Contents:
1. Spectra of Helium (He) Atom
2. Spectra of two valence electron systems
The students will be able to lean spectra of Helium and alkaline metals
and in general spectra of two electron system including their energy
levels and the fine structure
1. Spectra of Helium (He) Atom:
The Helium (He) atom has electronic configuration of 1s2. These two
electrons in the configuration are responsible for the valency of two and
hence in deciding the general characteristics of the optical spectrum. The
ground state Helium has two 1s electrons. Both the electrons have l = 0
and as n, l and ml are same; therefore, according to the Pauli Exclusion
Principle their spins are antiparallel.
Thus for 1s2, S and L are zero and hence J = 0 and the ground state is 1S0.
Now assuming that one of the two electrons is excited to the 2 p orbits
yielding a configuration 1s2p due to n =1 for the s electron and n = 2 for
the p electron. Pauli Exclusion Principle does not apply, as the two
electrons are non-equivalent. For the s and p electrons, l = 0 and 1,
respectively and thus L = 1.The state is represented by the symbol P. The
spins are either parallel or antiparallel giving S =1 and 0, respectively. As
the multiplicity is 2S+1, one gets triplet and singlet and for S = 0 the).
The J values for triplet are 2, 1, 0 and for singlet, it is 1. The possible
terms are therefore, 3P210 and 1P1.
In case, one of the two electrons is promoted to various excited
configuration, then the resulting states are singlet and triplet (The ground
state configuration leads only to singlet) and according to Hund’s rule,
the triplet states lie lower than the corresponding singlet states.
Normal
configuration
Excited
configuration
n Term
1s2 -- -- 1S0
1s2 1sns 2,3,4,…. 1S0 ,3S1
1s2 1snp 2,3,4,…. 1P1,3P210
1s2 1snd 3,4,5,…. 1D2,3D321
1s2 1snf 4,5,6… 1F3,3F432
The energy level diagram of Helium is shown in the figure. It depicts the
energies of the ground state and first few excited states of helium atom obtained
4P 4S
3S
2P
4P 4F
1S
Singlets
2S
4D
3P 3D
Triplets
4S
3S
3P
4D
3D
4F
2P
2S
from an analysis of the optical line spectra. The energy of the term increases in
the order of nS, nP, nD, nF,…….., for the same n. The lowest triplet state is
2 3S, and the lowest singlet state is 11S. The state 13S is not allowed because of
Pauli’s Exclusion Principle. The energy difference between the 11S and 23S is
quite large ~19.72 eV reflecting the tight binding closed shell electrons.
Helium atoms in singlet states (antiparallel spins) constitute parahelium and
those in triplet states (parallel spins) constitute orthohelium. The lowest state of
orthohelium, 23S does not combine with the ground state 11S. Those terms, that
cannot go to a lower state with the emission of radiation and correspondingly
cannot be reached from a lower state by absorption, are called metastable states.
The 21S state is also metastable, as the selection rule ∆l = ± 1 does not allow
any transition to 11S. The metastability of the 23S state is however, stronger than
that of 21S, as the transition 23S →11S would contradict the prohibition, of an
ortho-para transition as well as ∆l = ± 1. However, 23S → 11S transitions can be
caused by collision with other atoms or with the wall of a containing vessel that
distorts the electronic structure of the atom, and allow a change of spin to occur.
The transition from one energy state to another is governed by the selection
rule ∆L = ± 1, ∆S = 0, ∆J =0, ±1. Allowed transitions between different energy
levels gives rise to a large number of lines that can be put into two distinct
system of lines: a system of closely spaced triplets and a system of singlets. In
the emission spectrum of helium atom, lines of singlet and triplet system may be
grouped into four series each:
(i) Principle series: Two principle series for singlet are observed one is
from higher P state to normal 11S0 state and member of this series lie
in far ultraviolet region. This series also exists in absorption. Besides
this there exists another principle series whose spectral lines arise
from transitions from higher 1P states to the 21S state. This principle
series lie from infrared to near ultraviolet region. The members of the
triplet principle series are due to transitions from higher 3P state to
23S state. The triplet principal series lie from infrared to ultraviolet
region.
