Download Atomic structure

Atomic structure
Contents
1
Introduction
1
2
One-electron Systems
2.1 The hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Hydrogenic atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Constants and orders of magnitude . . . . . . . . . . . . . . . . . . . . . .
1
1
3
3
3
Many-electrons Systems
3.1 Hamiltonian . . . . . . . . . . . . . .
3.2 Central field approximation . . . . .
3.3 Spin and the Pauli principle . . . . .
3.4 Notation of electronic configurations
4
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4
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5
6
Corrections to the central field approximation
4.1 Relevance to the astrophysics of the ISM . . . .
4.2 Hamiltonian . . . . . . . . . . . . . . . . . . . .
4.3 L − S coupling, terms, levels, and Hund’s rules
4.4 Example: CI . . . . . . . . . . . . . . . . . . . .
4.5 Fine structure . . . . . . . . . . . . . . . . . . .
4.6 Hyperfine structure . . . . . . . . . . . . . . . .
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Introduction
Light is the principal vector of information to study the Universe. The essence of
observational astrophysics consists in extracting physics from electromagnetic waves.
The alterations of the wave along its path from the emitter to the receiver may serve
to probe the Universe or must be subtracted to retreive informations on the source of
photons. To these aims, a profound knowledge of the mechanisms for the production
of light appears essential. This is the goal of the first lecture which outrageously
summarizes the essential mechanisms that lead to the production of light by atoms and
molecules. Excellent textbooks at various levels cover this and related topics: Rybicki
& Lightman (1985); Bransden & Joachain (2003); Tennyson (2005).
2
One-electron Systems
2.1
The hydrogen atom
Hydrogen is the most abundant component of the Universe. It is also the simplest
neutral species, composed of one proton and one electron. From the quantum mechanics
postulates, the H system is fully described by a wave function Ψ which obeys the timedependent Schrödinger equation:
i~
∂Ψ
= HΨ(~r, t)
∂t
(1)
1
Table 1: Summary of the various quantum numbers for the H atom.
Quantum number Observable
Operator Wave Function
Eigenvalue
n
Energy level
H
Ψ
En
l
oam
`ˆ2
Ylml
l(l + 1)~2 , l ∈ N, l < n
ml
z-proj. of oam
`ˆz
Ylml
ml ∈ Z, |ml | ≤ l
s
Electron spin
ŝ2
σsms
s(s + 1)~2 , s = 1/2
ms
z-proj. of spin
sˆz
σsms
ms ∈ Z, |ms | ≤ s
For Hamiltonians which do not explicitely depend on time, stationary solutions are
obtained by separating time and space variables such that Ψ = Ψ(~r) exp(iEt/~). E is
interpreted as the total energy of the system and is solution of the time-independent
eigenvalue equation:
HΨ = EΨ
(2)
The spatial, non-relativistic Hamiltonian is given by
H =−
p̂2
ke2
~2 2 kZe2
∇ −
−
=
r
r
2e
m
2e
m
(3)
e is the reduced mass of the proton-electron system. In the case
where k = 1/4π0 and m
of the H atom, it is possible to solve exactly the Schrödinger equation, leading to spatial
wavefunctions of the form φ(r, θ, φ) = f (r)g(θ, φ). The intrinsic angular momentum of
the electron (spin) is described by the spin wave function σ(sz ). The complete hydrogenic
wave function Ψ(r, θ, φ, sz ) (also called spin-orbitals) is the product of the spatial and
spin wave functions:
1
Ψ(r, θ, φ, sz ) = Pnl (r)Ylml (θ, φ)σms (sz )
r
(4)
The entire state is thus characterized by five quantum numbers (QN) which are integers
or half-integers: n, l, ml , s, and ms .
• Energy level quantization: principal QN, n:
En = −
e 1
RH
13.6
m
= − 2 eV = −
R∞
2
me n2
n
n
• Quantization of the Orbital Angular Momentum (oam): orbital QN l.
