# Bragg-Primakoff Axion Photoconversion in Crystal Detectors

James B. Dent  Department of Physics, Sam Houston State University,
Huntsville, TX 77341, USA  Bhaskar Dutta  Mitchell Institute for Fundamental
Physics and Astronomy, Department of Physics and Astronomy, Texas A&M
University, College Station, TX 77845, USA  Adrian Thompson  Mitchell
Institute for Fundamental Physics and Astronomy, Department of Physics and
Astronomy, Texas A&M University, College Station, TX 77845, USA

###### Abstract

Axions and axion-like pseudoscalar particles with dimension-5 couplings to
photons exhibit coherent Primakoff scattering with ordered crystals at keV
energy scales, making for a natural detection technique in searches for solar
axions. We find that there are large suppressive corrections, potentially
greater than a factor of $\mathcal{O}(10^{3})$, to the coherent enhancement
when taking into account absorption of the final state photon. This effect has
already been accounted for in light-shining-through-wall experiments through
the language of Darwin classical diffraction, but is missing from the
literature in the context of solar axion searches that use a matrix element
approach. We extend the treatment of the event rate with a heuristic
description of absorption effects to bridge the gap between these two
languages. Furthermore, we explore the Borrmann effect of anomalous absorption
in lifting some of the event rate suppression by increasing the coherence
length of the conversion. We study this phenomenon in Ge, NaI, and CsI crystal
experiments and its impact on the the projected sensitivities of SuperCDMS,
LEGEND, and SABRE to the solar axion parameter space. Lastly, we comment on
the reach of multi-tonne scale crystal detectors and strategies to maximize
the discovery potential of experimental efforts in this vein.

††preprint: MI-HET-804

##  I Introduction

Axions and axion-like particles (ALPs) - potentially long-lived pseudoscalars
with weak couplings to the Standard Model (SM) that may have masses from the
sub-eV to the GeV - are central features in the landscape of solutions to the
strong CP problem Peccei and Quinn (1977); Wilczek (1978); Weinberg (1978);
Preskill  _et al._ (1983); Abbott and Sikivie (1983); Dine and Fischler
(1983); Battaglieri  _et al._ (2017), dark matter problem Marsh (2016); Arias
_et al._ (2012); Duffy and van Bibber (2009); Adams  _et al._ (2022), and in
the spontaneous breaking of generic global symmetries Chikashige  _et al._
(1981); Svrcek and Witten (2006); Goodsell and Ringwald (2010). In addition to
being dark matter candidates, axion-like particles in the keV to sub-eV mass
range produced in the sun are well motivated Sikivie (1983); Raffelt (1996).
Searches were carried out by several experimental collaborations by looking
for $a\to\gamma$ Primakoff conversion in solid crystal detectors, including
DAMA Bernabei  _et al._ (2004) (NaI), CUORE Li  _et al._ (2016, 2015) (TeO2),
Edelweiss-II Armengaud  _et al._ (2013), SOLAX Avignone  _et al._ (1998),
COSME Morales  _et al._ (2002), CDMS Ahmed  _et al._ (2009), and Majorana
Arnquist  _et al._ (2022) (Ge). Other upcoming experiments like SuperCDMS
Albakry  _et al._ (2022), LEGEND Abgrall  _et al._ (2021), and SABRE Antonello
_et al._ (2019) are projected to greatly expand coverage over the axion
parameter space and test QCD axion solutions to the strong CP problem in the
eV mass range. These experiments aim to take advantage of coherence in the
conversion rate when axions satisfy the Bragg condition, enhancing the
detection sensitivity by orders of magnitude relative to incoherent
scattering.

