Chapter 7 Atoms in a magnetic field

1
Chapter 7Atoms in a magnetic field
2
In chapter 6, we have seen that the angular
momentum or the associated magnetic moment of
atoms precess in the presence of magnetic field,
and it is directional quantised in z direction.
The interaction energy between the magnetic field
and the magnetic moment of the electrons in an
atom leads to a splitting of the energy terms,
which is described by the different possible
values of the magnetic quantum number.
This energy splitting can be determined in the
treatment of the Stern-Gerlach experiments, and
other types of experiments electron spin
resonance (ESR), nuclear magnetic resonance
(NMR), the Zeeman effect, and the Paschen-Back
effect.
3
Electron spin resonance
Electron spin resonance abbreviated ESR, and
sometimes EPR for electron paramagnetic
resonance. The method involves the production of
transitions between energy states of the
electrons which are characterised by different
values of the magnetic quantum number ms.
The spin of an electron has two possible
orientations in an applied magnetic field. They
correspond to two values of the potential energy.
The difference of the potential energy of these
two orientations
4
The transition frequencies are usually in the
range of microwave frequencies depending on the
strength of the applied magnetic field.
If a sinusoidally varying magnetic field B1
B1sin?t is applied in a direction perpendicular
to B0, transitions between the two states are
induced if the frequency ? ?/2? fulfils the
condition
Where ? 2.8026 1010 B0 Hz(tesla)-1
The frequency ? depends on the choice of the
applied magnetic field B0. For reasons of
sensitivity, usually the highest possible
frequencies are used, corresponding to the
highest possible magnetic fields. The fields and
frequencies used in practice are limited by
questions of technical feasibility. Usually,
fields in the range 0.1 to 1T are chosen. This
leads to frequencies in the GHz region
(centimeter waves).
5
Electron spin resonance
Schematic representation of the experimental
setup. The sample is located in a resonant cavity
between the pole pieces of an electromagnet. The
microwaves are generated by a klystron and
detected by a diode. To increase the sensitivity
of detection, the field B0 is modulated.
Below-left energy states of a free electron as
functions of the applied magnetic field.
Below-right signal U from the diode as a
function of Bo for resonance.
6
ESR spectrometers count as standard spectroscopic
accessories in many physical and chemical
laboratories. For technical reasons, usually a
fixed frequency is used in the spectrometers. The
magnetic field is varied to fulfil the resonance
condition and obtain ESR transitions. A
frequently used wavelength is 3cm (the so-called
X-band).

• ESR is utilised for
• Precision determinations of the gyromagnetic
  ratio and the g factor of the electron
• Measurement of the g factor of atoms in the
  ground state and in excited states for the
  purpose of analysing the term diagram
• The study of various kinds of paramagnetic
  states and centers in solid state physics and in
  chemistry molecular radicals, conduction
  electrons, paramagnetic ions in ionic and
  metallic crystals, colour centers.

7
Electron spin resonance was observed for the
first time in 1944 by the Russian physicist
Zavoisky. The analogous spin resonance of
paramagnetic atomic nuclei is seen under
otherwise identical conditions at a frequency
which is 3 orders of magnitude smaller, due to
the fact that nuclear moments are about a factor
of 1000 smaller than atomic magnetic moments the
corresponding frequencies are in the radio
frequency region. This nuclear magnetic resonance
(NMR) was observed in the solid state for the
first time in 1946 by Bloch and Purcell, nearly
10 years after it had first been used by Rabi to
measure the gyromagnetic ratio of nuclei in gas
atoms.
8
NMR (Nuclear Magnetic Resonance) and its
applications in medical diagnosis, atomic clock
and chemical shift
9
Hyperfine structure
The presence of the hyperfine structure was first
postulated by Pauli in 1924 as means of
explaining spectroscopic observations. In the
year 1934, Schüller further postulated the
existence of electric quadrupole moments in
nuclei.
Similar to electron, the atomic nuclei also
exists spins I and magnetic moments µI. The
interactions of these nuclear moments with the
electron leads to an additional splitting of the
spectral lines, which is the hyperfine structure.
