INVITED TALK
Simultaneous Estimations of Plasma Parameters using Quantitative Spectroscopy
(Plasma Devices LabMicrowave Tubes (MWT) Division,
CSIR-Central Electronics Engineering Research Institute (CEERI)Pilani-333031, Rajasthan, India
Joint ICTP-IAEA Workshop on Modern Methods in Plasma Spectroscopy, ICTP, Italy, March 23-27, 2015
*
Home Town:Agra
Taj MahalCSIR-CEERI, Pilani
**
1.Semiconductor• Hybrid Microcircuits• IC Design• MEMS and Microsensors• Nanotechnology & Devices• Photonics & Optoelectronics
3.Microwave Tubes• Gyrotron• Klystron• Magnetron• TWT• Cathodes• Plasma Devices
2.Electronic Systems•Agri-Electronics• Embedded System• Digital System• Power Electronics
CSIR
CEERI
Foundation was laid on 1953.Around 425 Permanent Employees
Pioneer Research Institute in India in the Field of Electronics Devices Tech.
CSIR-CEERI, Pilani, India Plasma Devices Lab
Scientist/TO Project FellowsDr. Ram Prakash, Group Leader Ms.Pooja Gulati, SRFDr. U. N. Pal, Sr. Sci. Mr. R. P. Lamba, QHFDr. Hasib Rahman, Sci. Fellow Ms. Nalini Pareek, QHFMr. Niraj Kumar , Sci. Mr. Aditya Sinha, PFMr. Mahesh Kumar, STO Mr. Varun Pathania, PFMr. B. L. Meena, STO Mr. Arvind Jadon, PF
AcknowledgementsPlasma Devices Group members
Plasma Devices Technology Activities
• VUV/UV Excimer Sources based on DBD Biomedical Applications (UV-B)Surface Treatment (VUV, UV-A)Water Purification (VUV, UV-C)
• High Power Plasma SwitchesThyratrons (25kV/1kA, 40kV/3kA) & Pseudospark (25kV/5kA, 20kV/20kA)
• Plasma Cathode Electron GunElectron Beam Sources (22kV/200A/cm2), (Sheet beam 20kV/1kA/cm2)
• Plasma Assisted Microwave Sources: PASOTRON (0.5MW) • Penning Discharge Devices
Ion Beam Sources (25keV Xe+ ion) and VUV Spectrometer-detector -system Calibration (22-106 nm)
Outline of the Presentation
Plasma Spectroscopy!
Why simultaneous plasma parameter measurements?
Simultaneous plasma parameter measurements usingCR-model
Least squire analysis
Opacity and diffusion analysis
Development of Penning Plasma Discharge Device forVUV-spectrometer calibration
Conclusion
Wavelength (Ao)
CII
HeI
Plasma Spectroscopy: Comment
This was due to the fact that it remains difficult to establish thelocal plasma conditions needed for the evaluation of passiveemission spectra.
“The Plasma Spectroscopy at some stage in its history had sufferedfrom misleading reputation of being primarily a subject, which hasits main goal of the identification of impurity species and respectivewavelengths” M. G. von Hellermann JET-P (94) 08
In most cases~ accurate knowledge of atomic properties, like emitted wavelength,transition probabilities, collisional cross-sections etc are required.
Spectrum Rich in information content
Ne Te Ti plasmamotion
concentration of different impurity species
4550 4600 4650 4700 4750 4800 4850 4900
1x1012
2x1012
3x1012
4x1012
5x1012
6x1012
7x1012
8x1012
shot no= 10015
Abso
lute p
hoton
inten
sity (
photo
ns. c
m-2.se
c-1. s
r-1)
Wavelength (A0)
Cumulative efforts CR model based codes ADAS, ALADDIN and CHIANTI etc.
