1)Find the Lande
g-factor for an atom in each of the following states:
^{}
^{3 }F _{3} , ^{3 }F _{2} and
^{3 }F _{4}
Solutions:
The Lande g-factor is:
g = 1 +
[J (J + 1) + S (S + 1) - L (L + 1)/ 2 J(J+1)]
For ^{3 }F _{3}: J = 3, S
= 1, L = 3
g = 1 +
[3 (3 + 1) + 1 (1 + 1) - 3 (3 + 1)/ 2 x3(3+1)]
g = 1 +
[12 + 2 - 12 / 24] = 1
+ 1/12 = 1.08
For ^{3 }F _{2}: J = 2, S
= 1, L = 3
g = 1 +
[2 (2 + 1) + 1 (1 + 1) - 3 (3 + 1)/ 2 x2(2+1)]
g = 1 +
[6 + 2 - 12/ 12] = 1 +
(-4/12) = 8/12= 2/3
For ^{3 }F _{4}: J = 4, S
= 1, L = 3
g = 1 +
[4 (4 + 1) + 1 (1 + 1) - 3 (3 + 1)/ 2 x3(3+1)]
g = 1 +
[20 + 2 – 12 / 24] = 1 +
5/12 = 1.41
2(a)
Assuming the L·S interaction to be much stronger than the
interaction with an external magnetic field, calculate the anomalous Zeeman
splitting of the lowest energy states:
^{2 }S _{1/2} , ^{2 }P _{1/2} and
^{2 }P _{3/2}
In
the hydrogen atom for a field of 0.05T
Present
a table with the results of the calculations showing the energy states in the
extreme left side column under ‘State’,
with the headers of the other columns:
L, S, J,
g
, M_{ J} , D E (in eV x 10 ^{-5 } )
Solution:
From the
example problem :
m _{J
}= m _{J} J / | J | = g (-e ħ
/2m) Ö J (J +
1) [ J/ Ö J (J +
1) ħ =
- e ħ /2m (g J)
The energy splitting is then given by:
D E =
- m _{J} B
= e ħ /2m g J·B = e ħ /2m g B J_{
z }= e ħ /2m g B M_{ J}
And we know already from quantum mechanics that:
M_{ J} = J, J – 1,……, -J + 1, - J
So that for a given field intensity B each energy
level will split into 2J + 1 sublevels with the amount of splitting determined
by the g –factor.
Then: D E = e ħ /2m g B M_{ J}
_{}
= (5.79 x 10^{-5} eV/T ) g (0.05T) M_{
J}
So, calculate the value of g for each energy state, i.e.
g = 1 + [J (J + 1) + S (S + 1) - L (L + 1)/ 2 J(J+1)]
Then substitute into the equation for D E with the correct value of M_{ J}
We obtain for the results table:
g = 1 + [J (J + 1) + S (S + 1) - L (L + 1)/ 2 J(J+1)]
Then substitute into the equation for D E with the correct value of M_{ J}
We obtain for the results table:
2(b) Given that: m _{s} = - e (2S)/ 2m and:
m _{L = }(-e (L)/ 2m)
Show in a vector diagram that m and J are not parallel.
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