ASSOCIATIE EURATOM-FOM
FOM-INSTITUUT VOOR PLASMAFYSICA
RIJNHUIZEN - NIEUWEGEIN - NEDERLAND
CYCLOTRON RADIATION FROM THERMAL AND NON-THERMAL ELECTRONS IN THE WEGA-STELLARATOR
by
H.W. Piekaar and W.R. Rutgers
Rijnhuizen Report 80-128 ASSOCIATIE EURATOM-FOM November 1980
FOM-INSTITUUT VOOR PLASMAFYSICA
RIJNHUIZEN - NIEUWEGEIN - NEDERLAND
CYCLOTRON RADIATION FROM THERMAL AND NON-THERMAL ELECTRONS IN THE WEGA-STELLARATOR
by
H.W. Piekaar and W.R. Rutgers'
Rijnhuizen Report 80-128
* Present address KEMA-Laboratory. Arnhem.
This work was performed as part of the research programme of the association agreement of Euratom and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver-Weten schappelijk Onderzoek" (ZWO)and Euratom. CYCLOTRON RADIATION FROM THERMAL AND NON-THERMAL ELECTRONS IN THE WEGA-STELLARATOR by H.W. Piekaar and W.R. Rutgers Association Euratom-FOM FOM-Instituut voor Plasmafysica Rijnhuizen, Nieuwegein, The Netherlands
ABSTRACT
Electron cyclotron radiation measurements on the WEGA- stellarator are reported. Emission spectra around 2to and
3oJce were measured with a far-infra-red spectrometer and InSb detectors. When the plasma loop voltage is high, runaway electrons give rise to intense broad-band emission. Runaway par ticles can be removed by increasing the plasma density. For low loop voltage discharges the electron temperature profile was deduced from thermal emission around 2w . In spite of the ce r low E-field, runaway particles are still created and pitch- angle scattered because "pe/w^^l. From non-thermal emission below 2u and 3u> the energy and number of non-thermal particles could be calculated, and was found to be in agreement with existing theories. I. INTRODUCTION
The WEGA-device is a small-aspect-ratio medium-sized stel- larator (R = 72 cm, a = 19 cm) with £ = 2, m = 5 helical windings on the vacuum vessel. A description of the experimental set-up is given in Ref. 1. Tokamak and stellarator discharges with the same value of the rotational transform i (at the limiter) can be produced. Titanium gettering of the wall of the vacuum vessel reduces the impurity content, and so the loop voltage, considerably. First results on lower hybrid heating experiments in the stella rator configuration are reported in Ref. 2. -I-
The stellarator is well-provided with diagnostic equipment as is shown in Fig. 1. This report deals with measurements of electron cyclotron radiation in the fre quency range of 70-140 GHz with a multi-channel far-infra red spectrometer (see photo 1). The equipment was installed and operated by a visiting team from the Institute for Plasma Physics in Nieuwegein, The Netherlands. Measurements were col lected during three short measuring campaigns in the winter of 78/79. The experimental set-up and some special problems of electron temperature measurements in a stellarator geometry are discussed in section II. In section III we present measure ments of electron cyclotron emission for different modes of machine operation as well as a temperature profile deduced from thermal emission at 2d) (ui is the electron gyrofrequency) . ce Apart from thermal emission around 2w , suprathermal radiation was detected below 2a> and 3w . In section IV the number and ce ce energy of these non-thermal electrons is inferred. Finally, in section V,these numbers are compared with theoretical ex pectations,followed by some concluding remarks in VI. Radiometer (4 Kanal) schneller GaseinlaB 1 Kanal \ Ho
Umladungsanaiyse 15 Kanal»
Positionsspuie
Limiler
Fig. 1. Diagnostic equipment on the WEGA-stellarator. (From: Jahresbericht 1978, Max-Planck Institut für Plasmaphysik, Garching). I I
A photograph showing the far-infra-red diagnostic system linked to the WEGA-stellarator. P polychromator; D screened box with InSb detectors; L light pipes to the diagnostic port. The coaxial cables passing in front of our system connect the HF-generators to the antennas. -4-
II. DIAGNOSTIC SYSTEM
The WEGA-machine is operated with a toroidal magnetic
field on axis of BQ = 1.44 tesla. Therefore, the electron
cyclotron frequency fQ = qB/ 2™ on axis is 40.3 GHz. When re- absorption of radiation is sufficiently large, the plasma emits like a black-body which means that the intensity is directly proportional to the (electron) temperature T if the electron velocity distribution function is (close to) a Max well ian. For WEGA plasma parameters as given in table I the optical depth is calculated to be larger than 1 for radiation at 2 ^ce (A is around 3.5 mm) for 64 cm < R < 74 cm, where R is the distance to the torus axis.
