Effect of secondary beam neutrals on MSE: theory

Transcription

1 Effect of secondary beam neutrals on MSE: theory S. Scott (PPPL) J. Ko, I. Hutchinson (PSFC/MIT) H. Yuh (Nova Photonics) Poster NP th Annual Meeting, DPP-APS Orlando, FL November 27

2 Abstract A standard calibration technique for Motional Stark Effect (MSE) diagnostics is to compare the polarization direction of Doppler-shifted Hα emission from a diagnostic neutral beam (DNB) that is fired into a gas-filled torus to the pitch angle inferred from known toroidal and vertical fields. However, the polarization direction of Hα emission from secondary beam neutrals that ionize, gyrate about field lines, and then charge exchange a second time differs from the polarization direction of the primary beam neutrals and thus confuses the calibration results. We compute the ratio of secondary-to-primary H emission, I s /I p, as a function of torus pressure for 5 kev hydrogen atoms in Alcator C-Mod. For helium gas, I s /I p is about unity at P=1 mtorr for the DNB in its new orientation: 7 o from perpendicular). The effect on the MSE calibration of Hα emission from these secondary beam neutrals is calculated by adding the Stokes vectors for all secondary-beam gyro angles whose Doppler shift lies within the MSE filter passband. The computed calibration error increases linearly with torus pressure and has distinct dependencies on MSE viewing geometry and pitch angle which are in qualitative agreement with recent measurements.

3 Motivation: with the DNB aimed perpendicular, we saw strong anomalies during beam-into-gas calibrations MSE measured angle (degrees) 2 1 Apply in-vessel calibration data to beam-into-gas calibration shots. Edge channels show signif Actual pitch angle (degrees) * An 'offset' exists for all channels at zero pitch angle icant curvature response. * Slope of measured response is greater than unity for all channels Slope of * measured Anomalies are response strongest is in greater the edge than channels unity for all channels.

4 Beam-into-Gas MSE Calibration is Confused by a Secondary Population of Beam Neutrals DNB V B = m/s MSE v beam = m/s Gas at 2mTorr: λ MFP = 1/nσ cx = 59 cm Fast ions move a distance = / = 221 cm for each cm of vertical drift. So fast ions drift only a distance 59/221 =.27 cm before they suffer a charge exchange event. Essentially 1% of the fast ions become neutrals again before they leave the MSE viewing footprint. The ions have random gyro angle when they charge exchange, thereby creating a fast neutral population with random gyro angle.

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7 But even after pivoting the DNB, some anomalies remain in beam-in-gas calibrations 1 γ FL = -1 o Error (degrees) γ FL = -7.4 o ch= Error (degrees) 1-1 Error (degrees) γ FL = +2 o Torus pressure (mtorr) * Measured pitch angles depend on torus pressure! * The amount of change with torus pressure is a function of the actual pitch angle and MSE viewing position * Also, the spectral intensity on the 'blue' side of unshifted Ha, which can only come from secondary beam ions that gyrate >18o, is reduced only modestly by pivoting the beam (see NP8.86).

8 Reducing residence time by pivoting the DNB: original calculation An early (but incorrect!) calculation suggested that pivoting the dnb by 7 o would reduce the population of secondary beam ions by a factor 4. Residence time τ φ for a beam ion to move out of the mse viewing footprint toroidally: τ φ = R MSE tan θ mse v cos γ F L For perpendicular injection of the dnb, the residence time of a beam ion is set by the time required for it to grad-b drift downward through the vertical extent of the mse footprint: τ B = h MSE v B The ratio of the residence times is = 2h MSE R MSE sin θ inj tan θ mse R pivot ρ beam For the edge mse channel, pivoting the dnb at 7 o provides an injection angle θ 9 o in the plasma, and thereby reduces the residence time by a factor of 4. This analysis assumes that once the dnb is pivoted, the shortest time constant for loss of a beam ion is motion along a field line. But alas, this isn t true... the beam ion can re-neutralize before it leaves the mse viewing footprint. A correct analysis (below) that treats loss of beam ions through both motion and re-neutralization shows that tilting the beam does reduce the ratio of secondary to primary beam neutral Hα emission, but by a smaller factor.

