- Email: [email protected]
Infrared characterization of a cascaded arc plasma
Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605} 634
Infrared characterization of a cascaded arc plasma Ram Raghavan, Philip W. Morrison Jr.* Department of Chemical Engineering, School of Engineering, Case Western Reserve University, 10900 Euclid, Cleveland, OH 44106-7217, USA Received 8 April 2000; accepted 21 May 2000
Abstract We have constructed a cascaded arc light source and characterized its infrared emission properties. Breaking down Ar gas #owing through the cascaded arc channel forms a wall-stabilized plasma. The properties of the plasma can be adjusted by varying the pressure (1}4 atm) and the current #owing through the arc (15}30 A). The infrared (2000}10,000 cm\) emission of the cascade arc plasma is measured using a Fourier transform infrared (FTIR) spectrometer. The infrared emission can be used to determine the electron density (n ) and the temperature (¹) of the plasma, assuming the plasma is in partial local thermodynamic equilibrium (PLTE). The plasmas at the highest currents and pressures satisfy the assumption of PLTE, while the plasmas at other conditions do not. This enables the calculation of new transition probabilities for the infrared transitions in the Ar plasma. The electron density of the cascaded arc Ar plasma at our operating conditions varies between 2 and 5;10 m\ while the temperature varies between 9500 and 11,500 K. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Cascade arc; Argon plasma; Fourier transform infrared spectroscopy (FTIR); Transition probabilities; Partial local thermodynamic equilibrium (PLTE)
1. Introduction In situ diagnostics using a Fourier transform infrared (FTIR) spectrometer are a way to quantify concentrations and temperatures of gas-phase species commonly associated with chemical vapor deposition (CVD) processes. For example, our research group has demonstrated that in situ FTIR can be used to measure temperatures and concentrations of CH and C H in hot "lament CVD
* Corresponding author. Tel.: #1-216-368-4238; fax: #1-216-368-3016. E-mail address: [email protected] (P. Morrison). 0022-4073/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 1 0 4 - 7
606
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
(HFCVD) reactors [1,2] as well as oxyacetylene torches for diamond growth [3]. The technique has been limited, though, due to the lack of a bright infrared light source. This leads to relatively low signal-to-noise ratios, which are inadequate for detecting low concentrations of transient radicals. Further, the current infrared light source used by the FTIR, commonly known as a globar, has a large aperture. This leads to a relatively large spot size (&10 mm) when the light is focused in the sample zone and thus makes it di$cult to perform accurate localized measurements. The motivation behind the current research is to develop a better light source for the FTIR spectrometer that is not only is very intense, but also has a very small aperture. The cascade arc satis"es these requirements. The cascade arc used for our measurements is a low-#ow (30 sccm) argon plasma. The plasma, when at equilibrium, has been shown to get as hot as 13,500 K [4]. As a result, the intensity of infrared emission from the cascade arc can be a factor of 20}200 higher than conventional globar at 1300 K [5]. Also, the aperture of the arc is 3 times smaller than the globar, which leads to better spatial resolution. There are various radiative processes which lead to two types of arc emission in the infrared: line and continuum. Line emission is a quantum phenomenon occurring when an electron falls from a higher excited state to a lower excited state; the di!erence in energy between the states is released as a photon. This results in a narrow `linea in the emission spectrum. This narrow line will be broadened, mainly due to interaction of the emitters with the surrounding electrons (Stark broadening) and atoms (resonance broadening) [6]. Continuum emission is a result of inelastic scattering of electrons in the presence of atoms and ions. When electrons decelerate in the vicinity of these atoms and ions, there is photon emission at all wavelengths. An advantage of the cascade arc is that it is a continuum emitter over a very large spectral range. Various authors (many from the Eindhoven University of Technology) have characterized the electrical and visible emission properties of the cascade arc. Kroesen et al. [7] describe the various processes occurring in the plasma and also set up transport equations to describe the system. They then solve these equations to obtain the electron temperature of the plasma using the continuity, momentum, and energy equations for each species in the plasma. Qing et al. [8] have characterized the e$ciency of an Ar/H plasma by calculating a simple power balance involving electric and heat #uxes. The e$ciency is just a ratio of the power dissipated by the arc to the input electric power. Increasing the Ar #ow through the arc increases the arc e$ciency but results in lowering the arc temperature. Adding hydrogen to the Ar also decreases the e$ciency of the plasma. Wilbers et al. [4,9] use the cascade arc as a visible light source for spectroscopic techniques but caution that instabilities can occur at high/pressures and currents. The operating pressure of their arc varies between 1 and 6 atm and the current between 20 and 70 A. The absolute intensities are calculated with respect to a tungsten ribbon lamp at 2667 K. These authors measure the intensity of their cascade arc at 2 bar to be an order of magnitude higher than a globar at 5000 cm\. Neutral particle densities and temperatures in the cascade arc plasma have also been calculated using optical absorption and emission spectroscopy [10}15]. Kock and his collaborators [16}18] use optical laser interferometry to measure electron densities, transition probabilities, and continuum emission coe$cients in these plasmas. The electron densities measured by Kock and his collaborators are very similar to those obtained by the researchers at the Eindhoven University of Technology. Although there is a wealth of literature about the cascade arc in the visible and ultraviolet regions [19}22], there is no detailed IR characterization of the arc. In this work, we have measured
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
607
the IR emission of the arc using a Fourier transform infrared (FTIR) spectrometer. The line and continuum emission of the Ar plasma will be used to determine the temperature and electron density of the plasma.
2. Apparatus A schematic of the cascade arc is shown in Fig. 1. The arc consists of water-cooled copper plates cascaded together, but insulated from each other, with a 4 mm hole machined through them. This con"guration is a classic design of a wall-stabilized arc [23]. There are three thoriated tungsten electrodes that serve as cathodes and a grounded end plate that serves as the anode. When the Ar gas #owing through the channel breaks down due to current #ow, a plasma results. The arc can be run under a variety of conditions by adjusting the argon #ow rate, the current #owing through the arc, and the pressure. In our system the #ow rate is kept constant at 30 sccm, the current of the arc is varied between 15 and 30 A, and the pressure is varied between 1 and 4 atm. As seen from Fig. 1, the plates have cooling water channels machined inside them and are insulated from one another using a boron nitride spacer held in place with polycarbonate disk and an O-ring. The boron nitride spacer, is 27 mm in diameter and 3.2 mm thick, and a C22 Viton O-ring provides the sealing surface. The polycarbonate disc has the same OD as the copper plate (70 mm) and an ID of 33.3 mm. The plates and their spacers are then sandwiched between the cathode plate and the anode plate. The cathode plate contains three symmetric 2% thoriated tungsten tips that are 1 mm in diameter and 10 mm long. Del Seal sections are welded permanently to the cathode and anode plate to facilitate the #ow of gases in and out of the system.
Fig. 1. Design of cascade arc. This design is derived from the work of Kroesen et al. [23].
608
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
The downstream end of the arc (anode plate end) has "ttings to attach a pressure gauge (Matheson 63-3112) with a range of 0}100 psi, a mass #ow controller (MKS Instruments 1259 B) with a range of 500 sccm, and a vacuum pump (GE Motors HP). The arc is water cooled by a manifold that consists of two branches, one branch for the tip holders and another branch for the plates. The total water #ow rate is 4 l/min. Finally, the ends of the arc are sealed o! using infrared transparent sapphire windows (with an IR cut-o! at 1500 cm\). The arc is then mounted on a custom built frame that allows easy handling of the arc itself. When the arc starts up under vacuum, the gas is very hot, and there are micron size particles of debris which strike the window if unprotected. In addition, the thermal gradients at the window during startup are severe enough to crack the window. To protect the window, a steel disk is placed on the inside surface of the window; a magnet holds the steel disk in place during arc start up. Once steady-state operating conditions are reached, the magnet is removed from the outside, and the steel disk slips out of the way. The optical components used to interface the arc to the spectrometer are shown in Fig. 2. The light from the arc is "rst collected by a 63 mm diameter, 50 mm focal length 903 parabolic mirror (Melles Griot). The mirror is mounted on a gimbal (Edmund Scienti"c) placed on a 60 cm long optical rail using a standard narrow optical carrier (Oriel Instruments). Although gold is the coating of choice in infrared re#ection optics due to its high re#ectivity, the mirror is rubidium coated which has a comparable re#ectivity. The collection mirror collimates the IR radiation, producing 50 mm diameter light beam that enters the emission port of a Bomem MB 157 FTIR spectrometer. The beamsplitter is only 25 mm in diameter, and hence only the central 25 mm of the collimated 50 mm beam is collected by the FTIR. The modulated signal then hits a 127 mm focal length mirror that focuses the light through the sample compartment of the FTIR, and a similar mirror in the detector optics collects the light. The etendue (optical throughput) of the collection optics is controlled by the detector size and its mirror and not by the arc size and its mirror (see the appendix). The detector used in the setup is an extended range DTGS (deuterated triglyciene sul"de) detector that is 1 mm in size. The Bomem MB 157 spectrometer has a maximum resolution of 2 cm\ when using the extended range DTGS detector. At this resolution, the instrument is capable of scanning at 12 scans/min. The experiments are all run at 25 scans at 2 cm\ resolution. Because of the arc's high intensity in the near IR, we use an attenuator in the
Fig. 2. Collection optics for infrared measurements of the arc.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
609
beam path to keep the centerburst of the spectrometer on scale and to prevent the DTGS detector from becoming non-linear. To convert the raw emission spectra to path-corrected radiance spectra, we calibrate the FTIR system with a 1300 K blackbody source [24]. For maximum sensitivity above 6500 cm\, we perform the blackbody calibration without the attenuator and then mathematically correct the arc emission spectra using the transmittance of the attenuator and the blackbody calibration. The circuit used to operate the arc consists of two power supplies. The "rst is a 1000 V, 2 A power supply (Advanced Energy MDX 2500 W Magnetron Drive) and the other is a 300 V, 30 A power supply (Sorenson DCR 300-33T). The 1000 V power supply is needed to ignite the plasma, since even at low pressures (20 Torr), signi"cant electric "elds (&1000 V) are required to breakdown argon. Once a discharge is created, however, large currents are necessary to sustain the arc. Using a circuit design similar to Kroesen et al. [23], a set of charged capacitors deliver a transient current as the circuit switches from the ignition supply to the high-current supply. If the capacitors were not present, a continuous arc will never form, as the breakdown caused by the high-voltage power supply would not last long enough for the high-current power supply to sustain it. The values of the capacitance are chosen in such a way that the RC time constant matches the response time of the high-current power supply (&13.5 ms). More details are available in a technical report from the Eindhoven University of Technology [23]. The circuit also contains high power, 1 resistors in series with the arc tips to ensure that there is always a positive resistance present because these plasma discharges have extremely non-linear curves and can have a negative `resistancea at times (d
610
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
The current can be changed while the arc is running. Care must be taken to adjust the current gradually, since rapid changes in current can shut down the arc. The pressure inside the arc chamber can also be increased gradually using the regulator of the compressed Ar tank feeding the arc but the current in the arc must remain fairly constant while increasing the pressure. The plasma has been operated at 16 di!erent conditions using pressures of 1, 2, 3, and 4 atm and currents of 15, 20, 25, and 30 A.
4. Partial local thermodynamic equilibrium (PLTE) in plasmas A typical emission spectrum from the arc showing the line and the continuum emission appears in Fig. 3. From Fig. 3, it is clear that the continuum intensity of the arc is roughly uniform at all wavelengths. At lower wavelengths (around 3000 cm\) the continuum intensity from the arc is similar to the emission of a globar (&1300 K blackbody), but is substantially brighter than the globar at higher frequencies ('6000 cm\). The spectra also contain very large line emissions due to the excited electronic states of Ar. Employing radiative exchange theory [25] and the observed line emission, we will estimate the plasma temperature (¹); the continuum emission will be used to estimate the electron density (n ). Radiative exchange theory provides a relationship between the radiance (or emission) and the temperature of a hot gas: R()"[1!()]BB(¹,)["]
W , m sr cm\
(1)
where R is the radiance (which depends on wavenumber (cm\)), is the transmittance, and BB is the Planck function (which depends on temperature and wavenumber). When we invoke radiative exchange theory, we are in e!ect assuming that the plasma is in local thermodynamic equilibrium (LTE) because we are using a local temperature (¹) to describe the system. However, we need to check whether LTE can be applied to our IR spectra. In fact, Timmermans et al. [26] have shown that plasmas similar to ours are not in LTE but in partial local thermodynamic equilibrium (PLTE). PLTE occurs when collisional processes establish a Boltzmann equilibrium among the excited states but are not su$cient to establish a Boltzmann equilibrium between the
Fig. 3. Radiance spectrum of an arc operating at 4 atm and 30 A. The largest peaks between 6000 and 9000 cm\ exceed 100 W/(m cm\ sr).
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
611
ground state and the excited states. Nevertheless, radiative exchange theory is still applicable, as shown below. A detailed balance for the various Ar states provides insight into how PLTE occurs. Consider the di!erent mechanisms populating/de-populating the "rst excited state of Ar for a simpli"ed three-level energy diagram. Referring to Fig. 4 and Table 1, an unsteady population balance on the "rst excited state results in dn " k n n # k n n #k n n n> #A n #k n>n > dt > > #JCARPML #JCARPML N?PRGAJC 0?BG?RGTC 0?BG?RGML CVAGR?RGML BCUCVAGR?RGML PCAMK@GL?RGML PCAMK@GL?RGML DPMK ?@MTC DPMK @CJMU DPMK ?@MTC #k n n #A n ). !( k n n # k n n > 'MLGX?RGML 1NMLR?LCMSQ #JCARPML #JCARPML #KGQQGML CVAGR?RGML RM LB BCUCVAGR?RGML CVAGRCB QR?RC RM EPMSLB QR?RC Similar equations will result for other excited states as well. Rosado's Ph.D. dissertation [27], gives values for the various collisional and radiative rates cited in Table 1. In the equation above, the three-particle recombination term (k n n n>) is observed to be very small compared to the other > terms in the detailed balance and can be ignored. From this detailed balance, and using the collisional and radiative rates, we can estimate relative populations of the various excited states. For the ground state, spontaneous emission is forbidden between the second excited state and the ground state, so the equivalent detailed balance becomes dn "k n n #k n n #A n #k n n>!n n (k #k #k ). > > > dt
Fig. 4. Simpli"ed energy level diagram for Ar.