Spectrum and Quantitative Plasma Spectroscopy
dxAN41I 2
1
x
x uluul
n=3
n=4
n=5n=6
n=�Ionization level
Hydrogen likeground state
Doubly excitedlevels
Helium like ground state
Continuum
1s2 (1S)
1s
1s2s (1S)
1s2s (3S)
1s2p (1P)1s2p (3P)
2p2 (1D) 2p2 (3P)
2s 2p (1P) 2s 2p (3P)
-- Photons cm-2 sec-1 sr-1
Spectrum and Quantitative Plasma Spectroscopy
1
uldeie
ullulue puNuNNAlNXlNN
tuN )()()()()()(
ulul
ululee AXNuSNuN )()(
The local temporal relaxation of the density N(u) of level u is described by,
Populating term
Depopulating term
For Simplifications: One can take some valid assumptions
Radialtransport:1D
urrr1
u
uu
NsourceK )(
Nu can be solved at various degrees of sophistications judged to be relevant.
Spectrum and Quantitative Plasma Spectroscopy
2). To a desired accuracy there is always some level above which theeffect of the radiative processes may be neglected.
4.) Compared with the relaxation time for the ground level populationthese others may regarded as instantaneous
3). Dominant population~ Ground state atoms, ions and metastablesand their some is constant
1). Electrons are Maxwellian
uldeie
ullulue puNuNNAlNXlNN
tuN )()()()()()(
ulul
ululee AXNuSNuN )()(
Main rate equation
will reduce to -----------------
Collisional Radiative (CR)-model and assumptions
Simplification in CR-model
]ANX)[u(R)1(N)1()1( 1u1u
e1u0edCR
])[(1)1( 11
111
1 uu
euu e
uCR ANXuRN
XSS
geie NNuRNNuRuN )()()( 10
and egCReiCRig NNSNN
dtdN
dtdN
from u=1 condition
Ng+Ni =Constant=NS,TotalFrom III appr.
From I, II & IV appr.
From IV appr.
Collisional Radiative (CR)-model solution
The quasi-steady-state solution for the dominant ground states will be,
Here R0 (u) and R1 (u) are the relative population coefficients and are complex functionof S, A, X, , , d which are function of electron temperature and density only.
The ADAS, ALADDIN, CHIANTI etc are few well-known codes to be used for rate coefficients
From u>1 condition
With an assumption of average electron density and temperature in an emissionlength 1 cm, the photon intensity from the CR-model,
)N~(N~CPE)N~(N~CPE)N~(N~CPE41)(I~ Memetastablegeexcitationiegrecombininul
where grecombininCPEexcitationCPE
represents the effective photon emission coefficients (photons cm3 sec-1).
The ADAS code derives PEC values for particular line ul after calculatingthe population distribution of levels.
Note: PEC’s are complicated functions of electron densities and temperatures.
Finding Ne, Te, Ni, Ng and Nm simultaneously
metastableCPEand,
Simultaneous plasma parameter measurements using CR-model
For VUV spectrometer-detector system calibration
Such calibrations are possible by branching ratio method or bysynchrotron radiation sources
A simple laboratory based method to calibrate a VUV spectrometerdetector systemRam Prakash et al. “Calibration of a VUV spectrograph by Collisional-Radiative modeling of a discharge plasma” J.Phys. B: At. Mol. Opt. Phys. 5 July 43 (2010) 144012(5pp). (Nominated for ENI International Award, Italy, 2011)
In this method, on the basis of experimentally observed intensities ofa number of spectral lines of helium in the visible region from aPenning discharge (PD) source , a large number of plasma parameters,such as, Ne, Te, Ni, Ng and Nm (2 3S) simultaneously estimated usingcollisional-radiative (CR) model of ADAS database.
These are used to obtain the absolute intensities of a few lines in thevacuum ultraviolet (VUV) region, which were compared with observedVUV spectrometer-detector system to obtain calibration factors.
Why simultaneous plasma parameter measurements?
A standard 1cm penning plasma source (VUV Source SD-01 from Jobin-Yvon,France) for wavelength calibration (i.e. from 100 Ǻ to 1700 Ǻ)Helium gas plasma was characterizedBoth visible and VUV spectra were recorded simultaneously.
Our model understanding
VUV spectrometer (Jobin-Yvon TGS 300, f 300 mmused with 290 g mm−1
grating, resolution 5 Ǻ)
Princeton Instrumentvisible spectrometer(resolution 2.5 Ǻ) fittedwith a CCD camera, whichhad been calibrated forabsolute intensitymeasurements.