Comparing to fine structure, the splitting
resulting from hyperfine structure is much
smaller, the measurement of which generally
requires an especially high resolution. The
angular momentum and magnetic moment of the
atomic nuclei
Where the quantum number I may be integral or
half-integral the nuclear magneton µN eh/2mp
is the unit of the nuclear magnetic moment.
10
The spin of atomic nuclei will directional
quantised in the presence of external magnetic
field. The z component of the nuclear magnetic
moment
Where the quantum number mI -I, -I1, , I, has
2I1 possibility. The maximum observable value of
µI is
For example for the hydrogen nucleus, the
proton µI(1H) 2.79µN I ½ gI 5.58 For
the potassium nucleus with mass number
40 µI(40K) -1.29µN I 4 gI -0.32 There
are also numberous nuclei with vanishing spins I
0. These nuclei do not contribute to the
hyperfine structure. Examples of this type of
nucleus are
11
Nuclear magnetic resonance
Energy levels for a nucleus with spin quantum
number I
Resonance frequency ? ?E / h
Nuclear magneton
12
A magnetic dipole moment (usually just called
magnetic moment) in a Magnetic field will have
a potential energy related to its orientation
with Respect to that field.
13
The Larmor frequency
The Larmor frequency of the electron spin is in
the microwave region, the Larmor frequency of
proton or other nucleus is three orders smaller.
14
Diagram of a simple nuclear spin resonance
apparatus
In 1946, Purcell and Bloch showed both
experimentally and theoretically that the
precessional motion of the nuclear spin is
largely independent of the translational and
rotational motion of the nucleus, and that the
method of NMR can be applied not only to free
atoms, but to atomic nuclei in liquids and solids.
15
The medical application Magnetic Resonance
Imaging (MRI)
Proton nuclear magnetic resonance detects the
presence of Hydrogens (protons) by subjecting
them to a large magnetic field to partially
polarize the nuclear spins, then exciting the
spins with properly tuned radio frequency (RF)
radiation, and then detecting weak RF radiation
from them as they relax from this magnetic
interaction. The frequency of this proton
signal is proportional to the magnetic field
to which they are subjected during this
relaxation process. an MRI image of a
cross-section of tissue can be made by producing
a well-calibrated magnetic field gradient across
the tissue so that a certain value of magnetic
field can be associated with a given location in
the tissue. Since the proton signal frequency is
proportional to that magnetic field, a given
proton signal frequency can be assigned to a
location in the tissue. This provides the
information to map the tissue in terms of the
protons present there. Since the proton density
varies with the type of tissue, a certain amount
of contrast is achieved to image the organs and
other tissue variations in the subject tissue.
16
MRI image
Since the MRI uses proton NMR, it images the
concentration of protons. Many of those protons
are the protons in water, so MRI is particularly
well suited for the imaging of soft tissue, like
the brain, eyes, and other soft tissue structures
in the head as shown above. The bone of the skull
doesn't have many protons, so it shows up dark.
Also the sinus cavities image as a dark region.
Bushong's assessment is that about 80 of the
body's atoms are hydrogen atoms, so most parts of
the body have an abundance of sources for the
hydrogen NMR signals which make up the magnetic
resonance image.
17
The setup of MRI
18
Be possible to image soft tissues Joint, brain,
and spinal cord
normal
knub
19
The gradient magnetic field
20
Two-dimensional map of the proton density
A rotating field gradient is used, linear
positioning information is collected along a
number of different directions. That information
can be combined to produce a two-dimensional map
of the proton densities. The proton NMR signals
are quite sensitive to differences in proton
content that are characteristic of different
kinds of tissue. Even though the spatial
resolution of MRI is not as great as a
conventional x-ray film, its contrast resolution
is much better for tissue. Rapid scanning and
computer reconstruction give well-resolved images
of organs.
21
the applications of MRI In 1999, 2,170 in
all of the world In 2002, 22,300 and 500 in
china (also several in Jinan) Usage, 60
million times in 2000 Cost 160 Advantage n
on-invasive, non-ionising radiation, and a high
soft-tissue resolution and discrimination in any
imaging plane.
