(10) electron spin & angular momentum coupling

Special Lectures on QM & SM
넫Special Lectures on Applied
Quantum and Statistical Mechanics
* March 2012 *
* Lecture Materials prepared by
Prof. Hyun M. Jang at POSTECH
* E-mail : hmjang@postech.ac.kr

(10) Electron Spin and
Angular-Momentum Coupling
in Multielectron Atoms

The wave-mechanical theory based
on the Schrödinger equation does not
account for the observed several
feature of atomic spectra.
( Ex 1 ) The wave-mechanical theory
predicts a single-line spectrum at λ =
656.3 nm for the electronic transition
between n = 2 and 3. However, the
spectrum actually shows two lines
0.14 nm apart each other (fine
structure).
( Ex 2 ) The normal Zeeman effect
predicts three spectral lines under a
B-field :
However, an atomic spectrum, more
often than not, consists of more than
three spectral lines (anomalous
Zeeman effect).
1
4
0
1
4
3
2
1



lo
lo
lo
mfor
m
eB
mfor
mfor
m
eB






(1) Electron Spin
In order to account for both the fine structure in spectral lines and the anomalous
Zeeman effect, Goudsmit and Uhlenbeck proposed that every electron has an
intrinsic angular momentum (thus, magnetic moment), called spin, whose
magnitude is the same for all electrons.
However, this proposition was open to serious objections : “In order to have the
observed angular momentum associated with the electron spin, so small an object
(r ≈10-16 m) would have to rotate with an equatorial velocity many times greater
than the velocity of light.”
The fundamental nature of electron spin was later confirmed by the famous
Dirac‟s development of relativistic quantum mechanics.
Spin Angular Momentum (S) :
where s is the intrinsic angular-momentum (spin) quantum number and = ½ .
Similar to the orbital angular-momentum vector, the spin angular-momentum
vector has the 2s+1 = 2 orientations specified by ms = +1/2 („spin up‟) and ms =
−1/2 („spin down‟).
L = extrinsic angular momentum, associated with the motion of the center of mass
S = intrinsic angular momentum, associated with the motion about the center of
mass
)1(.........
2
3
)1(   ssSS

The component Sz of the spin angular momentum of an electron along a magnetic
field in the z direction is determined by the spin magnetic quantum number (ms) :
z component of spin
angular momentum :
The spin magnetic dipole moment (μs) of an electron
is related to its spin angular momentum (S) by
the following relation :
where g is the g-factor (≈2 for spinning motion
and 1 for the orbital motion).
The possible components of μs along any axis,
say z axis, are therefore limited to
z component of
spin magnetic
moment :
where μB is the Bohr magneton.
)2(........
2
1
  sz mS
(3).....
2
SSμs
m
e
g
m
e

)4(.......)(
2
Bsz
m
e
 


* Pauli’s proposition :
* Spin ½ → electrons, protons, neutrons, quarks, * Spin 1 → photon
* Spin 2 → graviton
♦ Two eigenstates (basis vectors) : Spin-up
Spin-down
 

mmmLmmLcf
msmmsSmsssmsS
lz
ssszss


,)1()
)2(,)1(
22
22
with ms = -s, -s+1, ·····, s-1, s






"")
2
1
(
2
1
""
2
1
2
1
""
1
0
""
0
1






















represents
representswith

 (orthonormal eigenspinor or eigenvector for spin-up)
(orthonormal eigenspinor or eigenvector for spin-down)

(( Summary ))












2
)
2
1
(
2
1
)
2
1
()
2
1
(
2
1
2
)
2
1
(
2
1
2
1
4
3
)
2
1
(
2
1
)1
2
1
(
2
1
)
2
1
(
2
1
4
3
2
1
2
1
)1
2
1
(
2
1
2
1
2
1
2222
2222






zz
zz
SS
SS
SS
SS
''''''''
.,.,
ss
lsl
mmmmnnss
s
smlnmmln
ss
mmnmmnwith
ormwhere
m
ormmnmmnei












spinspacetotal
with
)1(  ssS

* Set of Quantum Numbers of an Atomic Electron
(2) Exclusion Principle
In 1925 Wolfgang Pauli discovered the fundamental principle that governs the
electronic configurations of atoms having more than one electron. His exclusion
principle states that
“No two electrons in an atom can exist in the same quantum state. Each
electron must have a different set of quantum numbers n, l, ml , ms.”
Name Symbol Possible Values Quantity Determined
Principal QN n 1, 2, 3, …… Electron energy
Orbital QN l 0, 1, 2, ….., n-1 Orbital angular-momentum
magnitude
(Orbital) Magnetic
QN
ml -l, …. 0 ……, +l Orbital angular-momentum
direction
Spin Magnetic QN ms -1/2, +1/2 Electron spin direction

