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 μ l and B. The electrons precess about the direction of the field, called Larmor precession. There is also a potential energy E=- μ l ·B.

Electron spin Electron spin was introduced by Uhlenbeck and Goudsmit in They proposed that the electron possesses an intrinsic angular momentum independent of any orbital angular momentum it might have. Four quantum numbers: n, l, m l, m s. The splitting of many spectral lines in a magnetic field can only be explained if the electron has a spin angular momentum s.

The total angular momentum & magnetic moment l s j μlμl μjμj μ μSμS

Fine structure All energy levels except the s states of one- electron 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 itself 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.

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.

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.

Lande g factor It is a measure of the ratio of the magnetic moment (in Bohr magnetons) to the angular momentum (in units of h bar). For the orbit angular momentum, j=l, g l =1. For spin angular momentum, j=s, g s =2. For the total angular momentum, g factor is given by:

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 3P  3S,is a doublet: D1=589nm, D2=589.6nm.

Alkali metals Selection rules: The first primary series: nP  2S, P: double substates, doublet, two lines are getting closer with n increased. The sharp secondary series: nS  2P, doublet, two lines are separated uniformly with n increased. The diffuse secondary series nD  2P, are triplet, D & P are both split into substates. 2 D 5/2  2 P 3/2, 2 D 3/2  2 P 3/2, 2 D 3/2  2 P 1/2; 2 D 5/2  2 P 1/2 (?)

Spectrum of alkali metals First primary series Sharp secondary series Diffuse secondary series

Stern-Gerlach experiment (I) Space quantization was first explicitly demonstrated by Stern & Gerlach in They directed a beam of neutral silver atoms from 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, m j =-1/2, +1/2, g j =2, m j g j =±1,

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 mgμ B B (2j+1 levels). Selection rules: