Syed Arshad Hussain
Lecturer
Department of Physics
Tripura University
►Doublet structure of Alkali spectra:
Careful examination of the spectra of alkali metals shows that each member of some of
the series are closed doublets. For example, sodium yellow line, corresponding to 3 p → 3s
transition, is a close doublet with separation of 6A0 while potassium (K) has a doublet separation
of 34A0 and so on. Further investigations show that only the S-terms are singlet, while all the
other terms P, D, F etc. are doublets. Such doublet structure in energy is observed for all the
atoms possessing a single valence electron i.e., in the outer most shell. Usually the doublet
spacing is small compared to the term difference (for Na the main D line is cantered at 5893A0;
D2 = 5890A0 and D1 = 5896A0) and hence it is called fine structure. To explain this feature
Uhlenback and Goudsmith first proposed the hypothesis of electron spin, which was later on
obtained as a natural consequence of Dirac’s relativistic theory.
Spin is essentially a quantum
phenomenon. The spin of the electron is found
1
1
to be h and S 2 = s ( s + 1) h 2 where s = , the
2
2
quantum number for spin.
(Li, Na, K, Rb, Cs, Fr)
Explanation of doublet structure of alkali atom:
The Hamiltonian for the lone electron of an alkali atom relative to the atomic core is
given by,
H0 =
p2
+ V ( r ) ............(1)
2µ
ur
P = momentum of the lone electron and
r
V r = Potential.
r
r. = Distance of the lone electron from the centre of the atomic core, i.e., nucleus.
Where,
()
Now considering the spin orbit interaction the total Hamiltonian is given by,
2
S-O
p
+V (r ) + H
............(2)
H=
2µ
Where H
S-O
= spin orbit interaction term
Alkali spectra
Syed Arshad Hussain
Lecturer
Department of Physics
Tripura University
r
r
An electron with orbital angular momentum l and spin s will behave a total angular
momentum
r r r
j =l+s
r
Which gives the quantum numbers for j as j = (l + s ),........., (l − s ) . Since for a single electron
1
1
1
1
1
s = or − ∴ j = l + and l − , for l = 0; j = only.
2
2
2
2
2
The coupling of spin with orbital angular momentum gives rise to the fine structure
splitting of spectral lines and it gives a full account of the doublet structure of alkali spectra. This
interaction is called spin orbit interaction.
The existence of electron spin and the doublet structure comes as a natural consequence
of the relativistic theory but classical electrodynamics gives a non-realistic description.r
r
Consider an electron of charge –e moving with velocity v at a distance r from the
nucleus of charge Ze (or an entire atomic configuration of effective charge Ze).
Viewed from electrons rest frame the entire atomic configuration (consisting of the
nucleus and the rest of the electronic charge cloud be taken as rigid core) is moving with a
velocity of – V and their effective field also moves with the same velocity. Associated with the
v
moving electric field arising from relativistic transformation equation. To first order in . the
c
r 1 r r
magnetic field is Β = E × v
c
ur
ur
and, E = −∇φ ; φ = scalar potential
r
1 ur
∴Β = − ∇Φ × v
c
1 ur r
= + ∇V × v
ec
where V = - eφ = potential.
Now for a spherical symmetric potential
r
ur
dV r
.r
∇V =
dr r
1 1 dV r r
.
r ×v
ec r dr
r
r
1 1 dV r ur
⎡ p = mv ⎤
.
r× p
=
⎣
⎦
mec r dr
r
r
u
r
1 1 dV
⎡l = r × p = angular momentum.⎤
.
l
=
⎣
⎦
mec r dr
∴Β =
Alkali spectra
Syed Arshad Hussain
Lecturer
Department of Physics
Tripura University
If the coordinate system is fixed with the atom the effective magnetic field is to be
1
multiplied by a factor . This is known as Thomas correction obtained from a spinning top
2
model of the electron.
1 1 dV r
.
∴Β =
l
2mec r dr
The interaction energy of the electronic magnetic moment with the magnetic field due to
the motion of charge particle is
uur r
S −O
H
= − µ s .Β
1 1 dV r r ⎡
e
e r⎤
=
.
.
l.s ⎢ µs = −
s
mc 2mec r dr
mc ⎥⎦
⎣
This equation holds in the rest frame of both electron and the nucleus as to first order in
v
, the energy is same. There is no term proportional to l 2 as in the rest frame of the electron;
c
there is no orbital magnetic moment.
r
r
r
r
Now we can write using quantum mechanical operator for l and s as hl and hs .
(
∴H
)
h 2 1 dV r r
.
l.s
2m 2 c 2 r dr
rr
=ξ ( r ) l.s ...........(2)
S −O
=
r
r
Where l and s are dimensionless vector operators,
h 2 1 dv
ξ (r ) =
.
2m 2c 2 r dr
The total Hamiltonian of the alkali atom considering spin orbit interaction is given by,
2
r
p
H=
+ V (r ) + ξ ( r ) l.s ...................(3)
2µ
For a pure Coulombic potential,
Ze 2
V (r ) = −
r
1 dV Ze 2
∴
= 3
r dr
r
h 2 Ze 2 1
∴ξ ( r ) =
.
