Chapter 6

Fine structure

Term scheme (Grotrian diagram) of the Sodium atom.

Double D lines (yellow)

+Ze

R r-e

-(Z-1)e

Fine structure in the optical spectra of atoms

Many of the lines in the spectra of alkali atoms are double, and are called doublets. They occur because all the energy terms En,l of atoms with single valence electrons, except for the s terms, are split into two terms. This splitting cannot be understood in terms of the theory discussed so far. It is fundamentally different from the lifting of orbital degeneracy discussed in chapter 5. If the orbital degeneracy has already been lifted, there must be a new effect involved, one which has not yet been taken into account. So far, we have not yet discussed the magnetic properties of atoms. To explain the doublet structure, the magnetic properties of atoms should not be neglected:

Main idea: A magnetic moment l is associated with the orbital angular moment; The electron also has a spin S, which is also associated with a magnetic moment s; The two magnetic moments l and s interact. They can be parallel or antiparallel to each other. The two configurations have slightly different binding energies, which leads to the fine structure of the spectrum.

A magnetic dipole

We know from electrodynamics that a circular electric current generates a magnetic dipole field. The magnetic dipole moment µl of a conducting loop is defined as:

I A = area

µl

AIl

A magnetic dipole

If we bring this magnetic dipole into a homogeneous magnetic field B, a torque is applied to the dipole:

B

The magnetic potential energy of the dipole:

cosBBVmag

I A = area

µl

The magnetic moment can be defined either in terms of the torque in the field or the potential energy.

Magnetic moment of the orbital motion

An electron moving in an orbit is equivalent to a circular electric current. The orbiting electron will do have a magnetic dipole moment. In atomic and nuclear physics, the magnetic moment is often defined as the torque in a uniform field of strength H (not of strength B).

AI

B

0

,

Where B = µ0 H, the induction constant µ0 = 410-7 Vs/Am

Calculation of the orbital moment

r

e T

rv

2

lm

el

2

The circulating electron has an angular momentum l and a magnetic dipole moment µl . For a negative charge, the vectors l and µl point in opposite directions.

For an electron, the charge q = -e, the velocity v moving in a circular orbit. If the time for a single revolution is T= 2/, a current:

2

e

T

qI

Calculation of the orbital moment

The magnetic moment of the circular current:

221 reIA

If we introduce the orbital angular momentum |l| = mvr = mr2, then the orbital angular momentum:

lm

e

02

Where m0 is the rest mass of electron. If the charge q is positive, the vectors µl and l point in the same direction; if it is negative, as with the electron, they point in opposite directions.

)1(2

)1( llh

lll

The magnitude of the magnetic moment of an orbit with the angular momentum quantum number l:

)1()1(22 00

llllm

el

m

eB

Where µB is called Bohr Magneton, the unit of magnetic moment in atoms:

224

0

10274078.92

Amm

eB

In atoms with a certain quantum state, the magnitude of the orbital angular momentum is determined by the orbital quantum number l:

lg Bl

Where gl is called g factor or Lande factor. It is dimensionless and here has the numerical value gl = 1. The g factor was introduced by Landé. In the presence of spin-orbit coupling, the coupling between the spin magnetic moment and orbital magnetic moment, the g factor was used to characterise the ratio of the magnetic moment and the total angular momentum. The g factor is defined as the ratio of the magnetic moment to the corresponding angular momentum in units of µB and ħ, respectively:

B

ll

lg

The orbital magnetic moment is related to the orbital angular momentum. It is also expressed:

If the gyroscope is tipped, the gimbals will try to reorient to keep the spin axis of the rotor in the same direction. If released in this orientation, the gyroscope will precess in the direction shown because of the torque exerted by gravity on the gyroscope.

Gyroscope and its precession

Precession and orientation in a magnetic field

An applied field with the magnetic flux density B0 acts on the orbital magnetic moment µl by trying to align the vectors and B0 parallel to one another, since the potential energy is a minimum in this orientation. The electrons, which are moving in their orbits, behave mechanically like gyroscopes and carry out the usual precession about the direction of the field. The precession frequency of the electron orbit, the Larmor frequency, is:

L

l l+l

B0

t

|l|sin l

BBg

l

B

lBll

L

sin

sin

sin

Where is called the gyromagnetic ratio. The Larmor frequency is independent of the angle .

Directional quantisationDirectional quantisation: The orientation of the angular momentum in space is not random. The solution of the Schrödinger equation implies that when one axis is established, one of the component of the angular momentum is quantised. The axis can be determined by a magnetic field. Therefore only discrete values of , the angle between B and l or µl, are allowed. The components of the angular momentum in z direction:

lz = mlħ, ml = 0, ±1,…, ±l

Where ml is the magnetic quantum number, and it can have 2l+1 different values. l is the orbital quantum number. The largest possible component of l in the z direction is lħ.The magnetic moment associated with the orbital angular momentum is correspondingly quantised. For its component in the z direction is:

Blzzl mlm

e

0

, 2

The experimental demonstration of the existence of a directional quantisation was provided by the Stern and Gerlach experiment.

