X-ray analysis
X-ray analysis
In a vacuum environment the behaviour of HV differs to other breakdown mechanisms. For an atmosphere like air, breakdown is initiated by ionising the surrounding medium. While vacuum there is no medium in which to ionise. Therefore, the source of ionisation has to come from the surface of the electrode. This occurs by microscopic sharp edges creating intense electric fields on the surface, locally heating it. The sharp point will eventually vaporise creating a cloud of material that can be ionised and allow the breakdown to occur across the gap, between HV surface and ground. Due to the low pressure across the gap, electrons are not slowed but accelerated due to the electric field. This means they have a very large kinetic energy when colliding with ground, interacting with the nuclei at the surface. There are two processes that will occur, either Bremsstrahlung or Auger effect.
Bremsstrahlung
Bremsstrahlung is the primary spectrum generated from X-rays in a vacuum discharge. This effect occurs as the electrons pass by nuclei, imparting energy to an electron in a shell which raises it to a different energy level. An electron will then drop back down into the previous shell, emitting a photon. This is demonstrated in the figure below. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases.
Auger effect
The Auger effect produces a discrete spectrum. This effect occurs as electrons collides with another in an inner electron shell, either kicking it out or pushing it into higher levels. If an Auger electron is kicked out then an outer shell electron will replace it, producing a photon. If the electron is pushed into a higher level it will either drop back down or another electron will replace it, again producing a photon in the process. In the figure below this effect is demonstrated. The Auger effect is discrete due to the given shell electron that the incident electron will collide with. The intensity of the x-rays emitted creates peaks at given frequency for the shells that are affected.
X-ray analysis for HV labor vacuum chamber
In order to describe the behaviour of the x-rays produced in vacuum an analytical form is required. As mentioned the x-rays are generated by Bremsstrahlung and the Auger effect.
The Auger effect has discrete x-ray transition lines that are well characterised. These transition lines can be found in the following NIST database: https://physics.nist.gov/PhysRefData/XrayTrans/Html/search.html. In our case the characteristic K lines in stainless steel and aluminium (vacuum chamber material) are the important ones. The KL and KM transitions have energy in the range of 1.4 - 1.5 kV for aluminium. In the case of stainless steel (Fe) KL, KM, and KN transitions have energy range from 6.2 - 7.1 kV. The intensity of the transition lines will be proportional to the current applied to the system.
In the case of Bremsstrahlung, a continuous spectrum is generated. The analytical form can be represented as Kramers' law:
[math]\displaystyle{ I(\lambda)d\lambda = K\left(\frac{\lambda}{\lambda_{min}}-1\right)\frac{1}{\lambda^2}d\lambda }[/math]
where [math]\displaystyle{ I }[/math] is the intensity, [math]\displaystyle{ K }[/math] is a constant proportional to the atomic number and applied current, and [math]\displaystyle{ λ }[/math] is the wavelength. The component [math]\displaystyle{ λ_{min} }[/math] = 1.24[math]\displaystyle{ /U }[/math] is known as the energy minimum (Duane-Hunt law), where [math]\displaystyle{ U }[/math] is the applied voltage. The intensity maximum can be represented by [math]\displaystyle{ I_{max}= KU^2 }[/math]. The full spectrum with the Auger effect and Bremsstrahlung is shown below for x-rays on a rhodium target with applied voltage of 60 kV.
As can be seen the Bremsstrahlung continuum is visible along with the characteristic K lines.
Only considering the Bremsstrahlung continuum, Kramers' law can be used to see what happens when independently varying the current and the voltage applied to the system. The first figure varies the voltage but keeps the applied current to 1 μA. The second figure varies the current but keeps the voltage to 50 kV.
As can be seen, changing the voltage extends the energy minimum and extrudes the spectrum around the peak. Therefore, increasing the voltage increases the integrated area of the continuum. In the case of the current, this increases the peak of the spectrum without moving the spectrum to the higher frequencies, hence, current directly increases the intensity of the spectrum. An interesting feature is that the current all cuts off at the same wavelength. This is due to the energy minimum (Duane-Hunt law), which is only dependent on the applied voltage. The other interesting feature is that even a small change in current contributes to a very direct increase in the x-ray spectrum compared to the applied voltage.
The cases stated previously did not take into account the affect of the vacuum chamber walls. As x-rays interact with matter, a reduction of the intensity occurs, called attenuation. The reduction may be caused by absorption or by deflection (scatter) of photons. This can be affected by different factors such as incident energy and atomic number of the absorber. The behaviour of this can be represented analytically, by the mass attenuation coefficient:
[math]\displaystyle{ \frac{\mu}{\rho} = x^{-1} \ln\left(\frac{I_0}{I}\right), }[/math]
where [math]\displaystyle{ μ }[/math] is the attenuation coefficient, [math]\displaystyle{ ρ }[/math] = density, mass thickness is given by [math]\displaystyle{ x = ρt }[/math] in which t is the thickness, [math]\displaystyle{ I_0 }[/math] = incident intensity, [math]\displaystyle{ I }[/math] = emerging intensity.
