Transcript lezioni iins

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).
5) Vibrational spectroscopy from molecules.
6) Incoherent inelastic scattering from
molecular crystals.
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.
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.
4) The neutron has a
magnetic moment, ideal for
studying static and dynamic
magnetic properties (not
discussed in what follows ).
Basic neutron properties
Mass: 1.67492729(28)  10−27 Kg
Mean lifetime:
885.7(8) s (if free)
Electric charge:
0e
Electric dipole moment:
<2.910−26 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
Very cold / Ultra-cold
Cold
Thermal
Hot
Epithermal / Resonant
Energy range (meV)
<0.5
0.5 - 5
5 - 100
100 - 103
>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 nr0 (always true for slow neutrons)
and dr0 (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.
Localized isotropic impact model
  2 2n  2 2N




U
(
r
)
(for rN  rn  r  0)
N   (rN , rn )  H 0  (rN , rn )  E  (rN , rn )

2 mN
 2mn

 a
 (rN , rn ) r 0  1   (rN ) ( s wave)
 r
The Schroedinger equation plus
condition are exactly equivalent to:
the
boundary


r  (rN , rn )
r 0 r
2  2
2  2
where : V (rn  rN ) 
a  (rn  rN ) 
b  (rn  rN )

mn
H 0  (rN , rn )  E  (rN , rn )  V (rn  rN ) lim
[Fermi pseudo - potential]
Tough equation… But it can be expanded
in power series of V: =0+1+2+… (if
na and da), where:
H 0 0  E 0  0

H 0 k  E k  V lim r k 1 
r 0 r
The Fermi approximation is identical to the well-known
first Born approximation:


r 0   V 0 r 0  V 0
r 0 r
H 0 1  E 1  V lim
QM text-book solution:
0 
expi k  rn   0 (rN ) (unperturbed state)

8

 Nucleus
1
3
neutron
expi k' rn 
1 
f (k ' , F ; k ,0)  F (rN ) (perturbation state)
3

rn
8

 Nucleus
1
neutron
where a spherical wave, modulated by the inelastic
scattering amplitude f(k’,F;k,0) has been introduced:
f (k ' , F ; k ,0)  
1 2mn
*




d
r
d
r
exp
i
k

k
'

r

(rN ) V (rn , rN ) 0 (rN ) 
n
N
n
F
2 

4 
 b  d R expi k  k '  R  F* (R ) 0 (R )  b  F exp(iQ  R )  0
and the following energy conservation balance and
useful definitions apply:
 2k 2
 2k '2
 2k 2  2k '2
 E0 
 EF 

   EF  E0 (energytransfer)
2mn
2mn
2mn
2mn
analogously one defines:
Q  k  k ' (momentumtransfer)
Slow neutron scattering from a nucleus
(k, E)
E0
EF
Qk-k’
E-E’=EF-E0
(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’:
n  I ( ,  , E' )  E' t
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:
 d 2 
I ( ,  , E ' ) r 2 J out ( ,  , E ' )




J in
J in
 ddE' θ, φ, E,E'
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).
Going back to our QM text book, one finds the
“recipe” for the neutron density current:



*
J
Im  
mn

which, applied to 0 and 1 (box-normalized, L3), gives:

J in  3 k
L mn
J out (Ω ) 
 k'
 k' 2
2
f
(
k
'
,
F
;
k
,
0
)

b  F exp(iQ  R ) 0
3
2
3
2
L mn r
L mn r

2
J out (Ω )   d J out ( , Ω )  J out ( E ' , Ω )  J out (Ω ) E  E ' EF  E0 
0
and finally, the neutron scattering fundamental equation:
0 F
2
 d  
k' 2


 b F exp(iQ  R) 0  E  E ' EF  E0 
 ddE'  θ, φ, E, E ' k
2
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:
 d 2 
k' 2


 b
 ddE'  θ, φ, E, E ' k

 F exp(iQ  R ) 0    EF  E0 
2
F
Finally, if the target is not at T=0, one should also consider
a statistical average (pI) over the initial nuclear states, I:
 d 2 
k'


 b2
 ddE'  θ, φ, E, E ' k

p
I
I
 F exp(iQ  R )  I    EF  EI 
2
F
E' 2 1
b 
S (Q,  )