9 Beam neutrals that ionize thru collisions with gas molecules accumulate in 'drift tubes' as they move along field lines DNB MSE field-of-view L MSE h MSE drift orbit w MSE r DNB field line L T γ FL θ d da drift tube ion birth position Angle of drift orbit differs slightly from that of field line: θ d = tan -1 ( v o sin θ inj sin γ FL + v D ) v o sin θ inj

10 Average density of protons in drift tube The volumetric birth rate R + for protons in the drift tube is: R + = n o (1) n He σ ion (1) v o where: n o (1) = population density of neutrals in the ground state σ ion (1) = ionization cross section for ionization v o = beam velocity. At typical torus pressures, only about 17% of the protons will be re-neutralized before leaving the drift tube. So to first order, the dominant process for loss of the protons out of the drift tube is their simple parallel motion along the drift orbit: R loss = n + v da. In steady state, we equate the volume-integrated birth rate and loss rate of the protons: n o (1) n He σ ion (1) v o L T aa = n + v o sin θ inj da n + = n o(1) n He σ ion (1) L T sin θ inj For typical conditions the beam ion density leaving the drift tube is about 1% of the beam neutral density.

11 Density of n=3 beam neutrals as drift orbit crosses MSE sightline There are two dominant processes for creating n = 3 hydrogen neutrals: Direct charge exchange of the proton into the n = 3 state; Charge exchange into the n = 1 state followed by collisional excitation into n = 3. These are of comparable magnitude. So for simplicity, evaluate only the birth rate due to charge exchange directly into the n = 3 state, and then multiply by a factor c (with c 2). The birth rate of hydrogen atoms in the n = 3 state becomes: R n=3 = c n + n He v o σ cx (3) where σ cx (3) is the cross section for charge exchange into the n = 3 state. There are two dominant loss processes for these n = 3 neutrals: radiative decay and simple linear motion of the neutrals out of the mse viewing volume. To first order, the loss of an n = 3 neutral out the sides of the mse viewing volume can be neglected, leaving only radiative decay as the loss mechanism. The dominant loss process is in fact the atoms linear motion through the sides of the viewing volume, which is a factor of 4 times faster than their radiative decay. But when an n = 3 hydrogen atom leaves one mse viewing volume, it typically just enters another one, and so it simply contributes to the Hα photon production in a different mse channel.

12 Equate the birth rate to the decay rate: n o (3) = c n + n He σ cx (3) v o ν 31 + ν 32 For typical conditions, n o (3) = n +. Finally, compute the rate I s at which Hα photons are generated by these n = 3 neutral hydrogen atoms in an MSE viewing volume. As before, neglect collisional de-excitation (which is small) and consider only the radiative decay rate: I s = n o (3) ν 32 h MSE w MSE L MSE A small detail: limiting behavior at small pitch angles. The length of the volume that contributes to the secondary beam emission is L MSE = h MSE / tan θ D. For small values of the pitch angle, θ D can be zero leading to an unphysical answer. This is resolved by remembering that the path length of the mse sightline in the torus (i.e. the distance L w from the intersection of the mse sightline with the dnb to the location where the mse sightline strikes the torus wall) is an upper bound L MSE. In Alcator, L w 1.5 m. L MSE = MIN( h MSE tan θ d, L w ) (1)

13 Ratio of secondary to primary Hα photons We want to compare the secondary emission intensity to primary emission mechanism: collisional excitation of the initial hydrogen beam. The rates for radiative decay are so much larger than the rates for collisional excitation, so only a small fraction of the beam is in an excited state so we will ignore multi-step processes. The dominant process for generating beam neutrals in the n = 3 state from beam neutrals in the n = 1 state is then simple collisional excitation, which generates a volumetric birth rate S 3 : S 3 = n o (1) n He σ ex (13) v o where σ ex (13) is the cross section for collisional excitation from the n = 1 to n = 3 state. The decay rate of these neutrals is the sum of the radiative decay into the n = 1 and n = 2 states. At steady state the birth rate must equal the decay rate and so the density of the primary beam neutrals in the n = 3 state is: n p o(3) = n o(1)n He σ ex (13)v o ν 31 + ν 32 (2) The viewing volume for these primary n = 3 neutrals is the intersection of the mse viewing footprint and the dnb cross section. So the total rate of generation of primary Hα photons, I p, is given by I p = n p o(3) h MSE w MSE (2r DNB ) ν 32 So the ratio of secondary to primary Hα emission intensity is: I s I p = n He σ cx (3)σ ion (1) σ ex (13) c L T 2r DNB MIN(L w, h MSE / tan θ d ) 2 sin θ inj