The Ylml (θ, φ) functions are called spherical harmonics, and are eigenfunctions of
the `ˆ2 operator, with eigenvalues (l + 1)~2 :
`ˆ2 Ylml = l(l + 1)~2 Ylml ,
l ∈ N,
l<n
• Electron intrinsic oam (spin): spin QN s = 1/2
The spin wave functions σsms are eigenfunctions of the ŝ2 operator, with eigenvalues s(s + 1)~2 :
ŝ2 σsms = s(s + 1)~2 σsms = 3/4~2 σsms
• Quantization of the z-projection of the oam: magnetic QN ml
The Ylml are eigenfunctions of the `ˆz operator, with eigenvalues ml :
`ˆz Ylml = ml ~Ylml ,
2
|ml | ≤ l
Table 2: Constants and orders of magnitude. We note k = 1/(4π0 ).
Constant name
Symbol Expression Value
Rydberg
R∞
α2 m2e c2 /2
109737.32 cm−1 or 13.6 eV
2
2
2
e c /2
H-Rydberg
RH
α m
109677.58 cm−1
Bohr radius
a0
~/(e
mαc)
5.29 ×10−11 m
2
Fine-structure constant α
ke /(~c)
≈ 1/137
• Quantization of the z-projection of the electron spin: QN ms
The functions σsms are eigenfunctions of ŝ with eigenvalues ms :
sˆz σsms = ms ~σsms ,
|ms | ≤ s
Note that the z-projection of an oamp
vector is always strictly smaller than the amplitude
of the oam vector itself, e.g. ml < l(l + 1). The eigenvalue En depends only on the
principal quantum number (QN) n. The constant that enters the expressions of En are
the fine-structure constant α and the Rydberg constant (RH or R∞ ).
Exercise
1. Show that each energy levels for an hydrogenic atom En are 2n2
degenerate.
2. Give a physical interpretation of R∞ and RH ?
3. Determine the “radius” a0 of the H atom using a classical description in which the orbital angular momentum is quantized.
2.2
Hydrogenic atoms
The solution for the H atom can be generalized to H-like atoms. Hydrogenic (or
hydrogen-like) atoms are ions which have a nucleus composed of Z protons (and Z
neutrons), and a single electron (e.g. He+ , Li++ , Be 3+ ). The Hamiltonian is changed to
account for the increased charge and mass of the nucleus: the electrostatic potential is
e = (me mA )/(me +
changed from V(r) = −e2 /r to −Ze2 /r, and the reduced mass becomes m
mA ) with mA = A uma. The eigenfunctions are unchanged, the degeneracy remains 2n2 .
The eigenvalues become
En = −
ec2
Z 2 α2 m
Z2 RA
=
−
eV
2n2
n2
(5)
e/me R∞ .
where RA = m
2.3
Constants and orders of magnitude
• Rydberg constant: 1 Ry = 13.6 eV
ev2 = ke2 /a0 and in the ground state, oam is m
ea0 v = ~.
• Bohr radius: virial gives m
2
2
eαc = ~.
This gives a0 = ~ /(ke
me ) and v = αc, or a0 m
3
e ≈ me ; k = 1/(4π0 ) ; a0 = 5.29 ×10−11 m
• reduced mass m
• fine-structure constant: α = ke2 /(~c) ≈ 1/137
3
Many-electrons Systems
3.1
Hamiltonian
We consider here an atom or ion made of a nucleus of charge Ze and N electrons. The
various terms entering the Hamiltonian include:
1. kinetic energy of each electron and their potential energy in the attractive electrostatic field of the nucleus
2. the repulsive electrostatic field between the electrons
3. for each electron, the magnetic interaction of the electronic spin with the orbital
angular momentum (spin-orbit interaction, see below)
4. smaller effects (spin-spin between e− , relativistic effects, nuclear corrections, ...)