Searching for solar axions via their coherent conversion in perfect crystals
was first treated by Buchmüller & Hoogeveen Buchmüller and Hoogeveen (1990)
using the Darwin theory of classical X-ray diffraction under the Bragg
condition Warren (1969). The authors also alluded to potential enhancements in
the signal yield when one considers the symmetrical Laue-case of diffraction
for the incoming ALP waves. Yamaji et al. Yamaji  _et al._ (2017) treated this
case thoroughly for the 220 plane of cubic crystals, also using the classical
theory, and included the effect of anomalous absorption, also known as the
Borrmann effect. It was shown by these authors that an enhancement to the
signal yield was possible, replacing the Bragg penetration depth
($L_{\textrm{bragg}}\sim 1$ $\mu$m) with the Borrmann-enhanced attenuation
length (ranging from 10 $\mu$m all the way to centimeter scales).

The effect of anomalous absorption of X-rays was first shown by Borrmann
Borrmann (1954), and theoretically explained by Zachariasen Zachariasen (1945,
1952) and other later authors (Battermann Batterman (1961, 1962), Hirsch
Hirsch (1952)). A quantum mechanical treatment was offered by Biagini Biagini
(1990, 1991) in which the Borrmann effect was explained by the interference of
statistical ensembles of the so-called $\ket{\alpha}$ and $\ket{\beta}$ Bloch
waves. There have been numerous modern studies that utilize the Borrmann
effect, notably as in photon-photon dissipation on Bragg-spaced arrays of
superconduncting qubits Poshakinskiy and Poddubny (2021), and in measuring
quadrupole transitions in X-ray absorption spectra Pettifer  _et al._ (2008).

Now, the calculation of the event rates expected for the Primakoff conversion
of solar axions coherently with a perfect crystal was treated in a more
traditional, particle physics-based approach in refs. Cebrian  _et al._
(1999); Bernabei  _et al._ (2001); Li  _et al._ (2015) and it was applied to
derive many of the constraints set by crystal-based solar axion experiments
including DAMA, CUORE, Edelweiss-II, SOLAX, COSME, CDMS, and Majorana
Demonstrator Bernabei  _et al._ (2004); Li  _et al._ (2016); Armengaud  _et
al._ (2013); Avignone  _et al._ (1998); Morales  _et al._ (2002); Ahmed  _et
al._ (2009); Arnquist  _et al._ (2022). However, absorption effects in Bragg
and Laue case diffraction were not considered in refs. Cebrian  _et al._
(1999); Bernabei  _et al._ (2001); Li  _et al._ (2015); indeed, when comparing
the event rates between these references and those presented in light-shining-
through-wall (LSW) experiments, which used the classical Darwin theory
approach (e.g. ref. Buchmüller and Hoogeveen (1990) and more recently ref.
Yamaji  _et al._ (2017)), there is a clear inconsistency. While the event
rates in the LSW literature only consider the coherent volume of the crystal
up to the relevant attenuation length ($\lambda\sim 1$ $\mu$m in the Bragg
diffraction case or $\lambda\lesssim 100$ $\mu$m in the Laue-case), the solar
axion searches have considered the whole volume of the crystal to exhibit
coherence. In this work, we show that such effects reduce the expected event
rates potentially up to the $\mathcal{O}(10^{3})$ level depending on the
assumed crystal size (and therefore, the assumed coherent volume enhancement)
and material. Although this may impact the existing sensitivities set by solar
axion searches in solid crystals, measures can be taken to optimize
suppression of the event rate due to absorption effects and recover some or
potentially all of the coherent volume.

In § II we re-derive the event rate formula for solar axion Primakoff
scattering under the Bragg condition, and in § III we discuss the anomalous
enhancement to the absorption length under the Borrmann effect and numerically
estimate the level of suppression in the coherent sum. In § IV we write down
the event rates for a perfect crystal exposed to the solar axion flux with and
without the absorption effects and discuss the relevant phenomenology. In § V
we project the impact on sensitivities with and without absorption effects for
SuperCDMS, LEGEND-200, LEGEND-1000, SABRE, and multi-tonne benchmark detector
setups and discuss possibilities to restore sensitivity from coherence in §
VI. Finally, in § VII we conclude and discuss further work.