14 c L T /(2r DNB ) is approximately unity because c 2 and L T = (1 2) r RNB. Only for very small drift angles θ d is the mininum set by L w rather than h MSE / tan θ d ; for an mse sightline that crosses the dnb at about R =.8 m, L w 1.7 m. and so the critical angle would be θ d = 1. o. In the typical limit where θ d > tan 1 (h MSE /L w ), and assuming E = 5 kev, h MSE =.3 m, we obtain I s I p =.13P sin θ inj tan θ d (3) where P is the torus pressure (helium) in mtorr. For typical values on Alcator C-Mod, (P = 1.3, θ inj = 1 o, θ d = 5 o ), the secondary emission intensity is comparable to the primary emission. (I s /I p = 1.1). The expected ratio of secondary-to-primary emission intensity varies linear with the torus pressure. This calculation includes all Hα photons generated along an mse sightline - irrespective of their Doppler-shifted wavelength. Pivoting the dnb may still reduce the beam-into-gas anomaly because the downstream secondary emission may be Doppler-shifted beyond the passband of the mse narrow ( λ = 1. nm) bandpass filters.

15 σ λ λ Process 1 22 m mtorr.2 mtorr (m) (m) Collisions with hydrogen gas ionization H o + H 2 H + +H CX H + + H 2 H o (any state) CX H + + H 2 H o (1s) CX H + + H 2 H o (2s+2p+3s+3p+3d) CX H + + H 2 H o (2s) CX H + + H 2 H o (3s) CX H + + H 2 H o (4s) CX H + + H 2 H o (2p) CX H + + H 2 H o (3p) CX H + + H 2 H o (3d) excitation H o (1) + H 2 H o (2s) (@25 kev) excitation H o (1) + H 2 H o (2p) (@25 kev) excitation H o (1) + H 2 H o (3s) (@35 kev) excitation H o (1) + H 2 H o (3p) (@35 kev) excitation H o (1) + H 2 H o (3d) (@35 kev) Radiative transitions radiative H o (2) H o (1) radiative H o (3) H o (1) radiative H o (3) H o (2) Collisions with hydrogen ions excitation H o (1) H o (2) 11. excitation H o (1) H o (3) 2. excitation H o (2) H o (3) 25. excitation H o (3) H o (4) 12. excitation H o (3) H o (5) 16. ionization 1 loss 28. ionization 2 loss 1. ionization 3 loss 21. de-excitation de-excitation Table 1: Cross sections and mean free paths for a 5 kev hydrogen neutral atom in hydrogen.

16 σ λ λ Process 1 22 m mtorr.2 mtorr (m) (m) Collisions with helium gas ionization H o + He H CX H + + He H o (any state) CX H + + He H o (1s) CX H + + He H o (2s+2p+3s+3p+3d) CX H + + He H o (2s) CX H + + He H o (3s) CX H + + He H o (4s) CX H + + He H o (2p) CX H + + He H o (3p) CX H + + He H o (3d) excitation H o (1) + He H o (2s) excitation H o (1) + He H o (2p) excitation H o (1) + He H o (3s) (@35 kev) excitation H o (1) + He H o (3p) (@35 kev) excitation H o (1) + He H o (3d) (@35 kev) Radiative transitions radiative H o (2) H o (1).7.7 radiative H o (3) H o (1) radiative H o (3) H o (2).7.7 Collisions with helium ions excitation H o (1) H o (2) 23. excitation H o (1) H o (3) 44. excitation H o (2) H o (3) 92. excitation H o (3) H o (4) 56. excitation H o (3) H o (5) 69. ionization 1 loss 14. ionization 2 loss 56. ionization 3 loss 12. de-excitation de-excitation Table 2: Cross sections and mean free paths for a 5 kev hydrogen neutral atom in helium.

