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A simple approach to deriving an electron energy distribution from an incoherent Thomson scattering spectrum
Spectrochimica Acta Part B 00 Ž2000. 1397᎐1410
A simple approach to deriving an electron energy distribution from an incoherent Thomson scattering spectrum 夽 Mao Huang 1, Kelly Warner, Scott Lehn, Gary M. HieftjeU Department of Chemistry, Indiana Uni¨ ersity, Bloomington, IN 47405, USA Received 2 February 2000; accepted 29 May 2000
Abstract A clear picture is given to illustrate how the electron energy distribution in a plasma can be directly related to an incoherent Thomson scattering spectrum when the electron density distribution is spherically symmetric in velocity space. Based on this relationship, a simple approach is outlined to derive the electron energy distribution unambiguously. The method involves plotting the differences in Thomson scattering intensities between adjacent wavelength channels as a function of the square of the wavelength shift. Thomson scattering spectra obtained from microwave-induced helium plasmas sustained at atmospheric pressure at forward powers of 100 and 350 W are used to derive electron energy distributions through this procedure. Significant deviations from a Maxwellian distribution were found under both conditions over the entire wavelength span covered in the experiment, corresponding to an electron energy range of 0.1᎐6.6 eV. Compared with a Maxwellian energy distribution, the portion of the electrons with energies between 2 and 6 eV appear to be shifted toward both lower and higher energy values in these plasmas. The ratios of the Thomson scattering intensity at the wavelength channel farthest from the center to the corresponding Maxwellian value indicate that the total number of electrons with energies larger than 6.6 eV is at least 1.8 and 3.1 times higher than that predicted by local thermodynamic equilibrium for plasmas at 100 and 350 W, respectively. In contrast to a Maxwellian distribution, electrons in the microwave-induced plasmas are not most concentrated at the center of velocity space, but reach a maximum in a spherical layer at a distance of approximately
夽
This paper was presented at the 2000 Winter Conference on Plasma Spectrochemistry, Fort Lauderdale, Florida, 10᎐15 January 2000. U Corresponding author. Tel.: q1-812-855-2189; fax: q1-812-855-0958. E-mail address: [email protected] ŽG.M. Hieftje.. 1 On leave from Chinese Academy of Sciences. 0584-8547r00r$ - see front matter 䊚 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 8 4 - 8 5 4 7 Ž 0 0 . 0 0 2 4 4 - 5
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4 = 107 cm sy1 from the space center. In addition, for both plasma power levels, the bulk of the electron velocity distribution function is slightly compressed and the maximum value is shifted toward lower velocity values compared with a Maxwellian distribution. 䊚 2000 Elsevier Science B.V. All rights reserved. Keywords: Electron energy distribution; Thomson scattering; Microwave-induced plasma; Non-Maxwellian distribution; Plasma diagnostics
1. Introduction Thomson scattering has been widely used as a powerful plasma diagnostic tool since the late 1960s, especially in the field of plasma physics and controlled nuclear fusion, due to the wellestablished theory w1᎐4x, experimental methods w5᎐7x and the availability of high power pulsed lasers. Beginning in the early 1980s, Thomson scattering also found applications in the study of analytical and industrial plasmas, such as inductively coupled plasmas ŽICPs. w8᎐23x, high voltage spark discharges w24x, microwave-induced plasmas ŽMIPs. w25᎐27x, glow discharges ŽGDs. w28,29x, electron cyclotron resonance ŽECR. processing plasmas and other industrial plasmas w30᎐37x. Assuming that the electron energy distribution is Maxwellian, the shape of a Thomson scattering spectrum is determined entirely by the scattering parameter ␣ w1᎐7x. When ␣ < 1, the scattering is termed incoherent; the scattering spectrum then reflects the thermal features of free electrons and can be used to measure plasma parameters such as the electron temperature ŽTe . and electron number density Ž ne .. When ␣ ) 1, the scattering is termed coherent or collective; the scattering spectrum is then dominated by ion features and can be used to determine ion temperature, ion plasma wave and other parameters. Incoherent Thomson scattering has been most commonly used in all of the aforementioned fields, while coherent Thomson scattering has been extensively employed in fusion plasmas w38᎐47x. If electrons follow a Maxwellian distribution, an incoherent Thomson scattering spectrum will be Gaussian in shape and therefore directly reflect the electron velocity distribution function in terms of per unit volume in velocity space w f Ž ¨ .x, which is also Gaussian in form. A plot of the logarithm of the scattering intensity vs. the
square of the wavelength shift will then produce a straight line, and the slope of this line will be inversely proportional to Te . In addition, the total scattering intensity will be proportional to ne . These two proportionality relationships form the basis of the simplest procedure for the use of Thomson scattering as an important plasma diagnostic technique w19᎐23x. It should be pointed out, however, that such a Gaussian-shaped Thomson scattering spectrum can be distorted for any of three reasons: first, the scattering might not be purely incoherent; in other words, the assumption ␣ < 1 might not hold. Second, there could be a deviation from a Maxwellian electron energy distribution. Third, anisotropy in the f Ž ¨ . might exist. In any of these three cases, the scattering spectrum will no longer serve as a direct reflection of the f Ž ¨ . and should be interpreted with caution, as will be detailed in Section 4 of this paper. ICPs sustained at atmospheric pressure are a good example of the first case mentioned above w18᎐20x. In most regions of the ICP, where Thomson scattering measurements are usually performed, the ␣ value is less than 1 but greater than 0.2 w23,24x, resulting in a slightly distorted spectrum with the central portion Žnear the incident laser wavelength. depressed compared with a Gaussian shape. If one then applies the conventional plotting procedure to the determination of Te and ne , Te will be over-estimated, while ne will be under-estimated. Fortunately, there exists an unambiguous relationship between each pair of Te and ne values and the shape and total intensity of the corresponding Thomson scattering spectrum. Consequently, look-up tables or a curvefitting technique can be used to retrieve real Te and ne values when a Maxwellian velocity distribution is assumed w18᎐20x.
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In contrast to the situation in atmosphericpressure ICPs, glow discharges, microwaveinduced plasmas, ECR processing plasmas and other industrial plasmas are more likely to scatter light in a purely incoherent manner, due to their much higher Te and much lower ne values. Unfortunately, Thomson scattering spectra obtained from these plasmas, in general, show even more serious distortion from a Gaussian shape. Since the scattering is incoherent, these distortions are probably caused by a non-Maxwellian electron energy distribution induced by either insufficiently frequent collisions, non-thermal heating mechanisms, or both. Unusual distortions in incoherent Thomson scattering spectra can be found in some fusion plasmas, especially in magnetic-confinement devices such as tokamaks and stellarators w45᎐51x, in which the plasma behavior is affected by strong magnetic fields, fast current pulses or disruptions. In general, these distortions cannot be explained by non-thermal features of the plasma, but can be attributed to anisotropy in velocity space, as will be discussed in Section 4. In this case, it is necessary for a Thomson scattering experiment to be performed in several viewing directions, allowing the differential scattering wave vector k to be parallel, perpendicular and oblique with respect to the magnetic field, for example, in order to probe the full f Ž ¨ .. In principle, the f Ž ¨ . can be uniquely determined from the full scattered intensity I Žk., but determining it in detail can be very difficult w52x. In an attempt to correctly interpret non-Gaussian-shaped incoherent Thomson scattering spectra, we offer in the present paper a clear picture to illustrate how electrons with the same energy contribute to intensities at different wavelength channels of a Thomson scattering detection system, and how electrons with different energies scatter light into the same wavelength channel. Based on this understanding, a simple approach is introduced to derive the electron energy distribution function ŽEEDF. directly from a Thomson scattering spectrum when an isotropic electron distribution in velocity space can be assumed. Results of non-Maxwellian EEDFs obtained from Thomson scattering spectra for atmospheric-pres-
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sure helium microwave-induced plasmas using this approach will be shown, and discussions of the results and a number of key points concerning the interpretation of non-Gaussian-shaped incoherent Thomson scattering spectra will be given.
2. Theoretical considerations For a given electron with a velocity ¨ in the viewing volume of a Thomson scattering system, the wavelength shift, ⌬ of electron-scattered light from the incident laser line, 0 , is proportional to the projected velocity of the electron along the differential scattering wave vector k, and can be calculated from a relationship of the form ⌬ s y 0 s Ž k ⭈ v . 02r Ž 2 c .
Ž1.
with k s k0 y ks
Ž2.
where is the wavelength, k0 and ks are the wave vector of the incident radiation and a vector along the observation direction with the same length, respectively, and c is the speed of light. The length of k is therefore k s 2sin Ž r2. k0 s 2 sin Ž r2. 2 r0 s 4 sin Ž r2. r0
Ž3.
where k0 is the length of k0 , and is the angle between k0 and ks . From Eqs. Ž1. and Ž3. one obtains ⌬ s 20 Ž ¨ krc . sin Ž r2.
Ž4.
where ¨ k is the projected velocity of the given electron along k. Eq. Ž4. can be written as ¨ k s c Ž ⌬r0 . r Ž 2sin Ž r2..
Ž5.
