideal measurement conditions for H are: k1<<k0k0Qfor any value ofE. Namely:

31. Overtones: excitations from the ground state (n=0) to states higher than the first (i.e. n=2,3…): considering that: one obtains:

32. 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. Example:fundamental and overtones in ZrH2,almost a harmonic oscillator (three-dimensional).

33. Anharmonicy Ideal vibrational model: set of decoupled harmonic oscillators (normalmodes). 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).

34. 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:

35. 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 molecularcrystals 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).

36. Similarly to what seen for the internal modes, an external structure factorfor the molecular lattice can be defined: making implicitly use of the decoupling hypothesis between internal and external modes:

37. 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:

38. 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

39. Example:hexamethylenetetramine (C6H12N4) on TOSCA

40. 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):

41. 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: In practice the powder spectrum will be aspherical average containing various modes i:

42. One can prove that a good approximation of the sphericalmeanis 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.

43. Example of the anisotropy importance in highly-oriented (>90%) polyethylene • –––––––––– c –––––––

44. Example: lattice modes in highly-oriented polyethylene simulated for TOSCA Qc(calc. by Lynch et al.)Q||c(calc. by Lynch et al.)

45. 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 betweensolids andliquids(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!

46. 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)].

47. E = Ei–EfQ = 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).

48. Why INS for soft matter? • Limitations of IR and Raman: selectionrules(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. • INSis alwayscomplementaryand often essential

49. i) Dispersion and acoustic modes collective modes dispersion:=j(q), con0<|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 areneeded (E>1 KeV): IXS. Acoustic modes:ac(|q|0)=cs|q|0. Thermal neutrons:(E=25.85 meV): |q|=35.2 nm-1.

50. 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 electricanharmonicity(distortions). iv) Spectral Interpretation Direct interpretation of the spectral line intensities: vibrationaleigenvectors(IR and Raman: f|D|i,f|P|i). Example: one-dimensional harmonic oscillator (at T=0):