17 Pure perpendicular beam injection The situation for pure perpendicular beam injection is qualitatively different because the parallel beam velocity is zero. Beam neutrals that ionize experience only the vertical drift velocity v D and the length of their drift tube is just the dnb radius. The length of their drift orbit inside the drift tube (including gyromotion) is large compared to the mean free path against charge exchange. The length of a beam ion s orbit as it drifts vertically through the height of the mse viewing volume ( 3 cm), is 6.5 meters which is still considerably greater than the mean free path against charge exchange. For typical calibration pressures (1-2 mtorr), we can make the approximation that all beam ions generated through ionization are locally lost to charge exchange. This yields R + R loss = n o (1) n He σ ion (1) v o = n + n He σcx tot v o. n + = n o (1) σ ion (1) σcx tot (4) To compute the birth rate R 3 of neutrals in the n = 3 state, evaluate the rate for direct charge exchange into the n = 3 state and multiply by the fudge factor c to account for charge exchange into n = 1 followed by collisional excitation into n = 3: R 3 = c n + n He v o σ cx (3) = c n o (1) n He v o σ cx (3)σ ion (1) σ tot cx (5)

18 To compute the particle density generated from this volumetric source, consider the two loss mechanisms: radiative decay and linear motion out of the region of interest. Because the geometry is simpler than the situation prevalent in the case of a beam with nonzero parallel velocity, we can calculate the n = 3 beam density including the both effects. Figure 1 illustrates a side view of the dnb. Beam ions that charge exchange acquire a random gyro angle, thus they are equally likely to be moving in the positive x-direction (shown in blue in the figure) as in the negative x-direction (red). x = 2 r DNB DNB x x= Figure 1: Geometry for calculating the n = 3 neutral density in the dnb for perpendicular injection. Neutrals born through charge exchange have equal probability of moving vertically up (blue) or down (red). Consider the population of the upwardly-moving and downwardlymoving neutrals separately, with corresponding neutral densities n u o(3) and n d o(3). Neutrals moving in the positive x direction all have speed v o and an average velocity in the x-direction given by v x = (v o /π) π o sin θdθ = 2 π v o. A particle balance in a differential region of thickness dx balances the birth rate, death rate due to re-combination, and flux across

19 the boundary. With appropriate boundary conditions, the solution is: R 3 n 3 (x = r DNB ) = 1 exp π(ν 31 + ν 32 ) r DNB (ν 31 + ν 32 ) 2v o We can define a radiative mean free path λ r 2v o /(π(ν 31 +ν 32 )) which is the vertical distance that a beam neutral moves before it suffers a radiative decay from n = 3 to either n = 2 or n = 1. There are two regimes, depending on the relative size of the beam and the radiative mean free path: λ r r DNB n 3 (x = r DNB ) = λ r r DNB n 3 (x = r DNB ) = ( π ) R 3 r DNB 2 v o (6) R 3 ν 31 + ν 32 (7) In Alcator c-mod we are in the short radiative mean-free-path limit (Eq. 7): there is a local balance between the birth rate of n = 3 neutrals and their radiative decay. The total photon production rate is just this n = 3 neutral density multiplied by the mse viewing volume and the radiative decay rate: I s ν = c n o (1) n He v 32 o ν 31 + ν 32 σ cx (3)σ ion (1) σcx tot h MSE w MSE 2r DNB A quantity of considerable interest is the ratio of secondary Hα emission from a rotated beam to that with a perpendicular beam. If the beam rotation angle is not too small, then I s I s h mse λ cx 1 2 sin θ inj tan θ D

20 Are we saved by the Doppler shift? The passband of mse optical filters is selected to pass the Dopplershifted π 3 line of the Hα mse spectrum. We want to compute the Doppler shift of non-local, secondary beam ions to determine whether their Hα emission will fall inside the mse passband. Local secondary neutrals ionize, re-neutralize, and radiate, all at the location of the intersection of the DNB and MSE sightline. Non-local secondary neutrals ionize off the horizontal midplane, then move along a field line until they cross the mse sightline, where they re-neutralize and radiate. DNB y lens Geometry for evaluating parallel velocity of non-local secondary beam neutrals θ b MSE lens R R δ s s R o MSE sightline DNB R φ φ ο x lens The beam ion will travel a toroidal angle φ = crosses the mse viewing sightline at the midplane. z R tan θ d before it

