# A high-resolution asymmetric von Hamos spectrometer for low-energy X-ray
spectroscopy at the CRYRING@ESR electron cooler

P. Jagodziński , D. Banaś, M. Pajek, A. Kubala-Kukuś,
Ł. Jabłoński, I. Stabrawa, K. Szary, D. Sobota
Institute of Physics, Jan Kochanowski University, Kielce, PL-25-406, Poland
&A. Warczak
Institute of Physics, Jagiellonian University, Kraków, PL-30-348, Poland
&A. Gumberidze, H.F. Beyer, M. Lestinsky, G. Weber
GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, D-64291, Germany
&Th. Stöhlker
GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, D-64291, Germany
GSI Helmholtz-Institut, Jena, D-07743, Germany
&M. Trassinelli
Institut des NanoSciences de Paris (INSP), CNRS, Paris, F-75005, France
<EMAIL_ADDRESS>
###### Abstract

We present research program and project for high-resolution wavelength-
dispersive spectrometer dedicated to low-energy X-ray spectroscopy at the
electron cooler of the CRYRING@ESR storage ring, which is a part of the
international Facility for Antiproton and Ion Research (FAIR) currently being
built in Darmstadt. Due to the unique shape of the electorn-ion recombination
X-ray source, resulting from the overlapping of the electron and ion beams in
the electron cooler, the spectrometer can work in the specific asymmetric von
Hamos (AvH) geometry. In order to completely eliminate the influence of
Doppler effect on the measured X-ray energies, two asymmetric von Hamos
spectrometers will be installed next to the dipole magnets on both sides of
the electron cooler to detect blue/red (0∘/180∘) shifted X-rays, e.g. emitted
in the radiative recombination (RR) process. The X-ray-tracing Monte-Carlo
simulations show that the proposed AvH spectrometer will allow to determine
with sub-meV precision, the low-energy X-rays (5-10 keV) emitted from stored
bare or few-electron heavy ions interacting with cooling electrons. This
experimental precision will enable accurate studies of the quantum
electrodynamics (QED) effects in mid-Z H- and He-like ions.

_Keywords_ Radiative recombination $\cdot$ highly charged ions $\cdot$ QED
effects $\cdot$ CRYRING@ESR electron cooler $\cdot$ asymmetric von Hamos
spectrometer $\cdot$ high-resolution X-ray spectroscopy $\cdot$ X-ray-tracing
Monte-Carlo simulations

##  1 Introduction

Radiative recombination (RR) is one of the most fundamental processes
occurring in collisions of highly charged ions with electrons. In this process
a free electron is captured into a bound state of an ion with simultaneous
photon emission which can be written as follows:

$A^{q+}+e(E)\rightarrow A^{(q-1)+}(n)+\hbar\omega.$ (1)

The emitted photon carries away the energy difference $\hbar\omega=E-E_{n}$
between the electron continuum state with the energy $E$ and the electron
bound state $E_{n}$, with principal quantum number $n$. The RR cross sections
generally increase for lower electron energies exhibiting for bare ions
$Z^{2}/nE$ scaling in the low-energy limit $(E<<E_{n})$ [1], where for few
electron ions the nuclear charge number Z (in electron charge units) has to be
replaced by an appropriate effective charge $Z_{eff}(q)$. Consequently, RR is
important for low n-states of highly charged high-Z ions interacting with low-
energy electrons and leads to emission of X-rays.The role of the relativistic
and high multipole effects for interpretation of RR X-ray experiments at high
electron energies, especially for heavy, highly charged ions has been
demonstrated and discussed in [2].

The radiative recombination of bare or few-electron ions with cold electrons
having very low energies, down to meV as it happens in the electron cooling
process in ion storage rings, gives unique possibilities to study the QED
effects in mid-Z and high-Z highly charged ions. In this case, the binding
energies of electrons in the final bound states can be precisely measured when
a high resolution X-ray spectroscopy is applied. Consequently, the emission of
X-rays from RR of highly charged ions with cold electrons in ion storage rings
gives access to study the QED effects in HCI with very high precision.

Despite of the fact that the quantum electrodynamics (QED) has been tested
with extremely high precision in hydrogen [3, 4], the QED effects [5, 6] for
large coupling constant, up to $Z\alpha\approx 0.7$ in uranium, are of
fundamental interest due to the nonperturbative nature of QED corrections in
this regime. In high-Z H-like ions, the higher order QED effects, which scale
with high powers of $Z$, become more important. For the Lamb shift the
dominating one-loop QED corrections, namely, the self-energy and vacuum
polarization scale as $(Z\alpha)^{4}$, while the two-loop corrections contain
terms which scale as $(\sim Z\alpha)^{5\div 7}$. For few-electron ions the QED
affects additionally the electron-electron interaction [7]. The finite nuclear
size effect, increasing strongly as $(Z\alpha)^{4+2(1-(Z\alpha)^{2})^{1/2}}$
[5, 7], limits a precision of determination of the Lamb shift in high-Z ions
due to the uncertainties in determination of nuclear charge distribution.

For this reason the studies of QED effects for mid-Z ions, in particular in
the $Z$ = 20-30 range, are of special interest because here the nuclear size
effect, which strongly decreases with $Z$, becomes small, while, on the other
hand, the relative precision for the determination of the Lamb shift is still
acceptable for a high energy resolution spectrometer. The precise measurements
of X-ray transitions, in the context of investigations of the QED effects in
mid-Z H- and He-like ions, were reported during the last decade [8, 9, 10, 11,
12, 13, 14, 15, 16].

For bare and H-like ions with $Z$ = 20-30, the 2$P_{1/2}$ – 1$S_{1/2}$ X-ray
transition are in the range of 4-10 keV. For these ions the calculated one-
loop Lamb shift for H-like ions is in the range 1.6-6.4 eV, being much larger
than the nuclear size effect contributing about 14-113 meV, while the two-loop
QED contribution [17] is in the range of 1-8 meV (see Tab. 1), respectively.

