Transcript NMR (Nuclear Magnetic Resonance) and its applications

Chapter 7
Atoms in a magnetic field
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.
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
B0
possible orientations in an applied
ms = 1/2
magnetic field. They correspond to
two values of the potential energy.
0

s  s(s  1)  B g s
s, z  ms B gs   12 gs B
ms = -1/2
The difference of the potential energy of these two orientations:
E  g s B B0
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:
E  h  g s B B0
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).
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.
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.
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.
NMR (Nuclear Magnetic Resonance)
and its applications in
medical diagnosis, atomic clock and
chemical shift
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:

I  I ( I  1),
gI N 
I 
I


Where the quantum number I may be integral or half-integral; the nuclear
magneton µN = eħ/2mp is the unit of the nuclear magnetic moment.
The spin of atomic nuclei will directional quantised
in the presence of external magnetic field. The z
component of the nuclear magnetic moment:
I , z  mI g I  N
Where the quantum number mI = -I, -I+1, …, I, has 2I+1 possibility.
The maximum observable value of µI is:
I
max
 IgI N
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:
4
2
40
56
88
188
208
238
He,126C,168O,20
Ca,26
Fe,38
Sr,144
Cd
,
Hf
,
Pb
,
48
72
82
92 U
Nuclear magnetic resonance
Energy levels for a nucleus with spin quantum number I
No field
Applied
magnetic field
mI = -1/2
energy
Resonance frequency:
 = E / h
E
mI = 1/2
gI N
hB
E 
, 
2

N 
e
,
2mP
Nuclear magneton
 B mP

 1836
 N me
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.
2e B 2  2  12 (5.79105 eV / T )(1T )
electron spin 

 1.76081011 s 1
16

6.5810 eV  s


 28.025GHz Larm or frequency
2
2 p B 2  2.79  (3.15108 eV / T )(1T )
8 1
 proton  spin 


2
.
6753

10
s
16

6.5810 eV  s


 42.5781MHz Larm or frequency
2
The Larmor frequency
Larmor/B
Particle Spin
/B
s-1T-1
Electron 1/2 1.7608 x 1011 28.025 GHz/T
Proton
1/2 2.6753 x 108 42.5781 MHz/T
Deuteron 1 0.4107 x 108 6.5357 MHz/T
1/2
3/2
1/2
1
1.8326 x 108 29.1667 MHz/T
0.7076 x 108 11.2618 MHz/T
1.0829 x 108 17.2349 MHz/T
0.1935 x 108
3.08 MHz/T
13C
1/2
0.6729 x 108
10.71 MHz/T
19F
1/2
2.518 x 108
40.08 MHz/T
Neutron
23Na
31P
14N
The Larmor frequency of the electron spin is in the microwave region, the
Larmor frequency of proton or other nucleus is three orders smaller.
Diagram of a simple nuclear spin resonance apparatus
Fixed magnetic
field B0
The radio-frequency
field B1
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.
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 wellcalibrated 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.
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.
The setup of MRI
Be possible to image soft tissues:
Joint, brain, and spinal cord
normal
knub
The gradient magnetic field
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.
the applications of MRI:
In 1999,
In 2002,
2,170 in all of the world
22,300
and 500 in china (also several in Jinan)
Usage,
~ 60 million times in 2000
Cost: ~ 160$
Advantage:
non-invasive, non-ionising radiation, and a high soft-tissue
resolution and discrimination in any imaging plane.
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.
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:
B  B0 (1  s )
where s is called the shielding factor or screening factor. The factor
s is small - typically 10-5 for protons and <10-3 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.
(vS  vR ) 10
d
vR
6
quoted as ppm
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


B0 (1  s )
2
  gyrom agnet
ic ratio
the chemical shift can also be expressed as:
(s R  s S )
d
106  (s R  s S ) 106
1s R
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:
d=1.23, (CH3)2; 2.16, CH3C=O;
2.62, CH2; 4.12, OH
The Caesium(Cs) atomic clock
— as a time and frequency standard
Cs has a nuclear spin I=7/2, and in the
atomic ground state, the angular momenta
of the electrons J=1/2;
The total angular momenta F=4 and F=3
The transition frequency used for the Cs
atomic clock corresponds to the transition
between the states:
F=3, mF=0, and F=4, mF=0
A portion of the term scheme of the Cs atom in the ground state as a function
of a weak applied magnetic field B0.
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.
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.
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.
Ordinary Zeeman effect
Without
Magnetic field
With magnetic field
Transverse observation
EB0 E⁄⁄B0 EB0
With magnetic field
Longitudinal observation
EB0 ,
circular
Ordinary Zeeman effect for the atomic Cd line at λ = 6438. With transverse
observation the original line and two symmetrically shifted components are
seen. Under longitudinal observation, only the split components are seen.
Transverse and longitudinal observation of
emission spectral lines in a magnetic field. S
is the entrance slit of the spectrometer.
Anomalous Zeeman effect
D1
D2
Without magnetic field
With magnetic field
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.
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.
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.
An oscillating electron is resolved into three component oscillators
e- oscillator
Component
2 and 3
Component 1
B0
2
3
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:
d 
e

B0  B B0
2m0

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:
mx  m02 x  ey B0  0
my  m02 y  exB0  0
mz  m02 z  0
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 = u0exp[i(0 – eB0/2m)t] and v = v0exp[i(0 + eB0/2m)t]
The component electron oscillators 2 and 3 thus emit or absorb circularly
polarised light with the frequency 0 ± d.
The frequency change has the magnitude:
d
e
d 