21 Secondary beam neutrals that cross an MSE sightline can be mapped back to their birth location in the DNB to compute their v. Δφ R R o E E D A B C B C D MSE DNB tan θ d = Δz / (R Δφ) MSE Δz B C D E A B C D E Elevation view θ d γ Fl DNB

22 For some sightlines, the major-radius at `birth of secondary neutrals lies outside the birth radius of the primary neutrals. DNB N P X Y MSE M D M D C N P B A X Y B C Z B C D A M N P X Y R Z= Δz B C D DNB A B C D M N P X θ d γ Fl Y Elevation view

23 The toroidal angle φ o at the point of intersection of the dnb and mse sightline is given by R o = R p tan θ b sin φ o +cos φ o tan θ b. The major radius R at a point along the mse sightline displaced from the dnb-mse intersection by a toroidal angle φ: R = R o cos(φ o δ) cos(φ o δ + φ) tan δ = x lens R o cos φ R o sin φ + y lens. We want to compute the difference in parallel velocity between primary beam neutral (at point A) and a non-local, secondary beam neutral at some point downstream along the mse sightline (e.g. point E). Toroidal angle at birth : φ b = φ o +β = sin 1 ( R o R sin(φ o + θ b ) ) θ b. R o = 87 R o = 7 φ = 6 o φ = 2 o φ = 6 o φ = 2 o φ o (degrees) o δ (degrees) R/R o z (cm) v /v p Table 3: Computed parallel velocity for two mse sightlines. Result: The parallel velocity of the non-local secondary beam neutrals differs from that of the primary beam neutrals by as much as 1%.

24 Doppler shift of gyrating secondary beam neutrals Having solved for the birth parallel velocity of a secondary beam ion, it is straightforward to figure out its gyrating velocity vector as it crosses the mse sightline in the midplane. The velocity component along the mse sightline (ŝ v) then determines the Doppler shift: c λ λ = v cos γ F L cos(φ δ) +v [ cos β sin(φ δ) sin β sin γ F L cos(φ δ) ] The corresponding Doppler shift of the primary beam neutral is λ p = v o c λ sin(θ b + δ). The MSE passband filters are chosen to pass this Doppler-shifted wavelength. For a core mse view at R o = 7 cm, the Doppler shift does not lie within the MSE passband for any value of the gyro angle beyond φ > 6 o. So most of the secondary beam emission is blocked by the bandpass filters for the core MSE channels. But for an edge mse channel at R o = 87 cm, the computed Doppler shift lies within the mse passband all the way out to φ = 2 o, and so the non-local, secondary beam neutrals should continue to confuse the mse beam-into-gas calibration measurements. This behavior is shown in Figure 3. The vertical scale represents the total range of gyro-angles that yield a Doppler shift inside the mse passband at a particular downstream location on a given mse sightline.

25 Note: R and b satisfy R b = and since m is constructed to be perp to b, if we construct the gyro orbit as a linear combination of R and m, then it must also be perp to b. gyro-orbit is perpendicular to local b B B φ z x α φ y z m b m R b m x sin φ sin γ FL ycosφ sin γ FL + z cos γ FL [x lens,y lens,z lens ] θ b DNB R B x = -B φ sin φ B y = B φ cos φ B z = B z v = v Rcosβ + v msinβ B φ = B o cos γ FL v= v b + v Rcosβ + v msinβ B θ = B o sin γ FL b = [x,y,z] = [-sin φ cos γ FL, cos φ cos γ FL, sin γ FL ]

26 Doppler shift (nm) 6 Ro=7 cm 5 Df = o 4 filter passband 3 Doppler shift (nm) Ro=8 cm Doppler shift (nm) gfl = 8o Ro=87 cm Gyro angle (degrees) 3 Doppler shift as a function of gyro-angle. The shaded region represents the passband of the mse filters, which are temperaturetuned to match the Doppler shift of the primary beam neutrals. Figure 2:

27 Accepted gyro angles (degrees) Accepted gyro angles (degrees) Accepted gyro angles (degrees) γ FL = 12o R o = 7 cm R o = 7 cm 82 8 γ FL = 8o γ FL = 5o φ (degrees) 8 75 Figure 3: Total range of gyro angles which result in a Doppler shift that lies within the mse passband of the mse filters for a variety of viewing sightlines.

