Department of Physical Electronics
Faculty of Science, Masaryk University, Brno
Physical laboratory 3
Task F Atomic and molecular optical
emission spectra
Tasks
1. Identify spectral lines emitted by electrode material vapours in an arc discharge and determine
their intensity. Calculate the temperature of the arc discharge from the Boltzmann
slope.
2. Determine the rotation temperature of the OH radical from the measured molecular spec-
trum.
Introduction
Substances excited to some of its higher levels can transmit their energy to their surroundings
in the form of radiation. The excitation can occur due to their non-zero temperature, chemical
reactions, collisions with fast electrons, energetic photons, radioactive radiation and many other
processes. If the substance is excited to high-enough levels, the emitted radiation can lie in the
visible region (approx. 400 – 700 nm) or in the region of even shorter wavelengths. The energy
is radiated in energy portions (quanta) hν = hc/λ due to spontaneous or stimulated transitions
of electrons from higher to lower energy levels. We can observe individual spectral lines in the
spectra of substances with discreet well-separated energy levels such as free atoms.1 If clusters
of lines with close energies exist in the structure of the substance, we can see groups of closelying
lines forming the so-called molecular bands characteristic for emission of materials formed
from several atoms. When the substance contains a large number of energy levels covering a
large energy interval and lying very close together, as is typical for condensed matter, the optical
emission spectrum will be continuous. Moreover, if this substance is in thermal equilibrium, the
spectrum can be close to that of a radiating black body.
The electromagnetic spectrum emitted by a substance carries a lot of information about the
said substance. Optical emission spectroscopy (OES) studies the visible light spontaneously emitted
by a substance. For example, OES is often used in plasma physics to determine particle
1
Even a seemingly simple spectral line can have different spectral shapes and, if it is actually an overlap of
spectral lines with nearly identical wavelengths, an inner structure. Also, magnetic and electric fields, collisions
with other particles and the lifetime of the excited state can also influence the spectral shape of a line.
Physical laboratory 3 manuals (version May 10, 2023) 2
concentrations (electrons, excited species or even ground-state species). OES belongs to noninvasive
methods of studying radiating objects. This means that we do not interfere with the
studied effect by the spectrometer. We only capture the intensity of the radiation emitted by the
studied object – we acquire the optical emission spectrum.
This laboratory is divided into two parts. In the first part, you will work with the spectrum of
free atoms (iron evaporated by an electric arc discharge). As previously described, a spectrum of
an atomic gas is composed of different spectral lines, each corresponding to an electron transition
between two exactly defined energy levels. In such a case, the calculation of the excitation temperature
of the atoms is the most basic application of OES. In the second part of the laboratory, you
will work with the spectrum of OH excited in a plasma. OH, being a diatomic particle, exhibits
a different type of optical spectrum. The spectrum of a discharge containing molecular gasses
is composed of (taking into account the limited resolution of a spectrometer) band structures
– spectral bands. Such bands form by overlapping lines corresponding to transitions between
levels whose energy is the sum of electronic, vibrational and rotational contributions. From the
intensities of the spectral lines, it is possible to calculate the different temperatures – excitation
temperature, vibration temperature and rotation temperature. As the rotational quanta are
very small, an equilibrium between the rotational states and the particles’ thermodynamic (temperature)
movement takes place effectively. That is why the rotation temperature (therefore a
temperature describing the excitation of rotational states) measured in the plasma approximately
corresponds to the translational temperature (the real temperature of neutral gas) in the plasma.
Atomic spectra
Spectral line intensity
The intensity of a spectral line of the wavelength λmn arising by an electron transition from the
upper level m of the energy Em the bottom n level of energy En can be written as
Imn =
hc
4πλmn
· Amn
gm
σ
· exp −
Em
kT
, (1)
where Em is the excitation energy of the upper level, Amn is the transition probability from the
m level to the n level, gm is the statistical weight of the upper energy level, σ is the total number
of states, h is the Planck’s constant, c is the speed of light, k is the Boltzmann constant and
T is the absolute temperature of the radiating body under thermodynamic equilibrium. The
energy difference between the upper and bottom levels governs the light quantum energy of the
corresponding frequency or wavelength:
(Em − En) = hνmn =
hc
λmn
. (2)
From the measurements, we can usually determine only the relative intensity of the spectral lines.