(ii) Sharp series: The sharp series arise due to transitions from S states to
the lowest P state. The lines of sharp series of singlet and triplet
system lie in visible and ultraviolet region.
(iii) Diffuse series: The diffuse series arise from transition between higher
D states to the lowest P state. The lines of this series for both singlet
and triplet system lie in visible and ultraviolet region.
(iv) Fundamental series: This series arise from transitions between higher
F states to the lowest D state. The lines of this series for singlet and
triplet system lie in infrared region.
Under high resolution, the lines of parahelium system are single and those of
orhtohelium show closely spaced triplet structure, that is very difficult to
resolve. The triple fine structure is due to spin-orbit interaction. The figure
shows 3P→3S transition.
3S1
3P2
3P1
3P0
3D3
3P2
3P1
3P0
3D1
3D2
Splitting of 3P210→3S1 and 3D321→3P210
state
3S state has one component 3S1 while 3P has three components 3P2, 3P1
and3P0. Three lines are observed according to selection rules. ∆L = ± 1,∆J =
0, ±1 and ∆S = 0, between these states.
The adjacent figure shows the fine structure of triplet diffuse series.
For triplet D we have S = 1, L = 2 and hence J = 3, 2, 1. According to
allowed selection rules, six transitions are possible. The complete spectrum
should consist of six lines. Normally, the very close spacing is not resolved
and only three lines are observed. For this reasons the spectrum is referred to
as compound triplet.
For He, the component separations 3P0-3P1 = 0.996 cm-1 and 3P1-
3P2 = 0.078
cm1. The terms of 23P of He are inverted with 3P0 component lying highest in
energy and 3P2 the lowest. Further there is breakdown of Lande interval rule.
This is due spin-spin interaction.
The order is inverted in both P and D state for He
2. Spectra of two valence electron systems
3S1
3P2
3P1
3P0
3D3
3P2
3P1
3P0
3D1
3D2
Splitting of 3P210→3S1 and 3D321→3P210
state
The electronic configuration of neutral atoms of group II elements (alkaline
elements) are given as
Elements Atomic
Number,Z
Configuration
Be 4 1s22s2
Mg 12 1s22s22p63s2
Ca 20 1s22s22p63s23p64s2
Zn 30 1s22s22p63s23p64s23d104s2
Sr 38 1s22s22p63s23p64s23d104s24p65s2
Cd 48 1s22s22p63s23p64s23d104s24p64d105s2
Ba 56 1s22s22p63s23p64s23d104s24p64d105s25p66s2
Hg 80 1s22s22p63s23p64s23d104s24p64d105s25p65d106s2
The last two equivalent electrons in the group II elements are responsible for the
general characteristics of the optical spectra. These elements in their ground
state have two equivalent s electrons in the nth (n = 2, 3, 4, 5 and 6) orbit and
therefore the ground state is n1S0. On exciting one of these two electrons to
higher s, p, d or f orbit one obtains a system of two non-equivalent electrons.
These systems gives singlet and triplet S, P, D and F state as described in the
case of helium.
The ground state configuration leads only to singlet states whereas each excited
configuration arising from the promotion if an electron gives rise to singlet and
triplet states with later lying lower than the corresponding singlet state.The
figure depicts the energies of the ground states and first few excited states of Sr
atom obtained from the analysis of the optical line spectra. This figure also
shows some of the transitions allowed by selection rules. (The fine structure
splitting is not shown).
7S 7S
6S
6S
6P
4D 4D
6P
7P 4F
5D
5S
5D
4F
5P
5P
Singlet Triplet
Allowed transitions between different energy levels give rise to a large number
of lines that can arise from transitions
(a) between closely spaced triplets and
(b) between a systems of singlet.
In the spectra of alkaline earth metals, the lines of tripled system and of
singlet system can be further classified into four chief series. These are:
(i) Principal series: This series arise from transition from higher P states
to the lowest S state.
(ii) Sharp series: This series arise from transition from higher S states to
the lowest P state.
(iii) Diffuse series: This series arise from transition from excited D states
to the lowest P state.
(iv) Fundamental series: This series arise because of transitions from
excited F states to lowest D state.
