Ch4 Fine structure of atoms

• Magnetic moments

• Spin of the electron

• Stern-Gerlach experiment

• Spectrum of the alkali atoms

• Spin-orbit coupling (interaction)

• The Zeeman effect

Some words

• Fine structure• Electron spin• Orbital motion• Magnetic moment• Alkali atoms• Valence • intrinsic • Magneton• Atomic state

• Spin-orbit coupling (interaction)

• Frame of reference• Larmor precession• Lande g factor• Alkali metals• diffuse• Stern-Gerlach experiment

• Zeeman’s effect

Magnetic moment of the orbital motion

• An electron moving in an orbit is equivalent to a circular current, which possesses a magnetic moment.

• The magnetic moment associated with the orbital angular momentum is quantised.

• An applied magnetic field B acts on the orbital magnetic moment by trying to align the vectors μl and B. The electrons precess about the direction of the field, called Larmor precession. There is also a potential energy E=- μl ·B.

)magnetonBohr (2

2 m

el

m

eB

Electron spin

• Electron spin was introduced by Uhlenbeck and Goudsmit in 1925. They proposed that the electron possesses an intrinsic angular momentum independent of any orbital angular momentum it might have. Four quantum numbers: n, l, ml, ms.

• The splitting of many spectral lines in a magnetic field can only be explained if the electron has a spin angular momentum s.

sm

egsss Ss

2 )1(||

The total angular momentum & magnetic moment

)ˆ

ˆˆ(

2

1

2

3

),cos(),cos(

,

2

22

j

lsg

Jg

sjlj

slj

j

Bj

slj

sl

l

s

j

μl

μjμ

μS

Fine structure• All energy levels except the s states of one-ele

ctron atoms are spit into two substates. This produces a doublet or multiplet structure of the spectral lines, namely, fine structure.

• It can not be explained by Coulomb interaction between the nucleus and the electrons. It results from a magnetic interaction between the orbital magnetic moment and spin magnetic moment of the electron, called spin-orbit coupling.

Spin-orbit coupling (I)• An electron revolving about a nucleus finds its

elf in a magnetic field produced by the nucleus which is circling about it in its own frame of reference.

• This magnetic field then acts upon the electron’s own spin magnetic moment to produce substates in terms of energy. S=1/2 could make some single state (n,l,m) split into two substates.

)1)((

])1()1([

4

)(

ˆˆ24

1

21

43

3

24

322

2

0

lll

lljj

n

cmZ

lsrcm

ZeBV

e

e

NucSls

Spin-orbit coupling (II)

• The coupling of two magnetic moments in an atom leads to an addition of the two angular momenta to yield a total angular momentum. j=l+s

• The total angular momentum has the magnitude of with j=|l±1/2| for a single electron system. The quantum number j is the quantum number of the total angular momentum .

• For a p electron, l=1, s=1/2, there are two states with j=1/2, 3/2, respectively.

)1( jj

The symbolism for atomic states• The atomic state is denoted by a symbol, and

the orbital angular momentum is indicated by upper case letters S,P,D,F.

• The primary quantum number n is written as an integer in front of the letter, and the total angular momentum quantum number j as a subscript.

• The multiplicity 2s+1 (the number of j values) is indicated by a superscript to the left of the letter.

)232111( 2 2 2/32

2/1212 /, /, j,lnPPln j

s

Lande g factor• It is a measure of the ratio of the magnetic mo

ment (in Bohr magnetons) to the angular momentum (in units of h bar).

• For the orbit angular momentum, j=l, gl=1. For spin angular momentum, j=s, gs=2.

• For the total angular momentum, g factor is given by:

)1( )1(

jjggmgjj B

j

jBjjjzBjj

)ˆ

ˆˆ(

2

1

2

32

22

j

lsg j

The alkali atoms• The alkali atoms have a weakly

bound outer electron, the so-called valence electron, and all other (Z-1) electrons are in closed shells.

• The yellow D line in the spectrum of the sodium (Na) atom, i.e. the transition 3P3S,is a doublet: D1=589nm, D2=589.6nm.

Alkali metals• Selection rules: • The first primary series: nP2S, P: double substa

tes , doublet, two lines are getting closer with n increased.

• The sharp secondary series: nS2P, doublet, two lines are separated uniformly with n increased.

• The diffuse secondary series nD2P, are triplet, D & P are both split into substates. 2D5/22P3/2, 2D

3/2 2P3/2, 2D3/2 2P1/2; 2D5/22P1/2(?)

0,1,,1 jl

Spectrum of alkali metals

First primary series

Sharp secondary series

Diffuse secondary series

Stern-Gerlach experiment (I)• Space quantization was first explicitly demons

trated by Stern & Gerlach in 1921. • They directed a beam of neutral silver atoms fr

om an oven through a set of collimating slits into an inhomogeneous magnetic field B which exerts a force on the magnetic moments.

• When the magnetic moment is parallel to B, it moves in the direction of increasing field strength, while an anti-parallel magnetic moment moves towards lower field strength.

Stern-Gerlach experiment (II)

• They found that the initial beam split into two distinct parts which correspond to the two opposite spin orientation in the magnetic field.

• For hydrogen atoms in ground state, n=1, l=0, j=s, mj=-1/2, +1/2, gj=2, mjgj =±1,

dz

dBgm

dz

dBF Bjjzz

Stern-Gerlach apparatus

The Zeeman effect• The total angular momentum J is the vector sum of

orbital angular momentum and spin angular momentum. J=L+S

• When there is no external magnetic field,the total angular momentum is conserved. When an external B is applied, J precesses about the direction of B, causing additional potential energy mgμBB (2j+1 levels).

• Selection rules:

jz mJjjJ )1(

1,0 m