Download The Origin of Atomic Emission Lines and the Structure of the Atom

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THE ORIGIN OF ATOMIC EMISSION LINES AND THE STRUCTURE OF THE ATOM
Stephen D Pastor
Department of Chemistry
Pace University
Pleasantville, NY 10570
[email protected]
Although Newton discovered in 1666 that the various wavelengths constituting white
light refracted at different angles when passed through a prism to give a colored
spectrum, the use of spectroscopy to identify elements and molecules in the solar system
and beyond developed late in the 19th century.1 Lockyer, for example, in his textbook
Elements of Astronomy published in 1871 devoted several pages to a discussion of the
analysis of spectra of the sun, planets, stars, and nebulae.2 The somewhat early
independent development of the sciences of physics and chemistry to that of atomic and
molecular spectroscopy has led to the sometimes confusing conventions used for the
designation of the ionization state of an atom and its corresponding emission line. The
OIII filter used to photograph particular emission lines of planetary and other emission
nebulae or visually enhance their telescopic detail provides a good example of differing
conventions used for ionization (or oxidation) states of an atom and its corresponding
emission lines.
In the convention adopted in the discipline of atomic spectroscopy,3 the unionized or
neutral (unoxidized) state of an atom is designated by the symbol of that element
followed by the Roman numeral I. For a neutral unionized oxygen atom the
corresponding symbol is OI. The unionized state of an oxygen atom, OI, has its full
complement of 8 electrons (orbital designations for the occupancy of the six electrons,
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the electron configuration, would be 1s22s22p4 where the numerical prefix is the principle
quantum number of the orbital; the lower case letter the type of orbital; and the
superscript is the number of electrons in that orbital, vide infra)4. The unionized state or
ground state of a hydrogen atom would be HI (electron configuration 1s1). Following
this convention, mono-ionized oxygen would be designated OII and mono-ionized
hydrogen is HII. The mono-ionized oxygen has only 7 electrons (1s22s22p3 electron
configuration) whereas the hydrogen atom has no electrons (only the nucleus remains,
which is simply a proton). Emission nebula are typically clouds of predominantly
ionized hydrogen atoms and are often called HII regions. HII regions can often be seen
visually with large amateur telescopes in galaxies such as Messier 33 in triangulum.
Doubly ionized oxygen is designated by the spectroscopist as OIII and has only 6
electrons (1s22s22p2 electron configuration). Due to the fact that both doubly-ionized
oxygen, OIII, and singly-ionized nitrogen, NII, have the same number of electrons (6
electrons in 1s22s22p2 orbitals), OIII and NII are isolectronic (they both have the same
number of electrons and electron configuration) and have similar fine structural levels in
their emission spectra, although the emission lines are expected to be of different energy
(nuclear charge is different and the resultant electron orbital energy levels will be
different).5
Standard notation in the chemical and physical sciences denote the neutral or unionized
state of an atom by the symbol of the element followed by a 0 in parentheses. The
neutral state of oxygen and hydrogen would be denoted as O(0) and H(0), respectively.
The first ionized state of hydrogen and oxygen following this convention would be
denoted H(I) and O(I), respectively. The doubly-ionized state of oxygen would be
denoted O(II). There are several other commonly accepted conventions found in the
chemical and physical literature for ionized states. Doubly-ionized (oxidized) oxygen
will commonly be found denoted in the literature as O(II), O2+, O++ or OII.
Returning now to our OIII filter, the symbol OIII represents doubly-ionized oxygen since
the context of the description is that of a spectral emission line filter, that is, the language
and discipline of spectroscopy. If reference was being made to the chemistry or physics
of doubly-ionized oxygen is would be correct to use the symbol O(II). Due to the fact
that OIII and O(II) both designate doubly-ionized oxygen, one may find both conventions
on the same page of a textbook dependent upon the context of the discussion. We will
see later that a further convention is used to designate whether a particular emission is
“allowed or forbidden,” vide infra.
Before we discuss the origin of atomic emission lines, we should answer the question
“why can hydrogen and other gaseous elements exist in nebulae and interstellar space as
free atoms rather than molecules as on earth?” The reaction of two hydrogen atoms, H.
(H denoting hydrogen and the dot denoting the unpaired electron; 1s1 orbital occupancy),
would be expected to form molecular hydrogen, H2 or H-H (where the line denotes a
bond between the hydrogens composed of two shared electrons of opposite spin, vida
infra).