If we neglect the terms (3) and (4) from the list above, the Hamiltonian is written as a sum,
over the electrons, of the kinetic and attractive Coulombian potential V(ri ) = kZe2 /ri ,
on the one hand, plus a term which is the sum of all the 2-particle repulsive potential
ke2 /ri j (i , j). The Hamiltonian may then be written as:
H=
N " 2
X
p̂
i=1
# X 2
ke
kZe2
+
−
ri
ri j
2e
m
i,j
(6)
The spatial part of the wave function of the system is an eigenfunction of the Hamiltonian
of Eq. 6 with eigenvalue E. Unfortunately, for such system the Schrödinger can not, in
general, be solved exactly because the separation of variables is not possible due to the
2-particle interaction term (1/ri j ).
3.2
Central field approximation
Under certain approximations, it is possible to solve for the time-independent Schrödinger
with the Hamiltonian given by Eq. 6. The idea consists in treating the 2-particle repulsive
sum as a perturbation to the sum over the 1-particle terms:
X
X
H = H0 + H 0 =
Hi + H 0 , |H 0 | |
Hi |
i
i
Hydrogenic wave functions ψai (i) are then eigenfunctions of each Hi with eigenvalues
E (we note ai = (ni li mli msi ) the set of QNs for electron i). The product Ψ(1, . . . , N) =
Qi N
ψai (i) would then be an eigenfunction of the unperturbed system1 with eigenvalue
i=1P
E = Ei . However, for large N, the sum of the 1/ri j terms might not well be negligeable
compared to the attractive term, in which case the interaction term can not be treated
as a perturbation. A two-step simplification is due to Slater and Hartree, and is called
the independent particle model:
1
Though not satisfying the Pauli exclusion principle (see below).
4
1. self-consistent field approximation: each electron feels an effective potential created
by the nucleus and screened by the N − 1 other electrons, V(ri ) = −k[Z − S(ri )]/ri
where S is the screening charge felt by electron i.
2. central field approximation: hVit ≡ V(r) i.e. spherical symmetry
This approximation allows to rewrite the full Hamiltonian as a central-field part plus a
perturbation:


#
N " 2
X
X 1 X S(r ) 
p̂
k[Z − S(ri )]e2
i 


H =
+ ke2 
−
−


ri
ri j
ri 
2e
m
i=1
i,j
= Hc + H ,
0
i
0
|H | |Hc |
where the screening charge S(ri ) is chosen such that the effect of H 0 is small. The
effective potential V(ri ) is also called Hartree-Fock potential. It is bracketed by the two
asymptotic behaviours: the screening charge is (N − 1)e at large distances, and 0 at
short radii, or equivalently, V → −k(Z − N + 1)e2 /r at large distances from the nucleus
(r → ∞) whilst V(r) → −kZe2 /r for r → 0. The resolution of the Schrödinger equation
is numerical and is done iteratively, as the potential depends on the positions of the
electrons, which in turn depend on the potential. A guessed potential serves to compute
the electrons positions, which are used to estimate the new potential...
Each electron wave function ψai (~
ri ) is solution of a central-field equation (similar to,
but noticeably different from, the hydrogenic case): ψai (~
ri ) = 1/ri Pni li (ri )Yli ml (θ, φ)σ(szi ).
i
The wave function ψai is called an electronic orbital. A global eigenfunction Ψc of the
unperturbed Hamiltonian is made of the products of the N orbitals. Here, the orbitals are
not the hydrogenic wave functions which correspond to the particular case V = −kZe2 /r
(r → 0). The consequence appears on the eigenvalue Ei of ψai , which now depends on
both ni and li (but not on ml because V(ri ) has spherical symmetry). The (unperturbed)
energy of the global system is thus
X
Ec =
Eni li
i
For each ni , there are ni values of li , and the z-projection of each oam li can take 2li + 1
values, mli = −li , . . . , li . The energy level Eni li of each individual electron is thus 2(2li + 1)
times degenerate where the factor of 2 comes from the spin projection msi = ±1/2.