##  II Coherence and Absorption

In order to show how photon absorption in coherent Bragg-Primakoff scattering
affects the event rate, it is worth going through a pedagogical review of what
we mean by coherent scattering and first assume that no absorption takes
place. For the reader who is familiar with coherence in neutrino scattering,
please refer to the approach illustrated by Bednyakov and Naumov Bednyakov and
Naumov (2021) in which coherent neutrino-nucleus scattering is calculated by
taking a sum over $N$ scattering centers in a nucleus.

Let $f(\vec{k},\vec{k}^{\prime})$ be the Primakoff scattering matrix element
for a single atomic target, for an incoming ALP 3-momentum $\vec{k}$ and
outgoing $\gamma$ 3-momentum $\vec{k}^{\prime}$. Written in terms of the
atomic form factor $F_{A}$,

$f=\mathcal{M}_{\textrm{free}}F_{A}(q)$ (1)

where $\mathcal{M}_{\textrm{free}}$ is the single-atomic scattering amplitude,
$q$ is the momentum transfer, with the angle of scattering defined by
$\vec{k}\cdot\vec{k}^{\prime}=E_{\gamma}k\cos 2\theta$, averaged over spins
and taken in the limit $k\gg m_{a}$, $m_{N}\gg k,E_{\gamma}$ Tsai (1986),

$|\braket{\mathcal{M}_{\textrm{free}}}|^{2}=\dfrac{8e^{2}g_{a\gamma}^{2}}{q^{4}}E_{\gamma}^{2}m_{N}^{2}k^{2}\sin^{2}2\theta$
(2)

for a nuclear mass $m_{N}$. The real atomic scattering form factor can be
taken from ref. Tsai (1974) which is defined such that $F_{A}(0)=Z$;

$F_{A}(q)=\dfrac{Zr_{0}^{2}q^{2}}{1+r_{0}^{2}q^{2}}$ (3)

for atomic number $Z$ and screening constant parameterization
$r_{0}=184.15e^{-1/2}Z^{-1/3}/m_{e}$, where $m_{e}$ is the electron mass.

Similarly, we sum over the $N$ scattering centers in a crystal;

$\mathcal{M}(\vec{k},\vec{k}^{\prime})=\sum_{j=1}^{N}f_{j}(\vec{k},\vec{k}^{\prime})e^{i(\vec{k}^{\prime}-\vec{k})\cdot\vec{r}_{j}}$
(4)

where $e^{i(\vec{k}^{\prime}-\vec{k})\cdot\vec{r}_{j}}$ is a phase factor that
comes from assuming plane wave solutions for the in and out states. This
assumption is key; for atomic scattering in vacuum, the eigenstates of the
final state photon should be a spectrum of plane waves.

If we square the total matrix element, we get

$\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}=\sum_{i=1}^{N}\mid
f_{i}\mid^{2}+\sum_{j\neq
i}^{N}\sum_{i=1}^{N}f_{j}^{\dagger}f_{i}e^{-i\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})}$
(5)

taking $\vec{q}\equiv\vec{k}-\vec{k}^{\prime}$. The first (diagonal) term is
the incoherent piece, while the second term is usually suppressed by the
average destructive interference of the phase factors. Using the Laue
diffraction condition Warren (1969),
$\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})=2\pi n$ for $n\in\mathbb{Z}$, then the
phase factor in the exponential goes to one and the scattering is coherent. In
this limit, the diagonal term is subdominant and the final matrix element
squared tends to $\mathcal{M}^{2}\to N^{2}f^{2}$ and we have full coherence.
See appendix B for a derivation of the event rate in full with this approach.

