VI) Electron Cyclotron Emissionweisen/Cours_fichiers/COURS_DIAG_09_5e.pdfVI) Electron Cyclotron...

VI) Electron Cyclotron Emission

Major diagnostic for Te profiles on most fusion devices.Provides info on suprathermal (HFS ECE on TCV).

Electron cyclotron frequency;

γme is the relativistic mass

,

Larmor radius

Power emitted by an electron accelerated by Lorentz force is given by Schott-Trubnikov formula (see Hutchinson, Principles of Plasma Diag-nostics):

This is a line spectrum with lines at the cyclotron harmonics:

ΩceeBγme----------=

γ 1

1 ve2 c2⁄–

--------------------------=

ρeve

Ωce----------=

ωmmΩce

1 β″ θcos–---------------------------=

VI) Electron Cyclotron Emission 15 April 2009 1

Plasma emissivity is obtained by integrating over distribution function:

For thermal electrons at harmonic m:

These lines are broadened by two main mechanisms: relativistic broaden-ing and Doppler broadening (collisional and radiative boadening are neg-ligible)

non-Gaussian shape

Gaussian

Δωωm--------⎠

⎞relat

Te

mec2-------------≅

Δωωm--------⎠

⎞Doppler

2Te

mec2------------- θcos≅

ω−ωm

j(ω)

0

Doppler, depends on θ

relativistic


Relation between emission and absorption

Transport of electron cyclotron emission is dominated by a balance of emission and reabsorbtion. Consider an e-m wave (or ray) propagating from a point S0 to S in a homogeneous medium of temperature T (Te in a plasma). It’s intensity I will vary according to

(All above depend on ω, even if not explicit. Units are Wm-2ssr-1)

When the τs>>1 the medium is optically thick and radiation is in thermal equilibrium with the emissive/absorbtive medium and I(s,ω)=B(T(s),ω),

where is the blackbody emis-

sion (Planck) and the last approximation is for hν<<T (Raleigh-Jeans for-mula), which is appropriate for electron cyclotron emission from fusion plasmas. The black body limit introduces a relation between emissivity and the absorption coefficient (Kirchhof’s law, true whatever the optical thickness is):

It follows for the absorption coefficient in a plasma:

is the ’optical thickness’

B ω T,( ) hν3

2πc2------------ 1hν T⁄( ) 1–exp------------------------------------- ν2T

2πc2------------≅⋅=

j ω T,( )α ω T,( )------------------- B ω T,( )=

αm ω Te,( )jm ω Te,( )B ω Te,( )-----------------------

Te

2mec2---------------⎝ ⎠⎜ ⎟⎛ ⎞ m 1–

∝=


JET

TCV


Optical thickness in a tokamak

When evaluating the optical thickness (or depth) in an inhomogeneous magnetic field, the number of res-onant electrons in the path leads to a term propor-tional to B/(dB/ds)∝R in a tokamak. For θ=π/2:

In contemporary experiments the first harmonic O mode and the second harmonic X-mode are generally optically thick, i.e. τ>3.

For X2, as used in TCV, τX2 ≈ 3.7×10-22neTeB/R [eV,m-3,T,m]

see e.g. Bornatici M. et al, Nuclear Fusion 23 (1983) 1153

P. Blanchard, EPFL thesis 2606 Bornatici M. et al, Plasma Physics 23 (1981) 1981


Optical thickness contours for modelled profiles in Te(0)-ne(0) parameter

space for X2, for a horizontal view through the TCV plasma midplane:

Observation of suprathermal/thermal electron populations:

Radiation from each of the population is partly absorbed when it passes through the region occupied by the other population.

Normally the thermal popu-lation is thick enough to screen suprathermal radia-tion for LFS Te measure-ments for θ=0.

τX2 αX2 R( ) Rd∫=

LFSHFS

R R(ω0)

α


Accessibility: limited by ε→0 (cutoff, see reflectometry, chapter 4):

For O1: ω>ωpe and for X2 ω>ωU

define a critical density above which radiation does not leave the plasma. Depends also on viewing geometry (HFS/LFS).

ωce ωce2 4ωpe++2------------------------------------------------=

0 1 2 3 4 5 6 7 8

x 1011

3

2

1

0

1

2

3

4

5

ω [rad/s]

N2 ⊥

ωlh

ωD

ωuh

ωG

ωpe

Ωce 2Ω

ce3Ω

ce

Mode X

Mode O

TCV, BT=1.45T, ne=3e19m-3

0.6 0.7 0.8 0.9 1 1.1 1.20

20

40

60

80

100

120

140

160

180

R [m]

Frequence [GHz]

2Ωce

Ωce

3Ωce

ωpe

ωuh

ωD

0.6 0.7 0.8 0.9 1 1.1 1.22

3

4

5

6

7

8x 10

19

R [m]

n crit [m3

]

P. Blanchard, thesis,2002

Localisation• 95% of the radiation comes from where ∫αds integrates up to 3.