H. + H. → H2
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However, for this reaction to proceed irreversibly requires a third body to carry off the
excess energy released upon bond formation. If two hydrogen atoms collide and a third
body is not present to carry off this excess energy, the two hydrogen atoms merely fly
apart.6 On earth the density of our atmosphere is high and a three-body collision is
statistically possible. Thus hydrogen atoms quickly combine to form molecular hydrogen
with the third body carrying off the excess energy. In contrast to the earth, the densities
of hydrogen atoms in interstellar regions can be a low as 103 m-3. At low densities, the
collision of one hydrogen atom with another may occur on average at intervals of 1013
seconds, almost a million years!7 The possibility of a three body collision would be
extremely low. Indeed, the formation of molecular hydrogen that is present in interstellar
space is suggested to be formed on the surface of interstellar dust grains.8
In 1859 Kirchoff put forth a series of postulates now known as Kirchhoff’s laws that
states 1) bright spectral lines (emission lines) are produced in a hot diffuse gas, 2)
adsorption lines are produced when a source of continuous light is placed in front of a
cool diffuse gas, and 3) a continuous spectra is produced by a hot dense gas or solid.9 To
the first approximation, atomic spectra involve emissions (or corresponding absorptions)
due to electronic transitions between orbitals of an atom (an electron jumping or falling
down between two orbitals.9 As we are interested in the origin of emission lines, we will
focus on the energy given off as light of a particular wavelength when an electron falls
from a higher to lower energy orbital of an atom. Molecular spectra involve transitions
between various vibrational and rotational levels of a molecule and are outside the scope
of our discussion.10
The hydrogen alpha filter is popular among astrophotographers. The hydrogen alpha
emission is denoted by the symbol for the element hydrogen followed by a Greek alpha,
Hα, which often is also denoted by a subscript, Hα. The origin of emission lines is the
falling of an electron from a higher energy orbital to a lower energy orbital with the
excess energy being given off as light or energy of a particular wavelength. The energy
of a particular orbital of hydrogen is quantized, that is, the orbital has a discrete energy
value that is determined solely by the quantum numbers of a particular orbital. Therefore
the difference in energy between two orbitals has a discrete value.11 It follows that the
energy given off when an electron falls from a higher energy orbital to a lower energy
must have a discrete value. The relationship between the value of the energy given off
and it’s corresponding frequency is E = hν, where E is the energy, h is Planck’s constant,
and ν is the frequency (sec-1). The corresponding wavelength of the emission is λ = c/ν,
where λ is the wavelength, ν is the frequency, and c is the speed of light.12 Substituting
the wavelength for frequency, we obtain E = hc/λ that shows there is an inverse
relationship between energy and wavelength, in other words, the shorter the wavelength
is of the emitted light the higher it is in energy. In common usage, the wavelength λ is
usually expressed in nanometers, nm , or the Angstrom unit, Å, and less often the
picometer, pm. A nanometer is equal to 1 x 10-9 meters or 1 x 10-7 centimeters, an
Angstrom is equal to 1 x 10-10 meters or 1 x 10-8 centimeters, and a picometer is equal to
1 x 10-12 meters or 1 x 10-10 centimeters; therefore 1 nm =10 Å = 1000 pm. Thus the Hα
emission line is at 656.2808 nm, 6562.808 Å, or 656280.8 pm.13
To understand the spectroscopy of the hydrogen atom, we need to have a basic
understanding of the general structure of an atom. We will start with the Bohr model of
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the atom which was developed at the turn of the 20th century. The Bohr model of the
atom consists of a positively charged nucleus with negatively charged electrons orbiting
around it. The nucleus of an atom is composed of positively charged protons and
uncharged neutrons. In a neutral unionized atom the total number of negatively charged
electrons equals the number of positively charged protons leading to a net charge of zero.
The atomic number of an element refers to the number of protons in the nucleus. The
atomic weight of an element or atom refers to the number of protons and neutrons in the
nucleus. Atoms with the same number of protons but a different number of neutrons are
called isotopes. The major isotope of hydrogen occurring in nature has an atomic number
and atomic weight of one. This means the hydrogen atom is composed of a single
electron in orbit around a proton (the major isotope of hydrogen has no neutrons). The
accepted symbol for the major isotope of hydrogen is 1H, where the superscript 1 is the
atomic weight. The second most abundant isotope of hydrogen has an atomic number of
1 and an atomic weight of 2, which means the atom is composed of an electron orbiting
around a nucleus composed of one proton and one neutron. This isotope is commonly
called deuterium and is designated 2H, and often because of its widespread use in the
sciences as D. The least common isotope of hydrogen in nature is called tritium and is
composed of an electron orbiting around a nucleus composed of one proton and two
neutrons, which is designated 3H. Thus 1H, 2H, and 3H are all isotopes of the element
hydrogen, and differ only in the number of neutrons. The chemical properties of an
element are determined by the atomic number (the number of electrons determines the
chemical properties of an element, which is equal to the atomic number in the neutral
atom) and all isotopes of an element have the same atomic number, and therefore, the
same chemical properties. The atomic weight of an isotope (its mass) will have an effect
on the energy levels of the orbiting electrons that will effect its spectral emission lines.