Measured helium spectral lines from the 1 cm penning plasma source
Visible Vacuum Ultra Violet
He I He II He I He II
3889.7 Ǻ (2s3S-3p3P0) 4685 Ǻ 522.2 Ǻ 303.9 Ǻ
3965.7 Ǻ (2s1S-4p1P0) 537 Ǻ
4714.8 Ǻ (2p3P0-4s3S) 584.4 Ǻ
4923.2 Ǻ (2p1P0-4d1D)
5049.0 Ǻ (2p1P0-3s1S)
5877.5 Ǻ (2p3P0-3d3D)
6680.0 Ǻ (2p1P0-3d1D0)
7067.6 Ǻ (2p3P0-3s3S)
7283.3 Ǻ (2p1P0-3s1S)
Our model understanding
Absolute intensities of visible spectrahas been obtained from Visiblespectrometer-detector system.
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 1111 41
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 2222 41
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 3333 41
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 4444 41
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 5555 41
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 6666 41
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 7777 41
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 8888 41
)N~(NC~PE)N~(NC~PE)N~(NC~PEI Memetageexctierecotexp 9999 41
where index 1-9 is used for nine different wavelengths of the observed He lines
This procedure gives unique solutions for Ng and Nm at given Ne and Te.
Calculated Ng and Nm corresponding to following nine spectral lines for each Neand Te using SVD technique under the quasi-neutral approximation . .
Simultaneous plasma parameter measurements using CR-model
ie NN ~~
Simultaneous plasma parameter measurements using CR-model
Created grid of Te and Ne.. The grid for Te was kept between 3 eV to150 eV at the step size of 0.5 eV and for density it was in between1x109 to 1x1013 cm-3.
Example: Grid points of PECexcitation from He I 7283.3 A0 line
Simultaneous plasma parameter measurements using CR-model
We then estimated the absolute intensities of He I 3889.7 , He I 3965.7 , He I4714.8 , He I 4923.2 , He I 5049.0 , He I 5877.5 , He I 6680.0 , He I 7067.6and He I 7283.3 lines at every positive values of , , , and .eN eTiN
~gN~
MN~
A mismatch parameter,
9
1
2
91i exp
calexp
III
The index ‘i ’ is summed over all the used wavelengths, here it is nine.
The minimum in the sigma shows the best fit values of experimentallyobtained intensities with the theoretically predicted intensities at certainvalues of , , , and .eN eT iN
~gN~
MN~
Variation of mismatch parameter ( ) for different Ne and Te values
Simultaneous plasma parameter measurements using CR-model
Simultaneous plasma parameter measurements using CR-model
Diff. pressures and fixed 40 mA discharge current
Sigma
3 x 10-2 mbar 0.20
1 x 10-2 mbar 0.16
7 x 10-3 mbar 0.16
4 x 10-3 mbar 0.20
1 x 10-3 mbar 0.23
4 x 10-4 mbar 0.41
Diff. currents and fixed 4 x 10-3 mbar pressure
Sigma
20 mA 0.20
50 mA 0.23
64 mA 0.15
80 mA 0.14
90 mA 0.15
100 mA 0.12
Simultaneous plasma parameter measurements using CR-model
Densities for varying discharge currentsat constant pressure of 4x10-3 mbar
Densities for varying dischargepressures at constant dischargecurrent 40 mA.
Simultaneous plasma parameter measurements using CR-model
Inferences:•The estimation of electron density is quite similar to the previousreporting's [P. X. Feng et al. Plasma Source Sci. Technol.12 (2003)p. 142-147].
•Though the temperature prediction is poorly determined, theabsolute intensity calculations and their fit with the all-experimental values were very good at different pressures anddischarge conditions. This gives us confidence to synthesizethe spectrum in the VUV region also.
•Imperfect estimation of temperature could be due to the nature ofthe PEC coefficients and their poor dependence on temperature forspecific case or due to re-absorption or diffusion problems.
VUV spectrometer calibration scheme
Simultaneous plasma parameter measurements using CR-model
Typical VUV spectrum for discharge current 100 mA and workingpressure 4x10-3 mbar
Simultaneous plasma parameter measurements using CR-model
Comparison of observed VUV spectrum and theoretical spectrum fordischarge current 100 mA and working pressure 4x10-3 mbar .