(2)
612
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Table 1 Processes occurring in plasmas. Ar is the ith excited state; refers to a trapping coe$cient. See [27] for values of the G parameters Process
Example
Parameters
Electron impact excitation Electron impact de-excitation Electron impact ionization Three-particle recombination Spontaneous emission
Ar #ePAr #e Ar #eQAr #e Ar #ePAr>#2e Ar #eQAr>#2e Ar PAr #h
k , k , k (m s\) k , k , k (m s\) k ,k ,k (m s\) > > > k ,k ,k (m s\) > > > A , A (s\) , : Trapping coe$cients which depend on the optical depth of the plasma
Radiative recombination
Ar>#ePAr #h
k , k , k (m s\) > > > > > >
For estimation purposes, we can assume a Boltzmann distribution (n /n "g /g e\#H I2) to H H calculate relative populations (n ) of the various excited states at energies E with degeneracies g . H H H Observing that our atmospheric and super-atmospheric Ar plasmas have electron temperatures around 1 eV (&11,600 K) with electron densities between 10 and 10 m\, we can then calculate order of magnitude estimates of the relative contributions of all the terms in the balance. We "nd that none of the terms in the ground state detailed balance can be neglected, so at steady state: k n #k n #(n /n )A #k n> > > . (3) n + k #k #k > For the "rst excited state, we "nd that radiative recombination to excited states (k n>n ) is > > negligible [27], so the detailed balance becomes dn "k n n #k n n #A n !n n (k #k #k ) > dt !A n . At steady state
(4)
k n #k n #(n /n )A "k n #k n #k n > - - - - - - (5) #(n /n )A . - From the order of magnitude calculations for each term, the dominant terms in the balance are the collisional excitation/de-excitation between the "rst and second excited states. Neglecting the smaller terms n k + . n k
(6)
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
613
Since k and k are rate constants that are linked by detailed reversibility, they are also related by the Boltzmann equation when the processes are at an equilibrium temperature ¹: k /k "g /g e\# \# I2. The detailed balance, therefore, shows that the collisional processes alone have driven "rst and second excited states of Ar into near equilibrium at a temperature ¹. This argument can be extended to the higher excited states as well. For instance, the signi"cant terms in a steady state detailed balance of the second excited state will be k n #k n "k n #k n . (7) But k n +k n (Eq. (6)) and so k n + . (8) k n Thus the excited states are related by detailed collisional reversibility and are in Boltzmann equilibrium described by a temperature ¹. We can now see how the ground state is related to the excited states using the relationship between the "rst and the second excited states. Substituting Eq. (6) into the ground state density expression (Eq. (3)), and noting the fact that n +10\n in our 1 eV plasma, we obtain k #k #k > GFFHFFI -\
n + &0.33;10\. (9) n k #(k k /k ) #10k #(A /n ) \ > > -\ - -\ -\ Compare this value to a Boltzmann calculation using E "11.65 eV, g "12, and ¹"1 eV: g e\# \# I2"1.05;10\. g These expressions show that the ground state is not at collisional equilibrium with the "rst excited state since n /n is approximately three times smaller than what is expected from Boltzmann equilibrium. Since the excited states are in collisional equilibrium with each other, but the ground state is not in equilibrium with the excited states, the plasma is said to be in partial local thermodynamic equilibrium (PLTE) [26]. One of the reasons for this `overpopulationa of the ground state is the radiative recombination term (k ), which is selectively populating only the > > ground state. Timmermans et al. [26] have also observed a transport process that also contributes to PLTE. Because there is a temperature gradient across the plasma channel, ions form at the center of the plasma (where it is the hottest) and then di!use to the wall where they recombine with electrons to form neutrals. These neutrals then di!use back to the center of the plasma to become excited/ionized again. Since the "rst excited state of Ar is 11.65 eV and the plasma temperature is only around 1 eV, the excitation of ground state Ar is a very slow process. Hence, the ground state can become additionally `overpopulateda with respect to the excited states due to this transport
614
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Fig. 5. Schematic Boltzmann diagram for `spectroscopically activea argon. The symbol corresponds to the mole fraction of `spectroscopically activea Ar.
process. If we call this di!usion rate D (m\ s\) and take it into account, the relationship between the ground state and the "rst excited then becomes k #k #k n > + , k #(k k /k )#(A /n )#10k #(D/n ) n > >
(10)
which is even smaller than the right-hand side of Eq. (9). Thus, this inward di!usion also contributes to deviations from LTE and produces PLTE. To summarize, radiative exchange theory and the use of the Planck function can be applied to the excited states of these plasmas but not to the ground state. In other words, the relative populations n /n of the excited states j and k have a Boltzmann distribution but not n /n and H I H n /n . Fig. 5 illustrates the impact on interpreting the IR spectra. The upper line is a Boltzmann I plot assuming all the states are at equilibrium with each other and the ground state. The slope is inversely proportional to the plasma temperature. In PLTE, the `temperaturea of the excited states is same as the upper line because they have a Boltzmann distribution. Their number densities are reduced by some factor, however, because they are not in Boltzmann equilibrium with the ground state since the ground state is `overpopulateda. Thus, the points for the excited states in PLTE are shifted down but have the same slope as before. The dashed extrapolation to the ground state (Fig. 5) represents the mole fraction of `spectroscopically active Ara that is consistent with the populations of the excited states. Thus the Ar has two components: (a) `spectroscopically activea Ar for which we can determine a temperature using radiative exchange theory and (b) `spectroscopically inactivea Ar which is ground state Ar which is `in excessa of PLTE. As a result, we can apply radiative exchange theory as long as we treat the partial pressure of the spectroscopically active Ar (P ) as an unknown parameter.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
615
The above argument presumes that only the ground state is `overpopulateda relative to the excited states. Rosado [27] has shown that the lower excited states are also slightly `overpopulateda which would cause curvature in the PLTE line of Fig. 5. We expect this to be a minor problem in our situation, however, since the IR spectrum of the plasma mainly results from transitions between high lying electronic levels (mostly third excited state and above) where the `overpopulationa has decreased to zero. Thus for relatively cool (&1 eV) plasmas, spectroscopically measuring plasma temperatures in the IR may be a more robust method than in the visible.
5. Temperature estimation from line radiation From radiative exchange theory [25], we know that R()"(1!())BB(¹, ),
(11)
where R is the measured intensity of line radiation, is the transmittance of the line, and BB is the Planck function. The Planck function is given by [4] W c ["] , (12) BB(¹, )" m sr cm\ exp(c /¹)!1 where c "1.191;10\ W/m sr (cm\) and c "1.4388 cm K. The transmittance has the form "e\-",
(13)
where OD is the optical depth of the plasma. Since our plasma is optically thin (OD;1), we can simplify Eq. (11) using the small argument approximation, R+OD * BB.
(14)
Using the notation of Rothman et al. [28], we know that the optical depth has the form OD"S(¹)G(! )n l, (15) where n "P /R¹, S(¹) is the line strength, G(! ) is the line shape function, is the line center, l the path length, and n is the number density of `spectroscopically activea argon. The line strength for the line at is given by I Ag !c E !c cm\ exp 1!exp ["] S(¹)" (16) 8c ¹ ¹ mol cm\
where I is the isotopic abundance of the species, A (1/s) is the transition probability for , g is the degeneracy of the upper state, c (cm/s) is the speed of light, and E (cm\) is the energy of the upper state. There are a number of sources from which we obtain transition probabilities, degeneracies, and transition energies for the observed lines [29}31]. The observed lines and their associated parameters (A, g , E , etc.) are seen in Table 2. Table 2 contains a total of 153 lines for Ar; there are no observable Ar> lines in our plasma. Please note that most of the A values are theoretical estimates with uncertainties of 30%.
616
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Table 2 Emission parameters for observed IR lines from Smith et al. [29] Transition (cm\)
Transition probability A (s\)
Con"guration of lower state
E upper (cm\)
Con"guration of upper state
Degeneracy of upper state g
2080.89 2116.93 2118.78 2120.82 2122.50 2131.84 2141.00 2141.72 2159.42 2160.98 2162.13 2193.46 2197.22 2213.54 2246.62 2270.12 2284.42 2284.81 2293.50 2608.85 2623.28 2637.01 2683.76 2689.18 2692.26 2696.43 2701.72 2737.42 2737.85 2740.33 2746.57 2753.88 2760.85 3003.19 3003.59 3016.73 3023.09 3040.56 3092.73 3100.18 3191.52 3226.20 3279.34 3279.74
1.65E#06 2.19E#04 5.72E#04 1.13E#04 5.79E#04 1.75E#05 9.73E#05 9.79E#05 1.46E#06 3.86E#05 5.46E#05 1.69E#06 2.82E#05 1.62E#06 8.88E#05 3.48E#05 2.12E#06 1.08E#06 1.37E#06 1.20E#06 5.21E#05 1.47E#06 3.91E#05 3.19E#06 2.67E#06 1.62E#06 1.05E#06 2.21E#05 2.63E#05 2.35E#06 4.33E#05 5.71E#05 2.13E#06 1.74E#05 2.41E#05 4.53E#05 1.09E#06 6.97E#04 1.81E#05 3.98E#05 2.54E#06 7.35E#04 4.06E#04 2.50E#04
5p 4d 5d 3d 4f 5p 5p 4d 5p 4f 4d 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5d 6p 5p 4d 5p 5p 4d 3d 5s 5p 3d 3d 5p 5s 3d 4f 4f
119023.6 122717.8 124051.6 116942.8 122329.7 120600.9 121011.9 121165.4 120619.0 123815.6 121068.7 120600.9 119760.2 119212.9 118906.6 119212.9 120753.5 119847.8 119444.8 119760.2 119566.0 121096.6 119683.1 121096.6 121161.3 119847.8 121161.3 126294.8 123903.2 119683.1 121653.2 121161.3 119760.2 121654.6 117151.3 116660.0 119683.1 118407.4 118459.6 119760.2 116660.0 116942.8 123468.0 123468.0
4d 5f 6f 5p 5d 4d 4d 6p 4d 5d 6p 4d 6s 4d 4d 4d 4d 4d 4d 6s 4d 6s 6s 6s 6s 4d 6s 7f 8s 6s 4f 6s 6s 4f 5p 5p 6s 5p 5p 6s 5p 5p 6d 6d
9 7 5 7 7 5 3 7 5 3 3 5 3 7 5 7 7 3 5 3 7 1 5 1 3 3 3 9 5 5 7 3 3 5 3 3 5 3 3 3 3 7 3 3
[2# H[1# H[0# H[2# [4# [1# [0# H[3# [0# [2# H[1# [1# [0# [2# [0# [2# [1# [0# [1# [1# [2# [0# [2# [1# [1# [1# [0# H[2# [2# [2# H[1# [1# [2# H[0# H[1# H[1# [0# H[1# H[1# [0# H[1# H[2# [1# [1#
H[3# [3# [2# [2# H[2# H[1# H[1# [2# H[2# H[1# [0# H[1# H[1# H[3# H[1# H[3# H[2# H[1# H[2# H[1# H[2# H[0# H[1# H[0# H[0# H[1# H[0# [3# H[1# H[1# [3# H[0# H[1# [2# [1# [0# H[1# [1# [0# H[1# [0# [2# H[0# H[0#
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634 Table 2 (continued ) Transition (cm\)
Transition probability A (s\)
Con"guration of lower state
E upper (cm\)
Con"guration of upper state
Degeneracy of upper state g
3282.77 3356.07 3415.22 3432.41 3467.04 3474.28 3482.83 3494.03 3508.07 3530.85 3536.32 3540.33 3545.80 3978.97 4171.35 4173.87 4192.60 4321.62 4521.07 4536.06 4642.51 4763.76 4803.83 4821.37 4821.77 4841.97 4849.22 4863.63 4916.43 4922.83 4981.47 4991.13 4992.23 5009.57 5044.67 5383.50 5424.62 5580.48 5580.51 5608.88 5730.66 5901.37 5972.06 6040.90 6051.66
7.21E#04 2.73E#06 1.13E#06 1.31E#06 6.38E#05 4.08E#06 1.30E#06 4.00E#06 3.45E#06 1.07E#06 1.50E#06 1.06E#06 2.80E#06 1.53E#06 3.59E#05 1.28E#06 1.60E#06 1.07E#05 1.07E#06 9.07E#04 1.62E#05 3.41E#05 4.81E#04 1.83E#05 1.59E#05 1.25E#06 5.45E#05 9.84E#05 1.29E#05 1.88E#05 1.45E#06 3.83E#05 5.54E#05 2.79E#05 7.43E#05 4.98E#05 1.31E#06 8.52E#05 8.27E#05 2.07E#06 1.37E#05 1.85E#06 3.00E#05 2.38E#06 1.86E#05
3dH[2# 5s H[1# 3dH[1# 5sH[0# 3dH[2# 5s H[1# 4dH[3# 5sH[0# 5s H[1# 5s H[1# 4dH[1# 5s H[1# 5sH[0# 3dH[3# 4p [0# 4dH[0# 3dH[3# 4p [0# 3dH[1# 4p [1# 4p [0# 4p [0# 3dH[1# 3dH[1# 3dH[1# 3dH[0# 4p [1# 4dH[2# 5p [0# 4dH[3# 3dH[2# 5s H[1# 3dH[0# 4dH[1# 3dH[1# 3dH[1# 3dH[1# 4p [1# 4p [1# 3dH[2# 4p [1# 4p [1# 4p [0# 3dH[1# 4p [1#
116999.3 116999.3 117563.0 118407.4 117183.6 116942.8 122695.7 118469.1 117151.3 116999.3 124137.2 117183.6 118407.4 116999.3 111667.8 122686.1 116942.8 111818.0 116660.0 111667.8 112138.9 111818.0 116942.8 120188.2 120188.6 116660.0 112138.9 125482.6 122479.4 124135.7 118407.4 118459.6 116660.0 124857.3 117183.6 120188.6 120229.8 111818.0 111667.8 120249.9 113020.4 112138.9 113468.5 120188.6 112138.9
5p [2# 5p [2# 5p [0# 5p [1# 5p [1# 5p [2# 5f [4# 5p [1# 5p [1# 5p [2# 5f [2# 5p [1# 5p [1# 5p [2# 3dH[0# 5f [1# 5p [2# 3d H[0# 5p [0# 3d H[0# 3d H[1# 3d H[0# 5p [2# 4f [1# 4f [1# 5p [0# 3d H[1# 6f [3# 7s H[1# 5f [3# 5p [1# 5p [0# 5p [0# 7f [1# 5p [1# 4f [1# 4f [2# 3d H[0# 3d H[0# 4f [3# 3d H[3# 3d H[1# 5s H[1# 4f [1# 3d H[1#
5 5 1 3 5 7 9 5 3 5 7 5 3 5 1 3 7 3 3 1 5 3 7 3 5 3 5 7 3 9 3 3 3 5 5 5 7 3 1 7 7 5 5 5 5
617
618
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Table 2 (continued ) Transition (cm\)
Transition probability A (s\)
Con"guration of lower state
E upper (cm\)
Con"guration of upper state
Degeneracy of upper state g
6082.33 6252.40 6287.71 6490.62 6513.20 6521.65 6533.36 6588.99 6644.25 6804.10 6823.91 6831.34 6848.04 7013.24 7093.46 7186.82 7229.52 7229.56 7230.92 7287.39 7308.72 7338.70 7351.29 7365.22 7381.21 7403.09 7456.98 7479.00 7509.28 7532.24 7556.00 7557.60 7565.67 7685.32 7715.93 7729.93 7808.69 8060.47 8090.83 8099.29 8228.27 8235.16 7843.31 7851.20 7870.45
5.29E#06 1.07E#06 8.36E#06 1.72E#06 1.76E#06 3.42E#04 7.08E#06 1.30E#06 4.70E#06 2.00E#06 7.36E#06 9.98E#06 9.55E#06 9.35E#06 5.23E#06 1.27E#07 2.68E#06 1.02E#06 3.50E#06 1.37E#07 1.08E#07 1.04E#07 1.51E#06 5.87E#06 5.39E#05 1.34E#07 1.65E#07 1.17E#07 1.28E#07 1.39E#07 4.94E#06 2.22E#06 1.08E#07 1.04E#07 1.03E#07 1.26E#07 4.47E#06 7.41E#06 1.19E#07 1.01E#06 9.11E#06 2.03E#06 2.43E#06 2.16E#06 5.74E#06
3d H[1# 4p [0# 3d H[1# 3d H[2# 3d H[2# 4p [2# 3d H[2# 4p [0# 4p [0# 3d H[2# 3d H[2# 3d H[2# 3d H[1# 3d H[2# 4p [0# 3d H[3# 3d H[3# 3d H[3# 4p [1# 4p [2# 4p [0# 4p [1# 4p [1# 4p [0# 4p [1# 4p [2# 3d H[3# 4p [1# 4p [1# 4p [1# 4p [1# 4p [2# 4p [0# 4p [1# 4p [0# 4p [1# 4p [2# 4p [1# 3d H[1# 4p [2# 3d H[2# 4p [1# 4p [1# 4p [2# 4p [0#
120230.1 114975.0 121654.6 120207.2 120229.8 112138.9 120249.9 113643.3 115366.9 120230.1 120249.9 121653.3 121653.2 121654.2 114147.7 120207.2 120249.9 120249.9 113468.5 112750.2 114805.1 113426.0 114641.0 114861.6 113468.5 113020.4 120207.1 113716.6 114641.0 114821.9 113643.3 113020.4 111667.8 114975.0 111818.0 114861.6 113426.0 114147.7 120229.8 113716.6 121654.2 115366.9 114975.0 113468.5 115366.9
4f [2# 5sH[0# 4f [2# 4f [4# 4f [2# 3d H[1# 4f [3# 5s H[1# 3dH[1# 4f [2# 4f [3# 4f [3# 4f [3# 4f [2# 3d H[1# 4f [4# 4f [3# 4f [3# 5s H[1# 3d H[3# 3dH[1# 3d H[2# 3dH[2# 5sH[0# 5s H[1# 3d H[3# 4f [4# 3d H[2# 3dH[2# 3dH[2# 5s H[1# 3d H[3# 3d H[0# 5sH[0# 3d H[0# 5sH[0# 3d H[2# 3d H[1# 4f [2# 3d H[2# 4f [2# 3dH[1# 5sH[0# 5s H[1# 3dH[1#
5 3 5 9 7 5 9 3 3 5 7 9 7 7 3 9 7 9 5 9 5 5 5 1 5 7 11 7 5 7 3 7 1 3 3 1 5 3 7 7 7 3 3 5 3
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
619
Table 2 (continued ) Transition (cm\)
Transition probability A (s\)
Con"guration of lower state
E upper (cm\)
Con"guration of upper state
Degeneracy of upper state g
8005.71 8025.99 8036.83 8253.80 8370.20 8370.60 8403.44 8412.04 8520.47 8530.46 8567.58 8632.93 8702.27 8717.88 8737.47 8770.57 8774.38 9023.72 9187.87 9366.37
1.08E#07 9.91E#06 8.82E#06 2.41E#06 9.17E#06 9.79E#06 1.00E#04 2.99E#06 1.14E#07 1.12E#06 4.48E#06 3.11E#06 1.89E#05 4.74E#05 1.90E#06 4.09E#06 3.99E#06 2.43E#06 5.39E#05 4.88E#06
4p 4p 4p 4p 3d 3d 4p 3d 3d 4p 4p 3d 4s 4p 4p 3d 4p 4p 4p 4p
113468.5 113643.3 112138.9 113716.6 120188.2 120188.6 114641.0 120230.1 120188.2 114147.7 114805.1 121653.3 104102.1 114805.1 114975.0 124137.4 114861.6 114641.0 114805.1 113468.5
5s H[1# 5s H[1# 3d H[1# 3d H[2# 4f [1# 4f [1# 3dH[2# 4f [2# 4f [1# 3d H[1# 3dH[1# 4f [3# 4p [0# 3dH[1# 5s H[0# 5f [2# 5s H[0# 3dH[2# 3dH[1# 5s H[1#
5 3 5 7 3 5 5 5 3 3 5 9 3 5 3 5 1 5 5 5
[2# [2# [0# [2# H[0# H[0# [1# H[0# H[0# [2# [1# H[3# H[0# [1# [1# H[1# [1# [2# [2# [0#
Since the lines are primarily Stark broadened, the lines are expected to have a Lorentzian line shape [6]. In fact, Lorentzian shapes "t the measured spectral lines satisfactorily. The line shape function for a Lorentzian has the form w/2 1 G(! )" ' ["] cm\. (17) (! )#(w/2) Here, w is the full width of the line in cm\. In all the calculations discussed below, a path length (l) of 1.6 cm is used. Although the arc channel is 3.6 cm long, not all light collected from the arc has that path length. Only the light coming from very close to the optical axis of the collection mirror would have traveled 3.6 cm. As discussed in the appendix, most of the other light would have traveled less than 3.6 cm. Using ray tracing and Eq. (14), we have calculated a path length of 1.6 cm. Combining Eqs. (14) and (15) gives R()"S(¹)G(! )l[P /R¹]BB(¹,). (18) The above equation shows that R() is a function of two unknown parameters: ¹ and P . Since R() is linear in P and non-linear in ¹, we can perform a least squares "t by guessing ¹, calculating a synthetic Ar emission spectrum (R ()), and then calculating the best "t value of
620
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Fig. 6. Least squares "tting of measured data for a 4 atm, 30 A arc by varying ¹ and calculating the corresponding best "t P . The number of spectral data points is 7882.