B0
2 4m0
For a magnetic field strength B0 = 1T, this yield the value:
d  1.4 1010 s 1  0.465cm1
For each spectral line with a given magnetic field B0, the frequency
shift d is the same. Theory and experiment agree completely.
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 s+ and s– light, where s stands for perpendicular
and the + and – signs for an increase and decrease of the frequency.
The s+ light is right-circular polarised, the s– 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.
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:


Vm j  (  j ) z  B0   m j g j  B B0 with m j  j , j  1,   j
B0, z
jz = mjħ
µj
j
µj,z = mjgjµB
Precession of j and µj about
the direction of the applied
field B0, j = l.
The (2j+1)-fold directional degeneracy is lifted in the
presence of the magnetic field, and then the term is
split into 2j+1 components. These are energetically
equidistant. The distance between two components
with mj = 1 is
E  g j B B0
For the ordinary Zeeman effect, the spin S = 0 and consider only
orbital magnetism. gj has a numerical value of 1. The frequency shift:
e
d 
B0
2m0
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.
The splitting diagram for a cadmium line
mj
2
1
0
-1
-2
1D
2
=6438
1P
1
0
-1
1
mj
-1
0
1
s-

s+
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.
R. A. Beth in 1936 found that the circular
polarised light quanta has not only the energy
but also the angular momentum.
s-, circular polarised photon
propagation

l 
s+, circular polarised photon
propagation

l 
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 -ħ
after the transition, the emitting light has angular momentum -ħ, and it
is thus circular polarised, which is s- light.
For mj = +1, the angular momentum of the system was changed +ħ
after the transition, the emitting light has angular momentum +ħ, and it
is thus circular polarised, which is s+ light.
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.
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.
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
( j ) j , z  m j g j B
The magnetic energy is
Vm j  ( j ) j , z B0
The number of splitting components in the field is given by mj and is
again 2j+1. 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:
Em j ,m j1  g j B B0
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
2P . For optical transitions, the selection rule is
3/2
again mj = 0, ±1. It yields 10 lines.
D1 line
2P
2S
1/2
mj mjgj
+1/2
-1/2
+1/3
-1/3
+1/2
+1
1/2
-1/2
s s
D2 line
2P
2S
3/2
mj mjgj
+3/2
+6/3
+1/2
-1/2
-3/2
+2/3
-2/3
-6/3
+1/2
+1
-1/2
-1
1/2
-1
ss  ss
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



 j   s  l  
B



( gl l  g s s )
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.
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):
 

 
(  j ) j  l cos(l , j )   s cos(s , j )
 
 
  B l (l  1) cos(l , j )  2 s( s  1) cos(s , j )




Vector model

S

3 j ( j  1)  s ( s  1)  l (l  1)
( j ) j 
B
2 j ( j  1)

L

J
 gj


L
1
2
J

S
1
2


S
( J ) J
j ( j  1)  B
The magnetic moment projected in j direction:

(  j ) j   g j B j / 

j ( j  1)  s( s  1)  l (l  1)
g j  1
2 j ( j  1)
The component of magnetic moment in z direction:
( j ) j , z  m j g j B
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.
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
Vms ,ml  (ml  2ms )B B0
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.
E  (ml  2ms )B B0
Term diagram and optical transitions
of Na atoms
glml+gsms
2
1
0
-1
-2
1
-1
(a) D1 and D2 lines of the neutral
Na atom; (b) the anomalous
Zeeman effect; (c) Pachen-Back
effect.
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.
homework
Pp220, 13.1, 13.3, 13.5, 13.8
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
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.
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.


L   li ,
i


S   si ,
i
  
J SL
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:
For example for a two-electron system like the He atom
The orbital angular momentum L of the atom:
  
L  l1  l2 ,

L  L( L  1)
L  l1  l2 , l1  l2  1,, l1  l2
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.
For the total spin angular momentum S:
  
S  s1  s2
with

S  S ( S  1) 
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:
  
J  L  S,
The quantum number J:
For S = 0, J = L;
For S = 1, J = L +1, L, L – 1

J  J ( J  1) 
singlet;
triplet
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:
n
2 S 1
LJ
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
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 = s 2 = 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.
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, …
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.
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.
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.
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.
  
j1  l1  s1 ;
  
j2  l 2  s2 ;



J   ji

with J  J ( J  1)
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.
Transition from LS coupling in light atoms to jj coupling in
heavy atoms in the series C – Si – Ge – Sn – Pb.
Magnetic moments of many-electron atom
In the case of LS coupling, the magnetic moment:



 J   L  S
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:
3J ( J  1)  S ( S  1)  L( L  1)
( J ) J 
  B  g J J ( J  1)  B
2 J ( J  1)

J ( J  1)  S ( S  1)  L( L  1)
gJ  1
2 J ( J  1)
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.

(J ) J , z  mJ g J B
With mJ = J, J - 1, … , -J
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
Hund’s rules.
Hund’s 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.
Rule #2: The term with maximum multiplicity lies
lowest in energy.
For example: in the electronic configuration p2, we expect
the order 3P < (1D, 1S)
The explanation of the rule lies in the effects of the spinspin interaction. Though often called by the name spinspin 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.
2
Space wavefunction
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 < 1D < 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.
3
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.
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 < 3P1 < 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.
Influence on the atomic energy levels
Hund’s rule #2
Hund’s rule #4
Hund’s rule #3
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
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.
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:
1) For each value of S, MS = mSi have the possible values S,
S-1, …, -S;
2) For each value of L, ML = mli have the possible values L,
L-1, …, -L;
3) 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.
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
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
How to determine the shell structures
and terms in experiments?
----- X-ray spectrum
homework
pp344
19.1, 19.4, 19.6, 19.7