Overview of Plasma Diagnostics (High Energy Density Plasmas)

Overview of plasma diagnostics (high energy density plasmas)

S.V. Lebedev

Imperial College London General comments

HEDP systems emit: • Visible, XUV, x-ray photons • Charged particles • Neutrons

Can be probed by similar particles

Small objects – high spatial and temporal resolution (10m, 1ns)

Principles of plasma diagnostics are mostly common for all types of plasmas

2 [email protected] May 2019 Outline

Electrical measurements: magnetic field and current

Laser probing: interferometry, density gradients

Emission: spectral lines, continuum

Thomson scattering

X-ray imaging

Proton probing

3 [email protected] May 2019 Magnetic field

Magnetic coil    E dl   B ds V  NAB C S RC after integrator: B(t)  V (t) NA int

Example of magnetic probe designed to measure magnetic field distribution inside the plasma More often a set of coils outside Always check absence of capacitive coupling! (rotation of coil by 1800 should give identical signal but of opposite polarity)

4 [email protected] May 2019 Magnetic field

Magnetic coil    E dl   B ds V  NAB C S RC after integrator: B(t)  V (t) NA int

dif. ampl. of oppositely wounded coils  (Bx, By, Bz) (100MHz) [Everson et.al, RSI 80, 113505 (2009)] 5 [email protected] May 2019 Magnetic field

Magnetic coil    E dl   B ds V  NAB C S RC after integrator: B(t)  Vint (t) NA B-probe

Magnetic probes in HEDP: • Usually positioned at some distance from the object – require extrapolation of the measured signal to the object (sometimes by many orders of magnitude) • Can be strongly affected by a “noise” from energetic particles generated in the system (can be tested using two oppositely-wound probes) Measured: 5 mT at 3cm from the coil Extrapolated: 800 T in the coil  800/5x10-3 = 1.6x105

Santos et al, NJP, 17, 6083051 (2015) 6 [email protected] May 2019 Rogowski coil

Gives total current through the loop, independent on current distribution.

(if the signal is not “too fast”, i.e.  > propagation time along the coil) I   n dAB dl B dl  0 I lA l

  nA0 I V    nA0 I

• n is number of turns per unit length, A is area of the coil cross-section • Induced voltage is proportional to dI/dt (V can be integrated before recording) • “Return wire” to exclude contribution from magnetic flux through the coil • needs electrostatic shielding

7 [email protected] May 2019 Probing by E/M waves

Plasma object

Laser beam Detector

Change of phase velocity Refraction of the beam (deflection, broadening) Rotation of polarisation plane (Faraday effect)

8 [email protected] May 2019 Refractive index measurements

Propagation of electro-magnetic waves   c   phase shift (interferometry) V   ph  k    refraction of the beam (schlieren and sadowgraphy)     rotation of polarisation plane (Faraday effect)    Refractive index of plasma without magnetic field:

2 21 2 2 2 2 p ne 3 1.1210  p  k c   (1 2 )  (1 ) ncr [ cm ]  2 ncr [ m]

Probing beam cannot propagate Cut-off densities: if the plasma density is above 337m (HCN laser) n = 1016cm-3 the critical density cr 19 -3 10.6m (CO2 laser) ncr = 10 cm 21 -3 1.06m (Nd, ) ncr = 10 cm . 21 -3 0.532m (Nd 2) ncr = 4 10 cm 9 [email protected] May 2019 Interferometry

Mach-Zehnder and Michelson configurations Signal on the detector:

2 2 2 2Er E p Et  (Er  E p )[1 2 2 cos()] (Er  E p )

2.0 intensity split: 50% / 50% 20% / 80% Equal intensities 1.5 5% / 95% of the probing  and reference 1.0 beams give the best visibility of Phase shift (typically ne<