Fig. 7. Comparing measured (4 atm, 30 A) and simulated spectra (¹"10,800 K, P "0.94 atm). The spectra have been baseline corrected for comparison: (a) 3000}4000 cm\ spectral region, (b) 7000}8000 cm\ spectral region.
P for that guess of ¹: S(¹ )G(! )lBB(¹ , ) R () "P 2 R¹ "P f (, ¹ ).
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
621
Therefore, R () f (, ¹ ) . P " J 2 [ f (, ¹ )] J The process is repeated for di!erent temperatures at 200 K intervals, and the temperature which results in the smallest least-squares error ([R()!R ()]) is the temperature of the plasma. The results of this process for a 4 atm, 30 A plasma is illustrated in Fig. 6. The plasma emssion simulated with the least-squares temperature estimate is compared to the measured emission in Fig. 7.
6. Electron density estimation 6.1. Continuum emission Once the temperature is calculated, the electron density can be estimated from the continuum radiation. The continuum emission is due to three sources, namely: free}free electron}ion interactions ( ); free}free electron neutral}atom interactions ( ); and free}bound electron}ion interac tions (). Using radiative exchange theory in the same way as before, the radiance due to the continuum emission can be written in a slightly di!erent form [32] R()"(1!e\C l 2 J)BB(¹, ),
(19)
where " # #. The individual emission coe$cients are given by [32]
!c n "1.632;10\ exp ¹ (¹
!c c , #1 exp 1# ¹ ¹
"1.026;10\n n ¹ Q(¹)
(20)
h 2k¹ where "1#0.1728 1# , E (z!1) h & !c n "1.632;10\ 1!exp . ¹ (¹
(21)
(22) (23)
The individual emission coe$cients have the units of W/(m cm\), n and n have the units of m\, and ¹ is in K. Here, Q(¹) (m) is the electron-neutral atom scattering cross-section calculated by Devoto [33]. The free}free Gaunt factor (dimensionless) is calculated using the formula of Menart et al. [34]. The factor z is the ionic state of the interacting ion (z"2 for singly ionized argon). (Note that the value of asymptotically converges to 1.2 W/(m cm\) as the emission frequency increases above 2000 cm\). is called the Biberman factor and is fairly independent of temperatures and wavelength. In our calculations, we assume that the Biberman factor is constant at a value of 1. Since the continuum is `#ata, and is relatively independent of temperature, the best
622
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
"t to the continuum can be calculated using a least-squares analysis by varying the only unknown factor, the electron density n . 6.2. Saha equilibrium We now have techniques to estimate the temperature (¹), electron density (n ), and the number density n "P /R¹ of `activea Ar assuming the plasma is in PLTE. If our assumption of PLTE is correct, then these parameters (¹, n , n ) should also be consistent with the Saha equation. To test our assumption, we use the measured ¹ and n from the line spectra calculations to calculate a new electron density n1 via the Saha equation. If this electron density is the same as that calculated from continuum emission, then our assumption of PLTE is valid. The Saha equation is [35]
2u>(¹) (2m k) E !E n>n " ¹ exp ! G h u (¹) k¹ n "2.4125;10
2u>(¹) E !E ¹ exp ! G ["] cm\, u (¹) k¹
(24)
where u> and u are the partition functions of singly ionized and neutral Ar [35], E is the lowering of the ionization potential, and all other terms have their usual meaning. There are several empirical and theoretical expressions for the lowering of ionization potential (E). We use Brunner's formula [35]
E(eV)"1.21;10\ (n (cm\)#2.5;10\
n (cm\) . ¹(K)
(25)
In these low-temperature, low electron density plasmas, we note that the IR spectra show no observable emission from Ar>, so we assume that there is a negligible amount of doubly or higher ionized argon ions. This observation means that the number density of ions is, in fact, equal to the number density of the electrons, i.e. n>"n . It must be noted that the electron density calculated from continuum emission is not very sensitive to variations in temperatures, but the electron density calculated from Saha equation depends strongly on the plasma temperature.
7. Results At the highest pressure and currents (4 atm, 25}30 A), the electron density calculated from Saha equation is in good agreement with the electron density estimated from continuum emission (Table 3). Table 3 However, the lower pressure data seems to suggest that the arc is not even in PLTE. In the case of the low current-low pressure plasmas (15 A, 1 atm), there is considerable disagreement between the electron density calculated from continuum emission and that calculated from the Saha equation. The PLTE theory is still not su$cient to explain the discrepancies in the low-current, low-pressure plasmas, probably because the low-pressure plasmas are not even in partial Boltzmann equilibrium. We can conclude that the high-pressure, high-current plasmas are
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
623
Table 3 Continuum and Saha electron densities compared. Line spectra are "tted assuming ¹ and P are adjustable (PLTE) Pressure (atm)
Current (A)
Temperature (K)
P (atm)
Mole fraction of Ar
Electron density from continuum (10 cm\)
Saha electron density (10 cm\)
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
15 15 15 15 20 20 20 20 25 25 25 25 30 30 30 30
10000 10000 10000 9900 10500 10400 10400 10400 10900 10800 10800 10700 11300 11000 10800 10900
0.08 0.18 0.27 0.68 0.09 0.21 0.35 0.69 0.11 0.23 0.37 0.77 0.11 0.28 0.43 0.84
0.08 0.09 0.09 0.17 0.09 0.11 0.12 0.17 0.11 0.11 0.12 0.19 0.11 0.14 0.14 0.21
2.69 2.21 2.17 2.40 3.18 2.63 2.55 3.06 3.28 3.30 3.22 3.76 3.58 3.90 3.87 4.63
0.57 0.82 1.01 1.47 0.91 1.29 1.90 2.36 1.38 1.88 2.39 3.24 1.93 2.46 2.62 4.04
in PLTE, but the atmospheric pressure, low current arcs are not in PLTE. As a result, the temperatures listed in Table 3 must be regarded as estimates except for the 4 atm, 25}30 A plasmas. 7.1. Calculation of new transition probabilities Having established that the 4 atm, 30 A is in PLTE, we can use the values of P and ¹ to determine new transition probabilities for 70 of the strongest transitions listed in Table 2. Putting in values for the constants in the line strength calculation at each line peak (Eq. (17)) leads to the formula
W K !c E P Ag . . exp R "99.67 J m cm\ sr atm s ¹ w¹
(26)
Since the 4 atm, 30 A plasma is closest to equilibrium, the temperature estimated from this plasma is deemed the most accurate. Based on this temperature, and since R is measured, new A values are calculated in one step from Eq. (26) and appear in Table 4. In many cases, considerable adjustment of the A values occurs, but this does not have much e!ect on the measured temperatures of the plasmas. Using the new A values and P from Table 3 we have re-calculated ¹ for the various plasmas using a non-linear regression package in STATGRAPHICS (Manugistics) using just the peak intensities of these 70 lines. We see that the temperature calculated from the non-linear
624
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Table 4 Modi"ed transition probabilities Center (cm\)
A (s\)
A (s\)
A /A
Center (cm\)
A (s\)
A (s\)
A /A
3190.0 3354.3 3471.9 3492.1 3506.2 4188.9 4763.6 4848.9 5424.0 5580.2 5580.2 5900.9 6039.0 6051.1 6082.5 6249.3 6286.9 6489.4 6510.6 6533.2 6585.7 6642.9 6803.6 6822.9 6830.5 6847.1 7011.2 7091.9 7185.7 7227.6 7227.6 7227.6 7286.4 7307.2 7337.4 7350.2 7361.5
2.54E#06 2.73E#06 4.08E#06 4.00E#06 3.45E#06 1.60E#06 3.41E#05 5.45E#05 1.31E#06 8.52E#05 8.27E#05 1.85E#06 2.38E#06 1.86E#05 5.29E#06 1.07E#06 8.36E#06 1.72E#06 1.76E#06 7.08E#06 1.30E#06 4.70E#06 2.00E#06 7.36E#06 9.98E#06 9.55E#06 9.35E#06 5.23E#06 1.27E#07 3.50E#06 2.68E#06 1.02E#06 1.37E#07 1.08E#07 1.04E#07 1.51E#06 5.87E#06
4.78E#06 3.74E#06 7.69E#06 5.81E#06 4.27E#06 2.39E#06 7.78E#05 1.06E#06 5.61E#06 2.04E#06 6.01E#06 4.24E#06 4.66E#06 5.36E#05 1.03E#07 3.99E#06 1.51E#07 2.47E#06 3.96E#06 1.15E#07 2.74E#06 8.92E#06 3.60E#06 1.36E#07 1.57E#07 1.50E#07 1.86E#07 9.66E#06 1.77E#07 5.74E#06 9.99E#06 7.77E#06 1.10E#07 1.30E#07 1.27E#07 2.67E#06 1.01E#07
0.53 0.73 0.53 0.69 0.81 0.67 0.44 0.52 0.23 0.42 0.14 0.44 0.51 0.35 0.51 0.27 0.55 0.70 0.45 0.61 0.48 0.53 0.55 0.54 0.64 0.64 0.50 0.54 0.72 0.61 0.27 0.13 1.25 0.83 0.82 0.56 0.58
7379.4 7401.4 7455.0 7477.5 7507.8 7530.8 7555.7 7555.7 7565.2 7681.6 7715.0 7726.9 7807.3 7840.5 7848.2 7868.6 8002.0 8022.4 8035.9 8058.4 8088.1 8097.4 8228.0 8233.4 8252.2 8367.9 8367.9 8518.5 8529.0 8565.9 8700.5 8716.2 8733.6 9021.9 9185.9 9361.3
5.39E#05 1.34E#07 1.65E#07 1.17E#07 1.28E#07 1.39E#07 4.94E#06 2.22E#06 1.08E#07 1.04E#07 1.03E#07 1.26E#07 4.47E#06 2.43E#06 2.16E#06 5.74E#06 1.08E#07 9.91E#06 8.82E#06 7.41E#06 1.19E#07 1.01E#06 9.11E#06 2.03E#06 2.41E#06 9.79E#06 9.17E#06 1.14E#07 1.12E#06 4.48E#06 1.89E#05 4.74E#05 1.90E#06 2.43E#06 5.39E#05 4.88E#06
6.83E#06 1.32E#07 1.97E#07 1.39E#07 1.65E#07 1.51E#07 1.78E#07 7.02E#06 9.34E#06 1.52E#07 9.70E#06 1.09E#07 6.98E#06 3.88E#06 1.26E#06 1.03E#07 1.51E#07 1.15E#07 5.67E#06 1.38E#07 9.18E#06 2.27E#06 6.00E#06 4.46E#06 3.81E#06 1.17E#07 1.95E#07 1.14E#07 6.65E#05 6.79E#06 4.11E#05 7.29E#05 3.86E#06 1.73E#06 9.92E#05 1.10E#07
0.08 1.02 0.84 0.84 0.77 0.92 0.28 0.32 1.15 0.69 1.06 1.15 0.64 0.63 1.72 0.56 0.71 0.87 1.56 0.54 1.30 0.45 1.52 0.46 0.63 0.84 0.47 1.01 1.68 0.66 0.46 0.65 0.49 1.40 0.54 0.44
regression analysis (Table 5) is very similar to the temperature calculated from the least squared analysis performed on all the 153 lines by varying P and ¹. The di!erences between the least-square temperature (¹ ) and the non-linear "t (¹ ) are, in fact, small even at low UQOP UJGL pressures and currents.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
625
Table 5 Comparing plasma temperatures obtained from least-squares analysis (¹ ) and non-linear regression analysis U (¹ ) with their 95% con"dence limits. The least-squares analysis uses 153 lines and literature A values (Table 3), U while the non-linear "t uses 70 lines and recalculated A values (Table 4) Pressure (atm)
Current (A)
Mole fraction
P (atm)
¹ (K) U
¹ (K) U
4 3 2 1 1 1 2
30 30 30 30 25 20 15
0.21 0.14 0.14 0.11 0.11 0.09 0.09
0.84 0.43 0.28 0.11 0.11 0.09 0.18
10900 10800 11000 11300 10900 10500 10000
10962$31 11069$28 11149$27 11472$26 11082$22 10689$20 10178$18
7.2. Analysis of new transition probabilities By comparing calculated line intensities using the new and old A values, we can determine how accurate these new A values are. Figs. 8, 9, and 10 are residual plots (R !R ) as
a function of the transition frequency using the old and new A values; their corresponding plasma temperatures come from the non-linear regression analysis. The residuals calculated with the new A values at 3 atm, 30 A are packed tightly around zero (Fig. 8(a)), while that is clearly not the case for the old A values (Fig. 8(b)). The new A values (Fig. 9(a)) predict intensities better than the old A values (Fig. 9(b)) for the 1 atm, 30 A plasma, but tend to have more negative deviations than positive deviations. In the lowest current plasma, however (Fig. 10), the residuals calculated from the new and the old A values deviate signi"cantly from zero (note change of scale in Fig. 10). This trend among Figs. 8}10 can be explained by the fact that the new A values are calculated assuming PLTE for the 4 atm, 30 A plasma, but the lower pressure and lower current plasmas are experiencing deviations from PLTE. Therefore, it predicts line intensities accurately for other plasmas at PLTE (like the 3 atm, 30 A plasma), but it does not do as well in predicting line intensities for low currents. This observation reinforces the conclusions made from the previous analysis that the high-pressure, high-current plasmas are close to PLTE, while the low-pressure low-current plasmas are not in PLTE. Closer examination of the Fig. 8(a) leads to some interesting observations. The new A values predict intensities for low-frequency transitions (3000}6000 cm\) better than higher frequency (6000}10,000 cm\). The low-frequency transitions occur between the highest excited states of Ar, while the higher frequency transitions occur between higher excited states and the "rst or second excited state. PLTE assumes that all the excited states are in equilibrium with each other, but Fig. 8(a) indicates that the highest excited states are more closely in equilibrium with each other, compared to the lower excited states ("rst and second). 7.3. Trends in n and ¹ We have now determined the electron density and the temperature of the plasma and have also established some primary trends in the data. Fig. 11 identi"es the trends in the electron densities
626
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Fig. 8. Residual plot (R !R ) vs. wavenumber for a 3 atm, 30 A plasma: (a) modi"ed A values
(¹"11,069 K), (b) original A values (¹"11,087 K). A values appear in Table 4.
and temperatures as a function of arc operating conditions. Remember the plasmas with lower currents are not in PLTE, and thus the temperature estimates at these conditions are somewhat suspect. The temperature at a given current remains almost constant over a range of pressures except when the arc is at 30 A (Fig. 11(a)). In contrast, the temperature is a strong function of current (Fig. 11(b)). The electron density on the other hand seems to depend both on the current and the pressure (Fig. 11(c), (d)). To help in visualizing the data better, we "t n and ¹ to an empirical polynomial model to generate contour plots of n and ¹ as a function of the input parameters pressure (P) and current (I)
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
627
Fig. 9. Residual plot (R !R ) vs. wavenumber for a 1 atm, 30 A plasma: (a) modi"ed A values
(¹"11,468 K), (b) original A values (¹"11,493 K). A values appear in Table 4.