For the relative intensity of the spectral line (in arbitrary units) we can write
Imn ∼
Amngm
λm
· exp −
Em
kT
. (3)
Which we can rewrite into
Imnλmn
Amngm
∼ exp −
Em
k T
(4)
and after using the logarithm rule into
ln
Imn · λmn
Amngm
∼ −
Em
kT
. (5)
Physical laboratory 3 manuals (version May 10, 2023) 3
We can experimentally measure the relative intensities of the spectral lines Imn and their wavelengths
λmn. For each line, we also know the Amngm (the product of the upper m level transition
probability and its statistical weight) and the excitation energy Em of the upper energy quantum
state m, which are tabulated.
If we make the Boltzmann plot
y = ln
Imnλmn
Amngm
= f (Em) , (6)
we can fit it by a line with the slope tan α = −1/kT, from which we can unambiguously determine
the absolute temperature of a radiating body T.
Acquisition of arc discharge spectrum
Grating spectrometers (fig. 1) are the most commonly used devices for the acquisition of the atomic
line spectra. The radiation is brought in by optical fibres often manufactured from silica glass to
transmit also near UV range radiation. The radiation passes through an entrance slit, and (after a
possible reflection from a mirror) it impacts a rotatable grating. The grating decomposes the light
and the diffracted radiation then (after another possible reflection from a mirror for prolonging of
the optical path and resolution enhancement) it goes to the detector. CCD camera, ICCD camera
or a photomultiplier are typically used detectors. The spectrometer settings and measured data
saving are usually done directly by a computer and a special electronic communication unit of the
spectrometer.
1
2
3
4
5
6
7
1
2
3
4
5
Figure 1: Schematic diagram of spectrum acquisition by a grating spectrometer. 1 - power source,
2 - arc discharge, 3 - spectrometer, 4 - optical fibre, 5 - computer.
Evaluation of measured spectrum
When evaluating the measured spectra, the first step is to identify the individual spectral lines.
Wavelengths of selected iron atom spectral lines are summarised in table 1. Keep in mind that
spectrometers do not measure the wavelengths absolutely precisely, and the measured wavelengths
can be shifted compared to their actual values.
Moreover, the effect of the background radiation and the dark current (i.e. current generated
by the detector while no light is going to the detector) need to be removed from the measurement.
A simple option how to subtract the background from the measured signal is by subtracting the
spectrum intensity measured in the close vicinity of the studied line where no part of the studied
Physical laboratory 3 manuals (version May 10, 2023) 4
λmn Em Amngm × 108
[nm] [eV] [s−1]
429.413 4.371 0.71
429.924 5.308 5.2
430.791 4.434 5.9
431.509 5.070 1.5
432.576 4.473 6.1
433.705 4.415 0.23
435.274 5.070 1.0
436.977 5.882 2.2
437.593 2.832 0.0094
438.357 4.312 7.7
440.475 4.371 4.4
441.512 4.415 2.8
442.731 2.851 0.0099
444.234 4.988 1.1
444.320 5.647 1.9
444.772 5.009 1.1
445.912 4.955 1.0
446.165 2.865 0.0052
446.655 5.606 5.3
447.602 5.614 5.4
448.217 2.875 0.0053
449.457 4.955 1.22
452.862 4.913 1.8
453.115 4.220 0.076
460.294 4.177 0.088
473.678 5.828 2.5
485.975 5.426 1.3
487.132 5.409 3.7
487.215 5.426 2.2
487.822 5.426 0.77
Table 1: Selected spectral line constants.
line (or any other line) lies from the line intensity itself. The relative line intensity can be then
calculated by integrating the measured signal throughout the whole line.
After identifying the lines and determining their intensity, it is possible to construct the Boltzmann
plot and calculate the excitation temperature of the radiating atoms from the slope of the
line.
Molecular spectra
Quantum description of a diatomic molecule state
Electron state classification
The electronic state of a free atom can be described by the quantum numbers L, S and J made
by summing up the moments l, s and j of all the electrons in the atom.