Thus although hydrogen and deuterium may have similar chemical properties, their
spectra would have different emission lines. The common isotope of oxygen has an
atomic number of 8 and an atomic weight of 16 that is composed of 8 electrons orbiting
around an nucleus composed of 8 protons and 8 neutrons, which is designated 16O. The
oxygen isotope 17O would be composed of 8 electrons orbiting around a nucleus of 8
protons and 9 neutrons. All elements are arranged by atomic number, which determines
their chemical properties, in the periodic table of the elements.
The orbitals of an atom in which the electrons reside are arranged in increasing energy
and are designated by the principle quantum number n. The lowest energy orbital is n =
1, the second highest in energy is n = 2, etc. The shape of an orbital is designated by the
letters s, p, d, and f or the quantum number “l” , where l = 0, 1, 2, or 3, in increasing
energy. The s orbital (l = 0) is spherical in shape and the p orbital (l = 1) is dumbbell
shaped. The electron itself also has a spin quantum number, ms, (conceptually clockwise
or counterclockwise) which is designated +½ or -½. The hydrogen atom will have in its
ground or unionized state an electron in an s orbital with n = 1, the shorthand orbital
occupancy designation being 1s1. The electron can have a spin of either +½ or -½ and in
the absence of an external magnetic field they are equal in energy (degenerate). Helium,
with an atomic number 2 and atomic weight of 4, has two electrons orbiting around a
nucleus composed of two protons and two neutrons. The electrons are in an s orbital with
n = 1 and the electrons have a spin of +½ or -½, with the orbital occupancy being 1s2. No
further electrons can be placed in that orbital because it would be a violation of the Pauli
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exclusion principle that states that no two electrons can have the same quantum numbers.
The two electrons in helium would have quantum numbers n = 1, l = 0, and ms = +½ and n
= 1, l = 0, and ms = -½, which differ by the spin of the electron.
When we deal with higher atomic numbers, the orbital classification becomes slightly
more complicated. For an atomic number greater than 2, the next electron would have to
go into an orbital with the principle quantum number n = 2. For example, lithium with
atomic number 3 would have an electron configuration of 1s22s1 and beryllium with
atomic number 4 would be 1s22s2. When it is time to fill a p orbital we need to consider a
new quantum number ml which refers to the orientation of the orbital in space. In a
Cartesian coordinate system with x, y, and z axes, the dumbbell shaped p orbitals can
orient about the three axes to generate a px, py, and pz (or quantum numbers ml = 1, 0, and
–1) orbitals. These three p orbitals are degenerate (of equal energy) in the absence of an
external magnetic field. The quantum numbers are derived from quantum mechanical
calculation. The interested reader can find more in depth coverage of the rules of filling
orbitals and quantum numbers in standard texts.4, 11, 14, 15-17 Before we leave this treatment
of the atom, it is important to realize that an electron traveling in an orbit has orbital
angular momentum (momentum is the product of velocity and mass, and orbital angular
momentum is the product of mass, velocity, and radius) and due to the spin of an
electron, spin angular momentum. Similar to the energy levels of orbitals, angular
momentum is quantized in the quantum theory of an atom i.e., the angular momentum
can have only a discrete set of values.18 The quantum numbers are now more accurately
defined as:19
1. “n” is the principle atomic number that determines the overall energy of an
atomic orbital.
2. “l” is the azimuthal quantum number which determines the shape of an orbital as
well as the orbital momentum of an electron in that orbital.
3. “ml” is the magnetic quantum number that determines the orientation of the
angular momentum in space.
4. “ms” is the spin quantum number that determines the direction of the electron
spin, or more correctly, the electron magnetic moment in space.
The possible values of the quantum numbers are summarized as follows:20
n = 1, 2, 3, 4, …
l = 0, 1, 2, 3, …, (n-1)
corresponding to s, p, d, f………
ml = 0, ±1, ±2, ±3, …, ±l
ms = +½ or -½
In a multi-electron system where there is interaction between electrons (for example,
electron-electron repulsion) as well as interaction of the electrons with the nucleus, the
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energy of an electron depends both upon n and l. The degeneracy of states with the
same principle (n) and azimuthal (l) quantum number but with different values of the
magnetic quantum number (ml) is removed in a magnetic field. The energy of, for
example, the px, py, and pz orbitals (or quantum numbers ml = 1, 0, and –1) is no longer
degenerate in a magnetic field, such as that found in a nebula.20
With this information we can assign the electron configuration of a carbon atom, which
has an atomic number of 6 and an atomic weight of 12. According to rules of filling
orbitals, the electron configuration would be 1s22s22p2, which would correspond to:
n = 1, l = 0, ms = +½
n = 1, l = 0, ms = -½
n = 2, l = 0, ms = +½
n = 2, l = 0, ms = -½
n = 2, l = 1, ml = 1, ms = +½
n = 2, l = 1, ml = 0, ms = +½
due to the fact that for l = 1, the allowed values of ml are 1, 0, and –1. In this case the
total spin of the electrons, designated S, is one (S = +1/2 + +1/2 =1). The multiplicity of
the state, which is 2S+1, is 3 and this is termed a triplet state. Note that the electrons in
the lower energy filled subshells (in this case n = 1, l = 0 and n = 2, l = 0) are always spin
paired (of opposite spins, that is for every +1/2 spin there is a –1/2 spin) and need not be
considered, only the outermost electrons (the valence electrons) need be considered (in
this case electrons in n = 2, l = 1). In this case, two of the electrons are unpaired.