22
Chemical applications of NMR
The frequency detected in NMR spectroscopy is
proportional to the magnetic field applied to the
nucleus. This would be a precisely determined
frequency if the only magnetic field acting on
the nucleus was the externally applied field. But
the response of the atomic electrons to that
externally applied magnetic field is such that
their motions produce a small magnetic field at
the nucleus which usually acts in opposition to
the externally applied field. This change in the
effective field on the nuclear spin causes the
NMR signal frequency to shift. The magnitude of
the shift depends upon the type of nucleus and
the details of the electron motion in the nearby
atoms and molecules. It is called a "chemical
shift". The precision of NMR spectroscopy allows
this chemical shift to be measured, and the study
of chemical shifts has produced a large store of
information about the chemical bonds and the
structure of molecules.
23
Chemical shift in NMR spectra
The effective magnetic field at the nucleus can
be expressed in terms of the externally applied
field B0 by the expression
where s is called the shielding factor or
screening factor. The factor s is small -
typically 10-5 for protons and lt10-3 for other
nuclei (Becker). In practice the chemical shift
is usually indicated by a symbol d which is
defined in terms of a standard reference.
24
Chemical shift
The signal shift is very small, parts per
million, but the great precision with which
frequencies can be measured permits the
determination of chemical shift to three or more
significant figures. The reference material is
often tetramethylsilane, Si(CH3)4, abbreviated
TMS. Since the signal frequency is related to the
shielding by
the chemical shift can also be expressed as
25
To determine the chemical bonding
A sample of a chemical Shift spectrum which is a
proton spectrum. The high-resolution peaks Can be
identified with the functional groups in the
radicals ?1.23, (CH3)2 2.16, CH3CO 2.62,
CH2 4.12, OH
26
The Caesium(Cs) atomic clock as a time and
frequency standard
Cs has a nuclear spin I7/2, and in the atomic
ground state, the angular momenta of the
electrons J1/2 The total angular momenta F4
and F3 The transition frequency used for the Cs
atomic clock corresponds to the transition
between the states F3, mF0, and F4, mF0
A portion of the term scheme of the Cs atom in
the ground state as a function of a weak applied
magnetic field B0.
27
The atomic beam resonance method of Rabi (1937)
Data curve from the Rabi atomic beam resonance
experiment. The intensity at the detector is at a
minimum when the homogeneous field B0 of magnet C
fulfils the resonance condition.
28
Zeeman effect
A splitting of the energy terms of atoms in a
magnetic field can be observed as a splitting of
the frequencies of transitions in the optical
spectra (or as a shift). A splitting of this type
of spectral lines in a magnetic field was
observed for the first time in 1896 by Zeeman.
The effect is small. Spectral apparatus of very
high resolution is required. These are either
diffraction grating spectrometers with long focal
lengths and a large number of lines per cm in the
grating, or else interference spectrometers,
mainly Fabry-Perot interferometers. With a
Fabry-Perot interferometers or with a grating
spectrometer of sufficient resolution, the
splitting in magnetic fields may be
quantitatively measured.
29
Fabry-Perot Interferometer
This interferometer makes use of multiple
reflections between two closely spaced partially
silvered surfaces. Part of the light is
transmitted each time the light reaches the
second surface, resulting in multiple offset
beams which can interfere with each other. The
large number of interfering rays produces an
interferometer with extremely high resolution
(106), somewhat like the multiple slits of a
diffraction grating increase its resolution.
30
Ordinary Zeeman effect
Ordinary Zeeman effect for the atomic Cd line at
? 6438?. With transverse observation the
original line and two symmetrically shifted
components are seen. Under longitudinal
observation, only the split components are seen.
31
Transverse and longitudinal observation of
emission spectral lines in a magnetic field. S is
the entrance slit of the spectrometer.
32
Anomalous Zeeman effect
D1 D2
The D lines of sodium. The D1 line splits into
four components, the D2 line into six in a
magnetic field. The wavelengths of the D1 and D2
lines are 5896 and 5889 ? the quantum energy
increases to the right in the diagram.