(Fig. 12). From Fig. 12(a) we can see that at temperatures below 10,500 K, the temperature of the plasma is mostly independent of pressure. To reach higher temperatures (around 11,000 K) at higher pressures, however, much more current has to be #owing through the plasma. This observation can be explained by the fact that at higher pressures, the plasma is denser, and hence more energy is required to heat it. Please remember that the temperatures have error bars around 100 K, and the statistical model does not take these measurement errors into account. The plots serve mostly to visualize trends in the data. The contour plots for n (Fig. 12(b)) suggest that at low (1 atm) and high (4 atm) pressures, less current is needed to achieve the same electron density (which mostly a!ect the continuum emission
628
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Fig. 10. Residual plot (R !R ) vs. wavenumber for a 2 atm, 15 A plasma: (a) modi"ed A values
(¹"10,178 K), (b) original A values (¹"10,223 K). A values appear in Table 4.
(Eqs. (19)}(23)). At low pressures, there are fewer neutral Ar and hence less energy is needed for ionization to occur. At very high pressure, however, the free paths of the various species in the plasma are smaller and hence there is a greater probability of the collisional processes that lead to ionization. The plots also indicate that at higher currents, a 4 atm plasma will have a higher electron density than a 1 atm plasma at the same current. This makes sense, as highpressure plasmas have more neutral Ar present and hence there is higher rate of ionization.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
629
Fig. 11. Electron densities and temperatures as a function of the di!erent operating pressures and currents.
Fig. 12. Contour plots of temperatures (a) and electron densities (b) plotted as a function of pressure and current.
630
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
8. Conclusions A cascade arc was built and operated at di!erent currents (15, 20, 25, and 30 A) and pressures (1, 2, 3, and 4 atm). The infrared emission (2000}10,000 cm\) of the arc was measured using a FTIR spectrometer. The path length of arc emission (1.6 cm) is di!erent from the actual geometric length (3.6 cm) of the arc, and this is established using ray tracing and radiative exchange theory (see the appendix). The line emission of the arc was used to calculate the temperature (¹) of the plasma while the electron density (n ) was calculated from the continuum emission. The electron density obtained from the continuum emission was compared to the electron density calculated from Saha equation to test the validity of the assumption of partial local thermodynamic equilibrium (PLTE). The Saha electron densities calculated using the PLTE temperature compares well with the electron densities obtained from continuum emission calculations for the high-pressure, high-current arcs (30 A, 3}4 atm). The Saha electron density and the continuum electron density are quite di!erent in the case of low current, lowpressure plasmas. These observations lead to the conclusion that the low-current, low-pressure plasmas are not even in PLTE, while the high-pressure, high-current arcs are at PLTE. Under the given range of operating conditions (4 atm5P51 atm; 30 A5I515 A) the plasma temperature varies between 9900 and 11,500 K, while the electron density varies between 2 and 5;10 m\. The line widths can be measured by "tting a Lorentzian line shape to each measured emission line. We observe that line widths vary with operating conditions. The A values used to simulate line spectra [29}31] have 30% uncertainties associated with them. New A values have been recalculated to "t the data better. Most of the A values change less than 30% while a few of them have to be changed by 30}50%. Changing the A values result in much better predictions of line intensities for plasmas close to PLTE. Because of the strong emission in the near IR, the total power output from the arc is 3 times that of the infrared source used by the FTIR (globar). As a matter of fact, without an attenuator the broadband light from the arc at the highest pressure and currents saturates the DTGS detector used in the measurements. At present, continuum intensity of the arc between 1500 and 6000 cm\ is similar to the globar. Higher currents would increase the continuum level of the arc and signi"cantly increase the intensity in the mid-IR.
Acknowledgements The authors gratefully acknowledge the "nancial support of the Defense University Research Instrumentation Program (Grant DAAH04-95-0060) and the National Science Foundation (CTS96-25095). The authors would also like to acknowledge technical discussions with Richard van de Sanden (Eindhoven University of Technology) as well as Tianming Bao, Cli! Hayman, Wayne Schmidt and Ram Wusirka for helping in the design and construction of the cascade arc. We would also like to thank Prof. Robert V. Edwards for his help with the statistical data analysis.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
631
Appendix. Derivation of e4ective path length Consider the amount of light collected by a parabloidal mirror emitted from an arc channel that is 3.6 cm long. Only a fraction of the light coming from very close to the optical axis of the collection mirror would have traveled the full distance. To calculate the e!ective path length (l ), we make the following assumptions: 1. The parabloid mirror used for collection has a 5 cm focal length and the mirror is positioned 50 mm away from the end of the cascade arc channel (Fig. 13). 2. The detector and its mirror limit the throughput of the optics (not the arc). The detector element used is 1 mm in diameter, and the mirror in front of the detector has a focal length of 16 mm. This leads to a magni"cation ratio between the arc and the detector of 50/16"2.66; therefore, a spot 1 mm in diameter at the detector is actually 2.66 mm in diameter at the cascade arc. To calculate the maximum solid angle collected from the arc (2(1!cos )), we note that the
throughput at the detector is equal to the throughput from the arc. Therefore, 2(1!cos )A "2(1!cos )A , where A and A are the areas of the detector element
and the arc, respectively. The ratio of these areas corresponds to the square of the ratio of the magni"cation factor. The collection angle of the optics at the detector ( ) is 0.588 rad, and hence we can calculate "0.22 rad.
3. To make the calculations easier, we treat the light coming out from the arc as two distinct types of rays depending on a critical angle (Fig. 13(b)). The light rays which have an angle less than
will have a path length given by ¸/cos (¸"36 mm). Any light rays which have an angle higher than the critical angle will have a path length signi"cantly less than the arc length: R/sin (R"4 mm). From Fig. 13(b), "tan\(2/36)"0.055 rad.
Fig. 13. (a) optics for collection of light from the arc, (b) critical angle determination.
632
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
4. Since the diameter of the arc is much smaller than the diameter of the collection optics, the arc is treated as an axial source, i.e. only the rays passing through the optical axis are considered. O!-axis radiation is assumed to depend on the angles the same way as the on-axis radiation. This simpli"es the calculation tremendously. Using the above assumptions and radiative exchange theory [25], we know that the radiance R is given by R"[1!e\?l]BB,
(A.1)
where is the absorption coe$cient and l is the distance the radiation has traveled. In the case of the arc optics, emission intensity is a function of the path length, which in turn is a function of the angle . Therefore,
¸ R( )" 1!exp ! cos
and
R R( )" 1!exp ! sin
BB when 0( (
BB.
Integrating over all solid angles gives the energy incident, E, on the collection optic. Therefore,
F
R( )2 sin d , where d"2 sin d assuming axial symmetry. Entering the two expressions for R( ) results in E" R( ) d"
E"
F
¸ 1!exp ! cos
BB2 sin d #
F
R 1!exp ! sin
F Using small argument approximation e\?6 +1! X and integrating
BB2 sin d .
E"2BB[! ¸ ln cos # R( ! )].
The e!ective path length can be calculated by comparing the E from the above equation to a value of E calculated assuming the path length l is independent of (l"¸"3.6 cm). We call this value of energy E : F E " [1!e\?*]BB2 sin d . Using the small argument approximation and integrating once again,
E "2BB ¸(1!cos ).
The e!ective path length can be calculated from a ratio of the two energies: 2BB[! ¸ ln cos # R( ! )]
" 2BB ¸(1!cos )
!ln cos #(R/¸)( ! ) K?V . " 1!cos
l " E ¸
E
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
633
From the cascade arc geometry (¸"3.6 cm, R"0.2 cm, "0.22 rad and "0.055 rad),
l "0.443 ¸Nl "1.6 cm. References [1] Morrison Jr PW, Taweechokesupsin O, Kovach CS, Roozbehani B, Angus JC. In situ infrared measurements during hot "lament CVD of diamond in a rotating substrate reactor. Diam Related Mater 1995;5:242}6. [2] Taweechokesupsin O. In situ gas phase measurements using infrared spectroscopy during hot "lament chemical vapor deposition of diamond. MS thesis, Case Western Reserve University, Cleveland, OH, 1996. [3] Szeto E. Analysis of the oxyacetylene #ame under diamond deposition conditions using Fourier Transform Infrared Spectroscopy. MS thesis, Case Western Reserve University, Cleveland, 1998. [4] Wilbers ATM, Kroesen GMW, Timmermans CJ, Schram DC. Characteristic quantities of a cascade arc used as a light source for spectroscopic techniques. Meas Sci Technol 1990;1:1326}32. [5] Haverlag M, Kroesen GMW, de Hoog FJ. In situ measurement of C F densities in an RF CF plasma using infrared absorption spectroscopy. In: D'Agostini R, editor. Proceedings of the 9th International Symposium on Plasma Chemistry. University of Bari, Bari, 1989. p. 441}4. [6] Griem HR. Plasma Spectroscopy. New York: McGraw-Hill, 1964. [7] Kroesen GMW, Schram DC, de Haas JCM. Description of a #owing cascade arc plasma. Plasma Chem Plasma Proc 1989;10:531}51. [8] Qing Z, de Graaf MJ, van de Sanden MCM, Otorbaev DK, Schram DC. Experimental characterization of a hydrogen/argon cascaded arc plasma source. Rev Sci Instr 1994;65:1469}71. [9] Wilbers ATM, Kroesen GMW, Timmermans CJ, Schram DC. The continuum emission of an arc plasma. JQSRT 1991;45:1}10. [10] Meeusen GJ, Ershov-Pavlov EA, Meulenbroeks RFG, van de Sanden MCM, Schram DC. Emission spectroscopy on a supersonically expanding argon/silane plasma. J Appl Phys 1992;71:4156}63. [11] Meulenbroeks RFG, van Beek AJ, van Helvoort AJG, van de Sanden MCM, Schram DC. Argon}hydrogen plasma jet investigated by active and passive spectroscopic means. Phys Rev E 1994;49:4397}406. [12] Meulenbroeks RFG, van der Heijden PAA, van de Sanden MCM, Schram DC. Fabry}Perot line shape analysis on an expanding cascaded arc plasma in argon. J Appl Phys 1994;75:2775}80. [13] Meulenbroeks RFG, Engeln RAH, Beurskens MNA, Pa!en RMJ, van de Sanden MCM, van der Mullen JAM, Schram DC. The argon}hydrogen expanding plasma: model and experiments. Plasma Sour Sci Technol 1995;4:74}85. [14] Buuron AJM, Otorbaev DK, van de Sanden MCM, Schram DC. A new absorption spectroscopy setup for the sensitive monitoring of atomic and molecular densities. Rev Sci Instr 1995;66:968}74. [15] Dahiya RP, de Graaf MJ, Severens RJ, Swelsen H, van de Sanden MCM, Schram DC. Dissociative recombination in cascaded arc generated Ar}N and N expanding plasma. Phys Plasmas 1994;1:2086}95. [16] Brunger M, Kock M. Electron density determination in an arc plasma by laser interferometry. Z Naturforsch 1975;30a:1560}2. [17] Baessler P, Kock M. An interferometric and spectroscopic study on a high-current argon arc. J Phys B: At Mol Phys 1980;13:1351}61. [18] Schnehage SE, Kock M, Schulz-Gulde E. The continuous emission of an argon arc. J Phys B: At Mol Phys 1982;15:1131}5. [19] Stuck D, Wende B. Photometric comparison between two calculable vacuum-ultraviolet standard radiation sources: Synchrotron radiation and plasma-blackbody radiation. J Opt Soc Am 1972;62:96}100. [20] Kaase H, Stephan HK. Calibration of radiation detectors in the VUV using a wall-stabilized Ar-cascade arc. Appl Opt 1979;18:2275}9. [21] Key PJ, Preston RC. Vacuum ultraviolet radiation scales: an accurate comparison between plasma blackbody lines and synchrotron radiation. Appl Opt 1977;16:2477}85. [22] Frost RM, Awakowicz P. Continuous calibration of a vacuum ultraviolet system from 65 to 125 nm by a cascade arc and comparison with the calibrated line radiation of a hollow cathode. Appl Opt 1997; 36:1994}2000.
634
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
[23] Kroesen GMW, van de Sande MJF, Wilbers ATM, Schram DC, Bisschops LA, Timmermans CJ, Heller HJ, Husken ABM. The implementation of a cascade arc. Technical Report VDF/NT 90-06. Eindhoven University of Technology, 1990. [24] Morrison Jr. PW, Haigis JR. In situ infrared measurements of "lm and gas properties during the plasma deposition of amorphous hydrogenated silicon. J Vac Sci Technol A 1993;11:490}502. [25] Siegel R, Howell JR. Thermal radiation heat transfer. Washington: Hemisphere Publishing Corporation, 1981. [26] Timmermans CJ, Rosado RJ, Schram DC. An investigation of Non-Equilibrium e!ects in thermal argon plasmas. Z Naturforsch 1985;40a:810}25. [27] Rosado RJ. An investigation of non-equilibrium e!ects in thermal argon plasmas. PhD thesis, Eindhoven University of Technology, Eindhoven, 1981. [28] Rothman LS, Rinsland CP, Goldman A, Massie ST, Edwards DP, Flaud JM et al. The HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 Edition JQSRT 1998;60:665}710. [29] Smith PL, Heise C, Esmond JR, Kurucz RL. Atomic spectral line database. CD-ROM 23 of R.L. Kurucz (http://cfa-www.harvard.edu/amdata/ampdata/kurucz23/sekur.html). Harvard Smithsonian Center for Astrophysics, 1995. [30] Wiese WL, Brault JW, Danzmann K, Helbig V, Kock M. Uni"ed set of atomic transition probabilities for neutral argon. Phys Rev A 1989;39:2461}71. [31] Wiese WL. Spectroscopic diagnostics of low temperature plasmas: techniques and required data. Spectrochim Acta 1991;46B:831}41. [32] de Regt JM, van Dijk J, van der Mullen JAM, Schram DC. Components of continuum radiation in an inductively coupled plasma. J Phys D 1995;28:40}6. [33] Devoto RS. Transport coe$cients of ionized argon. Phys Fluids B 1973;16:616}23. [34] Menart J, Heberlein J, Pfender E. Line-by-line method of calculating emission coe$cients for thermal plasmas consisting of monatomic species. JQSRT 1996;56:377}98. [35] Drawin HW, Felenbok P. Data for plasmas in local thermodynamic equilibrium. Paris: Gauthier-Villars, 1965.