It can be seen from Eq. Ž5. that for a given wavelength shift ⌬ Ži.e. for a given Thomsonscattering wavelength channel ., electrons with a velocity ¨ equal to or larger than ¨ k will have a
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Fig. 1. Velocity space Ž Vx , Vy , Vz . separated into spherical and planar layers, corresponding to electron groups with different energies and with different projected velocities along the differential scattering wavelength vector k. ¨ i , i and f Ž ¨ i . with i s 1, 2, . . . are electron velocity, the corresponding wavelength shift and the f Ž ¨ . at different layers or wavelength channels, respectively. See text for discussion.
chance to scatter light into that channel, depending on where they are located in velocity space, whereas electrons with a velocity lower than ¨ k will not be able to contribute to the intensity in that channel. Fig. 1 shows more clearly the relationship between the electron distribution in velocity space and the amount of light scattered into each wavelength channel defined by i with i s 1, 2, . . . For the sake of simplicity, the coordinate axes Vx , Vy and Vz have been chosen so that k coincides with Vx . Also, for brevity let us now use i instead of ⌬i to designate the wavelength shift corresponding to channel i . Let us now divide velocity space into different regions in two ways. First, the space is separated into a series of spherical shells, each with a thickness equivalent to the width of a wavelength channel; this approach is shown in Fig. 1. Second, the space is equally divided by a set of planes perpendicular to the Vx axis with the interval also equivalent to the width of a wavelength channel. The spherical and planar layers are defined by their corresponding velocity and projected veloc-
ity onto k, respectively, both of which are represented by the same symbol, ¨ n . Clearly, electrons in the spherical shell ¨ i will make a contribution to the Thomson scattering signal of every wavelength channel j , when ¨ j F ¨ i ; on the contrary, no electrons in the spherical layer ¨ j can contribute to the Thomson scattering signal in any wavelength channel i , when ¨ i ) ¨ j . In addition, the amount of light scattered by electrons in a spherical shell ¨ i into wavelength channel j is proportional both to the value of the f Ž ¨ . in that shell w f Ž ¨ i .x and to the volume of the ring defined by the region intercepted by the spherical shell ¨ i and the planar layer ¨ j Ž Vj,i .. The overall Thomson scattering intensity at that wavelength channel w I Ž j .x will then be proportional to the total amount of light scattered by all electrons located in the different shells Žhaving different energies. in the same planar layer ¨ j , i.e. ⬁
I Ž j . A Ý f Ž ¨ i . Vj,i
Ž6.
isj
As an example, let us calculate Vn, nq2 , which is the volume of the ring intercepted by planar layer ¨ n and spherical shell ¨ nq2 as shown in Fig. 1, where ABCD defines the cross-section of the ring in the Vx ᎐Vy plane. Any cross-section of the ring perpendicular to axis Vx is a planar ring with EG and FG as its outer and inner radii, respectively, as shown in the figure. Here, E and F lie anywhere on the curves AD and BC, respectively, but correspond to the same ¨ x coordinate. The area of this planar region can be calculated as Area s Ž EG2 y FG2 . s wŽ OE2 y OG2 . y Ž OF2 y OG2 .x s Ž OE2 y OF2 . s Ž ¨ nq 2 q ⌬¨r2 . 2 y Ž ¨ nq1 q ⌬¨r2 . 2 s wŽ ¨ nq 2 q ¨ nq1 q ⌬¨ .Ž ¨ nq2 y ¨ nq1 .x s w 2 ¨ nq 2 ⌬¨ x s 2 ¨ nq2 ⌬¨
Ž7.
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
I Ž n . y I Ž nq1 . A f Ž ¨ nq1 . Vn ,nq1 q f Ž ¨ n . Vn ,n
The volume of the region is then
Vn ,nq2 s
yf Ž ¨ nq 1 . Vnq1,nq1 s f Ž ¨ nq1 . 2 Vnq1 ⌬¨ 2
1 ¨ nq ⌬¨ 2 2 ¨ nq2 ⌬¨ d¨ s 2 ¨ nq2 Ž ⌬¨ . 2 1 ¨ ny ⌬¨ 2
qf Ž ¨ n . Ž ¨ n q ⌬¨r6 . ⌬¨ 2 yf Ž ¨ nq 1 . Ž ¨ nq1
H
Ž8. Eq. Ž8. indicates that for a given spherical layer ¨ i , the volume of each intercepted shell is the same if j - i, and is proportional to ¨ i . In other words, electrons in the spherical shell ¨ i scatter
the same amount of light into each wavelength channel for which ¨ j - ¨ i . The volume of the last intercepted region in the spherical layer Vi,i Ž ¨ i ., shown as the shaded portion in Fig. 1, can be calculated as follows, Vi ,i s
⌬¨
H0
½ž
1 ⌬¨ 2
¨i q
/ y ž ¨ y 12 ⌬¨ q x / 5 d x 2
2
i
¨
2
i
q Ž ¨ n q ⌬¨r6 . f Ž ¨ n .x ⌬¨ 2 Ž 11. The significance of Eq. Ž11. is that the difference between the Thomson scattering signals in two adjacent wavelength channels contains information that is solely related to the values of the EVDF Ž ¨ . corresponding to these two channels w f Ž ¨ n ., f Ž ¨ nq1 .x. For an infinitesimal ⌬¨ , Eq. Ž11. becomes Ž 12.
i
1 s ¨ i q ⌬¨ ⌬¨ 2 6
ž
q⌬¨r6 . ⌬¨ 2 s w Ž ¨ nq 1 y ⌬¨r6 . f Ž ¨ nq1 .
I Ž n . y I Ž nq1 . A ¨ nq1 f Ž ¨ nq1 . q ¨ n f Ž ¨ n .
⌬ H0 2 ¨ ⌬¨ y Ž2 ¨ y ⌬¨ . xy x 4 d x
s
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/
Ž9.
According to Eq. Ž6., the difference between the Thomson scattering signals in two adjacent wavelength channels ¨ n and ¨ nq 1 is I Ž n . y I Ž nq1 . A
⬁
⬁
isn
isnq1
Ý f Ž ¨i . Vn ,i y Ý
=Vnq 1 ,i s f Ž ¨ n . Vn ,n q
f Ž ¨i .
⬁
Ý
f Ž ¨ i . Vn ,i
isnq1
yf Ž ¨ nq 1 . Vnq1,nq1 y
⬁
Ý
f Ž ¨i .
isnq2
= Vnq 1,i
Ž 10.
Note that on the right hand side of Eq. Ž10., Vn,i in the first summation is equal to Vnq1,i in the second, as has been verified by Eq. Ž8., so all the terms in the first summation are canceled by those in the second summation except the first term f Ž ¨ nq 1 .Vn,nq1. Therefore, applying Eqs. Ž8. and Ž9., we have
Note that ¨ f Ž ¨ . is exactly the form of an energy distribution function of electrons wEEDFx with an isotropic velocity distribution in velocity space w f Ž ¨ .x, regardless of whether or not the electrons obey a Maxwellian distribution. This point can be understood as follows. The number of electrons with velocity between ¨ and ¨ q d¨ is 4 ¨ 2 f Ž ¨ .d¨ . As the electron energy Es Ž1r2. m¨2 , a velocity change d¨ corresponds to an energy change d E s m¨ d¨ , which means that for a constant d¨ the corresponding d E is not constant but proportional to ¨ . In other words, the abovementioned number of electrons corresponds to an energy interval which is proportional to ¨ . Therefore, the EEDF should be proportional to that number divided by ¨ to keep d E constant, resulting in the form of ¨ f Ž ¨ . for the EEDF. As a result, Eq. Ž12. indicates that the difference between the Thomson scattering signals in two adjacent wavelength channels is a direct reflection of the value of the EEDF at the energy corresponding to the wavelength between the two channels, as long as the bandwidth of the wavelength channel is sufficiently small. If, however, the wavelength channel is relatively wide, then the signal difference I Ž n . y IŽ nq 1 . is proportional to an average value of
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the EEDF at the energy corresponding to the wavelength shift between the two channels. Clearly, the EEDF can be obtained simply by plotting the differences in Thomson scattering signals between adjacent wavelength channels as a function of the corresponding energy, which is in turn proportional to the square of the wavelength shift. So plotting I Ž n . y I Ž nq1 . vs. the square of wŽ n q nq1 .r2x will yield the shape of the EEDF. Normalizing the plot to the measured electron number density ne will give absolute values of the EEDF. In addition, the electron velocity distribution function EVDF is proportional to ¨ 2 f Ž ¨ ., whereas I Ž n . y IŽ nq1 . is proportional to ¨ f Ž ¨ .. As a result, the Thomson scattering spectrum also directly provides information about the EVDF, as shown in Fig. 2, where the shaded area is a measure of the EVDF value at a velocity corresponding to Ž n q nq1 .r2. For a wavelength channel of infinitesimal width, I Ž n . y I Ž nq1 . is proportional to the derivative of the profile of the Thomson scattering spectrum. Therefore, an alternative approach to deriving the EEDF, EVDF and f Ž ¨ . is to use the following relationships: EEDFA yI⬘ Ž . , when ) 0; I⬘ Ž . , when - 0
Ž 13.
EVDFA y I⬘ Ž .
Ž 14.
f Ž ¨ . A yI ⬘ Ž . r
Ž 15.
where I⬘Ž . is the derivative of the Thomson scattering spectrum with respect to wavelength shift. Obviously, the resolutions of all of the distribution functions obtained using the abovementioned approaches depend strongly on the resolution of the wavelength channels employed in the experiment.
