Comparison of experimental and theoretical fast ion slowing-down times in DIII-D

Short deuterium beam pulses are injected into the D III–D tokamak to study the variation of beam slowingdown time with temperature and density. The slowing-down time is inferred from the rate of decay of the d(d,n)3He neutron emission. To within ∼30%, the results are consistent with Sivukhin's classical theory. The short beam pulses,are also useful for measurements of the central deuterium density.

Analysis of tokamak plasmas routinely assumes classical Coulomb coupling between different plasma species. For example, the conclusion that ion thermal conduction is anomalous in beam heated tokamaks [1] hinges on the assumption that the beam power is deposited in the ion channel classically. Only a few experiments have tested the assumption of classical coupling between species. Beam ion spectra measured by charge exchange agreed with spectra computed from classical theory [2,3], although discrepancies of a factor of two were observed in the fits. A higher impurity temperature than hydrogen temperature in low density, * Permanent address: University of California, Irvine, CA 92717, USA.
beam heated PLT plasmas was explained by using classical coupling arguments [4]. Measurements of the time evolution of the 15 MeV proton emission indicated that 0.8 MeV 3 He ions [5] and fast wave heated 3 He minority ions [6] slowed down at roughly the expected rate. On T-10, the time evolution and magnitude of the drop in ion temperature during electron cyclotron heating was consistent with the expected reduction in power flow from the electrons [7]. Although each of these experiments was consistent with classical theory, their accuracy was such that deviations as large as 50%, or more, from theory may have escaped detection. Perhaps the best quantitative check of beam energy loss is from studies of the rate of decay of the neutron emission following deuterium beam injection [8,9]. This emission comes from the d(d,n) 3 He reaction between the fast ions and the plasma deuterons. Since the neutron measurement is volume averaged, ±10% uncertainties in the profiles of electron temperature and density have a relatively small effect on the expected rate of decay; hence, the interpretation of the measurements is straightforward. The major weakness of the previous studies [8,9] is that the decay in neutron emission was measured immediately after beam injection, when the classical slowing-down time T S [10] was rapidly changing due to changes in T e and n e . In a preliminary study, Kim et al. attempted to improve the accuracy of the neutron technique by injecting short beam pulses into the D-III tokamak [11].
In our experiment, we have studied the deceleration of beam ions by measuring the decay in neutron NUCLEAR FUSION, Vol.28, No. 10 (1988) emission following the injection of ~2 ms pulses of deuterium beams into steady state deuterium plasmas in the D III-D tokamak. The total energy injected during the beam pulse was only ~3 % of the plasma stored energy, and, hence, density and temperature of the discharge were not perturbed (T e changed by <5%), and the slowing-down time could be studied under constant conditions. In previous work [8,9], the velocity distribution before deceleration was the steady state fast ion distribution established during beam injection. Our experiment was characterized by the important simplification that the beam pulse length was much shorter than the slowing-down time. Thus, the fast ion velocity distribution at the end of the beam pulse essentially consisted of ions with a discrete spectrum of velocities corresponding to the original spectrum of fast neutrals at E b , E b /2, and E b /3 injected by the beam. Since the cross-section for the d(d,n) 3 He reaction is a steeply increasing function of energy [12], the contribution of half and one-third energy ions to the neutron emission rates was relatively small; so the slowing-down of an essentially monoenergetic population of fast ions could be observed, giving a more precise comparison with theory. The previous studies [8,9] found that the decay of the neutron emission is consistent with classical predictions to within a factor of two. We find that the decay in emission agrees with classical theory to within 30%.
A deuterium probe beam was injected into steady state divertor plasmas with T e = 0.6-2.2 keV and n e = (l-12)X 10 13 cm~3. The central (r/a « 0.3) electron temperature was measured by absolutely calibrated Thomson scattering [13] and electron cyclotron emission [14] diagnostics with an accuracy of ~20%. Within experimental uncertainties, no systematic discrepancies between the two temperature diagnostics were observed. The line averaged electron density was measured by a CO 2 interferometer. The profiles of electron density and temperature were measured by Thomson scattering. Many of the plasmas were heated by hydrogen beams (P b £ 10 MW), and the highest densities were obtained in H-mode [15] plasmas. With the exception of the lowest temperature plasmas, the discharges had sawteeth. Most of the data were obtained at B t = 2.1 T and the plasma current varied from I p = 0.5 to 2.0 MA. Impurity levels were generally modest (Z eff $ 2). The typical hydrogen concentration (n h /n d = 30%) was estimated from the ratio of H a to D a emission.