Now consider interactions of the final state $\gamma$ with the crystal
lattice, including the absorption and scattering effects. Pragmatically, we
modify the plane wave solutions of the final state photon to that of one in a
dielectric medium,

$\vec{k}^{\prime}\to\bar{n}\vec{k}^{\prime},\,\,\,\bar{n}=n-i\kappa,$ (6)

where $\bar{n}$ is the complex index of refraction with real part $n$ and
imaginary part $\kappa$. Making this modification, we have

$\displaystyle e^{i\bar{n}\vec{k}^{\prime}\cdot(\vec{r}_{i}-\vec{r}_{j})}$
$\displaystyle\to
e^{in\vec{k}^{\prime}\cdot(\vec{r}_{i}-\vec{r}_{j})}e^{-\frac{\mu}{2}|\hat{k}^{\prime}\cdot(\vec{r}_{j}-\vec{r}_{i})|},$
(7)

The absorption coefficient $\mu$ (which can also be expressed in terms of
attenuation length or mean free path $\lambda=1\mu$) is related to the
imaginary part of the index of refraction through $\mu\equiv
2\kappa|\vec{k}|$. Conceptually, this factor encodes the effect of a reduced
coherent interference amplitude between any two scattering centers, since a
photon plane wave sourced at one scattering center will have been attenuated
after reaching another scattering center.

We note that Eq. 6 and Eq. 7 are heuristic modifications, since the attenuated
plane wave solution is not a true eigenstate of the interaction Hamiltonian,
but rather a simple ansatz made to estimate the phenomenology of absorption.
For further convenience, we use
$z_{ij}\equiv\mid\hat{k^{\prime}}\cdot(\vec{r}_{i}-\vec{r}_{j})\mid$ and
$\lambda=1/\mu$. We then have

$\displaystyle\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}$
$\displaystyle=\sum_{i=1}^{N}\mid f_{i}\mid^{2}$ $\displaystyle+$
$\displaystyle\sum_{j\neq
i}^{N}\sum_{i=1}^{N}f_{j}^{\dagger}f_{j}e^{-i\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})}e^{-z_{ij}/(2\lambda)}$
(8)

After using the Laue diffraction condition
$\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})=2\pi n$ and several manipulations of
the sum, we find that

$\displaystyle\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}$
$\displaystyle\gtrsim f^{\dagger}f\sum_{j\neq i}^{N}\dfrac{\lambda
L_{x}L_{y}N}{V}$ $\displaystyle\gtrsim
f^{\dagger}fN^{2}\dfrac{\lambda}{L_{z}}$ (9)

Comparing the proportionailty in Eq. II to the usual result $\propto N^{2}$,
we see that the coherent volume is $V\times\lambda/L_{z}$, and the total
scattering rate is suppressed by a factor $\lambda/L_{z}$, and is now more
consistent with Darwin theory calculations Buchmüller and Hoogeveen (1990);
Yamaji  _et al._ (2017).

This inequality above is strictly a lower limit because, as we will show in §
III, the suppression to the coherent sum by the absorptive sum, which we label
as $I$,

$I\equiv\sum_{j\neq i}^{N}\sum_{i=1}^{N}e^{-z_{ij}/(2\lambda)},$ (10)

may be mitigated under certain conditions. Therefore, the suppression factor
$\lambda/L_{z}$ serves as a pessimistic guiding estimate, but in principle we
should compute the sum in Eq. 10 explicitly.

Figure 1: Crystallographic planes cut by the $hkl=220$, 224, and 333
reciprocal lattice vectors in the FCC lattice, and their basis vectors.

##  III Anomalous Absorption and the Borrmann Effect

The suppression to the event rate can be alleviated by considering the
anomalous enhancement to the absorption depth or mean free path $\lambda$,
which, in crystallographic diffraction, is not strictly proportional to the
the inverse photon cross section multiplying into the material number
desnsity, $1/(n\sigma)$.

Take for instance ref. Yamaji  _et al._ (2017) in which the authors have found
that for the Laue-case conversion of ALPs, the attenuation length is modified
as

$L_{\text{att}}\to L_{\alpha/\beta}\equiv
2L_{\text{att},\alpha/\beta}\bigg{(}1-\exp\bigg{(}-\dfrac{L}{2L_{\text{att},\alpha/\beta}}\bigg{)}\bigg{)}$
(11)

where $L_{\text{att},\alpha/\beta}=\dfrac{L_{\text{att}}}{1\mp\epsilon}$ and
$\epsilon$ is a ratio involving the imaginary parts of the scattering form
factor. These modifications come from the anomalous dispersion or anomalous
absorption effect, or the Borrmann effect. It is an effect that occurs for so-
called “Bloch waves” $\alpha$ and $\beta$ that form in the crystal, discussed
further in refs. Biagini (1990, 1991).

