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some doc improvements
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Expand Up @@ -155,69 +155,34 @@ X-ray flux calculations for ionization chambers and photodiodes
---------------------------------------------------------------------

Gas-filled ionization chambers are widely used as X-ray detectors. They are
simple to use, inexpensive, and can be highly linear in estimating the
photon flux over many orders of magnitude. X-rays entering a chamber
filled with an inert gas (typically He, N2, or one of the noble gases, or a
mixture of these) will be partially absorbed by the gas, with the strong
energy dependence shown above. By adjusting the composition of the gas,
nearly any fraction of the incident X-ray beam can be absorbed at a
particular X-ray energy, making these ideal detectors to sample the
intensity of an X-ray beam incident on a sample, while attenuating only a
fraction of the beam.
simple to use, inexpensive, and can give highly linear measures of photon flux
over many orders of magnitude. X-rays entering a chamber filled with an inert
gas (typically He, N2, or one of the noble gases, or a mixture of these) will
be partially absorbed by the gas, with the strong energy dependence shown
above. By adjusting the composition of the gas, nearly any fraction of the
incident X-ray beam can be absorbed at a particular X-ray energy, making these
ideal detectors to sample the intensity of an X-ray beam incident on a sample,
while attenuating only a fraction of the beam.

Some of the X-rays in the gas will be absorbed by the photo-electric effect
which will *ionize* the gas, generating free electrons and energetic ions. Te
which will *ionize* the gas, generating free electrons and energetic ions. The
first ionization event will generate an electron-ion pair with the energy of
the X-ray minus the binding energy of the core electron. The high-energy
electron and ion pair will further ionize other gas molecules. With an
electric potential (typically on the order of 1 kV /cm) across the plates of
the chamber, a current can be measured that is proportional to the X-ray
energy and fluence of the X-rays.
the chamber, a current is generated that is proportional to the X-ray energy
and fluence of the X-rays.

In addition to the photo-electric absorption, X-rays can be attenuated by gas
molecules in an ion chamber by incoherent (Compton) or coherent (Rayleigh)
scattering processes. The coherent scattering will not generate any electrons
in the gas, but will elastically scatter X-rays out of the main beam.
Incoherent scattering will generate some current, though not all (and
typically only a small portion) of the incident X-ray energy is given to an
electron to generate a current. Compton scattering gives a distribution of
energies to the scattered electron depending on the angle of scattering. The
median energy of electrons generated by Compton scattering X-rays of energy
:math:`E` at 90 degrees will be

.. math::
E_{median} = E / (1 + m_ec^2 / E)
For X-rays of 10 keV, :math:`E_{median}` is about 192 eV. For 20 keV X-rays,
it will be 750 eV, and for 50 keV X-rays, it will be 4.5 keV. Because the
angular distribution of Compton scattering is not uniform, these median values
over-estimate the amount of energy transferred to the scattered electron by a
small amount that increases with energy. The mean energy of the
Compton-scattered electron can be found by integrating the Klein-Nishina
distribution. Since these values depend only on the incident X-ray energy,
these calculations have been done and the values tabulated in the
`Compton_energies` table in the XrayDB sqlite database.

Although the energy transferred to the electron by Compton scattering is much
less than by the photo-electric process the contribution can be important.
This is especially true for low-Z gas molecules such as He and N2 at
relatively high energies (10 keV and above) for which incoherent scattering
becomes much more important than photo-electric absorption, as shown above
for C. That is, for accurate estimates of fluxes from ion chamber currents at
energies about 20 keV or so, the contribution from Compton scattering should
be included. For photo-diodes (typically made of Si), the Compton scattering
cross-section exceeds the photo-electric cross-section about 56 keV.

Effective Ionization Potentials of gases and semiconductors
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The process of converting the X-ray generated current into X-ray fluence
involves several steps. The energy from a single X-ray-generated
electron is converted into a number of electron-ion pairs given by the
*effective ionization potential* of the gas. These are reasonably
well-known values (see :cite:`Knoll2010`) that are all between 20 and 40
eV, given in the :ref:`Table of Effective Ionization Potentials
<xray_ionpot_table>`.
involves several steps. The energy from a single X-ray-generated electron is
converted into a number of electron-ion pairs given by the *effective
ionization potential* of the gas. These values are available from a few
sources and range between 20 and 40 eV, given in the :ref:`Table of Effective
Ionization Potentials <xray_ionpot_table>`.

.. index:: Table of Effective Ionization Potentials
.. _xray_ionpot_table:
Expand All @@ -228,63 +193,118 @@ eV, given in the :ref:`Table of Effective Ionization Potentials
those supported by the functions :func:`ionization_potential` and
:func:`ionchamber_fluxes`.