The probe beam injected 1.7 MW of 74 keV deuterium neutrals at either an angle of 47° with respect to the plasma current at the magnetic axis or at 63° (Fig. 2 of Ref. [16]). The beam current rose in ~ 0.1 ms and fell in 0.01 ms. Typically, the probe beam injected 33% full-energy neutrals (power fraction), 35% half-energy neutrals, and 32% one-third-energy neutrals (as determined by in situ Doppler shift spectroscopy). The time evolution of the neutron emission was measured by using uncollimated plastic and ZnS ( 6 Li) scintillators with a temporal resolution of < 0.1 ms [8,17].
The effect of a short neutral beam pulse on the neutron emission is shown in Fig. 1. The neutron rate I n rises linearly until the end of the beam pulse, and then decays approximately exponentially for several e-foldings with a time constant r n . The neutron emission associated with the probe beam is due to beam-plasma interactions. Since the beam velocity (u b = 2.7 X 10 8 cm-s" 1 ) is large compared to the ion thermal velocity (i> th = 4 X 10 7 cm -s" 1 ) and plasma rotation (u rot < 5 X 10 6 c m s " 1 ) , the fusion reactivity essentially depends on the velocity of the beam ions alone. As the beam ions slow down, the fusion reactivity decreases and the neutron emission falls. For the high density H-mode plasmas, thermonuclear reactions were comparable in magnitude to the beamtarget reactions. (In low density plasmas, the beamtarget emission was one to two orders of magnitude larger than the thermonuclear emission.) Nevertheless, except in a few cases where a sawtooth caused a large reduction in thermonuclear emission during the decay of the beam-target emission, accurate measurements of r n were still possible. In these high density discharges, r n (~4 ms) was much shorter than the sawtooth period (~ 15 ms).  The observed decay times agree with classical theory (Fig. 2). To obtain the theoretical prediction, the neutron emission produced by decelerating, monoenergetic beam ions was calculated numerically. The code employed analytic fits to the measured temperature and density profiles and an analytic fit to the fusion cross-section of the form [12] A 2 / [ ( A 4 -A 3 E ) a = Deceleration of the beam ions was computed by using Sivukhin's formula for <d\V/dt> [18] including three ion species (hydrogen, deuterium, carbon). Inclusion of hydrogen (n h /n d = 30%) decreases r n by approximately 10%, which improves the agreement with experiment. Stix's asymptotic expansion of Sivukhin's formula [19] yields similar results for r n . The expression given by Strachan et al. [8] 1 -5 1 ? ri (1) T^ is the slowing-down time on electrons and E n is the energy at which the fusion reactivity has fallen to e" 1 of its value at E b .) The ~27% contribution of half-energy ions to the neutron emission was included in the calculation. Since a 36 keV ion decelerates 2.4 times faster than a 72 keV ion, inclusion of halfenergy ions reduces r n by approximately 16%. The code neglects energy diffusion (test particle approximation). To estimate the accuracy of this approximation, we use the Green's function solution for the beam distribution function given by Goldston [3] and approximate the fusion cross-section by the Gamow form [a a exp(-A/v)]. We find that energy diffusion increases the neutron emission by a factor where v is the mean velocity of the beam ions at time t and 7 is the energy diffusion term given in Ref. [3]. Evaluating Eq. 2 at t = r n , we find that the test particle approximation results in a ~10% underestimate of r n . A sensitivity analysis indicates that uncertainty in the beam deposition profile (which was not measured in these plasmas) and other profile uncertainties yield 15% uncertainty in the theoretical prediction. The weak dependence of r n on profile effects is confirmed by the observation that a moderate amplitude (AT e /T e « 10%) sawtooth caused an imperceptible (< 5%) change in the decay of I n in plasmas where the thermonuclear emission was negligible. Uncertainty in the electron temperature measurement accounts for the largest uncertainty (~ 20%) in the prediction. Uncertainty in the hydrogen concentration contributes an additional 10% uncertainty in the theory. Except at the shortest decay times where the beam pulse duration is no longer short compared to T n , the experimental measurements of r n are accurate to better than 10%. For variations in r n of over a decade, the data are consistent with theory. Averaged over the data, the ratio of measured decay time to computed decay time i s T experiment/ r theoiy = 0 g 5 ± 0 j 5 Although electron drag dominates in all cases (E b > E crit ), both ion and electron drag must be included in the theoretical prediction to obtain an excellent fit to the data.