Putting the electrons in two different orbitals minimizes electron-electron repulsion and
this represents the ground state (lowest energy state) electronic configuration. This
electron placement is in accordance with Hund’s rule or the rule of maximum multiplicity,
which states that each orbital of a given energy level (in this case n = 2 and l = 2) is first
occupied by a single electron.4, 21
As an illustration of Hund’s rule, the orbitals of the carbon atom can be filled according
to the Pauli exclusion principle as follows
n = 1, l = 0, ms = +½
n = 1, l = 0, ms = -½
n = 2, l = 0, ms = +½
n = 2, l = 0, ms = -½
n = 2, l = 1, ml = +1, ms = +½
n = 2, l = 1, ml = +1, ms = -½
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The total spin, designated by S, of the electrons in the highest energy orbitals is zero (S =
+1/2 + -1/2 = 0) and the multiplicity (2S+1) is 1, which is termed a singlet state. In a
singlet state all the electrons are spin paired, that is, for every +1/2 spin there is a –1/2
spin (In the case where S = ½ and the multiplicity is 2, the term is called a doublet state).
However, according to Hund’s rule or the rule of maximum multiplicity, each orbital of a
given energy level is first occupied by a single electron (the pairing of electrons takes
energy). Hence , the triplet electron configuration is expected to have the lowest energy.
This singlet state of carbon is a valid electron configuration, but it is not the lowest
energy state of the carbon atom.
Summarizing:
•
A singlet state has all electrons paired (for every +1/2 spin there is a –1/2 spin)
and has a total spin of S = 0 and a multiplicity (2S+1) of 1.
•
A doublet state has one unpaired electron and has a total spin of S = ½ and a
multiplicity of 2.
•
A triplet has two unpaired electrons and has a total spin of S = 1 and a multiplicity
of 3.
A neutral oxygen atom, O(0), has 8 electrons with an electron configuration 1s22s22p4
whose electron configuration in accordance with the Pauli exclusion principle and
Hund’s rule is:
n = 1, l = 0, ms = +½
n = 1, l = 0, ms = -½
n = 2, l = 0, ms = +½
n = 2, l = 0, ms = -½
n = 2, l = 1, ml = +1, ms = +½
n = 2, l = 1, ml = 0, ms = +½
n = 2, l = 1, ml = -1, ms = +½
n = 2, l = 1, ml = +1, ms = -½
The total spin of this electron configuration of oxygen is one, S = 1, and the multiplicity
(2S+1) is 3, which corresponds to a triplet ground state.
The hydrogen atom is the simplest case to consider because it is not a multi-electron
system. The electron configuration is 1s1 corresponding to n = 1, l = 0, ms = +½. The
electron in this ground state upon interaction with a high energy source (x-ray, collision
with a high velocity particle, high energy electromagnetic irradiation) can be excited into
a higher energy orbital with n = 2, 3, 4, 5, … or higher. The electron can fall back to the
ground state orbital n = 1 from the n = 2, 3, 4, 5, … or higher orbitals with the emission
of a photon of a particular wavelength (remember the energy of these orbitals is
quantized). This series of emission lines (as well as the corresponding absorption lines)
is called the Lyman series, which are in the ultraviolet region of the electromagnetic
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spectrum.22 The electrons need not fall immediately from the upper state to the ground
state, but can cascade down from higher-energy orbitals to lower-energy orbitals.
The emission of photons due to an electron falling from the n = 3, 4, 5, … or higher
orbitals to the n = 2 orbital is called the Balmer series, which are in the visible region of
the electromagnetic spectrum. The emission of photons due to an electron falling from
the n = 4, 5, 6 …or higher orbitals to the n =3 orbital is called the Paschen series, which
is in the infrared region of the electromagnetic spectrum. The line with the longest
wavelength in each series is designated α, the second longest wavelength is β, and so
forth.23 The Lyman spectral series is designated Lα, Lβ,… or Ly-α, Ly-β,… (the Greek
letters are also often seen as a subscript, Lα, Lβ, …). The Balmer series is usually
represented by Hα, Hβ, … or Hα, Hβ, …).