Table 1:  Theoretical predictions of 2$P_{1/2}$ – 1$S_{1/2}$ energy transition, Lamb Shift of 1S1/2 level and its contributions in H-like mid-Z ions (20 $\leqslant~{}\textrm{Z}~{}\leqslant$ 30) [18, 19]. All values are in eV. Element | Theoretical energy | Lamb | Self | Vac | High | Nuc | Rel
---|---|---|---|---|---|---|---
| of 2$P_{1/2}$ – 1$S_{1/2}$ | Shift | En | Pol | Order | Size | Rec
Ca (Z = 20) | 4101.8 | 1.634 | 1.748 | -0.129 | 0.0007 | 0.014 | 0.0005
Ti (Z = 22) | 4968.4 | 2.259 | 2.423 | -0.188 | 0,001 | 0.024 | 0.0007
V (Z = 23) | 5433.3 | 2.626 | 2.820 | -0.225 | 0.001 | 0.029 | 0.0007
Fe (Z = 26) | 6956.0 | 3.973 | 4.282 | -0.365 | 0.002 | 0.053 | 0.001
Ni (Z = 28) | 8078.2 | 5.096 | 5.507 | -0.491 | 0.003 | 0.076 | 0.002
Zn (Z = 30) | 9287.1 | 6.431 | 6.958 | -0.646 | 0.004 | 0.113 | 0.002

##  2 High resolution X-ray measurements at CRYRING@ESR storage rings

The ion storage rings equipped with an electron cooler offer unique
experimental possibilities to study the RR of bare and few-electron high-Z
ions with electrons. An electron cooler provides an intense beam of cold
electrons which is merged with the ion beam circulating in the ring. The
electrons, whose average velocity matches the average ion velocity $\langle
v_{e}\rangle=\langle v_{ion}\rangle$ for cooling condition, are used to cool
the stored ion beam by elastic electron-ion collisions, but can also be
employed as an electron target to study the RR at low relative energies down
to the meV range. More precisely, in the electron cooler the electron
velocities in the ion frame have the so-called flattened Maxwellian
distribution characterized by transverse ($T_{\perp}$) and longitudinal
($T_{\parallel}$) electron beam temperatures with $T_{\parallel}\ll
T_{\perp}$. At cooling condition the RR rates are determined by the electron
transverse temperature, typically $kT_{\perp}\sim$ 100 meV, which is
determined by the cathode temperature in the electron gun. Generally, the RR
rates for bare ions recombining with free electron having flattened velocity
distribution scales as $\alpha_{RR}\sim Z^{2}/(kT_{\perp})^{1/2}$ [1].

Figure 1:  General view of the CRYRING showing 0∘ and 180∘ ports located on
the axis of the electron cooler [20].

Since at the CRYRING@ESR storage ring (see Fig. 1) the adiabatic expansion of
the electron beam in the cooler is applied [21, 22], a reduction of the
transverse electron beam temperature by two orders of magnitude, down to
$kT_{\perp}\sim$ 1 meV [22] is reached. Taking into account that the electron
density can be conserved for expanded electron beam [21, 22], this results in
an increase of the RR rates by an order of magnitude. It is important to note
that the relative electron-ion velocity in the electron cooler can be detuned
from its ,,cooling” zero-value for a short time, which allows to measure a
dependence of both the radiative (RR) and dielectronic (DR) recombination on
the relative electron energy. This is of particular interest for DR studies,
which can deliver independent information on the electron beam temperatures in
the electron cooler. The RR in storage rings can be studied experimentally by
detecting the downcharged ions separated by the dipole magnet next to the
electron cooler. In this case only the total RR rates, i.e. summed up from
$n=$1 to some $n_{cut}$, which is limited by the field ionization effect in
the dipole magnet, can be accessed. The counting of downcharged ions is also
applied in DR studies aimed to observe narrow DR resonances, which also carry
information on the electronic structure of the ion. In order to make the RR
experiments state-selective, the RR photons, i.e. X-rays in case of mid- or
high-Z ions, have to be observed. For bare ions stored in the ring the RR
X-ray spectra measured in the electron cooler consist of a series of RR lines
from recombination of free electrons to the bound states, e.g. K-RR lines,
L-RR lines, etc., and secondary X-rays from the Lyman, Balmer and higher
series, due to the radiative deexcitation cascades following the initial
population of higher bound states by RR process. It is important to note that
the radiative cascades effectively increase the intensities of characteristic
X-rays, by as much as an order of magnitude for Lyman transitions [23].

Figure 2:  Scheme of the CRYRING@ESR showing the asymmetric von Hamos (AvH)
spectrometers installed at the 0∘ and 180∘ ports located on the axis of the
electron cooler [20]. Note that in this drawing the spectrometers are rotated
by 90∘ (positions and dimensions of the spectrometers are not to scale).

The X-ray RR experiments at the electron cooler of the ESR storage ring at GSI
have been performed with bare Au79+ and U92+ ions [24, 25, 26, 27, 28], aiming
at testing of QED effects. In these experiments, both the K-RR and Lyman
X-rays were measured using energy-dispersive semiconductor detectors. These
experiments have demonstrated the importance of the relativistic and QED
effects in H-like and He-like high-Z ions. In particular, the ground-state
Lamb shift was measured [28] with a high precision setting one of the
strongest tests of QED for high-Z H-like ions. In other experiment [29] at the
electron cooler a dependence of RR rate upon the relative electron energy was
measured in a state-selective manner for U92+ ions recombining with electrons
in the relative energy range of $\pm$1 eV. This experiment showed that even
for ultimately-low-energy collisions the relativistic effects are important
[29, 30] and, additionally, it demonstrated a systematic enhancement of the
measured RR rates for low n-states with respect to the standard RR
calculations [1, 31]. This finding opens new questions for interpretation of
the RR enhancement effect [32] and, on the other hand, it demonstrates that
the observed RR enhancement is an important factor which can improve
statistics and thus the precision of high-resolution X-ray spectroscopy
applied to determine accurately the RR X-ray transition energies.

Up to now, the high-resolution wavelength-dispersive X-ray spectroscopy has
been applied at the ESR only at the internal gas target. In these experiments
the FOcusing Compensated Asymmetric Laue (FOCAL) [33, 34] and Johann-type [35]
X-ray crystal diffraction spectrometers have been used. At the internal gas
target of the ESR, the precision of the high-resolution measurements of X-ray
transition energies and, consequently of the Lamb shift, are influenced by the
uncertainties in the determination of the observation angle ($\theta_{lab}$)
and relativistic ion velocity $\beta=v_{ion}/c$. As we present in the
following section, these drawbacks do not appear for the proposed AvH
spectrometer observing the X-rays on the axis of the electron cooler
simultaneously at $\theta_{lab}$ = 0∘ and $\theta_{lab}$ = 180∘, which are
red- and blue- Doppler shifted photons, respectively (Fig. 2).

##  3 Concept of the Asymmetric von Hamos spectrometer (AvH)

As shown by X-ray-tracing Monte-Carlo simulations, which are described in more
details below (see also Ref. [36, 37]), the X-rays emitted from the electron
cooler at CRYRING@ESR storage ring can be measured with a high energy
resolution, of the order of 100 meV, by using a wavelength-dispersive X-ray
spectrometer exploiting a diffraction of photons on a crystal to spread out
the incident radiation with respect to its wavelengths on a position sensitive
X-ray detector. For low-energy X-rays (5-10 keV) the reflection-type
spectrometers based on X-ray diffraction on a flat or curved crystal geometry
of Johann [38], Johansson [39] or von Hamos [40, 41] can be used. The flat-
crystal spectrometers generally have a very simple construction, but they are
characterized by low efficiency due to the non-focusing X-ray optics. The
curved crystal geometry results in improvement of the spectrometer efficiency
due to focusing of the diffracted X-rays, but can cause a loss of energy
resolution due to not well controlled deformation and waviness of the curved
crystal [42]. Consequently, the von Hamos geometry, which combines a flat
crystal properties in the dispersive plane (influencing resolution) with the
focusing capabilities of X-rays in the perpendicular plane (influencing
intensity) [40, 41], seems to be the optimal solution to measure the X-rays
from the CRYRING@ESR electron cooler.