 (k pdl  krdl) 0.0  0510 n  Line density corresponding to one fringe:  [ 1 e 1]dl  n dl   e 337m (HCN laser) n L = 6.6.1014cm-2 c ncr 2cncr e 10.6m (CO laser) n L = 2.1.1016cm-2 Number of fringes: 2 e . 17 -2  1.06m (Nd, ) neL = 2.1 10 cm  18 F   4.4610 [m]  2 nedl . 17 -2 2 [cm ] 0.532m (Nd 2) neL = 4.2 10 cm For “cw” plasmas need to exclude vibrations (two wavelength interferometry) 10 [email protected] May 2019 Interferometry  Refractive index for neutral gases

fi,k is an oscillator strength for i to k transition 2 fi,k ni 1 (2 e / m) ik is the frequency of transition a     2 2 i,k ( ik  ) ni is the density of atoms in state i For optical frequencies far from spectral lines ( is polarizibility): Phase shift: 2   (1)dl a 1 2 na   Number of fringes due to electrons and atoms: Atom 2   2 He 1.3.10-24 cm3 F   4.461014   n dl   n dl [cm] e a . -24 3 2  H2 5 10 cm •different sign of phase shift for electrons and atoms Air 1.1.10-23 cm3 [e.g. for H =0 at n /n =2.2% (0.56% for He) @ 6943Å] Ar 1.10-23 cm3 2 e a •different dependence on  Al 4.4.10-23 cm3

11 [email protected] May 2019 Interferometry

Mach-Zehnder and Michelson configurations Signal on the detector:

2 2 2 2Er E p Et  (Er  E p )[1 2 2 cos()] (Er  E p )

2.0 intensity split: 50% / 50% 20% / 80% Equal intensities 5% / 95% 1.5 of the probing  1.0 and reference beams give the

0.5 best visibility of Phase shift (typically ne<

0.0  (k dl  k dl) 0510  p r n  Line density corresponding to one fringe:  [ 1 e 1]dl  n dl   e 337m (HCN laser) n L = 6.6.1014cm-2 c ncr 2cncr e 10.6m (CO laser) n L = 2.1.1016cm-2 Number of fringes: 2 e . 17 -2  1.06m (Nd, ) neL = 2.1 10 cm  18 F   4.4610 [m]  2 nedl . 17 -2 2 [cm ] 0.532m (Nd 2) neL = 4.2 10 cm For “cw” plasmas need to exclude vibrations (two wavelength interferometry) 12 [email protected] May 2019 Interferometric imaging

Wide probing and reference beams to produce 2-D image of plasma Misalignment between the beams ()  d reference fringes ( d = / [2.sin(/2)] ) in the regions without plasma

Fringe shift due to plasma  map of neL Practical lower limit ~1/10 – 1/30 of fringe

Shear interferometer (best for small plasma objects)

Magnified beam is used as a reference

Numbers show fringe-shift in a particular position

M. Tatarakis et al, PoP, 5, 682 (1998) 13 [email protected] May 2019 Simple Example: Interferogram of 8 Wire Tungsten Array

Initial wire position

Ablation Stream

Axis Analysis method: [Swadling et al. ‐ PoP 20, 022705 (2013)] Trace experimental and background fringes: Number fringes, interpolate and subtract

Number Interpolate Subtract Phase Fringes Phase Fields More Complex example – 32 wire Al array Results published ‐ Swadling et al. ‐ Physics of Plasmas 20, 022705 (2013). More Complex example – 32 wire Al array Results published ‐ Swadling et al. ‐ Physics of Plasmas 20, 022705 (2013). Schlieren and shadow imaging

Transverse density gradients deflect the probing beam

Light source schlieren shadow y shape   x dl Knife plasma edge Image plane Deflection angle: Knife to measure 1-D gradients  Intensity distribution d 1 d pin-hole for “bright-field” schlieren  dl  n dl I  d 2 d 2    e  L  dl dy 2ncr dy  2 2  disk for “dark-field” schlieren I dx dy 