The transition of an electron falling from n = 3 to n = 2 gives the common Hα emission
line, whereas an electron falling from n = 4 to n = 2 gives the Hβ emission line. The
emission of an electron falling from n = 4 to n = 2 gives off more energy (shorter
wavelength photon) than the corresponding fall from n = 3 to n = 2, which is consistent
with the shorter wavelength of the Hβ emission line. The Hα emission (6563 Å) is red,
the Hβ emission (4861 Å) is turquoise, the Hγ emission (4340 Å) is blue, and the Hδ
emission (4102 Å) is violet. Using Bohr’s quantization of angular momentum, the
following equation can be derived for the allowed energy of an atom:24
En = hc/λ = (-13.6 eV)(1/n2)
where En is the energy of an electron in an orbital of principle quantum number n, c is the
speed of light, h is Planck’s constant, and λ is the wavelength of light corresponding to
that energy. Thus both the energy emitted by an electron and the corresponding
wavelength of a photon emitted on going from a high energy orbital to a low energy
orbital would be:
ΔE = Ehigher
energy orbital –
Elower energy orbital
ΔE = hc/λ = (-13.6 eV)(1/n2) High – (-13.6 eV)(1/n2)Low
The wavelength of a photon emitted by a hydrogen atom in which the electron drops
from a n = 3 orbital (higher energy state) to a n = 2 level (lower energy state) is:
hc/λ = (-13.6 eV)(1/32) – (-13.6 eV)(1/22)
12400 eV.Å/λ = (-13.6 eV)(1/32) – (-13.6 eV)(1/22)
λ = 6565 Å
which is in close agreement with the measured Hα emission at 6563 Å.
The assignment of the atomic orbitals responsible for the emission lines of the doubly
oxidized oxygen atom, OIII, is not as straightforward as the case of HI, because OIII is a
multi-electron system. In the case of a multi-electron system, the electron-electron
repulsion, orbital angular momentum, and the magnetic moment of the spinning electron
must be taken into account. Instead of electron transitions between orbitals, the electron
transitions take place between energy states or terms of an atom. The allowed (or
forbidden) electron transitions between various terms or energy states of an atom, which
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are characterized by their maximum orbital angular momentum, can be used to account
for the observed spectra of atoms.
In a multi-electron system where electron-electron repulsion is much larger than the
interaction of the orbital angular momentum with the spin angular momentum, the energy
states or terms can be described using a Russell-Saunders or L-S coupling scheme. 25-27
Both individual orbital angular momentum (“l”) as well as individual spin angular
momentum (ms) couple strongly. This condition is the case for light atoms with atomic
numbers up to bromine. In these multi-electron systems the orbital angular momentum
and electron spins combine to give a set of new quantum numbers, namely:
L = total orbital angular momentum number = Σ l = sum of the individual orbital angular
momentum of an atom.
S = total atomic spin quantum number.
The states corresponding to values of L are as follows:
L
Designation
0
S
1
P
2
D
3
F
4
G
5
H
If we consider the hydrogen atom, l = 0 , Σ l = 0, ms = ½, Σ ms = ½, and the multiplicity
2S+1 is 2, and therefore we have a 12S state, where the number proceeding the term
symbol is the principle quantum number n and the superscript is the multiplicity. A term
symbol is an abbreviated description of the energy, angular momentum, and spin
multiplicity of an atom in a particular energy state. For a hydrogen atom where the
electron has been excited into a 2p orbital (1s22p1 that is designated a p1 state), l = 1 , Σ l
= 1, ms = ½, Σ ms = ½, and the multiplicity 2S+1 = [2(1/2) + 1] is 2 and therefore we
have a 22P term symbol.
Looking at our previous example carbon, we see that for the electronic configuration
1s22s22p2:
n=2
l=1
ml = +1, 0, -1
ms = +1/2 or –1/2
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which leads to six possible combinations of ml and ms as we have seen previously,
namely:
n = 1, l = 0, ms = +½
n = 1, l = 0, ms = -½
n = 2, l = 0, ms = +½
n = 2, l = 0, ms = -½
n = 2, l = 1, ml = 1, ms = +½
n = 2, l = 1, ml = 0, ms = +½
However, the individual orbital angular momentum and spin angular momentums do
combine in a multi-electron system to form the new quantum numbers:
ML = Σml = total orbital angular momentum along a given axis (although we have
not discussed these summations, the orbital angular momentums are vectors, that
is, the momentum has a numerical value and a direction).
MS = Σms = total spin angular momentum
The allowed values for ML and MS are the same as for the quantum numbers l and ms.
In a similar fashion, the total angular momentum of an electron is the resultant of the
orbital angular momentum vector and the electron-spin angular momentum vector. The
total angular quantum number to the term or energy state of a multi-electron system is:
J = total angular momentum quantum number = L + S, L + S –1, L + S –2, …⏐LS⏐, or altogether 2S+1 different values)
Thus for carbon, a p2 state with two valence electrons in a p orbital, the maximum L = Σl
= 2, maximum ML = Σml =2, and MS = Σms = (+1/2) + (+1/2) = 1. We can then
construct a microstate table for all the possible microstates of a system using the allowed
values of ML and Ms. The numbers represent the values of ml and the right hand
superscript the values of ms. Note that microstates cannot be composed of ml and ms
values that are identical, because that is a violation of the Pauli exclusion principle.
