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

First Elements of Thermal Neutron Scattering Theory (I)

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).4) Inelastic scattering from fluids (intro).

5) Vibrational spectroscopy from molecules.5) Vibrational spectroscopy from molecules.

6) Incoherent inelastic scattering from 6) Incoherent inelastic scattering from molecular crystals.molecular crystals.

7) Some applications to soft matter.7) Some applications to soft matter.

0) Introduction

Why neutron scattering (NS) from condensed matter?

Nowadays NS is relevant in physics, material science, chemistry, geology, biology, engineering etc., being highly complementary to X-ray scattering.

E. Fermi C. G. Shull B. N. Brockhouse

What is special with NS?1) Neutrons interact with nuclei and not with their electrons (neglecting magnetism). Ideal for light elements, isotopic studies, similar-Z elements, and lattice dynamics.

2) Neutrons have simultaneously the right and E, matching the typical distance and energy scales of condensed matter.

4) The neutron has a magnetic moment, ideal for studying static and dynamic magnetic properties (not discussed in what follows ).

3) Weakly interacting with matter due to its neutrality, then: (a) small disturbance of the sample, so linear response theory always applies; (b) large penetration depth for bulky samples; (c) ideal for extreme condition studies; (d) little radiation damage.

Basic neutron properties

Mass: 1.67492729(28) 10−27 KgMean lifetime: 885.7(8) s (if free)

Electric charge: 0 eElectric dipole moment: <2.910−26 e·cm

Magnetic moment: −1.9130427(5) μN

Spin: 1/2

Neutron wave-mechanical properties

Interested only in slow neutrons (E<1 KeV), where:

E=mv2/2 (m=1.675·10-27 Kg) and

=h/(mv)Using the wave-vector (k=2/), one has:

E(meV)=81.81 (Å)-2=2.072 k(Å-1)2

=5.227 v(Km/s)2=0.08617 T(K)

The slow neutron “zoology”

(a version of)

Name Energy range (meV)

Very cold / Ultra-cold <0.5

Cold 0.5 - 5

Thermal 5 - 100

Hot 100 - 103

Epithermal / Resonant >103

1) Neutron scattering from nuclei

The neutron-Nucleus interaction

1) Short ranged (i.e. 10-15 m).

2) Intense (if compared to e.m.).

3) Spin-dependent.

4) Complicated (even containing non-central terms).

Example: D=p+n, toy model (e.g. rectangular potential well)

width: r0=2·10-15 m

depth: Vr=30 MeV

binding energy: Eb=2.23 MeV

Coulomb equivalent (p+p) energy: EC=0.7 MeV

The slow neutron-Nucleus system

Good news: if nr0 (always true for slow neutrons) and dr0 (d: size of the nuclear delocalization) we do not need to know the detail of the n-N potential for describing the n-N system! Two quantities (r0 and the so-called scattering length, a) are enough.

)wave()(1),(

0)(for),(),(),()(22

nNnNnN0nNNN

rEHUmm

rrrrrrrrr

Localized isotropic impact model

The Schroedinger equation plus the boundary condition are exactly equivalent to:

potential]-pseudo[Fermi

)( :where

),(lim)(),(),(

NnnNnN0

rrrrrr

rrrrrrrr

Tough equation… But it can be expanded

in power series of V: =0+1+2+… (if

na and da), where:

The Fermi approximation is identical to the well-known first Born approximation:

110 lim

QM text-book solution:

where a spherical wave, modulated by the inelastic scattering amplitude f(k’,F;k,0) has been introduced:

state)ion(perturbat)()0,;,'(

state)ed(unperturb)(exp8

Nucleus

neutron

Nucleus

neutron

and the following energy conservation balance and useful definitions apply:

transfer)(momentum'

:defines oney analogousl

transfer)(energy2

N0NnN*FnNn2

)exp()()('exp

)(),()('exp2

1)0,;,'(

RQRRRkkR

rrrrrkkrrkk

Slow neutron scattering from a nucleus

(k, E)

(k’, E’)

Measurable quantity: number of scattered neutrons, n detected in the time interval t, in the solid angle between and +, and between and +, having an energy ranging between E’ and E’+ E’:

tEEIn ')',,(

Qk-k’ E-E’=EF-E0

Scattering problem: how is I(,,E’) related to the intrinsic target properties [i.e. to i(RN)]?