Example: (If an imaging system between the object and the . -3 =2 10 rad, =532nm Bow-shock at detection plane is used, then . 19 -3 an obstacle L is the plasma length) nedl = 1.5 10 cm

Jet boundary shock

Supersonic plasma jet 19 [email protected] May 2019 Example: plasma jet – obstacle interaction

Supersonic magnetized plasma jet interacts with an obstacle (experiments on MAGPIE facility at Imperial College):

Obstacle 18 -3 (d=0.5mm) ne ~ 10 cm , Vjet ~ 100km/s,  = 5.10-3rad, =532nm

Bow-shock

Schlieren image highlights the positions of sharp gradients: bow-shock and collimation shock at the jet boundary

Plasma jet Interferogram allows measurements of the electron density in the regions where interference fringes are traceable

Flow direction 20 [email protected] May 2019 Faraday rotation

Linearly polarised wave is superposition of two circularly polarised. Difference in refractive indexes of the two waves leads to rotation of polarisation plane Rotation of polarisation plane:   e n B dl e n     e  e B dl  1/ 2   2mec ncr (1 ne / ncr ) 2mec ncr

17 2    2.2610  n 3 B dl [rad ] [cm]  e[cm ] [G]

Example:

19 -3 5 0 ne=10 cm , B=10 G (10T), =532nm, L=1cm =>  =3.67

Magnetic field can be found if the density distribution is known Convenient to combine with interferometry

M. Tatarakis et al, PoP, 5, 682 (1998) 21 [email protected] May 2019 Faraday rotation

( at ± β from  extinction )

Detector Probe beam Faraday Polariser Self-emission sensitivity intensity rotation angle angle I x, y 2  2 SE  I  s  I  sin  I S   s x, y  I S x, y  sin x, y    B  B  2    1 1 I B x, y IS x,y I S x,y tan x, y   sin    I B x,y I B x,y  2  I S x, y 2 

G. Swadling et al, RSI, 85, 11E502 (2014) 22 [email protected] May 2019 Example of Faraday rotation measurements

Interaction with conducting obstacle Polarimetry images: two channels

23 [email protected] May 2019 Abel inversion

Many diagnostics measure a particular plasma parameter integrated along probing path. For a cylindrically symmetric object it is possible to reconstruct radial distribution from several chord integrals

a rdr Chord integral of e.g. F(y)  f (r)dx  2 f (r) 2 2 phase shift, rotation angle y (r  y ) or radiative emissivity

1 a dF dy f (r)    2 2 Radial profile of quantity F  r dy (y  r )

Accuracy of reconstruction strongly depends on errors in dF/dy  need to have sufficiently large number of chords and low level of “noise”

24 [email protected] May 2019 Outline

Emission: spectral lines, continuum

25 [email protected] May 2019 Continuum radiation

Free-free (bremsstrahlung) Free-bound (recombination on level n) j() is spectral power 2 0.5 h / kTe 4 3/ 2 (In h )/ kTe j ff ( ) ~ neni Zi T e j fb ( ) ~ neni Zi T e per unit frequency divide by 2 if need j()!

Temperature measurements Density (Zeff) measurements from absolute from the slop of continuum intensity (for h << kTe – in visible ) if radiation (h > kTe – typically temperature is known from X-ray spectrum) (weak dependence on temperature)

2 2 2 ne Zeff n n Z  n Z j() ~  e i i e eff T 1/ 2 i

need to chose spectral region free of lines

26 [email protected] May 2019 Continuum radiation

Black-body spectrum Radiation spectrum of tungsten Z-pinch Cuneo et al, PoP 2001  3 h j ( ) ~ bb eh / kT 1

 1.191020 j( )  [W / cm2 / A / ster]  12395    5  ATeV  A (e 1)

Transmission Grating Spectrometer Silicon PIN detectors

Hohlraum temperature on Z source grating

Differential filtering

Filtered PCD detectors source

27 [email protected] May 2019 Relative intensities of spectral lines

-1 h ki Aki is transition probability for spontaneous emission [s ] j  Akink -3 4 nk is number of atoms (ions) in the upper state [cm ]

f is the “oscillator strength”, usually used in spectroscopy 15 gi fik ik Aki  6.6710 2 gk A Need to know how population of different states depends on plasma parameters!