Therefore, the microstate +1+,+1+ is not allowed because both electrons have the same
identical quantum numbers.
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Microstate Diagram of OIII
MS
+1
0
-1
ML
+1+, +1-
2
1
+1+, 0+
+1+, 0- +1-, 0+
+1-, 0-
0
+1+, -1+
+1+, -1- -1+, +1-
+1-, -1-
0 -, 0 +
-1+, 0+
-1
-1+, 0- -1-, 0+
-1-, 0-
-1+, -1-
-2
So rather than six possible values, we see that we have 15 possible microstates. We can
remove from this microstate diagram a 1D term (Maximum ML = Maximum L = 2,
Maximum S = MS = 0, and multiplicity is 2S+1 = 1). There are 5 microstates for a 1D
term and all must have a maximum MS = 0. The microstates for a 1D term are:
MS
+1
0
-1
ML
2
+1++1-
1
+1+, 0-
0
+1+, -1-
-1
-1+, 0-
-2
-1+, -1-
which leaves the following microstates:
MS
+1
0
-1
1
+1+, 0+
+1-, 0+
+1-, 0-
0
+1+, -1+
-1+, +1-
+1-, -1-
ML
2
0 -, 0 +
-1
-2
-1+, 0+
-1-, 0+
-1-, 0-
12
We can remove from the microstates remaining in the diagram a 3P term (Maximum ML
= Maximum L = 1, Maximum S = MS = +1, and multiplicity is 2S+1 = 3). There are 9
microstates for a 3P term and all must have a maximum MS = 1. The microstates for a 3P
term are:
MS
+1
0
-1
1
+1+, 0+
+1-, 0+
+1-, 0-
0
+1+, -1+
-1+, +1-
+1-, -1-
-1
-1+, 0+
-1-, 0+
-1-, 0-
0
-1
ML
2
-2
leaving the following microstate:
MS
+1
ML
2
1
0
0 -, 0 +
-1
-2
We can remove from this microstate diagram a 1S term (Maximum ML = Maximum L =
0, Maximum S = MS = +0, and multiplicity is 2S+1 = 1). There is 1 microstate for a 1S
term which has a maximum MS = 0.
Hund’s rule can now be restated for collections of microstates or terms:
1. The term having the largest multiplicity is most stable (lowest in energy)
2. For several terms having the same multiplicity, the term with the largest value of
L is most stable.
3. For terms with the same values of L and S, the term with the lowest energy is
dependent upon J. If the orbital shell is less than half full, states with a small
value of J lie lower in energy. The reverse is true for orbital shells more than half
full.
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Applying these rules for carbon, the energy ranking of these terms is:
Highest Energy
1
1
S or 21S
D or 21D
3
P or 23P
Lowest Energy
The complete term symbol involves the adding the value of J for each term that is equal
to the sum of S + L, which is the coupling of spin and orbital angular momenta that is
known as spin-orbital coupling. For the 1S state, L and S are equal to zero, therefore J =
0. For the 1D term, L = 2 and S = 0, therefore J = 2. For the 3P term, L = 1 and S = +1,
therefore J = 2, 1, and 0 (remember J = L + S, L + S –1, L + S –2, …..⏐L-S⏐, or altogether
2S+1 different values). Thus the application of Hund’s rule for a shell less than half
filled:
Highest Energy
1
1
S0 or 21S0
D2 or 21D2
3
P2 or 23P2
3
P1 or 23P1
3
P0 or 23P0
Lowest Energy
Thus we have the term symbols, which are an abbreviation for the energies of different
electron configurations, which arise because of different electron interactions such as
electron-electron repulsion and spin-orbital coupling. An electron in an excited state falls
between different energy states or terms that leads to the observed spectrum of the atom.
It follows that the 1S0 and 2D2 are higher in energy than the 3P terms, but a single electron
configuration cannot be identified, and that the energy of these states cannot be
determined by simple rules. Furthermore, the interaction of each state with an external
magnetic field can lead to the splitting of energy states leading to the fine structure seen
in atomic spectra.5, 23 If both the energy of certain states is shifted in an applied external
magnetic field and the energy of the separation between states is proportional to the field
strength H (for low values of H), the observed splitting is called Zeeman Splitting. The
magnitude of Zeeman splitting in spectra can provide information about magnetic fields
on the solar surface and stars.23
14
An electron is not free to cascade down between different energy states, that is,
transitions between only certain states are allowed. These rules of what transitions are
allowed are called selection rules. Three important ones for our discussion are:
1. Transitions for which ΔS ≠ 0 are forbidden, that is, transitions involving a change
in the number of unpaired electrons are not allowed.