E
Target property only
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:

 d 2 
 d 

  dE' 


 d  θ, φ, E 0
 ddE'  θ, φ, E, E '
b
2
p 
I
I
F
(for EF  EI  E )
E  E F  EI
 F exp(iQ  R )  I
E
2
Giving up to the selection of the scattered neutron direction ()
too, one writes the s. cross-section:
 4 b 2
for E  0 (bound c.s.)
 d 
2
 ( E )   d 


2

4

b
 d  θ, φ, E
mn2
for E   (free c.s.)
Neutron scattering from an extended system
 I,F (rN )   I,F (R1 , R 2 ,...,R N ) (many body st at es)
2  2
2  2
b  (rn  rN ) 
mn
mn
N
 b  (r
j1
j
n
 R j ) (comb- like pot ent ial)
So, is everything so easy in NS? No, not quite…
Neutrons and incoherence
“Real-life” neutron scattering (from a set of
nuclei with the same Z)
 d 2   d 2 
 d 2 

  


 
 ddE'   ddE' COH  ddE'  INC
2

d
where:   

 ddE' 

COH
E '  COH 1
 S (Q,  )
E 4
 d 2 
E '  INC 1

 
 Sself (Q,  )
E 4
 ddE'  INC
scattering laws defined for a many-body
system (set of nuclei with the same Z) as:

S (Q,  ) 
N
p 
I
I

Sself (Q,  ) 
N
F
 exp(iQ  R ) 
F
N
j1
 p 
I
I
F
2
N
j1
j
I
   EF  EI 
 F exp(iQ  R j )  I    EF  EI 
2
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:
2
N
 b exp(iQ  R )
j
j1

j
spin  position
N
N
 b exp(iQ  R )
j1
j
b
j
spin  position
for j  j' : bj bj'


 for j  j' : bj bj'
spin
spin
j'1
 bj bj
 bj
j'
exp(iQ  R ' j' )
spin  position
spin
spin
 b2
bj'
spin
 b 
2
The incoherence origin (rigorous theory):
2
 TOT  4 bˆ ;  COH  4 bˆ 
2
 INC
2
2

 4 bˆ  bˆ 


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:
B ˆ
ˆ
b  A  σˆ  i N
2
This implies the existence of b+ and b- (if iN>0) for
any isotope (N, Z), respectively for iN½:

1
A
(iN  1)b   iNb 
2iN  1


2
B
b  b
2iN  1

After some algebra, only for unpolarized neutrons
and nuclei, one can write:
1
iN , M N ; sn , mn bˆ iN , M N ; sn , mn  A;

2(2iN  1) M N ,m n
2
B
1
2
2
ˆ
iN , M N ; sn , mn b iN , M N ; sn , mn  A 
iN (iN  1)

2(2iN  1) M N ,m n
4
With various isotopes (cj ) one gets:

j
TOT:

N, j j
(iN, j  1)b  i b
ˆ
b   cj
2iN, j  1
j
 2
j
Deuterium
 2
j
(iN, j  1)(b )  iN, j (b )
2
ˆ
b   cj
2iN, j  1
j
Hydrogen
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
High-Q spatial incoherence
(rearranging…)
 d 2 
 d 2 
 COH

  


 ddE'  DIS  ddE' COH  INC
 d 2   TOT

 
 ddE'   INC
 d 2 


 ddE'  INC
 d 2 
 d 2 

  

 ddE'  INC  ddE'  DIS
Incoherent approximation
 d 2   TOT  d 2 

 

 for   0
 ddE'   INC  ddE'  INC

S (Q,  )  Sself (Q,  )  S dist (Q,  )  0
When does it apply in a crystal?
Q  2
u
2 1/ 2
d2
Practical example : D2SO4 (T=10 K)
d(DO)=0.091 nm
u21/2(D)=0.0158 nm
2u21/2/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:

I (Q, t )   d expit  S (Q,  )


I self (Q, t )   d expit  Sself (Q,  )

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):
1
I (Q, t ) 
N
 exp iQ  R (0)expiQ  R (t )
j
k
j,k
1
I self (Q, t )   exp iQ  R j (0)expiQ  R j (t )
N j
I dist (Q, t )  I (Q, t )  I self (Q, t )
So far we have dealt only with a pure monatomic
system (set of nuclei with the same Z).
But what about “real-life” samples
(e.g. chemical compounds)?
Sum over ”s” distinct species (concentration c[s]):
 d 2 
k' 1



 ddE' COH k 2
 d 2 
k' 1

 
 ddE'  INC k 2

(s, z)


dt
exp

i

t
c
[
s
]
c
[
z
]
b
b
I
(Q, t )

s z


s
z



(s)


dt
exp

i

t
c
[
s
]
b

b
I

s
self (Q, t )