Local Thermodynamic Equilibrium (LTE). n g E k  k exp( k ) Boltzmann distribution n0 g0 kT  j A g E 1  1 1 exp( ) Temperature from the ratio of line intensities j2 A2 g2 kT

1/ 2 3  T   E  3 17     LTE is a good approximation at high density, when ne[cm ] 910      EH   EH  collisional transitions dominate over radiative

(EH 13.6eV ) Griem, H.R., “Plasma spectroscopy” 28 [email protected] May 2019 Relative intensities of spectral lines

Coronal Equilibrium dn k Collisional excitation balanced by radiative de-excitation  nen0 1kV    nk Akj  0 dt jk n n  V  k  e 0k  f (T ) Population of levels depends on temperature n0  Akj j Collisional Radiative models (equilibrium or time-dependent)

Fractional abundances of Al ions Line ratio (Al) as function of Te and Ni J. Davis et al, LPB 2001 J.P. Apruzese et al, JQSRT 1997

29 [email protected] May 2019 X-ray spectrum of a 600eV Al plasma

Effect of density on emission spectrum from 3mm diameter Al plasma •Lines and continuum •Saturation at blackbody level •Line “inversion” at high densities due to self-absorption

J. Apruzese, NRL (2001)

30 [email protected] May 2019 X-ray spectrum of a 600eV Al plasma

J. Apruzese, NRL (2001)

31 [email protected] May 2019 Spectral line broadening

Doppler broadening  Line profile for Maxwellian distribution: Ion temperature from   2  2 2     (  0 ) c  8 mi  1/ 2   Ti[eV ] 1.6810   I( )  I( 0 )exp  2 2     mH  0   2VT  0 

Also for measurements of directed velocity e.g. for H at Ti=100eV  ~ 3.7Å Problem - choice of line (in X-rays  is too small)

Stark broadening (in electric field produced by neighbouring particles)

e Stark-broadened hydrogen lines (Balmer) Characteristic E field: E  ~ n2/3 r 2 i

Characteristic  (FWHM) 2/3  n[cm3 ] (for H ( =4861 Å)):  [A]  0.4  1/ 2  14   10  15 -3 e.g.  ~ 1.8 Å for ni=10 cm

32 [email protected] May 2019 Zeeman splitting

  (M g  M g )2 4.71013 B Usually difficult to find a suitable [ A] 2 2 1 1 [ A] [G] spectral line! ( / ~  – hard in X-rays ) Example: Li beam excited by dye laser

33 [email protected] May 2019 Zeeman splitting

2 2 2 2 2 13 Al III 4s-4p ( S1/2 – P1/2 vs S1/2 – P3/2 )  [ A]  (M 2 g2  M1g1)[ A] 4.710 B[G] Different sensitivity to B-field: Zeeman splitting – B-field. Example: laser plasma in magnetic field Stark broadening – ne

S. Tessaring et al., PoP, 18, 093301 (2011) 34 [email protected] May 2019 Outline

Thomson scattering

35 [email protected] May 2019 Thomson scattering (optical)

Scattering of e/m radiation by single electron

d 2 2  re sin ( ) laser d   8  r 2  6.651025 cm2 3 e

scattering geometry ki, ks are wave-vectors of incident and scattered radiation

ki ks V is velocity vector of an electron V       k momentum ki  mV  ks  m(V  V )    mV 2 m(V  V )2 ki energy  i    s   2  2 k  k   s i Change in frequency          V (k  k )  V k is determined by the k  2 i sin( )  2 k sin( ) s i s i c 2 i 2 component of velocity || to vector k