2. Allowed transition have ΔL = 0, ±1 (L = 0 → L = 0 excluded)
3. ΔJ = 0, ±1 (J = 0 → J = 0 excluded)
Transitions that violate these rules are forbidden. However, the quantum mechanical
rules governing transitions are often violated. This is particularly true for spectra in
nebulae where the lifetimes of metastable states are long compared to here on earth, vide
ante. Thus we can state that allowed transitions are strong and forbidden transitions are
weak. However, the shear volume or mass of these metastable energy states in nebulae
can lead to strong emissions being detected. In many cases the conditions in nebulae are
not favorable for forming states from which allowed transitions can occur. Collisions
with free low energy electrons may only have enough energy to excite the atom to lower
excited energy states where allowed transitions to the ground state cannot occur. The
nebular density is low and collisional deactivation of these metastable states is unlikely,
hence the forbidden transition is almost certain to occur.28
With this background, let us return to the OIII atom, which has the electronic
configuration 1s22s22p2. This is the same electronic configuration as a carbon atom and
has the same term symbols, namely:
Highest Energy
1
1
S0 or 21S0
D2 or 21D2
3
P2 or 23P2
3
P1 or 23P1
3
P0 or 23P0
Lowest Energy
The energy of these terms is expected to be different a priori from that of carbon. Do to
the fact that the nuclear charge is different for 16O than 12C, we would expect the
emission lines to be of different wavelength.
In OIII a weak transition of an electron between 1S0 and 1D2 gives rise to an emission line
at λ 4363 Å. Although the spin state does not change in this transition (ΔS = 0), the
angular momentum change is greater than 1 (ΔL = 2). The electron transition between 1S0
and 1D2 is therefore a forbidden transition. When referring to a forbidden transition or
emission line the symbol for the atom with its ionization number is placed in brackets, in
15
this case [OIII]. The forbidden emission line of doubly ionized oxygen is denoted by
4363[OIII] or 4363 Å [OIII]. The two strong observed [OIII] emissions are also
forbidden transitions between 1D2 → 3P2 and 1D2 → 3P1 because a change in the number
of unpaired electrons occurs (ΔS ≠ 0). The emissions of OIII corresponding to electron
transitions between 1D2 → 3P2 and 1D2 → 3P1 are observed at 5007 Å [OIII] and 4959 Å
[OIII], respectively.5,28 Interestingly, the intensity (I) ratio of I(4363)/[I(4959) +
I(5007)] provides a measure to the number of OIII ions excited to levels S and D from P
by electron collision. The ratio of the number of ions in level S to that in D is
proportional to the temperature and provides a means to find electron temperatures in
gaseous nebula.5 As an interesting aside, due to the fact that the forbidden emission lines
of doubly excited oxygen where not observed on earth (the best vacuums on earth cannot
compete with the vacuum in interstellar space), the forbidden lines of [OIII] were
originally thought to belong to an unknown element, which was called nebulium.28
In our discussion we have considered a Russell-Saunders (or L-S coupling scheme) where
electron-electron repulsion is stronger than the interaction of the orbital angular
momentum with the spin angular momentum. We examined whether an electron
transition was allowed or forbidden between various terms or energy states of an atom
based upon certain selection rules. These selection rules were for electric dipole
emissions. We did not discuss an additional selection rule which requires that transitions
occur between even and odd energy levels, which relates to the symmetry of the
orbitals.29 Selection rules for magnetic dipole and electric quadrupole emissions were not
considered, which may be allowed even when the selection rule criteria are not meet for
electric dipole emission, albeit with much smaller transition probabilities. The case of jj
coupling where the interaction of orbital angular momentum with spin angular
momentum is greater than electron-electron repulsion leads to a new set of selection rules
(in this case j = l + s and J = Σj). All of these cases may need to be considered for a full
interpretation of a particular spectrum.30,31,32,33,34 Indeed, the coupling of the spin of an
electron with the nuclear spin of a proton in a hydrogen atom leads to two energy states
and results in the observed 21-centimeter radio frequency radiation, which is a powerful
tool for probing galactic structure and the interstellar medium.35,36
Before we close, let us return one more time to the origin of the Hα emission line. The
origin of the Hα emission line in the Balmer series was attributed to the energy released
when an electron falls from the n = 3 to the n = 2 orbital of HI, which is indeed the case.
The alert reader may have noticed that Hα emission is found in HII regions (emission
nebulae), but by definition HII is mono-ionized and has no electrons! So what is the
origin of the Hα emission line? Most of the hydrogen gas in emission nebulae is ionized
by the strong ultraviolet radiation of stars imbedded or forming in the nebulae. The
ionization process releases electrons into the interstellar medium of the nebula. When the
nuclei of the hydrogen atoms (remember HII is a proton, vide ante) collide with and
capture an electron, the electron cascades down the orbital levels of the HI atom (the
addition of an electron to HII results in the HI ionization state). This electron capture
process by HII to give an HI atom with an electron in a higher energy orbital is the source
of the emission lines observed in nebulae.3
16
Footnotes:
1. For a discussion of early historical developments, see White, H. E. Introduction to
Atomic Spectra, McGraw-Hill: New York, 1934, chp 1
2. Lockyer, N. J. Elements of Astronomy; D. Appleton and Co.: New York, 1871, pp
263–271.
