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#Production

- [Control room](http://buon.fjfi.cvut.cz/roperation/tasks/STUDENTS/1214OndrejFicker/Level_I/index.php)

## Logbook

- XX1114: Plánování experimentu se skupinou EXJF
- 111214: HXR intro setup
- XX0215: Setting NI_basic
- XX0315: Role of power supply voltage


#Runaway electrons in tokamak Golem

Runaway electrons (RE) are supra-thermal particles which are present in tokamaks at some circumstances. They can be perceived if there is some accelerating force stronger than friction forces. Such accelerating force is usually electric force and the friction force is caused by Coulomb collisions with plasma particles. The Coulomb interaction can be indeed considered as a friction force in plasma because it consist of many small contributions from surrounding particles (Fokker-Planck/Landau approximation). For higher energies of a test particle, this friction force goes down as $\propto 1/v^{-3}$. For some constant accelerating force, this means that there is a limit velocity for the collisional slowing down and faster particles can be accelerated to very large velocities. Electrons that fulfill this condition are called runaway electrons. Although these particles are strongly accelerated, their maximum energy is limited by the relativistic effects - energy losses by the synchrotron radiation and in the case of Golem namely by the confinement. The RE leave the plasma volume sooner or later, as the magnetic field fails to confine them. The impact of the focused RE electron beam on the wall and other internal structers may be extremely devastating for the tokamak of ITER size. Therefore, the reliable control and mitigation system of the RE beam must be developed and RE electrons are studied in almost every tokamak.

## Theory

The theory of RE generation, acceleration, transport and mitigation was not yet fully understood and it is the object of complex modelling efforts. Only the basic concept is introduced here. The RE are generated either by the direct acceleration caused by external field (primary mechanism) or by the momentum transition during close collisions (secondary mechanism). The first mechanism is dominant in Golem due to the short discharges and high resistivity. The secondary mechanism, called avalanche, simply does not have enough time to significantly contribute to the runaway population. The primary mechanism is also called Dreicer mechanism, after H. Dreicer [Dreicer,1959], one of the first theoreticians to study this field. The mechanism could be described by this simple equation of motion for test particle in plasma

$$
\frac{\partial v}{\partial t}-\frac{e}{m_{e}}E=eE_{D}\Psi(x),
$$

where $e$ is the charge of the electron $E$ is the accelerating field $E_{D}$ is so called Dreicer field which is defined in further text and $\Psi(x)$ is Chandrasekhar function of argument $v/v_{Te}$ - the velocity of the particle normalised to the thermal velocity of the plasma.

![*The graph of the Chandrasekhar function that describes the dependence of the Coulombic friction force on the velocity of the particle. The constant acceleration by electric force is dispalyed as the red line.  All the electrons that have the parallel component of velocity ($v_{\parallel E}$) in region III are runaway electrons.*](/TrainingCourses/FTTF/2014-2015/OndFick/Chan.png "The graph of the Chandrasekhar function that describes the dependence of the Coulombic friction force on the velocity of particle. The constant acceleration by electric force is dispalyed as the red line.  All the electrons that have the parallel component of velocity ($v_{\parallel E}$) in region III are runaway electrons.") 


The Dreicer field is the value of the electric field in which all the electrons of plasma become RE. It corresponds to the maximum of Chandrasekhar function and it is defined by the relation

$$
E_{D}=\frac{n_{e}e^{3}}{4\pi\epsilon_{0}^{2}kT_{e}}\ln\Lambda,
$$

where $n_{e}$ is the plasma density, $T_{e}$ is the plasma temperature and $\ln\Lambda$ Coulomb logarithm. If the $kT_{e}$ term is replaced by $m_{e}c^{2}$, the equation would define the critical field. This is the minimal value of field for the runaway presence. In experiment the limiting value is usually higher due to the temperature dependence.

The secondary generation is negligible in our case due to the short length of the discharge and week confinement of RE.
The density of runaway electrons in so-called runaway region at given time could be estimeated as the integral of distribution function from critical velocity to infinity. Using this calculation, the discharges with large RE produciton may be identified. The other way is to use the analytical runaway rate (e.g. Kruskal-Bernstein rate) that gives directly the RE density when integrated.

## Golem and RE

Tokamak Golem, as a small device with high loop voltage, is quite suitable for the RE electron creation. The breakdown voltage usually exceeds 10 V and the loop voltage during the discharge is not much lower. This is caused mainly by the high density of impurities in the chamber. Therefore, the typical discharge is accompanied by the RE electrons creation. These runaways usually originate in the breakdown, albeit they are continuously accelerated during the discharge. Their further fate depends namely on the magnetic field. If the field is strong enough with respect to the kinetic energy the electrons may be confined. Otherwise, the accelerated electrons hit the wall after they gain enough energy. For the low magnetic field case, the loss of first runaways is observed 2-3 ms after the breakdown. However, also the discharges with strong emission just before the discharge termination were observed. It is shown further that those electrons may be created in this late phase of the discharge due to the low density and high electric field.


