Scattering Theory - Invariant

1
Scattering Theory - Invariant
(Porod (1951)) Q ?I(S) dS (1/(2p)3)?I(q)
dq total scattering over the whole of
reciprocal space
2
Scattering Theory - Invariant
(Porod (1951)) Q ?I(S) dS (1/(2p)3)?I(q)
dq total scattering over the whole of
reciprocal space Q ?p (0) lt?2gt V
(see derivation in Roe, 1.5)
3
Pair distribution fcn
Single atomic species (Roe, Sect.
4.1) Short range order Atom environments vary
- can only get an avg picture Define PDF g(r) to
describe structure Can get PDF from scattering
data
4
Pair distribution fcn
Single atomic species (Roe, Sect.
4.1) Short range order Atom environments vary
- can only get an avg picture Define PDF g(r) to
describe structure of atoms, on avg, in
dr n2 (r) dr, and g(r) n2 (r)/ ltngt where
ltngt avg density of atoms
atom
dr
r
5
Pair distribution fcn
Single atomic species (Roe, Sect.
4.1) Short range order g(r) radial
distribution fcn when material is amorphous
(isotropic)
6
Pair distribution fcn
Single atomic species I(q) A(q) 2 (b ?
exp(iq rj) )(b ??exp(-iq rk)) N b2
b2 ???? exp(-iq r jk) rjk rj - rk
N b2 N b2 ? n2 (r) exp(-iq
rk) dr
N
N
j1
k1
N
j1
j?k
V
7
Pair distribution fcn
Single atomic species I(q) A(q) 2 (b ?
exp(iq rj) )(b ??exp(-iq rk)) N b2
b2 ???? exp(-iq r jk) rjk rj - rk
N b2 N b2 ? n2 (r) exp(-iq
rk) dr N b2 N b2 ? n2 (r) - ltngtexp(-iq
rk) dr N b2 ?ltngtexp(-iq rk) dr N b2 N b2
ltngt ? g(r) - 1exp(-iq rk) dr N b2 ltngt ?(q)
N
N
j1
k1
N
j1
j?k
V
8
Pair distribution fcn
Single atomic species I(q) A(q) 2 (b ?
exp(iq rj) )(b ??exp(-iq rk)) N b2
b2 ???? exp(-iq r jk) rjk rj - rk
N b2 N b2 ? n2 (r) exp(-iq r)
dr N b2 N b2 ? n2 (r) - ltngtexp(-iq r) dr
N b2 ?ltngtexp(-iq r) dr N b2 N b2 ltngt ?
g(r) - 1exp(-iqr) dr N b2 ltngt
?(q) unobservable - ignore Define
interference fcn i(q) (I(q) - N b2)/ N b2
N
N
j1
k1
N
j1
j?k
V
9
Pair distribution fcn
Single atomic species N b2 N b2 ltngt ? g(r)
- 1exp(-iqr) dr N b2 ltngt ?(q) unobservab
le - ignore Define interference fcn i(q)
(I(q) - N b2)/ N b2 Then i(q) ltngt ? g(r) -
1exp(-iqr) dr And get g(r) - 1 from inverse
Fourier transform of i(q) g(r) - 1 1/ltngt
?i(q) exp(iqr) dq
V
V
V
10
Pair distribution fcn
Simple polymer (Cs Hs) Need gCC( r), gHH(r),
gCH(r) gCH(r) nCH(r) /ltnHgt ltnHgt
avg density of H atoms
11
Pair distribution fcn
Simple polymer (Cs Hs) Need gCC( r), gHH(r),
gCH(r) gCH(r) nCH(r) /ltnHgt ltnHgt
avg density of H atoms gCH(r) gHC(r)
12
Pair distribution fcn
Simple polymer I(q) ? N? b?2 ? N? b? ?
b? ? n?? (r) exp(-iqr) dr m different types of
atoms (for C H, m2) ?????( 1.m)? denote
atom types N? of ???????
m
m
m
?1
?1
?1
V
13
Pair distribution fcn
Simple polymer I(q) ? N? b?2 ? N? b? ?
b? ? n?? (r) exp(-iqr) dr m different types of
atoms (for C H, m2) ?????( 1.m)? denote
atom types N? of ??????? N? N x?????
ltn?gt ?? ltngt x?? x????????????? of
??????? Then I(q) N ? x? b?2 N ltngt? ? x?
x? b? b? ? g??(r) -1 exp(-iqr) dr
m
m
m
?1
?1
?1
V
m
m
m
V
?1
?1
?1
14
Pair distribution fcn
Simple polymer I(q) N ? x? b?2 N ltngt? ?
x? x? b? b? ? g??(r) -1 exp(-iqr) dr m
different types of atoms (for C H,
m2) ?????( 1.m)? denote atom types N?
of ??????? N? N x????? ltn?gt ?? ltngt x??
x????????????? of ??????? Then i(q) ltngt ?
g(r) - 1exp(-iqr) dr And get g(r) - 1 from
inverse Fourier transform of i(q) g(r) - 1
1/ltngt ?i(q) exp(iqr) dq
m
m
m
V
?1
?1
?1
V
15
Pair distribution fcn
Simple polymer I(q) N ? x? b?2 N ltngt? ?
x? x? b? b? ? g??(r) -1 exp(-iqr) dr i(q)
? g(r) - 1exp(-iqr) dr If w? x?b?/ ?
x?b? then ? g(r) exp(-iqr) dr ? ? w? w? ?
g?? (r) exp(-iqr) dr
m
m
m
V
?1
?1
?1
V
m
?1
m
m
V
V
?1
?1
16
Pair distribution fcn
Simple polymer I(q) N ? x? b?2 N ltngt? ?
x? x? b? b? ? g??(r) -1 exp(-iqr) dr i(q)
? g(r) - 1exp(-iqr) dr If w? x?b?/ ?
x?b? then ? g(r) exp(-iqr) dr ? ? w? w?
?g?? (r) exp(-iqr) dr Can get g(r), but not
separate g?????N????????????????????????.
m
m
m
V
?1
?1
?1
V
m
?1
m
m
V
V
?1
?1