(3) Symmetry of the Wave Function
The Pauli exclusion principle is a quantum phenomenon, which is
ultimately connected with the symmetry of ψ. Consider the Schrödinger
equation for the two-electron atom like He :
……. (5)
where r1 and r2 denote the coordinates of the two electrons, 1 and 2,
respectively.
Consider what happens if we exchange the positions of the two electrons.
Since the electrons are identical, the probability density is unchanged,
namely,
……. (6)
♦ Two possibilities exist from Eq. (6) :
* Symmetric ψ : ψS (r1, r2) = ψS (r2, r1) ……. (7)
* Anti-symmetric ψ : ψA(r1, r2) = −ψA(r2, r1) ……. (8)
    ),(),(),(,
2
,
2
21212121
2
2
2
21
2
1
2
rrErrrrVrr
m
rr
m
 

.
1
11
r
asdefinedis



2
12
2
21 ),(),( rrrr  

Let “a” denotes the quantum state a (say, n, l, ml, ms= + ),
whereas “b” represents the quantum state b (say, n, l, ml, ms= − ).
Then, considering the exclusion principle, we have two possible
combinations :
…….. (9)
…….. (10)
(i) Exchanging particles 1 and 2 leaves ψs unaffected, while it reverses
the sign of ψA.
(ii) For ψs, both particles 1 and 2 simultaneously exist in the same state
with a = b. → contradict to the exclusion principle.
(iii) For ψA, if we set a = b, then ψA= 0. → consistent with the
exclusion principle.
(iv) Therefore, ψA is an adequate description for a system of more than
one electron.
2
1
2
1
 )1()2()2()1(
2
1
babaS  
 )1()2()2()1(
2
1
babaA  

For N different electrons (fermions) with m distinct quantum states, ψ (i.e., ψA) can
be generalized using the following type of determinant (Slater determinant) :
(4) Fermions and Bosons
)(......)2()1(
.................................
)(.......)2()1(
)(.......)2()1(
!
1
N
N
N
m
mmm
bbb
aaa
A



 
Fermions Bosons
Spin Odd half-integral spin (1/2, 3/2,
5/2, ….)
0 or integral spin
Symmetry ψA (anti-symmetric w.r.t. an
exchange of any pair)
ψS (symmetric w.r.t. an
exchange of any pair)
Exclusion
Principle
apply does not apply
Statistics Fermi-Dirac Distribution Bose-Einstein Distribution
Examples p, n, e (elementary particles
that constitute matters)
photon, graviton, μ-meson (field
particles), α particle

(5) Shell Structure of Many-Electron Atoms
The atomic shells are denoted by the following scheme :
n = 1 2 3 4 5
K L M N O
Electrons that share a certain value of l in a given shell (n) are said to
occupy the same subshell.
(Na) 1s2 2s2 2p6 3s1
2s subshell → n = 2, l = 0; 2p subshell → n = 2, l = 1, etc.
Each subshell is characterized by the orbital quantum number, l.
(i) For a given l, there are (2l + 1) different ml (∵ ml = 0, ±1, ····· ±l).
(ii) For a given ml, two possible ms (± ).
(iii) ∴Each subshell can contain a maximum of 2(2l + 1) electrons.
2
1

(iv) The maximum number of
electrons a shell (n) can hold is
represented by
2 {1 + 3 + 5 + ····· + 2(n - 1) + 1}
= 2 {1 + 3 + 5 + ····· + (2n - 1)}
Quantum mechanics predicts the
general rule for the filling order in
which the energy levels are filled
in multielectron atoms. This order
is represented in the right-hand side
figure.
However, this filling order is slightly
different from the order of energy for
filled subshells that can be written as :
E1S < E2S < E2P < E3S < E3P < E3d <
E4S < E4P < E4d < E4f < E5S < E5P <
E5d < E5f < E6S < ······
 


1
0
max )12(2
nl
l
lN

(6) Spin-Orbit Coupling
An electron circles an atomic
nucleus, as viewed from the frame
of reference of the nucleus. From
the electron‟s frame of reference,
the nucleus is circling the electron
(right-hand side figure).
The magnetic field the electron
experiences as a result is directed upward from the plane of the orbit.
This internal B-field then exerts a torque on the spinning electron. The
interaction between the electron‟s spin magnetic moment and this internal
B-field leads to the phenomenon of spin-orbit coupling.
The interaction energy (USO) of a magnetic dipole moment μ in a field
B is given by USO = − μ·B = − μB cosθ ………. (11)
where θ = the angle between μ and B.