A magnetic moment arises from the movement of every electron in the shell of the atom or
a molecule. The magnetic interaction between the spin and orbital momenta of the individual
Physical laboratory 3 manuals (version May 10, 2023) 5
electrons lead to their mutual intertwining. Such spin-orbital interactions can be approximately
described, for example, by the Russell-Saunders spin-orbital coupling.
A model of a free atom with the Russell-Saunders coupling can be even extended to describe the
electronic state of a diatomic molecule. The orbital and spin momenta corresponding to individual
electrons are coupled into resultant orbital L and spin S momenta of the whole molecule. If the
charges of the atomic cores are small, the vectors of L and S are lying in the direction of the
electric field they create, and the spin-orbital coupling is very weak.
The resultant orbital momentum of the electrons carries out a precession movement around
the axis between the two atomic cores, and the projection of this moment onto this axis can have
only Λ values. The quantum number Λ can be
Λ = 0, 1, 2, ... (7)
All electron states with Λ > 0 are twice degenerate. This corresponds to two possible rotation
directions of the momentum rotation with the same projection onto the inter-core axis.
We describe the states with Λ=0, 1, 2, 3, 4... using the terms Σ, Π, ∆, Φ, Γ... analogically to
a free atom.
The coupling of the magnetic moment S into the inter-core axis is caused by a magnetic field
created by the moving electrons, unlike in the previous case, where the cause was the electric field
between the atomic cores in the molecule. The resultant spin momentum carries out precession
movement around the axis between the two atomic cores, and the projection of this moment onto
this axis can have only the values of Σ . The quantum number Σ is an analogue to the quantum
number s in a free atom and can take on values of
Σ = S, S − 1, ..., −S (8)
The total number of the molecule spin momentum orientations corresponds to the multiplicity of
the state (2S+1). This number is typically shown in the left superscript above the term symbol
of the energy state – for example 3Π.
The projection of the total momentum arising from the spin-orbital coupling onto the inter-core
axis can take on values of Ω , where the quantum number Ω can be
Ω = Λ + Σ. (9)
If Λ=1, Σ=1, 0, -1, then Ω=2, 1, 0. This is generally denoted as 3Π2, 3Π1 a 3Π0. Due to the
influence of the spin-orbital coupling, states with different values of Ω have different energies. This
results in the state of 3Π being a triplet.
The energy states of a free atom can be described by the quantum numbers L, S and J
analogically to the quantum numbers Λ, Σ and Ω of a diatomic molecule. However, such a set
of these three numbers is not enough to describe a molecule’s electronic state and to introduce
the transition selection rules. We need to add one (in the case of a heteronuclear molecule) or
two (for a homonuclear molecule) characteristics considering the symmetry of the electron wave
function ψe.
For a diatomic molecule, it is necessary to establish if its ψe is symmetrical or antisymmetrical
with respect to any plane containing the inter-core axis. Symmetrical states are denoted by a
right superscript +, any antisymmetrical ones by a right superscript −. These symbols are only
used for the Σ states. States with Λ > 0 are always twice degenerated in this regard.
For a homonuclear molecule, it needs to be established if it is symmetrical or antisymmetrical
with respect to the centre mass of the molecule. Symmetrical states are denoted by a g right
subscript and those antisymmetrical by right subscript u.
Selection rules
A free atom excited to a higher energy state than its ground state can relax back to its ground
state by radiating one or more photons. The change in the electronic states while radiating a
photon is governed by selection rules giving the probability of all of the many possible transitions.
Physical laboratory 3 manuals (version May 10, 2023) 6
Transition examples
n Selection rule allowed forbidden
1 ∆Λ = 0, ±1 Σ − Σ, Π − Σ, ∆ − Π ∆ − Σ, Φ − Π, Φ − Σ
2 ∆S = 0 3Σ−3Σ,3Π−3Σ 3Σ−1Σ,3Π−1Σ
3 + ↔ −, + ↔ +, − ↔ − Σ+ − Σ+, Σ− − Σ− Σ+ − Σ−, Σ− − Σ+
4 g ↔ u, u ↔ u, g ↔ g Σ+
u − Σ+
g , Πg − Σ+
u Σ+
g − Σ+
g , Πu − Σ−
u
Table 2: Selection rules for a spontaneous electron transition between two states derived from the
dipole approximation for Hund’s coupling a in a diatomic molecule.