2
s
2
s
where I(s)self(Q,t) is the so-called self intermediate
scattering function for the sth species:
1
(s)
I self (Q, t ) 
Ns
Ns

j1

 

exp  iQ  R sj (0) exp iQ  R sj (t )
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,):
1
(s)
Sself (Q,  ) 
2

(s)


dt
exp

i

t
I
self (Q, t )


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 sz),
or the cross intermediate scattering function for
the s,zth pair of species (if sz):
N
(s, z)
I (Q, t ) 
Ns N z

Ns N z
 
j
k
and S(s,z) (Q,) is the so-called total inelastic
structure factor for the sth species (if sz), or
the cross inelastic structure factor for the
s,zth pair of species (if sz):
S
(s, z)

s
z
exp

i
Q

R
(
0
)
exp
i
Q

R

j
k (t )
1
(Q, ) 
2

(s, z)


dt
exp

i

t
I
(Q, t )


The total contains a “distinct” plus a “self”
terms, while the cross only a “distinct” term.
Coherent sum rules

n

from:  d  n S (Q,  )  (i) n n I (Q, t )
t

t 0

0th )  d S (Q, )  S (Q)  Static structurefactor

where:
1
S (Q)   pI I
N I

N
 exp(iQ  R

) exp(iQ  R j ) I
k, j
1 )  d  S (Q,  ) 
st
k
2
 Q
2M
2
 ER  Recoil Energy
Incoherent sum rules


from:  d  Sself (Q,  )  (i) n I self (Q, t )
t

t 0
n
n
n

0th )  d Sself (Q,  )  1  Normalization


1 )  d  S self (Q,  ) 
st
2
 Q
2M

2
 ER  Recoil Energy

2nd )  d   ER  Sself (Q,  )   2 v  Q 
2
2

 Kineticenergy: Ek
MN 2

v
2

4

3
3rd )  d   ER  Sself (Q,  ) 
2
2
M

 Q   UQ
i
i
j
j
i, j
 Laplacianof t he pot ent ialU

4 th )  d   ER  Sself (Q,  )   4 v  Q 
4
4



2
4 
 2 U  Q  Square gradient of the potentialU
M
Detailed balance


S (Q,ω)  exp  kBT S (Q,  )


Sself (Q,ω)  exp  kBT Sself (Q,  )
from the microscopic reversibility principle:
p
n
m exp iQ  R1  n     Em  En  
m
n exp iQ  R1  m     En  Em  
2
m,n
p
2
n, m
2
pm
pn
m expiQ  R1  n    Em  En  q.e.d.

pn
m,n
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.
Generalized scattering law (sometimes used for
mixed systems)
1
(Q,  ) 
2
(
b
)
 m
m

dt
 2 exp i t 
  bn bn' exp  iQ  R n (0)exp iQ  R n' (t )
n, n'
Generalized self scattering law (sometimes
used for mixed systems)

1
dt
 self (Q,  ) 
exp i t 

 INC, m  2
m
   INC, n exp  iQ  R n (0)exp iQ  R n (t )
n
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):
exp X  exp

1
2
X
2

Three-dimensional crystalline lattice (N cells and r
atoms in the elementary cell: “l” and “d” indexes):
R l.d (t )  l
 d  ul,d (t )
equilibrium
Harmonicity (expansion of ul,d(t) in normal
modes; e.g. phonon “s”; quantized):

u l,d (t ) 
2 NM d
  e
3Nr
s 1
1 / 2
s
s, d
as expi q  l  ist 

 e a exp i q  l  ist 

s, d

s
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 q1BZ (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:
 d 2 
k'



bd bd' expiQ  d'd  l'

 ddE' COH Nk d l',d'

dt
 2 exp i t  exp iQ  u0,d (0)expiQ  ul',d' (t )
and using the Bloch theorem together with
the commutation rules: eAeB=eA+Be[A,B]/2,
one writes:
exp iQ  u 0,d (0)expiQ  u l',d' (t )