3. Non-Maxwellian electron energy distributions in a helium microwave-induced plasma Thomson scattering data published in an ear-
Fig. 2. A Thomson scattering spectrum with I Ž n . and I Ž nq1 . being the intensities at any two adjacent wavelength channels n and nq1 , respectively. The shaded area is proportional to the average value of the EVDF at the wavelength shift Ž n q nq 1 .r2.
lier paper were used to derive electron energy distributions by means of the approach described in the preceding section wsee Figs. 4, 5, 12 and 13 in Huang et al. w25x for detailsx. These Thomson scattering data, obtained from a helium-supported microwave plasma torch ŽMPT., yielded the electron energy distributions shown in Figs. 3 and 4. The helium plasmas corresponding to Figs. 3 and 4 were operated at forward powers of 100 and 350 W and observed at heights of 4 and 7 mm above the top of the MPT, respectively. The electron temperatures shown in the figure captions were determined by the Maxwellian distributions that best fit the bulk of the scattering spectrum, and were used to calculate the corresponding Maxwellian EEDFs for comparison. Both the derived EEDFs and the Maxwellian EEDFs were normalized to the electron number density measured by Thomson scattering. Figs. 5᎐8 show the corresponding EVDF and f Ž ¨ . for both sets of MPT conditions. All the data in these figures were also normalized to the measured ne . Significant deviations from a Maxwellian distribution can be clearly seen in all of the distribution functions. The departures appear to be more
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
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Fig. 3. Electron energy distribution function EEDF Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 100 W and at an observation height of 4 mm above the top of the torch. Ž`. Corresponding EEDF values from a Maxwellian distribution with measured Te s 21 500 K and ne s 6 = 1013 cmy3. Data are normalized to the measured ne .
significant for the plasma at a forward power of 350 W than at 100 W. In general, the population of electrons with energies between 2 and 6 eV is
depressed, while electrons with energies between 0.5 and 2 eV are overpopulated Žsee Figs. 3 and 4.. More importantly, yet not obvious in these
Fig. 4. Electron energy distribution function EEDF Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 350 W and at an observation height of 7 mm above the top of the torch. Ž`. Corresponding EEDF values from a Maxwellian distribution with measured Te s 20 000 K and ne s 6.5= 1013 cmy3 . Data are normalized to the measured ne .
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Fig. 5. Electron velocity distribution function EVDF Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 100 W and at an observation height of 4 mm above the top of the torch. Ž`. Corresponding EVDF values from a Maxwellian distribution with measured Te s 21 500 K and ne s 6 = 1013 cmy3 . Data are normalized to the measured ne .
Fig. 6. Electron velocity distribution function EVDF Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 350 W and at an observation height of 7 mm above the top of the torch. Ž`. Corresponding EVDF values from a Maxwellian distribution with measured Te s 20 000 K and ne s 6.5= 1013 cmy3 . Data are normalized to the measured ne .
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
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Fig. 7. Electron distribution in velocity space f Ž ¨ . Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 100 W and at an observation height of 4 mm above the top of the torch. Ž`. Corresponding f Ž ¨ . values from a Maxwellian distribution with measured Te s 21 500 K and ne s 6 = 1013 cmy3 . Data are normalized to the measured ne .
Fig. 8. Electron distribution in velocity space f Ž ¨ . Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 350 W and at an observation height of 7 mm above the top of the torch. Ž`. Corresponding f Ž ¨ . values from a Maxwellian distribution with measured Te s 20000 K and ne s 6.5= 1013 cmy3. Data are normalized to the measured ne .
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figures, the total number of electrons with energies equal to or larger than 6.6 eV is dramatically increased for both conditions in comparison with that predicted by a Maxwellian distribution. This point can be clarified by comparing the Thomson scattering intensity at the last channel Žthe greatest wavelength shift. with the value calculated from a Maxwellian distribution. This very last channel was centered 5.0 nm from the incident laser wavelength, which corresponds to an electron energy of 6.6 eV or an electron velocity of 1.53= 108 cm sy1 . The intensity of Thomson scattering in the last wavelength channel for the plasma at 100 W forward power is higher than its corresponding Maxwellian value by a factor of 1.8 Žsee Fig. 13 in Huang et al. w25x., whereas that with 350 W of forward power is 3.1-fold higher than its corresponding Maxwellian value Žsee Fig. 12 in Huang et al. w25x.. In other words, electrons in these plasmas with energies G 6.6 eV scatter 1.8 and 3.1 times more light, respectively, into all Thomson scattering wavelength channels than Maxwellian distributions would predict, because their contribution to each wavelength channel is the same, as has been proven in Section 2. Since the amount of scattered light is proportional to the number of electrons that take part in the scattering, the number of electrons with energy G 6.6 eV in these plasmas is therefore some 1.8and 3.1-fold higher than the Maxwellian values, respectively. Moreover, the tail portion of the EEDF in microwave-induced plasmas might extend still further in comparison with the Maxwellian distribution, a trend already apparent in Figs. 3᎐8 Ži.e. the calculated distribution decreases much more slowly with E or ¨ in the wing portions than does the Maxwellian distribution.. Accordingly, the above estimation of the number of high-energy electrons is probably conservative, which might explain in part why microwave-induced helium plasmas are highly efficient in the excitation of non-metallic elements, especially the halogens. Clearly, obtaining details of the EEDF for electron energies larger than 6.6 eV is impossible with the current data and will require Thomson scattering at even greater wavelength shifts to be
probed with high signal-to-noise ratio and wavelength resolution. Concerning the f Ž ¨ ., it is interesting to note, as shown in Figs. 7 and 8, that the electrons in microwave-induced plasmas are not most concentrated in the center of velocity space, as are electrons that possess a Maxwellian distribution. Instead, they reach a maximum in a spherical layer at a distance of some 4 = 107 cm sy1 from the space center. Of course, the occurrence of the peak in the f Ž ¨ . might be, in part, also attributable to the loss of electron population at higher velocities, as can be seen in the tail portions in Figs. 7 and 8.
4. Discussion The Thomson scattering data used in this paper were collected at 16 wavelength channels, each of which was 0.3 nm wide, and the wavelength shift between any two adjacent channels was also 0.3 nm. This wavelength resolution should be good enough to detect a substantial deviation from a Maxwellian distribution. The systematic error caused by the spectral response of the detection system could be more serious than any error due to the width of the wavelength channel. For this reason, a single detector was used to measure Thomson signals at all different channels by scanning the wavelength of the monochromator, and the spectral response was calibrated using a tungsten ribbon lamp. As a result, the systematic error should be less than 1%. As for the noise on the measured Thomson scattering signals, the relative standard deviation was approximately 2᎐5%, depending on the wavelength shift. The larger the wavelength shift, the higher the relative standard deviation. Since the deviations from Maxwellian distributions shown in Figs. 3᎐8 can be as large as 20᎐30%, also depending on the wavelength shift, errors in the measurement should not be a dominant factor that causes these deviations. As described in Section 2 and clearly shown in Fig. 1, the intensity of an incoherent Thomson scattering spectrum at any wavelength shift cannot be greater than that at any shorter wavelength shift, when the f Ž ¨ . is isotropic. In other
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
words, the value of the derivative of an incoherent Thomson scattering spectrum should be no greater than zero for positive wavelength shifts, and no less than zero for negative wavelength shifts. Of course, when the signal-to-noise ratio is poor, an experimental Thomson scattering profile might appear very noisy, resulting in some positive derivative values for positive wavelength shifts, or some negative derivative values for negative wavelength shifts. In this real world situation, the EEDF obtained using the derivative of a Thomson scattering spectrum will include some negative values. To eliminate this error, appropriate curvesmoothing procedures should be applied to the Thomson scattering spectrum before the EEDF is derived from it using Eqs. Ž13. ᎐ Ž15.. In this sense, directly taking the differences between signals of adjacent wavelength channels as a function of the wavelength squared is probably a better way to obtain the EEDF, as proposed in Section 2, instead of taking the derivative of the Thomson scattering profile. Any irregularities in the incoherent Thomson scattering spectrum, such as satellites, dips or a shift of the peak in a Gaussian profile w48᎐51x that cause anomalous derivative values and that cannot be accounted for solely by a low signal-tonoise ratio should be interpreted as a reflection of anisotropy of the f Ž ¨ .. Imagine, for example, that for some reason electrons in the planar layer ¨ n in velocity space, shown in Fig. 1, have an unusually low concentration. In this situation, a dip might appear in the Thomson scattering spectrum at a wavelength shift n . On the contrary, if for some reason electrons gather in that layer, then a satellite might emerge at that wavelength shift. When a full incoherent Thomson scattering spectrum looks fairly symmetric, but the central peak is not located at the incident laser wavelength, the plasma must be drifting at a certain velocity. In this case, Eq. Ž5. can be used to determine the projected component of the drift velocity along k, with ⌬ as the wavelength shift of the central peak. To determine the three-dimensional drift velocity vd , Thomson scattering spectra with k along three different directions are
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generally required. If the plasma is known to drift in a given direction, the Thomson scattering measurement can then be designed so that k is coincides with ¨ d , or at least so k is not perpendicular to ¨ d , in order to acquire complete information about the f Ž ¨ .. It is well known that for electrons with a Maxwellian distribution, the incoherent Thomson scattering spectrum has the same shape as the f Ž ¨ ., i.e. Gaussian. It is, therefore, said that incoherent Thomson scattering spectra directly reflect the f Ž ¨ .. However, this perfect coincidence is unique to Maxwellian distributions, which can be clarified as follows. A Gaussian shaped incoherent Thomson scattering spectrum can be expressed in the form I Ž . A eyC
2
Ž 16.