The concept of double differential scattering cross-section (d2/d/dE’) has to be introduced:

where Jin is the current density of incoming neutrons (i.e. neutrons per m2 per s), all exhibiting energy E. Analogously, for the outgoing neutrons, one could write: Jout(,,E’)= r

-2 I(,,E’) (spectral density current).

d )',,()',,(

E'E,φ,θ,

Going back to our QM text book, one finds the “recipe” for the neutron density current:

')(),'(),()(

)exp('

)0,;,'('

EEEEΩJΩEJΩJdΩJ

outoutoutout

which, applied to 0 and 1 (box-normalized, L3), gives:

and finally, the neutron scattering fundamental equation:

')exp('

'EE,φ,θ,

EEEEibk

for the transition from the nuclear ground state 0 to the excited state F, with the constraint: E-E’=EF-E0.

Summing over all the possible nuclear excited states F, one has to explicitly add the energy conservation:

)exp('

'EE,φ,θ,

Finally, if the target is not at T=0, one should also consider a statistical average (pI) over the initial nuclear states, I:

onlypropertyTarget

)exp('

'EE,φ,θ,

EEipbk

where the inelastic structure factor or scattering law S(Q,) has been defined. Giving up to the neutron final energy (E’) selection, one writes the single differential s. cross-section:

)for('

''EE,φ,θ,Eφ,θ,

EEEdEd

Neutron scattering from an extended system

So, is everything so easy in NS? No, not quite…

Giving up to the selection of the scattered neutron direction () too, one writes the s. cross-section:

s.)c.(freefor4

s.)c.(bound0for4)( 2

Eφ,θ,

potential) like-(comb)(2

states)body(many),...,,()(

N21FI,NFI,

Neutrons and incoherence

“Real-life” neutron scattering (from a set of nuclei with the same Z)

where: ),(4

' self1INC

scattering laws defined for a many-body

system (set of nuclei with the same Z) as:

IjFIself

)exp(),(

S(Q,) is obvious, but where does Sself(Q,) come from? From the spins of neutron (sn,mn) and nucleus (IN,MN), so far neglected! b depends on IN+sn

- e.g. full quantum state for a neutron-nucleus pair:|k’,sn,mn; F, IN, MN

How does it work? Assuming randomly distributed neutron and nuclear spins, one can have a simple idea of the phenomenon:

spinj'spinjspinj'j

spinjjspinj'j

positionspin

1j'j'j'

positionspin

)'exp()exp(

bbbbbj'j

The incoherence origin (rigorous theory):

ˆ4;ˆ4

since b is actually not simply a number, but is the scattering length operator acting on |iN, mN (nucleus) and on |sn, mn (neutron) spin states:

2ˆ iσ

This implies the existence of b+ and b- (if iN>0) for any isotope (N, Z), respectively for iN½:

,;,ˆ,;,)12(2

;,;,ˆ,;,)12(2

m,MnnNN

m,MnnNNnnNN

AmsMibmsMii

After some algebra, only for unpolarized neutrons and nuclei, one can write:

jjN,jjN,j

)())(1(ˆ

bibicb

Important case: hydrogen

(protium H, iN=1/2): b+=10.85 fm, b-=-47.50 fm

TOT=82.03 b, COH=1.7583 b

(deuterium D, iN=1): b+=9.53 fm, b-=0.98 fm

TOT=7.64 b, COH=5.592 b

Deuterium

Hydrogen

With various isotopes (cj ) one gets:

High-Q spatial incoherence

(rearranging…)

0),(),(),(

0for''

distself

QQQ SSS

Incoherent approximation

When does it apply in a crystal?When does it apply in a crystal?

Practical example : D2SO4 (T=10 K)

d(DO)=0.091 nmu21/2(D)=0.0158 nm2u21/2/d2=11.9 nm-1

|Qinc|100 nm-1

2) Time-correlation functions

S(Q,) and Sself(Q,) are probe independent, i.e. they are intrinsic sample properties. But what do they mean?

Fourier-transforming the two spectral functions, one defines I(Q,t) and Iself(Q,t), the so-called intermediate scattering function and self intermediate scattering function:

),(exp),(

selfself

StidtI

After some algebra (e.g. the Heisenberg representation), one writes I(Q,t) and Iself(Q,t) as time-correlation functions (with a clearer physical meaning):

),(),(),(

)(iexp)0(iexp1

selfdist

jjjself

tItItI

kj )(iexp)0(iexp1

),( tN

tI RQRQQ

So far we have dealt only with a pure monatomic system (set of nuclei with the same Z).

tIbbsctdtk

tIbbzcsctdtk

),(][iexp2

),(][][iexp2

(s)self

z)(s,zs

Sum over ”s” distinct species (concentration c[s]):