The apparent calibration factors at 584, 537 and 522 Å lines can be inferred as ~3x1010,~2 x1011 and ~1x1012 per count
Sensitivity Curve
Obtained calibration factors for different wavelengths in the VUV region atdifferent discharge currents and constant pressure.
Simultaneous plasma parameter measurements using CR-model
Further Cross-CheckCalculated intensities change with Te in the range of 20-60 eV when the otherparameters are kept fixed.
There is no significant change in the intensities and their ratios in the VUV region forlarger range of temperatures.
Simultaneous plasma parameter measurements using CR-model
FFFurthththe CCCr Cross CChChChe kkck
Least Square Calculations qLeast Square Calculations
To simplify the problem recombination process is neglected above 3 eVIt is a good approximation for plasma to be taken purely ionizing above 3 eV
The SVD produces a solution that is the best approximation in the leastsquire sense
To cross-checked these results we used least squire fit procedure also
Least Square Calculations
The intensity equation can be written as
Memetastablegeexcitationtheor N~NC~PEN~NC~PEI41
where Itheor gives the theoretically estimated intensities for allnine lines. Now we can define a merit function,
9
1
2
___exp
9
1
2__exp
~~~~41
jMejmetastablegejexcitationjt
jjtheorjt NNCPENNCPEIIIS
where Iexpt_j represents the corresponding experimental intensities for all ninelines. If
jej_excitation ANC~PE jej_metastable BNC~PEand
The above equation will reduce to,
9
1
2
41
jMjgjj_texp N~BN~AIS
Least Square Calculations
The merit function S should be minimum with respect to Ng and Nm for thebest-fit values. This could be achieved by,
0gNS 0
iNS
and
After simplification, we gotdcccbcaNm
12
2
1121~
221
211
cdccbdaN~g
where,9
11 4
1j
j_texpj IAa9
11 4
1j
j_texpj IBb9
1
21 4
1j
jAc9
12 4
1j
jj BAc9
1
2
41
jjBd
Again we derived the values of and for the set values of the grid points for Tebetween 3 eV - 150 eV at the step size of 0.5 eV and for Ne in between 1 x 10 9 - 1 x 1013 cm-3.
gN~
MN~
Used same mismatch technique to find the plasma parameters
For discharge current 100 mA and pressure 4x10-3 mbar
The obtained results are exactly similar to the SVD technique
Least Squire Calculations
Jalaj Jain, Ram Prakash, et al., J. Theo. & Appl. phys. Vol 9 (2015) 25-31
Opacity and Diffusion Analysis
Optically thin plasma model: Very near homogeneous along the line-of-sight and remainsin steady-state for the duration of the observation.
Optically thick plasma: The re-absorption (opacity) of the photons may cause the wronginformation about the estimated plasma parameters [K Behringer and U Fantz, New J. Phys. 2, 23 (2000)]
Used ADAS database to calculate opacity affected photon emissivity coefficientson the basis of escape factor methodology.
Basically inclusion of escape factors causes a net reduction in Einstein’s A coefficients, i.e., itmodifies the atomic transition probability values. The effect of self-absorption not onlychanges the net emergent flux but also changes the population of excited states and hencethe effective rate coefficients.
A diffusion time scale ~10-4 sec is also taken into account in order to introduce the diffusionof metastable states to the wall. The diffusion time scale was estimated for our geometryusing radius of the cylinder (0.5 cm), zero order Bessel's function and the diffusion coefficientfor helium.
Opacity and Diffusion Effects on Spectral Lines
Inclusion of opacity in the observed spectral lines through PECs and addition of diffusion ofneutrals and metastable state species in the CR-model code improves the electrontemperature estimation in the simultaneous measurement.
Jalaj Jain, Ram Prakash, et al., J. Theo. & Appl. phys. Vol 9 (2015) 25-31
Penning Plasma DischargeSourcePennnnnnnSouuuuuuu
niiiiiing PlPlPlPlPlPlasma DiDiDiDiDiDischhhhhhargennnning Plasma Dischargerning Plasma Dischargerce
Ram Prakash, Gheesa lal Vyas, Jalaj Jain, JitendraPrajapati, Udit Narayan Pal, Malay Bikas Chowdhuri andRanjana Manchanda Rev. Sci. Instrum, Vol. 83, 4December (2012) 123502(1-7).