TABLE I
Machine parameters Plasma parameters B = 1.44 tesla 25 kA (stellarator) o (50 kA, tokamak) 1=2, m=5 helical windi igs 2-3 volt XH.W. " 14° kA loop R =0.72 meter Z 3 o eff = a,. = 0.15 m 100 IBS lim pulse a , = 0.19 m T 450 eV wall eo T. 125 eV io n = 2.4'1019 m~3 eo Si = 12-13 cm
Because of the low electron temperature, Doppler and relativistic line broadening are smaller than 1% for obser vation nearly perpendicular to the B-field. On the other hand, line broadening of the order of 20% occurs due to the varia tion of the magnetic field across the discharge tube. In the tokamak mode of operation the poloidal B-field is so small that the total B-field has roughly the 1/R dependence of the
toroidal field BT. Along the line of sight (22*5° downwards) through our diagnostic port the position where the plasma is emitting 2w radiation is therefore easily calculated f rom + B(r) ^ l/(rcos22^ RQ)> where r is the distance to the minor axis of the torus. For the stellarator mode of operation the -5- poloidal field cannot be neglected and was calculated by com puter*. Lines of constant (B* + B2) are not at all vertical in this case as is shown in Fig. 2. The gradient in the B-field in the outer part of the torus is smaller, and in the inner part larger, than in the tokamak mode. The flux surfaces are no longer circular as is indicated in the same figure. Cyclotron radiation is collected with a quartz lens and a 2.5 cm diameter circular waveguide acting as a light-pipe. During the last experiments this lens was replaced by a fused quartz vacuum window because the lens was covered by a layer of titanium. Titanium gettering was used to reduce the impurity content of the plasma and to reduce the density increase during lower hybrid heating experiments. A grid polarizer (Cambridge Physical Sciences) could be inserted between vacuum window and entrance to the light-pipe system. The setup of the four-channel grating spectrometer and the calibration source to analyze the cyclotron emission is shown schematically in Fig. 3. An optical switch is used to compare the plasma emission with radiation from a microwave noise tube (T=11.000 K) or a klystron (f = 69 GHz). The spectrometer was calibrated over the whole frequency band with the noise tube, whereas the klystron was used for a day-to-day check of the spectrometer setting and the detector sensitivities. The klystron and the calorimeter (TRW) were also used to measure the transmission of the system. The attenuation from plasma up to the detectors turned out to reach 17 dB due to some misalignment of the light-pipes, the gaps for the vacuum window and the optical shutter, and the low trans mission of the spectrometer at long wavelengths. The determina tion of the absolute value of the plasma emissivity is not ex pected to be better than 3 dB. The four-channel grating spectrometer is of the Czerny- Turner type with blazed gratings (cut at an angle of 25°, A../d * 0.825) as dispersive elements. Gratings with a groove distance of 4 mm and 3.25 mm were used for the frequency band of 70-100 GHz and 90-140 GHz respectively. The frequency cali bration was performed by measuring radiation from a 4-mm and a 2-mm wavelength klystron in first and second order with four different gratings. The dispersion curves for the four channels are shown in Fig. 4.
* The authors are indebted to Drs. G. Pacher and M. Lipa for providing us with these calculatioi5. -6-
lines of constant B flu* surface
vacuum vassal steilarator windings i.d. 38 cm
toroidal field coil to polychromator
Fig. 2. Topology of the magnetic field for standard steilarator operation. The dotted areas indicate the plasma volumes seen by the four detectors for a given setting of the grating spectrometer.