The total scattering form factor can be decomposed into the real and imaginary
parts Chantler (2000);

$f=f^{0}+\Delta f^{\prime}+i\Delta f^{\prime\prime}$ (12)

where $f^{0}$ is the atomic form factor, usually given as the Fourier
transform of the charge density;

$f^{0}(q)\equiv\int d^{3}\vec{x}\rho(\vec{x})e^{i\vec{q}\cdot\vec{x}}$ (13)

The second term in the real part of the form factor is the anomalous form
factor $\Delta f^{\prime}$, and $\Delta f^{\prime\prime}$ is the imaginary
part of the form factor associated with absorption. From Batterman Batterman
(1961, 1962), the anomalous absorption due to the Borrmann effect modifies the
absorption coefficient $\mu_{0}=1/\lambda$ as

$1/\lambda=\mu_{\text{eff}}=\mu_{0}\bigg{[}1-\dfrac{F^{\prime\prime}(hkl)}{F^{\prime\prime}(000)}\bigg{]}$
(14)

Here $F^{\prime\prime}(hkl)$ is the combination of structure function and
imaginary form factor, $F^{\prime\prime}(hkl)=S(hkl)\Delta f^{\prime\prime}$.
The ratio in the second term of the expression is the Borrmann parameter,
usually denoted as $\epsilon$111In ref. Yamaji  _et al._ (2017), they use
$\kappa$.. More explicitly, studies by Wagenfield have related the Borrmann
parameter to the quadrupole photoelectric cross section Wagenfeld (1987,
1966); Persson and Efimov (1970);

$\epsilon\equiv
D\bigg{(}1-2\sin^{2}\theta_{B}\frac{\sigma^{Q}}{\sigma_{\textrm{PE}}}\bigg{)}\frac{|S(h,k,l)|}{|S(0,0,0)|}$
(15)

where $D$ is the Debye-Waller factor accounting for thermal vibrations in
anomalous absorption, $D=e^{-Bs^{2}}$ where $s=\sin\theta/\lambda$ and $B$ is
a temperature-dependent constant. The Debye-Waller factors for cryogenic
temperatures can be found in ref. Peng  _et al._ (1996) as well as fits to
$\Delta f^{\prime\prime}$ for several pure materials of interest.
Equivalently, we can express the Borrmann factor in terms of the imaginary
form factor $\Delta f^{\prime\prime}$ and the quadrupole form factor $\Delta
f^{\prime\prime}_{Q}$ (which obeys the selection rules $\ell=\ell^{\prime}\pm
2$);

$\epsilon\equiv D\bigg{(}1-2\sin^{2}\theta_{B}\frac{\Delta
f^{\prime\prime}_{Q}}{\Delta
f^{\prime\prime}}\bigg{)}\frac{|S(h,k,l)|}{|S(0,0,0)|}$ (16)

and $\Delta f^{\prime\prime}$ is more explicitly written as Wagenfeld (1987)

$\Delta
f^{\prime\prime}=\sum_{\ell^{\prime},m^{\prime}}\sum_{n,\ell,m}\dfrac{\pi\hbar^{2}}{m_{e}}\bigg{|}\int\psi_{f}^{*}(\boldsymbol{r})\boldsymbol{\varepsilon}_{0}\cdot\boldsymbol{\nabla}e^{i\boldsymbol{k}\cdot\boldsymbol{r}}\psi_{i}(\boldsymbol{r})d^{3}r\bigg{|}^{2}$
(17)

While fits to this form factor can be found in ref. Peng  _et al._ (1996), we
can also usefully relate it to the vectorial form factor defined in ref.
Catena  _et al._ (2020) and calculated using the DarkARC (Python) or DarkART
(C++) codes;

$\Delta f^{\prime\prime}(k)=\pi\hbar^{2}m_{e}|\boldsymbol{f}_{1\to
2}(k)|_{\texttt{DarkARC}}^{2}$ (18)

For more discussion and example functional forms of the Borrmann parameter,
see appendix D.