Infrared characterization of a cascaded arc plasma Ram Raghavan, Philip W. Morrison Jr.* Department of Chemical Engineering, School of Engineering, Case Western Reserve University, 10900 Euclid, Cleveland, OH 44106-7217, USA Received 8 April 2000; accepted 21 May 2000
Abstract We have constructed a cascaded arc light source and characterized its infrared emission properties. Breaking down Ar gas #owing through the cascaded arc channel forms a wall-stabilized plasma. The properties of the plasma can be adjusted by varying the pressure (1}4 atm) and the current #owing through the arc (15}30 A). The infrared (2000}10,000 cm\) emission of the cascade arc plasma is measured using a Fourier transform infrared (FTIR) spectrometer. The infrared emission can be used to determine the electron density (n ) and the temperature (¹) of the plasma, assuming the plasma is in partial local thermodynamic equilibrium (PLTE). The plasmas at the highest currents and pressures satisfy the assumption of PLTE, while the plasmas at other conditions do not. This enables the calculation of new transition probabilities for the infrared transitions in the Ar plasma. The electron density of the cascaded arc Ar plasma at our operating conditions varies between 2 and 5;10 m\ while the temperature varies between 9500 and 11,500 K. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Cascade arc; Argon plasma; Fourier transform infrared spectroscopy (FTIR); Transition probabilities; Partial local thermodynamic equilibrium (PLTE)
1. Introduction In situ diagnostics using a Fourier transform infrared (FTIR) spectrometer are a way to quantify concentrations and temperatures of gas-phase species commonly associated with chemical vapor deposition (CVD) processes. For example, our research group has demonstrated that in situ FTIR can be used to measure temperatures and concentrations of CH and C H in hot "lament CVD
* Corresponding author. Tel.: #1-216-368-4238; fax: #1-216-368-3016. E-mail address: [email protected] (P. Morrison). 0022-4073/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 1 0 4 - 7
606
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
(HFCVD) reactors [1,2] as well as oxyacetylene torches for diamond growth [3]. The technique has been limited, though, due to the lack of a bright infrared light source. This leads to relatively low signal-to-noise ratios, which are inadequate for detecting low concentrations of transient radicals. Further, the current infrared light source used by the FTIR, commonly known as a globar, has a large aperture. This leads to a relatively large spot size (&10 mm) when the light is focused in the sample zone and thus makes it di$cult to perform accurate localized measurements. The motivation behind the current research is to develop a better light source for the FTIR spectrometer that is not only is very intense, but also has a very small aperture. The cascade arc satis"es these requirements. The cascade arc used for our measurements is a low-#ow (30 sccm) argon plasma. The plasma, when at equilibrium, has been shown to get as hot as 13,500 K [4]. As a result, the intensity of infrared emission from the cascade arc can be a factor of 20}200 higher than conventional globar at 1300 K [5]. Also, the aperture of the arc is 3 times smaller than the globar, which leads to better spatial resolution. There are various radiative processes which lead to two types of arc emission in the infrared: line and continuum. Line emission is a quantum phenomenon occurring when an electron falls from a higher excited state to a lower excited state; the di!erence in energy between the states is released as a photon. This results in a narrow `linea in the emission spectrum. This narrow line will be broadened, mainly due to interaction of the emitters with the surrounding electrons (Stark broadening) and atoms (resonance broadening) [6]. Continuum emission is a result of inelastic scattering of electrons in the presence of atoms and ions. When electrons decelerate in the vicinity of these atoms and ions, there is photon emission at all wavelengths. An advantage of the cascade arc is that it is a continuum emitter over a very large spectral range. Various authors (many from the Eindhoven University of Technology) have characterized the electrical and visible emission properties of the cascade arc. Kroesen et al. [7] describe the various processes occurring in the plasma and also set up transport equations to describe the system. They then solve these equations to obtain the electron temperature of the plasma using the continuity, momentum, and energy equations for each species in the plasma. Qing et al. [8] have characterized the e$ciency of an Ar/H plasma by calculating a simple power balance involving electric and heat #uxes. The e$ciency is just a ratio of the power dissipated by the arc to the input electric power. Increasing the Ar #ow through the arc increases the arc e$ciency but results in lowering the arc temperature. Adding hydrogen to the Ar also decreases the e$ciency of the plasma. Wilbers et al. [4,9] use the cascade arc as a visible light source for spectroscopic techniques but caution that instabilities can occur at high/pressures and currents. The operating pressure of their arc varies between 1 and 6 atm and the current between 20 and 70 A. The absolute intensities are calculated with respect to a tungsten ribbon lamp at 2667 K. These authors measure the intensity of their cascade arc at 2 bar to be an order of magnitude higher than a globar at 5000 cm\. Neutral particle densities and temperatures in the cascade arc plasma have also been calculated using optical absorption and emission spectroscopy [10}15]. Kock and his collaborators [16}18] use optical laser interferometry to measure electron densities, transition probabilities, and continuum emission coe$cients in these plasmas. The electron densities measured by Kock and his collaborators are very similar to those obtained by the researchers at the Eindhoven University of Technology. Although there is a wealth of literature about the cascade arc in the visible and ultraviolet regions [19}22], there is no detailed IR characterization of the arc. In this work, we have measured
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
607
the IR emission of the arc using a Fourier transform infrared (FTIR) spectrometer. The line and continuum emission of the Ar plasma will be used to determine the temperature and electron density of the plasma.
2. Apparatus A schematic of the cascade arc is shown in Fig. 1. The arc consists of water-cooled copper plates cascaded together, but insulated from each other, with a 4 mm hole machined through them. This con"guration is a classic design of a wall-stabilized arc [23]. There are three thoriated tungsten electrodes that serve as cathodes and a grounded end plate that serves as the anode. When the Ar gas #owing through the channel breaks down due to current #ow, a plasma results. The arc can be run under a variety of conditions by adjusting the argon #ow rate, the current #owing through the arc, and the pressure. In our system the #ow rate is kept constant at 30 sccm, the current of the arc is varied between 15 and 30 A, and the pressure is varied between 1 and 4 atm. As seen from Fig. 1, the plates have cooling water channels machined inside them and are insulated from one another using a boron nitride spacer held in place with polycarbonate disk and an O-ring. The boron nitride spacer, is 27 mm in diameter and 3.2 mm thick, and a C22 Viton O-ring provides the sealing surface. The polycarbonate disc has the same OD as the copper plate (70 mm) and an ID of 33.3 mm. The plates and their spacers are then sandwiched between the cathode plate and the anode plate. The cathode plate contains three symmetric 2% thoriated tungsten tips that are 1 mm in diameter and 10 mm long. Del Seal sections are welded permanently to the cathode and anode plate to facilitate the #ow of gases in and out of the system.
Fig. 1. Design of cascade arc. This design is derived from the work of Kroesen et al. [23].
608
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
The downstream end of the arc (anode plate end) has "ttings to attach a pressure gauge (Matheson 63-3112) with a range of 0}100 psi, a mass #ow controller (MKS Instruments 1259 B) with a range of 500 sccm, and a vacuum pump (GE Motors HP). The arc is water cooled by a manifold that consists of two branches, one branch for the tip holders and another branch for the plates. The total water #ow rate is 4 l/min. Finally, the ends of the arc are sealed o! using infrared transparent sapphire windows (with an IR cut-o! at 1500 cm\). The arc is then mounted on a custom built frame that allows easy handling of the arc itself. When the arc starts up under vacuum, the gas is very hot, and there are micron size particles of debris which strike the window if unprotected. In addition, the thermal gradients at the window during startup are severe enough to crack the window. To protect the window, a steel disk is placed on the inside surface of the window; a magnet holds the steel disk in place during arc start up. Once steady-state operating conditions are reached, the magnet is removed from the outside, and the steel disk slips out of the way. The optical components used to interface the arc to the spectrometer are shown in Fig. 2. The light from the arc is "rst collected by a 63 mm diameter, 50 mm focal length 903 parabolic mirror (Melles Griot). The mirror is mounted on a gimbal (Edmund Scienti"c) placed on a 60 cm long optical rail using a standard narrow optical carrier (Oriel Instruments). Although gold is the coating of choice in infrared re#ection optics due to its high re#ectivity, the mirror is rubidium coated which has a comparable re#ectivity. The collection mirror collimates the IR radiation, producing 50 mm diameter light beam that enters the emission port of a Bomem MB 157 FTIR spectrometer. The beamsplitter is only 25 mm in diameter, and hence only the central 25 mm of the collimated 50 mm beam is collected by the FTIR. The modulated signal then hits a 127 mm focal length mirror that focuses the light through the sample compartment of the FTIR, and a similar mirror in the detector optics collects the light. The etendue (optical throughput) of the collection optics is controlled by the detector size and its mirror and not by the arc size and its mirror (see the appendix). The detector used in the setup is an extended range DTGS (deuterated triglyciene sul"de) detector that is 1 mm in size. The Bomem MB 157 spectrometer has a maximum resolution of 2 cm\ when using the extended range DTGS detector. At this resolution, the instrument is capable of scanning at 12 scans/min. The experiments are all run at 25 scans at 2 cm\ resolution. Because of the arc's high intensity in the near IR, we use an attenuator in the
Fig. 2. Collection optics for infrared measurements of the arc.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
609
beam path to keep the centerburst of the spectrometer on scale and to prevent the DTGS detector from becoming non-linear. To convert the raw emission spectra to path-corrected radiance spectra, we calibrate the FTIR system with a 1300 K blackbody source [24]. For maximum sensitivity above 6500 cm\, we perform the blackbody calibration without the attenuator and then mathematically correct the arc emission spectra using the transmittance of the attenuator and the blackbody calibration. The circuit used to operate the arc consists of two power supplies. The "rst is a 1000 V, 2 A power supply (Advanced Energy MDX 2500 W Magnetron Drive) and the other is a 300 V, 30 A power supply (Sorenson DCR 300-33T). The 1000 V power supply is needed to ignite the plasma, since even at low pressures (20 Torr), signi"cant electric "elds (&1000 V) are required to breakdown argon. Once a discharge is created, however, large currents are necessary to sustain the arc. Using a circuit design similar to Kroesen et al. [23], a set of charged capacitors deliver a transient current as the circuit switches from the ignition supply to the high-current supply. If the capacitors were not present, a continuous arc will never form, as the breakdown caused by the high-voltage power supply would not last long enough for the high-current power supply to sustain it. The values of the capacitance are chosen in such a way that the RC time constant matches the response time of the high-current power supply (&13.5 ms). More details are available in a technical report from the Eindhoven University of Technology [23]. The circuit also contains high power, 1 resistors in series with the arc tips to ensure that there is always a positive resistance present because these plasma discharges have extremely non-linear curves and can have a negative `resistancea at times (d
610
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
The current can be changed while the arc is running. Care must be taken to adjust the current gradually, since rapid changes in current can shut down the arc. The pressure inside the arc chamber can also be increased gradually using the regulator of the compressed Ar tank feeding the arc but the current in the arc must remain fairly constant while increasing the pressure. The plasma has been operated at 16 di!erent conditions using pressures of 1, 2, 3, and 4 atm and currents of 15, 20, 25, and 30 A.
4. Partial local thermodynamic equilibrium (PLTE) in plasmas A typical emission spectrum from the arc showing the line and the continuum emission appears in Fig. 3. From Fig. 3, it is clear that the continuum intensity of the arc is roughly uniform at all wavelengths. At lower wavelengths (around 3000 cm\) the continuum intensity from the arc is similar to the emission of a globar (&1300 K blackbody), but is substantially brighter than the globar at higher frequencies ('6000 cm\). The spectra also contain very large line emissions due to the excited electronic states of Ar. Employing radiative exchange theory [25] and the observed line emission, we will estimate the plasma temperature (¹); the continuum emission will be used to estimate the electron density (n ). Radiative exchange theory provides a relationship between the radiance (or emission) and the temperature of a hot gas: R()"[1!()]BB(¹,)["]
W , m sr cm\
(1)
where R is the radiance (which depends on wavenumber (cm\)), is the transmittance, and BB is the Planck function (which depends on temperature and wavenumber). When we invoke radiative exchange theory, we are in e!ect assuming that the plasma is in local thermodynamic equilibrium (LTE) because we are using a local temperature (¹) to describe the system. However, we need to check whether LTE can be applied to our IR spectra. In fact, Timmermans et al. [26] have shown that plasmas similar to ours are not in LTE but in partial local thermodynamic equilibrium (PLTE). PLTE occurs when collisional processes establish a Boltzmann equilibrium among the excited states but are not su$cient to establish a Boltzmann equilibrium between the
Fig. 3. Radiance spectrum of an arc operating at 4 atm and 30 A. The largest peaks between 6000 and 9000 cm\ exceed 100 W/(m cm\ sr).
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
611
ground state and the excited states. Nevertheless, radiative exchange theory is still applicable, as shown below. A detailed balance for the various Ar states provides insight into how PLTE occurs. Consider the di!erent mechanisms populating/de-populating the "rst excited state of Ar for a simpli"ed three-level energy diagram. Referring to Fig. 4 and Table 1, an unsteady population balance on the "rst excited state results in dn " k n n # k n n #k n n n> #A n #k n>n > dt > > #JCARPML #JCARPML N?PRGAJC 0?BG?RGTC 0?BG?RGML CVAGR?RGML BCUCVAGR?RGML PCAMK@GL?RGML PCAMK@GL?RGML DPMK ?@MTC DPMK @CJMU DPMK ?@MTC #k n n #A n ). !( k n n # k n n > 'MLGX?RGML 1NMLR?LCMSQ #JCARPML #JCARPML #KGQQGML CVAGR?RGML RM LB BCUCVAGR?RGML CVAGRCB QR?RC RM EPMSLB QR?RC Similar equations will result for other excited states as well. Rosado's Ph.D. dissertation [27], gives values for the various collisional and radiative rates cited in Table 1. In the equation above, the three-particle recombination term (k n n n>) is observed to be very small compared to the other > terms in the detailed balance and can be ignored. From this detailed balance, and using the collisional and radiative rates, we can estimate relative populations of the various excited states. For the ground state, spontaneous emission is forbidden between the second excited state and the ground state, so the equivalent detailed balance becomes dn "k n n #k n n #A n #k n n>!n n (k #k #k ). > > > dt
Fig. 4. Simpli"ed energy level diagram for Ar.