The selection rules valid for the electronic state transitions of a diatomic molecule derived by
the electric dipole approximation, including examples of allowed and forbidden transitions, are
shown in table 2.
The validity of the selection rule no. 2 in table 2 decreases with the increase of the core
charge of the molecule. For example, a triplet-singlet transition is strictly forbidden for molecular
hydrogen. However, for example, the a3Π − X1Σ+ forbidden transition can be weakly observed
in the CO molecule.
Diatomic molecule spectrum
Diatomic molecule as a vibrating rotor
The total energy of a diatomic molecule can be separated into mechanical energy and internal
energy. The mechanical energy is the sum of the kinetic energy of the translation movement and
the potential energy. The internal energy is composed of, amongst others, the energy connected
to the electronic state, rotation energy and vibration energy. The energy related to the rotation
and vibration of the molecule is quantised, unlike the kinetic energy.
If the interaction between the vibration and rotation can be neglected, the energy of the
molecule E in a given electronic-vibration-rotation state is the sum of the energies corresponding
to the electronic state (Ee), vibration state (Ev(ν)) and rotation state (Er(N)):
E = Ee + Ev(ν) + Er(N). (10)
The vibration energy of a diatomic molecule is proportional to the vibration quantum number
ν taking on non-negative integer values. The vibration energy in the anharmonic oscillator
approximation can be written as
Ev(ν) = hcGν = hc[ωe(ν +
1
2
) − xeωe(ν +
1
2
)2
+ yeωe(ν +
1
2
)3
...] (11)
The vibration wavenumber ωe, the anharmonicity constants xe, ye and also the vibration term
Gν in this equation are dependent on the electronic state of the vibrating molecule.
The rotation energy of a diatomic molecule is proportional to the rotation quantum number N
that, just like ν, takes on non-negative integer values. The rotation energy in the nonrigid rotor
approximation can be written as
Er(N) = hc[BνN(N + 1) − DeN2
(N + 1)2
+ ...] (12)
where Bν and De are rotation constants corresponding to the given electronic state of the molecule.
As the mean atomic core distance ans well as the mean momentum of the molecule increase with
increasing vibration quantum number ν, the rotation constants depend on the vibration state of
the molecule. Rotation constant Bν can be written as
Bν = Be − αe(ν +
1
2
), (13)
Physical laboratory 3 manuals (version May 10, 2023) 7
Transition examples
n Selection rule allowed forbidden
1 ∆ν unlimited Π(ν = 3) - Π(ν = 1) -
2 ∆N = 0, ±1 Σ(N = 9) - Σ(N = 10) Π(N = 5) - Σ(N = 2)
Table 3: Selection rules for changes of the rotation and vibration quantum number for simultaneous
spontaneous electron-vibration-rotation transition.
where Be and αe are tabulated spectroscopic constants depending on the electronic state of the
molecule.
Electronic relaxations of a molecule by a spontaneous transition of an electron from a higher
energy state to a lower one are governed by transition rules shown in table 2. Based on this
table, it is possible to predict lower electronic states for any given upper state. The rotation and
vibration state of the molecule can also change during the electronic relaxation of the molecule.
Corresponding selection rules, including examples, are shown in table 3. In the optical spectrum,
instead of the classical atomic lines, we can observe molecular rotation-vibration bands. Given
high enough resolution in these bands, we can find the vibration sequences, vibration progressions
and rotation R, P and Q branches. The principle behind these structures is shown in figures 2
and 3.
Interpretation of molecular potential energy curves
Figure 4 shows two potential energy curves corresponding to two different states of an I2 molecule.
The 1Σ state is the ground state and the 3Π state is the lowest excited state. The description of
these states and curves using the spectral terms reveals that the 3Π state is a metastable state
(meaning it has a long lifetime) as the transition from this state to the ground state is against
emission vibration
sequences
absorption and emission
vibration progressions
Figure 2: Formation principle of the vibration progressions and sequences.