33
The Zeeman effect results from the splitting of
energy states with the interaction of the
resultant angular momentum and external magnetic
fields. If the resultant angular momentum is
composed of both spin and orbital angular
momentum, one speaks of the anomalous Zeeman
effect. The normal Zeeman effect describes
states in which no spin magnetism occurs,
therefore with pure orbital angular momentum. In
these states, at least two electrons contribute
in such a way that their spins are coupled to
zero. Therefore, the normal Zeeman effect is
found only for states involving several (at least
two) electrons.
34
Explanation of the Zeeman effect from the
standpoint of classical electron theory
The ordinary Zeeman effect may be understood to a
large extent using classical electron theory, as
it was shown by Lorentz shortly after its
discovery. In the model, the emission of light
by an electron whose motion about the nucleus is
interpreted as an oscillation. The radiation
electron is treated as the electron by three
component oscillators according to the rules of
vector addition component oscillator 1
oscillators linearly, parallel to the direction
of B0 oscillators 2 and 3 oscillate circularly
in opposite senses and in a plane perpendicular
to the direction of B0. This resolution into
components is allowed, since any linear
oscillation may be represented by the addition of
two counterrotating circular ones.
35
An oscillating electron is resolved into three
component oscillators
Without the magnetic field B0, the frequency of
all the component oscillators is equal to that of
the original electron, namely ?0. With the field
B0 component 1, parallel to B0, experiences no
force. Its frequency remains unchanged. It emits
light which is linearly polarised with its E
vector parallel to the vector B0. The circularly
oscillating components 2 and 3 are accelerated or
slowed down by the effect of magnetic induction,
depending on their direction of motion. Their
circular frequencies are increased or decreased
by an amount
36
Calculation of the frequency shift for the
component oscillators
Without the applied magnetic field, the circular
frequency of the component electrons is ?0. The
Coulomb force and the centrifugal force are in
balance. In a homogeneous magnetic field B0
applied in the z direction, the Lorentz force
acts in addition. In Cartesian coordinates, the
following equations of motion are then valid
For component 1, z z0exp(i?0t), the frequency
remains unchanged. For component 2 and 3, we
substitute u x iy and v x iy. The
equations have the following solutions u
u0expi(?0 eB0/2m)t and v v0expi(?0
eB0/2m)t The component electron oscillators 2
and 3 thus emit or absorb circularly polarised
light with the frequency ?0 ??.
37
The frequency change has the magnitude
For a magnetic field strength B0 1T, this yield
the value
For each spectral line with a given magnetic
field B0, the frequency shift ?? is the same.
Theory and experiment agree completely.
38
For the polarization of the Zeeman components, we
find the following predictions component
electron oscillator 1 has the radiation
characteristics of a Hertzian dipole oscillator,
oscillating in a direction parallel to B0. In
particular, the E vector of the emitted radiation
oscillates, and the intensity of the radiation is
zero in the emitted radiation oscillates parallel
to B0. This corresponds exactly to the
experimental results for the unshifted Zeeman
component. It is also called the ? component (?
for parallel). If the radiation from the
component electron oscillators 2 and 3 is
observed in the direction of B0, it is found to
be circularly polarised observed in the
direction perpendicular to B0, it is linearly
polarised. This is also in agreement with the
results of the experiment. This radiation is
called ? and ? light, where ? stands for
perpendicular and the and signs for an
increase and decrease of the frequency. The ?
light is right-circular polarised, the ? light
is left-circular polarised. The direction is
defined relative to the lines of the B0 field,
not relative to the propagation direction of the
light.
39
Description of the ordinary Zeeman effect by the
vector model
Both ordinary and anomalous can be described by a
complete quantum mechanical treatment, which we
will not discuss here. For simplicity, we employ
the vector model. The angular momentum vector j,
and the magnetic moment µj, precess together
around the field axis B0. The additional energy
of the atom due to the magnetic field is then
Precession of j and µj about the direction of
the applied field B0, j l.