B(z)

m = 00

m = -1

m = -22

m = 1

m = 22zL

6L

For example： 2l

2,,0 ZL

6 )12(2)1( llL

lZ mL

lml ,,2,1,0

BBlzl m )2,1,0(,

Spin and magnetic moment of the electron

Electron spin was introduced by Uhlenbeck and Goudsmit in 1925 to explain spectroscopic observations. In fact, the splitting of many spectral lines in a magnetic field can only be explained if the electron has a spin angular momentum s.

Goudsmit on the discovery of electron spin.mht

What is the electron spin.mht

The spin is the intrinsic nature of electron, which has its own spin angular momentum s and associated magnetic moment µs.

The spin angular momentum of the electron:

)1(sss

s=1/2 is the spin quantum number. The associated spin magnetic moment:

sm

egss

02

gs is the so-called g factor of the electron, and its value has been empirically

determined to be gs = 2.0023. Dirac showed in 1928 that the spin of the electron is a necessary consequence of a relativistic quantum theory (the Schrödinger theory is non-relativistic). The g factor gs = 2 could also be thus derived. The slight difference between the predicted value of 2 and the empirical value can only be understood if the interaction of the electron with its own radiation field is taken into account through quantum electrodynamics.

Schematically shown the spin and magnetic moment of the electron

Spin

Charge

Mass

MagneticMoment

)1(sss

-e

m0

sm

egss

02

Directional quantisation of the spin in the magnetic field

Similar to the orbital angular momentum and magnetic moment, the orientation of the spin angular momentum is also directionally quantised in the presence of an external magnetic field B. The spin can only have two orientations in a defined z axis: parallel and antiparallel (or spin up or spin down). Its components in this defined z direction are

sz = ms ħ with ms = ±1/2;

ms is the magnetic quantum number of the spin.The orientation of the associated magnetic moment is also directional quantised. The z component is

µs,z = -gsmsµB.

The electron spin has two possible orientation in a magnetic field in the z direction. These are characterise

d by the quantum number ms = ±1/2

B, z

4/3s

ms = +1/2

ms = -1/2

21zs

21zs

Similar to the orbital motion, the gyromagnetic ratio of the electron was defined as the ratio between the magnetic moment and the angular moment of the electron:

02m

eg

s ss

The relation between gyromagnetic ratio and g factor:

Bsss gm

eg

02

It will see that the easiest and most definitive way to calculate the magnetic properties of atoms is often to make use of measurements of the ratio or g.

Detection of directional quantisation by Stern and Gerlach

In 1921, the deflection of atomic beams in inhomogeneous magnetic fields made it possible:

The experimental demonstration of directional quantisation and The direct measurement of the magnetic moments of atoms.

Stern and Gerlach experiment

Left panel: the atomic beam passes through an inhomogeneous magnetic field. One observes the splitting of the beam into two components. Right panel: observed intensity distribution of an atomic beam with and without an applied magnetic fiel

d. In the first experiments of Stern and Gerlach, it was a beam of silver atoms which was generated in an atomic beam furnace and collimated by a series of slits. Later, hydrogen atoms form a gas discharge were also used.

Without the field, the vectors of the magnetic moments and angular momenta of the atoms are randomly oriented in space. In a homogeneous field, these vectors precess around the field direction z. The inhomogeneous field exerts an additional force on the magnetic moments. The direction and magnitude of this force depends on the relative orientation between the magnetic field and the magnetic dipole. A magnetic dipole which is oriented parallel to the magnetic field moves in the direction of increasing field strength, while an antiparallel dipole moves towards lower field strength. A dipole which is perpendicular to the field does not move.

Inhomogeneous B

The collimated beam passes through a highly inhomogeneous magnetic field, with the direction of the beam perpendicular to the direction of the field and of the gradient. The directions of the field and gradient are the same.

The deflection force can be derived from the potential energy in the magnetic field:

BVmag

,cos

dz

dB

dz

dB

z

VF z

magz

Where is the angle between the magnetic moment and the direction of the field gradient.In classical mechanics, any orientation is allowed. Atoms with magnetic moments perpendicular to the filed gradient are not deflected. Those in which the vectors are parallel are deflected the most, and all possible intermediate values can occur. In the classical picture one thus expects a continuum of possible deflections. With H and Ag atoms, however, two rather sharp peaks were observed on the detector.