3. Seeds, M. A. Foundations of Astronomy,4th edition, Wadsworth: Belmont, CA:
1997, pp 204–205.
4. Hiller, L. A.; Herber, R. H. Principles of Chemistry; McGraw-Hill: New York,
1960, chp 1.
5. See Chaisson, E. J. in Frontiers of Astrophysics; Avrett, E. H., editor; Harvard
University Press: Cambridge, 1976, pp 260 –267.
6. Hartquist, T. W.; Williams, D. A. The Chemically Controlled Cosmos,
Cambridge University Press: Cambridge, 1995, pp 24-25.
7. Hartquist, T. W.; Williams, D. A. The Chemically Controlled Cosmos,
Cambridge University Press: Cambridge, 1995, pp 9-11.
8. For a detailed discussion on the role of dust on chemistry in interstellar space, see
Dust and Chemistry in Astronomy, Millar, T, J.; Williams, D. A., editors; Institute
of Physics Publishing: Bristol, 1993.
9. Carroll, B. W.; Ostlie, D. A. An Introduction to Modern Astrophysics; AddisonWesley: Reading, MA, 1996, chp 5.
10. Seeds, M. A. Foundations of Astronomy, 4th edition; Wadsworth: Belmont, CA:
1997, pp 211-212.
11. Huheey, J. E.; Keiter, E. A.; Keiter, R. L. Inorganic Chemistry: Principles of
Structure and Reactivity, 4th edition ; HarperCollins College Publishers: New
York, 1993, pp 18-20.
12. Turro, N. J, Modern Molecular Photochemistry; University Science Books:
Sausalito, CA, 1991, pp 8-10.
13. Lang, Astrophysical Formulae, 2nd edition, Springer-Verlag: New York, 1980.
14. Miessler, G. L.; Tarr, D. A. Inorganic Chemistry; Prentice Hall: Englewood
Cliffs, NJ, 1991.
15. Pauling, L. The Nature of the Chemical Bond; 3rd edition; Cornell University
Press: Ithica, 1960.
16. Bowser, J. R. Inorganic Chemistry; Brooks/Cole Publishing: Pacific Grove, CA,
1993.
17. For a more rigorous introduction, see Denaro, A. R. A Foundation for Quantum
Chemistry; Halsted Press-Wiley: New York, 1975.
18. Tripler, P. A. Physics for Scientists and Engineers, Vol 2, 3rd edition: Worth
Publishers: New York, 1991, pp 882-883.
17
19. Ref. 14, p 28.
20. Ref. 17, p 127-128.
21. Daniels, F.; Alberty, R. A. Physical Chemistry, 3rd edition; Wiley: New York,
1966, pp 458-461
22. a) Ref. 1, p 14. b) Ref.. 9, pp 134-135.
23. Harwitt, M. Astrophysical Concepts, 2nd edition, Springer-Verlag: New York,
1988, pp chp 7.
24. For a complete treatment, see ref. 9, chp 5.
25. a) For a introduction, see ref. 16, chp 1. b) Ref. 14, pp 41-48. c)For a rigorous
discussion, see ref. 1, chp 12.
26. Jolly, W. L. Modern Inorganic Chemistry; McGraw-Hill: New York, 1984, pp
532-534.
27. Ref. 15, pp 580-588.
28. Aller, L. H. Atoms, Stars, and Nebula, 3rd edition; Cambridge: Cambridge, 1991,
pp 215-220.
29. Unsöld, A.; Baschek, B. “The New Cosmos” ; Springer: Berlin, 2002, chp 7.
30. Cowley, C.; Wiese,W. L.; Fuhr, J.; Kuznetsova, L.A. in “Allen’s Astrophysical
Quantities 4th edition”; Cox, A. N., editor; AIP press/Springer: New York, 2000,
chp 4 and references therein.
31. Cotton, F. A. “The Chemical Applications of Group Theory 5th edition”; Wiley,
1971, chp 9.
32. Ref. 1, chp 12.
33. Semat, H. “Introduction to Atomic Physics”; Rinehart: New York, 1947, chp 6.
34. Flurry Jr., R. L. “Quantum Chemistry” Prentice Hall: Englewood Cliffs, NJ, 1983,
chp 8 and references therein.
35. Ref. 28, pp 232-233.
36. Ref. 30, pp 78-79.
Latin translations:
vide infra
See below
vide ante
See before
a priori
From first principles