40
The (2j1)-fold directional degeneracy is lifted
in the presence of the magnetic field, and then
the term is split into 2j1 components. These are
energetically equidistant. The distance between
two components with ?mj 1 is
For the ordinary Zeeman effect, the spin S 0
and consider only orbital magnetism. gj has a
numerical value of 1. The frequency shift
The magnitude of the splitting is thus the same
as in classical theory. For optical transitions,
the selection rule ?mj 0, 1. From quantum
theory one also obtains the result that the
number of lines is always three the ordinary
Zeeman triplet.
41
The splitting diagram for a cadmium line
Splitting of the ? 6438? line of the neutral Cd
atom, transition 1P1 1D2, into three
components. The spins of the two electrons are
antiparallel and thus compensate each other,
giving a total spin S 0. The splitting is equal
in each case because only orbital magnetism is
involved.
42
R. A. Beth in 1936 found that the circular
polarised light quanta has not only the energy
but also the angular momentum.
43
Based on the conservation of the angular momentum
for the system of electrons and light quanta
For ?mj 0, the angular momentum of the system
was not changed after the transition, the
emitting light has no angular momentum, and it is
thus linearly polarised, which is ? light. For
?mj -1, the angular momentum of the system was
changed -h after the transition, the emitting
light has angular momentum -h, and it is thus
circular polarised, which is ?- light. For ?mj
1, the angular momentum of the system was
changed h after the transition, the emitting
light has angular momentum h, and it is thus
circular polarised, which is ? light.
44
The anomalous Zeeman effect
In general case, the atomic magnetism is due to
the superposition of spin and orbital magnetism,
which results the anomalous Zeeman effect. The
term anomalous Zeeman effect is historical, and
is actually contradictory, because this is the
normal case. In cases of the anomalous Zeeman
effect, the two terms involved in the optical
transition have different g factors, because the
relative contributions of spin and orbital
magnetism to the two states are different. The g
factors are determined by the total angular
momentum j and are therefore called gj factors.
The splitting of the terms in the ground and
excited states is therefore different, in
contrast to the situation in the normal Zeeman
effect. This produces a larger number of spectral
lines.
45
The relation between the angular momentum J, the
magnetic moment µJ and their orientation with
respect to the magnetic field B0 for strong
spin-orbit coupling.
The angular momentum vectors S and L combine to
form J. J and uJ are not coincide.
46
For the transitions of the Na D lines, three
terms involved, namely the 2S1/2, the 2P1/2 and
the 2P3/2, the magnetic moments in the direction
of the field are
The magnetic energy is
The number of splitting components in the field
is given by mj and is again 2j1. The distance
between the components with different values of
mj the so-called Zeeman components is no
longer the same for all terms, but depends on the
quantum numbers l, s, and j
47
Experimentally, it is found that gj 2 for the
ground state 2S1/2, 2/3 for the state 2P1/2 and
4/3 for the state 2P3/2. For optical transitions,
the selection rule is again ?mj 0, 1. It
yields 10 lines.
D1 line
D2 line
48
Magnetic moments with spin-orbit coupling
In anomalous Zeeman splitting, other values of gj
than 1 or 2 are found. The gj factor links the
magnitude of the magnetic moment of an atom to
its total angular momentum. The magnetic moment
is the vector sum of the orbital and spin
magnetic moments
The directions of the vectors µl and l are
antiparallel, as are those of the vectors µs and
s. In contrast, the directions of j and µj do not
in general coincide. This is a result of the
difference in the g factors for spin and orbital
magnetism.
49
The magnetic moment µj resulting from vector
addition of µl and µs precesses around the total
angular momentum vector j, the direction of which
is fixed in space. Due to the strong coupling of
the angular momenta, the precession is rapid.
