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Transcript
Chemistry 431
Lecture 9
Spin-orbit coupling
NC State University
Spin-orbit Hamiltonian
The magnitude of spin-orbit coupling is measured
spectroscopically as a splitting of spectral lines.
The Hamiltonian is:
HSO = 1 hcA j( j + 1)) – l(l
( + 1)) – s(s
( + 1))
2
Where l is the orbital angular momentum quantum
number and s is the spin quantum number.
The total angular moment is j = l + s and A is the
magnitude of the spin-orbit coupling in wavenumbers.
HSO = 1 hcA (l +s)(l + s + 1) – l(l + 1) – s(s + 1)
2
= 1 hcA l 2 + s 2 + ls + sl + l + s – l 2 – l – s 2 – s) = hcA l⋅s
2
Spin-orbit
p
Hamiltonian
The magnitude of the spin orbit coupling can be
calculated in terms of molecular parameters by the
substitution
2
Zα
hcA L⋅S=
< 13 >L⋅S
2
r
where α is the fine structure constant α = e2/hc4πε0,
which is a dimensionless constant (α
( = 1/137.037).
1/137 037)
L and S are operators.
Z is an effective atomic number
number.
The spin orbit coupling splitting can be calculated from
E SO =
Ψ H SOΨ dτ = Z 2
2(137)
*
LS
L⋅S
Ψ 3 Ψ dτ
r
*
Spin-orbit Hamiltonian
This can be evaluated using the above identity that
can be recast
2
2
2
1
L⋅S= J –L –S
2
to give an spin-orbit coupling energy in terms of
molecular parameters
Z
1
ESO = 1 j( j + 1) – l(l + 1) – s(s + 1)
2
2(137) 2 r 3
where
1 =
r3
Ψ
*
1 Ψdτ
r3
Spin-orbit Hamiltonian
We can evaluate this integral explicity for a given
atomic orbital. For example for Ψ210 we have
3/2
–Zr/2a 0
Ψ 210 = 1 aZ Zr
e
cos θ
a
0
4 2π 0
so that the integral is
1 = 1 Z
32π a 0
32
r3
π
2π
5
dφ
0
cos 2θsinθdθ
0
0
which integrates to
2
a
< 13 > = 1 aZ 2π 2 0
32π 0
3 Z
r
5
2
or Z3/24 in atomic units.
∞
=
1 Z
24 a 0
3
r 2e Zr/a 0 13 r 2dr
r
Spin-orbit Hamiltonian
Therefore we have
Therefore,
3
Z
< 13 > = 3 3
r
n a ol(l + 1/2)(l + 1)
Therefore, in general the spin-orbit splitting is given by
4
j( j + 1) – l(l + 1) – s(s + 1)
Z
ESO =
2l(l + 1/2)(l + 1)
2(137) 2ao3n 3
Note
N
t th
thatt the
th spin-orbit
i
bit coupling
li iincreases as th
the
fourth power of the effective nuclear charge Z, but
only as the third power of the principal quantum
number n. This indicates that spin orbit-coupling
interactions are significantly
g
y larger
g for atoms that
are further down a particular column of the periodic
table.
The Sodium D Line
One notable atomic spectral line of sodium vapor is
the so-called D-line, which may be observed directly
as the sodium flame-test line and also the major light
output of low-pressure
low pressure sodium lamps (these produce
an unnatural yellow. The D-line is one of the classified
Fraunhofer lines.
lines Sodium vapor in the upper layers of
the sun creates a dark line in the emitted spectrum of
electromagnetic
g
radiation byy absorbing
g visible light
g in
a band of wavelengths around 589.5 nm. This wavelength corresponds to transitions in atomic sodium in
which
hi h the
th valence-electron
l
l t
ttransitions
iti
ffrom a 3s
3 tto 3p
3
electronic state.
The Splitting of the D Line
Closer examination
e amination of the visible
isible spectr
spectrum
m of atomic
sodium reveals that the D-line actually consists of
two lines called the D1 and D2 lines at 589
589.6
6 nm
and 589.0 nm, respectively. The splitting between
These lines arises because of spin-orbit
p
coupling.
p g
The constant A is usually given in cm-1. For Na, it is
11.5 cm-1. Na has one unpaired electron (s = ½).
If we consider
id the
th s -> p transition,
t
iti
th
then ffor the
th
excited state, p, we have l = 1. Thus, j = 3/2 or ½.
Practical calculations using
the Spin-orbit Hamiltonian
The two
t o energy
energ le
levels
els can be calc
calculated
lated in terms of
the constant A.
E3/2 = 1 A 3/2(3/2 + 1) – 1(1 + 1) – 1/2(1/2 + 1) = 1 A
2
2
E1/2 = 1 A 1/2(1/2 + 1) – 1(1 + 1) – 1/2(1/2 + 1) = – A
2
The energy difference
Th
diff
between
b t
the
th lines
li
iis 3/2A
3/2A.
Thus, the energy splitting for Na is 17.3 cm-1.