Figure 3:  Sketch of the asymmetric von Hamos (AvH) spectrometer adopted for
long-linear X-ray source in the electron cooler, where: $L_{s}$ \- source-to-
crystal distance, $L_{d}$ \- crystal-to-detector distance, $R_{C}$ \- crystal
curvature, $\theta_{B}$ \- Bragg angle.

In design of a crystal spectrometers to be installed at the electron cooler of
CRYRING@ESR the unique features of X-ray source must be taken into account.
Namely, the X-ray source at electron cooler is linear, which is defined by an
overlap of the ion and electron beams, with effective dimensions of about 800
mm [43] $\times$ 1 mm. For such extended X-ray source to be accepted by the
X-ray diffraction spectrometer, which usually prefers a small-size X-ray
source to achieve high energy resolution, we decided to design a spectrometer
in non-standard von Hamos geometry.

In a standard von Hamos spectrometer [40, 41] the photons emitted from a small
(close to point-like) X-ray source are diffracted at the Bragg angle
$\theta_{B}$ on the crystal, which is flat in the dispersive plane and
cylindrically bent to the radius $R_{C}$ in the focusing plane, and are
measured at the focal point by a position-sensitive X-ray detector. In the
standard configuration of the von Hamos spectrometer, the source-to-crystal
$L_{s}$ and crystal-to-detector $L_{d}$ distances are equal:
$L_{s}=L_{d}=R_{C}/\sin\theta_{B}$, and both the X-ray source and the
detector, are located on the same axis. In the asymmetric von Hamos geometry
proposed for the X-ray measurements in CRYRING@ESR electron cooler, the RR
photons are emitted from a long ($\sim$ 800 mm) linear X-ray source located
about 4 m away from a possible on-axis location of the diffraction crystal
($L_{s}$). That large $L_{s}$ distance results from a fixed location of the
0∘/180∘ ports at the electron cooler available for installation of the
spectrometers. Therefore the source-to-crystal and crystal-to-detector
distances are in following relationship: $L_{s}\gg L_{d}=R_{C}/\sin\theta_{B}$
(see Fig. 3). In this non-standard von Hamos geometry, a high energy
resolution of the spectrometer remains unchanged, mainly, due to its ,,on-
axis” location and a small divergence ($\sim$ 10 mrad) of the X-ray photons
hitting the crystal. In order to cover the 5-10 keV X-ray energy range, the
asymmetric von Hamos spectrometer will be equipped with a 100 $\times$ 100 mm2
Si(111) crystal, cylindrically bent to a radius of $R_{C}$ = 1 m. For the 3rd
order of diffraction the spectrometer will cover the Bragg angle range
$\theta_{B}$ = 36∘-83∘. The photons diffracted from a crystal will be
registered by AdvaPIX TPX3 [44] silicon detector which consist of a matrix of
individual pixels (see Fig. 4). Both crystal and detector will be mounted on
independent parallel rails.

Figure 4:  Scheme of the asymmetric von Hamos (AvH) spectrometr (top view).

Each rails will be equipped with 4-axes motorized manipulators, allowing for
optimization of the crystal/detector angles and positions. The dispersive
plane of the asymmetric von Hamos spectrometer will be oriented horizontally.
The stepping precision of the linear and rotational movements will be 1 $\mu$m
and 0.001∘, respectively, which is enough for precise adjustment of the
positions and angles of the diffraction crystal and X-ray detector. The
proposed AdvaPIX TPX3 detector is sensitive to the position and time of the
registered X-ray photon. The detector matrix is a square with a side of 256
pixels of sizes 55 $\times$ 55 $\mu$m2. The signal processing guarantees time
resolution of 1.6 ns. The timing capability of the detector with AdvaPIX TPX3
redout [44] will be important to eliminate the continuous X-ray background
from the intense electron beam generating bremsstrahlung. This will be
achieved by counting the coincidences between the X-rays and downcharged ions,
separated from the main ion beam in the dipole magnet and detected by a
particle detector.

The AvH spectrometer will be able to work also in standard von Hamos geometry
mode, which allow for off-line calibration of the spectrometer and checking
the influence of crystal quality on the instrumental energy resolution. The
off-line configuration will use a 30 keV electron gun delivering intense
electron beam of a diameter down to 150 micrometers to excite the
characteristic K$\alpha$ X-rays from Fe target. The electron gun and the
2-axes target manipulator will be mounted on 6-way cross at 90 degrees angles
relative to each other, whereas the detector will be installed on extended
fixed-angle arm set up on the position and angle corresponding to the Fe
K$\alpha$ fluorescence line (see Fig. 4).

Figure 5:  Scheme of the installation of the asymmetric von Hamos
spectrometers at the CRYRING@ESR electron cooler.

The energy of photons observed in laboratory frame and emitted by a moving
source is modified by the Doppler effect. To eliminate the influence of the
Doppler effect on measurement of the X-rays emitted from the electron cooler
of CRYRING@ESR, a pair of the asymmetric von Hamos (AvH) X-ray spectrometers
will be installed before and behind the dipole magnets of the electron cooler
section (Fig. 5) in order to detect the blue- and red-shifted X-rays emitted
along the electron-cooler axis. For AvH spectrometers mounted on the electron
cooler axis, the measured blue- ($\theta_{lab}$ = 0∘) and red- ($\theta_{lab}$
= 180∘) Doppler shifted X-rays in the laboratory frame have the following
energies:

$E_{b,r}=E_{0}[(1\pm\beta)/(1\mp\beta)]^{1/2},$ (2)

where $E_{0}$ is the X-ray transition energy of interest (in the moving ion
frame). From this expressions it can be found that when the energies of blue-
($E_{b}$), and red- ($E_{r}$) shifted X-rays are measured, the X-ray energy in
the emitter frame $E_{0}$ can be expressed as:

$E_{0}=(E_{b}E_{r})^{1/2}.$ (3)

This means that the X-ray transition energy in the moving ion frame can be
determined from the measured energies of blue- and red-shifted photons,
without the influence of the relative ion velocity $\beta$, which usually
introduce sizeable uncertainties to the Doppler effect correction.

