* This article is closely associated with a previous entry. The author assumes reader understand basic definition of physical plasma as the fourth state of matter.
Figure 1: Applying magnetic field towards the plasma causes its shape to distort due to Lorentz force acting on moving charged particles.
It was a while ago I took a spectrum of the UVC germicidal lamp without its waterproof quartz glass. Its discrete sharp peaks can be described by hallmark of the old quantum theory - light emitted due to electronic transitions between discrete energy levels within the gaseous atoms contained inside the tube. Within the past hundred years, physicists has worked out many useful formulae to relate measurable parameters of a gas discharge spectra such as intensity and its associated wavelength with physical mechanisms that is happening inside the tube at the atomic level. Here, I have worked out some crude but important characterization of the plasma when the UVC germicidal lamp is energized.
Figure 2: Spectrum of UVC germicidal lamp with tagged wavelengths for 17 strong peaks
After plotting the numerical data into a spectrum (above), I have listed down the wavelength and intensity for 27 individual peaks which I had identified its origin thanks to NIST online database. Interestingly, the peaks was the "fingerprint" of not one but two different chemical elements (as low pressure gas) inside the tube. As mentioned in previous post, the beautiful cyan glow and strong UVC emission at 253.7 nm are the principal character of electrically excited low pressure mercury-vapour, hence finding peaks corresponding to mercury atoms is not surprising. The second interesting element was argon which I presume it was the "filler" noble gas when they used to purge the tube before vacuuming down and filling it up with a small quantity of mercury vapour. All the peaks are of neutral, i.e. Hg (I) and Ar (I) atoms.
Out of 27 peaks, 10 belongs to Hg (I) transitions while the rest are Ar (I). Finding the peaks was easy with common graphing tool (I used Origin 7.0). To make further analysis clear, I shall begin by separating this study into a few components:
1. Calculation of Electron Temperature via Boltzmann Plot
2. Voigt Profile Fitting of Spectral Peak to Determine Broadening Mechanism
3. Estimation of Plasma (absolute) Temperature
4. Comparing Peak Intensity Ratio for Self-Absorption Mechanism
5. Conclusion on Plasma Diagnostics With Respect To Electrical Characteristics
ELECTRON TEMPERATURE AND BOLTZMANN PLOT
You can think of plasma as a "hot soup" of randomly colliding electrons and ions. The electron temperature of plasma allows us to gauge the "hotness" or more specifically the statistical kinetic energy of electrons bumping randomly to each other. Because electrons are so much lighter compared to ions, they transfer energy in different rates when they collide with ions than itself. As such, for the value of electron temperature to stay valid, it is important that the plasma have reached a state where all its particles, i.e. electrons, atoms and ions has reached a common temperature known as Local Thermal Equilibrium (LTE). Otherwise, this physical quantity is known as "excitation temperature".
This temperature can be calculated through the famous Boltzmann plot. In essence, it assumes two emission energy levels of the same atomic population, Ei and Ek are in thermal equilibrium at temperature T, given by:
Equation 1
where N1 and N2 denotes number density and g1 and g2 are the statistical weights respectively. The total number density (I skipped a few algebraic derivation steps) can be simplified to:
Equation 2
At LTE, the intensity for each spectral peak is given by:
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Equation 3
where Aji is the Einstein coefficient for upper transition probability. Rearranging equation 3 by taking natural logarithm on both sides for linearising the expression (y = mx+c) we arrive at:
Equation 4
where the gradient, (-1/kT) tells the electron temperature T, of the plasma. This is the Boltzmann plot where the x and y parameter of this linear expression can be obtained both experimentally (intensities and wavelengths) and from NIST standards (energy levels, Einstein coefficients and statistical weights).