+----------------------+----------------+
| gas/materia name(s) | potential (eV) |
+======================+================+
| hydrogen, H | 36.5 |
+----------------------+----------------+
| helium, He | 41.3 |
+----------------------+----------------+
| nitrogen, N, N2 | 34.8 |
+----------------------+----------------+
| oxygen, O, O2 | 30.8 |
+----------------------+----------------+
| neon, Ne | 35.4 |
+----------------------+----------------+
| argon, Ar | 26.4 |
+----------------------+----------------+
| krypton, Kr | 24.4 |
+----------------------+----------------+
| xenon, Xe | 22.1 |
+----------------------+----------------+
| air | 33.8 |
+----------------------+----------------+
| methane, CH4 | 27.3 |
+----------------------+----------------+
| carbondioxide, CO2 | 33.0 |
+----------------------+----------------+
| silicon, Si | 3.68 |
+----------------------+----------------+
| germanium, Ge | 2.97 |
+----------------------+----------------+
+-----------------------+----------------+
| gas/material name(s) | potential (eV) |
+=======================+================+
| hydrogen, H | 36.5 |
+-----------------------+----------------+
| helium, He | 41.3 |
+-----------------------+----------------+
| nitrogen, N, N2 | 34.8 |
+-----------------------+----------------+
| oxygen, O, O2 | 30.8 |
+-----------------------+----------------+
| neon, Ne | 35.4 |
+-----------------------+----------------+
| argon, Ar | 26.4 |
+-----------------------+----------------+
| krypton, Kr | 24.4 |
+-----------------------+----------------+
| xenon, Xe | 22.1 |
+-----------------------+----------------+
| air | 33.8 |
+-----------------------+----------------+
| methane, CH4 | 27.3 |
+-----------------------+----------------+
| carbondioxide, CO2 | 33.0 |
+-----------------------+----------------+
| silicon, Si | 3.68 |
+-----------------------+----------------+
| germanium, Ge | 2.97 |
+-----------------------+----------------+

From this table, we can see that the absorption (by photo-electric effect) of 1
X-ray of energy 10 keV will eventually generate about 300 electron-ion pairs.
That is not much current, but if :math:`10^8 \,\rm Hz` X-rays are absorbed per
second, then the current generated will be around 5 nA. Of course, the
thickness of the gas or more precisely the length of gas under ionizing
potential will have an impact on how much current is generated. The
photo-current will then be amplified and converted to a voltage using a current
amplifier, and that voltage will then recorded by a number of possible means.
Note that while the ion chamber itself will be linear over many orders of
magnitude of X-ray flux (provided the potential between the plates is high
enough - typically in the 1 kV/cm range to efficiently collect all the charged
particles before the recombine), a current amplifier at a particular setting of
sensitivity will be linear only over a couple orders of magnitude (typically
between output voltage of 0.05 to 5 V). Because of this, the sensitivity of
the current amplifier used with an ion chamber needs careful attention.
X-ray with energy 10 keV will generate about 300 electron-ion pairs. That is
not much current, but if :math:`10^8 \,\rm Hz` X-rays are absorbed per second,
then the current generated will be around 5 nA. Of course, the length of the
gas or more precisely the length of gas under ionizing potential will have an
impact on how much current is generated. The photo-current generated can be
amplified and converted to a voltage using a current amplifier, and that
voltage will then recorded by a number of possible mean: a voltage-to-frequency
generator and a digital counter is a common method for integrated current for a
specific amount of time, but other sampling methods can also be used.

An ion chamber can be linear over many orders of magnitude of X-ray flux,
provided the potential between the plates is high enough - typically in the 1
kV/cm range to efficiently collect all the charged particles before the
recombine. As an important practical note, a typical current amplifier at a
particular setting of sensitivity will be linear only over a limited range
(often over an output voltage of 0.02 to 5 V). Because of this, the
sensitivity of the current amplifier used with an ion chamber needs careful
attention to avoid saturation and maintain sensitivity.

A photo-diode works in much the same way as an ionization chamber. X-rays
incident on the diode (typically Si or Ge) will be absorbed and generate a
photo-current that can be collected. Typically PIN diodes are used, and
with a small reverse bias voltage. Because the electrons do not need to
escape the material but generate a current transported in the
semiconductor, the effective ionization potential is much lower - a few
times the semiconductor band gap instead of a few time the lowest
core-level ionization potential. The current generated per X-ray will be
larger than for an ion chamber, but still amplified with a current
amplifier in the same way as is used for an ion chamber. Generally, diodes
are thick enough that they absorb all incident X-rays.
photo-current that can be collected. Typically PIN diodes are used, and a
small reverse bias voltage is often applied. Because the electrons do not need
to escape the material but generate a current transported in the semiconductor,
the effective ionization potential is much lower - a few times the
semiconductor band gap instead of a few time the lowest core-level ionization
potential. The current generated per X-ray will therefore be larger than for
an ion chamber, and will also generally have a much faster response time. The
generated current will still measured in the same manner as a gas-filled
ionization typically using a current amplifier and integrating counter. Of
course, the thickness of the diode is difficult to adjust. The active length of
diodes are typically a few hundred microns, which is often thick enough to
absorb nearly all the incident X-rays.