klystron or noise tube
hollow mirror light guides 1" diameter
vacuum windows X-U/r-V U switch .i-.i quartz lens or window
shutter box screened box with low noise pre-amplifiers 1 spectrometer (evacuated)
Fig. 3. Schematic of our mm-wave diagnostic system. -7-
The wavelength of the klystrons was measured to high accuracy with a Fabry-Perot interferometer. Therefore, the wavelength calibration of the spectrometer is better than 1%. The resolu tion A/A A was also measured. For the 4-mm grating the resolution is roughly proportional co the square root of the wavelength instead of proportional to the wavelength as we measured before for shorter wavelength. The reason seems to be the limited num ber of grooves and the large focal spot. The etendue of the spec trometer is Adfl=2*10~s m2 sr. The radiation is detected with InSb crystals at a temperature of 2.5 kelvin in a magnetic field of 1 tesla. The responsivity is typically 500 V/W. The noise level, referred to the preamplifier input, is 2 uV at a band width of 30kHz . With the above-mentioned etendue and attenua tion and a bandwidth of 2.5 GHz a typical number for the minimum detectable emission is 3»10~12 watt/m2 sr Hz or about 100 eV at a B-field of 1.5 tesla. This is equivalent to a noise level of only 12.5 eV at B = 3 tesla and the same frequency resolution. A reduction in the transmission loss of the spectrometer and the light-pipe system by at least a factor five must be possible, which would reduce the noise level of this FIR spectrometer to only a few eV for high B-field machines.
-" -
- ,X X ,wX X - S*y*S*s* X = 2115 pm /yV/ * d = 4000 pm — d 32 / /\// * " 50 pm f*/} 0&- 2450 pm S/y ' X = 4330 pm '//// x d= 4000 pm '///
0 1000 2000 3000 4000
»- number of steps grating drive
Fig. 4. The dispersion curves for the four frequency outputs from the far-infra-red monochromators/ measured with two klystrons and three different gratings in first and second order. -ö-
III. EXPERIMENTAL RESULTS
Emission measurements were started in the tokamak mode of operation. The level of emission greatly depends on plasma density and impurity content because the machine is operated close to a regime in which high energetic electrons (runaways) are easily created. In Fig. 5 the loop voltage V, the plasma current I and the average density n are shown as a function of time for a tokamak shot in which the density was slowly decreasing for t ^ 50 ms follower by a 15% increase later in time. Plasma emission measured with three InSb detectors is shown in the same figure. The emission starts to grow at t = 25 ms when the density falls below 1,65*10l5 m~3 and is soon far above the thermal level (10 IJV for channel III). From many shots under similar plasma conditions and different settings of the gratii.g spectrometer we found that the frequency spectrum of this non-thermal emission was rather flat. This points to radiation from runaway electrons as the large relativistic broadening and the frequency downshift of emission from (mildly) relativistic electrons can smear out the frequency spectrum. When the density rises again, for t > 55 ms, the emission decreases, presumably due to the strong reduction in the runaway creation rate. Similar obser vations were done during HF-heating experiments as shown in Fig. 6. The line density nearly doubles from t = 44 ms to t = 54 ms, i.e. during the time the HF-generators for plasma heating at the lower hybrid frequency are on. The flux of hard X-rays illustrates the buildup of a hot electron component for t > 25 ms, the removal of hot particles during HF-heating, and the reappearance of this component for t > 65 ms. Note that the hard X-ray flux has its maximum at t = 5 7 ms whereas the synchrotron radiation attains its peak even before t = 51 ms. The reason is that for the hard X-rays to be emitted the hot electrons have to hit the discharge vessel first (the position of the detector is at a great distance from the limiter). -9-
o » —_i o- . ->~
W IC »-• >
I
-»- t (ms)
100 -
i i - -r > _ - 200 " /- > ƒ .i /' 100 - X^ "\_/\ o _-> 1— . i * •
125
-»• t (ms)
Fig. 5. Upper figure: Loop voltage V, plasma current I and average
density ne as a function of time for WEGA-tokaraak.. Lower figure: Synchrotron radiation at f = 111 GHz (channel II), 119 GHz (III) and 128 GHz (IV). -10-
3 U
3
3
Li
3
3 U
Firj. 6. Line density, hard X-ray signal and synchrotron radiation at 101 GHz (channel II), 108 GHz (III) and life GHz ;iV) as a function of time for a tokamak shot with HF-heatinrj. Note a shift of the base line after a strong burst o^ radi/»- tion during breakdown (t ^ 12 ns). -11-
The synchrotron radiation measurement is strongly per turbed by very intense emission during breakdown at t = 11 ms. Saturation of the detectors is followed by a shift of the base line to an unknown, negative, level. To block this unwanted emission, e fast iris shutter, which can be opened within about 4 ms, was used in front of the spectrometer (see Fig. 3). Problems with interference were also present during heating . The pick-up, 10-100 uV, due to the HF-heating pulse,penetrated our screened box enclosing the detectors and amplifiers. To determine the zero-level before and after the HF-heating pulse, •we therefore repeated our measurements opening the shutter at t = 10 ms and t = 60 ms. Thermal emission around OJ = 2OJ could be measured for stellarator discharges (Fig. 7), as for them the hard X-ray flux is generally negligible.