While a dedicated study of the Borrmann parameter would require the
calculation of the photoelectric quadrupole cross section $\sigma^{Q}$,
Borrmann parameters for germanium crystal are already reported in the
literature. We use the form factors derived in ref. Batterman (1961) to
estimate the Borrmann effect for each reciprocal lattice plane, giving us an
anomalous attenuation length along the direction of travel of photons inside
the detector $I(\vec{k},\vec{G})$. We tabulate these and the corresponding
values of $\epsilon$ in Table 1 and plot the Borrmann parameters for Ge, Si,
CsI, and NaI crystals in Fig. 9.

$(hkl)$ | Ge | Si
---|---|---
$\epsilon$ |  $\lambda$ [$\mu$m] |  $\epsilon$ |  $\lambda$ [$\mu$m]
111 |  0.69 |  34.2 |  0.68 |  96.26
220 |  0.95 |  229.69 |  0.91 |  357.44
131 |  0.66 |  31.08 |  0.63 |  82.62
400 |  0.91 |  122.95 |  0.85 |  209.55
133 |  0.64 |  28.77 |  0.59 |  74.85
422 |  0.88 |  86.13 |  0.81 |  158.64
440 |  0.85 |  67.99 |  0.77 |  131.92
$(hkl)$ | NaI | CsI
$\epsilon$ |  $\lambda$ [$\mu$m] |  $\epsilon$ |  $\lambda$ [$\mu$m]
200 |  0.92 |  181 |  0.53 |  27.03
220 |  0.86 |  104.3 |  0.27 |  17.47
222 |  0.82 |  76.84 |  0.13 |  14.55
400 |  0.77 |  62 |  0.04 |  13.28
420 |  0.73 |  52.48 |  0 |  12.66
440 |  0.62 |  37.1 |  -0.05 |  12.13
600 |  0.58 |  34.12 |  -0.05 |  12.11
Table 1: Anomalous coefficients $\epsilon(hkl)$ and attenuation lengths
$\lambda(E_{\gamma},hkl)$ in Ge, Si (diamond cubic), NaI, and CsI (FCC) due to
the Borrmann effect were computed using the imaginary form factor with
$E_{\gamma}=3$ keV. At this energy the normal attenuation length is $\sim 10$
$\mu$m. See also Fig. 9.

The absorptive part of the coherent sum that remains after the Laue condition
is met is

$I(\vec{k},\vec{G})\equiv\sum_{j\neq
i}^{N}\sum_{i=1}^{N}e^{-\frac{(\vec{k}-\vec{G})}{|\vec{k}-\vec{G}|}\cdot(\vec{r}_{i}-\vec{r}_{j})/(2\lambda)}$
(19)

which, when the Bragg condition is met, is strictly a function of $\vec{k}$
and $\vec{G}$ since the mean free path $\lambda$ can be related via Eq. 14.
Taking the Ge lattice as an example, with lattice constant $d=5.657$ Å, we
evaluate $I(\vec{k},\vec{G})$ numerically by constructing a lattice of $N$ Ge
atoms. Since computing the full sum for a real crystal of centimeter length
scale would require a huge number of evaluations $(\propto N^{2})$, we take a
sparse sampling of $N$ atoms across the physical crystal volume such that the
sum is computationally feasible. The sum can then be evaluated in increments
of increasing $N$ to test for convergence. We find that a lattice of around
$N\simeq 10^{4}$ atoms in a cubic geometry is enough to obtain a convergent
error of around 5%. Some evaluations of $I(\vec{k},\vec{G})$ as a function of
varying mean free path $\lambda$ are shown in Fig. 2 for several choices of
scattering planes $\vec{G}$ and incoming wavevectors $\vec{k}$.