3.3
Spin and the Pauli principle
The state of each electron is fully described by a wave function, called spin-orbital, of the
form
Ψnlml ms = ψai σi
which consists in the product of the orbital and the spin parts. The question is “how
do we build, from the individual wave functions, a wave function describing the full system
?”. Products of the form Ψ(1, . . . , N) = ψa (1)ψb (2) . . . ψz (N) are solutions (a is the set
of QN for the electron 1, etc). However, the Pauli principle postulates that the total
wavefunction must be antisymmetric through the interchange of any two identical
Fermions. One consequence is the exclusion principle which states that no two identical
particles can occupy the same spin-orbital, or that no two fermions can have the same
5
set of quantum numbers. This principle implies that for a system of identical fermions,
the wave function of the system must be antisymmetric in the combined space and
spin coordinates of the particles, that is, Ψ changes sign if the coordinates (both spatial
and spin) of two electrons are interchanged. A simple manner to construct the global
wave function which satisfies complete antisymmetry is through the use of the Slater
determinant.
3.4
Notation of electronic configurations
The central field approximation allows to determine the electronic configuration of a
system, i.e. the n, l values of all the electrons. The dependence of the En,l on l stems
from the screening effect of the electrons. Clearly, as n and l increase, En,l increases
since large n, l correspond to large radii r, and electrons in these orbitals do not feel
much the attractive nucleus potential; for a given l, En,l increases with n. For a given n,
En,l increases with l since larger l correspond to larger centrifugal forces. Additionally,
En,l increases with n + l. These simple behaviours allow to understand the Mendeleiev
periodic table. The total energy of the system being the sum of the En,l , it is fully determined by the electronic configuration. The notation for such electronic configurations
is nlN with N the number of electrons in the orbital n, l. The orbital QN l is coded with
letters (s, p, d, f, g, h for l = 0, 1, 2, 3, 4, 5); a shell corresponds to n (also noted K, L, M,
N...), a sub-shell to l. Since the values ml and ms are not required for the assignment
of an electronic configuration, the degeneracy di of each subshell Eni li must include the
number of ways the Ni electrons can be arranged amongst the νi = 2(2li + 1) possible
states of this subshell: di = νi !/(νi ! (νi − Ni )!. The total degeneracy g, or statistical weight,
of a given configuration is then the product of the di for each subshell.
Exercise
Write the electronic configuration of the ground state of neutral and
ionized carbon CI and CII, and for SI, SIII. Compute the statistical
weight g of each configuration.
4
4.1
Corrections to the central field approximation
Relevance to the astrophysics of the ISM
The non-relativistic, unperturbed, Hamiltonian Hc , leads to approximate solutions for
hydrogenic or complex atoms. Still, high-orders terms are neglected in the process.
These terms stem from the fact that the full system involves several angular momenta
which must be added since only the total angular momentum is a constant of motion. The vector model gives a simple interpretation of how angular momenta coupling
works (Cowan 1981). Though weak, these interactions are crucial, and it is not possible
to overemphasized their importance in astrophysics. Of the many high-order corrections that may be added to Hc , we will concentrate ourselves on essentially two, which
lead the fine and hyperfine structures of atoms. An important application of atomic
fine structure is the λ158µm line of ionized carbon CII, the importance of which will
be recognized when discussing the thermodynamics of the interstellar medium. Unvaluable informations on the physical state of the ISM has been brought with the COBE
6
satellite which surveyed the whole sky in the fine structure lines of CII and NII at 158
and 205µm respectively. Fine-structure lines will also be discussed in the context of
molecules. When relevant, taking into account the hyperfine structure allow precise
determinations of the physical conditions of the gas (density, temperature). Interactions
of particles (atoms or molecules) with external magnetic fields, through the Zeeman effect, are often the only way to estimate the strength of the magnetic fields in the dense
regions where stars form.