Therefore only the time average of its projection
on j can be observed, since the other components
cancel each other in time. This projection (µj)j
precesses in turn around the B0 axis of the
applied magnetic field B0. In the calculation of
the magnetic contribution to the energy Vmj, the
projection of µj on the j axis (µj)
50
Vector model
The magnetic moment projected in j direction
The component of magnetic moment in z direction
51
The paschen-Back effect
For the Zeeman effect, the splitting of spectral
lines in a magnetic field hold for weak
magnetic fields. weak means that the splitting
of energy levels in the magnetic field is small
compared to fine structure splitting or in other
words, the spin-orbit coupling is stronger than
the coupling of either the spin or the orbital
moment alone to the external magnetic
field. When the magnetic field B0 is strong
enough so that the above condition is no longer
fulfilled, the splitting picture is simplified.
The magnetic field dissolves the fine structure
coupling. L and s are, to a first approximation,
uncoupled, and process independently around B0.
The quantum number for the total angular momentum
j, thus loses its meaning. This limiting case is
called the Paschen-Back effect.
52
The Pachen-Back effect
The components of the orbital (µl)z and spin
(µs)z moments in the field direction are now
individually quantised. The corresponding
magnetic energy is
The splitting of the spectral lines
In a strong magnetic field B0, the spin S and
orbital L angular momenta align independently
with the field B0. A total angular momentum J is
not defined.
53
Term diagram and optical transitions of Na atoms
(a) D1 and D2 lines of the neutral Na atom (b)
the anomalous Zeeman effect (c) Pachen-Back
effect.
54
Question 1 Why is the 4D1/2 term not split in a
magnetic field? Explain this in terms of the
vector model. Question 2 Calculate the angle
between the total and the orbital angular momenta
in a 4D3/2 state.
55
homework

• Pp220, 13.1, 13.3, 13.5, 13.8

56
Many-electron atoms

• Possible electronic configuration
• Angular momentum coupling
• Magnetic moments of many-electron atoms
• Electronic configuration and atomic term scheme
  ground state, excited states

57
Angular momentum coupling
In the one-electron system, the individual
angular momenta l and s combine to give a
resultant angular momentum j. In many-electron
atoms, there is a similar coupling between the
angular momenta of different electrons in the
same atom. These angular momenta are coupled by
means of magnetic and electric interactions
between electrons in the atom. They combine
according to specific quantum mechanical rules to
produce the total angular momentum J of the atom.
The vector model provides insight into the
composition of the angular momentum.
Since the total angular momentum of an atom is
equal to zero in closed shell, in calculating the
total angular momentum of an atom, it is
therefore necessary to consider only the angular
momenta of the valence electrons, i.e. the
electrons in non-filled shells.
There are two limiting cases in angular momentum
coupling the LS coupling, and jj coupling.
58
LS coupling (Russell-Saunders coupling)
For many-electron atoms if the spin-orbit
interactions (si li) between the spin and
orbital angular momenta of the individual
electrons i are smaller than the mutual
interactions of the orbital or spin angular
momenta of different electrons coupling (li lj)
or (si sj), the orbital angular momenta li
combine vectorially to a total orbital angular
momentum L, and the spins combine to a total spin
S. L couples with S to form the total angular
momentum J.
59
LS coupling gives a good agreement with the
observed spectral details for many light atoms.
For heavier atoms, another coupling scheme called
j-j coupling provides better agreement with
experiment. The vector model
60
For example for a two-electron system like the He
atom
The orbital angular momentum L of the atom
The quantum number L determines the term
characteristics L 0, 1, 2, indicates S, P,
D, terms. It should be noted here that a term
with L 1 is called a P term but this does not
necessarily mean that in this configuration one
of the electrons is individually in a p state.