##  4 X-ray-tracing Monte-Carlo simulations

A straightforward procedure to determine the main characteristics of the
proposed AvH X-ray spectrometer, as well as to achieve better understanding of
its optical properties, is the application of X-ray-tracing simulation. The
Monte-Carlo method is especially well suited to study both the properties of
the complex optical instruments, as well as for efficient simulations of the
experiments, in particular, with moving ion beams, where due to complexity of
the calculations a fully analytical treatment is impractical. Therefore, to
simulate measurements with the AvH spectrometer we used the Monte-Carlo X-ray-
tracing code, which was developed to calculate the intensity distribution of
diffracted X-rays on the 2D-detector for various X-ray spectrometer geometries
and crystals [36, 45, 37]. This X-ray-tracing code is written in C++ using Qt
framework [46]. The code tracks the trajectory of each photon emitted randomly
from the X-ray source and checks whether it is diffracted by the crystal and
registered by a detector. In case when the photon reaches the crystal surface,
the diffraction process described by the crystal rocking curve (from XOP2.4
[47, 48, 49]) is considered. The photon diffracted by the crystal is then
traced on its way to the 2D-detector (see Fig. 6).

Figure 6:  Geometry of the asymmetric von Hamos spectrometer shown in its
dispersive plane. In this configuration, the source-to-crystal distance
$L_{s}$ is larger than the crystal-to-detector distance $L_{d}$, which is
fixed by the crystal curvature $R_{C}$ and the selected Bragg angle
$\theta_{B}$ according to $L_{d}=R_{C}/\sin\theta_{B}$.

A shape of the X-ray source was assumed according to the conditions of the
discussed experiment where the photons are emitted from the space defined by
overlapping of the ion and electron beams. The simulation starts from
generation of six random numbers, namely, two photon emission angles in a
spherical coordinate system, three coordinates of a point of photon emission,
and the photon energy taking into account a natural linewidth of the
transition given by a Lorentz energy distribution. The uniform random number
generators based on a Mersenne Twister algorithm [50] and transformations were
used to generate the random numbers having desired distributions. For X-rays
emission from a moving source, both the energy and emission angle observed in
laboratory frame are subjected to the Lorentz transformation [51]. Therefore
the photon transition energy in the moving ion frame $E_{0}$ is related to
photon energy in the laboratory frame $E_{lab}$ by the Doppler formula. The
isotropic distribution of emitted photons in the moving ion frame is also
transformed to anisotropic distribution of photons in laboratory frame. Each
photon emitted from the X-ray source is traced on its path to a crystal. A
possible diffraction point of a photon at the crystal is determined by solving
the system of equations for a straight line photon path and a cylindrically
bent crystal surface. Additionally, the traced diffraction events have to be
confined to the assumed dimensions of the crystal [36]. For each calculated
diffraction point on a crystal surface, the angle $\theta$ between photon
direction and crystallographic planes is calculated. The difference
$\Delta\theta=\theta-\theta_{B}$, where $\theta_{B}$ is the Bragg angle
described by Bragg’s law [52], determines the probability of X-ray diffraction
according to the dynamical theory described by the diffraction profile called
the crystal rocking curve [53]. In the treatment of Darwin [53, 54] this
approach accounts for the effects of X-ray refraction, absorption and multiple
reflections in a macroscopic perfect crystal which leads to a corrected form
of the Bragg angle [55]. In this approach, the diffracted photons are
scattered with a certain probability in a narrow angular range of
$\Delta\theta$ expressed as the angular distribution of X-rays diffracted from
the crystal surface. The photon diffracted from the crystal was traced on
their way to the X-ray detector. The point of recording the photon on a
detector was calculated by solving the system of equations describing a point
of intersection of a straight line diffracted photon path and a plane
representing the detector surface. In this way a 2D-distribution of photons
registered by the position-sensitive detector is obtained, which is further
used to obtain the energy resolution of the discussed spectrometer.

##  5 Results and discussion

The simulated 2D-distribution of photons emitted from moving ions, diffracted
by the crystal and detected by the position sensitive detector installed in
the asymmetric von Hamos spectrometer is shown in Fig. 7. The observed pattern
of detected photons is curved in the focusing plane, which is caused by the
crystal bending. The energy of photons in the emitter frame is $E_{0}$ = 5400
eV. For ion kinetic energy $E_{ion}$ = 5 MeV/u corresponding to ion velocity
(in units of speed of light) $\beta$ = 0.1 and for observation angle
$\theta_{lab}$ = 0∘ (blue-shifted photons), the photon energy in the
laboratory frame is $E_{b}$ = 5989.239 eV. This pattern was used to derive a
one-dimensional energy profile of the detected X-rays. This was achieved by
projecting the events on the dispersion plane taking into account a curvature
of the simulated pattern, which was described by the 2nd order polynomial
fitted to the simulated distribution. Such projection, which is the energy
spectrum of detected X-rays, is shown in the Fig. 8 for different ion beam
diameters, including beam size $\phi$ = 0.5 mm corresponding to
2D-distribution in Fig. 7. In Fig. 8 simulated energy profiles for Si(111)
crystal in 3rd diffraction order, bent to the radius of $R_{C}$ = 1000 mm and
set to the Bragg angle $\theta_{B}$ = 82.078∘ for different ion beam diameters
($\phi$) in the range of 0.5-2.0 mm are presented. The FWHM widths of these
profiles can be interpreted as the energy resolution of the spectrometer,
because in the simulations the natural broadening of the emission line
($\Gamma$) was not included. We found that for X-rays energy $E_{0}$ = 5.4
keV, the energy resolution varies between 68-250 meV.

Figure 7:  Simulated 2D-distribution of X-rays hitting the position-sensitive
X-ray detector for 5.4 keV photons (in the ion frame) diffracted by the
Si(111) for crystal 3rd order of diffraction for the ion beam of 5 MeV/u
kinetic energy and 0.5 mm diameter. Figure 8:  Simulated line profiles of 5.4
keV photons (in the ion frame) emitted at forward angle ($\theta_{lab}$ = 0∘),
diffracted by a cylindrically bent Si(111) crystal and registered by position-
sensitive X-ray detector. Fitted energy resolutions (FWHM) are shown in the
figure for different ion beam diameters $\phi$. Figure 9:  Calculated total
energy resolution of asymmetric von Hamos spectrometer for Si(111) crystals
and for photon energy range of 5-9 keV (in the ion frame) for 5 MeV/u ion beam
of 1 mm diameter. The values next to the points on the graph correspond to the
photon energy in eV in the laboratory frame.