where C is a constant. Combining Eqs. Ž15. and Ž16. yields f Ž ¨ . A yI ⬘ Ž . r s 2CeyC r s 2CeyC 2
2
Ž 17. Comparing Eqs. Ž16. and Ž17., it can be seen that the f Ž ¨ . is indeed identical in shape to I Ž .. This coincidence, however, will no longer hold when electrons possess a non-Maxwellian distribution. For example, if an incoherent Thomson scattering spectrum can be expressed in the form I Ž . A ay b , when 0 F F arb; I Ž . s 0, when ) arb
Ž 18.
where a and b are constants, as shown in Fig. 9, the corresponding f Ž ¨ . should be f Ž ¨ . A yI ⬘ Ž . r s br , when 0 F F arb; 0, when ) arb Ž 19. or f Ž ¨ . A c1 br¨ , when 0 F ¨ F c1 arb; when ¨ ) c1 arb
Ž 20.
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
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Fig. 9. A Thomson scattering spectrum in the form: I Ž . A a y b, when 0 F F arb; I Ž . s 0, when ) arb.
where c1 s ¨r is a constant for a particular experiment and can be determined by Eq. Ž5., as shown in Fig. 10. In this case, the f Ž ¨ . looks entirely different from I Ž . in shape. If a theoretical model assumes that the plasma consists of two electron groups, each of which possesses its own Maxwellian distribution defined by a different temperature Žusually a high temperature and a low temperature. w53x, then the incoherent Thomson scattering spectrum is the sum of two Gaussian profiles in the form I Ž . A a1e
2
yC 1
q a2 e
yC 1 2
Ž 21.
where a1 , a2 , C1 , and C2 are constants. The corresponding f Ž ¨ . should be f Ž ¨ . A y I⬘ Ž . r A a1C1eyC 1 q a2 C2 eyC 2 2
s C1 a1eyC 1 q a2
ž
2
C2 yC 1 2 e C1
2
/
Ž 22.
Fig. 10. f Ž ¨ ., derived from the Thomson scattering spectrum shown in Fig. 9, in the form: f Ž ¨ . A c1 br¨ , when 0 F ¨ F c1 arb; 0, when ¨ ) c1 arb.
It can be seen by comparing Eq. Ž22. with Eq. Ž21. that the shape of the f Ž ¨ . is different from I Ž ., although both electron groups possess Maxwellian distributions. In addition, because both electron groups scatter light to all wavelength channels, the Thomson scattering spectrum should be interpreted with caution. For example, the temperatures of the two electron groups can not be obtained by simply plotting the logarithmic scattering intensity vs. the wavelength shift squared at the central portion Žfor the low Te . and at the tail portion Žfor the high Te ., respectively. Instead, a curve fitting technique based on the form of Eq. Ž21. should be employed to determine a1 , C1 , a2 and C2 , where a1 and a2 are related to the weights of the two Maxwellian components, and C1 and C2 are related to the temperatures of the two electron groups, respectively.
5. Conclusions A picture is given, in which velocity space is separated into a series of spherical shells as well as a series of planar layers, to clearly illustrate how electrons with different energies scatter light into the same wavelength channel, and how electrons with the same energy scatter light into different wavelength channels in an incoherent Thomson scattering experiment. It has been proven that electrons with a given energy scatter the same amount of light into all other wavelength channels that correspond to energies smaller than the given electron energy, but make no contributions to wavelength channels with corresponding energies larger than the given electron energy. Therefore, the difference in Thomson scattering intensities between any two adjacent wavelength channels is related solely to the EEDF values at the energies corresponding to the wavelength shifts of these two channels. Based on this realization, a procedure is outlined to derive the EEDF simply by plotting the difference between the intensities of two adjacent wavelength channels as a function of the square of the wavelength shift. Absolute values of the
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
EEDF can then be obtained by normalizing the plot to the measured electron number density. The EEDFs of microwave-induced helium plasmas sustained at atmospheric pressure at forward powers of 100 and 350 W, respectively, show significant deviations from Maxwellian distributions over the entire wavelength range, which is equivalent to electron energies of 0.1᎐6.6 eV. A portion of the electrons with energies in the range of approximately 2᎐6 eV seem to be shifted towards both lower and higher energy values. The ratio of the Thomson scattering intensity measured at the greatest wavelength shift to the calculated Maxwellian value can be considered to be a minimum estimation of the ratio of the total number of electrons with energies larger than the energy corresponding to that channel Ži.e. 6.6 eV for the present work. to the Maxwellian value. Accordingly, high energy electrons Ži.e. EG 6.6 eV. in these plasmas are overpopulated by a factor of at least 1.8 and 3.1 for forward powers of 100 and 350 W, respectively, compared with Maxwellian distributions. Such an overpopulation of high energy electrons explains in part the high efficiency of helium microwave-induced plasmas in the excitation of non-metal elements, especially the halogens. The f Ž ¨ . values obtained show that electrons in these plasmas are not most abundant at the center of velocity space, but reach a maximum in a spherical layer at a distance of some 4 = 107 cm sy1 from the space center. In addition, the bulk of the EVDF is slightly compacted and the maximum value is shifted toward lower velocity values for both microwave plasma conditions, compared with a Maxwellian distribution. The coincidence that an incoherent Thomson scattering spectrum has the same shape as that of the f Ž ¨ . is unique to a Maxwellian electron distribution. Therefore, incoherent Thomson scattering spectra from plasmas with non-Maxwellian electron distributions do not directly reflect the f Ž ¨ ., and should be interpreted with caution. Furthermore, satellites, dips or other irregularities, which cause anomalous derivative values Žsee Section 4. of the scattering intensity profile, should not appear in an incoherent Thomson
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scattering spectrum, if the f Ž ¨ . is assumed to be isotropic. If such irregularities occur, they could be attributable to a low signal-to-noise ratio or to anisotropy in velocity space.
Acknowledgements This research was supported by the Department of Energy through Grant DE-FG0298ER14890. References w1x E.E. Salpeter, Phys. Rev. 120 Ž1960. 1528. w2x J.A. Fejer, Can. J. Phys. 38 Ž1960. 1114. w3x J.P. Dougherty, D.T. Farley, Prog. R. Soc. 259A Ž1960. 79. w4x M.N. Rosenbluth, N. Rostoker, Phys. Fluids 5 Ž1962. 776. w5x D.E. Evans, J. Katzenstein, Rep. Prog. Phys. 32 Ž1969. 207. w6x J. Sheffield, Plasma Scattering of Electromagnetic Radiation, Academic Press, New York, 1975. w7x J.J. Kunze, in: W. Lochte-Holtgreven ŽEd.., The Laser as a Tool for Plasma Diagnostics in Plasma Diagnostics, North-Holland, Amsterdam, 1968. w8x M. Huang, S.A. Lehn, E.J. Andrews, G.M. Hieftje, Spectrochim. Acta Part B 52 Ž8. Ž1997. 1173. w9x N.N. Sesi, D.S. Hanselman, P. Galley, J. Horner, M. Huang, G.M. Hieftje, Spectrochim. Acta Part B 52 Ž1. Ž1997. 83. w10x N.N. Sesi, G.M. Hieftje, Spectrochim. Acta Part B 51 Ž13. Ž1996. 1601. w11x J.M. de Regt, F.P.J. de Groote, J.A.M. van der Mullen, D.C. Schram, Spectrochim. Acta Part B 51 Ž11. Ž1996. 1371. w12x J.M. de Regt, R.D. Tas, J.A.M. van der Mullen, B. van der Sijde, J. Quant. Spectrosc. Radiat. Transfer 56 Ž1. Ž1996. 67. w13x J.A.M. van der Mullen, J.M. de Regt, J. Anal. Chem. 355 Ž5-6. Ž1996. 532. w14x J.M. de Regt, J.A.M. van der Mullen, D.C. Schram, Phys. Rev. E: Stat. Phys. Plasmas, Fluids, Relat. Interdiscip. Top. 52 Ž3-B. Ž1995. 2982. w15x J.M. de Regt, R.A.H. Engeln, F.P.J. de Groote, J.A.M. van der Mullen, D.C. Schram, Rev. Sci. Instrum. 66 Ž5. Ž1995. 3228. w16x D.S. Hanselman, N.N. Sesi, M. Huang, G.M. Hieftje, Spectrochim. Acta Part B 49 Ž5. Ž1994. 495. w17x M. Huang, D.S. Hanselman, P. Yang, G.M. Hieftje, Spectrochim. Acta Part B 47 Ž6. Ž1992. 765. w18x M. Huang, P. Yang, D.S. Hanselman, C.A. Monnig, G.M. Hieftje, Spectrochim. Acta Part B 45 Ž4r5. Ž1990. 511.