4.2
Hamiltonian
The central field approximation leaves the values of ml and ms unspecified and, as
noted before, each electron energy level is 2(2l + 1) times degenerate. This is where the
perturbative term of the Hamiltonian, H 0 , comes in:
H 0 = Hoo + Hso + Hss + . . .
(7)
It includes an orbit-orbit term which includes the difference between the actual (nonspherical) Coulombian repulsion between the electrons and the averaged Hartree-Fock
potential V(r). The spin-orbit term is due to magnetic interaction. The spin-spin term
describes the Pauli exclusion principal and has no classical interpretation.
To determine the perturbed eigenfunctions, one needs to know whether all the above
terms are of the same order, or not. In most cases, the spin-spin term is negligeable
compared to the o-o and s-o terms. We can then distinguish three cases:
1. |Hoo | |Hso |: this is known as j − j coupling which applies for large systems
(larger than iron typically).
2. |Hoo | ∼ |Hso |: intermediate coupling. We do not treat it here as in most cases, we
will be dealing with small atoms or molecules.
3. |Hoo | |Hso |: this is known as L − S (or Russell-Saunders) coupling. This is the
most frequent case in astrophysics, particularly for small to intermediate values
of Z.
4.3
L − S coupling, terms, levels, and Hund’s rules
In the case of L − S coupling, the Hamiltonian reads
H = Hc + Hoo
(8)
In the L − S coupling scheme, the individual orbital angular momenta of the electrons
~li are added to form the total oam ~L = P ~li . Similarly, the individual spin angular
i
~ = P s~i . Finally, both are combined to form ~J = ~L + S.
~ In
momenta s~i are added to form S
i
the absence of coupling between L̂ and Ŝ (such as spin-orbit, see below), the Hamiltonian
commutes with the three total angular momenta operators Ĵ, L̂, and Ŝ. The rules for
coupling angular momenta are such that on adding two angular momenta ~j1 and ~j2 ,
with QN j1 and j2 , the QN of the total oam ~j = ~j1 + ~j2 can take the values
j = j1 + j2 , j1 + j2 − 1, . . . , | j1 − j2 | − 1, | j1 − j2 |
and its projection can take the values2
m j = m1 + m2 = −j, −j + 1, . . . , j − 1, j
2
There are min(2 j1 + 1, 2 j2 + 1) and (2 j1 + 1)(2j2 + 1) = 2 j + 1 possible values for j and m j .
7
When more than two oam are considered, they must be combined by pairs, although in
any order.
The eigenfunctions of the Hamiltonian are |ηLSM J i where η is an eigenfunction of
Hc and describes the electronic configuration, and the the two QNs L and S describe
the perturbation Hoo . The eigenfunction |ηLSM J i that is, a configuration with a specific
value of L and S, is called a term. The number of terms for a configuration depends on
the number of electrons in the open (sub)shells3 . Terms are noted
2S+1 p
L
(9)
where 2S + 1 is the multiplicity, and p = (−1) i li the parity.
According to the coupling rules above, there are 2min(L, S) + 1 levels associated
with a term, and each level (characterized by a value of J) further degenerates into
(2L + 1)(2S + 1) states – which correspond to the 2M J + 1 possible values of J. In the
~ an electronic configuration only gives rise to
absence of coupling between ~L and S,
terms. The spin-orbit coupling, which gives rise the levels of a given term, will be
described below.