In order to investigate the performance of the AvH spectrometer in a wide
photon energy range of 5-10 keV, the variation of the spectrometer energy
resolution was also simulated as a function X-ray energies. The results for
ion beam size of $\phi$ = 1 mm is shown in Fig. 9. In the range of 5-10 keV
X-ray energies the energy resolution of the AvH spectrometer varies, from 0.12
eV to 2.2 eV and can be further improved down to about 70 meV by reducing ion
beam diameter to 0.5 mm. However a location of centroid of a line profile can
be determined with much better precision, in particular, when the profile is
known (e.g. from simulation). By fitting such profiles, the precision of
determination of a location of its centroid, which is in fact the X-ray
transition energy of interest to be measured, can be further improved down to
sub-meV range for optimized ion beam size at photon energies about 5 keV. Such
precision for the centroids of measured X-ray profiles was estimated by
fitting the analytical model profile Asym2Sig [56] available in OriginLab
software [57], which reproduces the simulated line profile. Consequently, the
performed simulations show that the proposed asymmetric von Hamos spectrometer
can measure the energies of X-ray transitions with very high precision, down
to sub-meV.

In this case, the energy of X-ray transition can be measured at 1-2 ppm
precision, meaning that the QED effects can be studied at a high precision
level. It is worth noting here that the expected high precision of the order
of few seb-meV, approaches the ,,natural” linewidth of K-RR lines, which for
the free-bound electron transitions to the ground state is given only by the
transverse electron beam temperature being in the CRYRING@ESR electron cooler
of the order of 1 meV. This shows that for the discussed case the expected
precision for determination of X-ray energies nearly reaches the ultimate
,,natural” limit set by the transverse temperature of the electron beam [58].

Figure 10:  Simulated contributions to the total energy resolution of AvH
spectrometer: diameter size of the source (red), crystal rocking curve (green)
and detector pixel size (blue).

The factors affecting the total spectrometer energy resolution are the angular
divergence of the X-ray beam diffracted from the crystal, the source diameter
and the spatial resolution of the position sensitive detector. Based on Monte-
Carlo simulations these contributions were determined and are shown in Fig.
10. The crystal rocking curve contributes at the level of 0.055-0.095 eV, the
detector pixel size contribution varies between 0.008-0.079 eV for X-ray
energy range from 5 keV to 10 keV. The main contribution to the total energy
resolution is caused by the transverse size of the source, which contributes
at the level of 0.1-1.7 eV. The latter factor has a decisive influence on
total energy resolution of AvH spectrometer.

Figure 11:  Simulated Fe $K\alpha_{1,2}$ X-ray emission lines (points) for 3rd
order of diffraction with the Si(111) crystal with curvature radius of $R_{C}$
= 1000 mm. The solid line stand for the fit of the Fe $K\alpha_{1,2}$ lines
with the Voigt function. FWHM is the total energy resolution resulting from
the convolution of the Gaussian instrumental broadening (wG) with the
Lorentzian natural shape of the transition (wL).

The energy resolution of the spectrometer is also influenced by crystal
waviness caused by its bending. This is a consequence of Hooke’s law according
to which the curvature in one direction induces a deformation in the
perpendicular direction [59]. The proposed asymmetric von Hamos spectrometer
will be equipped in fixed-angle arm (see Sec. 3), which allows to work in
standard von Hamos geometry mode. This configuration was designed to calibrate
the spectrometer as well as to check for possible crystal imperfection
including its waviness, which affect the spectrometer energy resolution.
Consequently, the diffraction crystal imperfection will be controlled and
optimized experimentally.

The Monte-Carlo simulations were also used to determine the instrumental
energy resolution of AvH, which depends only on spectrometer components as a
quality and waviness of diffraction crystal and X-ray detector, but is
independent on the X-ray source properties. The instrumental energy resolution
was checked for standard von Hamos geometry configuration available in the
spectrometer (see Fig. 4). The simulated line profiles of X-rays of Fe
$K\alpha_{1}$ and Fe $K\alpha_{2}$ transitions, having natural linewidth
$\Gamma_{1}$ = 1.6 eV and $\Gamma_{2}$ = 2.33 eV, respectively [60], detected
by the position sensitive detector is shown in the Fig. 11. The simulated
energy spectrum of iron $K\alpha$ X-ray lines exhibits clear symmetry, which
is characteristic for standard von Hamos geometry. The simulated spectrum was
analyzed by a least-squares fitting procedure using Voigt function being a
convolution of the Lorentzian profiles, describing the natural X-ray
transition widths, with the Gaussian function which can be interpreted as the
experimental broadening of the spectrometer. For this simulation, a size of
the X-ray source, defined by the electron beam spot on a target, generated by
the electron gun, was 150 $\mu$m in diameter. In fitting, the Lorentzian
widths of the $K\alpha_{1,2}$ lines were kept fixed at the values reported by
Campbell and Papp [60], while the parameters of the peak centroid and
intensity, instrumental Gaussian width and linear background were used as free
fitting parameters. The simulations show that the Gaussian contribution for Fe
$K\alpha_{1}$ line is 0.145 eV. It should be thus noted that, the instrumental
contribution to the energy resolution is greater for described standard von
Hamos geometry as compared to the case of asymmetric geometry discussed in the
paper. It is caused by a bigger divergence of the photons emitted from
calibrating target excited by the electron gun, which is 55 mrad as compared
with 12 mrad for the RR X-ray photons emitted from the electron cooler. This
calculations demonstrate that a quality of diffraction crystal, influencing
the instrumental energy resolution of the instrument can be controlled with Fe
$K\alpha$ X-rays in the AvH spectrometer.

Table 2:  K-shell RR rate coefficients for the flattened electron beam of transverse temperature $kT_{\perp}$ = 1 meV calculated for selected bare ions with Z = 20-30 [1]. Ion | K-shell binding energy [keV] | K-RR rate [ph/s]
---|---|---
Ca20+ | 5.470 | 8.1$\times$105
Ti22+ | 6.626 | 9.8$\times$105
Fe26+ | 9.278 | 1.4$\times$106
Ni28+ | 10.775 | 1.5$\times$106
Zn30+ | 12.389 | 1.8$\times$106

In order to estimate the expected count rate of the K-shell RR photons in the
planned experiment using the designed asymmetric von Hamos spectrometer, the
needed K-shell RR rate coefficients were calculated following Ref. [1]. For
example, for 108 of mid-Z bare ions stored in the ring and an electron beam
density in the electron cooler of 107 cm-3 [22], the expected emission rate of
K-shell RR X-rays for bare mid-Z ions, e.g. Ca20+ ions, is of the order of 106
photons per second (see Tab. 2). The estimated efficiency of the asymmetric
von Hamos spectrometer calculated on the basis of Monte-Carlo X-ray-tracing
simulations as the ratio of the photons number emitted by the source in the
direction the crystal to the number of photons registered by the detector is
at the level of 1 $\times$ 10-7. Therefore, the K-RR X-ray detection count
rate is expected to be of the order of 100 photons/hour. This shows that the
AvH spectrometer will allow to measure the QED effects for mid-Z ions with
sub-meV accuracy, being thus sensitive to two-loop QED effects.