One interesting outcome for plotting this is that from the data points is that we can determine if the spectrum is "optically thin" through the degree of linearity, i.e. the points coincide a straight line. Optically thick spectra is typically associated with dense plasma where some part of its light has been re-absorbed into the plasma itself, rendering the spectra unreliable for temperature analysis. From tedious work finding all the parameters for Boltzmann plot, I managed to do it for both mercury and argon:
Figure 3: Boltzmann plot for argon lines (top) and mercury lines (bottom). While the data is chaotic for argon's case, data for mercury seems to exhibit linearity where the expected gradient enables calculation of the electron temperature.
Even though argon has 17 emission lines, it is immediately obvious that the Boltzmann plot for argon lines yields chaotic results for a linear fit. Fortunately, there seems to be a linear relationship for mercury lines and I have used a linear fitting software which gave me a gradient of -1.43 with error of 0.35. This yields electron temperature at (0.699 ± 0.37) eV that is 8112 K for people not in plasma physics and 8384.7 degrees Celsius for non-physicists. The temperature is in the order of magnitude similar to discharge lamps reported in various literature elsewhere.
PEAK PROFILE FOR BROADENING MECHANISM
The spectrum for plasma (refer to figure 2) contains discrete sharp peaks which I have briefly mentioned as the result of electronic transitions within trillions of atoms of the gas contained in the germicidal lamp. Following the quantum theory, ideally for each electronic transition, the atom emits one particle of light, the photon, with a fixed wavelength and should result an infinitely thin line. However, experiments should find that no matter how "pure" a plasma is, we will always see the peak profile having a broadened "waist", like a mountain. So what was broadening it?
Broadly, there are three primary mechanisms causing this effect here listed with decreasing significance:
1. Doppler effect coming from light emitted from moving particles within the plasma which reveal itself as spectral line with a Gaussian profile.
2. Stark effect coming from energy levels (especially those close to the continuum) of an atom being "disturbed" by electric fields of nearby moving ions and/or externally applied E-field, this effect reveal itself in spectral line of Lorenzian profile.
3. Intrinsic property of quantum mechanics itself. i.e. the Heisenberg Uncertainty Principle, which states that any electrically excited atoms having a finite lifetime will have slight indeterminate energy level, causing the light emitted from atoms having so slightly a shorter or longer wavelengths with one another. An effect I believe too insignificant to observe based on the resolution of our experiment set-up.
For simplicity sake, I have chosen the brightest peak of the spectrum, i.e. the line at 253.7 nm for profile analysis. As we magnify the wavelength scale, we will see the "line-like spike" resolve into a "mountain". There are a few types of fitting available to extract information regarding its shape. For further analysis, I used a combination of both Gaussian and Lorenzian profile fitting - the Voigt profile.
Figure 4: Voigt profile fitting for the brightest spectral peak at 253.7 nm. Red line indicates the fitting.
Using a Voigt fitting, Origin 7 provides me two w parameters for defining the typology of the "mountain" shape. A ratio of wG and wL (0.00381/1.91006) gives us 0.002 which is significantly less than 0.5 - the threshold for Lorenzian profile, so Gaussian it is.
Figure 5: First order derivative of fitted Voigt profile. The peaks enables calculation of FWHM value.
Identifying the peak having a Gaussian profile has an advantage. Other than confirming the dominant broadening mechanism as the result of Doppler effect, a simple equation based on Maxwellian distribution can be used to estimate the absolute temperature of the plasma by knowing the profile's full wave half maximum (FWHM) value. We will discuss that in the next sub-chapter but for the moment, I have worked out the FWHM via first order derivative of said Voigt profile.
The positive and negative peaks of Voigt derivative correspond to the half-width value Δλ1/2, where |254.28 - 252.51| = 1.77 nm.