The regions in which the lines of these four series of singlet and triplet system
lie for the Sr element
Element Principle
series
Sharp series Diffuse series Fundamental
series
Sr (Singlet) VIS,UV IR,VIS IR,VIS VIS
Sr (Triplet) IR,VIS VIS,UV IR,UV VIS,UV
Each series converge towards shorter wavelength to fit the wavenumber of the
spectral line to hydrogen like formula
�̅� = 𝑅𝑍2 (1
𝑛𝑒" 2 −
1
𝑛𝑒′ 2)
are successful only if non integral quantum number are used. For each spectral
series the term T’’ = RZ2/𝑛𝑒′′2 is constant and is the wavenumber of series limit.
ne = n - µ, is the effective quantum number where n is an integer and µ is a
fraction called quantum defect that is zero for non-penetrating orbits. (As in
case of alkali atoms, the penetration of various possible electron orbits into the
atom core is measured by the deviation of the term values from those of
hydrogen). Two methods are used to measure the penetration
(i) by modifying the nuclear charger in the term value expression for hydrogen,
(ii) modifying the principal quantum number.
These leads to term values for (i) as
T = 𝑅(𝑍−𝜎)2
𝑛2
and for (ii) as
T = 𝑅𝑍0
2
(𝑛− µ)2
Where 𝜎 is a screening constant, Z0 is the effective nuclear charge when the
electron is well outside the core. From the spectrum, ignoring fine structure, it is
observed that:
(i) Triplet sharp and diffuse series have one common limit and singlet
sharp and diffuse another.
(ii) The frequency difference between common limit of triplet sharp and
diffuse series and triplet principal series is equal to the frequency of
the first member of the principal series (Rydberg-Schuster law).
(iii) The frequency difference between the common limit of the triplet
sharp and diffuse series and the triplet fundamental series is equal to
the frequency of the first member of the triplet diffuse series (Runge
law).
(iv) The same is true even if fine structure is taken into account.
(v) (i), (ii), and (iii) holds for singlet series also.
Fine structure
The examination of alkaline spectra reveals that lines of various triplets’
series have a structure known as fine structure. It is observed that:
(i) All members of the principal series are composed of three lines with
decreasing separation with increasing frequency, and approach a
single limit.
(ii) All member of sharp series are composed of three lines with the same
separation, and approach a triple limit.
(iii) All members of diffuse and fundamental series contain six lines, three
strong and three satellites and approach a triple limit.
The fine structure in triplet series is due to splitting of various energy levels
by spin-orbit interaction as described previously in case of alkali atoms. In case
of alkaline metals, in triplet levels we have S = 1 and L can take values 0, 1, 2
and 3 for S, P, D and F states, respectively. For S state, S = 1 L = 0, therefore, J
= 1. For 3S1 there is no splitting of the level. For P state, L = S =1, therefore J =
2, 1, 0 and P state is split into three components with different J values.
Similarly, D and F states, represented by 3D321 and 3F432, respectively splits into
three components each.
In the triplet principal series transitions take place from 3P210 to lowest 3S1 level
as shown in the figure and according to selection rules
∆S = 0, ∆L = ±1, ∆J = 0,±1
three transitions take place.
The spacing between the transitions depends on the splitting of P level which
decreases with increasing n as per the equation.
Δ�̅� =𝑅𝐴𝛼2𝑍4
𝑛3𝑙(𝑙 + 1)
For large value of n, the splitting of the P level is negligible and as a result, one
observes a single transition in series limit.
Fig. shows the fine structure transition for sharp series that take place from
different excited triplet S states to lowest triplet P state.
3S1
3P2
3P1
3P0
Figure shows the Fine structure transitions for sharp series
As a result of allowed transitions, three lines are observed. The spacing between
the lines of triplet is governed by the splitting of the lowest triplet P state. As
the sharp series triplet separation is produced by the splitting of the lower 3P210
level on which all the transitions terminates, therefore it remains constant.