##  6 Summary

In this paper, the research program and design of a high energy resolution
asymmetric von Hamos spectrometer for low energy (5-10 keV) X-ray spectroscopy
experiments at CRYRING@ESR electron cooler is presented. Using Monte-Carlo
X-ray-tracing simulations the main characteristics of the spectrometer were
obtained. Performed simulations show that the QED effects in mid-Z ($Z=$
20-30) highly charged ions can be studied with an accuracy of sub-meV for
photon about 5 keV enabling sensitivity to two-loop QED contributions.

## Acknowledgements

This research was funded in whole or in part by National Science Centre,
Poland, under Grant No. DEC-2022/06/X/ST2/00453. For the purpose of Open
Access, the author has applied a CC-BY public copyright licence to any Author
Accepted Manuscript (AAM) version arising from this submission.

## References

  * [1] M. Pajek and R. Schuch.  Radiative recombination of bare ions with low-energy free electrons.  Phys. Rev. A, 45:7894–7905, 1992.
  * [2] J. Eichler and Th. Stöhlker.  Radiative electron capture in relativistic ion–atom collisions and the photoelectric effect in hydrogen-like high-z systems.  Phys. Rep., 439:1–99, 2007.
  * [3] M. Niering, R. Holzwarth, J. Reichert, P. Pokasov, Th. Udem, M. Weitz, T. W. Hänsch, P. Lemonde, G. Santarelli, M. Abgrall, P. Laurent, C. Salomon, , and A. Clairon.  Measurement of the Hydrogen 1${S}$-2${S}$ transition frequency by phase coherent comparison with a microwave cesium fountain clock.  Phys. Rev. Lett., 84:5496–5499, 2000.
  * [4] Ch. G. Parthey, A. Matveev, J. Alnis, B. Bernhardt, A. Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Predehl, Th. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, Ch. Salomon, P. Laurent, and T. W. Hänsch.  Improved measurement of the hydrogen 1${S}$-2${S}$ transition frequency.  Phys. Rev. Lett., 107:203001, 2011.
  * [5] J. S. Brodsky and P. J. Mohr.  Quantum Electrodynamics in Strong and Supercritical Fluids.  Springer, Berlin, 1978.
  * [6] I. Lindren.  QED effects in strong nuclear fields.  Phys. Scr., T80A:133–140, 1999.
  * [7] V. M. Shabaev, A. N. Artemyev, and V. A. Yorokhin.  QED and nuclear effects in high-Z few-electron atoms.  Phys. Scr., T86:7–15, 2000.
  * [8] C. T. Chantler, D. Paterson, L. T. Hudson, F. G. Serpa, J. D. Gillaspy, , and E. Takács.  Absolute measurement of the resonance lines in heliumlike vanadium on an electron-beam ion trap.  Phys. Rev. A., 62:042501, 2000.
  * [9] D. F. Anagnostopoulos, D. Gotta, P. Indelicato, and L. M. Simons.  Low-energy x-ray standards from hydrogenlike pionic atoms.  Phys. Rev. Lett., 91:240801, 2003.
  * [10] H. Bruhns, J. Braun, K. Kubiček, J. R. Crespo López-Urrutia, and J. Ullrich.  Testing QED screening and two-loop contributions with He-like ions.  Phys. Rev. Lett., 99:113001, 2007.
  * [11] C. T. Chantler, J. M. Laming, D. D. Dietrich, W. A. Hallett, R. McDonald, and J. D. Silver.  Hydrogenic lamb shift in iron Fe25+ and fine-structure Lamb shift.  Phys. Rev. A., 76:042116, 2007.
  * [12] P. Beiersdorfer.  Precision energy-level measurements and QED of highly charged ions.  Can. J. Phys., 87:9–14, 2009.
  * [13] J. Gillaspy, C. T. Chantler, D. Paterson, L. Hudson, F. Serpa, and E. Takacs.  First measurement of Lyman alpha x-ray lines in hydrogen-like vanadium: results and implications for precision wavelength metrology and tests of QED.  J. Phys. B, 43:074021, 2010.
  * [14] P. Amaro, S. Schlesser, M. Guerra, E.-O. Le Bigot, J.-M. Isac, P. Travers, J. P. Santos, C. I. Szabo, A.Gumberidze, and P. Indelicato.  Absolute measurement of the relativistic magnetic dipole transition energy in heliumlike argon.  Phys. Rev. Lett., 109:043005, 2012.
  * [15] C. T. Chantler, M. N. Kinnane, J. Gillaspy, L. Hudson, A. T. Payne, L. F. Smale, A. Henins, J. Pomeroy, J. Tan, J. Kimpton, E. Takacs, and K. Makónyi.  Testing three-body quantum electrodynamics with trapped ti20+ ions: evidence for a Z-dependent divergence between experiment and calculation.  Phys. Rev. Lett., 109:153001, 2012.
  * [16] K. Kubiček, P. Mokler, V. Mäckel, J. Ullrich, and J. R. Crespo López-Urrutia.  Transition energy measurements in hydrogenlike and heliumlike ions strongly supporting bound-state QED calculations.  Phys. Rev. A, 90:032508, 2014.
  * [17] V. Yerokhin, P. Indelicato, and V. Shabaev.  Two-loop self-energy correction in high-Z hydrogenlike ions.  Phys. Rev. Lett., 91:073001, 2003.
  * [18] Peter J. Mohr.  Energy levels of hydrogen-like atoms predicted by quantum electrodynamics, 10$\leq$Z$\leq$40\.  Atomic Data and Nuclear Data Tables, 29(3):453–466, 1983.
  * [19] W.R. Johnson and Gerhard Soff.  The lamb shift in hydrogen-like atoms, 1$\leq$Z$\leq$110\.  Atomic Data and Nuclear Data Tables, 33(3):405–446, 1985.