ESTIMATION OF PLASMA TEMPERATURE
As mentioned before, to characterize a plasma which composed of different types of particles, have to account two different temperatures. While we have characterized the electron temperature with Boltzmann plot, there is another class of particles - ions, which are heavier and moves slower relative to electrons and their temperature can be related to the absolute temperature of the plasma based on FWHM for Doppler broadening given by:
Δλ1/2 = 2λ SQRT (2kT ln 2 / mc^2)
Equation 5
where λ is the peak wavelength, m is the atomic mass and c is the speed of light. T is the absolute temperature where by rearranging the equation, I have calculated it to be 1.84E10 degree Kelvin. Which is rather nonsensical. For comparable results with our previously obtained electron temperature (plasma temperature should be slightly "less hotter" than electron temperature but LTE demands them to be as close as possible) the FWHM value should be in the order of 0.001 nm, which is far beyond the resolving capabilities of my spectrometer set-up.
I would argue such absurdities are due to equation 5, which depends heavily on the FWHM value which was limited experimentally by my spectrometer slit width, not until we analyze FWHM for each 27 peaks using precision spectrometer (which is quite a lot of work), I cannot be convinced that the absolute temperature falls in that order of magnitude. Naturally, because the plasma temperature differs so greatly with the electron temperature, I cannot conclude the plasma is even in LTE.
PEAK INTENSITY RATIO FOR SELF-ABSORPTION MECHANISM
There is however another way to determine the "usability" of the spectrum by finding its optical "thickness" with experimental intensity ratio between two emission lines that has the same upper energy level, described by:
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Equation 6
we can eliminate the exponential term since E2 = E1, and therefore leads to:
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Equation 7
where the left hand side (L.H.S) can be obtained via experiment intensity and right hand side (R.H.S) through NIST standards. With that, I have came to using three Hg (I) lines of 404.11 nm, 435.34 nm and 545.94 nm having identical upper energy level at 62350.325 cm^-1 corresponding to electron shell 3s at 5d(10)6s7s configuration, calculated their LHS and RHS values to be 2.57 against 1.43 and 3.57 against 2.46. Both differ by at least 1.1 suggesting optically thick plasma. This can be interpreted as self-absorption of photons by the atoms inside the plasma and the rate of absorption differs between different spectral lines depending on factors like energy level lifetime and charge densities etc.
CONCLUSION
To be honest, eventually diagnosing the plasma as optically thick was rather disappointing considering the amount of work spent on gathering spectral data from NIST and the associated analysis. Nonetheless we learned that the plasma confined within the germicidal lamp's quartz tube contains argon. Even though no conclusion can be drawn on the state of LTE for the plasma, the electron "excitation" temperature was found to be in the order of magnitude expected for a discharge lamp, that is at around 8000 K based on Boltzmann plot. The plasma temperature was found to be in the order of 10,000,000,000 K is unrealistic, but it was due to experimental limitation vis-a-vis the design of the spectrometer (thus limiting the FWHM resolution). It could also be due to some other broadening mechanisms which I have overlooked.
On optical thickness, I am tempted to suggest the difference between LHS and RHS terms from equation 7 arise from self-absorption are due to the electrical characteristics of the plasma. The germicidal lamp is most probably energized by alternating voltages which cause the particles in the plasma to "move" differently when it is energized by static DC sources. To understand these character of a plasma in detail, I recommend further work on a system that enables me to fix the state of vacuum and gas inside a chamber where we can manipulating the electrical properties of plasma to study its spectral changes although such work will be painstakingly time consuming.
REFERENCES
[1] N.M. Shaikh, B. Rashid, S. Hafeez, Y Jamil and M.A. Baig, "Measuremement of Electron Density and Temperature of Laser Induced Zinc Plasma", 2006, J. Phys D: Appl. Phys., pp.1384
[2] C. Aragon, J.A. Aguilera, "Characterization of Laser Induced Plasmas by Optical Emission Spectroscopy: A Review of Experiments and Methods", 2008, Spectrochimica Acta B, pp.893
[3] A.D.Giacomo, V.A.Shakhatov, O.De Pascale, "Optical Emission Spectroscopy and Modeling of Plasma Produced by Laser Ablation of Titanium Oxides", 2001, Spectrochimica Acta B, pp.753