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w19x M. Huang, G.M. Hieftje, Spectrochim. Acta Part B 44 Ž8. Ž1989. 739. w20x M. Huang, G.M. Hieftje, Spectrochim. Acta Part B 44 Ž3. Ž1989. 291. w21x M. Huang, K.A. Marshall, G.M. Hieftje, Anal. Chem. 58 Ž1. Ž1986. 207. w22x M. Huang, K.A. Marshall, G.M. Hieftje, Spectrochim. Acta Part B 40 Ž9. Ž1985. 1211. w23x M. Huang, G.M. Hieftje, Spectrochim. Acta Part B 40 Ž10᎐12. Ž1985. 1387. w24x A. Scheeline, M.J. Zoellner, Appl. Spectrosc. 38 Ž2. Ž1984. 245. w25x M. Huang, D.S. Hanselman, Q. Jin, G.M. Hieftje, Spectrochim. Acta Part B 45 Ž12. Ž1990. 1339. w26x J. Jonkers, J.M. de Regt, J.A.M. van der Mullen, H.P.C. Vos, F.P.J. de Groote, E.A.H. Timmermans, Spectrochim. Acta Part B 51 Ž11. Ž1996. 1385. w27x J. Jonkers, L.J.M. Selen, J.A.M. van der Mullen, E.A.H. Timmermans, D.C. Schram, Plasma Source Sci. Technol. 6 Ž4. Ž1997. 533. w28x H.J. Wesseling, B. Kronast, Europhys. Lett. 33 Ž4. Ž1996. 273. w29x K. Muraoka, K. Uchino, M.D. Bowden, Plasma Phys. Control. Fusion 40 Ž7. Ž1998. 1221. w30x W. Cronrath, M.D. Bowden, K. Uchino, K. Muraoka, H. Muta, M. Yoshida, J. Appl. Phys. 81 Ž5. Ž1997. 2105. w31x M.D. Bowden, T. Okamoto, T. Sakoda, K. Uchino, K. Muraoka, M. Maeda, Y. Manabe, M. Kitagawa, T. Kimura, 5th International Symposium on Laser-Aided Plasma Diagnostics, Ž1991. 230. w32x M.D. Bowden, F. Kimura, H. Muta, K. Uchino, K. Muraoka, M. Maeda, J. Vac. Sci. Technol. A 11 Ž6. Ž1993. 2893. w33x J.M. de Regt, R.D. Tas, J.A.M. van der Mullen, D.C. Schram, J. Phys. D: Appl. Phys. 29 Ž6. Ž1996. 1489. w34x M.D. Bowden, M. Kogano, Y. Suetome, T. Hori, K. Uchino, K. Muraoka, J. Vac. Sci. Technol A 17 Ž2. Ž1999. 493. w35x T. Hori, M.D. Bowden, K. Uchino, K. Muraoka, Appl. Phys. Lett. 69 Ž24. Ž1996. 3683. w36x T. Hori, M.D. Bowden, K. Uchino, K. Muraoka, M. Maeda, J. Vac. Sci. Technol. A 14 Ž1. Ž1996. 144.
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A simple approach to deriving an electron energy distribution from an incoherent Thomson scattering spectrum 夽 Mao Huang 1, Kelly Warner, Scott Lehn, Gary M. HieftjeU Department of Chemistry, Indiana Uni¨ ersity, Bloomington, IN 47405, USA Received 2 February 2000; accepted 29 May 2000
Abstract A clear picture is given to illustrate how the electron energy distribution in a plasma can be directly related to an incoherent Thomson scattering spectrum when the electron density distribution is spherically symmetric in velocity space. Based on this relationship, a simple approach is outlined to derive the electron energy distribution unambiguously. The method involves plotting the differences in Thomson scattering intensities between adjacent wavelength channels as a function of the square of the wavelength shift. Thomson scattering spectra obtained from microwave-induced helium plasmas sustained at atmospheric pressure at forward powers of 100 and 350 W are used to derive electron energy distributions through this procedure. Significant deviations from a Maxwellian distribution were found under both conditions over the entire wavelength span covered in the experiment, corresponding to an electron energy range of 0.1᎐6.6 eV. Compared with a Maxwellian energy distribution, the portion of the electrons with energies between 2 and 6 eV appear to be shifted toward both lower and higher energy values in these plasmas. The ratios of the Thomson scattering intensity at the wavelength channel farthest from the center to the corresponding Maxwellian value indicate that the total number of electrons with energies larger than 6.6 eV is at least 1.8 and 3.1 times higher than that predicted by local thermodynamic equilibrium for plasmas at 100 and 350 W, respectively. In contrast to a Maxwellian distribution, electrons in the microwave-induced plasmas are not most concentrated at the center of velocity space, but reach a maximum in a spherical layer at a distance of approximately
夽
This paper was presented at the 2000 Winter Conference on Plasma Spectrochemistry, Fort Lauderdale, Florida, 10᎐15 January 2000. U Corresponding author. Tel.: q1-812-855-2189; fax: q1-812-855-0958. E-mail address: [email protected] ŽG.M. Hieftje.. 1 On leave from Chinese Academy of Sciences. 0584-8547r00r$ - see front matter 䊚 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 8 4 - 8 5 4 7 Ž 0 0 . 0 0 2 4 4 - 5
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4 = 107 cm sy1 from the space center. In addition, for both plasma power levels, the bulk of the electron velocity distribution function is slightly compressed and the maximum value is shifted toward lower velocity values compared with a Maxwellian distribution. 䊚 2000 Elsevier Science B.V. All rights reserved. Keywords: Electron energy distribution; Thomson scattering; Microwave-induced plasma; Non-Maxwellian distribution; Plasma diagnostics
1. Introduction Thomson scattering has been widely used as a powerful plasma diagnostic tool since the late 1960s, especially in the field of plasma physics and controlled nuclear fusion, due to the wellestablished theory w1᎐4x, experimental methods w5᎐7x and the availability of high power pulsed lasers. Beginning in the early 1980s, Thomson scattering also found applications in the study of analytical and industrial plasmas, such as inductively coupled plasmas ŽICPs. w8᎐23x, high voltage spark discharges w24x, microwave-induced plasmas ŽMIPs. w25᎐27x, glow discharges ŽGDs. w28,29x, electron cyclotron resonance ŽECR. processing plasmas and other industrial plasmas w30᎐37x. Assuming that the electron energy distribution is Maxwellian, the shape of a Thomson scattering spectrum is determined entirely by the scattering parameter ␣ w1᎐7x. When ␣ < 1, the scattering is termed incoherent; the scattering spectrum then reflects the thermal features of free electrons and can be used to measure plasma parameters such as the electron temperature ŽTe . and electron number density Ž ne .. When ␣ ) 1, the scattering is termed coherent or collective; the scattering spectrum is then dominated by ion features and can be used to determine ion temperature, ion plasma wave and other parameters. Incoherent Thomson scattering has been most commonly used in all of the aforementioned fields, while coherent Thomson scattering has been extensively employed in fusion plasmas w38᎐47x. If electrons follow a Maxwellian distribution, an incoherent Thomson scattering spectrum will be Gaussian in shape and therefore directly reflect the electron velocity distribution function in terms of per unit volume in velocity space w f Ž ¨ .x, which is also Gaussian in form. A plot of the logarithm of the scattering intensity vs. the
square of the wavelength shift will then produce a straight line, and the slope of this line will be inversely proportional to Te . In addition, the total scattering intensity will be proportional to ne . These two proportionality relationships form the basis of the simplest procedure for the use of Thomson scattering as an important plasma diagnostic technique w19᎐23x. It should be pointed out, however, that such a Gaussian-shaped Thomson scattering spectrum can be distorted for any of three reasons: first, the scattering might not be purely incoherent; in other words, the assumption ␣ < 1 might not hold. Second, there could be a deviation from a Maxwellian electron energy distribution. Third, anisotropy in the f Ž ¨ . might exist. In any of these three cases, the scattering spectrum will no longer serve as a direct reflection of the f Ž ¨ . and should be interpreted with caution, as will be detailed in Section 4 of this paper. ICPs sustained at atmospheric pressure are a good example of the first case mentioned above w18᎐20x. In most regions of the ICP, where Thomson scattering measurements are usually performed, the ␣ value is less than 1 but greater than 0.2 w23,24x, resulting in a slightly distorted spectrum with the central portion Žnear the incident laser wavelength. depressed compared with a Gaussian shape. If one then applies the conventional plotting procedure to the determination of Te and ne , Te will be over-estimated, while ne will be under-estimated. Fortunately, there exists an unambiguous relationship between each pair of Te and ne values and the shape and total intensity of the corresponding Thomson scattering spectrum. Consequently, look-up tables or a curvefitting technique can be used to retrieve real Te and ne values when a Maxwellian velocity distribution is assumed w18᎐20x.