  * [20] M. Lestinsky, V. Andrianov, B. Aurand, V. Bagnoud, D. Bernhardt, H. Beyer, S. Bishop, K. Blaum, A. Bleile, At. Borovik Jr., F. Bosch, C.J. Bostock, C. Brandau, A. Bräuning-Demian, I. Bray, T. Davinson, B. Ebinger, A. Echler, P. Egelhof, A. Ehresmann, M. Engström, C. Enss, N. Ferreira, D. Fischer, A. Fleischmann, E. Förster, S. Fritzsche, R. Geithner, S. Geyer, J. Glorius, K. Göbel, O. Gorda, J. Goullon, P. Grabitz, R. Grisenti, A. Gumberidze, S. Hagmann, M. Heil, A. Heinz, F. Herfurth, R. Hess, P.-M. Hillenbrand, R. Hubele, P. Indelicato, A. Källberg, O. Kester, O. Kiselev, A. Knie, C. Kozhuharov, S. Kraft-Bermuth, T. Kühl, G. Lane, Yu.A. Litvinov, D. Liesen, X.W. Ma, R. Märtin, R. Moshammer, A. Müller, S. Namba, P. Neumeyer, T. Nilsson, W. Nörtershäuser, G. Paulus, N. Petridis, M. Reed, R. Reifarth, P. Reiss, J. Rothhardt, R. Sanchez, M.S. Sanjari, S. Schippers, H.T. Schmidt, D. Schneider, P. Scholz, R. Schuch, M. Schulz, V. Shabaev, A. Simonsson, J. Sjöholm, Ö. Skeppstedt, K. Sonnabend, U. Spillmann, K. Stiebing, M. Steck, T. Stöhlker, A. Surzhykov, S. Torilov, E. Träbert, M. Trassinelli, S. Trotsenko, X.L. Tu, I. Uschmann, P.M. Walker, G. Weber, D.F.A. Winters, P.J. Woods, H.Y. Zhao, and Y.H. Zhang.  Physics book: CRYRING@ESR.  Eur. Phys. J. Spec. Top., 225:797–882, 2016.
  * [21] H. Danared, G. Andler, L. Bagge, C.J. Herrlander, J. Hilke, J. Jeansson, A. Källberg, A. Nilsson, A. Paál, K. Rensfelt, U. Rosengård, J. Starker, and M. af Ugglas.  Electron cooling with an ultracold electron beam.  Phys. Rev. Lett., 72:3775–3778, 1994.
  * [22] H. Danared, A. Källberg, G. Andler, L. Bagge, F. Österdahl, A. Paál, K. Rensfelt, A. Simonsson, O. Skeppstedt, and M. A. Ugglas.  Studies of electron cooling with a highly expanded electron beam.  Nucl. Inst. Meth. A, 441:123–133, 2000.
  * [23] M. Pajek and R. Schuch.  X-rays from recombination of bare ions with electrons in an electron cooler.  Nucl. Inst. Meth. B, 98:165–169, 1995.
  * [24] H. Beyer, D. Liesen, F. Bosch, K. Finlayson, M. Jung, O. Klepper, R. Moshammer, K. Beckert, H. Eickhoff, B. Franzke, F. Nolden, P. Spädtke, M. Steck, G. Menzel, and R. D. Desllates.  X rays from radiative electron capture of free cooling electrons for precise lamb-shift measurements at high Z: Au78+.  Phys. Lett. A, 184:435–439, 1994.
  * [25] D. Liesen, H. Beyer, K. Finlayson, F. Bosch, M. Jung, O. Klepper, R. Moshammer, K. Beckert, H. Eickhoff, B. Franzke, F. Nolden, P. Spädtke, M. Steck, G. Menzel, and R. Deslattes.  X-rays from radiative electron capture of highly-charged heavy ions in an electron cooler.  Z. Phys. D, 30:307–313, 1994.
  * [26] H. Beyer, G. Menzel, D. Liesen, A. Gallus, F. Bosch, R. Deslattes, P. Indelicato, T. Stöhlker, O. Klepper, R. Moshammer, F. Nolden, H. Eickhoff, B. Franzke, and M. Steck.  Measurement of the ground-state lambshift of hydrogenlike uranium at the electron cooler of the esr.  Z. Phys. D, 35:169–175, 1995.
  * [27] A. Gumberidze, Th. Stöhlker, D. Banaś, K. Beckert, P. Beller, H. F. Beyer, F. Bosch, X. Cai, S. Hagmann, C. Kozhuharov, D. Liesen, F. Nolden, X. Ma, P. H. Mokler, A. Oršić-Muthig, M. Steck, D. Sierpowski, S. Tashenov, A. Warczak, and Y. Zou.  Electron-electron interaction in strong electromagnetic fields: The Two-Electron Contribution to the ground-state energy in He-like Uranium.  Phys. Rev. Lett., 92:203004, 2004.
  * [28] A. Gumberidze, T. Stöhlker, D. Banaś, K. Beckert, P. Beller, H. Beyer, F. Bosch, S. Hagmann, C. Kozhuharov, D. Liesen, F. Nolden, X. Ma, P. Mokler, M. Steck, D. Sierpowski, and S. Tashenov.  Quantum electrodynamics in strong electric fields: the ground-state lamb shift in hydrogenlike uranium.  Phys. Rev. Lett., 94:223001, 2005.
  * [29] D. Banaś, M. Pajek, A. Surzhykov, T. Stöhlker, C. Brandau, A. Gumberidze, C. Kozhuharov, H. Beyer, S. Böhm, F. Bosch, M. Czarnota, S. Chatterjee, J.-Cl Dousse, S. Fritzsche, S. Hagmann, D. Liesen, P. Mokler, A. Müller, A. Kumar, R. Reuschl, D. Sierpowski, U. Spillmann, J. Szlachetko, S. Tashenov, S. Trotsenko, P. Verma, and A. Warczak.  Subshell-selective x-ray studies of radiative recombination of u92+ ions with electrons for very low relative energies.  Phys. Rev. A, 92:032710, 2015.
  * [30] R. Reuschl, A. Gumberidze, C. Kozhuharov, U. Spillmann, S. Tashenov, T. Stöhlker, and J. Eichler.  State-selective x-ray studies of radiative recombination into bare and h-like uranium at threshold energies.  Phys. Rev. A, 77:032701, 2008.
  * [31] H. Bethe and E. Salpeter.  Quantum Mechanics of One- and Two-Electron Systems.  Springer, Berlin, 1957.
  * [32] M. Hörndl, S. Yoshida, and J. Burgdörfer K. Tőkési.  Enhancement of low energy electron-ion recombination in a magnetic field: Influence of transient field effects.  Phys. Rev. Lett., 95:243201, 2005.
  * [33] H. F. Beyer, T. Gassner, M. Trassinelli, R. Hess, U. Spillmann, D. Banaś, K.-H. Blumenhagen, F. Bosch, C. Brandau, W. Chen, Chr. Dimopoulou, E. Förster, R. E. Grisenti, A. Gumberidze, S. Hagmann, P.-M. Hillenbrand, P. Indelicato, P. Jagodziński, T. Kämpfer, Chr. Kozhuharov, M. Lestinsky, D. Liesen, Y. A. Litvinov, R. Loetzsch, B. Manil, R. Märtin, F. Nolden, N. Petridis, M. S. Sanjari, K. S. Schulze, M. Schwemlein, A. Simionovici, M. Steck, Th. Stöhlker, C. I. Szabo, S. Trotsenko, I. Uschmann, G. Weber, O. Wehrhan, N. Winckler, D. F. A. Winters, N. Winters, and E. Ziegler.  Crystal optics for precision x-ray spectroscopy on highly charged ions-conception and proof.  J. Phys. B: At. Mol. Opt. Phys., 48:144010, 2015.