The attenuation coefficient is well measured for a given element and available here: https://physics.nist.gov/PhysRefData/XrayMassCoef/tab3.html. The following figures are the attenuation coefficient for three different materials: aluminium, iron, and lead. These materials well be the only ones considered due to materials used for the vacuum chamber walls and for radiation shielding.
As can be seen, with increasing energy the attenuation decreases. The spike features in the figures occur due to the transition energy for a given electronic shell structure of that element. The [math]\displaystyle{ μ_{en} }[/math] is the mass energy-absorption coefficient.
Considering all the above, a simulation for the spectrum that would be emitted in the BeamEDM and n2EDM vacuum chambers can calculated. It should be noted that only the Bremsstrahlung continuum will be considered as the Auger effect is in the lower energy range which is the most attenuated.
In order to determine the radiation dose the power of the spectrum is required. The power of the generated x-rays can be approximated by [math]\displaystyle{ P_X }[/math] = radiation yield*[math]\displaystyle{ i }[/math]*[math]\displaystyle{ V }[/math]. The radiation yield is the conversion of energy from the electron to generating the x-rays, for various materials the values can be found here: https://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html. The relation to dose from power is: [math]\displaystyle{ 1 nW = 36 μSv/hr/μA }[/math] for radiation absorbed by a body of 100 kg (water), equivalent dose. The following determination of the spectrum also assumes that the x-rays are uniform, but in reality its highly directional. Another aspect to determine the dose that has to be considered is the solid angle subtended from the source to the body. This is taken into account in the following calculations.
To start with, the BeamEDM setup. Considering only the 4 mm thick aluminium vacuum chamber wall in which a breakdown creates x-rays on the surface of aluminium electrodes, only attenuated by the vacuum chamber, with an applied voltage of 100 kV and current of 1 μA. Using Kramer's law can produce the following spectrum:
This shows the spectrum is attenuated by approximately 90%, resulting in a emitted intensity of 9.6%. This calculates as a power of 135 μW for the incident x-rays, with the attenuated power of 13 μW. Therefore, the emitted x-ray dose is 234 μSv/hr/μA. This calculation is only considering the path were the x-rays are emitted directly towards the vacuum chamber, not passing through the electrodes. The same can be looked at for the n2EDM setup. In this case a stainless steel chamber with 4 mm thickness is used. The voltage applied is 200 kV with a current of 1 μA.
In this case the initial spectrum is much higher then the previous one due to the increased voltage. The attenuated spectrum is 3.3% of the initial, therefore, the incident power is 446 μW, with an attenuated power of 15 μW. Calculating through this would result in a dose of 264 μSv/hr/μA.
The average background rate is a few mSv per year, approximately 0.2 μSv/hr on average. Therefore, the dose stated above is much larger then the average background. The previous numbers considered a 1 m distance from the source (standing next to the vacuum chamber), hence, the solid angle is much larger so the dose is much higher then an average user would experience operating HV from outside the Faraday cage. These calculations above do not take into account the presence of electrodes, therefore, this was also performed. The same conditions as the previous BeamEDM simulation as above is used here.
After attenuation the intensity is reduced by 0.8%, therefore, power is 1.1 μW so a dose of 2.8 μSv/hr/μA. This is significantly better but if a breakdown test is done in an empty chamber this will not apply. Therefore, some form of shielding will be required. In order to check the effect of adding shielding to the outside of the Faraday cage, the previous simulations were performed again adding in the attenuation due to the presence of the 1 mm thick lead layer.
In the BeamEDM figure, the intensity is reduced to 0.1% of the incident power, while the second one, the n2EDM setup, the incident intensity is reduced to 0.2%. This calculates as 2 μSv/hr/μA and 14 μSv/hr/μA respectively at 1 m distance, i.e. the lead layer. In the case of the n2EDM setup at a distance of 2 m, the dose drops to 0.5 μSv/hr/μA.
For completeness a simulation with the n2EDM setup but with a lead layer increased to 5 mm was also performed.
In this case the incident intensity is attenuated to 0.000015%. This translate to a dose at 1 m of 1.2 nSv/hr/μA.
Finally for a check of the radiation levels from x-ray induced breakdowns at ILL on beamtime, the BeamEDM simulation was performed but with 5 mm mu-metal shielding and aluminium electrodes.
At a distance of 1 m, the incident intensity is reduced to 0.5%, hence, a dose of 1.9 μSv/hr/μA, far below the level to be expected due to the neutron beam.
Summary
- It has been calculated that the radiation levels in the n2EDM or BeamEDM setups will produce a large dose, above background. This is a risk and can be mitigated with the installation of shielding.
- The minimum shielding required is >1 mm thick in order to mitigate the radiation levels to that of the background rate.
- The current should be limited to <1 μA, otherwise the dose numbers stated here will increase linearly which could result in higher radiation doses.
- The radiation levels due to breakdowns in vacuum during beamtime is far below the level expected due to neutrons.
Reference material
Papers
- S. J. B. Reed (2005). Electron Microprobe Analysis and Scanning Electron Microscopy in Geology. Cambridge University Press. p. 12. ISBN 978-1-139-44638-9.
- File:Auger effect.pdf
- File:An underappreciated radiation hazard from high voltage electrodes in vacuum.pdf
- File:Production of x-rays and interactions of x-rays with matter.pdf