61
For the total spin angular momentum S
The spin quantum number S ½ ½ 1 or S
½ - ½ 0
The interaction between S and the magnetic field
BL, which arises from the total orbital angular
momentum L, results in a coupling of the two
angular momenta L and S to the total angular
momentum J
The quantum number J For S 0, J
L singlet For S 1, J L 1, L, L
1 triplet
62
In the general case of a many-electron system,
there are 2S 1 possible orientations of S with
respect to L, i.e. the multiplicity of the terms
is 2S 1. The complete nomenclature for terms or
energy states of atoms
For many-electron systems, the possible
multiplicities For two electrons S 0 S
1 singlet triplet For three electrons S
½ S 3/2 doublet Quartet For four
electrons S 0 S 1 S 2 singlet
triplet Quintet For five electrons S ½ S
3/2 S 5/2 doublet Quartet Sextet
63
Atomic terms of He atom
If both electrons are in the lowest shell 1s2,
they have the following quantum numbers n1 n2
1, l1 l2 0, s1 s2 ½ The resulting
quantum numbers for the atom L 0, S 0, ms1
-ms2, J 0, the singlet ground state 1S0 Or
L 0, S 1, ms1 ms2, J 1, the triplet state
3S1, which is forbidden by the Pauli principle.
64
If the atom in the electron configuration 1s2s,
we have the following quantum numbers n1 1,
n2 2, l1 l2 0, s1 s2 ½ , The resulting
quantum numbers L 0, S 0, J 0, the
singlet state 1S0 Or L 0, S 1, J 1, the
triplet state 3S1
In the same way, the states and term symbols can
be derived for all electron configurations 1s2p,
1s3d, 2p3d,
65
The selection rule ?L 0, ?1 ?S 0 ?J 0,
?1.
Term scheme of the He atom. Some of the allowed
transitions are indicated. There are two term
system, between which radiative transitions are
forbidden.
66
Term diagram for the nitrogen. Nitrogen has a
doublet and a quartet systems. The electronic
configuration of the valence electrons is given
at the top.
67
Term diagram for the carbon. Carbon has a singlet
and a triplet systems. The electronic
configuration of the valence electrons is given
at the top.
68
jj coupling
jj coupling is the case for coupling of electron
spin and orbital angular momenta is larger
compared to the interactions (li lj) and (si
sj) between different electrons. It occurs mostly
in heavy atoms, because the spin-orbit coupling
for each individual electron increases rapidly
with the nuclear charge Z.
69
In jj coupling, a resultant orbital angular
momentum L is not defined. There are therefore no
term symbols S, P, D, etc. one has to use the
term notation (j1, j2) etc.. The number of
possible states and the J values are the same as
in LS coupling. A selection rule for optical
transitions ?J 0, ?1, and a transition from J
0 to J 0 is forbidden. Purely jj coupling
is only found in very heavy atoms. In most cases
there are intermediate forms of coupling
(intermediary coupling), which the
intercombination between terms of different
multiplicity is not so strictly forbidden.
70
Transition from LS coupling in light atoms to jj
coupling in heavy atoms in the series C Si Ge
Sn Pb.
71
Magnetic moments of many-electron atom
In the case of LS coupling, the magnetic moment
The total moment µJ precesses around the
direction of J, and the observable magnetic
moment is only that component of µJ which is
parallel to J
72
In one of chosen direction z, the only possible
orientations are quantised and they are described
by the quantum number mJ, depending on the
magnitude of J.
With mJ J, J - 1, , -J
73
Atomic ground states
The possible electronic configurations of the
atoms, concerning to the quantum numbers n and l,
are governed by Pauli principle.
The atomic term scheme, including of the ground
state and the excited states, related to the
energetic order of the states with different
values of ml and ms and the combination of the
angular momenta of individual electrons to form
the total angular momentum of the atom.
There are several rules for the energetic
ordering of the electrons within the subshells in
addition to the Pauli principle.
In LS coupling, the angular momenta are governed
by Hunds rules.
74
Hunds rules
Rule 1 Full shells and subshells contribute
nothing to the total angular momenta L and
S. Rule 2 The term with maximum multiplicity
lies lowest in energy. Rule 3 For a given
multiplicity, the term with the largest value of
L lies lowest in energy. Rule 4 For atoms with
less than half-filled shells, the level with the
lowest value of J lies lowest in energy.
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Rule 2 The term with maximum multiplicity lies
lowest in energy.
For example in the electronic configuration p2,
we expect the order 3P lt (1D, 1S)
The explanation of the rule lies in the effects
of the spin-spin interaction. Though often called
by the name spin-spin interaction, the origin of
the energy difference is in the coulomb repulsion
of the electrons.