1
2
2
 exp  Q  u 0,d (0)  Q  u 0,d' (t )
2

 exp Q  u 0,d (0)Q  u l',d' (t )
and then:
 d 2 
k'



bd bd' expiQ  d'd  l '

 d dE'  COH Nk d l',d'

dt
exp Bd (Q,0)/2  Bd' (Q,0)/2 
exp i t  expBd,l',d' (Q, t )
2

where the static term is the Debye-Waller
factor, whose exponent is:

B0,d (Q,0) 
2 NM d

Q  e d,s
s
s
2
2ns  1  2Wd (Q)
with ns number of thermally activated phonons;
while the dynamic term contains:

Bd,l',d' (Q, t ) 
2 N M d M d'

s
(Q  e d,s )(Q  e*d',s )
s
n
s
1
exp[ist  iq s  (d'd  l' )]  ns exp[ist  iq s  (d'd  l' )]
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):
1
2
expBd,d',l' (Q, t )  1  Bd,d',l' (Q, t )  Bd,d',l' (Q, t )  ...
2
1
n
 Bd,d',l' (Q, t )  ...
n!
one gets for the first term, 1, an elastic contribution:
 d 2 


 N 1  bd bd' expiQ  d'd  l '
d l',d'
 ddE'  COH, elast
exp Bd (Q,0)/2  Bd' (Q,0)/2  
Integrating over E’ and making use of the reciprocal
lattice () sum rule: l exp(iQl) = 83vcell-1 (Q-),
one obtains the well-known Bragg law:
8 3
 d 

bd bd' expiQ  d'd exp Wd (Q)  Wd' (Q)



Nvcell d,d'
 d  COH,
elast

2
Fn ( Q ) : nuclear unit -cell structure factor
   (Q  τ ) (Bragg peaks)
τ
Neutron powder diffraction pattern from Li2NH
(plus Al container) at room temperature.
Abscissa: d=2Q -1
One-phonon coherent
contribution
If ER,d(Q)</s-1 [the lightest Md] then:
expBd,d',l' (Q, t )1  Bd,d',l' (Q, t )
and one obtains the single phonon (created or
annihilated) d. d. coherent cross section:
 d 2 
k'



bdbd' expiQ  d'd  l'

 ddE' COH, 1 Nk d l',d'

dt
exp Wd (Q)  Wd' (Q) 
exp i t Bd,l',d' (Q, t )
2

Plugging the equation for Bd,d’,l’(Q,t) into
the one-phonon coherent d. d. crosssection and
performing the Fourier
transforms and the reciprocal lattice
sums, one gets:
 d  
bd
k ' 8
1



s 
expiQ  d  Wd (Q)Q  e d,s 

Md
d
 ddE'  COH, 1 k 2 Nvcell s
2
3
2
 ns  1    s   Q-q s -τ  (1  phonon, j (q), creation  0)
τ
 d 
bd
k ' 8
1



s 
expiQ  d  Wd (Q)Q  e d,s 

Md
d
 ddE' COH, 1 k 2 Nvcell s
2
3
 ns    s   Q  q s -τ  (1  phonon, j (q), annihilation   0)
τ
2
Dispersion
Curve
practical 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 zone
in a face-centered cubic lattice
(f.c.c.)
1st Brillouin zone
f.c.c.
face-centered cubic lattice
One-phonon incoherent
contribution
Bloch
theorem:
(d)

 INC exp iQ  R l,d (0)expiQ  R l,d (t ) 
l,d
(d)
 N   INC
exp Bd (Q,0)expBd (Q, t )
d
where the static term is the Debye-Waller
factor, whose exponent is:

Bd (Q,0) 
2 NM d

s
Q  e d,s
s
2
2ns  1
with ns number of thermally activated phonons;
while the dynamic term contains:

Bd (Q, t ) 
2 NM d

s
Q  ed,s
s
 ns exp ist 
2
n
s
 1 expist 
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):
1
2
expBd (Q, t )  1  Bd (Q, t )  Bd (Q, t )  ...
2
1
n
 Bd (Q, t )  ...
n!
if ER,d(Q)</s-1 then:
expBd (Q, t )  1  Bd (Q, t )
and one obtains the single phonon (creation or
annihilation) d. d. incoherent cross section:
 d  
k' 
1


 
 ddE'  INC, 1 k d 4 2 NM d
2
(d)
INC

Q  e d,s
2
s
s
  ns  1    s  ns    s exp Bd (Q,0)
Density of (phonon) states: density probability
for a phonon of any kind with frequency between
 and +d:
1
g ( ) 
3rN
3rN
 (   )
s
s
The single-phonon incoherent d. d. cross section
(creation or annihilation) becomes more simply:
(d)
 d 2 
k '  INC
Q2


 
 ddE'  INC, 1 k d 4 2M d

r ed (  )
2
g(  )
{1  exp[ (k BT ) ]}
1
exp 2Wd (Q)
where:
1  exp  (k T ) 
1
B
1
 n( ) for   0 (annih.)