Compton scattering and Ion Chamber Current
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In addition to photo-electric absorption, X-rays can be attenuated by gas
molecules in an ion chamber by incoherent (Compton) or coherent (Rayleigh)
scattering processes. The coherent scattering will not generate any electrons
in the gas, but will elastically scatter X-rays out of the main beam. On the
other hand, incoherent scattering will generate some current, though typically
only a small portion of the incident X-ray energy is given to a scattered
electron. In fact, Compton scattering has a distribution of energies given to
the scattered electron depending on the angle of scattering, so that the energy
of the scattered electron is

.. math::
E_{e} = E_{\gamma} - E_{\gamma} / [1 + E_{\gamma}/(m_ec^2) (1- \cos\theta)]
where :math:`E_{\gamma}` is the incident X-ray energy, and :math:`\theta` is the
scattering angle. From this, it easy to estimate the median energy of
electrons generated by Compton scattering X-rays of energy :math:`E` at 90
degrees will be

.. math::
E_{\rm median} = E_{\gamma} / (1 + m_ec^2 / E_{\gamma})
(recall that :math:`1 - 1/(1+x) = 1 / (1+1/x)`). For X-rays of 10 keV,
:math:`E_{\rm median}` is about 192 eV. For 20 keV X-rays, it will be 750 eV,
and for 50 keV X-rays, it will be 4.5 keV. Because of the angular distribution
of Compton scattering is not uniform, these median values over-estimate the
amount of energy transferred to the scattered electron by a small amount that
increases with energy. The mean energy of the Compton-scattered electron can
be found by integrating the Klein-Nishina distribution. Since these values
depend only on the incident X-ray energy, these calculations have been done and
the values tabulated in the `Compton_energies` table in the XrayDB sqlite
database.

Although the energy transferred to the electron by Compton scattering is much
less than by the photo-electric process the contribution can be important.
This is especially true for low-Z gas molecules such as He and N2 at
relatively high energies (10 keV and above) for which incoherent scattering
becomes much more important than photo-electric absorption, as shown above
for C. That is, for accurate estimates of fluxes from ion chamber currents at
energies about 20 keV or so, the contribution from Compton scattering should
be included. For photo-diodes (typically made of Si), the Compton scattering
cross-section exceeds the photo-electric cross-section about 56 keV, and so
should also be included for high-energy X-ray measurements.


Ion Chamber Flux calculations
Expand Down Expand Up @@ -332,20 +352,22 @@ both electrons and ions using the effective ionization potential above:
.. math::
C_{\rm photo} = 2 q_e E I_{\rm photo} / V_{\rm eff}
where :math:`q_{e}` is the electron charge (1.6e-19 C), :math:`E` is the incident
X-ray energy (in eV), :math:`I_{\rm photo}` is the flux (in Hz), and
:math:`V_{\rm eff}` is the effective ionization potential for the gas. The
leading 2 comes because both electrons and ions are typically counted for the
current from an ion chamber. It is sometimes useful to add a Frisch mesh grid
to collect the slower ions and shunt them so as to not count that portion of
the current, and thereby give the ion chamber a faster time response. In that
case, the current will be half of the value given above.

The coherent (Rayleigh) scattering produces no electrons, but the incoherent
(Compton) scattering does. The energy of the Compton-scattered electron varies
with both X-ray energy and scattering angle, as does the probability of
scattering. Integrating over all angles gives the mean electron energy, which
we use to obtain the current from the incoherent scattering:
where :math:`q_{e}` is the electron charge (:math:`1.6\times10^{-19} \rm{C}`),
:math:`E` is the incident X-ray energy (in eV), :math:`I_{\rm photo}` is the
flux (in Hz), and :math:`V_{\rm eff}` is the effective ionization potential for
the gas. The leading 2 comes because both electrons and ions are typically
counted for the current from an ion chamber. It is sometimes useful to add a
Frisch mesh grid to collect the slower ions and shunt them so as to not count
that portion of the current, and thereby give the ion chamber a faster time
response. In that case, the current will be half of the value given above.

As discussed above, the coherent (Rayleigh) scattering produces no electrons,
but the incoherent (Compton) scattering does, and the energy of the the
Compton-scattered electron varies with both X-ray energy and scattering angle,
as does the probability of scattering. Integrating over all angles (and
assuming the ion chamber is large enough to stop the scattered electrons) gives
the mean electron energy, which we use to obtain the current from the
incoherent scattering:

.. math::
C_{\rm incoh} = 2 q_e E_{\rm mean} I_{\rm incoh} / V_{\rm eff}
Expand All @@ -354,7 +376,6 @@ where :math:`E_{\rm mean}` is the mean energy of Compton-scattered
electron (approximately, but slightly less than the :math:`E_{\rm median}`
value above.


The current from an ion_chamber is typically measured as a voltage generated by
a current-to-voltage amplifier. The measured voltage will have a gain or
sensitivity in units of `A/V`. The goal is typically to calculate the flux
Expand Down

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