The Instrumental Function of the X-ray Imaging Crystal Spectrometer on Alcator C-Mod

Transcription

1 The Instrumental Function of the X-ray Imaging Crystal Spectrometer on Alcator C-Mod M. Bitter, K. W. Hill, B. Stratton, S. Scott Princeton Plasma Physics Laboratory, Princeton, NJ, USA A. Ince-Cushman, M. Reinke, J. E. Rice MIT Plasma Science and Fusion Center, Cambridge, MA, USA P. Beiersdorfer, M. F. Gu Lawrence Livermore National laboratory, Livermore, CA, USA S. G. Lee NFRC, Korea Basic Science Institute, Daejeon, Korea C. Broennimann, E. F. Eikenberry DECTRIS Ltd, 5232 Villigen-PSI, Switzerland Also see: NP (K. W. Hill, et al.); PO (A. Ince-Cushman, et al.) 49 th Annual Meeting of the Division of Plasma Physics, Orlando, Florida, Nov , 2007

2 ABSTRACT A new high-resolution X-ray imaging crystal spectrometer was implemented on Alcator C-Mod for Doppler measurements of the radial profiles of the ion temperature and toroidal plasma rotation velocity. The spectrometer consists of two spherically bent crystals and high count rate PILATUS II semi-conductor diode arrays; and it records spatially and time resolved spectra of He-like and H-like argon. The paper presents analytical and numerical calculations of the instrumental function and describes its effects on the observed spectral line profiles. Work supported by DOE contracts: DE-AC02-76CHO3073, DE-FC02-99ER54512, W-7405-Eng-48, and DOE Initiative for Plasma Diagnostic Developments, Contract-1083

3 Introduction The imaging x-ray crystal spectrometer on Alcator C-Mod consists of spherically bent crystals and PILATUS semiconductor diode arrays in the Johann configuration. It is now routinely producing T i and v toroidal profiles with good time (< 20 ms) and spatial (1 cm) resolution. Such a diagnostic is also planned for ITER. In this poster, we derive the instrumental function, or Johann error, from geometrical considerations and discuss its effects on the observed line profiles. Outline: Instrumentation Spectra and deduced T i and v tor profiles The Johann error and its effects

4 Background: Conventional Johann Spectrometer Spectrometer with cylindrically bent crystal and 1-D, position-sensitive, multi- wire proportional counter is widely used for Ti(0) measurements on tokamaks but does not provide spatial resolution

5 Imaging Properties (Astigmatism) of a Spherical Crystal A point source on the Rowland circle has two images at F m and F s, formed by the meridional (red) and sagittal (blue) rays (Astigmatism) For a Bragg angle of 45, F s is at infinity, so a parallel bundle of X-rays emanating from the plasma is focused to a point on the detector. The imaging is rotationally symmetric about the normal of the crystal.

6 Imaging X-ray Crystal Spectrometer Instrumentation 3 detectors for He-like argon emission cover nearly entire vertical extent of plasma 1 detector for H-like emission views core region He-like Crystal Gate Valve B-port H-like Crystal He-like Detectors H-like Detector Spectrometer Layout Pilatus X-ray Detector Module

7 Spectra of He-like and H-like Argon are recorded from the entire 72 cm high plasma and a 20 cm high central region

8 Instrument Parameters Spherically bent Crystals: He-like Argon: (102)-quartz, 2d spacing = Å, radius of curvature = 1444 mm, Bragg angle = for line w at Å. H-like Argon: (102)-quartz, 2d spacing = Å, radius of curvature = 1385 mm, Bragg angle=54.87 for Lα line at Å. PILATUS II detector characteristics: Overall size: 35 mm x 85 mm (195 pixels x 487 pixels) Pixel size: mm x mm Maximum count rate: 1 MHz per pixel Readout time: 3 ms 20 bit digitization Typical spatial resolution: 48 chords with ~1 cm radial resolution Typical time resolution: ms

9 3 PILATUS II Detectors Provide Continuous Spatial Coverage of He-like Ar Spectra Bottom C-Mod Plasma (Height =72 cm) Core Crystal Detector Top

10 First Spatially-Resolved Spectra on C-Mod April 13, 2007

11 Waveforms C-Mod Shot: with ICRF heating and argon injection

12 Raw Data: He-like Argon Spectra (un-inverted)

13 T i Increases and Profile Broadens during H-mode Phase with ICRF heating Un-inverted

14 Un-inverted V tor Changes during ICRF heating

15 Central Apparent T i from He-like Ar lower than from H- like Ar: Suggests Hollow Emission Profiles Un-inverted

16 The Johann Error The Johann focusing error depends on the crystal size, the arrangement of the detector plane, and the wavelength. Its effects are a distortion and a shift to shorter wavelengths of the observed (or apparent) line profile, which vary with the ion temperature. This shift must be taken into account for measurements of rotation velocities. The Johann error can be derived from geometrical considerations and is depicted in the following Figures for the spectral lines w and z. Since, as a result of the Johann error, there is not a one-to-one correspondence between the wavelengths and coordinates of the detector points, the definition of a wavelength scale on the detector is a non-trivial requirement. The most appropriate definition of a wavelength scale is obtained by assigning to each wavelength λ the point of intersection of the detector plane with the circle of radius Rcos(θ) about the center of the crystal sphere, where θ is the corresponding Bragg angle and R the radius of curvature of the crystal. With this definition, the wavelength scale depends only on R and θ (i.e. the 2dspacing of the crystal) and is independent of potentially varying experimental parameters, such as the crystal size.