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
In contrast to the situation in atmosphericpressure ICPs, glow discharges, microwaveinduced plasmas, ECR processing plasmas and other industrial plasmas are more likely to scatter light in a purely incoherent manner, due to their much higher Te and much lower ne values. Unfortunately, Thomson scattering spectra obtained from these plasmas, in general, show even more serious distortion from a Gaussian shape. Since the scattering is incoherent, these distortions are probably caused by a non-Maxwellian electron energy distribution induced by either insufficiently frequent collisions, non-thermal heating mechanisms, or both. Unusual distortions in incoherent Thomson scattering spectra can be found in some fusion plasmas, especially in magnetic-confinement devices such as tokamaks and stellarators w45᎐51x, in which the plasma behavior is affected by strong magnetic fields, fast current pulses or disruptions. In general, these distortions cannot be explained by non-thermal features of the plasma, but can be attributed to anisotropy in velocity space, as will be discussed in Section 4. In this case, it is necessary for a Thomson scattering experiment to be performed in several viewing directions, allowing the differential scattering wave vector k to be parallel, perpendicular and oblique with respect to the magnetic field, for example, in order to probe the full f Ž ¨ .. In principle, the f Ž ¨ . can be uniquely determined from the full scattered intensity I Žk., but determining it in detail can be very difficult w52x. In an attempt to correctly interpret non-Gaussian-shaped incoherent Thomson scattering spectra, we offer in the present paper a clear picture to illustrate how electrons with the same energy contribute to intensities at different wavelength channels of a Thomson scattering detection system, and how electrons with different energies scatter light into the same wavelength channel. Based on this understanding, a simple approach is introduced to derive the electron energy distribution function ŽEEDF. directly from a Thomson scattering spectrum when an isotropic electron distribution in velocity space can be assumed. Results of non-Maxwellian EEDFs obtained from Thomson scattering spectra for atmospheric-pres-
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sure helium microwave-induced plasmas using this approach will be shown, and discussions of the results and a number of key points concerning the interpretation of non-Gaussian-shaped incoherent Thomson scattering spectra will be given.
2. Theoretical considerations For a given electron with a velocity ¨ in the viewing volume of a Thomson scattering system, the wavelength shift, ⌬ of electron-scattered light from the incident laser line, 0 , is proportional to the projected velocity of the electron along the differential scattering wave vector k, and can be calculated from a relationship of the form ⌬ s y 0 s Ž k ⭈ v . 02r Ž 2 c .
Ž1.
with k s k0 y ks
Ž2.
where is the wavelength, k0 and ks are the wave vector of the incident radiation and a vector along the observation direction with the same length, respectively, and c is the speed of light. The length of k is therefore k s 2sin Ž r2. k0 s 2 sin Ž r2. 2 r0 s 4 sin Ž r2. r0
Ž3.
where k0 is the length of k0 , and is the angle between k0 and ks . From Eqs. Ž1. and Ž3. one obtains ⌬ s 20 Ž ¨ krc . sin Ž r2.
Ž4.
where ¨ k is the projected velocity of the given electron along k. Eq. Ž4. can be written as ¨ k s c Ž ⌬r0 . r Ž 2sin Ž r2..
Ž5.
It can be seen from Eq. Ž5. that for a given wavelength shift ⌬ Ži.e. for a given Thomsonscattering wavelength channel ., electrons with a velocity ¨ equal to or larger than ¨ k will have a
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M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
Fig. 1. Velocity space Ž Vx , Vy , Vz . separated into spherical and planar layers, corresponding to electron groups with different energies and with different projected velocities along the differential scattering wavelength vector k. ¨ i , i and f Ž ¨ i . with i s 1, 2, . . . are electron velocity, the corresponding wavelength shift and the f Ž ¨ . at different layers or wavelength channels, respectively. See text for discussion.
chance to scatter light into that channel, depending on where they are located in velocity space, whereas electrons with a velocity lower than ¨ k will not be able to contribute to the intensity in that channel. Fig. 1 shows more clearly the relationship between the electron distribution in velocity space and the amount of light scattered into each wavelength channel defined by i with i s 1, 2, . . . For the sake of simplicity, the coordinate axes Vx , Vy and Vz have been chosen so that k coincides with Vx . Also, for brevity let us now use i instead of ⌬i to designate the wavelength shift corresponding to channel i . Let us now divide velocity space into different regions in two ways. First, the space is separated into a series of spherical shells, each with a thickness equivalent to the width of a wavelength channel; this approach is shown in Fig. 1. Second, the space is equally divided by a set of planes perpendicular to the Vx axis with the interval also equivalent to the width of a wavelength channel. The spherical and planar layers are defined by their corresponding velocity and projected veloc-
ity onto k, respectively, both of which are represented by the same symbol, ¨ n . Clearly, electrons in the spherical shell ¨ i will make a contribution to the Thomson scattering signal of every wavelength channel j , when ¨ j F ¨ i ; on the contrary, no electrons in the spherical layer ¨ j can contribute to the Thomson scattering signal in any wavelength channel i , when ¨ i ) ¨ j . In addition, the amount of light scattered by electrons in a spherical shell ¨ i into wavelength channel j is proportional both to the value of the f Ž ¨ . in that shell w f Ž ¨ i .x and to the volume of the ring defined by the region intercepted by the spherical shell ¨ i and the planar layer ¨ j Ž Vj,i .. The overall Thomson scattering intensity at that wavelength channel w I Ž j .x will then be proportional to the total amount of light scattered by all electrons located in the different shells Žhaving different energies. in the same planar layer ¨ j , i.e. ⬁
I Ž j . A Ý f Ž ¨ i . Vj,i
Ž6.
isj
As an example, let us calculate Vn, nq2 , which is the volume of the ring intercepted by planar layer ¨ n and spherical shell ¨ nq2 as shown in Fig. 1, where ABCD defines the cross-section of the ring in the Vx ᎐Vy plane. Any cross-section of the ring perpendicular to axis Vx is a planar ring with EG and FG as its outer and inner radii, respectively, as shown in the figure. Here, E and F lie anywhere on the curves AD and BC, respectively, but correspond to the same ¨ x coordinate. The area of this planar region can be calculated as Area s Ž EG2 y FG2 . s wŽ OE2 y OG2 . y Ž OF2 y OG2 .x s Ž OE2 y OF2 . s Ž ¨ nq 2 q ⌬¨r2 . 2 y Ž ¨ nq1 q ⌬¨r2 . 2 s wŽ ¨ nq 2 q ¨ nq1 q ⌬¨ .Ž ¨ nq2 y ¨ nq1 .x s w 2 ¨ nq 2 ⌬¨ x s 2 ¨ nq2 ⌬¨
Ž7.
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
I Ž n . y I Ž nq1 . A f Ž ¨ nq1 . Vn ,nq1 q f Ž ¨ n . Vn ,n
The volume of the region is then
Vn ,nq2 s
yf Ž ¨ nq 1 . Vnq1,nq1 s f Ž ¨ nq1 . 2 Vnq1 ⌬¨ 2
1 ¨ nq ⌬¨ 2 2 ¨ nq2 ⌬¨ d¨ s 2 ¨ nq2 Ž ⌬¨ . 2 1 ¨ ny ⌬¨ 2
qf Ž ¨ n . Ž ¨ n q ⌬¨r6 . ⌬¨ 2 yf Ž ¨ nq 1 . Ž ¨ nq1
H
Ž8. Eq. Ž8. indicates that for a given spherical layer ¨ i , the volume of each intercepted shell is the same if j - i, and is proportional to ¨ i . In other words, electrons in the spherical shell ¨ i scatter
the same amount of light into each wavelength channel for which ¨ j - ¨ i . The volume of the last intercepted region in the spherical layer Vi,i Ž ¨ i ., shown as the shaded portion in Fig. 1, can be calculated as follows, Vi ,i s
⌬¨
H0
½ž
1 ⌬¨ 2
¨i q
/ y ž ¨ y 12 ⌬¨ q x / 5 d x 2
2
i
¨
2
i
q Ž ¨ n q ⌬¨r6 . f Ž ¨ n .x ⌬¨ 2 Ž 11. The significance of Eq. Ž11. is that the difference between the Thomson scattering signals in two adjacent wavelength channels contains information that is solely related to the values of the EVDF Ž ¨ . corresponding to these two channels w f Ž ¨ n ., f Ž ¨ nq1 .x. For an infinitesimal ⌬¨ , Eq. Ž11. becomes Ž 12.
i
1 s ¨ i q ⌬¨ ⌬¨ 2 6
ž
q⌬¨r6 . ⌬¨ 2 s w Ž ¨ nq 1 y ⌬¨r6 . f Ž ¨ nq1 .
I Ž n . y I Ž nq1 . A ¨ nq1 f Ž ¨ nq1 . q ¨ n f Ž ¨ n .
⌬ H0 2 ¨ ⌬¨ y Ž2 ¨ y ⌬¨ . xy x 4 d x
s
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/
Ž9.
According to Eq. Ž6., the difference between the Thomson scattering signals in two adjacent wavelength channels ¨ n and ¨ nq 1 is I Ž n . y I Ž nq1 . A
⬁
⬁
isn
isnq1
Ý f Ž ¨i . Vn ,i y Ý
=Vnq 1 ,i s f Ž ¨ n . Vn ,n q
f Ž ¨i .
⬁
Ý
f Ž ¨ i . Vn ,i
isnq1
yf Ž ¨ nq 1 . Vnq1,nq1 y
⬁
Ý
f Ž ¨i .
isnq2
= Vnq 1,i
Ž 10.