## Diagnostics

### New HXR scintilator detector

In the autumn 2014, new NaI(Tl) scintillation detector with photomultiplier (PMT) and HV power source was installed for regular usage. Unlike the borrowed detector that was used in previous years, this one is powered by negative voltage. No additional amplifier or other device is needed and the signal could be directly acquired by the NI Standard/NI basic system. The negative voltage powering is less effective in detecting the individual pulses and the detecor works in integration regime during the discharge. Therefore spectroscopic measurements will be harder with the new detector, yet the exact time of HXR emmision and the radiated power and absorbed dose may be estimated. The output signal is negative, however it is turned to possitive values for postprocessing and plotting. Due to the different signal properties was necessary to develop new routines. For detailed information about photomultipliers, see \href{http://www2.pv.infn.it/~debari/doc/Flyckt_Marmonier.pdf}{external}.

![*The photo of the HXR scintillation PMT detector and HV power source with short description of important parts.*](/TrainingCourses/FTTF/2014-2015/OndFick/HXR_detector.jpg  "The photo of the HXR scintillation PMT detector and HV power source with short description of important parts.")

### Fast data acquisition

In order to get the best time resolution possible for measurements with the detector mentioned above, new data acquisition PC was utilised for HXR measuments. This National Instruments' acqusition system has the maximum sampling frequency 250 MHz. In the following figures, the signal of $^{137}Cs$ gamma radition is displayed, it was measured using the maximum frequency that allows acquisition of approx. 33 ms. From detailed pictures, it is obvious that the decay time of typical peak is in the order of 100 $\mu\mathrm{s}$. The rise time of the peak (proportional to the light decay time of scintillation crystal) is bellow 1 $\mu\mathrm{s}$.  This correspondes to the tabelated value for primary decay time of NaI(Tl) crystal (250 ns - Saint Gobain) quite well.

![*The photo of the PC "NI_basic" with fast data acquisition card.*](/TrainingCourses/FTTF/2014-2015/OndFick/NI_basic.jpg  "The photo of the PC "NI_basic" with fast data acquisition card.*")

![*The signal of HXR detector exposed to the $^{137}Cs$  $\gamma$ source and aquired by the NI 250 MHz card, acquisition time 33\,ms.*](/TrainingCourses/FTTF/2014-2015/OndFick/NI_Cs_signal_full.png  "The signal of HXR detector exposed to the $^{137}Cs$  $\gamma$ source and aquired by the NI 250 MHz card, acquisition time 33\,ms")


![*The detail of typical peak in $^{137}Cs$ signal measured by HXR detector.*](/TrainingCourses/FTTF/2014-2015/OndFick/Cs_peak_detail.png "The detail of typical peak in $^{137}Cs$  signal measured by HXR detector")


![*The detail of the edge of the peak in $^{137}Cs$  gamma  radiation signal measured by HXR detector.*](/TrainingCourses/FTTF/2014-2015/OndFick/Cs_peak_rise_time_detail.png "The detail of the edge of the peak in $^{137}Cs$  gamma  radiation signal measured by HXR detector")



### Calibration

The calibration of the HXR scintillation detector in energy is done with the help of gamma radiating isotopes $^{60}Co$ and $^{137}Cs$. The gamma quanta generated by the nuclear transitions coresponds to energies 1.1732 MeV and 1.3325 MeV for $^{60}Co$ and 662 keV for $^{137}Cs$. When the spectrum of particular isotope is measured, these energies are highlighted in the form of clear peak of full energy absorbtion (all the energy of gamma photon was converted to the visible light in scintillator and then to electric signal in PMT). An example of the $^{137}Cs$ + $^{60}Co$ spectra measured by the HXR detector and NI basic for 8 seconds using the sampling frequency 1 MHz is in the picture below.

![Energy spectrum of both $^{60}Co$ and $^{137}Cs$ radiation sources, the energy axis is based just on the large $Cs$ peak however also smaller peaks of $Co$ may be observed near corresponding eneries. This means that linear response of the detector can be assumed in this energy region.](/TrainingCourses/FTTF/2014-2015/OndFick/Hist_to_GOLwiki_cal.png  "Energy spectrum of both $^{60}Co$ and $^{137}Cs$ radiation sources, the energy axis is based just on the large $Cs$ peak however also smaller peaks of $Co$ may be observed near corresponding eneries. This means that the response of the detector is linear in this energy region.")