In the case of the spin magnetic moment of the electron :
μ cosθ = μSZ = ± μBohr = ± μB ……….. (12) Eq. (4)
∴ For the spin-orbit coupling : USO = ± μBB ………. (13)
where
The magnetic interaction energy (USO) for an electron in the 2p
state was estimated to be 2.3ⅹ10-5 eV ( = 3.7 ⅹ10-24 J ).
* See pp. 248 – 249 of Beiser.
Here, the upper level
corresponds to
where the lower level
corresponds to
,
2
3
2
1
1  slj
.
2
1
2
1
1  slj
External B-field의 인가 없이도 electron
에 대해 상대적으로 운동하는 proton에
기인한 internal B-field 에 의해 energy
splitting 이 일어남.
Two closely spaced
emission line in the
2p → 1s transition
meB 2/

(7) Total Angular Momentum J
Each electron in an atom has a certain orbital angular
momentum (L) and a certain spin angular momentum
(S), both of which contribute to the total angular
momentum (J).
For simplicity, let us consider an atom having a
single electron outside the closed inner shell (i.e., group
I elements of the periodic table, H, Li, Na, K, Cs, etc.).
Since the total angular momentum and the magnetic
moment of a closed shell are zero, the outer electron‟s J
is given by
( ex ) If l = 0, j = +1/2 only.
The component (Jz) of J in the z direction is given by
* Two possible orientations for an one-outer electron atom :
(i) j = l + s = l +1/2 so that j > l.
(ii) j = l - s = l -1/2 so that j < l.






sZ
mllmlZ
mlml
mS
ssS
YmYL
YllYL




,)1(
,
,)1(
22
22
)16(.......02/1
)15(........)1(
)14(.......



lsljwhere
jj J
SLJ
)17(.......,1,.......,1, jjjjmwithmJ jjZ
 
If we require
then j should be l ±1/2.
(Ref) S. Gasiorowicz, “Quantum
Physics, 2nd ed.” pp. 253-259
(Wiley; 1996).
slsl mmslmmsl jjJ  22
)1( 

* The two ways in which
L and S can be added
to form J when l = 1
and s = ½.
* For one-outer electron :
( Ex ) What are the possible
orientations of J for j = 3/2
and j = ½ states that
correspond to l = 1 ?
* For j = 3/2,
Eq. (17) →
mj = − 3/2, − 1/2, + ½, + 3/2
* For j = ½,
Eq. (17) → mj = − 1/2, + ½


)1(
,
2
3
)1(,)1(


jj
ssll
J
SL

* If there is no external magnetic
field, the total angular momentum
J is conserved in magnitude and
direction. L and S precess about
the direction of their resultant J.
* If there is an external magnetic
field B present, then J precesses
about the direction of B while L
and S continue precessing about
J. The precession of J about B is
what gives rise to the anomalous
Zeeman effect. → Different
orientations of J involve slightly
different energies in the presence
of B.

(8) LS (Orbital-Spin) Coupling
For closed shells, all orbitals with the same n and l values are doubly occupied
by s = ±1/2 and do not contribute to L, S and J values. However, for partly filled
shells, it is not clear from a configuration specification alone how the electrons
with given n and l are distributed among the different possible ml and ms values.
If we consider an open shell with k electrons, we can associate an orbital
angular momentum Li and a spin angular momentum Si with the i th electron.
The label i is introduced only for convenience and we have to remember that the
electrons are truly indistinguishable.
♦ total orbital angular momentum :
♦ total spin angular momentum :
♦ total angular momentum :
This scheme is called “L-S coupling or Russell-Saunders coupling” and is
valid for lighter atoms with Z (atomic number) ≤ 40.
Alternatively, one could couple together the orbital and spin angular momenta
of each electron to obtain a total angular momentum for each electron.
The coupling scheme outlined in Eq. (21) is called “j-j coupling” and is the more
appropriate scheme for Z ≥ 40.
)18(.......
1