The question then arises about how the energy of the various terms of a given electronic configuration depend on L and S. Some indicative (exact only for the ground
states) rules, valid only in L − S coupling, and known as Hund’s rules, explain satisfactorily the energy ordering in most cases of relevance for the ISM. In precedence order,
these rules are:
P
1. terms with highest multiplicity 2S + 1 have the lowest energy
2. for a given configuration and spin S, terms with higher L have lower energy
3. for atoms with less than half-filled shells (e.g. 2p2 ), terms with lowest J have lowest
energy
4. for atoms with more than half-filled shells (e.g. 2p4 ), terms with highest J have
lowest energy
4.4
Example: CI
Neutral carbon (CI) is suffienctly important in the ISM physics that we consider it as an
interesting worked-out example. The electronic ground state configuration is 1s2 2s2 2p2 ,
and only the two (equivalent) electrons in the 2p subshell need be considered. We wish
to determine the terms arising in the L − S coupling. The two electrons have l1 = l2 = 1
and s1 = s2 = 1/2, leading to L = 0, 1, and 2, and S = 0 or 1. We thus have 6 terms. The
S = 0 value leads to the 1 S, 1 P, and 1 D terms. The S = 1 value leads to the terms 3 S, 3 P,
and 3 D.
The 3 D term has L = 2, which allows five values of ML from −2 to 2, and ML = ±2
are obtained for ml1 = ml2 = −1 and 1. Identically, the S = 1 value allows MS = −1, 0,
and 1, and the MS = ±1 values correspond to ms1 = ms2 = −1/2 and 1/2. Hence, the term
3 D includes some states for which the two electrons have identical m and m quantum
s
l
numbers (for example: ml1 = ml2 = −1 andms1 = ms2 = −1/2). According to the Pauli
principle, this would be possible only if the two electrons do not belong to the same
shell (i.e. n1 , n2 ), which is not the case for CI. Hence, this term gives rise to states that
Note that to determine the spectroscopic terms corresponding to a given electronic configuration η, only
the electrons in incomplete subshells (“optically active”) are relevant, as filled subshell have automatically
L = S = 0.
3
8
1D
1D
2
6392
9850
6363
O0
2
9823
C0
o
o
9809 A
6300 A
3P
3P
0
146 µm
3P
0
3P
1
370 µm
1
63 µm
3P
C+
2
3P
609 µm
2
2P
3/2
2P
1/2
158 µm
Figure 1: The fine-structure energy levels for several atoms of relevance in the ISM.
Wavelengths of the allowed radiative transitions are given.
are forbidden, and this term is forbidden. There are methods to determine the terms
rejected by the Pauli principle. A rule of thumb (e.g. Tennyson 2005) is that L + S must
be even to satisfy the Pauli principle. In the case of CI, only the 1 S, 1 D, and 3 P thus
remain. Table 3 gives the allowed terms for various electronic configurations with two
electrons with l = 0 and 1.
According to the Hund’s rules above, we find that the 3 P term has the lowest energy
(highest multiplicity), then 1 D, and the 1 S is the highest. Within the 3 P term, there are
three levels (anticipating the section below on fine-structure coupling) corresponding
to J = 0, 1, and 2, and Hund’s rule #3 above says that 3 P0 < 3 P1 < 3 P2 .
Exercise
1. Show that filled shells and subshells have L = S = 0, and hence
only give rise to the 1 S term.
2. Consider a system with two optically active electrons in two
different shells n and n0 , for which the electronic configuration
is npn0 p.In the L − S coupling scheme, give the list of the finestructure spectroscopic terms corresponding to this electronic
state.
3. We now consider an atom where the two electrons belong to the
same shell, e.g. the ground state of the SIII ion or CI. Write the
ground-state electronic configuration of CI and SIII. Show that
there are less terms than previously found, which now reduce to
1 S, 1 D, and 3 P.
9
4.5
Fine structure
The fine-structure coupling results from the coupling of the orbital angular momentum
of the electron ~l with its intrinsic angular momentum ~s. Hence, the fine-structure is
often refered to as l − s (or spin-orbit) coupling. The full treatment of the fine structure is
based on the relativistic Hamiltonian. Physically, the l − s coupling has its origin in the
coupling between the magnetic field, created by the orbiting electron, with the magnetic
moment of the electron. The latter was first evidenced by the Stern & Gerlach (1922)
experiment and was interpreted as the evidence of an intrinsic angular momentum:
~ e = −ge µB~s = −9.3(−24) J/T,
µ
µB = e~/2me ,
ge ≈ 2
In the electron rest-frame, the electric field associated to the varying potential V(r) is
~ =
moving with a velocity ~
v and is therefore equivalent to an internal magnetic field B
~ C > 0.