The Pauli principle requires that the total
wavefunction be antisymmetric. A symmetric spin
state forces an antisymmetric spatial state where
the electrons are on average further apart and
provide less shielding for each other, yielding a
lower energy.
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Space wavefunction
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Rule 3 For a given multiplicity, the term with
the largest value of L lies lowest in energy.
For example in the configuration p2, we expect
the order 3P lt 1D lt 1S.
The basis for this rule is essentially that if
the electrons are orbiting in the same direction
(and so have a large total angular momentum) they
meet less often than when they orbit in opposite
directions. Hence their repulsion is less on
average when L is large.
These influences on the atomic electron energy
levels is sometimes called the orbit-orbit
interaction. The origin of the energy difference
lies with differences in the coulomb repulsive
energies between the electrons.
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For large L value, some or all of the electrons
are orbiting in the same direction. That implies
that they can stay a larger distance apart on the
average since they could conceivably always be on
the opposite side of the nucleus. For low L
value, some electrons must orbit in the opposite
direction and therefore pass close to each other
once per orbit, leading to a smaller average
separation of electrons and therefore a higher
energy.
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Rule 4 For atoms with less than half-filled
shells, the level with the lowest value of J lies
lowest in energy.
For example since p2 is less than half-filled,
the three states of 3P are expected to lie in the
order 3P0 lt 3P1 lt 3P2.
When the shell is more than half full, the
opposite rule holds (highest J lies lowest).
The basis for the rule is the spin-orbit
coupling. The scalar product S L is negative if
the spin and orbital angular momentum are in
opposite directions. Since the coefficient of S
L is positive, lower J is lower in energy.
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Influence on the atomic energy levels
Hunds rule 2
Hunds rule 4
Hunds rule 3
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Identical particle the electrons have the same
rest mass, charge and spin, and can not be
identified in quantum mechanics. Equivalent
electrons electrons with the same quantum
numbers n and l, or the electrons in the same
shell and subshell. Non-equivalent electrons
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The complete schemes for atoms correspond to a
particular electron configuration and to a
certain type of coupling of the electrons in
non-filled shells. The energetic positions of
these terms are uniquely determined by the
energies of interaction between the nucleus and
electrons and between the electrons themselves.
Quantitative calculations are extremely
difficult, because atoms with more than one
electron are complicated.
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The possible atomic terms for a given electron
configuration 1) only the electrons in open
shells must be considered 2) each electron is
characterised by the four quantum numbers n, l,
ml and ms (a set of quantum numbers)

• To derive all the possible terms (LS coupling for
  example), all the possible variations of the
  couplings have to be considered
• For each value of S, MS ?mSi have the possible
  values S, S-1, , -S
• For each value of L, ML ?mli have the possible
  values L, L-1, , -L
• When the electrons are completely decoupled by a
  strong magnetic field (according to Ehrenfest),
  the individual electrons are quantised according
  to ml l, l-1, , -l and ms ½.

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The complete term scheme
For non-equivalent electrons (LS coupling) ss
sp sd pp pd dd
1S, 3S 1P, 3P 1D, 3D 1S, 1P, 1P, 3S, 3P, 3D 1P,
1D, 1F, 3P, 3D, 3F 1S, 1P, 1D, 1F, 1G, 3S, 3P,
3D, 3F, 3G
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For equivalent electrons, less terms
p2, p4 P3 d2, d8 d3, d7 d4, d6 d5
1S, 1D, 3P 4S, 2P, 2D 1S, 1D, 1G, 3P, 3F 2P, 2D,
2F, 2G, 2H, 4P, 4F 1S, 1D, 1F, 1G, 1I, 3P, 3D,
3F, 3G, 3H, 5D 2S, 2P, 2D, 2F, 2G, 2H, 2I, 4P,
4F, 4D, 6S
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How to determine the shell structures and terms
in experiments? ----- X-ray spectrum
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homework

• pp344
• 19.1, 19.4, 19.6, 19.7