Conclusions from Stern and Gerlach experiments

There is a directional quantisation. There are only discrete possibilities for the orientation relative to a filed B0, in this case two, parallel and antiparallel.

In general this method provides observed values for atomic magnetic moments if the magnitude of the field gradient is known.

The s electron has an orbital angular momentum l = 0 and one observes only spin magnetism.

For all atoms which have an s electron in the outermost position, the angular momenta and magnetic moments of all inner electrons cancel each other, which gives the same value for the deflection force. One measures only the effect of the outermost s electron.

Like gyroscopes, atoms maintain the magnitude and direction of their angular momenta in the course of their motion in space.

Fine structure and spin-orbit coupling

Fine structure: all energy terms except of the s state are split into two substates, which produces a doublet or multiplet structure of the spectral lines.

The fine structure cannot be explained with the Coulomb interaction between the nucleus and the electrons. It results from a magnetic interaction between the orbital magnetic moment and the intrinsic moments of the electron, the spin-orbital coupling. Depending on whether the two moments are parallel or antiparallel, the energy term is shifted somewhat.

The coupling of the magnetic moments leads to an addition of the two angular momenta to yield a total angular momentum.

Conclusions for the spin-orbit coupling

l and s add to give a total angular momentum j with the vector model j = l + s, and j will precess in the magnetic field.

j

l

s

z, B

jz

j has the magnitude )1( jj with j = |l ± s| and s = ±1/2.

Here j is a new quantity: the quantum number of the total angular momentum.

Similar to l, there is also a directional quantisation for j. The z component must obey the condition:jz = mjħ, mj = j, j-1,…, -j (2j+1 possibility)For example, a state with j = 3/2 is fourfold degenerate.

A magnetic moment µj is associated with j;

For optical transitions, a selection rule Δj = 0 or ±1 is valid; however, a transition from j = 0 to j = 0 is always forbidden. This selection rule may be considered to be an empirical result, derived from the observed spectra.

Calculation of spin-orbit splitting in the Bohr model

To calculate the energy difference between the parallel and antiparallel orientations of the orbital angular momentum and the spin, the simple Bohr model will be used.

The motion of the electron around the nucleus generates a magnetic field Bl at the site of the electron. This field interacts with the magnetic moment of the electron. To determine the magnitude of this magnetic field, we borrow from relativity theory and assume that the electron is stationary and that the nucleus moves instead.

The magnetic field of the moving charge +Ze is found from the Biot-Savart law to be

lv

(µs)z

r+-

l

v

(µs)z

-r+

-

lmr

Zerv

r

ZeBl

03

030

4)]([

4

Where the angular momentum

rvmvmrl

00

The magnetic field which is generated by the relative motion of the nucleus and the electron is thus proportional and parallel to the orbital angular momentum of the electron. We still require the back transformation to the center-of-mass system of the atom, in which the nucleus is essentially at rest and the electron orbits around it. A factor ½ occurs in this back transformation, the so-called Thomas factor, which can only be justified by a complete relativistic calculation. The particle in its orbit is accelerated, and from the viewpoint of the proton, the rest system of the electron rotates one additional time about its axis during each revolution around the orbit. The back transformation is therefore complicated and will not be calculated in detail here.

lmr

ZeBl

03

0

8

The magnetic moment of the electron process about the magnetic field Bl produced by the orbital motion.

l µs

r+

s

Bl, zThe interaction energy between the spin and the orbital field, with gs = 2 :

)(8

)(2

2

320

02

0

,

lsrm

Ze

Bsm

e

BV

l

lssl

To get the order of magnitude, we set Z = 1, r = 0.1 nm, and obtain Vl,s 10-4 eV. The field produced by the orbital motion at the position of the electron is about 1 tesla = 104 Gauss.

The energy of spin-orbit interaction can be written in the form:

),cos(22, slsl

asl

aV sl

Where a is the spin-orbit coupling constant 30

20

2

8 rm

Zea

The scalar product l · s may be expressed in terms of the vectors l and s by using the law of cosines:

)]1()1()1([2

)(2

222

2,

sslljja

slja

V sl

s

l

j = s+lThis form is directly measurable by determination of the doublet structure in the optical spectra.

If we use the radius rn of the nth Bohr radius as a rough approximation for r:

02

2204

mZe

nrn

6

4

n

Za

We must remember, however, that there are no fixed orbits in the quantum theoretical description of the atom. Therefore it is necessary to replace r

–3 by the corresponding quantum theoretical average

dVrr )/(/1 323

Where is the wavefunction and dV the volume. Then we obtain:

)1)(( 213

4

llln

Za

The energy terms with considering of spin-orbit coupling?