36 [email protected] May 2019 Thomson scattering

Scattering by plasma Total electric field in the scattered wave is the sum of electric k i ks fields scattered from a large number of small plasma cells

j (each with density n =n +δn and phase shift φ )  e e e j     E ~ ( n   n )exp( i ) ~ n exp( i ) n  exp( i ) 0 (no scattering from s  e e j  e  j  e  j  uniform media) j j  j   

*  Scattering signal is determined by correlation of I s ~ Es Es  ~  nk n j  exp i( k  j )  jk density fluctuations in plasma

Plasma quasi-neutrality (Debye shielding):

D < D – density fluctuations from thermal motion

D > D – from collective modes (waves)

37 [email protected] May 2019 Scattering parameter

k waves scattered from two cells (1 and 2), separated by ki ks distance  along scattering k-vector, have a path difference  : 1   2     2 sin( )  i for   with k  2k i sin( )  2 k 2 2   2   compare  with   scattering parameter   D

 1/ 2 4 1 1.0810  n 3    i   [ cm ]  e[ cm ]    k D 4 D sin/ 2 sin( / 2 )  Te[ eV ] 

<<1 ( << D) => scattering on fluctuations occurring on spatial scales << D , thermal fluctuations inside Debye sphere ( incoherent scattering )

>>1 ( >> D) => scattering on collective modes, from electrons which motions are correlated ( coherent scattering )

38 [email protected] May 2019  Thomson scattering  Thomson scattering spectral density function [ Froula et al. “Plasma scattering of   electromagnetic radiation”, 2010]    T  Sk ,  (  s  i )    S(k, )  Se (k , )  Si (k,)

~ electron distribution Qualitatively: function “Ion” and “electron” terms in the total scattering - EM are scattered by the electrons (ions are too heavy)! “ion” term

“electron” term Debye shielding in plasma Electron motions are followed by “electron cloud” (+e): Fast motions, large frequency shifts, “electron” term “ion” term

“electron” term Ion motions shielded by electron cloud (-e/2) and by “ion cloud” (-e/2) (not to scale) Slow motions, small frequency shifts, “ion” term

39 [email protected] May 2019 Thomson scattering

 1 Incoherent scattering: fluctuations are determined by  1 k thermal motion of electrons D (low density, high temperature, large scattering angles)     Sk,  f (V ) (  kV )dV - f(V) is the electron distribution function e  m   m 2  S  n e exp e  - for Maxwellian distribution e e 2  2  2 Tek  2Tek    1/ 2 3 e.g.,  = 25nm for Nd laser (532nm) at 900  6.610 Te[ eV ] sin( / 2 )  (for plasma with Te=100eV)

Relativistic effects at high T

Possibility to measure electron distribution function (e.g. presence of fast electrons)

40 [email protected] May 2019 Thomson scattering

 1  1 Coherent scattering: kD scattering from test electron is balanced by scattering Total scattering (frequency from the shielding (electron) cloud  Se => 0 integrated)  1 for a test ion scattering is from the shielding electron Se  2 1 cloud  Si dominates (“scattering on ions”)  (high density, low temperature, small scattering angles)  Z 4 Si  2  2 2  (1 )1  (ZTe /Ti ) A. Dangor et al, JAP v.64 (1988)  Ti  i ~ k  e M i

Narrow ion feature centred on Ion wings due to ion broad electron spectrum. acoustic resonance Electron peaks give plasma density:   1/ 2   2 k T 3k T  2 2 2   k B e  B i  ( )  p (1 3/ )  2  1 mi mi  41 [email protected] May 2019 Thomson scattering

Experimental considerations Amplitude of the signal (lower limit on density)

PS 10 18 3 ~ ne T L ~ 10 for ne ~ 10 cm PLaser

Plasma light (upper limit on density)

2 PS  ne Pff , fb  ne

Scattering geometry – plasma light is collected from a larger volume than scattered