(2)
612
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Table 1 Processes occurring in plasmas. Ar is the ith excited state; refers to a trapping coe$cient. See [27] for values of the G parameters Process
Example
Parameters
Electron impact excitation Electron impact de-excitation Electron impact ionization Three-particle recombination Spontaneous emission
Ar #ePAr #e Ar #eQAr #e Ar #ePAr>#2e Ar #eQAr>#2e Ar PAr #h
k , k , k (m s\) k , k , k (m s\) k ,k ,k (m s\) > > > k ,k ,k (m s\) > > > A , A (s\) , : Trapping coe$cients which depend on the optical depth of the plasma
Radiative recombination
Ar>#ePAr #h
k , k , k (m s\) > > > > > >
For estimation purposes, we can assume a Boltzmann distribution (n /n "g /g e\#H I2) to H H calculate relative populations (n ) of the various excited states at energies E with degeneracies g . H H H Observing that our atmospheric and super-atmospheric Ar plasmas have electron temperatures around 1 eV (&11,600 K) with electron densities between 10 and 10 m\, we can then calculate order of magnitude estimates of the relative contributions of all the terms in the balance. We "nd that none of the terms in the ground state detailed balance can be neglected, so at steady state: k n #k n #(n /n )A #k n> > > . (3) n + k #k #k > For the "rst excited state, we "nd that radiative recombination to excited states (k n>n ) is > > negligible [27], so the detailed balance becomes dn "k n n #k n n #A n !n n (k #k #k ) > dt !A n . At steady state
(4)
k n #k n #(n /n )A "k n #k n #k n > - - - - - - (5) #(n /n )A . - From the order of magnitude calculations for each term, the dominant terms in the balance are the collisional excitation/de-excitation between the "rst and second excited states. Neglecting the smaller terms n k + . n k
(6)
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
613
Since k and k are rate constants that are linked by detailed reversibility, they are also related by the Boltzmann equation when the processes are at an equilibrium temperature ¹: k /k "g /g e\# \# I2. The detailed balance, therefore, shows that the collisional processes alone have driven "rst and second excited states of Ar into near equilibrium at a temperature ¹. This argument can be extended to the higher excited states as well. For instance, the signi"cant terms in a steady state detailed balance of the second excited state will be k n #k n "k n #k n . (7) But k n +k n (Eq. (6)) and so k n + . (8) k n Thus the excited states are related by detailed collisional reversibility and are in Boltzmann equilibrium described by a temperature ¹. We can now see how the ground state is related to the excited states using the relationship between the "rst and the second excited states. Substituting Eq. (6) into the ground state density expression (Eq. (3)), and noting the fact that n +10\n in our 1 eV plasma, we obtain k #k #k > GFFHFFI -\
n + &0.33;10\. (9) n k #(k k /k ) #10k #(A /n ) \ > > -\ - -\ -\ Compare this value to a Boltzmann calculation using E "11.65 eV, g "12, and ¹"1 eV: g e\# \# I2"1.05;10\. g These expressions show that the ground state is not at collisional equilibrium with the "rst excited state since n /n is approximately three times smaller than what is expected from Boltzmann equilibrium. Since the excited states are in collisional equilibrium with each other, but the ground state is not in equilibrium with the excited states, the plasma is said to be in partial local thermodynamic equilibrium (PLTE) [26]. One of the reasons for this `overpopulationa of the ground state is the radiative recombination term (k ), which is selectively populating only the > > ground state. Timmermans et al. [26] have also observed a transport process that also contributes to PLTE. Because there is a temperature gradient across the plasma channel, ions form at the center of the plasma (where it is the hottest) and then di!use to the wall where they recombine with electrons to form neutrals. These neutrals then di!use back to the center of the plasma to become excited/ionized again. Since the "rst excited state of Ar is 11.65 eV and the plasma temperature is only around 1 eV, the excitation of ground state Ar is a very slow process. Hence, the ground state can become additionally `overpopulateda with respect to the excited states due to this transport
614
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Fig. 5. Schematic Boltzmann diagram for `spectroscopically activea argon. The symbol corresponds to the mole fraction of `spectroscopically activea Ar.
process. If we call this di!usion rate D (m\ s\) and take it into account, the relationship between the ground state and the "rst excited then becomes k #k #k n > + , k #(k k /k )#(A /n )#10k #(D/n ) n > >
(10)
which is even smaller than the right-hand side of Eq. (9). Thus, this inward di!usion also contributes to deviations from LTE and produces PLTE. To summarize, radiative exchange theory and the use of the Planck function can be applied to the excited states of these plasmas but not to the ground state. In other words, the relative populations n /n of the excited states j and k have a Boltzmann distribution but not n /n and H I H n /n . Fig. 5 illustrates the impact on interpreting the IR spectra. The upper line is a Boltzmann I plot assuming all the states are at equilibrium with each other and the ground state. The slope is inversely proportional to the plasma temperature. In PLTE, the `temperaturea of the excited states is same as the upper line because they have a Boltzmann distribution. Their number densities are reduced by some factor, however, because they are not in Boltzmann equilibrium with the ground state since the ground state is `overpopulateda. Thus, the points for the excited states in PLTE are shifted down but have the same slope as before. The dashed extrapolation to the ground state (Fig. 5) represents the mole fraction of `spectroscopically active Ara that is consistent with the populations of the excited states. Thus the Ar has two components: (a) `spectroscopically activea Ar for which we can determine a temperature using radiative exchange theory and (b) `spectroscopically inactivea Ar which is ground state Ar which is `in excessa of PLTE. As a result, we can apply radiative exchange theory as long as we treat the partial pressure of the spectroscopically active Ar (P ) as an unknown parameter.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
615
The above argument presumes that only the ground state is `overpopulateda relative to the excited states. Rosado [27] has shown that the lower excited states are also slightly `overpopulateda which would cause curvature in the PLTE line of Fig. 5. We expect this to be a minor problem in our situation, however, since the IR spectrum of the plasma mainly results from transitions between high lying electronic levels (mostly third excited state and above) where the `overpopulationa has decreased to zero. Thus for relatively cool (&1 eV) plasmas, spectroscopically measuring plasma temperatures in the IR may be a more robust method than in the visible.
5. Temperature estimation from line radiation From radiative exchange theory [25], we know that R()"(1!())BB(¹, ),
(11)
where R is the measured intensity of line radiation, is the transmittance of the line, and BB is the Planck function. The Planck function is given by [4] W c ["] , (12) BB(¹, )" m sr cm\ exp(c /¹)!1 where c "1.191;10\ W/m sr (cm\) and c "1.4388 cm K. The transmittance has the form "e\-",
(13)
where OD is the optical depth of the plasma. Since our plasma is optically thin (OD;1), we can simplify Eq. (11) using the small argument approximation, R+OD * BB.
(14)
Using the notation of Rothman et al. [28], we know that the optical depth has the form OD"S(¹)G(! )n l, (15) where n "P /R¹, S(¹) is the line strength, G(! ) is the line shape function, is the line center, l the path length, and n is the number density of `spectroscopically activea argon. The line strength for the line at is given by I Ag !c E !c cm\ exp 1!exp ["] S(¹)" (16) 8c ¹ ¹ mol cm\
where I is the isotopic abundance of the species, A (1/s) is the transition probability for , g is the degeneracy of the upper state, c (cm/s) is the speed of light, and E (cm\) is the energy of the upper state. There are a number of sources from which we obtain transition probabilities, degeneracies, and transition energies for the observed lines [29}31]. The observed lines and their associated parameters (A, g , E , etc.) are seen in Table 2. Table 2 contains a total of 153 lines for Ar; there are no observable Ar> lines in our plasma. Please note that most of the A values are theoretical estimates with uncertainties of 30%.
616
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Table 2 Emission parameters for observed IR lines from Smith et al. [29] Transition (cm\)
Transition probability A (s\)
Con"guration of lower state
E upper (cm\)
Con"guration of upper state
Degeneracy of upper state g
2080.89 2116.93 2118.78 2120.82 2122.50 2131.84 2141.00 2141.72 2159.42 2160.98 2162.13 2193.46 2197.22 2213.54 2246.62 2270.12 2284.42 2284.81 2293.50 2608.85 2623.28 2637.01 2683.76 2689.18 2692.26 2696.43 2701.72 2737.42 2737.85 2740.33 2746.57 2753.88 2760.85 3003.19 3003.59 3016.73 3023.09 3040.56 3092.73 3100.18 3191.52 3226.20 3279.34 3279.74
1.65E#06 2.19E#04 5.72E#04 1.13E#04 5.79E#04 1.75E#05 9.73E#05 9.79E#05 1.46E#06 3.86E#05 5.46E#05 1.69E#06 2.82E#05 1.62E#06 8.88E#05 3.48E#05 2.12E#06 1.08E#06 1.37E#06 1.20E#06 5.21E#05 1.47E#06 3.91E#05 3.19E#06 2.67E#06 1.62E#06 1.05E#06 2.21E#05 2.63E#05 2.35E#06 4.33E#05 5.71E#05 2.13E#06 1.74E#05 2.41E#05 4.53E#05 1.09E#06 6.97E#04 1.81E#05 3.98E#05 2.54E#06 7.35E#04 4.06E#04 2.50E#04
5p 4d 5d 3d 4f 5p 5p 4d 5p 4f 4d 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5p 5d 6p 5p 4d 5p 5p 4d 3d 5s 5p 3d 3d 5p 5s 3d 4f 4f
119023.6 122717.8 124051.6 116942.8 122329.7 120600.9 121011.9 121165.4 120619.0 123815.6 121068.7 120600.9 119760.2 119212.9 118906.6 119212.9 120753.5 119847.8 119444.8 119760.2 119566.0 121096.6 119683.1 121096.6 121161.3 119847.8 121161.3 126294.8 123903.2 119683.1 121653.2 121161.3 119760.2 121654.6 117151.3 116660.0 119683.1 118407.4 118459.6 119760.2 116660.0 116942.8 123468.0 123468.0
4d 5f 6f 5p 5d 4d 4d 6p 4d 5d 6p 4d 6s 4d 4d 4d 4d 4d 4d 6s 4d 6s 6s 6s 6s 4d 6s 7f 8s 6s 4f 6s 6s 4f 5p 5p 6s 5p 5p 6s 5p 5p 6d 6d
9 7 5 7 7 5 3 7 5 3 3 5 3 7 5 7 7 3 5 3 7 1 5 1 3 3 3 9 5 5 7 3 3 5 3 3 5 3 3 3 3 7 3 3
[2# H[1# H[0# H[2# [4# [1# [0# H[3# [0# [2# H[1# [1# [0# [2# [0# [2# [1# [0# [1# [1# [2# [0# [2# [1# [1# [1# [0# H[2# [2# [2# H[1# [1# [2# H[0# H[1# H[1# [0# H[1# H[1# [0# H[1# H[2# [1# [1#
H[3# [3# [2# [2# H[2# H[1# H[1# [2# H[2# H[1# [0# H[1# H[1# H[3# H[1# H[3# H[2# H[1# H[2# H[1# H[2# H[0# H[1# H[0# H[0# H[1# H[0# [3# H[1# H[1# [3# H[0# H[1# [2# [1# [0# H[1# [1# [0# H[1# [0# [2# H[0# H[0#
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634 Table 2 (continued ) Transition (cm\)
Transition probability A (s\)
Con"guration of lower state
E upper (cm\)
Con"guration of upper state
Degeneracy of upper state g
3282.77 3356.07 3415.22 3432.41 3467.04 3474.28 3482.83 3494.03 3508.07 3530.85 3536.32 3540.33 3545.80 3978.97 4171.35 4173.87 4192.60 4321.62 4521.07 4536.06 4642.51 4763.76 4803.83 4821.37 4821.77 4841.97 4849.22 4863.63 4916.43 4922.83 4981.47 4991.13 4992.23 5009.57 5044.67 5383.50 5424.62 5580.48 5580.51 5608.88 5730.66 5901.37 5972.06 6040.90 6051.66
7.21E#04 2.73E#06 1.13E#06 1.31E#06 6.38E#05 4.08E#06 1.30E#06 4.00E#06 3.45E#06 1.07E#06 1.50E#06 1.06E#06 2.80E#06 1.53E#06 3.59E#05 1.28E#06 1.60E#06 1.07E#05 1.07E#06 9.07E#04 1.62E#05 3.41E#05 4.81E#04 1.83E#05 1.59E#05 1.25E#06 5.45E#05 9.84E#05 1.29E#05 1.88E#05 1.45E#06 3.83E#05 5.54E#05 2.79E#05 7.43E#05 4.98E#05 1.31E#06 8.52E#05 8.27E#05 2.07E#06 1.37E#05 1.85E#06 3.00E#05 2.38E#06 1.86E#05
3dH[2# 5s H[1# 3dH[1# 5sH[0# 3dH[2# 5s H[1# 4dH[3# 5sH[0# 5s H[1# 5s H[1# 4dH[1# 5s H[1# 5sH[0# 3dH[3# 4p [0# 4dH[0# 3dH[3# 4p [0# 3dH[1# 4p [1# 4p [0# 4p [0# 3dH[1# 3dH[1# 3dH[1# 3dH[0# 4p [1# 4dH[2# 5p [0# 4dH[3# 3dH[2# 5s H[1# 3dH[0# 4dH[1# 3dH[1# 3dH[1# 3dH[1# 4p [1# 4p [1# 3dH[2# 4p [1# 4p [1# 4p [0# 3dH[1# 4p [1#
116999.3 116999.3 117563.0 118407.4 117183.6 116942.8 122695.7 118469.1 117151.3 116999.3 124137.2 117183.6 118407.4 116999.3 111667.8 122686.1 116942.8 111818.0 116660.0 111667.8 112138.9 111818.0 116942.8 120188.2 120188.6 116660.0 112138.9 125482.6 122479.4 124135.7 118407.4 118459.6 116660.0 124857.3 117183.6 120188.6 120229.8 111818.0 111667.8 120249.9 113020.4 112138.9 113468.5 120188.6 112138.9
5p [2# 5p [2# 5p [0# 5p [1# 5p [1# 5p [2# 5f [4# 5p [1# 5p [1# 5p [2# 5f [2# 5p [1# 5p [1# 5p [2# 3dH[0# 5f [1# 5p [2# 3d H[0# 5p [0# 3d H[0# 3d H[1# 3d H[0# 5p [2# 4f [1# 4f [1# 5p [0# 3d H[1# 6f [3# 7s H[1# 5f [3# 5p [1# 5p [0# 5p [0# 7f [1# 5p [1# 4f [1# 4f [2# 3d H[0# 3d H[0# 4f [3# 3d H[3# 3d H[1# 5s H[1# 4f [1# 3d H[1#
5 5 1 3 5 7 9 5 3 5 7 5 3 5 1 3 7 3 3 1 5 3 7 3 5 3 5 7 3 9 3 3 3 5 5 5 7 3 1 7 7 5 5 5 5
617
618
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Table 2 (continued ) Transition (cm\)
Transition probability A (s\)
Con"guration of lower state
E upper (cm\)
Con"guration of upper state
Degeneracy of upper state g
6082.33 6252.40 6287.71 6490.62 6513.20 6521.65 6533.36 6588.99 6644.25 6804.10 6823.91 6831.34 6848.04 7013.24 7093.46 7186.82 7229.52 7229.56 7230.92 7287.39 7308.72 7338.70 7351.29 7365.22 7381.21 7403.09 7456.98 7479.00 7509.28 7532.24 7556.00 7557.60 7565.67 7685.32 7715.93 7729.93 7808.69 8060.47 8090.83 8099.29 8228.27 8235.16 7843.31 7851.20 7870.45
5.29E#06 1.07E#06 8.36E#06 1.72E#06 1.76E#06 3.42E#04 7.08E#06 1.30E#06 4.70E#06 2.00E#06 7.36E#06 9.98E#06 9.55E#06 9.35E#06 5.23E#06 1.27E#07 2.68E#06 1.02E#06 3.50E#06 1.37E#07 1.08E#07 1.04E#07 1.51E#06 5.87E#06 5.39E#05 1.34E#07 1.65E#07 1.17E#07 1.28E#07 1.39E#07 4.94E#06 2.22E#06 1.08E#07 1.04E#07 1.03E#07 1.26E#07 4.47E#06 7.41E#06 1.19E#07 1.01E#06 9.11E#06 2.03E#06 2.43E#06 2.16E#06 5.74E#06
3d H[1# 4p [0# 3d H[1# 3d H[2# 3d H[2# 4p [2# 3d H[2# 4p [0# 4p [0# 3d H[2# 3d H[2# 3d H[2# 3d H[1# 3d H[2# 4p [0# 3d H[3# 3d H[3# 3d H[3# 4p [1# 4p [2# 4p [0# 4p [1# 4p [1# 4p [0# 4p [1# 4p [2# 3d H[3# 4p [1# 4p [1# 4p [1# 4p [1# 4p [2# 4p [0# 4p [1# 4p [0# 4p [1# 4p [2# 4p [1# 3d H[1# 4p [2# 3d H[2# 4p [1# 4p [1# 4p [2# 4p [0#
120230.1 114975.0 121654.6 120207.2 120229.8 112138.9 120249.9 113643.3 115366.9 120230.1 120249.9 121653.3 121653.2 121654.2 114147.7 120207.2 120249.9 120249.9 113468.5 112750.2 114805.1 113426.0 114641.0 114861.6 113468.5 113020.4 120207.1 113716.6 114641.0 114821.9 113643.3 113020.4 111667.8 114975.0 111818.0 114861.6 113426.0 114147.7 120229.8 113716.6 121654.2 115366.9 114975.0 113468.5 115366.9
4f [2# 5sH[0# 4f [2# 4f [4# 4f [2# 3d H[1# 4f [3# 5s H[1# 3dH[1# 4f [2# 4f [3# 4f [3# 4f [3# 4f [2# 3d H[1# 4f [4# 4f [3# 4f [3# 5s H[1# 3d H[3# 3dH[1# 3d H[2# 3dH[2# 5sH[0# 5s H[1# 3d H[3# 4f [4# 3d H[2# 3dH[2# 3dH[2# 5s H[1# 3d H[3# 3d H[0# 5sH[0# 3d H[0# 5sH[0# 3d H[2# 3d H[1# 4f [2# 3d H[2# 4f [2# 3dH[1# 5sH[0# 5s H[1# 3dH[1#
5 3 5 9 7 5 9 3 3 5 7 9 7 7 3 9 7 9 5 9 5 5 5 1 5 7 11 7 5 7 3 7 1 3 3 1 5 3 7 7 7 3 3 5 3
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
619
Table 2 (continued ) Transition (cm\)
Transition probability A (s\)
Con"guration of lower state
E upper (cm\)
Con"guration of upper state
Degeneracy of upper state g
8005.71 8025.99 8036.83 8253.80 8370.20 8370.60 8403.44 8412.04 8520.47 8530.46 8567.58 8632.93 8702.27 8717.88 8737.47 8770.57 8774.38 9023.72 9187.87 9366.37
1.08E#07 9.91E#06 8.82E#06 2.41E#06 9.17E#06 9.79E#06 1.00E#04 2.99E#06 1.14E#07 1.12E#06 4.48E#06 3.11E#06 1.89E#05 4.74E#05 1.90E#06 4.09E#06 3.99E#06 2.43E#06 5.39E#05 4.88E#06
4p 4p 4p 4p 3d 3d 4p 3d 3d 4p 4p 3d 4s 4p 4p 3d 4p 4p 4p 4p
113468.5 113643.3 112138.9 113716.6 120188.2 120188.6 114641.0 120230.1 120188.2 114147.7 114805.1 121653.3 104102.1 114805.1 114975.0 124137.4 114861.6 114641.0 114805.1 113468.5
5s H[1# 5s H[1# 3d H[1# 3d H[2# 4f [1# 4f [1# 3dH[2# 4f [2# 4f [1# 3d H[1# 3dH[1# 4f [3# 4p [0# 3dH[1# 5s H[0# 5f [2# 5s H[0# 3dH[2# 3dH[1# 5s H[1#
5 3 5 7 3 5 5 5 3 3 5 9 3 5 3 5 1 5 5 5
[2# [2# [0# [2# H[0# H[0# [1# H[0# H[0# [2# [1# H[3# H[0# [1# [1# H[1# [1# [2# [2# [0#
Since the lines are primarily Stark broadened, the lines are expected to have a Lorentzian line shape [6]. In fact, Lorentzian shapes "t the measured spectral lines satisfactorily. The line shape function for a Lorentzian has the form w/2 1 G(! )" ' ["] cm\. (17) (! )#(w/2) Here, w is the full width of the line in cm\. In all the calculations discussed below, a path length (l) of 1.6 cm is used. Although the arc channel is 3.6 cm long, not all light collected from the arc has that path length. Only the light coming from very close to the optical axis of the collection mirror would have traveled 3.6 cm. As discussed in the appendix, most of the other light would have traveled less than 3.6 cm. Using ray tracing and Eq. (14), we have calculated a path length of 1.6 cm. Combining Eqs. (14) and (15) gives R()"S(¹)G(! )l[P /R¹]BB(¹,). (18) The above equation shows that R() is a function of two unknown parameters: ¹ and P . Since R() is linear in P and non-linear in ¹, we can perform a least squares "t by guessing ¹, calculating a synthetic Ar emission spectrum (R ()), and then calculating the best "t value of
620
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Fig. 6. Least squares "tting of measured data for a 4 atm, 30 A arc by varying ¹ and calculating the corresponding best "t P . The number of spectral data points is 7882.