Neodymium (Nd2Fe14B)
Development of Penning Plasma Discharge Device
Device Specifications
Stainless Steel (304L) Vacuum ChamberThe chamber dimensions 200 mm x 200 mm x 450 mmPlasma size approximately 50 mm x 50 mm x 50 mmDeveloped in three different anode configurations,
1. Single Anode Ring2. Double Anode Ring3. Rectangular Anode
Vacuum Base Pressure: 1x10-6 mbarWorking Pressure: 10-5 mbar to 10-2 mbar.Used Gases: Helium, Argon, Neon.Power: 2.5 kV, 1 A DC power supplyAxial magnetic field: 1kGThe discharge and VUV emission is made for steady operation
(Continuous source)
Double anode ring configuration has beenobserved & optimized as efficient sourceof visible and VUV light radiation simultaneously.
X-axis length = 50 mmY-axis cross section 30 mm 50 mmZ-axis cross section 18 mm 50mm
12 mm is aligned on x-axis
Anode optimization (Diagnostics)
(Visible spectroscopy) Langmuir ProbeVUV spectroscopy
At fixed pressure 6X10-3mbar and fixeddischarge current 5 mA.
At fixed pressure 6X10-3mbar
At fixed pressure 6X10-3mbar
Discharge Current 15 mA
Double Ring Single Ring Rectangular Anode
Working gas pressure (mbar)
9.0x10-4 1.0x10-3 9.0x10-4 1.0x10-3 9.0x10-4
(He I 667.81nm/He I 728.13nm)
1.40 1.26 1.05 1.03 1.21
Predicted density (cm-3)
~2 x1011 ~1x1011 ~2 x1010 ~2x1010 ~9 x1010
Predicted density (cm-3) LP
~2.5 x1011 ~1.2x1011 ~2.1 x1010 ~2.6x1010 ~9.5 x1010
Studies on Penning Plasma Discharge Device
AMAZE Simulation Software VORPAL PIC Simulation
After 1 sec
Anode Optimization (Simulations)
VORPAL PIC SimulationAfter 1 sec
Observed Spectra
Working pressure 2x10-3 mbar and applied voltage 400 V
Observed VUV Spectral Range: 20 nm - 110 nm
Visible and VUV Spectra of pure helium at working pressure 7x10-3 mbar, applied voltage 1.0kV
Observed Visible Spectral Range: 400 nm - 750 nm
501.
56H
eI
VUV spectral range useful for calibration
At working pressure 7x10-3 mbar, applied voltage 1.5kV &discharge current 13 mA in the double anode ring penningdischarge arrangement
200 250 300 350 400 450 500 550 600107
108
109
1010
1011
VU
V C
alib
ratio
n Fa
ctor
s
VUV Wavelengths (Angstrom)
Ne~Ni=3x1011 cm-3
Ng~2.90x1013 cm-3
Nm = 5.66x107 cm-3
Appl. V0=1.5kV, Pressure=7x10-3 mbarTe=3.0 eV, Gas=Helium,
Application of simultaneous measurements in Tokamak Divertor Plasma
Simultaneous fit of the deuterium Balmer series D , D , D , D line intensities with ADAScollisional-radiative (CR) model data in ASXED upgrade tokamak during ohmic dischargeshas been used
During detached divertor conditions, the plasma becamerecombining and temperature dropped from 7-10 eV to ~1.5 eV inthe outer divertor region
Ram Prakash et al., IPP 10/31 Sep, 2006
485.5 486.0 486.5 487.0
0
1x1020
2x1020
3x1020
4x1020
5x1020
6x1020
7x1020
(g)
AUG21249Time=2.254-2.354 s VOL2
Inte
nsity
(Ph
m n
m s
)
Wavelength (nm)
D
H 486.15
418 420 422 424 426 428 430 432 434 4360.0
5.0x1019
1.0x1020
1.5x1020
2.0x1020
(j)
AUG21249Time=2.254-2.354 s VOL2
BII 419.48
Inte
nsi
ty (
Ph
m-2
nm
-1 s
-1)
Wavelength (nm)
D +H 433.94
CII 426.73
BIII triplet424.29,424.35424.37
Ca I 422.67CIIIOII BD 1-1BD 0-0
CD 400 402 404 406 408 410 412 414
0
1x1020
2x1020
3x1020
(l)
AUG21249Time=2.254-2.354 s VOL2
C III 406 .794C III 406 .891
O II 407 .586O II 407 .215
C III 407 .030
Mo I 4 05.1 2
BII 412.19
Inte
nsity
(Ph
m-2
nm
-1 s
-1)
Wavelength (nm)
D +H 410.06
CII 402.11
ShotNo.