Note that on the right hand side of Eq. Ž10., Vn,i in the first summation is equal to Vnq1,i in the second, as has been verified by Eq. Ž8., so all the terms in the first summation are canceled by those in the second summation except the first term f Ž ¨ nq 1 .Vn,nq1. Therefore, applying Eqs. Ž8. and Ž9., we have
Note that ¨ f Ž ¨ . is exactly the form of an energy distribution function of electrons wEEDFx with an isotropic velocity distribution in velocity space w f Ž ¨ .x, regardless of whether or not the electrons obey a Maxwellian distribution. This point can be understood as follows. The number of electrons with velocity between ¨ and ¨ q d¨ is 4 ¨ 2 f Ž ¨ .d¨ . As the electron energy Es Ž1r2. m¨2 , a velocity change d¨ corresponds to an energy change d E s m¨ d¨ , which means that for a constant d¨ the corresponding d E is not constant but proportional to ¨ . In other words, the abovementioned number of electrons corresponds to an energy interval which is proportional to ¨ . Therefore, the EEDF should be proportional to that number divided by ¨ to keep d E constant, resulting in the form of ¨ f Ž ¨ . for the EEDF. As a result, Eq. Ž12. indicates that the difference between the Thomson scattering signals in two adjacent wavelength channels is a direct reflection of the value of the EEDF at the energy corresponding to the wavelength between the two channels, as long as the bandwidth of the wavelength channel is sufficiently small. If, however, the wavelength channel is relatively wide, then the signal difference I Ž n . y IŽ nq 1 . is proportional to an average value of
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the EEDF at the energy corresponding to the wavelength shift between the two channels. Clearly, the EEDF can be obtained simply by plotting the differences in Thomson scattering signals between adjacent wavelength channels as a function of the corresponding energy, which is in turn proportional to the square of the wavelength shift. So plotting I Ž n . y I Ž nq1 . vs. the square of wŽ n q nq1 .r2x will yield the shape of the EEDF. Normalizing the plot to the measured electron number density ne will give absolute values of the EEDF. In addition, the electron velocity distribution function EVDF is proportional to ¨ 2 f Ž ¨ ., whereas I Ž n . y IŽ nq1 . is proportional to ¨ f Ž ¨ .. As a result, the Thomson scattering spectrum also directly provides information about the EVDF, as shown in Fig. 2, where the shaded area is a measure of the EVDF value at a velocity corresponding to Ž n q nq1 .r2. For a wavelength channel of infinitesimal width, I Ž n . y I Ž nq1 . is proportional to the derivative of the profile of the Thomson scattering spectrum. Therefore, an alternative approach to deriving the EEDF, EVDF and f Ž ¨ . is to use the following relationships: EEDFA yI⬘ Ž . , when ) 0; I⬘ Ž . , when - 0
Ž 13.
EVDFA y I⬘ Ž .
Ž 14.
f Ž ¨ . A yI ⬘ Ž . r
Ž 15.
where I⬘Ž . is the derivative of the Thomson scattering spectrum with respect to wavelength shift. Obviously, the resolutions of all of the distribution functions obtained using the abovementioned approaches depend strongly on the resolution of the wavelength channels employed in the experiment.
3. Non-Maxwellian electron energy distributions in a helium microwave-induced plasma Thomson scattering data published in an ear-
Fig. 2. A Thomson scattering spectrum with I Ž n . and I Ž nq1 . being the intensities at any two adjacent wavelength channels n and nq1 , respectively. The shaded area is proportional to the average value of the EVDF at the wavelength shift Ž n q nq 1 .r2.
lier paper were used to derive electron energy distributions by means of the approach described in the preceding section wsee Figs. 4, 5, 12 and 13 in Huang et al. w25x for detailsx. These Thomson scattering data, obtained from a helium-supported microwave plasma torch ŽMPT., yielded the electron energy distributions shown in Figs. 3 and 4. The helium plasmas corresponding to Figs. 3 and 4 were operated at forward powers of 100 and 350 W and observed at heights of 4 and 7 mm above the top of the MPT, respectively. The electron temperatures shown in the figure captions were determined by the Maxwellian distributions that best fit the bulk of the scattering spectrum, and were used to calculate the corresponding Maxwellian EEDFs for comparison. Both the derived EEDFs and the Maxwellian EEDFs were normalized to the electron number density measured by Thomson scattering. Figs. 5᎐8 show the corresponding EVDF and f Ž ¨ . for both sets of MPT conditions. All the data in these figures were also normalized to the measured ne . Significant deviations from a Maxwellian distribution can be clearly seen in all of the distribution functions. The departures appear to be more
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
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Fig. 3. Electron energy distribution function EEDF Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 100 W and at an observation height of 4 mm above the top of the torch. Ž`. Corresponding EEDF values from a Maxwellian distribution with measured Te s 21 500 K and ne s 6 = 1013 cmy3. Data are normalized to the measured ne .
significant for the plasma at a forward power of 350 W than at 100 W. In general, the population of electrons with energies between 2 and 6 eV is
depressed, while electrons with energies between 0.5 and 2 eV are overpopulated Žsee Figs. 3 and 4.. More importantly, yet not obvious in these
Fig. 4. Electron energy distribution function EEDF Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 350 W and at an observation height of 7 mm above the top of the torch. Ž`. Corresponding EEDF values from a Maxwellian distribution with measured Te s 20 000 K and ne s 6.5= 1013 cmy3 . Data are normalized to the measured ne .
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M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
Fig. 5. Electron velocity distribution function EVDF Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 100 W and at an observation height of 4 mm above the top of the torch. Ž`. Corresponding EVDF values from a Maxwellian distribution with measured Te s 21 500 K and ne s 6 = 1013 cmy3 . Data are normalized to the measured ne .
Fig. 6. Electron velocity distribution function EVDF Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 350 W and at an observation height of 7 mm above the top of the torch. Ž`. Corresponding EVDF values from a Maxwellian distribution with measured Te s 20 000 K and ne s 6.5= 1013 cmy3 . Data are normalized to the measured ne .
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
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Fig. 7. Electron distribution in velocity space f Ž ¨ . Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 100 W and at an observation height of 4 mm above the top of the torch. Ž`. Corresponding f Ž ¨ . values from a Maxwellian distribution with measured Te s 21 500 K and ne s 6 = 1013 cmy3 . Data are normalized to the measured ne .
Fig. 8. Electron distribution in velocity space f Ž ¨ . Ž'. derived from the Thomson scattering spectrum of an atmospheric-pressure helium MIP sustained on an MPT at a forward power of 350 W and at an observation height of 7 mm above the top of the torch. Ž`. Corresponding f Ž ¨ . values from a Maxwellian distribution with measured Te s 20000 K and ne s 6.5= 1013 cmy3. Data are normalized to the measured ne .
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M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
figures, the total number of electrons with energies equal to or larger than 6.6 eV is dramatically increased for both conditions in comparison with that predicted by a Maxwellian distribution. This point can be clarified by comparing the Thomson scattering intensity at the last channel Žthe greatest wavelength shift. with the value calculated from a Maxwellian distribution. This very last channel was centered 5.0 nm from the incident laser wavelength, which corresponds to an electron energy of 6.6 eV or an electron velocity of 1.53= 108 cm sy1 . The intensity of Thomson scattering in the last wavelength channel for the plasma at 100 W forward power is higher than its corresponding Maxwellian value by a factor of 1.8 Žsee Fig. 13 in Huang et al. w25x., whereas that with 350 W of forward power is 3.1-fold higher than its corresponding Maxwellian value Žsee Fig. 12 in Huang et al. w25x.. In other words, electrons in these plasmas with energies G 6.6 eV scatter 1.8 and 3.1 times more light, respectively, into all Thomson scattering wavelength channels than Maxwellian distributions would predict, because their contribution to each wavelength channel is the same, as has been proven in Section 2. Since the amount of scattered light is proportional to the number of electrons that take part in the scattering, the number of electrons with energy G 6.6 eV in these plasmas is therefore some 1.8and 3.1-fold higher than the Maxwellian values, respectively. Moreover, the tail portion of the EEDF in microwave-induced plasmas might extend still further in comparison with the Maxwellian distribution, a trend already apparent in Figs. 3᎐8 Ži.e. the calculated distribution decreases much more slowly with E or ¨ in the wing portions than does the Maxwellian distribution.. Accordingly, the above estimation of the number of high-energy electrons is probably conservative, which might explain in part why microwave-induced helium plasmas are highly efficient in the excitation of non-metallic elements, especially the halogens. Clearly, obtaining details of the EEDF for electron energies larger than 6.6 eV is impossible with the current data and will require Thomson scattering at even greater wavelength shifts to be
probed with high signal-to-noise ratio and wavelength resolution. Concerning the f Ž ¨ ., it is interesting to note, as shown in Figs. 7 and 8, that the electrons in microwave-induced plasmas are not most concentrated in the center of velocity space, as are electrons that possess a Maxwellian distribution. Instead, they reach a maximum in a spherical layer at a distance of some 4 = 107 cm sy1 from the space center. Of course, the occurrence of the peak in the f Ž ¨ . might be, in part, also attributable to the loss of electron population at higher velocities, as can be seen in the tail portions in Figs. 7 and 8.