Fig. 7. Comparing measured (4 atm, 30 A) and simulated spectra (¹"10,800 K, P "0.94 atm). The spectra have been baseline corrected for comparison: (a) 3000}4000 cm\ spectral region, (b) 7000}8000 cm\ spectral region.
P for that guess of ¹: S(¹ )G(! )lBB(¹ , ) R () "P 2 R¹ "P f (, ¹ ).
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
621
Therefore, R () f (, ¹ ) . P " J 2 [ f (, ¹ )] J The process is repeated for di!erent temperatures at 200 K intervals, and the temperature which results in the smallest least-squares error ([R()!R ()]) is the temperature of the plasma. The results of this process for a 4 atm, 30 A plasma is illustrated in Fig. 6. The plasma emssion simulated with the least-squares temperature estimate is compared to the measured emission in Fig. 7.
6. Electron density estimation 6.1. Continuum emission Once the temperature is calculated, the electron density can be estimated from the continuum radiation. The continuum emission is due to three sources, namely: free}free electron}ion interactions ( ); free}free electron neutral}atom interactions ( ); and free}bound electron}ion interac tions (). Using radiative exchange theory in the same way as before, the radiance due to the continuum emission can be written in a slightly di!erent form [32] R()"(1!e\C l 2 J)BB(¹, ),
(19)
where " # #. The individual emission coe$cients are given by [32]
!c n "1.632;10\ exp ¹ (¹
!c c , #1 exp 1# ¹ ¹
"1.026;10\n n ¹ Q(¹)
(20)
h 2k¹ where "1#0.1728 1# , E (z!1) h & !c n "1.632;10\ 1!exp . ¹ (¹
(21)
(22) (23)
The individual emission coe$cients have the units of W/(m cm\), n and n have the units of m\, and ¹ is in K. Here, Q(¹) (m) is the electron-neutral atom scattering cross-section calculated by Devoto [33]. The free}free Gaunt factor (dimensionless) is calculated using the formula of Menart et al. [34]. The factor z is the ionic state of the interacting ion (z"2 for singly ionized argon). (Note that the value of asymptotically converges to 1.2 W/(m cm\) as the emission frequency increases above 2000 cm\). is called the Biberman factor and is fairly independent of temperatures and wavelength. In our calculations, we assume that the Biberman factor is constant at a value of 1. Since the continuum is `#ata, and is relatively independent of temperature, the best
622
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
"t to the continuum can be calculated using a least-squares analysis by varying the only unknown factor, the electron density n . 6.2. Saha equilibrium We now have techniques to estimate the temperature (¹), electron density (n ), and the number density n "P /R¹ of `activea Ar assuming the plasma is in PLTE. If our assumption of PLTE is correct, then these parameters (¹, n , n ) should also be consistent with the Saha equation. To test our assumption, we use the measured ¹ and n from the line spectra calculations to calculate a new electron density n1 via the Saha equation. If this electron density is the same as that calculated from continuum emission, then our assumption of PLTE is valid. The Saha equation is [35]
2u>(¹) (2m k) E !E n>n " ¹ exp ! G h u (¹) k¹ n "2.4125;10
2u>(¹) E !E ¹ exp ! G ["] cm\, u (¹) k¹
(24)
where u> and u are the partition functions of singly ionized and neutral Ar [35], E is the lowering of the ionization potential, and all other terms have their usual meaning. There are several empirical and theoretical expressions for the lowering of ionization potential (E). We use Brunner's formula [35]
E(eV)"1.21;10\ (n (cm\)#2.5;10\
n (cm\) . ¹(K)
(25)
In these low-temperature, low electron density plasmas, we note that the IR spectra show no observable emission from Ar>, so we assume that there is a negligible amount of doubly or higher ionized argon ions. This observation means that the number density of ions is, in fact, equal to the number density of the electrons, i.e. n>"n . It must be noted that the electron density calculated from continuum emission is not very sensitive to variations in temperatures, but the electron density calculated from Saha equation depends strongly on the plasma temperature.
7. Results At the highest pressure and currents (4 atm, 25}30 A), the electron density calculated from Saha equation is in good agreement with the electron density estimated from continuum emission (Table 3). Table 3 However, the lower pressure data seems to suggest that the arc is not even in PLTE. In the case of the low current-low pressure plasmas (15 A, 1 atm), there is considerable disagreement between the electron density calculated from continuum emission and that calculated from the Saha equation. The PLTE theory is still not su$cient to explain the discrepancies in the low-current, low-pressure plasmas, probably because the low-pressure plasmas are not even in partial Boltzmann equilibrium. We can conclude that the high-pressure, high-current plasmas are
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
623
Table 3 Continuum and Saha electron densities compared. Line spectra are "tted assuming ¹ and P are adjustable (PLTE) Pressure (atm)
Current (A)
Temperature (K)
P (atm)
Mole fraction of Ar
Electron density from continuum (10 cm\)
Saha electron density (10 cm\)
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
15 15 15 15 20 20 20 20 25 25 25 25 30 30 30 30
10000 10000 10000 9900 10500 10400 10400 10400 10900 10800 10800 10700 11300 11000 10800 10900
0.08 0.18 0.27 0.68 0.09 0.21 0.35 0.69 0.11 0.23 0.37 0.77 0.11 0.28 0.43 0.84
0.08 0.09 0.09 0.17 0.09 0.11 0.12 0.17 0.11 0.11 0.12 0.19 0.11 0.14 0.14 0.21
2.69 2.21 2.17 2.40 3.18 2.63 2.55 3.06 3.28 3.30 3.22 3.76 3.58 3.90 3.87 4.63
0.57 0.82 1.01 1.47 0.91 1.29 1.90 2.36 1.38 1.88 2.39 3.24 1.93 2.46 2.62 4.04
in PLTE, but the atmospheric pressure, low current arcs are not in PLTE. As a result, the temperatures listed in Table 3 must be regarded as estimates except for the 4 atm, 25}30 A plasmas. 7.1. Calculation of new transition probabilities Having established that the 4 atm, 30 A is in PLTE, we can use the values of P and ¹ to determine new transition probabilities for 70 of the strongest transitions listed in Table 2. Putting in values for the constants in the line strength calculation at each line peak (Eq. (17)) leads to the formula
W K !c E P Ag . . exp R "99.67 J m cm\ sr atm s ¹ w¹
(26)
Since the 4 atm, 30 A plasma is closest to equilibrium, the temperature estimated from this plasma is deemed the most accurate. Based on this temperature, and since R is measured, new A values are calculated in one step from Eq. (26) and appear in Table 4. In many cases, considerable adjustment of the A values occurs, but this does not have much e!ect on the measured temperatures of the plasmas. Using the new A values and P from Table 3 we have re-calculated ¹ for the various plasmas using a non-linear regression package in STATGRAPHICS (Manugistics) using just the peak intensities of these 70 lines. We see that the temperature calculated from the non-linear
624
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Table 4 Modi"ed transition probabilities Center (cm\)
A (s\)
A (s\)
A /A
Center (cm\)
A (s\)
A (s\)
A /A
3190.0 3354.3 3471.9 3492.1 3506.2 4188.9 4763.6 4848.9 5424.0 5580.2 5580.2 5900.9 6039.0 6051.1 6082.5 6249.3 6286.9 6489.4 6510.6 6533.2 6585.7 6642.9 6803.6 6822.9 6830.5 6847.1 7011.2 7091.9 7185.7 7227.6 7227.6 7227.6 7286.4 7307.2 7337.4 7350.2 7361.5
2.54E#06 2.73E#06 4.08E#06 4.00E#06 3.45E#06 1.60E#06 3.41E#05 5.45E#05 1.31E#06 8.52E#05 8.27E#05 1.85E#06 2.38E#06 1.86E#05 5.29E#06 1.07E#06 8.36E#06 1.72E#06 1.76E#06 7.08E#06 1.30E#06 4.70E#06 2.00E#06 7.36E#06 9.98E#06 9.55E#06 9.35E#06 5.23E#06 1.27E#07 3.50E#06 2.68E#06 1.02E#06 1.37E#07 1.08E#07 1.04E#07 1.51E#06 5.87E#06
4.78E#06 3.74E#06 7.69E#06 5.81E#06 4.27E#06 2.39E#06 7.78E#05 1.06E#06 5.61E#06 2.04E#06 6.01E#06 4.24E#06 4.66E#06 5.36E#05 1.03E#07 3.99E#06 1.51E#07 2.47E#06 3.96E#06 1.15E#07 2.74E#06 8.92E#06 3.60E#06 1.36E#07 1.57E#07 1.50E#07 1.86E#07 9.66E#06 1.77E#07 5.74E#06 9.99E#06 7.77E#06 1.10E#07 1.30E#07 1.27E#07 2.67E#06 1.01E#07
0.53 0.73 0.53 0.69 0.81 0.67 0.44 0.52 0.23 0.42 0.14 0.44 0.51 0.35 0.51 0.27 0.55 0.70 0.45 0.61 0.48 0.53 0.55 0.54 0.64 0.64 0.50 0.54 0.72 0.61 0.27 0.13 1.25 0.83 0.82 0.56 0.58
7379.4 7401.4 7455.0 7477.5 7507.8 7530.8 7555.7 7555.7 7565.2 7681.6 7715.0 7726.9 7807.3 7840.5 7848.2 7868.6 8002.0 8022.4 8035.9 8058.4 8088.1 8097.4 8228.0 8233.4 8252.2 8367.9 8367.9 8518.5 8529.0 8565.9 8700.5 8716.2 8733.6 9021.9 9185.9 9361.3
5.39E#05 1.34E#07 1.65E#07 1.17E#07 1.28E#07 1.39E#07 4.94E#06 2.22E#06 1.08E#07 1.04E#07 1.03E#07 1.26E#07 4.47E#06 2.43E#06 2.16E#06 5.74E#06 1.08E#07 9.91E#06 8.82E#06 7.41E#06 1.19E#07 1.01E#06 9.11E#06 2.03E#06 2.41E#06 9.79E#06 9.17E#06 1.14E#07 1.12E#06 4.48E#06 1.89E#05 4.74E#05 1.90E#06 2.43E#06 5.39E#05 4.88E#06
6.83E#06 1.32E#07 1.97E#07 1.39E#07 1.65E#07 1.51E#07 1.78E#07 7.02E#06 9.34E#06 1.52E#07 9.70E#06 1.09E#07 6.98E#06 3.88E#06 1.26E#06 1.03E#07 1.51E#07 1.15E#07 5.67E#06 1.38E#07 9.18E#06 2.27E#06 6.00E#06 4.46E#06 3.81E#06 1.17E#07 1.95E#07 1.14E#07 6.65E#05 6.79E#06 4.11E#05 7.29E#05 3.86E#06 1.73E#06 9.92E#05 1.10E#07
0.08 1.02 0.84 0.84 0.77 0.92 0.28 0.32 1.15 0.69 1.06 1.15 0.64 0.63 1.72 0.56 0.71 0.87 1.56 0.54 1.30 0.45 1.52 0.46 0.63 0.84 0.47 1.01 1.68 0.66 0.46 0.65 0.49 1.40 0.54 0.44
regression analysis (Table 5) is very similar to the temperature calculated from the least squared analysis performed on all the 153 lines by varying P and ¹. The di!erences between the least-square temperature (¹ ) and the non-linear "t (¹ ) are, in fact, small even at low UQOP UJGL pressures and currents.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
625
Table 5 Comparing plasma temperatures obtained from least-squares analysis (¹ ) and non-linear regression analysis U (¹ ) with their 95% con"dence limits. The least-squares analysis uses 153 lines and literature A values (Table 3), U while the non-linear "t uses 70 lines and recalculated A values (Table 4) Pressure (atm)
Current (A)
Mole fraction
P (atm)
¹ (K) U
¹ (K) U
4 3 2 1 1 1 2
30 30 30 30 25 20 15
0.21 0.14 0.14 0.11 0.11 0.09 0.09
0.84 0.43 0.28 0.11 0.11 0.09 0.18
10900 10800 11000 11300 10900 10500 10000
10962$31 11069$28 11149$27 11472$26 11082$22 10689$20 10178$18
7.2. Analysis of new transition probabilities By comparing calculated line intensities using the new and old A values, we can determine how accurate these new A values are. Figs. 8, 9, and 10 are residual plots (R !R ) as
a function of the transition frequency using the old and new A values; their corresponding plasma temperatures come from the non-linear regression analysis. The residuals calculated with the new A values at 3 atm, 30 A are packed tightly around zero (Fig. 8(a)), while that is clearly not the case for the old A values (Fig. 8(b)). The new A values (Fig. 9(a)) predict intensities better than the old A values (Fig. 9(b)) for the 1 atm, 30 A plasma, but tend to have more negative deviations than positive deviations. In the lowest current plasma, however (Fig. 10), the residuals calculated from the new and the old A values deviate signi"cantly from zero (note change of scale in Fig. 10). This trend among Figs. 8}10 can be explained by the fact that the new A values are calculated assuming PLTE for the 4 atm, 30 A plasma, but the lower pressure and lower current plasmas are experiencing deviations from PLTE. Therefore, it predicts line intensities accurately for other plasmas at PLTE (like the 3 atm, 30 A plasma), but it does not do as well in predicting line intensities for low currents. This observation reinforces the conclusions made from the previous analysis that the high-pressure, high-current plasmas are close to PLTE, while the low-pressure low-current plasmas are not in PLTE. Closer examination of the Fig. 8(a) leads to some interesting observations. The new A values predict intensities for low-frequency transitions (3000}6000 cm\) better than higher frequency (6000}10,000 cm\). The low-frequency transitions occur between the highest excited states of Ar, while the higher frequency transitions occur between higher excited states and the "rst or second excited state. PLTE assumes that all the excited states are in equilibrium with each other, but Fig. 8(a) indicates that the highest excited states are more closely in equilibrium with each other, compared to the lower excited states ("rst and second). 7.3. Trends in n and ¹ We have now determined the electron density and the temperature of the plasma and have also established some primary trends in the data. Fig. 11 identi"es the trends in the electron densities
626
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Fig. 8. Residual plot (R !R ) vs. wavenumber for a 3 atm, 30 A plasma: (a) modi"ed A values
(¹"11,069 K), (b) original A values (¹"11,087 K). A values appear in Table 4.
and temperatures as a function of arc operating conditions. Remember the plasmas with lower currents are not in PLTE, and thus the temperature estimates at these conditions are somewhat suspect. The temperature at a given current remains almost constant over a range of pressures except when the arc is at 30 A (Fig. 11(a)). In contrast, the temperature is a strong function of current (Fig. 11(b)). The electron density on the other hand seems to depend both on the current and the pressure (Fig. 11(c), (d)). To help in visualizing the data better, we "t n and ¹ to an empirical polynomial model to generate contour plots of n and ¹ as a function of the input parameters pressure (P) and current (I)
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
627
Fig. 9. Residual plot (R !R ) vs. wavenumber for a 1 atm, 30 A plasma: (a) modi"ed A values
(¹"11,468 K), (b) original A values (¹"11,493 K). A values appear in Table 4.
