The Zeeman Effect & Electron Spin

Question 1

What is the zeeman effect?

Answer 1

Where an external magnetic field “lifts the degeneracy” of the l-states in the H-like atom: single energy level splits to 2l+1 levels

Question 2

What does a 1D graph of ω look like without and with an applied B-field and what are its characteristics?

Answer 2

Zero B-field, single spike in middle centred at ω0. With applied B-field, 3 spikes with middle one at ω0 and one either side at +/- ω(L), the Larmor frequency.

Question 3

What is the equation for the Larmor frequency, ω(L)?

Answer 3

ω(L) = eV/2m(e)

Question 4

What is the equation for ΔE, the energy change of the split energy levels?

Answer 4

ΔE = +/-ћω(L) = +/- μ(B)*B, where μ(B) = eћ/2m(e) = Bohr Magneton

Question 5

What do the split spectra always comprise of? Why is this?

Answer 5

3 lines even if the state has l >= 2. Due to a second selection rule: Δm(l) = 0, +/- 1

Question 6

What is the equation for the z-component of the magnetic moment of the electron?

Answer 6

μ(z) = -e/2m(e) Lz

Question 7

What is the splitting of a spectral line into 3 frequencies (or wavelengths)?

Answer 7

The Normal Zeeman Effect

Question 8

What is the anomalous zeeman effect?

Answer 8

Where spectra have more complicated patterns of splitting (not just simple +/- ω(L) rule)

Question 9

What is spin?

Answer 9

Spin is another form on angular momentum intrinsic to particles.

Question 10

What values can the spin quantum number take?

Answer 10

Half-integer values

Question 11

What is the equation for the spin angular momentum component along the z-axis?

Answer 11

s(z) = +/- 1/2 *ћ

Question 12

What is the equation for the magnitude of the spin angular momentum s?

Answer 12

s = ћ*sqrt(s(s+1)) = sqrt(3)/2 *ћ

Question 13

What is the equation for the total angular moment J?

Answer 13

J = L + S, where L is the orbital angular momentum and S is the spin angular momentum.

Question 14

How can we define a total angular momentum quantum number?

Answer 14

|J| = ћ*sqrt(j(j+1)), j = | l+/- 1.2 |

Question 15

What is the equation for the z component of the total angular moment J(z)?

Answer 15

J(z) = ћ*m(j), where m(j) = 0, +/-1, ….. +/- j

Question 16

What happens when j = l+1/2 and when j = l-1/2?

Answer 16

j=l+1/2: L and S have a parallel z-component, j=l-1/2: L and S have anti-parallel z-component.

Question 17

What are the new selection rules for the total angular momentum quantum number j?

Answer 17

Δj = +/- 1, Δm(j) = 0, +/-1

Question 18

In the Stern-Gerlach experiment, how were the Silver atoms given kinetic energy?

Answer 18

Provided by thermal energy from the furnace used to sublimate the material.

Question 19

What was done with the silver after it was given KE in the Stern-Gerlach experiment/

Answer 19

Beam collimated and travels along flight tube, all in vacuum. Normally, beam would simply splat on to a detector in a single stripe due to the shape of the collimator.

Question 20

What was used in the Stern-Gerlach experiment to creat an inhomogeneous magnetic field?

Answer 20

Pair of specifically shaped magnetic pole pieces inserted across flight tube to generate the B-field.

Question 21

What was the surprising result of the Stern-Gerlach experiment? Why did this happen?

Answer 21

Beam split into two rather than an odd number 2l+1 stripes. This is explained by hypothesising intrinsic angular momentum of the atom with half-integer spin, so that 2s+1 = 2.

Question 22

What is the equation for ΔLi, the change in angular momentum for a small element of mass and charge?

Answer 22

ΔLi = ri x Δpi, where ri is the radius of the circle around the axis

Question 23

What does each element of mass and each element of charge contribute to for a classical charged entity?

Answer 23

Each element of mass contributes to the angular momentum. Each element of charge contributes to the magnetic moment.

Question 24

What is the equation for L(total)?

Answer 24

L(total) = sum over i of ri x Δpi

Question 25

What is the equation for μ(total)?

Answer 25

μ(total) = sum over i of Δqi/2mi *ri x Δpi

Question 26

What is a simple way of writing the magnetic moment μ?

Answer 26

μ = -q/2m *gL, where q is the total charge and m is the total mass and g is a dimensionless quantity called Gyromagnetic ratio.

Question 27

What is the equation for the total magnetic moment vector for an electron?

Answer 27

μ = -e/2m(e) *(L +gs), where s is the spin

Question 28

What can we learn from looking at the electrons frame of reference in the nucleus? What is the phenomena called?

Answer 28

The electron ‘sees’ a current loop due to the nucleus. The B-field due to the ‘orbiting’ nucleus interacts with the electrons spin magnetic moment. Called spin-orbit coupling.

Question 29

What is the equation for the spin-orbit interaction energy, V(mag)?

Answer 29

V(mag) = -μ(s).B, where B is the B-field

Question 30

Since spin can have 2 possible orientations, what does this mean for the degeneracy?

Answer 30

The spin-orbit coupling can lift the degeneracy of spin states (can make an up magnetic moment or a down magnetic moment depending on the spin).

Question 31

What is meant by fine structure?

Answer 31

Where external magnetic fields and SOC give rise to changes in optical spectra.