Figure shows the fine structure transitions for the triplet diffuse series that take
place from different excited 3D levels to lowest 3P level. Each of 3P and 3D level
splits into three levels. As a result of allowed selection rules, six transitions are
possible. The spacing between the transitions depends on the splitting of 3P and
3D levels. For large value of n, the spacing between 3D321 levels diminishes and
3P2
3P1
3P0
3D1
3D2
3D3
3S1
3P2
3P1 3P0
therefore 3D321 levels are not resolved. As a result of this only three transitions
are observed whose spacing depends on the splitting of 3P210 levels. Same
situation exists for the fundamental series. However, now for large value of n,
the spacing between the transitions depends on the splitting of 3D321 levels,
which is constant.
As for sharp and diffuse series the transitions terminate at lowest 3P level
therefore, for large value of n, the spacing between transitions is same for both
sharp and diffuse series.
Qualitatively, the relative intensity can be obtained by rules given for alkali
atom spectra. Let us apply those rules for the members of diffuse series.
For transitions (a) ∆J = + 1 with J = 3,
(b) ∆J = 0, J = 2
(c) ∆J = + 1, J = 2
(d) ∆J = -1, J = 1
(e) ∆J = 0, J = 1 and
(f) ∆J = +1, J = 1.
For all these transitions ∆ L has the same value. Thus transition ‘a’ is the
strongest and ‘d’ is the weakest. The intensities of other transitions are
intermediate between the intensities of ‘a’ and ‘d’.
The transitions in order of decreasing intensities are a >c >f >b >e >d.
Assignment 1.
Calculate the Rydberg denomionators for the first term value of the Principle
series of sodium. The wavenumber of the transition is 16973.7 cm-1. The R for
Na is 109734 cm-1
Write down the spectral term
T= R/ne2
So ne2=109734/16973.7=254
Assignment 2.
The first ionization potential of sodium is two fifth that of hydrogen. Calculate
the effective nuclear charge of the sodium atom as far as the 3s electron is
concerned.
For hydrogen
Z = 1, n = 1.
Therefore using En=-RZ2/n2 E1 = -R cm-1 = - 13.6 eV. The ionization energy
of hydrogen is 13.6eV.
Now 𝐼𝐻
𝐼𝑁𝑎=
(𝑅𝑍𝑒𝑓𝑓2 𝑛2⁄
[𝑅𝑍𝑒𝑓𝑓2 𝑛2]𝑁𝑎⁄
For Na, n = 3 and for hydrogen Zeff = 1. Putting (INa/IH) = 2/5 in the above
equation
Z2 eff =1.897
Assignment 3.
The principal and sharp series for the Li atom converge to continua at 43487
and 28583 cm-1 respectively. Calculate the quantum defect for the common term
in each of the series (RLi = 109729 cm-1).
The common term for sharp and principal series is 22P term. The wavenumber
position of this term is 28535 cm-1 and we know T= R/ne2
Therefore ne2=R/T= 109729/28583 =3.83896
So, ne= 1.9593
Μ= n- ne= 2-1.9593=.0407
Assignment 4
The principal and sharp series for the sodium atom converge to continua at
41450 and 24477 cm-1, respectively. Calculate the ionization potential of Na.
The wavenumber corresponding to series limit of principal series give the
position of normal ground state 32S of Na. Hence ionization energy corresponds
to 41450cm-1 as 1 eV= 8065 cm-1. Thus 41450 cm-1 is equal (41450/8065) eV =
5.139eV.
Summary:
A two-electron atom or helium-like ion is a quantum mechanical system
consisting of one nucleus with a charge of Ze and two electrons. This is
the first case of many-electron systems where the Pauli Exclusion
Principle plays a central role. It is considered as a three body problem.
The important two-electron atoms are:
Hydrogen anion (Z=1) Helium atom (Z=1) Lithium ion (Z=1)
Beryllium ion(Z=1) Boron ion (Z=1)
The optical spectrum of the two electron atom has two systems of lines:
A para system of single lines, and an ortho system of triplets (closely
spaced group of three lines).
The energy levels in the atom for the single lines are indicated
by 1S0 1P1
1D2 1F3 etc., and for the triplets, some energy levels are
splitted: 3S1 3P2
3P1 3P0
3D3 3D2
3D1 3F4
3F3 3F2
Alkaline earths and Mercury also have spectra with similar features, due to the
two outer valence electron.