k
i
iLL
)19(.......
1



k
i
iSS
)20(.......SLJ 
)21(..........,..........,
1
111 

k
i
ikkk JJJSLJSL


)1(
)1(*


iii
iii
ss
llwhere
S
L

Let us concentrate on the Russell-Saunders coupling. For a group of k
electrons, the possible values of the total spin angular-momentum quantum
number are given by :
……. (22)
Then, the magnitude of the total spin angular-momentum :
…….. (23)
Similarly, the possible values of the total orbital angular-momentum
quantum number L are obtained by writing :
L = l1 + l2 ······ + lk, l1 + l2 + ······ + lk – 1, ·········· ≥ 0 ….. (24)
and ……. (25)
* Eqs. (22) and (24) can be understood in terms of their z components of total
angular momenta, MS and ML.
)(
2
1
,,2
2
,1
2
,
2
)(0,,2
2
,1
2
,
2
.0,1, 2121
koddfor
kkk
kevenfor
kkk
ssssssS kk



)1(  SSS
)1(  LLL

(ex 1) for three electrons (l1 = 1, l2 = 1, l3 = 1) : L = 3, 2, 1, 0 and
(ex 2) for one f electron (l1 = 3), and two p electrons (l2 = l3 = 1) :
L = 5, 4, 3, 2, 1 and
If one of li is larger than the others, the minimum value is that given by
orienting the other angular momenta to oppose it.
Then, the possible total angular-momentum quantum numbers are
obtained by writing :
……… (26)
(ex 3)
(ex 4) Find the possible values of J of two atomic electrons whose orbital
angular-momentum quantum numbers are l1 = 1 and l2 = 2.
The possible values of L are given by Eq. (24). For two-electron case,
Eq.(24) reads :
Eq.(22) → S = 1, 0
.
2
1
,
2
3
S
.
2
1
,
2
3
S
SLSLSLJ  ,,1,
2
3
,
2
5
,
2
7
,
2
9
:
2
3
,3  JSLfor
.1,2,3,,1, 212121  LllllllL

The possible values of J are given by the LS coupling scheme, viz,
Here
.0,1,2,3,4,,1,  JSLSLSLJ


6)1(
2)1(
222
111


ll
ll
L
L
For l1 = 1, l2 = 2, we have :
1,2,3.,..........,1, 122121  L
32L 6L
2L

(9) Atomic Term Symbol
Let us look at carbon (C) atom in the ground-state configuration
(1s2 2s2 2p2). In this configuration, 1s and 2s electrons make up the
closed shell and the 2s subshell, respectively. Thus, the only electrons
that can contribute to nonzero values of L, S and J are the two 2p electrons.
Without considering the Pauli exclusion principle, we have 15 distinct
possible states for a given p2 configuration :
l1=l2=1, s1 = s2 = . See “Atoms & Molecules,” by M. Karplus and R.N.
Porter (W.A. Benjamin, Inc., 1970), chapter 4.
Thus, a term symbol that characterizes the entire quantum state of a given
atom had been proposed and is represented by the following symbol :
……. (27)
(i) L represents the total orbital angular-momentum quantum number :
L = 0 1 2 3 4 5
S P D F G H
J
S
Ln 12 
2
1

(ii) The superscript denotes the multiplicity of the state,
which is the number of different possible orientations of L and S
( thus, the number of different possible values of J vector ).
Recall ……. (26)
when S = 0, J = L only, the multiplicity = 1 (singlet state)
when S = , J = , the multiplicity = 2 (doublet state)
when S = 1, J = L+1, L or L-1, the multiplicity = 3 (triplet state)
∴ Multiplicity (Ω) = 2S + 1
(iii) The total angular-momentum quantum number J is used as a subscript.
SLSLSLJ  .,..........,1,
2
1
2
1
L

(ex 1) the ground state of Na : 3 2S1/2
the first excited state of Na : 3 2P1/2
Find the possible quantum numbers, n, l, j and mj of the outer electron.
* for 3 2S1/2 ; n = 3, l = 0, j = ½, mj = ±
* for 3 2P1/2 ; l = 1, Eq. (26) → → ∴ two possible j values.
(ex 2) Is it possible for a 2 2P5/2 state to exist ?
is impossible !
(10) Hund Rules
It is known that there could be 5 distinct possible states for a given p2
configuration of carbon atom (1s2 2s2 2p2).
They are : 3P2,1,0
1D2
1S0
2
1
2
1
2
3
orj 
2
1
,
2
1
,
2
1
,1,3)(
))17.((
2
3
,
2
1
,
2
1
,
2
3
,
2
3
,1,3)(


j
j
mjlniior
Eqmjlni
2
5
2
3
,
2
1
1  JJL
* We use capital letters for
L and J since this state
would involve more than
one state.