.............(4)
2m 2 c 2 r 3
( )
Now the spin orbit term can be considered as perturbation term. The contribution of the
spin orbit term can be calculated by considering its expectation value. (here we consider only the
first order correction)
Alkali spectra
Syed Arshad Hussain
Lecturer
Department of Physics
Tripura University
S −O
∴∆E = ψ H
ψ
.............(5)
Where ψ is the ground state wave function of the effective one particle system.
r
In an atom the total angular momentum j is always conserved even if individual
r
r
S −O
l and s may not be conserved. Hence we can work in a representation or system where H
is
r
r
diagonal. Also we choose nl j m j representation, where individual l and s are combined to
r
form the conserved quantity j
Thus,
∆ E = ψ n ljm j H
S −O
ψ n ljm j
rr
= ψ n l ( r ) ξ ( r ) ψ n l ( r ) ψ lsjm j l . s ϕ lsjm j
rr
..........................(5 ).
= ζ n l l .s
W h ere,
ζ nl = ψ nl (r ) ξ (r ) ψ nl (r )
∞
=
∫ Rnl ( r )ξ ( r ) r
2
2
dr
0
=
Ze2h2
2 2
2m c
=
Alkali spectra
2
0
1
2 2
r3
Ze h
1
∫ Rnl (r ) r 3 r
2 2
2m c
We have,
∞
2
dr
Syed Arshad Hussain
Lecturer
Department of Physics
Tripura University
Z3
=
;
⎛ 1⎞ 3
3
r3
n l ( l + 1) ⎜ l + ⎟ a0
⎝ 2⎠
1
∴We can write, ζ nl = R yα 2
Where, R y =
me4
2h 2
a0 =
h2
me 2
Z4
⎛ 1⎞
n3l ( l + 1) ⎜ l + ⎟
⎝ 2⎠
= Rydberg constant and
e2
= fine structure constant.
hc
rr
To calculate the angular term l.s we proceed as follows:
r r 1 uur2 ur2 uur2
l.s = ⎡ j − l − s ⎤
⎦
2⎣
α=
rr
rr
∴ ψ lsjm j l.s ψ lsjm j = l.s
1 2 2 2
j −l − s
2
1
= ⎡⎣ j ( j + 1) − l ( l + 1) − s ( s + 1) ⎤⎦
2
1⎡
3⎤
1
= ⎢ j ( j + 1) − l ( l + 1) − ⎥
Qs =
2⎣
4⎦
2
1
1
Here for a given l , j = l + and l − .
2
2
1
1
Thus each level split into two sublevels with j = l + and l − .
2
2
=
Alkali spectra
Syed Arshad Hussain
Lecturer
Department of Physics
Tripura University
For $j = l +
1
2
rr
1 ⎡⎛ 1 ⎞ ⎛ 3 ⎞
3⎤
l.s = ⎢⎜ l + ⎟ ⎜ l + ⎟ − l ( l + 1) − ⎥
2 ⎣⎝ 2 ⎠ ⎝ 2 ⎠
4⎦
1⎡
3l 1
3
3⎤
= ⎢l 2 + + l + − l 2 − l − ⎥
2⎣
2 2 4
4⎦
l
=
2
1
For $j = l −
2
r
1 ⎡⎛ 1 ⎞ ⎛ 3 ⎞
3⎤
l.s$ = ⎢⎜ l + ⎟ ⎜ l + ⎟ − l ( l + 1) − ⎥
2 ⎣⎝ 2 ⎠ ⎝ 2 ⎠
4⎦
1⎡ 2 1 2
3⎤
l − −l −l − ⎥
⎢
2⎣
4
4⎦
1
= − ( l + 1)
2
Thus the energy levels are given by,
1⎞
l
⎛
E1 ⎜ $j = l + ⎟ = E0 + ζ nl and
2⎠
2
⎝
( l + 1)
1⎞
⎛
E2 ⎜ $j = l − ⎟ = E0 − ζ nl
2⎠
2
⎝
=
The separation is given by,
⎛ 1⎞
∆E = E1 − E2 = ζ nl ⎜ l + ⎟
⎝ 2⎠
Thus the splitting of energy levels i.e., separation between the two levels for each value
1
of l is ( 2l + 1) ζ nl . This is the so called fine structure of spectral lines for spin orbit interaction
2
of energy levels.
Now we may draw the energy level diagram for the non relativistic theory then the
correction to the energy levels due to relativistic variation of mass and at last correction to the
energy level due to the presence of spin orbit interaction.
Alkali spectra
Syed Arshad Hussain
Lecturer
Department of Physics
Tripura University
1
1
Now for l = 0, j = ± but j can never be –ve . Hence j = + . Therefore 1s level doesn’t split up
2
2
for ( S-O) interaction.
1 3
For l = 1, j = ,
2 2
5 3
l = 2, j = ,
2 2
7 5
l = 3, j = , and so on
2 2
Thus except l = 0 (which has no S − O interaction) the other energy levels split up into two
levels (doubles) as shown below due to S − O interaction.
The mechanism responsible to the doublet splitting is S − O interaction. Each energy
1⎞
1
⎛
level has got a spin multiplicity ⎜ 2s+ ⎟ = 2. + 1 = 1 + 1 = 2
2⎠
2
⎝
Alkali spectra
Syed Arshad Hussain
Lecturer
Department of Physics
Tripura University
In spectroscopic notation, each energy level is represented as,
2 s +1
n
Lj
n = Principal quantum number.
L = orbital angular momentum quantum number.
J = Total angular momentum quantum number = l + s
2 s + 1 = spin multiplicity.
l = 0, 1, ……….n – 1
ml = -l………. +l
j=l+s
j = (l + s), (l +s – 1), ………… (l – s).
Alkali spectra