first elements of thermal neutron scattering theory (ii) daniele colognesi istituto dei sistemi...

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

QQ

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

QQ

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

QQ

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.