open
closed
i —1 > 40 1 ® / - > 20 - I^_ J f 0 V ^«Hjhjt M* ****** HF . WM 1 •
open
closed
125
Pig. V a. 2coce emission from the centre of the plasma column for WEGA-stellarator. The shutter is opened at t = 10 ms. Idem, with t = 60 ms. -12-
By scanning with the grating spectrometer, emission spectra were measured for stellarator as well as for tokamak discharges. A temperature profile for the stellarator discharge, as deduced from thermal emission around 2OJ , is shown in ce Fig. 8. The signals from the most sensitive channel from seven consecutive plasma shots were used. Figure 9 gives a collection of data from two different channels collected in about 35 plasma shots. From these measurements we conclude that the electron temperature profileshows a rather flat top. 750 Fig. 8. Electron temperature profile deduced from the 2ioce emission collected by one detector (shot number 11170-11177). The solid line is given by 0L) T = T / 1 + r-0.042 e eo
"T • 1 1
> - x detector 2 - 1) 600 • detector 3 Fig. 9. 01 "" r0 =0.06 m Electron temperature profile 400 • ^a-3 - 1 for shot number 11204-11239. - '* *\ V ' 200 •\ . - / s X
n 1 .. , ...1 i •10 0 5 r (10~2m) -13-
The central temperature of 450 + 50 eV is close to the value measured by soft X-rays and Thomson scattering, showing that the absolute calibration of the system is quite good. The temperature profile shows an inward shift of 4 + 1 cm of the axis of the discharge, which is slightly more than to be expected from the stellarator flux surfaces (2 cm, see Fig. 2). With regard to the heating aspect of the experiments no significant increase in temperature was noted after the HP-heating pulse. In general we found a slight reduction in emission during HF for the standard stellarator case, for which the T -profiles are given in Figs. 8 and 9, but some times also a higher emission was measured. However, the number of measurements with low interference from the gene rators is too limited to draw definite conclusions. Apart from thermal emission around 2OJ the neasured
frequency spectrum also shows emission maxima below 2?Dce and 3u> An example is given in Fig. 10. Because the frequency is outside the band for thermal emission from non-relativistic electrons, the radiation is emitted by suprathermal electrons. The frequency is down-shifted by a factor A - (v2/c2). A full analysis of the creation and radiation of these mildly rela- tivistic electrons is given in the next chapter.
6 non-thermal thermal i 5 emission
4 3 H
2
1
0 70 80 90 100 ^ frequency (GHz)
Fig. 10. Frequency spectrum of the second harmonic, clearly showing a thermal and a non-thermal part. -14-
IV. THE SUPRATHERMAL EMISSION SPECTRUM
For the analysis of the suprathermal part of the emission spectrum we closely follow the assumptions of Celata and Boyd 3) concerning the energy distribution function of the runaway particles. The supr at hernial distribution function in the formulation used in their computer code is characterized n by the parameters G , run> and the pitch angle 9
f(G) = C n„exp(-e/6W6, (1) run and has a low energy cutoff, 6 (Fig. 11). C is a constant.
aexp - (C/£0)
•-€
Pig. 11. Suprathermal distribution function.
The suprathermal particles of density nrun are assumed to be distributed spatially uniform in a channel of a certain width centred at r = 0. All suprathermals are assumed to have the same pitch angle, 6 (the angle between the electron velocity and the magnetic field),
Finding £ • The frequency at which the peaks of the spectrum appear depends only on 6 • That this is so, can be seen from an analytical calculation of the emission coefficient j^fx), where x = f/f and I the harmonic number. ce 3) for If HU+2)(l - (x2A2)) sin29/U+l) << 1, we have emission perpendicular to the magnetic field VX) = 3? + j?