Te(eV)
error%
Ne (m-3) error %
(m-2)
error%
(m-2)
error%
21258 7.87 10.8 7.86x1019 13.6 3.03x1016 10.6 4.54x1018 150.0
21279 7.33 29.9 6.21x1019 21.5 1.60x1016 31.1 3.82x1018 176.3
21320 10.3 21.5 7.72x1019 21.6 1.43 x1016 24.1 1.20x1018 Independent
21322 1.54 15.2 1.26x1020 16.6 2.63 x1019 106.5 4.69x1018 28.0
21325 1.62 8.9 2.06x1020 14.9 1.73 x1019 57.2 2.87x1019 23.2
21327 1.56 4.4 2.47x1020 6.6 3.39 x1019 28.3 5.41 x1019 10.8
xN~ ixN~ g
Estimated plasma parameters in the ohmic series discharges along with low-density H-mode discharge
Table V: Obtained recombining and ionizing terms in the ohmic series discharges
ShotNo.
photons m3
sec-1Ionizing term (Ph. m-2 sec-1)
21258 2.83x10-18 6.74 x1018
21279 2.94x10-18 2.92 x1018
21320 4.49x10-18 4.96 x1018
21322 6.39 x10-21 2.12 x1019
21325 6.74 x10-21 2.40 x1019
21327 4.61 x10-21 3.86 x1019
excitationCPE )xN~(N~CPE geexcitation grecombininCPE
photons m3
sec-1Recombining term (Ph. m-2 sec-1)
1.01 x10-21 3.59 x1017
1.13 x10-21 2.69 x1017
6.64 x10-22 6.15 x1016
1.51 x10-20 8.94 x1019
1.39 x10-20 8.23 x1019
1.49 x10-20 1.99 x1020
)xN~(N~CPE iegrecombinin
Ram Prakash et al., IPP 10/31 Sep, 2006
Further application of simultaneous measurements Conclusion
A simple method to infer large number of plasma parameterssimultaneously from a Penning Plasma discharge (PPD) source has beendeveloped.
The electron density, electron temperature, ground-state atom and iondensities and also the triplet metastable state (2 3S) density are theparameters estimated.
The derived plasma parameters are then used to obtain the absoluteintensities of a few lines in the vacuum ultraviolet (VUV) region.
This has been compared with the observed VUV spectral lines for whichintensity calibration was not available which facilitates to determine thecalibration factors for a few VUV spectral lines.
It is seen that the inclusion of opacity in the observed spectral lines throughCR-model based photo emission coefficients (PECs) and addition of diffusionof neutrals and metastable state species in the CR-model code improves theelectron temperature estimation in the simultaneous measurement.
Multi-gases analysis and Non-Maxwellian electron consideration need to bestudied further for large data point calibration curve.
Students:Mr. G. L. Vyas, Mr. Jalaj Jain, Ms. Bishu Agarwal
Colleagues:Dr. Vinay Kumar, IPR, GandhinagarDr. P. Vasu, IPR, GandhinagarDr. M. B. Chowdhuri, IPR, GandhinagarMrs. R. Manchanda, IPR, GandhinagarDr. Udit Narayan Pal, CSIR-CEERI, Pilani
Collaborators:Prof. R. Dux & Prof. Kurtz Behringer IPP, GermanyProf. H. P. Summers & Prof. Martin O'Mullane, JET, UKDr. R-E-H Clark, IAEA, AustriaDr. M. Goto, NIFS, Japan
Acknowledgements
Thanks