(Fig. 12). From Fig. 12(a) we can see that at temperatures below 10,500 K, the temperature of the plasma is mostly independent of pressure. To reach higher temperatures (around 11,000 K) at higher pressures, however, much more current has to be #owing through the plasma. This observation can be explained by the fact that at higher pressures, the plasma is denser, and hence more energy is required to heat it. Please remember that the temperatures have error bars around 100 K, and the statistical model does not take these measurement errors into account. The plots serve mostly to visualize trends in the data. The contour plots for n (Fig. 12(b)) suggest that at low (1 atm) and high (4 atm) pressures, less current is needed to achieve the same electron density (which mostly a!ect the continuum emission
628
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
Fig. 10. Residual plot (R !R ) vs. wavenumber for a 2 atm, 15 A plasma: (a) modi"ed A values
(¹"10,178 K), (b) original A values (¹"10,223 K). A values appear in Table 4.
(Eqs. (19)}(23)). At low pressures, there are fewer neutral Ar and hence less energy is needed for ionization to occur. At very high pressure, however, the free paths of the various species in the plasma are smaller and hence there is a greater probability of the collisional processes that lead to ionization. The plots also indicate that at higher currents, a 4 atm plasma will have a higher electron density than a 1 atm plasma at the same current. This makes sense, as highpressure plasmas have more neutral Ar present and hence there is higher rate of ionization.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
629
Fig. 11. Electron densities and temperatures as a function of the di!erent operating pressures and currents.
Fig. 12. Contour plots of temperatures (a) and electron densities (b) plotted as a function of pressure and current.
630
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
8. Conclusions A cascade arc was built and operated at di!erent currents (15, 20, 25, and 30 A) and pressures (1, 2, 3, and 4 atm). The infrared emission (2000}10,000 cm\) of the arc was measured using a FTIR spectrometer. The path length of arc emission (1.6 cm) is di!erent from the actual geometric length (3.6 cm) of the arc, and this is established using ray tracing and radiative exchange theory (see the appendix). The line emission of the arc was used to calculate the temperature (¹) of the plasma while the electron density (n ) was calculated from the continuum emission. The electron density obtained from the continuum emission was compared to the electron density calculated from Saha equation to test the validity of the assumption of partial local thermodynamic equilibrium (PLTE). The Saha electron densities calculated using the PLTE temperature compares well with the electron densities obtained from continuum emission calculations for the high-pressure, high-current arcs (30 A, 3}4 atm). The Saha electron density and the continuum electron density are quite di!erent in the case of low current, lowpressure plasmas. These observations lead to the conclusion that the low-current, low-pressure plasmas are not even in PLTE, while the high-pressure, high-current arcs are at PLTE. Under the given range of operating conditions (4 atm5P51 atm; 30 A5I515 A) the plasma temperature varies between 9900 and 11,500 K, while the electron density varies between 2 and 5;10 m\. The line widths can be measured by "tting a Lorentzian line shape to each measured emission line. We observe that line widths vary with operating conditions. The A values used to simulate line spectra [29}31] have 30% uncertainties associated with them. New A values have been recalculated to "t the data better. Most of the A values change less than 30% while a few of them have to be changed by 30}50%. Changing the A values result in much better predictions of line intensities for plasmas close to PLTE. Because of the strong emission in the near IR, the total power output from the arc is 3 times that of the infrared source used by the FTIR (globar). As a matter of fact, without an attenuator the broadband light from the arc at the highest pressure and currents saturates the DTGS detector used in the measurements. At present, continuum intensity of the arc between 1500 and 6000 cm\ is similar to the globar. Higher currents would increase the continuum level of the arc and signi"cantly increase the intensity in the mid-IR.
Acknowledgements The authors gratefully acknowledge the "nancial support of the Defense University Research Instrumentation Program (Grant DAAH04-95-0060) and the National Science Foundation (CTS96-25095). The authors would also like to acknowledge technical discussions with Richard van de Sanden (Eindhoven University of Technology) as well as Tianming Bao, Cli! Hayman, Wayne Schmidt and Ram Wusirka for helping in the design and construction of the cascade arc. We would also like to thank Prof. Robert V. Edwards for his help with the statistical data analysis.
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
631
Appendix. Derivation of e4ective path length Consider the amount of light collected by a parabloidal mirror emitted from an arc channel that is 3.6 cm long. Only a fraction of the light coming from very close to the optical axis of the collection mirror would have traveled the full distance. To calculate the e!ective path length (l ), we make the following assumptions: 1. The parabloid mirror used for collection has a 5 cm focal length and the mirror is positioned 50 mm away from the end of the cascade arc channel (Fig. 13). 2. The detector and its mirror limit the throughput of the optics (not the arc). The detector element used is 1 mm in diameter, and the mirror in front of the detector has a focal length of 16 mm. This leads to a magni"cation ratio between the arc and the detector of 50/16"2.66; therefore, a spot 1 mm in diameter at the detector is actually 2.66 mm in diameter at the cascade arc. To calculate the maximum solid angle collected from the arc (2(1!cos )), we note that the
throughput at the detector is equal to the throughput from the arc. Therefore, 2(1!cos )A "2(1!cos )A , where A and A are the areas of the detector element
and the arc, respectively. The ratio of these areas corresponds to the square of the ratio of the magni"cation factor. The collection angle of the optics at the detector ( ) is 0.588 rad, and hence we can calculate "0.22 rad.
3. To make the calculations easier, we treat the light coming out from the arc as two distinct types of rays depending on a critical angle (Fig. 13(b)). The light rays which have an angle less than
will have a path length given by ¸/cos (¸"36 mm). Any light rays which have an angle higher than the critical angle will have a path length signi"cantly less than the arc length: R/sin (R"4 mm). From Fig. 13(b), "tan\(2/36)"0.055 rad.
Fig. 13. (a) optics for collection of light from the arc, (b) critical angle determination.
632
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
4. Since the diameter of the arc is much smaller than the diameter of the collection optics, the arc is treated as an axial source, i.e. only the rays passing through the optical axis are considered. O!-axis radiation is assumed to depend on the angles the same way as the on-axis radiation. This simpli"es the calculation tremendously. Using the above assumptions and radiative exchange theory [25], we know that the radiance R is given by R"[1!e\?l]BB,
(A.1)
where is the absorption coe$cient and l is the distance the radiation has traveled. In the case of the arc optics, emission intensity is a function of the path length, which in turn is a function of the angle . Therefore,
¸ R( )" 1!exp ! cos
and
R R( )" 1!exp ! sin
BB when 0( (
BB.
Integrating over all solid angles gives the energy incident, E, on the collection optic. Therefore,
F
R( )2 sin d , where d"2 sin d assuming axial symmetry. Entering the two expressions for R( ) results in E" R( ) d"
E"
F
¸ 1!exp ! cos
BB2 sin d #
F
R 1!exp ! sin
F Using small argument approximation e\?6 +1! X and integrating
BB2 sin d .
E"2BB[! ¸ ln cos # R( ! )].
The e!ective path length can be calculated by comparing the E from the above equation to a value of E calculated assuming the path length l is independent of (l"¸"3.6 cm). We call this value of energy E : F E " [1!e\?*]BB2 sin d . Using the small argument approximation and integrating once again,
E "2BB ¸(1!cos ).
The e!ective path length can be calculated from a ratio of the two energies: 2BB[! ¸ ln cos # R( ! )]
" 2BB ¸(1!cos )
!ln cos #(R/¸)( ! ) K?V . " 1!cos
l " E ¸
E
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
633
From the cascade arc geometry (¸"3.6 cm, R"0.2 cm, "0.22 rad and "0.055 rad),
l "0.443 ¸Nl "1.6 cm. References [1] Morrison Jr PW, Taweechokesupsin O, Kovach CS, Roozbehani B, Angus JC. In situ infrared measurements during hot "lament CVD of diamond in a rotating substrate reactor. Diam Related Mater 1995;5:242}6. [2] Taweechokesupsin O. In situ gas phase measurements using infrared spectroscopy during hot "lament chemical vapor deposition of diamond. MS thesis, Case Western Reserve University, Cleveland, OH, 1996. [3] Szeto E. Analysis of the oxyacetylene #ame under diamond deposition conditions using Fourier Transform Infrared Spectroscopy. MS thesis, Case Western Reserve University, Cleveland, 1998. [4] Wilbers ATM, Kroesen GMW, Timmermans CJ, Schram DC. Characteristic quantities of a cascade arc used as a light source for spectroscopic techniques. Meas Sci Technol 1990;1:1326}32. [5] Haverlag M, Kroesen GMW, de Hoog FJ. In situ measurement of C F densities in an RF CF plasma using infrared absorption spectroscopy. In: D'Agostini R, editor. Proceedings of the 9th International Symposium on Plasma Chemistry. University of Bari, Bari, 1989. p. 441}4. [6] Griem HR. Plasma Spectroscopy. New York: McGraw-Hill, 1964. [7] Kroesen GMW, Schram DC, de Haas JCM. Description of a #owing cascade arc plasma. Plasma Chem Plasma Proc 1989;10:531}51. [8] Qing Z, de Graaf MJ, van de Sanden MCM, Otorbaev DK, Schram DC. Experimental characterization of a hydrogen/argon cascaded arc plasma source. Rev Sci Instr 1994;65:1469}71. [9] Wilbers ATM, Kroesen GMW, Timmermans CJ, Schram DC. The continuum emission of an arc plasma. JQSRT 1991;45:1}10. [10] Meeusen GJ, Ershov-Pavlov EA, Meulenbroeks RFG, van de Sanden MCM, Schram DC. Emission spectroscopy on a supersonically expanding argon/silane plasma. J Appl Phys 1992;71:4156}63. [11] Meulenbroeks RFG, van Beek AJ, van Helvoort AJG, van de Sanden MCM, Schram DC. Argon}hydrogen plasma jet investigated by active and passive spectroscopic means. Phys Rev E 1994;49:4397}406. [12] Meulenbroeks RFG, van der Heijden PAA, van de Sanden MCM, Schram DC. Fabry}Perot line shape analysis on an expanding cascaded arc plasma in argon. J Appl Phys 1994;75:2775}80. [13] Meulenbroeks RFG, Engeln RAH, Beurskens MNA, Pa!en RMJ, van de Sanden MCM, van der Mullen JAM, Schram DC. The argon}hydrogen expanding plasma: model and experiments. Plasma Sour Sci Technol 1995;4:74}85. [14] Buuron AJM, Otorbaev DK, van de Sanden MCM, Schram DC. A new absorption spectroscopy setup for the sensitive monitoring of atomic and molecular densities. Rev Sci Instr 1995;66:968}74. [15] Dahiya RP, de Graaf MJ, Severens RJ, Swelsen H, van de Sanden MCM, Schram DC. Dissociative recombination in cascaded arc generated Ar}N and N expanding plasma. Phys Plasmas 1994;1:2086}95. [16] Brunger M, Kock M. Electron density determination in an arc plasma by laser interferometry. Z Naturforsch 1975;30a:1560}2. [17] Baessler P, Kock M. An interferometric and spectroscopic study on a high-current argon arc. J Phys B: At Mol Phys 1980;13:1351}61. [18] Schnehage SE, Kock M, Schulz-Gulde E. The continuous emission of an argon arc. J Phys B: At Mol Phys 1982;15:1131}5. [19] Stuck D, Wende B. Photometric comparison between two calculable vacuum-ultraviolet standard radiation sources: Synchrotron radiation and plasma-blackbody radiation. J Opt Soc Am 1972;62:96}100. [20] Kaase H, Stephan HK. Calibration of radiation detectors in the VUV using a wall-stabilized Ar-cascade arc. Appl Opt 1979;18:2275}9. [21] Key PJ, Preston RC. Vacuum ultraviolet radiation scales: an accurate comparison between plasma blackbody lines and synchrotron radiation. Appl Opt 1977;16:2477}85. [22] Frost RM, Awakowicz P. Continuous calibration of a vacuum ultraviolet system from 65 to 125 nm by a cascade arc and comparison with the calibrated line radiation of a hollow cathode. Appl Opt 1997; 36:1994}2000.
634
R. Raghavan, P.W. Morrison Jr. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 605}634
[23] Kroesen GMW, van de Sande MJF, Wilbers ATM, Schram DC, Bisschops LA, Timmermans CJ, Heller HJ, Husken ABM. The implementation of a cascade arc. Technical Report VDF/NT 90-06. Eindhoven University of Technology, 1990. [24] Morrison Jr. PW, Haigis JR. In situ infrared measurements of "lm and gas properties during the plasma deposition of amorphous hydrogenated silicon. J Vac Sci Technol A 1993;11:490}502. [25] Siegel R, Howell JR. Thermal radiation heat transfer. Washington: Hemisphere Publishing Corporation, 1981. [26] Timmermans CJ, Rosado RJ, Schram DC. An investigation of Non-Equilibrium e!ects in thermal argon plasmas. Z Naturforsch 1985;40a:810}25. [27] Rosado RJ. An investigation of non-equilibrium e!ects in thermal argon plasmas. PhD thesis, Eindhoven University of Technology, Eindhoven, 1981. [28] Rothman LS, Rinsland CP, Goldman A, Massie ST, Edwards DP, Flaud JM et al. The HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 Edition JQSRT 1998;60:665}710. [29] Smith PL, Heise C, Esmond JR, Kurucz RL. Atomic spectral line database. CD-ROM 23 of R.L. Kurucz (http://cfa-www.harvard.edu/amdata/ampdata/kurucz23/sekur.html). Harvard Smithsonian Center for Astrophysics, 1995. [30] Wiese WL, Brault JW, Danzmann K, Helbig V, Kock M. Uni"ed set of atomic transition probabilities for neutral argon. Phys Rev A 1989;39:2461}71. [31] Wiese WL. Spectroscopic diagnostics of low temperature plasmas: techniques and required data. Spectrochim Acta 1991;46B:831}41. [32] de Regt JM, van Dijk J, van der Mullen JAM, Schram DC. Components of continuum radiation in an inductively coupled plasma. J Phys D 1995;28:40}6. [33] Devoto RS. Transport coe$cients of ionized argon. Phys Fluids B 1973;16:616}23. [34] Menart J, Heberlein J, Pfender E. Line-by-line method of calculating emission coe$cients for thermal plasmas consisting of monatomic species. JQSRT 1996;56:377}98. [35] Drawin HW, Felenbok P. Data for plasmas in local thermodynamic equilibrium. Paris: Gauthier-Villars, 1965.