2)1-1 -a —x .21* 2 1 -22, I 1 -£- sin 21" e cos2 6 (2) (£-1)'2 extra ord -15- where a = ra c2/6 . o o Setting (d/dx) jo(x) = 0 for each mode in order to find the peaks of the spectrum, we find for the extraordinary mode
a (l-x2/«- ) = 2x3/£ (3)
Experimentally, we find a non-thermal peak at 72 GHz and also at 103.5 GHz (Fig. 12) . Substituting x = 72/42.1 for 1 = 2, we obtain a = 9.5 and 6 =54 keV. For x = 103.5/42.1 we get o 3 a = 10.5 and 6 o = 49 keV 3 in this case)
3 <0
110 120 130 140 *- f (G HZ)
Fig. 12. Frequency spectrum of WEGA-tokamak showing the second and the third harmonic. Measurements with channel II and III are denoted with crosses and dots. Findir}2_9. If 6 is known, the ratios of the specific intensities at the peaks of the harmonics can be used to find 9. The expression for the ratio of peak specific I / I t n s intensities is given in the appendix and for 2 ' 3 ^ ^ expression reduces to _ . _ , _ {1+0.269 (cos28-l.333 sin29) + 0.187 sin^e} 1 /T1 = i./bc sin 2Qü , {1+0.328 (cos26-l.875 sin26) + 0.482 sin'e}
9 I2/I3 where we also used the fact that a * 10. Experimentally, see Fig. 12, we observed 10° 55.9 20 14.6 that I2/Io * 2. We conclude from the table that for WEGA, the non-thermal 30 7.0 emission is caused by electrons having 40 4.3 pitch angles significantly larger than 50 3.1 50°. 60 2.4 70 1.96 -16-
FindingJ—^m n i . To find nru n we first return to Eq. £1) and determine the constant C by requiring the integral over f (6) between 6 and 6,. to be equal to n . For that we *—«LJ |1 A. Ul 1 need 6 and 6,,. Experime->tally, we observe no suprathermal radiation anymore below 70.8 GHz. For suprathermals of 50-54 keV, 70.8 GHz corresponds to suprathermals at a location where f = 70.8/1.71 = 4\.4 GHz, i.e. about 1 cm outward from the ce location of the thermal emission peak. The suprathermal en hancement starts to be observed at 78.5 GHz. Assuming this to be coming from the nearest possible source, i.e. 1 cm outward from the location of the thermal emission peak, we arrive at x = 78.5/41.4, implying, Eq. (3) , ot = 32.6 and G = G =15 keV. o co The maximum energy g » an electron acquires if it is freely accelerated under an applied field, will depend on the energy confinement time T , and is given by4)
2 2 e,, = Te(E/Ec) (xcv0) , (4)
where v is the electron-electron collision frequency, given by
fl vQ = q"n fcn A/25.8 /? e^ /ïn^(kT) , (5)
and Ec,the critical electric field, by
3 2 Ec - nq Hn A/kT 4TT e . (6)
!9 3 Inserting for WEGA TC= 3.6 ms*, T = 500 eV, n = 2.4xl0 m~ , 1 1 E = 0.63 VnT , £n A = 16.1, we obtain Ec = 20.3 VirT , v = 105 s"1, and €,, = 63 keV. If we now perform the inte gration, we find for the constant C in the distribution function, 2.22, permitting us to determine n by calculating I - (x, 6 , 8, n ) and comparing this with the experimental findings.
*) Note that the energy confinement time is a function of the radius and that the value of 3.6 ms is somewhat larger than the overall value of 2.4 ms quoted in Ref. 2. -17-
Insertion of x = 1.71, G =52 keV, 1=2, and 6 = 70° o in Eq. (A.4) (see appendix) leads to
38 j?JSTm = 1.06X10" u cen run
For j°T we find 0.035 j|T> Experimentally, we found the intensity of the non-thermal peak at 72 GHz to be equal to the intensity on the thermal peak of the second harmonic at 84 GHz:
2 3 2 2 1 1 -1 I.t.h (2d)c e ) = 1.85x4a)ce k Te /8TT C Wm~ rad" s" sr , (7) as the power in the ordinary mode is experimentally found to be at most 85% of that of the extraordinary mode. Due to the fact that the electron cyclotron radiation barely exceeded the noise level, we performed polarization measurements only at peak emission, i.e. at the peak of the thermal second harmonic cyclotron emission. It is found that the polarization varies between 8% and 20%, where the degree of polarization is defined by the ratio.