Physical laboratory 3 manuals (version May 10, 2023) 8
R-branch P-branch Q-branch
Figure 3: Principle of formation of the R, P and Q rotation branches for the case of J=N.
the ∆S = 0 selection rule. Only the vibration states are usually shown in the potential energy
curves. The rotation states are not. The base state curves in not in the shape of a parabola.
Therefore, the anharmonic oscillator approximation was used for its calculation. More than 80
vibration states correspond to the base state of the I2 molecule. De is the dissociation energy of
the base state. If such energy is introduced to the I2 molecule, the energy of its vibrations is so
high that the molecule tears apart into two I atoms (it dissociates) in the 2P3/2 states. Lower
energy is needed for the dissociation of the I2 molecule in its excited 3Π state. If this state is
dissociated, two I atoms are created, and one of them is in the 2P3/2 state and the other one in the
2P1/2 state. Yet more information can be derived from the potential energy curves. By using the
Franck-Condon principle, it is possible to determine what vibration states will be preferentially
excited by electron impact, what transitions will be the most intensive in the optical spectrum,
etc.
Rotation line intensity
The rotation quanta are much smaller than the vibration or electronic quanta. Therefore also
rotation states will be excited if a vibration state is excited. The intensity of a rotation line is
given by the following formula 14.
In υ J
n υ J = Cn υ
n υ ˜ν4
SJ J e−
Bυ N (N +1)hc
kT (14)
In υ J
n υ J is the spectral line intensity, Bυ is the rotation constant of the upper vibration state,
N is the rotation quantum number of the upper state. J is the quantum number for the total
momentum. N = J − 1/2 for the case of the laboratory measurements. Both the numbers J
and N are used in molecular spectroscopy. In some works, these different quantum numbers can
be mistaken, leading to errors in the measured data evaluation. C is a constant with the same
value for all rotation lines of a single branch, ˜ν4 is the wavenumber of the studied line, SJ J is the
H¨onl-London intensity factor of the given transition, k is the Boltzmann constant, h the Planck
constant, c is the speed of light and T the sought for temperature. By performing the logarithm
of 14 we will bet the Boltzmann plot – a linear dependence of ln
In υ J
n υ J
˜ν4SJ J
on N (N + 1).
Physical laboratory 3 manuals (version May 10, 2023) 9
Figure 4: Potential energy curves of the I2 molecule.
ln
In υ J
n υ J
˜ν4SJ J
= −
Bυ hc
kT
N (N + 1) + const (15)
The rotation temperature can be calculated from the slope of the Boltzmann plot.
N J SJ J λ[nm]
1 3/2 0.563 307,843
2 5/2 1.065 307.996
4 9/2 2.100 308.326
5 11/2 2.640 308.512
6 13/2 3.160 308.733
Table 4: Rotation quantum numbers, intensity factors and wavelengths of selected spectral lines
of the OH radical.
Experimental procedure
1. The task is to evaluate the atomic spectrum of iron vapours measured in an arc discharge
and the molecular spectrum of OH radical measured in a non-isothermic low-temperature
plasma. Derive the corresponding temperature (excitation in the case of iron atoms and
rotation in the case of OH radicals) from both spectra. Correct lines need to be identified,
the background needs to be subtracted, and the relative intensity of the lines needs to be
calculated. This data is used in the Boltzmann plot, and the temperature is derived from the
Physical laboratory 3 manuals (version May 10, 2023) 10
slope of the plot. You can use hereafter described Span and Lifbase for the data evaluation.
A brief manual for using these programs, for example, on the OH radical is shown in the
following points. Iron atoms do not require Lifbase simulations. Their treatment in Span is
the same. The data needed for the evaluation of the Boltzmann plot of the OH radical can
be found in point 5.
2. Open the *.spc file with the radical spectrum in Span 1.7. The OH A2Σ+ → X2Π transition
was measured. The software can be freely downloaded from http://physics.muni.cz/
~zdenek/span/. The measured spectra are typically shifted for up to several nanometres.
An additional step is therefore needed for correct data analysis. In our case, we can use the
OH radical spectrum simulated in Lifbase 2.1 for correct spectral line identification. The
simulated spectra give true wavelengths. This step is not needed in the case of the iron
atoms, where the lines are much more separated.