## Dose estimation

The new detector is not fast enough to enable the measurements of energies of single particle. However, as the detector is in the integration regime, it is possible to estimate the overall radiation dose (deposited energy) during the discharge. To calculate this quantity, we have to relate the integral of the signal to some well defined amout of energy - for example integral of $^{137}Cs$ characteristic peak. However, it is not possible to choose suitable voltage for both $^{137}Cs$ calibration and measuments during the discharge as the signal is orders of magnitude larger due to the integration effect during the discharge. Therefore, the area of the Cs peak must be scaled down to voltages where the peak starts not to be observable in the signal.

### Peak area and height vs. PS voltage

Let us begin with determining of the dependence of $^{137}Cs$ characteristic peak height on the supplied volatage, the data and the fit are displayed below.

![*The measured signal corresponding to the characteristic $^{137}Cs$ gamma radiation measured by HXR scintillation detector for different voltages on the PMT, fitted exponential function*](/TrainingCourses/FTTF/2014-2015/OndFick/Ucal2.png "The measured signal corresponding to the characteristic $^{137}Cs$ gamma radiation measured by HXR scintillation detector for different voltages on the PMT, fitted exponential function")

Using this fit, the following functional dependence to scale the Cs (662 keV) peak was determined as

$$
U_{Cs}=\exp(0.011*(U_{PS}-789.4))
$$

However, the exponential fit could overestimate the values in the low suplly voltage region (which is of interest). To check this issue, the logarithms of signal values were fitted lineary, resulting koeficitients are almost the same ($U_{Cs}=\exp(0.0125*(U_{PS}-792.4))$). The asymptotic error of the parameters is below 3% in both cases. With this function we can estimate the signal induced by 662 keV radiation even for low voltage supplies (e.g. 300 V) where the signal is not directly observable. During the experiment, the signal is integrated, so we do need the scaling of peak area and not just the peak height. As the width of the peak is approximatelly the same for arbitrary height of the peak, the functional dependence of the peak area on the peak height is probably close to linear, this assuption is supported by the following figure, where x coordinate of points is the peak height and y coordinate is the area. All the peaks are from Cs radiation source signal measured with one voltage value of the power source.

![*The peak area of the characteristic $^{137}Cs$ gamma radiation as a linear function of peak height*](/TrainingCourses/FTTF/2014-2015/OndFick/HeigthTOarea.png "The peak area of the characteristic $^{137}Cs$ gamma radiation as a linear function of peak height")

The fit takes a form $Area=0.039*Height$. Note that this multiplicative constant is for **time axis in milliseconds** to enable faster estimation, however one have to remember this. Now we have all the ingrediences to design the algorithm for rough estimation of the dose.

$$
D_{shot}[nGy]=\frac{\int HXR_{shot,ms}}{U_{Cs}(U_{PS})*0.039*m_{C}}*662*1.6\cdot 10^{-7},
$$

where $\int HXR_{shot,ms}$ is the time (ms) integrated HXR signal, $U_{Cs}(U_{PS})$ is the height of Cs peak for given power source voltage, $m_{C}$ is the weigth of the scintillation crystal (0.38 kg for our detector) and the remaining numbers are related to constants derived in previous text or to unit conversion. We should remind that this relation is valid only for current set-up - this means that the detector is connected via both NI basic and TEXTONIX osciloscope via T-connection. The connection of the osciloscope significantly decreases the signal on NI basic! One should also pay attention to the offset of the signal which arises due to the magnetic field connected with inductive current drive. the offset is negative for the flipped signal (positive peaks) and have to be substracted. This will be taken into account in following versions.

Notice that the dose is usually in the order of nGy, as is shown in the next paragraph. This is by far below every safety limit and it corresponds to $\sim 10^{4}$ particle of average energy 1 MeV absorbed by the detector head.

### Example shots

In the following figures, the HXR signal and the integrated dose is displayed for two different discharges. Notice that there is dynamic offset in the HXR data and therefore only the part of the dose below the peaks is due to radiation, the rise at the beginning and rise/drop at the end is artificial. In the first case the dose is approximately 0.1 nGy and i the second case with strong emission during the termination phase the dose is 0.35 nGy. These are very rough estimations, however, the real value should not more than one order of magnitude larger, unless some systematic error is present in the calculation.

