10. Angular Momentum in Atoms

Question 1

State the equation for the magnetic moment of an electron

Answer 1

μ_e = -e/(2m_e) gS
where g is the gyromagnetic ratio ~2.0032
S is the electron spin

Question 2

State the equation for the total angular momentum, J

Answer 2

*J* = *L* + *S*
Total = orbital + spin

Question 3

State the equation for the total magnetic moment, μ_T

Answer 3

*μ_T* = *μ_orb* + *μ_spin*
Total  = orbital + spin

Question 4

State the symbol for the angular momentum quantum number

Answer 4

j

Question 5

State the symbol for the angular momentum projection quantum number

Answer 5

m_j

Question 6

State the equation for the total angular momentum, J in terms of h_bar and j

Answer 6

J | = h_bar sqrt( j(j+1) )

Question 7

State the equation for the z component of the angular momentum, J in terms of h_bar and m_j

Answer 7

J_z = h_bar * m_j

Question 8

What is j and m_j equal to?

Answer 8

j = | l + 1/2 |
m_j = -(l+1/2) ... -1/2, 1/2, ... (l + 1/2)

Question 9

State the two selection rules that we need to know

Answer 9

Δl = ± 1
Δm_l = 0, ± 1

Question 10

See page 5 of document for spin orbit coupling diagram

Answer 10

Do it

Question 11

For the spin orbit coupling diagram, what does the nucleus orbiting the electron produce?

Answer 11

A current loop -> effective magnetic field B_eff

Question 12

State the equation for the interaction energy due to the magnetic moment in the field

Answer 12

ΔE = - μ_e * B_eff

Question 13

What orientations does electron spin have?

Answer 13

up or down

Question 14

State the symbol of the interaction energy for electrons with different spins

Answer 14

up - +ve

down - -ve

Question 15

What is spin orbit coupling proportional to?

Answer 15

Z^4

Question 16

Where does the Z^4 come from?

Answer 16

Z - more charge
Z^3 - higher orbits
Both together create a stronger B_eff

Question 17

What is “fine structure”?

Answer 17

When SOC and external magnetic fields lead to small splittings in atomic energy levels