Solutions To Atomic Physics Problems

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:

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.