In general, the simbolism is n 2S+1LJ. The upper case letters S, L and J apply to several-electron atoms, while the corresponding lower case letters apply to single-electron atoms. Where S(or s) is spin quantum number, L(or l) is the orbital angular momentum quantum number, and J(or j) is the total angular momentum quantum number.

for a state in which the valence electron has the quantum numbers n = 2, l = 0, j = ½.

2 2S1/2

2 2P1/2

2 2P3/2

For states in which the valence electron has the quantum number n = 2, l = 1, j = ½ or 3/2, respectively.

The spin-orbit interaction splits each level into two. For example the sodium D lines, the 3P state splits into 3P1/2 and 3P3/2 states.

For 3P1/2 state, s = ½, l = 1, j = ½, Vl,s = -a;For 3P3/2 state, s = ½, l = 1, j = 3/2, Vl,s = a/2.

3P 3P1/2

3P3/2

a/2

-a

For D state splits into D3/2 and D5/2 states

For 3D3/2 state, s = ½, l = 2, j = 3/2;For 3D5/2 state, s = ½, l = 2, j = 5/2.

Summarize for the fine structure of one-electron states:

Interaction of the electron with the orbital momentum or the orbital moment splits each level into two.

For s terms there is no splitting, because there is no magnetic field with which the spin can align itself.

Levels with higher values of the quantum number j have higher energies.

The fine structure splitting Vl,s is proportional to the fourth power of the nuclear charge Z.

The slitting is greatest for the smallest principal quantum number n.

Level scheme of the Alkali atoms

Term scheme for alkali atoms, i.e. one-electron states, including the spin-orbit splitting. The terms are displaced with respect to those of the H atom. The fine structure splitting decreases with increasing values of n and l. A few allowed transitions are indicated.The selection rules:

l = ±1, j = ±1, 0

Optical transitions are thus allowed only if the angular momentum changes. The total angular momentum angular j, however, can remain the same. This would happen if the orbital angular momentum and the spin changed in opposite directions.

The first principal series of the alkali atoms arises from transitions between the lowest n2S1/2 term (i.e. n = 2, 3, 4, 5, 6 for Li, Na, K, Rb, Cs) and the P terms 2P1/2 and 2P3/2. Since the s terms are single-valued, one sees pairs of lines. The same holds for the sharp secondary series, which consists of transitions between the two lowest P terms and all higher 2S1/2 lines. The lines of the diffuse secondary series, however, are triple, because both the P and the D terms are double.

2D5/2

2D3/2

2P3/2

2P1/2

Allowed transitions

forbidden transitions

Fine structure in the hydrogen atomSince the wavefunctions of the H atom are known explicitly, its fine structure can be exactly calculated. For the H atom, both the relativity correction and the spin orbit interaction are small compared to the energy terms En,l, but the two are of comparable magnitude:

En,l,j = En,l + Erel + El,s

The two correction terms, the one for the relativistic mass change Erel and the other for the spin-orbit coupling El,s, together give the fine structure correction EFS = Erel + El,s. The complete calculation was carried by Dirac.

22

4

3

2/1

1Z

njn

EE n

FS

Where the Sommerfeld fine structure constant:

)4

(4

20

0

2

ec

orc

e

The energy shift of H atom with respect to the calculated energy terms En,l is of the order of 2, i.e. (1/137)2, and is thus difficult to measure. According to Dirac’s calculation, the fine structure energy of the H atom depends only on j, not on l. this means that terms with differing l (for the same n) and the same j have the same energies. They are energetically degenerate.

The fine structure energies of heavier atoms are larger and are thus easier to observe. Their calculations, however, are far more difficult, because the exact calculation of the wavefunctions of atoms with more than one electron is far more complex.

The Lamb shift

In the years 1947-1952, Lamb and Retherford showed that even the relativistic Dirac theory did not completely describe the H atom. They used the methods of high-frequency and microwave spectroscopy to observe very small energy shifts and splitting in the spectrum of atomic hydrogen. In other words, they used the absorption by H atoms of electromagnetic radiation from high-frequency transmitters or Klystron tubes they could, in this way, observe energy difference between terms with the same j, namely differences of 0.03cm-1 — this corresponds to a difference of 900 MHz — between the terms 22S1/2 and 22P1/2.

The Lamb shift: fine structure of the n = 2 level in the H atom according to Bohr, Dirac and quantum electrodynamics taking into account the Lamb shift. The j degeneracy is lifted.

Levels with the same quantum numbers n and j, but different l, are not exactly the same. Rather, all S1/2 terms are higher than the corresponding P1/2 terms by an amount equal to about 10% of the energy difference (P3/2 -P1/2), and the P3/2 terms are higher than the D3/2 terms by about 2‰ of (D5/2 – D3/2)

homework

Pp202, 12.1, 12.2, 12.3, 12.5, 12.7, 12.9