• In the case of high optical thickness (τ>>3) this can be a narrow layer near the cold resonance for LFS observation.

• For the same frequency, with τ>>3, HFS and LFS observation correspond to (slightly) displaced observation regions.

Detection of ECE radiation• Fourier transform spectrometer: large band, can cover 2-3 harmonics

Michelson interferometer, mechanical scanning of the length of one the arms. Each ω component produces fringes with different period in Δx. A Fourier

transform of I(Δx) provides E(ω) up to a scale factor.Low time resolution (~tens of ms). JET uses an absolutely calibrated instru-ment of this kind.

• Grating spectrometer: fairly high resolution, good sensitivity

• (Super)-heterodyne: highest sensitivity and resolution. Uses one or two levels of heterodyne mixing to select a frequency band. (used on most fusion devices, including JET and TCV)

Δx , reciprocating mirror

fixed mirror

beamsplitter detector

Δx

Ιhigh low ω


• Beam transport:- Oversized open waveguide with or without focussing element- Fundamental waveguide with low divergence antenna and optional focuss-ing element (mirror) as in TCV (fig. below)- Large distances in oversized waveguides to reduce losses

• TCV currently has two fixed-channel radiometers. One is generally used for HFS measurements (sensitive to suprathermal electrons) and the other for LFS. For each, there are two antennae, which are chosen depending on the plasma position. The LFS midplane view is better collimated, because it uses a plastic lens, allowing a narrow focus in the plasma.

• In addition, an orientable mirror system (like for ECH) on the mid-plane, is available. It could be used for studies of the fast electron population (θ≠0).

• TCV has a correlation ECE system, currently equipped with two fre-quency scannable YIG filters. They allow to position observation channels at arbitrary distance within the plasma, for correlation stud-ies (MHD, turbulence). See V. Udinstev’s crpplocal pages.

62 64 66 68 70 723

2

1

0

1

2

3

4

R [cm]

Z [cm]

Tile

Horn antenna

Elliptical mirror

0.6 0.8 1 1.20 .8

0 .6

0 .4

0 .2

0

0.2

0.4

0.6

0.8

Z [m]

R [m]

Vacuum interface & transition to oversized WG

LFS system

HFS system


Schematic of one of the super-heterodyne HFS radiometers on TCV. 24 channels, 78.5-114.5 GHz (0.63m<R<0.91m for BT=1.43T)channel bandwidth ΔfIF=0.75GHz, ‘video’ bandwidth ΔfV=40 kHz

� �

� � � � � � � � � � � � � � � � �� ! " � ! � � � " � � � �� ! � � � � � � � � ! " � ! � � " � � � � �� " � � � � � � � � � " � + � � � � �

� � � " � ! � � � � ! � � � � ! � � � " � � " ! � " � � � � � �� " � ! � � � � � � � � � � � � � � � � � � � � � � � " � ! � � � ( � ! ��

� � � � � � ! " ! � � � � � ! � " � � � � � � ! � � � � � � � � � � / � � � � � � � � � � � � ! " � ! � � � � � � � �� ! " � � �

� � � � � � � � � � � � � � � 4


Sensitivity• Single mode waveguide has ‘diffraction-limited’ étendue:

, whatever the shape of the antenna pattern.

Can be understood from uncertainty principle ΔxΔk≥2π by noting

that Ω≥(Δk/k)2.

• If single mode receiver sees only blackbody radiation (B) at tempera-ture T>>hν (Rayleigh Jeans limit), it will receive a power per unit

frequency bandwidth

• A practical radiometer only receives a fraction of this because of cou-pling losses throughout the system.

• As long as statistical noise from the source dominates over receiver noise, measurement noise is limited by Bose-Einstein statistics of blackbody radiation (exercise 6.1):

~ 10-2 for typical conditions.