  * [34] T. Gassner, M. Trassinelli, R. Hess, U. Spillmann, D. Banaś, K.-H. Blumenhagen, F. Bosch, C. Brandau, W. Chen, Chr. Dimopoulou, E. Förster, R. Grisenti, A. Gumberidze, S. Hagmann, P.-M. Hillenbrand, P. Indelicato, P. Jagodziński, T. Kämpfer, Chr. Kozhuharov, M. Lestinsky, D. Liesen, Y. Litvinov, R. Loetzsch, B. Manil, R. Märtin, F. Nolden, M. Petridis, M. Sanjari, K. Schulze, M. Schwemlein, A. Simionovici, M. Steck, Th. Stöhlker, C. Szabo, S. Trotsenko, I. Uschmann, G. Weber, O. Wehrhan, N. Winckler, D. Winters, N. Winters, E. Ziegler, and H. F. Beyer.  Wavelength-dispersive spectroscopy in the hard x-ray regime of a heavy highly-charged ion: the 1s lamb shift in hydrogen-like gold.  New J. Phys., 20:073033, 2018.
  * [35] M. Trassinelli, A. Kumar, H. F. Beyer, P. Indelicato, R. Märtin, R. Reuschl, Y. S. Kozhedub, C. Brandau, H. Bräuning, S. Geyer, A. Gumberidze, S. Hess, P. Jagodziński, C. Kozhuharov, D. Liesen, U. Spillmann, S. Trotsenko, G. Weber, D. F. A. Winters, and Th. Stöhlker.  Observation of the $2p_{3/2}\rightarrow~{}2s_{1/2}$ intra-shell transition in He-like uranium.  Europhys. Lett., 87:63001, 2009.
  * [36] P. Jagodziński, M. Pajek, D. Banaś, H.F. Beyer, M. Trassinelli, and Th. Stöhlker.  Ray-tracing simulations of spherical Johann diffraction spectrometer for in-beam X-ray experiments.  Nucl. Inst. Meth. A, 753:121, 2014.
  * [37] P. Jagodziński, M. Pajek, D. Banaś, A. Kubala-Kukuś, J. Szlachetko, M. Cotte M, and M. Salomé.  Properties of polycapillary optics dedicated to low-energy parallel-beam wavelength-dispersive spectrometers for synchrotron-based x-ray fluorescence study.  Optics Express, 29(17):27193–27211, 2021.
  * [38] H. H. Johann.  Die Erzeugung lichtstarker Röntgenspektren mit Hilfe von Konkavkristallen.  Zeitschrift für Physik, 69:185–206, 1931.
  * [39] T. Johansson.  Über ein neuartiges, genau fokussierendes Röntgenspektrometer.  Zeitschrift für Physik, 82:507–528, 1933.
  * [40] L. von Hamos.  Röntgenspektroskopie und Abbildung mittels gekrümmter Kristallreflektoren. I. Geometrisch-optische Betrachtungen.  Annalen der Physik, 409:716–724, 1933.
  * [41] L. von Hamos.  Röntgenspektroskopie und Abbildung mittels gekrümmter Kristallreflektoren. II. Beschreibung eines fokussierenden Spektrographen mit punktgetreuer Spaltabbildung.  Annalen der Physik, 411:252–260, 1934.
  * [42] S. Takagi.  Dynamical theory of diffraction applicable to crystals with any kind of small distortion.  Acta Cryst., 15:1311–1312, 1962.
  * [43] R. Schuch, D.R. DeWitt, H. Gao, S. Mannervik, W. Zong, and N.R. Badnell.  Recombination studies of Highly Charged Ions in the cooler ring CRYRING.  Phys. Scr., T73:114–118, 1997.
  * [44] AdvaPIX.  http://www.advacam.com/camera/advapix.
  * [45] P. Jagodziński, J. Szlachetko, J.-Cl Dousse, J. Hoszowska, M. Szlachetko, U. Vogelsang, D. Banaś, T. Pakendorf, A. Meents, J. A. van Bokhoven, A. Kubala-Kukuś, M. Pajek, and M. Nachtegaal.  A DuMond-type crystal spectrometer for synchrotron-based X-ray emission studies in the energy range of 15-26 keV.  Rev. Sci. Instrum., 90:063106, 2019.
  * [46] Qt documentation.  https://www.qt.io/product/framework.
  * [47] M. S. del Río and R. J. Dejus.  Xop v2.4: Recent developments of the x-ray optics software toolkit.  Proc. of SPIE, 8141:814115, 2011.
  * [48] M. Sánchez del Río, C. Ferrero, and V. Mocella.  Computer simulation of bent perfect crystal diffraction profiles.  Proc. SPIE, 3151:312–323, 1997.
  * [49] M. Sánchez del Río.  Ray tracing simulations for crystal optics.  Proc. SPIE, 3448:230–245, 1998.
  * [50] M. Matsumoto and T. Nishimura.  Mersenne Twister: A 623-dimensionally equidistributed uniform pseudo-random number generator.  ACM Transactions on Modeling and Computer Simulation, 8:3–30, 1998\.
  * [51] J. Eichler and W.E. Meyerhof.  Relativistic Atomic Collision.  Academic Press, San Diego, 1995.
  * [52] W. H. Bragg and W. L. Bragg.  The reflection of X-rays by crystals.  Proc. R. Soc. London Ser. A, 88:428–438, 1913.
  * [53] B. E. Warren.  X-ray diffraction.  Dover Publications Inc., New York, 1969.
  * [54] W. H. Zachariasen.  Theory of X-ray diffraction in crystals.  Dover Publications, New York, 1967.
  * [55] A. H. Compton and S. K. Allison.  X-rays in Theory and Experimentn.  Van Nostrand, New York, 1935.
  * [56] OriginLab - Asym2Sig function.  https://www.originlab.com/doc/Origin-Help/Asym2Sig-FitFunc.
  * [57] OriginLab Software for Graphing & Analysis.  https://www.originlab.com/.
  * [58] M. J. Pushie, I.J. Pickering, M. Korbas, M.J. Hackett, and G.N. George.  Elemental and chemically specific x-ray fluorescence imaging of biological systems.  Chem. Rev., 114(17):8499–8541, 2014.
  * [59] M. Krisch, A. Freund, G. Marot, and L. Zhang.  Study of dynamically bent crystals for x-ray focusing optics.  Nucl. Inst. Meth. A, 305:208, 1991.
  * [60] J. L. Campbell and T. Papp.  Widths of the atomic K-N7 levels.  At. Data Nucl. Data Tables, 77:1, 2001.