3. Download Lifbase 2.1 (https://www.sri.com/contact/form/lifbase-o) and run it. Input
the desired molecule, OH (A-X) in our case, in the Simulation window. The wavelengths
are given in angstr¨om, so set the simulated interval to 3050Å– 3140Å. Use the resolution
of 0.15Å, tick the box for a thermalised systems and select L’(Air) in Units. Run the
simulation by clicking the green button. In the simulated spectrum, determine (according
to the wavelengths from table 4) the spectral lines with the quantum numbers 1, 2, 4, 5 and
6 necessary for the temperature calculation.
4. Identify the same 5 lines in the measured spectra using the simulated spectrum. In Span
1.7 set the studied molecule: Database → Select Molecules → OHr.
Have only this molecule ticked as the presence of more molecules or atoms could be very
confusing. Select the spectral line by a left mouse click and assign a correct identification to
the line from the selection. Span 1.7 shows you the J quantum numbers in the selection (do
not confuse these with N ), simplifying the work. If you have at least one line identified, you
can further ease your work by shifting (correcting) the spectrum to the correct wavelength
interval by going to Spectrum → Correct Wavelengths → From Lines in Graph. Then
identify the rest of the five spectral lines and repeat the wavelength correction. You can
verify your result by adding the tabulated spectrum of the OH radical to the background by
Graph → Background → Add... → From database (minimum atomic and molecular
line intensity = 0).
5. Determine the intensities of the selected five lines In υ J
n υ J with the rotation quantum numbers
N of 1, 2, 4, 5 a 6 (or J : 3/2, 5/2, 9/2, 11/2 a 13/2) from the spectrum. Subtract the
spectral background from their intensities. Determine their wavelengths and the wavenumbers
˜ν4. The value of the rotation constant for the upper vibration state is Bυ = 1696.6 m−1.
Determine the rotation temperature from the Boltzmann plot by using the herein shown formulae
and tabulated constants. Verify your calculated temperature by calculation in Span
1.7: Analysis → Temperature → From Graph. The possible (small) difference in the
calculated temperatures can arise from e.g. different spectroscopic constants.
Summary of the spectrum treatment in Span 1.7
• Open Span 1.7
• Open the spectrum file by File → Open → ’OH-XX.SPC’, while XX is the number of your
assigned spectrum.
• Select the OHr molecule in the database: Database → Select Molecules → OHr
• Select a chosen spectral line by a left click. Select spectral line table will open. Find
the correct line with its parameters closest to the ones in table 4, e.g. OHRI 307.84400 1
3/2. Confirm your selection by clicking OK.
Physical laboratory 3 manuals (version May 10, 2023) 11
• Shift the spectrum to the correct wavelength interval: Spectrum → Correct Wavelengths
→ From Lines in Graph.
• Identify the remaining lines.
• Verify your result by adding the tabulated OH radical spectrum to the background: Graph
→ Background → Add... → From database → OK.
(minimum atomic and molecular line intensity = 0).
• Correct the background noise level to zero for determining the line intensities: Spectrum →
Correct Intensities → Manually in Graph → OK. Shift the spectrum by the up and
down arrows. To turn off the correction mode: Spectrum → Correct Intensities →
Manually in Graph.
• Further line information can be fund in: Analysis → Peak Intensities → From Graph,
where the first column shows the wavelengths of the selected lines, the secnd and third
column shows the line boundaries on the x axis and the fourth column shows the intensity.
• Verify the calculated rotation temperature by Analysis → Temperature → From Graph.
You can find more information about Span 1.7 in Help.
References
[1] U. Fantz: Basics of plasma spectroscopy, Plasma Source Sci. Technol. 15 (2006) S137
[2] J. Janča a kolektiv, Obecná fyzika IV., 1989
[3] P. Slavíček, Měření fyzikálních parametrů vysokofrekvenční plazmové trysky, Dizertační práce,
1999
[4] Geoffrey V. Marr, Plasma spectroscopy, Elsevier, 1968
[5] K.P. Huber, G. Herzberg, Molecular spectra and molecular structure - IV. constants of diatomic
molecules, Van Nostrand, 1979