![*The HXR signal and the dose absorbed by the detector during shot 18970 (relatively low emission)*](/TrainingCourses/FTTF/2014-2015/OndFick/Golem_DOSE_18970.png "The HXR signal and the dose absorbed by the detector during shot 18970 (relatively low emission)")

![*The HXR signal and the dose absorbed by the detector during shot 18979 (relatively high emission)*](/TrainingCourses/FTTF/2014-2015/OndFick/Golem_DOSE_18979.png "The HXR signal and the dose absorbed by the detector during shot 18970 (relatively high emission)")



## Former experiments

In the past years, the measurements of HXR energy using borrowed detectors were conducted and it was found that most of the photons emitted are around 1 MeV. Also the dependence of mean HXR intensity on the fuel pressure before the initiation of the discharge was determined. The result is liearly decreasign with increasing pressure. Electron density is proportional to this pressure, however the role of gas recycling is not taken into acount and therefore the linear decrease could not be directly linked with RE production. It is better to use direct interferometric density measurement that was installed lately. 

## Vacuum acceleration model - estimation of ultimate energy limit

The loop voltage in Golem is very high, therefore RE can gain energy much faster than in other tokamaks. To estimate the kinetic energy increase on the timescale of the discharge, it is possible to use very rough vacuum acceleration estimation. The calculation for shot 18970 is displayed in following figure. We suppose that RE is born just during the breakdown and it is accelerated by electric field on axis (derived from evolution loop voltage). In the moment of HXR emmision, the energy of such electron would be something like 5 MeV. However the energy of electron in plasma is several times smaller due to the Coulombic friction force, thus there is no need to be afraid of particles with energies of many MeV if the RE leave the plasma within few milliseconds.

![*Plasma current, loop voltage, HXR signal and estimated energy of parcticle accelerated in vacuum by the given field.*](/TrainingCourses/FTTF/2014-2015/OndFick/Re_discharge18970.png "Plasma current, loop voltage, HXR signal and estimated energy of parcticle accelerated in vacuum by the given field.")

### Outward drift velocity - how long the acceleration could last

Accoring to paper [Guan et. al., 2010], the outward drift velocity of RE may be calculated as
$$
v_{D}=\frac{qE}{B_{T}},
$$
where $q$ is the safety factor, $E$ is electric field and $B_{T}$ toroidal electric field. For Golem parameters this means loss of RE few miliseconds (2-5) after generation, which is in agreement with experiment for RE generated during the breakdown.

## Maximum number of RE regarding $I_{p}$

If we assume that the velocity of RE is near the speed of light (i.e. $3 \cdot 10^{8}$ m/s) we can estimate the number of RE necessary to carry some fraction of plasma current using realtion $I=en_{RE}v_{RE}S_{a}$. If we assume that RE current is 1 kA (not the case of Golem) we get the estimation of runaway electron density $n_{RE}=10^{14}\mathrm{m^{-3}}$. Reasonable value of current born by the RE is 10 A, which corresponds to density just $n_{RE}=10^{12}\mathrm{m^{-3}}$. This means that the fraction of runaway electron to thermal electrons is very low (something like one millionth).



## When the RE are created

In Golem, there are two types of the discharges with respect to the nature of runaway electron losses (and generation as we will show). 

### Breakdown

Plasma breakdown is definitelly the most suitable phase for RE genreatio, as the voltage is high a electron density is low and starts to increase. The figure bellow shows the typical stabilised shot with RE emission during the "flat-top". The third graph of the figure displays runaway rate according to Kruskal and Bernstain, the light violet curve is integral of this rate (overall number of RE being generated) and also different method - estimation of number of electrons in runaway region (this does not cover time development, only status at given time). From this graphs it is obvious, that RE are realy created only during the breakdown and the generated density of RE is $n_{e}\sim 10^{8}-10^{9} \mathrm{m^{-3}}$. However these estimates are sensitive to the value of temperature and impurity level $Z_{eff}$ (for details se Diploma thesis of author in references). For the case of this calculation $T_{e}=2 \mathrm{eV}$ and $Z_{eff}=5$.

![*Plasma current, loop voltage, electron density, estimated RE growth rate and RE density, HXR signal for shot 18970 with stable plasma position.*](/TrainingCourses/FTTF/2014-2015/OndFick/Re_discharge_RE_est18970.png "Plasma current, loop voltage, electron density, estimated RE growth rate and RE density, HXR signal for shot 18950 with stable plasma position.")

### Position instability

If addition plasma position stabilisation coils are not used, the Golem plasma is usually radially (and/or vertically) unstable, this sometimes leads to partial termination of the plasma and "second breakdown" because the loop voltage is still high. However, during the partial termination the plasma density is very small and the conditions for RE generation are even better than during the breakdown and this results to almost imidiate RE/HXR emission. This interpretation is in agreement with the prediction in figure below. Values $T_{e}=2 \mathrm{eV}$ and $Z_{eff}=5$ were used for this calculation again.