(Iex - Iord)/(Iex + I°rd) . (8)
For WEGA, insertion of 450 eV and 1.44 T leads to I., = 1.52xl0-12 Wm~2 rad'1 s"1 sr"1. tn As the optical depth T at the centre is 1.87, the inclusion of the effect of reflectivity hardly changes the intensity of the thermal radiation. The radiation of the runaway electrons, however, is considerably enhanced by reflections I = £l£- j (co) , (9) R l-re'T ST where r denotes the reflection coefficient and where we have also taken into account the fact that the radiation at 72 GHz is partly absorbed by the plasma at location +10cm. -18-
From the density and the temperature profiles we estimate T <_ 35 eV and n ^ 0.4><1015 m~3 for that location, which leads to an optical depth ' of i = 0.03. The reflectivity r is calculated from Costley's model , assuming a transfer coefficient p of 0.15, defined as the fraction of one electromagnetic mode converted into the other mode at each reflection. According to their analysis
I = 1 - r + rp (10) jord rp
The experimentally deduced polarization values were between 8 and 20% (average over 6 discharges was 8%; the maximum value reached, 20%). From this we conclude that r ranges between 0.93 and 0.97. If we now equate the current due to the runaways, I , to Itu» we arrive at an estimate of n - For r = 0.93 and r = 0.97 we arrive at 15 3 15 3 n ru,n_ = 1.91 x 10 m~ and nru n =l.l5xl0 m" respectivec - ly. For Ar, the width of the channel assumed to be uniform ly filled with runaways, we inserted 2 cm. Expressed as a fraction of the total density of 2.4 x io" m"3,this amounts to lO"" and 5><10~5 respectively. The current associated with the runaways should be in the order of
n xqxv(e)xTrx(Ar/2)2 * 15 A. run ^ -19-
V. NUI4BERS DERIVED FROM THE THEORY OF RUNAWAY ELECTRONS
ï^e_runawa^_rate. The runaway rate 7 can be calculated from the expression7' n 3(Z +D/16 /(V1)Ec ^c Y = 0.42 v (Ec/E) exp E 4E
(11) -1 For T = 500 eV, JtnA = 16.1, Z = 3, we obtain E = 20.3 Vm e e c v = 101 3" ~ . Ij.fi thuute: Jloo p voltage is 2.7 volt, the E equals 2 .7/2TT*0.685 = 0.63 Vin-1. Insertion in expression (11) leads to
f+ = 2.13xl0"3 s-1
7 In the steady state, the runaway density n is given by ')
•G„ ^ 7;2 E c 12 nrun/n = ^ [WF h • < >
Insertion of the relevant data leads to a fractional runaway population of the order of 1.1*10~5. It should be realized
that this ratio is quite sensitive to Ec/E.
Z!}f_i2ÏIê2êI2ï_£H£2ll- Electrons can be accelerated freely if their velocity is larger than a critical velocity, v . 7) The critical energy is given by '
2 Wc = n q HnA/4-ne^E . (13)
For WEGA parameters, Wc = 14.8 keV. The distribution function develops a long tail in the opposite direction of the E-field. The particles reach the above-mentioned 60 keV. In spite of the low value of E/E = 1/30, the distribution function is unstable because the density is so high. Due to heavy Landau damping by tha bulk, there is a minimum parallel phase velocity v for the unstable waves. As a result, only those runaways with energies exceeding (w /w )2(E /E)kT , can -20-
be in cyclotron resonance with the unstable waves and can participate in turbulent pitch-angle-diffusion 4) . For WEGA also this formula leads to 15 keV.
3!!}Ë_E!:£E?ëD^i9üiÊE_§Q§E2Y* Electrons with velocities smaller than the above-mentioned v do not suffer any pitch-angle- scattering. The decrease in parallel energy for v.. > v causes backward diffusion while leaving the distribution with v.. < v unchanged. By conservation of the number of resonant runaways, and by invoking energy conservation since 4) pitch-angle-scattering preserves the particle energy , we can estimate the resulting average perpendicular energy. For v , i.e. the parallel energy before pitch-angle-scattering, we take the above derived 63 keV, leading to v /v = 2.05. J o m The energ^yJ between v o and v m is thus found from
v (1/(vo~ vm»f vJdv|| = 2-55 vm ' (14) vm
After pitch-angle-scattering we have
> (1/{vb"vm),j (2vl +vll)dvll ' (15) v m
As v.. will now be equal to Vj_ + v , the integration can be per formed and the result set equal to Eq. (14) , which leads to the relation
V V 3 1 0 55 v v + 55 ( bK/ mJ - Ï^JvJom ~ - ( v/bmm> J- = 0.
The solution is v./v = 1.85 or, expressed in energy, J}mv£ =53 keV.