(s)self )(iexp)0(iexp

1),( t

NtI RQRQQ

But what about “real-life” samples (e.g. chemical compounds)?

where I(s)self(Q,t) is the so-called self intermediate

scattering function for the sth species:

),(exp2

1),( (s)

self(s)self tItidtS QQ

and where S(s)self(Q,) is the so-called self

inelastic structure factor for the sth species. Properties similar to those of Sself(Q,):

The coherent part is slightly more complex

I(s,z)(Q,t) is the so-called total intermediate scattering function for the sth species (if sz), or the cross intermediate scattering function for the s,zth pair of species (if sz):

z)(s, )(iexp)0(iexp),( tNN

NtI RQRQQ

and S(s,z) (Q,) is the so-called total inelastic

structure factor for the sth species (if sz), or the cross inelastic structure factor for the s,zth pair of species (if sz):

),(exp2

1),( z)(s,z)(s, tItidtS QQ

The total contains a “distinct” plus a “self” terms, while the cross only a “distinct” term.

Coherent sum rules

factorstructureStatic)(),()0th

QQ SSd

EnergyRecoil2

),()1 R

jk,jkII )exp()exp(

1)( RQRQQ iip

where:

nnn ),()(),(:from

iSd QQ

ionNormalizat1),()0 selfth

EnergyRecoil2

),()1 R

selfst

22self

2 :energy Kinetic

Incoherent sum rules

selfn ),()(),(:from

iSd QQ

potentialtheofLaplacian

2),()3

ji,jjii2

potentialtheofgradientSquare

44self

Detailed balance

),(exp),(

selfself

from the microscopic reversibility principle:

q.e.d.exp

EEnimp

EEminp

EEnimp

Analogous proof for the scattering law

The zoo of excitations…

…and their (|Q|-E) relationships

3) Inelastic scattering

from crystals

Scattering law from a many-body system: analytically solved only in few cases (e.g. ideal gas, Brownian motion, and regular crystalline structures, with a purely harmonic dynamics.

n'n,n'nn'n

)(iexp)0(iexp

exp2)(

Generalized scattering law (sometimes used for mixed systems)

nnnnINC,

mmINC,

)(iexp)0(iexp

Generalized self scattering law (sometimes used for mixed systems)

In a harmonic crystal the time correlation functions are exactly solvable in terms of phonons due to the Bloch theorem for a 1D harmonic oscillator (X is its adimensional coordinate):

221expexp XX

Three-dimensional crystalline lattice (N cells and r atoms in the elementary cell: “l” and “d” indexes):

)()( dl,mequilibriu

l.d tt udlR

Harmonicity (expansion of ul,d(t) in normal modes; e.g. phonon “s”; quantized):

tiiaNM

1sssds,

with es,d polarization versor, a†s (as) creation

(annihilation) operator of the sth phonon with s frequency and 2|q|-1 wavelength.

Collective index “s”: {qx,qy,qz,j} with q1BZ (first Brillouin zone: N points). “j” labels the phonon branches (3 acoustic e 3r-3 optic).

Polarizations: 2 transverse and 1 longitudinal.

Total: 3Nr d.o.f.

Dispersion curves: s=j(q)

Coherent scattering

Plugging the equations for Rl,d(t) and ul,d(t) (i.e. phonon quantization) into the coherent d. d. cross-section, one gets:

)(exp)0(expexp2

''exp'

d',l'd0,

d d',l'd'd

tiitidt

and using the Bloch theorem together with the commutation rules: eAeB=eA+Be[A,B]/2, one writes:

)()0(exp

)()0(2

)(exp)0(exp

d',l'd0,

),(expexp2

)/20,()/20,(exp

''exp'

d',l'd,d'd

d d',l'd'd

tBtidt

and then:

)(2122

)0,( ds

dd0, Q

eQQ Wn

where the static term is the Debye-Waller factor, whose exponent is:

with ns number of thermally activated phonons; while the dynamic term contains:

)]''(exp[)]''(exp[

*s,d'sd,

d',l'd,

lddqlddq

itiniti

Coherent elastic scattering

Phonon expansion

Expanding exp[Bd,d’,l’(Q,t)] in power series, one gets a sum of terms with n phonons (created or annihilated):

...),(!