Need to suppress scattering from the walls, windows and optics or separate by time-of-flight for short laser pulses

42 [email protected] May 2019 Thomson scattering

Experimental considerations

Spatially resolved scattering using fibre- optics and imaging spectrometer Measurements of plasma flow velocity and plasma temperature (MAGPIE facility at Imperial College) [Harvey-Thompson et al., PRL 2012 & PoP 2012] 43 [email protected] May 2019 Outline

X-ray imaging

44 [email protected] May 2019 X-ray imaging

Pin-hole camera q Magnification M  Object Pinhole Detector p

Geometrical  M 1 Band-pass filter resolution      is pin-hole size)  M  pq λq 1 “Diffraction”   1.22  d  M Imploding Z-pinch resolution

Detector: X-ray film (time-integrated) or MCP camera (time-resolved)

energy 2 Detector irradiation:  unit area q2

45 [email protected] May 2019 X-ray radiography

Point-projection q  p Magnification M  p q p Geometrical   M 1 resolution      M 

( is the source size)

Detector: film (time-integrated) or MCP

Point-source backlighter

Area backlighter (requires high uniformity)

46 [email protected] May 2019 X-ray spectroscopy and radiography with spherical crystals

Radiography Spectroscopy with 1D spatial resolution

monochromatic radiography

Spectrum of Al Z-pinch on Magpie Plasma jet and imploding capsule on Z D. Sinars et al., RSI, v.75 (2004)

h 47 [email protected] May 2019 Probing with ion beams

Deflection of ions by electric or magnetic fields B, E For small angles: θ Ei q  B dl[T m] sin( )    B dl  7 2mi Ei A E[MeV ]

q 1 tan( )   E dl   E dl [MV / mm]     2Ei 2Ei[MeV ] Incident beam Ionisation

Tl+ MCF: probing by heavy ions (A=200, E~300keV)

Measurements of electric potential in plasma: Plasma •position of ionisation from trajectory

Tl+ •potential from change of energy Tl++ 48 [email protected] May 2019 Proton probing

Proton beam from laser irradiated foil (~MeV with broad energy spectrum) Registration on stack of radiohromic films, to provide energy resolution

• Point-projection imaging with MeV energy protons • Proton attenuation is typically negligible, deflection by E- and B-fields • Broad energy spectrum of protons. • Different layers in the film stack correspond to different proton energy – can be used for time-resolved imaging (ps time-of-flight delay)

49 [email protected] May 2019 Proton probing

Macchi et al., RMP, v.85 (2013)

50 [email protected] May 2019 Proton probing

Cecchetti et al., Phys. Plas. v.16 (2009) Proton beam from laser irradiated foil (broad energy spectrum) Registration on stack of radiohromic films, to provide energy resolution

E / M fields: Kugland et al., RSI 83, 101301 (2012) Rygg, et al., Science v.319 (2008)

15MeV D3He fusion protons (mono-energetic)

51 [email protected] May 2019 Bibliography

Books “Principles of plasma diagnostics”, I.H. Hutchinson “Laser-aided diagnostics of plasmas and gases”, K. Muraoka, M. Maeda “Plasma diagnostics”, W. Lochte-Holtgreven “Plasma diagnostic techniques”, R.H. Huddlestone & S.I. Leonard “Plasma scattering of electromagnetic radiation”, D.H. Froula, S.H. Glenzer, N.C. Luhmann, J. Sheffield (2010)

Review articles “Instrumentation of magnetically confined fusion plasma diagnostics”, N.C. Luthmann & W.A. Peebles, Rev. Sci. Instruments, v.55, 279 (1984) “Diagnostics for fusion reactor conditions”, E. Sindoni & C. Warton, eds. (Varenna School on plasma physics, 1978)

Proceedings of conferences “High Temperature Plasma Diagnostics”, published in journal Rev. Sci. Instruments (every two years), include review articles on the present state of different diagnostic techniques

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