![*Plasma current, loop voltage, electron density, estimated RE growth rate and RE density, HXR signal for shot 18970 with unstable plasma position.*](/TrainingCourses/FTTF/2014-2015/OndFick/Re_discharge_RE_est18950.png "Plasma current, loop voltage, electron density, estimated RE growth rate and RE density, HXR signal for shot 18950 with unstable plasma position.")

The two types described above can be combined, of course. The position instability may be also repeated several times using kicks of fast stabilisation field.

### Discharge without RE 

Regarding the two types of discharges described above, a question arises: Why there are discharges without RE with approximatelly the same parameters as in the first type? The predicted density of RE for discharges without any HXRs measured is indeed near $10^{8}\mathrm{m^{-3}}$. There can be more reasons: RE generation maybe suppresed (faster increase of density?), RE energy is disipated due to the close collisions and good confinement or there are some RE lost (due to instabilities or turbulences?) before they gain enough energy to be detected (several tens of keVs). Large statistcal analysis might help to solve this issue, however the sett-up must be routinelly used for several month to enable this analysis. To this moment, it seems that impurity level and magnetic field are the most important parameters. High magnetic field ($>0.4T$) generaly means that the emission during the "flat-top" is not observed and together with stable position it is possible to obtain discharges with no RE even for high loop voltage and high impurity level operation.

## Suggestions for operation without RE


* Better vacuum conditions - less impurities
(new vacuum vessel to prevent leakages, more efficient vacuum pumping)
* Good wall conditioning scheme - short glow after 5 discharges?
* High fuel pressure / plasma electron density operation
* Lower loop voltage (should be automatic in pure plasma)
* Radial and vertical stabilistation to prevent RE generation due to the density drop ("second breakdown")
* As large toroidal field as possible to secure good confinement

# Conclusions

In this report, the basic theoretical backgroud regarding RE electron generation in small tokamaks, diagnostic tools and experimental data measured on tokamak Golem were described. The energies of runaway electrons reached due to acceleration in loop voltage and the lack of the collisional friction force were estimeted to be near 1 MeV, while the number of density of RE electrons is probably $10^{7}-10^{12} \mathrm{m}^{-3}$, based on several different methods of estimation. The calculated dose absorbed by the HXR detecor is usually less than 1 nGy with precision approximatelly one order of magnitude. Therefore operation of Golem is reatively safe. The absorbed dose during high HXR production shot is 0.5 nGy (in few ms), while the typical dose power due to the background is 66 nGy/h. The dose power is much larger for Golem operation, however, due to the short duration of radiation, the overall effect is negligible in comparison with background radiation or lungs X-ray examination. In order to operate without RE, it is recommended to make the vacuum conditions better and operate with high plasma electron density and high toroidal magnetic field.

# Attached scripts

* [GolemHXR_hist.py](/TrainingCourses/FTTF/2014-2015/OndFick/GolemHXR_hist.py) - Generates histogram of calibration data 
* [GolemDOSE.py](/TrainingCourses/FTTF/2014-2015/OndFick/GolemDOSE.py) - Calculation of the absorbed dose
* [Golem_exp.py](/TrainingCourses/FTTF/2014-2015/OndFick/Golem_exp.py) - Displays important parameters and estimate of RE density
* [Golem_exp_acc.py](/TrainingCourses/FTTF/2014-2015/OndFick/GolemHXR_hist.py) - Estimates RE acceleration in vacuum and electric field given by loop voltage

## References
* [Web of knowledge](http://apps.webofknowledge.com/)
* [DREICER, H. *Electron and ion runaway in a fully ionized gas, I*. Physical Review, vol. 115, 1959]
* [GUAN, X.; QIN H.; FISH N.J. *Phase-space dynamics of runaway electrons in tokamaks* Physics of Plasmas 17, 092502, 2010]
* [KULHÁNEK, P. \textit{Úvod do teorie plazmatu}. AGA, Praha, 2013.](http://www.aldebaran.cz/studium/fpla.pdf)
* [Flyckt, S.O.; Marmonier, C. PHOTOMULTIPLIER TUBES - principles and applications - Photonis, France](http://www2.pv.infn.it/~debari/doc/Flyckt_Marmonier.pdf)
* [Saint Gobain NaI(Tl) documentation](http://www.crystals.saint-gobain.com/uploadedFiles/SG-Crystals/Documents/NaI%28Tl%29%20Data%20Sheet.pdf)
* [Kocmanová, L. Runaway electrons in tokamak and their detection, Master thesis, 2013]