The_instabilitY_threshold.For the low-density regime of tokamaks the threshold condition for the instability of the slow modes is given by 8)
Ymax(kll ,min} - vei ' (16) -21- where k?k„ ru r u) -v n _ ÏÏ 1 Y ce _^e run ai . (17) max 4 k3 Vo ce — Pe
According to Mok also a high-density regime should exist in which instabilities are excited by runaway beams. Assuming (17) to be apllicable also for this regime, we obtain for WEGA
Y'ma x » (nru n/n) up e„ . (18)
Here u> = 2.65X1011, while (n /n) >_ l.lxlO-5, consequently,
6 Y'ma x —> 2.9xi0 .
Calculation of v . leads to ei 3 v . = 0.67x10-* n(CTT KnA s-i = 1>34xl05 s-i . ei ; Te(eV) 2 17.25
The threshold condition Eq. (16) for this instability is thus clearly exceeded for these conditions.
9) The_k-values. The bump-in-tail formation is predicted to occur for parallel velocities close to
vm = (uce/V (vthe/kzXDe> ' (19) where A is the bulk electron Debye length. Upper and lower limits on k are given by the inequality
kzvthe i u i k2C ' (20) the left-hand inequality corresponding to Landau damping on the bulk electrons and the right-hand one being given by the condition that the wave must be in resonance with the run aways. Taking u ^ w , we arrive at
kzz ,maitia*x (cnrl> = 21i and kz ^«,Hn ,min(cn, "1) = 9- -22-
The theoretical analysis can be summarized as follows '. For the above values of (E/E ) the runaway creation rate is large enough to create an extended runaway tail. In these conditions, waves with CJ < CJ < w can grow unstable pe ce through the anomalous Doppler effect (u - nu - k v = G, n < 0) and will lead to an isotropization of the electron dis tribution function in the vicinity of the parallel phase velocity. This isotropization of £ (v) results in the formation of a bump in the runaway tail which can excite waves with u = ÜJ k /k through the Cerenkov effect; the net result of the instability is to pitch-angle-scatter high-energy electrons. -2J-
VI. DISCUSSION
It has been shown that electron temperature profiles can be deduced from 2w -measurements in a st^llarator with a rota- ce tional transform i = 0.11. It can be calculated that for t >0.2 the gradient in the magnetic field dB/dR in the outer part of the toroidal vacuum chamber is too small to permit a local tem perature determination by the analysis of its electron cyclotron radiation. Due to the relatively low toroidal magnetic field (1.44 T) the equivalent noise level of this temperature diagnostic was rather high (100 eV) . A field of 2.5 T would bring the noise level down, due to more intense emission, to 15 eV (WEGA, phase II). The measured electron temperature is in agreement with the values deduced from soft X-ray emission- However, during HF-heating a temperature increase of several hundred eV was 2) measured by the latter system , whereas no significant heating was observed from the cyclotron emission. An explanation may be found in the fact that cyclotron radiation , when viewed inward along a major radius in case of optically thick plasmas, sees only low-energy electrons, in our case electrons having ener gies less than 1 keV. The X-ray system, on the other hand, measures emission between 2 and 10 keV, i.e., in the tail of the electron energy distribution function. Consequently, if an additional hot tail is created during the HF-pulse, cyclotron radiation will have to be collected on the high-B field side (inside) of the torus in order that radiation of tail particles can be observed, as only then the radiation from these parti cles will not be reabsorbed by the thermal bulk radiating at the same frequency at a larger radius. Thus, to check the char acter of the electron energy distribution by its electron cyclotron emission, one has to compare the emission radiated inward as well as outward along a major radius. Emission from runaway electrons could be identified be cause their downshift was so large that emission occurred out side the frequency band for thermal emission of the bulk par ticles. The plasma conditions, in particular a low electric field and a high ratio of electron cyclotron frequency ever electron plasma frequency , differ fro» the conditions under which normally runaway effects occur in tokamaks ' {i.e. low density, high E-fields). Synchrotron emission measurements turn out to be a very sensitive way of determining the number and energy of runaway particles.
The authors are indebted to Pre . T. Consoli and C.N. Braams for the possibility to work under a Euratom mobility contract in the "Centre d'Etude Nucléaire" in Grenoble, to Prof. H. de Kluiver for his interest in this work and his participation in one of the measuring campaigns, and to J.J.L. Caarls for his help in setting up and calibrating the FIR spectrometer. Finally, we wish to express our gratitude to all members of the WEGA-team for their hospitality and com radeship, and to Prof. F. Lngelmann for critically reading the manuscript. This work was performed under the Euratom-FOM associa tion agreement with financial support from ZWO and Euratom.