 n( )  1 for   0 (creat.)
weak point (“”), i.e. the meaning of the
averaged eigenvector:
ed ( )
2
1
2

e

d,s
3rNg ( )  ωs  Δ
The separation from Q is rigorous only in cubic
lattices:
Q  e 
2
d,s
s
Q2
2

ed ( )
3
otherwise one has the isotropic approximation.
In addition, using this approximation, one
proves that:
Q2 2
Bd (Q,0)  2Wd (Q) 
ud
3
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:
Gd (  )  r e d (  )
u d2
2
g(  )

  
3 Gd ( )
d

coth

2M d 0 
 2k BT 
(d)
 d 2 
k '  INC
Q2


 
 ddE'  INC, 1 k d 4 2M d
from
which:
Gd (  )
 Q2 2 

exp 
u d 
1
{1  exp[ (kBT ) ]}  3

Density of states
projected on H
practical example:
lithium hydride LiH
DDM13, (Dyck & Jex, 1981)
SM-IV, (Verble, Warren & Yarnell, 1968)
TOSCA-II (Colognesi et al., 2002)
0.05
LiH,
Optic
-1
ZH() (meV )
0.04
0.03
—————— longitudinal
—————— transverse
0.02
0.01
0.00
70
80
90
100
110
120
 (meV)
130
140
150
Multiphonon incoherent
contributions
Coherent multiphonon terms are too complex and not
very useful (e.g. for powders: Bredov approximation).
Here only incoherent terms. Definition:
 d 2 
 d 2 



 
 ddE'  INC, Mult  ddE'  INC
 d 2 

 
 ddE'  INC, 1
remembering that:
 d  
k'

 
 ddE'  INC k
2

dt
 2 exp i t 
(d)
 INC

exp Bd (Q,0)expBd (Q, t )
4
d
and that:
1
2
expBd (Q, t )  1  Bd (Q, t )  Bd (Q, t )  ...
2
1
n
 Bd (Q, t )  ...
n!
one gets for the first term, 1, an elastic contribution:
 d 2 
k'



 ddE'  INC, Elas k

dt
 2 exp i t 
(d)
(d)
 INC
k '  INC

exp 2Wd (Q)  
exp 2Wd (Q)  
4
k d 4
d
not to be confused with the incoherent s. d. crosssection:

 d 2 
 d 
 dE'

   
 d  INC 0  ddE'  INC
For the second term one gets, B(Q,t), the single
phonon contribution (1, created or annihilated)
already known:
(d )
 d 2 
k '  INC Q 2


 
 ddE'  INC, 1 k d 4 2M d

Gd (  )
{1  exp[ (k BT ) ]}
1
exp 2Wd (Q)
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:

dt
~
~
~
n
Bd (Q,  )  Bd (Q,  )  ... Bd (Q,  )
0 2 exp i t  Bd (Q, t )  



~
 Bd (Q,  )

n times
n
where:
Gd (  )
Q 2
Q 2
~
Bd (Q,  ) 

f d ( )
2M d  {1  exp[ /(kBT )]} 2M d
we obtain:
 d  

k'




 ddE'  INC,  n k d 4
(d)
INC
2
 Q 


 2M d 
2
n
n

f d ( )

exp 2W (Q)
n!
d
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 f d ( ) :
2 M d u d2
Ad 
;
3
3
1
md 

A
d ;
2
2M d u d
vd 
2
M d u d2
Ek
d
Properties
of fd()
 md2
Then we have:
 f d ( )
n
2




  nmd  
1
1
  exp

2nvd 
2 nvd  md 

n

Not always appropriate…
3.0
self(Q,) (a.u.)
2.5
2.0
1.5
1.0
0.5
0.0
0
20
40
60
80
100
120
140
160
180
E (meV)
-D-glucose at T=19 K,
example from TOSCA