A θ φ,( ) Ω λ2=d4π∫

p ω T,( ) B ω T,( )A θ φ,( ) Ωd∫= kBT=

ΔTrmsT---------------

ΔfVΔfIF----------=


Example: Sawteeth in thermal TCV plasma (P. Blanchard, thèse 2602, FSB, 2002)

HFS system, ECH, no ECCD, calibrated against Thomson scattering

� �

1.5

2

2.5

1.5

2

2.5

1.5

2

2.5Tec

e [ke

V]

1.5 1.505 1.51 1.515 1.52

1.5

2

2.5

t [s]

1)

2)

3)

4)

� � �

1)

2)3)4)

� �

1.504 1.505 1.506 1.507 1.5081200

1400

1600

1800

2000

2200

2400

t [s]

Tec

e [eV

]

t1 t2 t3

� � �

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1500

1000

1500

2000

2500

3000

R [m]

TE

CE [

eV]

t3

t2

t1

Rinv

� � �

� � � � � � � � � � � � � � � � � � � � " # � % � � ) + � � � - � � � � � ) � - ) 0 1 � - % � 4 " " % ' ( * , � - � - �+ � % � . - � 1 2 4 ( * 6 , � - � - � 7 � + � 9- � . � � � 0 % � � % ) � % - ; � � ) . � � � % + � � � ( * > , � - � - � + � % � % - ; � �) . � � � % + � � � ( * ? , � - � - � � ) � � % + ) � % - ; � � ) . � � � % + � � � ( � @ � � A # % 0 + � � - � � � � + � % � � � � # � � � � � ) - � ) � � � ) � - # � + � � � � � + # - � � - � ) 1 + 1 - � - � � ( � B � � C � % - � ) � + + " � � ) . � � ) � �) + 1 � # � � % ) � � 1 - � - � � ) � % - ) � � " 9 � % ( # % " � % * � , # % � � � - � � ) � 1 - � - � * , � � + 1 � � ) * , ) � 1 - � - � * ? , ( C � � � % � � + � " # + � � ) � Q � 0 + � � � + - � � � + % # % 0 + � � 0 � + # % � E � +% - ) � - � � ) � " # 0 % - � � % � � � + � % � - E � � % � H � (


Example: Suprathermal ECE in TCV produced by X2 ECCD and X3 ECE

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

2

4

6

8

10

12

14

16

18

t [s]

Temperature [keV]

φLX2=0o

φLX2=+20o

φLX2=2 0o

X3

Abs

orbe

d fra

ctio

n

Ohmic X2 X2,X3

X2, modulated X3

X2,X3 X2

–60 –40 –20 0 20 40 600

1

2

3

4

5

6

7

Tece/T TS

X2+X3

X2

X2 local toroidal angle [deg]X2 local toroidal angle

dots: Thomson scattering


VII) Particle diagnostics

Heavy Ion Beam Probe• One of the few diagnostics capable of measuring plasma potential

• Measurements of density and potential fluctuations

• Future: measurements of vector potential → poloidal flux

• Principle: inject singly ionised heavy ions (E~200keV in TJ-II and T10)analyse escaping double-ionised secondaries

VII) Particle diagnostics 15 April 2009 14

• Accelerator 100-500keV, <1mA, for small research device, Cs, Tl ...

A++e-→A2++2e-

• Directional energy analyser to observe Aq from small sample volume provides plasma potential at ionisation location:

Wsec=Wpri+(qsec-qpri)Φsv

• Secondary beam intensity related to ne at sample volume

Isec=IprineΔlσeffqsec/qpri

• Ipri from attenuation calculation. Relative fluctuations do not require Ipri, but some ‘contamination’ due to line integral effects along trajec-tories exists.

• Nearby (cm) sample volumes provide turbulence k,ω spectra by cor-relation, as well as cross-phase between vr=Ep/BT and ne fluctua-tions.

• Toroidal displacement of secondary related to poloidal flux at sample volume.

From Wootton & Schoch, Varenna 1991 course andworkshop on diagnostics


• Primary and secondary beams need to be steered by controlled elec-trostatic sweep plates to compensate for ambient magnetic fields. Pri-mary beam impact monitored by in-vessel detector

Split energy analyser with 4 quadrants* control detector input sweep plates from left/right imbalance. * control high voltage at top plate to keep beam up/down centered.

• The high voltage VA at the top plate allowing beam centering is pro-portional to the ion energy Wsec

• Analyser can be equipped with several quadrant detectors for correla-tion measurements

• By varying injection energy and direction a large portion of a cross section can be accessed.

VA

-50

0

50

100

150

0 0.1 0.2 0.3 0.4 0.5 0.6 0.