~ 2 . The amplitude of the magnetic interaction is ∆Es−o = −µ~e · B
~ = C ~s · `,
−~
v × E/c
The interaction energy (sign and amplitude) thus depends on ~l · ~s: the s-o interaction
thus breaks the level degeneracy and the level energy now depends on ml and ms .
The calculation (e.g., Cowan 1981; Bransden & Joachain 2003) leads to the following
expression for the energy correction due to s-o coupling:
∆En = |En |
α2 Z2
α2 Z4
= RA 3
∼ α2 En ∼ 10−3 − 10−5 En
nl(l + 1)
n l(l + 1)
(10)
In the L − S coupling scheme, the spin-orbit term correction is small with respect to
the Coulombian corrections. The energy levels of each term in the L−S coupling scheme
were (2L + 1)(2S + 1) times degenerate, as the projections of the total oam and spin, ML
and MS , were left unspecified. When taking into account the spin-orbit coupling, the
~ couple to form the total angular momentum
two angular momenta ~L and S
~
~J = ~L + S
(11)
The degeneracy is then partly removed because the Hamiltonian does no longer commute with L̂ and Ŝ. Each term 2S+1 L thus splits into a certain number of fine-structure
levels, the energy of which now depends on the total oam quantum number J, instead
of L and S. The correct set of quantum numbers are then L, S, J, M J = ML + MS . The
complete wave function is thus characterized by the electronic configuration solution η
of Hc plus the set of four QNs LSJM J , or |ηLSJM J i. The eigenvalues do not depend on M J
hence the fine structure levels are still 2J + 1 times degenerate. The 2J + 1 fine-structure
levels are noted
2S+1 p
LJ
(12)
The physical idea behind the spin-orbit coupling is the following. The individual
electronic oam’s are not conserved because of the interactions between the electrons.
As the spin-orbital interaction is small, the ~li ’s and ~si ’s are added to form total oam’s ~L
~ which are again not constants of motion. On the contrary, the total oam ~J must
and S,
be conserved for an isolated system (i.e. no external field). In the vector model, the
~ vectors precess about ~J, so that their projections on an arbitrary axis (usually
~L and S
z) are not conserved. The ~J vector itself rotates about the z-axis and its direction is
therefore not constant. However, the z-projection of ~J is conserved and M J is a good
~
quantum number. In usual cases encountered in astrophysics, the interaction of ~L and S
10
Table 3: Possible terms for electronic configurations (nl)k , for l = 0 and 1.
Configuration
ns
ns2
np, np5
np2 , np4
np3
np6
S=0
S = 1/2
S=1
S = 3/2
2S
1S
2P
1 S, 1 D
3P
2 P, 2 D
4S
1S
is weak (perturbation) and they precess slowly about ~J, so that their magnitude remain
well-defined, but not their direction. The full set of good quantum numbers is thus L,
S, J, M J = ML + MS .
Exercise
Make a classical estimate of the intrinsic magnetic momentum of an
electron in an atom, µe , of the internal magnetic fields B, and of the
spin-orbit splitting ∆E. Compute B for the Na doublet.