28 Doppler shift (nm) 6 Ro=7 cm 5 Df = o 4 filter passband 3 Doppler shift (nm) Ro=8 cm Doppler shift (nm) gfl = 8o Ro=87 cm Gyro angle (degrees) 3 Figure 3: Doppler shift as a function of gyro-angle. The shaded region represents the passband of the mse filters, which are temperature-tuned to match the Doppler shift of the primary beam neutrals.

29 Accepted gyro angles (degrees) Accepted gyro angles (degrees) Accepted gyro angles (degrees) γ FL = 12o R o = 7 cm R o = 7 cm 82 8 γ FL = 8o γ FL = 5o φ (degrees) 8 75 Figure 4: Total range of gyro angles which result in a Doppler shift that lies within the mse passband of the mse filters for a variety of viewing sightlines.

30 Calculate effect of secondary emission assuming it is 'local' Blue-shifted contribution to spectrum due to fast-ion gyro motion Channel 8 Pitch angle =. E o 4 2 A range of gyro angles yield Doppler- shifted wavelengths that pass thru the MSE filters Intensity 1..5 E 1/2 E 1/2 Doppler shift (nm) -2-4 MSE filter passband Gyro-angles that contribute to MSE signal Wavelength (nm) Ion gyro angle (degrees) * Assume some ratio of secondary to primary Hα emission intensity. * Compute which gyro angles (Doppler shifts) survive the MSE filter. * Then compute polarization angle of composite (primary+secondary) emission for comparison with MSE measurements. * Can also compute the entire spectrum (no filter) for comparison with spectroscopic measurements.

31 2 1 z lens =. Ch= Measured pitch angle (degrees) Actual pitch angle (degrees) Figure 13: Computed effect of secondary beam neutrals on the pitch angles measured by mse channel (edge channel), assuming: (a) perpendicular dnb; and (b) first mse lens positioned in the torus horizontal midplane. Colored curves represent different ratios of the secondary to primary Hα photon rate, I s /I p.

32 3 2 I s / I p =.5 z lens =. Ch = Effect of filter width, ch=, Is/Ip=.5 Measured pitch angle (degrees) nm Actual pitch angle (degrees) Figure 14: Computed effect of secondary beam neutrals on pitch angles measured by mse channel (edge channel) assuming (a) perpendicular sc dnb; (b) first mse lens positioned in the torus horizontal midplane; (c) ratio of secondary to primary beam Hα photons =.5. Curves are labelled according to the asssumed value of the passband of the mse bandpass filter, in nanometers.

33 Computed effect of `secondary' beam Hα emission on polarization angle measured by MSE 2 Measured pitch angle 1-1 Channel.1 I s / I p = Actual pitch angle * S-shaped function of actual pitch angle. * 'Offset' at zero pitch angle. * Shift in MSE-measured angle can be many degrees for I s / I p >.1

34 Conclusions Secondary emission of by beam neutrals that ionize, gyrate about field lines, re-neutralize and then emit an Hα photon can seriously confuse beam-into-gas MSE calibrations. Unmistakable signatures of such secondary emission are: Polarization angles measured by MSE vary ~linearly with torus pressure; Observation of intensity on the blue side of the spectrum (when the DNB is pointed away from the MSE optics) A simplified atomic physics / geometry model predicts that secondary beam neutrals contribute significantly to the chord-integrated light intensity even even after DNB is pivoted by 7 o. But for many MSE channels, the Doppler shift of the secondary beam emission puts its wavelength outside the passband of the optical filters, which significantly reduces its `pollution effect during beam-into-gas calibrations.

35 Conclusions: effect on MSE measure angles A simplified, local model of the effect of secondary beam emission predicts: Significant changes to pitch angle even with small intensity of secondary neutrals, e.g. 2 o changes for 1% emission due to secondary neutrals. Details of the effect of secondary neutrals is affected by the vertical 5 o tilt angle of the MSE sightlines: Difference between apparent angle and true angle increases as one moves away from a pitch angle of -5 o ; The sign of the difference reverses direction for pitch angles < -5 o versus > -5 o. The generated error varies smoothly with MSE viewing location, being strongest in the edge channels. The generated error also varies strongly with the actual pitch angle. These trends can be compared against actual measurements (see next poster).