Er (V/cm)

ρ

HIBP measurements

neoclassicalTJ-II

ρ


•

• Two-point correlation measurements at r1,r2:* Divide the signals (ne or Φ fluctuations) into j=1 to M time win-

dows with preferably 2N samples* Fast Fourier Transform on each window -> A1j(ω) and A2j(ω)

* Autocorrelation theorem says Pω(r1,r2)=⟨A1j(ω)* A2j(ω)⟩j is cross-power spectrum (FT of temporal cross-correlation function). Average taken over the M time windows

HIBP in TEXT

in Diagnostics forContemporary FusionExperiments, Course &Workshop, Varenna 1991

Wootton & Schoch


• k estimated from 2 point correlations:

Example from TEXT:


• Same measurements allow local particle flux to be estimated:

• Early TEXT result suggest electrostatic turbulence important for par-ticle transport:

The right figure shows that the particle flux is important near the edge anddrops off sharply towards the interior of the plasma. This is expected since thesource of neutrals is located near the edge. Neutrals from outside the plasmacannot penetrate very far.


HIPB measurement of local poloidal flux (dream)

Plasma poloidal field causes a toroidal displacement of the primary and secondary beams (a few cm). Since analyser position is fixed, this has to be compensated for by toroidal steering of the primary beam.

In an axisymmetrical field (tokamak) there is a simple relation between the vector potentials at the injection, ionisation and detection points and the beam momentum. It results from the conservation of canonical momentum:

= constant of motion

, in particular in a tokamak and for the poloi-dal flux passing between flux surfaces 1 & 2 (full turn definition of ψ):

Hence conservation of canonical momentum can be written in more famil-iar terms:

The conservation law relates the source (s), ionisation (i) and detection (d) points by:

and neglecting the momentum change due to electron impact ionisation, with the charge, 2e , of the secondary ion explicited:

Pφ mRvφ qRAφ+=

B A∇×= Bp Aφeφ( )∇×=

ψ2 ψ1 Aφeφ( )∇× SdS∫=– Aφeφ dl 2π R2Aφ2 R1Aφ1–( )=⋅∫°=

Pφ mRvφ 2πqψ+=

mRsvφs 2πqψs+ mRivφi 2πqψi+=

mRivφi 2 2πeqψi⋅+ mRdvφd 2 2πeψd⋅+=


The two equations combine:

or

• Since there are deflectors between the injection and detection points, additional known terms, depending on the deflection voltages appear in the above equation.

• ψd and ψs do not depend on the precise distribution of current in the plasma and can be provided by a flux map of the entire experiment, using the known coil currents and magnetic measurements.

• vφd can be determined from a two/point measurement along the sec-ondary trajectory, for instance at the entrance at at the exit of an energy analyser, allowing simultaneous electrical and magnetic potential measurements.

• A key issue is the effect of the toroidal field ripple (loss of axisyme-try) experienced by the particles as they pass between the toroidal field coils on their ways in and out of the vacuum chamber. However the full 3D magnetic field distribution outside the plasma is known from magnetic diagnostics and hence on trajectories in outside the plasma (on the way in and out) can be calculated in principle exactly. A solution for minimizing ripple may be to make the ions pass very close to the plane of symmetry between adjacent TF coils, where dB/dφ~0.

• In the late nineties a reconstruction of the equilibrium using the posi-tion at which secondaries arrived after passing the plasma was achieved in TEXT (G J Schwefelberger et al, Rev. Sci. Instrum, 69, 3828 , 1998). This sheme leads to a mixture of local and trajectory-intergrated contributions, which must be unfolded.

2πψi2πm

e----------- Rdvφd Rsvφs–( ) 4πψd 2– πψs+=

ψi ψd ψd ψs– 2πme

----------- Rdvφd Rsvφs–( )+=–


Momentum balance of fast Tl ions in T10 (conceptual study, Febr. 2008)

β1,β2 are electrostatic steering plates, which ensure that the ion arrive at the detector with the right angle. They ’donate’ angular momentum to the ions. Ripple is strongest near TF coils, just outside plasma. Below, trajectories with and without ripple.


Figure below shows how the important quantities (poloidal flux ψ, angu-lar momentum l, canonical momentum C (all expressed in units of flux) , vary along the ion trajectory.

Note the contributions to l provided by the steering plates, note also that when the ripple is included a ’parasitic’ contribution to l comes from the ripple region (near TF coils). This leads to an erroneous estimate for ψ, denoted ψt. The meaning of ψt is simply an effective ψ, calculated from Bz and BR , as encountered by the ion and hence dependent on the ion trajectory, since these two fields contain components produced by the TF coils. C here, in the ripple cases, also has a slightly different meaning, since it includes ψt , rather than ψ.

This effect must be accurately calculated/calibrated for a measurement of axisymetrical part of the poloidal flux, i.e. the part not including contributions from the TF coils. It’s possible, if on the detection side, the ion position is known at two points, for instance. Successful demonstration of local poloidal flux measurements using HIBP would represent a breakthrough.


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