17 Johann Focusing Error for λ w = Å The Johann focusing error EB for the wavelength λ w can be visualized by rotating the black quadrangle ABCD about point A into the red or blue quadrangles. The point B is the intersection point of the detector plane with the circle Rcos(Θ w ) about point A and, by definition, assigned to λ w. Note, B is an accumulation point for photons of λ w and there are no photons with λ w on the right (long-wavelength side) of B.

18 Johann Focusing Error for λ z = Å The Johann focusing error for the wavelength λ z can be derived in the same way as for λ w if the crystal is divided in two unequal parts which extend from C to the edges of the crystal. The left part, from C to the left edge of the crystal, causes a larger Johann error than the smaller, right part of the crystal. The Johann errors from both crystal parts must be considered. Note, the intersection point D of the detector plane with the circle Rcos(Θ z ) about A is, by definition, assigned to λ z.

19 Numerical Results for a Gaussian centered at λ w = Å We assume a Gaussian profile: (1) exp["(c /v i # " # w # w ) 2 ], where c /v i = T i (kev ) for the spectral line w of He-like argon. With our definition for the wavelength scale on the detector, (1) can be expressed in coordinates w (in units of mm) on the detector, where # w is assigned to w = 0 (2) P(w) = const L where const = 1 2 $ % and % = 1 exp[" T i (kev) w(mm)2 ] T i (kev) and L is the length of the crystal. The observed (apparent) line profile on the detector is then given by (3) F(u) = P(w) Q(u " w) dw where Q(u " w) = 1 A 1 u " w u & u"eb with A = cos(') 2 R is the distribution of photons of a particular wavelength, which - by definition of our wavelength scale - is assigned to the coordinate w, over the detector due to the Johann error.

20 Numerical Results for a Gaussian centered at λ w = Å Equation (3) can be rewritten as (3 ) F(u) = const A L where # = T i (kev) and integrated by parts, so that u 1 $ exp["# w 2 ] dw, u " w u"eb (4) F(u) = F 1 (u) " F 2 (u) with (5) F 1 (u) = const exp["# (u " EB) 2 ] (6) F 2 (u) = 2 const u EB # $ w u " w exp["# w 2 ] dw} u"eb Numerical results for expressions (4) - (6) are shown in the following Figures.

21 Numerical Results for a Gaussian centered at λ w = Å Figure (a): Apparent Line Profile F(u) (a) Figure (b): Component F1(u) represents a Gaussian shifted by the Johann Error EB Figure (c): Component F2(u) Crystal Length, L (mm) = 64 Johann Error, EB (mm) = (b) Enter Ti (kev) = 2.0 Ti_obs (kev) = u_max (mm) = d_lambda (A) = (c) Due to the Johann error the center of the apparent line profile F(u) is not at u = 0 but shifted to u = mm.

22 Numerical Results for a Gaussian centered at λ z = Å The calculation of the Johann Errors for spectral line z is analogue to that for spectral line w, except that we must now calculate the Johann errors for two unequal crystal parts. Crystal Length, L (mm) = 64 Johann-Errors: EJ1_0 = EJ2_0 = Enter Ti(keV) = 1 Ti_obs(keV) = u_max (mm) = d_lambd (A) = Without Johann error, the center of the observed profile would be at u_max (mm) = Note, the shift and T i_obs for the line z are significantly larger than for line w.

23 Apparent distance between lines w & z vs Ti The separation of the apparent profiles of spectral lines w and z varies with the ion temperature as a result of the Johann error

24 He-like Argon Spectrum w x y z

25 Ion Temperature Profile vs Time

26 Toroidal Rotation Velocity Profile vs Time

27 Ion Temperature Profiles at different Times During Locked Mode Period

28 Apparent distance between lines w & z at different Times During Locked Mode Period

29 Apparent distance between lines w & z The separation of the apparent profiles of spectral lines w and z as a result of the Johann error

30 Summary The new imaging x-ray crystal spectrometer on Alcator C-Mod is contributing to C-Mod physics program by providing T i and v toroidal profiles with good time (<20 ms) and spatial (1 cm) resolution. Such a diagnostic is also planned for ITER. Development of data analysis software is in progress. Analytical expressions for the instrumental function, or Johann error, have been derived from geometrical considerations. Effects of the rocking curve have been neglected in the present calculations and will be determined from the convolution of the rocking curve with our expression for the Johann error in forthcoming calculations. The instrumental function, or Johann error, causes distortions and line shifts of the observed (apparent) line profiles. These instrumental effects must be taken into account for the analysis of ion temperature and plasma rotation velocity profiles.