The dependence of r n on the injection angle was studied by injecting pulses from two beam sources with different orientations into the same plasma 200 ms apart. On a subsequent discharge, the timing of the pulses was reversed. It was found that for NUCLEAR FUSION, Vol.28, No. 10 (1988) LETTERS n e = 1.2 X 10 13 cm" 3 and n e = ( 2 -3 ) X 10 13 cm" 3 (interferometer data were unavailable for the second discharge), the time evolution of the emission is virtually identical for the two orientations. Calculations of the source rate of full-energy ions with the ONETWO code [20] predict that, at n e = 1.2 X 10 13 cm" 3 , the deposition profiles for both beam orientations should peak very strongly near the magnetic axis (FWHM ^0 . 1 5 r/a). If the calculated profiles are used, the predicted difference in r n for the two orientations is 5%, in good agreement with experiment.
Pulsed deuterium injection is also useful for measuring the central deuterium density. The beam-plasma reaction rate is given by where n b and n d are the beam and deuterium densities, respectively, and (aw) is the fusion reactivity averaged over the distribution functions. Immediately after the beam pulse, the beam ions have scarcely decelerated so that / • n b n d d?

(4)
The reactivity av is evaluated at v = v^: + Ci v th -c 2 v rot , where Ci and c 2 are constants of 0(1). Since the injection velocity of full-energy ions v in j is known accurately and is much larger than v th and v rot , av can be evaluated accurately. The number of full-energy beam ions N b is known accurately from calibration of the beam power. Defining a central deuterium density n d as the average value of n d at the radius of the beam ions, fi d = / n b n d d ? / / n b d?", Eq. (4) yields to test the validity of this technique. In one study, we switched working gases from deuterium to hydrogen in ohmically heated plasmas. After ~8 hydrogen discharges, the measured deuterium concentration n d /ff e fell to 57 ± 11% of its previous level, consistent with spectroscopic measurements of the H a and D a emissions. In a second experiment, the dependence of n d /n~e on the plasma current I p in H° ->• D + , H-mode plasmas was studied. The inferred deuterium density n d scaled linearly with electron density for n e = (3-12) X 10 13 cm" 3 . In these plasmas, Z eff was sufficiently low (Z eff < 2, with carbon and nickel being the dominant impurities [21 ]•) that fuelling by hydrogen gas from the beamlines was primarily responsible for dilution of n d /n e and so a linear dependence of n d on n~e is not unexpected. In the light of the uncertainty in the neutron calibration, the absolute magnitude of n d (4 X 10 13 cm" 3 for n e = 9.0 X 10 13 cm" 3 ) was also reasonable.
In conclusion, the use of short deuterium beam pulses has permitted accurate measurements of beam slowing-down time in Ohmic and beam heated D III-D plasmas. Accurate relative measurements of the central deuterium density have also been made. The slowingdown time agrees with classical theory to within 30%. ABSTRACT. The magnetic divertor structure in a modular stellarator has been predicted to change with the addition of an external vertical magnetic field. The diverted flux is expected to emerge predominantly on the inside of the torus when a vertical magnetic field which shifts the magnetic surfaces towards the inside is applied, and vice versa. The prediction has been verified using probe arrays located in the divertor regions. In addition to the redistribution of diverted particle flux, the distance a field line travels from the separatrix until it reaches the wall has been predicted to increase for each of the magnetically altered cases. The formation of magnetic island chains beyond the separatrix has been found to be responsible for the increase.

INTRODUCTION
Magnetic divertors have been employed in many devices to control impurity reflux and improve plasma parameters [1][2][3][4]. Unfortunately, the diverted plasma flows may intersect such necessary obstructions as diagnostics and antennas before reaching an area where it is feasible to collect the diverted plasma, which results in enhanced impurity generation close to the central plasma. This can present a significant problem in stellarators where a naturally occurring divertor structure often distributes particle fluxes over a large portion of the device [5]. The work reported here shows that in a stellarator geometry the addition of a small vertical magnetic field can dramatically alter the locations of plasma flow out of the coil volume. This letter discusses the predicted changes in the divertor structure which occur as a result of the application of various external magnetic fields to the Interchangeable Module Stellarator (IMS) [6], and experimental evidence confirming the alterations to the divertor structure.