4.6
Hyperfine structure
So far, the relativistic nature of nucleons has been neglected. The proton/neutron
has been treated as an electric monopole through its (effective) charge. It has also
been assumed to be point-like and infinite mass with respect to the electron. Relaxing
these assumptions lead to two types of hyperfine perturbations: isotope shifts for
the latter, and hyperfine structure for the former. We restrict ourselves to hyperfine
structure effects, which are responsible for fine–structure-level splitting, and which are
of fundamental importance to the astrophysics of the ISM. The hyperfine structure
effects amounts to the nucleus possessing an intrinsic angular momentum or nuclear
spin ~I, which couples with the total electronic angular momentum:
~ = ~J + ~I
F
(13)
Along the same line as for the fine-structure of the H atom, the nuclear spin gives rise
~ associated with the
to a magnetic dipole µ~N which will interact with the magnetic field B
relative orbital motion of the electrons and nucleus. This term is thus ∝ ~L · ~I and will be
zero if the orbital angular momentum L is zero. In addition to this spin-orbit term, there
is another interaction term to be taken into account, which is the interaction between
the electronic and nuclear magnetic dipoles µ~e and µ~N . The latter term is proportional
to their scalar product, and gives rise to a hyperfine effect even for L = 0 states.
After taking into account these hyperfine structure effects, the energy of a given
(LJ) fine-structure level now depends on the quantum number F. Each fine-structure
level splits into hyperfine structure components, labelled by F (the possible values of
F are |I − J|, |I − J| + 1, . . . , I + J). There are min[(2I + 1), (2J + 1)] hyperfine components.
11
Figure 2: Cowan (1981)
These components form a hyperfine structure multiplet . Each component is 2F + 1fold degenerate since their energy does not depend on MF . Application of an external
magnetic field would evidence these hyperfine levels.
The two contributions (s-o and dipole-dipole interactions) are of the same order of
magnitude, which we may obtain by considering the amplitude of the s-o coupling. The
magnetic momentum of the electron is of the order of the Bohr’s magneton, µB = e~/2me ,
whilst the magnetic momentum of the nucleus is of the order of e~/2mp ≈ µB /1836.
~ will be ≈ 2000 times weaker than the fine-structure
Hence the spin-orbit term ∝ µ~N · B
interaction. In the fundamental electronic state of the H atom for instance, this term is
zero since L = 0 in this case. The dipole-dipole interaction however leads to
Es,i =
4 µ0 ge µB gp µN ~
~s · i
3 2π
a3
(14)
0
where s = 1/2 and i = 1/2 for the proton. Recalling that f ( f + 1) = i(i + 1) + 2~s ·~i + s(s + 1)
with f = 0 or 1 gives ~s · ~i = −3/4 and 1/4 respectively such that:
δEhfs = 9.43(−25) × [1/4 − (−3/4)] J = 0.04738 cm−1 = 5.9 µeV
ν = 1420.4057517667MHz
12
References
Bransden, B. & Joachain, C. 2003, Physics of Atoms and Molecules, 2nd edn. (Benjamin
Cummings)
Cowan, R. D. 1981, The theory of atomic structure and spectra (University of California
Press)
Rybicki, G. B. & Lightman, A. P. 1985, Radiative Processes in Astrophysics (Wiley-VCH)
Tennyson, J. 2005, Astronomical Spectroscopy: An Introduction To The Atomic
And Molecular Physics Of Astronomical Spectra (Imperial College Press Advanced
Physics Texts) (Imperial College Press)
13
Index
angular momentum
intrinsic, 2
orbital, 2
Bohr radius, 3
central field approximation, 4
electronic orbital, 5
fine-structure
constant, 3
level, 10
line, 6
Hartree-Fock potential, 4
Hund’s rules, 8
hydrogenic wave function, 2
hyperfine structure
component, 11
multiplet, 11
j-j coupling, 7
L-S coupling, 7
magnetic quantum number, 2
multiplicity, 7
Pauli principle, 5
quantum number, 2
Rydberg constant, 3
self-consistent field approximation, 4
spectroscopic notation
level, 7
state, 7
term, 7
spin, 2
spin-orbital, 2, 5
statistical weight, 6
vector model, 6
14