The One-Electron Atom
127
Example 3
A beam of sodium atoms with velocity 10
3
m/s, moves along the magnetic poles
over a distance of 0.15 m, in the x-direction. We determine the separation between
the two components of the beam, at a distance of 0.6 m from the magnet, given
that the magnetic induction between the poles varies in the z-direction as
B = (1 – 100z) W/m
2
(4.114)
The force on the atoms is
F z = ±
(100)
2 e
e
m
= ± 9.28 × 10
–22
N
(4.115)
If t 1 is the time taken by the atoms to traverse the poles and t 2 is the time
taken to move from the magnets to the plane of observation, the separation
between the two components of the beam is
∆z =
2
1
1 2
| |
| |
1
2 2
z
z
n
n
F
F
t
tt
m
m
+
(4.116)
Since t 1 ≈ 1.5 × 10
–4
s and t 2 ≈ 6.0 × 10
–4
s, one gets
∆z = 4.9 × 10
–3
m
(4.117)
Example 4
Some general properties of fine structure lines are now enumerated.
1. Transitions from the level with principal quantum number n > 1, to the
ground state give rise to doublets corresponding to
np 1/2 → 1s 1/2 , np 3/2 → 1s 1/2
(4.118)
2. Transitions from the level with principal quantum number n > 2 to the level
with n = 2, gives rise to seven lines corresponding to
np 1/2, 3/2 → 2s 1/2
ns 1/2 → 2p 1/2, 3/2
(4.119)
nd 3/2 → 2p 1/2, 3/2
nd 5/2 → 2p 3/2
3. If n 0 ≥ 2, where n 0 is the principal quantum number of the final state,
every increase in n 0 increases the number of fine structure line by 6.
These additional lines correspond to
127
Example 3
A beam of sodium atoms with velocity 10
3
m/s, moves along the magnetic poles
over a distance of 0.15 m, in the x-direction. We determine the separation between
the two components of the beam, at a distance of 0.6 m from the magnet, given
that the magnetic induction between the poles varies in the z-direction as
B = (1 – 100z) W/m
2
(4.114)
The force on the atoms is
F z = ±
(100)
2 e
e
m
= ± 9.28 × 10
–22
N
(4.115)
If t 1 is the time taken by the atoms to traverse the poles and t 2 is the time
taken to move from the magnets to the plane of observation, the separation
between the two components of the beam is
∆z =
2
1
1 2
| |
| |
1
2 2
z
z
n
n
F
F
t
tt
m
m
+
(4.116)
Since t 1 ≈ 1.5 × 10
–4
s and t 2 ≈ 6.0 × 10
–4
s, one gets
∆z = 4.9 × 10
–3
m
(4.117)
Example 4
Some general properties of fine structure lines are now enumerated.
1. Transitions from the level with principal quantum number n > 1, to the
ground state give rise to doublets corresponding to
np 1/2 → 1s 1/2 , np 3/2 → 1s 1/2
(4.118)
2. Transitions from the level with principal quantum number n > 2 to the level
with n = 2, gives rise to seven lines corresponding to
np 1/2, 3/2 → 2s 1/2
ns 1/2 → 2p 1/2, 3/2
(4.119)
nd 3/2 → 2p 1/2, 3/2
nd 5/2 → 2p 3/2
3. If n 0 ≥ 2, where n 0 is the principal quantum number of the final state,
every increase in n 0 increases the number of fine structure line by 6.
These additional lines correspond to