4. Discussion The Thomson scattering data used in this paper were collected at 16 wavelength channels, each of which was 0.3 nm wide, and the wavelength shift between any two adjacent channels was also 0.3 nm. This wavelength resolution should be good enough to detect a substantial deviation from a Maxwellian distribution. The systematic error caused by the spectral response of the detection system could be more serious than any error due to the width of the wavelength channel. For this reason, a single detector was used to measure Thomson signals at all different channels by scanning the wavelength of the monochromator, and the spectral response was calibrated using a tungsten ribbon lamp. As a result, the systematic error should be less than 1%. As for the noise on the measured Thomson scattering signals, the relative standard deviation was approximately 2᎐5%, depending on the wavelength shift. The larger the wavelength shift, the higher the relative standard deviation. Since the deviations from Maxwellian distributions shown in Figs. 3᎐8 can be as large as 20᎐30%, also depending on the wavelength shift, errors in the measurement should not be a dominant factor that causes these deviations. As described in Section 2 and clearly shown in Fig. 1, the intensity of an incoherent Thomson scattering spectrum at any wavelength shift cannot be greater than that at any shorter wavelength shift, when the f Ž ¨ . is isotropic. In other
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
words, the value of the derivative of an incoherent Thomson scattering spectrum should be no greater than zero for positive wavelength shifts, and no less than zero for negative wavelength shifts. Of course, when the signal-to-noise ratio is poor, an experimental Thomson scattering profile might appear very noisy, resulting in some positive derivative values for positive wavelength shifts, or some negative derivative values for negative wavelength shifts. In this real world situation, the EEDF obtained using the derivative of a Thomson scattering spectrum will include some negative values. To eliminate this error, appropriate curvesmoothing procedures should be applied to the Thomson scattering spectrum before the EEDF is derived from it using Eqs. Ž13. ᎐ Ž15.. In this sense, directly taking the differences between signals of adjacent wavelength channels as a function of the wavelength squared is probably a better way to obtain the EEDF, as proposed in Section 2, instead of taking the derivative of the Thomson scattering profile. Any irregularities in the incoherent Thomson scattering spectrum, such as satellites, dips or a shift of the peak in a Gaussian profile w48᎐51x that cause anomalous derivative values and that cannot be accounted for solely by a low signal-tonoise ratio should be interpreted as a reflection of anisotropy of the f Ž ¨ .. Imagine, for example, that for some reason electrons in the planar layer ¨ n in velocity space, shown in Fig. 1, have an unusually low concentration. In this situation, a dip might appear in the Thomson scattering spectrum at a wavelength shift n . On the contrary, if for some reason electrons gather in that layer, then a satellite might emerge at that wavelength shift. When a full incoherent Thomson scattering spectrum looks fairly symmetric, but the central peak is not located at the incident laser wavelength, the plasma must be drifting at a certain velocity. In this case, Eq. Ž5. can be used to determine the projected component of the drift velocity along k, with ⌬ as the wavelength shift of the central peak. To determine the three-dimensional drift velocity vd , Thomson scattering spectra with k along three different directions are
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generally required. If the plasma is known to drift in a given direction, the Thomson scattering measurement can then be designed so that k is coincides with ¨ d , or at least so k is not perpendicular to ¨ d , in order to acquire complete information about the f Ž ¨ .. It is well known that for electrons with a Maxwellian distribution, the incoherent Thomson scattering spectrum has the same shape as the f Ž ¨ ., i.e. Gaussian. It is, therefore, said that incoherent Thomson scattering spectra directly reflect the f Ž ¨ .. However, this perfect coincidence is unique to Maxwellian distributions, which can be clarified as follows. A Gaussian shaped incoherent Thomson scattering spectrum can be expressed in the form I Ž . A eyC
2
Ž 16.
where C is a constant. Combining Eqs. Ž15. and Ž16. yields f Ž ¨ . A yI ⬘ Ž . r s 2CeyC r s 2CeyC 2
2
Ž 17. Comparing Eqs. Ž16. and Ž17., it can be seen that the f Ž ¨ . is indeed identical in shape to I Ž .. This coincidence, however, will no longer hold when electrons possess a non-Maxwellian distribution. For example, if an incoherent Thomson scattering spectrum can be expressed in the form I Ž . A ay b , when 0 F F arb; I Ž . s 0, when ) arb
Ž 18.
where a and b are constants, as shown in Fig. 9, the corresponding f Ž ¨ . should be f Ž ¨ . A yI ⬘ Ž . r s br , when 0 F F arb; 0, when ) arb Ž 19. or f Ž ¨ . A c1 br¨ , when 0 F ¨ F c1 arb; when ¨ ) c1 arb
Ž 20.
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
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Fig. 9. A Thomson scattering spectrum in the form: I Ž . A a y b, when 0 F F arb; I Ž . s 0, when ) arb.
where c1 s ¨r is a constant for a particular experiment and can be determined by Eq. Ž5., as shown in Fig. 10. In this case, the f Ž ¨ . looks entirely different from I Ž . in shape. If a theoretical model assumes that the plasma consists of two electron groups, each of which possesses its own Maxwellian distribution defined by a different temperature Žusually a high temperature and a low temperature. w53x, then the incoherent Thomson scattering spectrum is the sum of two Gaussian profiles in the form I Ž . A a1e
2
yC 1
q a2 e
yC 1 2
Ž 21.
where a1 , a2 , C1 , and C2 are constants. The corresponding f Ž ¨ . should be f Ž ¨ . A y I⬘ Ž . r A a1C1eyC 1 q a2 C2 eyC 2 2
s C1 a1eyC 1 q a2
ž
2
C2 yC 1 2 e C1
2
/
Ž 22.
Fig. 10. f Ž ¨ ., derived from the Thomson scattering spectrum shown in Fig. 9, in the form: f Ž ¨ . A c1 br¨ , when 0 F ¨ F c1 arb; 0, when ¨ ) c1 arb.
It can be seen by comparing Eq. Ž22. with Eq. Ž21. that the shape of the f Ž ¨ . is different from I Ž ., although both electron groups possess Maxwellian distributions. In addition, because both electron groups scatter light to all wavelength channels, the Thomson scattering spectrum should be interpreted with caution. For example, the temperatures of the two electron groups can not be obtained by simply plotting the logarithmic scattering intensity vs. the wavelength shift squared at the central portion Žfor the low Te . and at the tail portion Žfor the high Te ., respectively. Instead, a curve fitting technique based on the form of Eq. Ž21. should be employed to determine a1 , C1 , a2 and C2 , where a1 and a2 are related to the weights of the two Maxwellian components, and C1 and C2 are related to the temperatures of the two electron groups, respectively.
5. Conclusions A picture is given, in which velocity space is separated into a series of spherical shells as well as a series of planar layers, to clearly illustrate how electrons with different energies scatter light into the same wavelength channel, and how electrons with the same energy scatter light into different wavelength channels in an incoherent Thomson scattering experiment. It has been proven that electrons with a given energy scatter the same amount of light into all other wavelength channels that correspond to energies smaller than the given electron energy, but make no contributions to wavelength channels with corresponding energies larger than the given electron energy. Therefore, the difference in Thomson scattering intensities between any two adjacent wavelength channels is related solely to the EEDF values at the energies corresponding to the wavelength shifts of these two channels. Based on this realization, a procedure is outlined to derive the EEDF simply by plotting the difference between the intensities of two adjacent wavelength channels as a function of the square of the wavelength shift. Absolute values of the
M. Huang et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 1397᎐1410
EEDF can then be obtained by normalizing the plot to the measured electron number density. The EEDFs of microwave-induced helium plasmas sustained at atmospheric pressure at forward powers of 100 and 350 W, respectively, show significant deviations from Maxwellian distributions over the entire wavelength range, which is equivalent to electron energies of 0.1᎐6.6 eV. A portion of the electrons with energies in the range of approximately 2᎐6 eV seem to be shifted towards both lower and higher energy values. The ratio of the Thomson scattering intensity measured at the greatest wavelength shift to the calculated Maxwellian value can be considered to be a minimum estimation of the ratio of the total number of electrons with energies larger than the energy corresponding to that channel Ži.e. 6.6 eV for the present work. to the Maxwellian value. Accordingly, high energy electrons Ži.e. EG 6.6 eV. in these plasmas are overpopulated by a factor of at least 1.8 and 3.1 for forward powers of 100 and 350 W, respectively, compared with Maxwellian distributions. Such an overpopulation of high energy electrons explains in part the high efficiency of helium microwave-induced plasmas in the excitation of non-metal elements, especially the halogens. The f Ž ¨ . values obtained show that electrons in these plasmas are not most abundant at the center of velocity space, but reach a maximum in a spherical layer at a distance of some 4 = 107 cm sy1 from the space center. In addition, the bulk of the EVDF is slightly compacted and the maximum value is shifted toward lower velocity values for both microwave plasma conditions, compared with a Maxwellian distribution. The coincidence that an incoherent Thomson scattering spectrum has the same shape as that of the f Ž ¨ . is unique to a Maxwellian electron distribution. Therefore, incoherent Thomson scattering spectra from plasmas with non-Maxwellian electron distributions do not directly reflect the f Ž ¨ ., and should be interpreted with caution. Furthermore, satellites, dips or other irregularities, which cause anomalous derivative values Žsee Section 4. of the scattering intensity profile, should not appear in an incoherent Thomson
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scattering spectrum, if the f Ž ¨ . is assumed to be isotropic. If such irregularities occur, they could be attributable to a low signal-to-noise ratio or to anisotropy in velocity space.
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