...),(2

1),(1),(exp

nl',d'd,

2l',d'd,l',d'd,l',d'd,

tBtBtB

one gets for the first term, 1, an elastic contribution:

)/20,()/20,(exp

''exp'

d d',l'd'd

elastCOH,

ibbNdEd

Integrating over E’ and making use of the reciprocal lattice () sum rule: l exp(iQl) = 83vcell

-1 (Q-), one obtains the well-known Bragg law:

)peaksBragg()(

)()(exp'exp8

factorstructure cell-unitnuclear:)(

d'dd'd,

d'dcell

elastCOH,

WWibbNvd

Neutron powder diffraction pattern from Li2NH (plus Al container) at room temperature. Abscissa: d=2Q -1

One-phonon coherent contribution

If ER,d(Q)</s-1 [the lightest Md] then:

),(1),(exp l',d'd,l',d'd, tBtB QQ

and one obtains the single phonon (created or annihilated) d. d. coherent cross section:

),(exp2

)()(exp

''exp'

d',l'd,d'd

d d',l'd'd

tBtidt

Plugging the equation for Bd,d’,l’(Q,t) into the one-phonon coherent d. d. cross-section and performing the Fourier transforms and the reciprocal lattice sums, one gets:

)0creation ),( phonon,1(1

)(exp2

)0onannihilati),(phonon,1(

)(exp2

DispersionCurvepractical example: lithium hydride LiD (cubic, Fm3m, i.e. NaCl type) with r=2 j=1,2,3

Red: Acoustic

Blue: Optic

Full: Transverse

Dash: Longitudinal

First Brillouin zonein a face-centered cubic lattice

(f.c.c.)

face-centered cubic lattice

1st Brillouin zone f.c.c.

One-phonon incoherent contribution

),(exp)0,(exp

)(iexp)0(iexp

d(d)INC

dl,dl,dl,

(d)INC

Bloch theorem:

where the static term is the Debye-Waller factor, whose exponent is:

2)0,( n

with ns number of thermally activated phonons; while the dynamic term contains:

Single phonon and density of states

Expanding exp[Bd(Q,t)] in power series, one gets a sum of terms with n phonons (created or annihilated):

...),(!

...),(2

1),(1),(exp

tBtBtB

if ER,d(Q)</s-1 then:

),(1),(exp dd tBtB QQ

and one obtains the single phonon (creation or annihilation) d. d. incoherent cross section:

)0,(exp1

(d)INC

Density of (phonon) states: density probability for a phonon of any kind with frequency between and +d:

)(2exp]})(exp[1{

(d)INC

The single-phonon incoherent d. d. cross section (creation or annihilation) becomes more simply:

where:

(creat.)0

(annih.)0for

)()(exp1

weak point (“”), i.e. the meaning of the averaged eigenvector:

1)( ee

The separation from Q is rigorous only in cubic lattices:

2sd, )(

otherwise one has the isotropic approximation.

In addition, using this approximation, one proves that:

dd 3)(2)0,( uQQ

link between the exponent of the Debye-Waller factor

and the mean square displacement of the d species.

It is often used the density of states projected on d:

)()()(

(d)INC

]})(exp[1{

from which:

70 80 90 100 110 120 130 140 150

LiH, Optic

DDM13, (Dyck & Jex, 1981) SM-IV, (Verble, Warren & Yarnell, 1968) TOSCA-II (Colognesi et al., 2002)

Density of states projected on Hpractical example: lithium hydride LiH

—————— transverse—————— longitudinal

Multiphonon incoherent contributions

MultINC,

Coherent multiphonon terms are too complex and not very useful (e.g. for powders: Bredov approximation). Here only incoherent terms. Definition:

(d)INC

),(exp)0,(exp4

remembering that:

...),(!

...),(2

1),(1),(exp

tBtBtB

and that:

one gets for the first term, 1, an elastic contribution:

(d)INC

ElasINC,

)(2exp4

')(2exp

not to be confused with the incoherent s. d. cross-section:

''0 INC

For the second term one gets, B(Q,t), the single phonon contribution (1, created or annihilated) already known:

)(2exp]})(exp[1{

d dINC

While for the (n+1)th term, one gets Bn(Q,t), a contribution with n phonons (created and/or annihilated). Using the convolution theorem:

timesn

...),(~

),(exp2

QBQBQBtQBtidt

where::

)(2)]}/(exp[1{

we obtain:

)(2exp!

(d)INC

Self-convolution shifts and broadens fd(), but blurs its details too…

Sjölander approximation: [fd()]n is replaced by an appropriate Gaussian (same mean and variance ):

:)(for

Then we have:

Properties

of fd()

0 20 40 60 80 100 120 140 160 1800.0

E (meV)

Not always appropriate…

-D-glucose at T=19 K, example from TOSCA

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