REFERENCES
1. R. Fritsch et al., Proc. 9th Symp. on Fusion Technology, Garmisch Partenkirchen (1976) 287. 2. F. Söldner et al., Proc. 9th Eur. Conf. on Contr. Fusion and Plasma Phys., Oxford (1979) 5. 3. CM. Celata and D.A. Bo*;d, Nucl. Fusion j_7 (1977) 735. See also the excellent review paper on runaway electrons, H. Knoepfel and D.A. Spong, Nucl. Fusion lj> (1979) 785. 4. C.S. Liu, Y.C. Mok, K. Papadopoulos, F. Engelmann and M. Bornatici, Phys. Rev. Lett. 39_ (1977) 701. 5. F. Engelmann and M. Curatolo, Nucl. Fusion 1_3 (1973) 497. 6. A.E. Costley, R.J, Hastie, 6.W.M. Paul and J. Chamberlain, Phys. Rev. Lett. 3J (1974) 758. 7. Y.C. Mok, Thesis University of Maryland, 1978, Preprint 910P001. 8. M. Bornatici, F. Engelmann, C.S. Liu, Y.C. Mok and K. Papadopoulos, Proc. Workshop on Plasma Transport, Heating and MHD Theory, Varenna (1^*77) 7. 9. P. Brossier, Nucl. Fusion 18 (1978) 1069. -25-
APPENDIX
3) The specific intensity for a slab of width Ar equals 2 2 I = j(w)Ar, where j(u) = £f (£)m £c o> /w at f = f Ö8./Y» Ï is the relativistic y= V /1-v2/c2. The parameter £ is defined by
f (JÜ n(o>) = £6 ce - ü) (A.l) where the coefficient of spontaneous emission n(w) is given by
2 2 °° 2 2 ce q U) r- 0jjjj(£0±) + 61 J^ (£61) Z .- (A.2) nCa,)^-^ I ord. extraord. 8TT eQc «,= 1
From the delta function we find
62 =v7cz = 1 - x2z /£//> z2 , while 6,1 = B cos 0 and 3 •= £ sin 6 . Keeping only the first terms of the Bessel function and its derivatives, we have
,-2A 2£-2 /. x2^ Z~l , 2A 2 B1 [1 - -rj) sin 6fl cos26J2+sin26J'2 = u-n :2
2 2 2 &U + 2) . 2Q] ^ f. X ' e. sin 6 1 + 1—==— cos' 2U+1) >> I2
,2 , U+2) zsin29 ^ £U+l)sin2i - cosz90 + '~ ~<— " f (A.3) 23U+1) 23U+2) J
Combining Eqs. (A.l), (A.2) and (A.3), we arrive at an expression for jg_,(u>) ,
2 2\ ,e , . 2- V*-* X 1 - sin ö * bl U-l)!2 -26-
.? "* 2 £(£+2)sin 6 1 - 2 2 2 2 £ sin e (£+2) sin 9 1- 1-- 2(£+l) J 2(1+1) 3 2 2 (£+1)
-(G/G ) 2 £{£+4)sin29 q n COJ Cn e (A.4) 3 } ^ o ce run 2 (£ + 2) 8TT2E Ê o o
where G = m c'(£/x-l). The contribution of the ordinary mode equals
,2 2 2 2 U-TT)OOS2 (l - 2L) sin 6 cos ' 2(4+1)
2 X : (£+2)£sin e(l- ) 2 2 2 2 2 ? x2 £ sin 6 (£+2) sin 6 £(£+4)sin 9 1 2l£+ï) + ^ "F"' 2(^+1) _| 23(£+l) 23 (£+2)
x (A.5) JST(w)
From (A.3) it follows that the ratio of peak specific intensities is
X' £(£+2) • 2, 2£+l 1 + 1 - 2n 32.+1 005 ° " 2Ü+T)- san f 2£ sin 26xaxexe a e J I 3£+2 «,+l ^£+1 (H+l) *£+! 2 (£+1) (£+3) 1+ 1- cos e - (JWJ 2(£+2) sin'9
£2sin26 ( (£+2) 2sin26 £(£+4)sin29' .1 ** 2(2.41) 23 (£+1) + 23 (£+2) x2 X£+l (£+l)2sin2G (2+3)2sin26 (2+1) (£+5)sin26 1 - a (£+l): 2(4+2) 2 (£+2) 2*(£+3)
(A.6)