Then, it is necessary to find their relative energy in order to choose which
term characterizes the ground state. This can be done by a set of simple
rules, called Hund’s rules.
* Rules (a) and (b) arise from the electron-electron interaction, while rule (c) is a
consequence of the spin-orbit (magnetic) interaction (See Page 14 of this chapter).
* In the above rules, S and L can be effectively replaced by their z component
quantum numbers, MS and ML, respectively.
(a) The terms are ordered according to their S values, the term with
maximum S being most stable and the stability decreasing with
decreasing S. Thus, the ground state has maximum spin multiplicity.
(b) For a given value of S, the state with maximum L is most stable.
(c) For given S and L, the minimum J value is most stable if there is
an open shell that is less than half-full and the maximum J is most
stable if the subshell is more than half-full.

Here
ML = ml1 + ml2 + ·····
MS = ms1 + ms2 + ·····
and J = L+S, L+S-1, ····· .SL 

ML = ml1 + ml2 + ·····, MS = ms1 + ms2 + ·····, and J = L+S, L+S-1, ····
where ML and MS are the quantum numbers for the z component of the total orbital
and spin angular momentum, respectively. We further have the relations :
* Ms = S, S-1, S-2, ……., -S (2S+1 values of MS for a given S)
* ML = L, L-1, L-2, ……., -L (2L+1 values of ML for a given L)
(( Example ))
Oxygen atom has 2s22p4 electronic configuration outside the filled 1s shell.
Find the term symbol for its ground-electronic state ?
(i) According to the Rule (a), the maximum S or MS (i.e., parallel spins) is preferred.
MS = ½ + ½ + ½ - ½ = 1. (See the bottom Table in the previous page.)
∴ S = 1. → ∴ Ω = 2S+1 = 3
(ii) p-orbital ↔ l = 1 → The possible value of ml = 0, ±1
According to the Rule (b), the maximum L or ML is preferred.
ML = (+1)+(0)+(−1)+(+1) = 1. (See the bottom Table in the previous page.)
∴ L = 1 → ∴ P symbol
(iii) J = L+S, L+S-1, ·····
∴ possible J = 2, 1, 0
2p4 configuration ↔ more than half-full. → ∴ maximum J is preferred (Rule (c)).
∴ J = 2. → 3P2
.SL 
.SL 

(11) Fine Structure of Electronic Energy Levels
Consider one-electron atom for simplicity.
…….. (11)
…….. (28)
Quantum mechanically :
……. (29)
…….. (30)
A factor of ½ is introduced because of the Thomas precession.
J = L + S ………. (14)
J2 = (L + S)·(L + S) = L2 + S2 + 2 L·S
…….. (31)
where s = ½.
Bμ '
soH
LB 32
04
1
rcm
e


LS





 322
0
2
'' 1
82
1
rcm
e
HH soso

Bohr
m
e
m
e

2

ee μSμ
   )1()1()1(
22
1 2
222
 sslljjSLJ

SL

…...... (32)
………… (33)
where En = −E1 /n2 = − 13.6 eV/n2
Similarly, the relativistic correction gives the following expression :
……… (34)
∴ Combining Eq. (33) with Eq. (34), one obtains the final energy levels
of hydrogen atom including the fine-structure effect :
…….. (35)
where α ≡ 55.4 eV/2mc2.
3
0
3
)()1()
2
1
(
11
nalllr 

 
 












)1()21(
43)1()1(
)()1()21(2
)1()1()1(1
8
2
2
3
0
2
22
0
2
''
lll
lljjn
mc
E
nalll
sslljj
cm
e
EH
n
soso










 3
)21(
4
2 2
2
'
l
n
mc
E
E n
r















4
3
21
1
6.13
2
2
2
j
n
nn
eV
Enj


The fine structure (LS coupling + relativistic correction) effect now breaks
the degeneracy in l (i.e., for a given n, the different allowed values of l do
not all carry the same E.).
The energy is actually determined by n and j.
Thus, the azimuthal eigenvalues for orbital and spin angular momentum
(ml and ms) are no longer “good quantum numbers.”
The spin and orbital angular momenta are not separately conserved.
The “good” quantum numbers are n, l, s, j and mj.

(10) electron spin & angular momentum coupling

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