Figure 2: The absorption factor $I(\vec{k},\vec{G})$ as a function of the mean
free path $\lambda=1/\mu$ for a crystal of cubic volume with side length 5 cm.

One interesting phenomenon that can be seen in Fig. 2 is that there are
certain choices of $\vec{k}^{\prime}=\vec{k}-\vec{G}$ such that
$\vec{k}^{\prime}\cdot(\vec{r}_{i}-\vec{r}_{j})=0$. In this special
circumstance, while many of the terms in the coherent sum will tend to zero
with decreasing $\lambda$, the terms where this dot product is zero will
survive. What this means physically is that the plane in which
$\vec{r}_{i}-\vec{r}_{j}$ lies will avoid the decoherence from absorption as
long as it remains orthogonal to $\vec{k}^{\prime}$. This relation can be made
more apparent by considering the dot product under the Bragg condition;

$\displaystyle\hat{k}^{\prime}\cdot(\vec{r}_{i}-\vec{r}_{j})$
$\displaystyle=\bigg{(}\frac{\vec{G}}{2\vec{k}\cdot\hat{G}}-\frac{\vec{G}}{k}\bigg{)}\cdot(\vec{r}_{i}-\vec{r}_{j})=0$
(20)

where we take $\hat{k}=(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)$,
solving this equation for $\theta$ in the $hkl=400$ case gives

$\theta=\cot^{-1}\left(\frac{n_{x}\cos(\phi)-n_{y}\sin(\phi)}{n_{z}}\right)+\pi
c_{1}$ (21)

for $n_{x},n_{y},n_{z},c_{1}\in\mathbb{Z}$. This defines a family of lattice
points that remain in the absorption sum $I$ even in the limit $\lambda\to 0$,
resulting a lower bound on $I$ as shown for some example choices of $\hat{k}$
in Fig. 2. This effect is similar in nature to the Laue-case diffraction
enhancements where the photoconversion occurs down the scattering planes,
minimizing the absorption, as studied in ref. Yamaji  _et al._ (2017).

In Fig. 3 the absorption factor $I$ is shown for the plane $\vec{G}(1,1,1)$ as
a function of azimuthal and polar angles of the incoming axion momentum
$\theta,\phi$ under the Bragg condition. This fixes $k=E_{\gamma}$ for a given
$(\theta,\phi)$, and therefore the attenuation length $\lambda$ given by Eq.
14. We see a two prominent features of mitigated absorption in the $S$-shaped
band (tracing out a great circle on the 2-sphere), where (i) $I\to 1$ as these
$(\theta,\phi)$ combinations correspond to larger energies where the photon
absorption cross section falls off as we move further into the $S$, and (ii)
there is a jump discontinuity in the $S$-band due to an absorption edge in the
photoelectric cross section for germanium at around 11 keV.

Figure 3: The absorption factor $I(\vec{k},\vec{G})$ for the $hkl=111$ plane
as a function of the incoming ALP direction $\hat{k}(\theta,\phi)$ when the
Bragg condition is satisfied and the mean free path is given by the Borrmann
anomalous absorption coefficient. Here we take a crystal of cubic volume with
side length 5 cm. The solid white lines trace sample paths of the daily solar
angle in January (solid), March (dashed), June (dash-dotted), and September
(dotted) at the latitude and longitude of the Gran Sasso site.

##  IV Event Rates

Figure 4: Solar ALP scattering rates in a 250 cm3 Ge crystal, comparing the
rates with full volume coherence to ours with anomalous absorption effects
included through the absorption factor $I(\vec{k},\vec{G})$. Here we fix the
coupling $g_{a\gamma}=10^{-8}$ GeV-1, energy resolution to be $\Delta=1.0$ keV
(for $E_{ee}<6$ keV) and $\Delta=1.5$ keV (for $E_{ee}>6$ keV).

The event rate for Primakoff coherent scattering with a perfect crystal worked
out in Cebrian  _et al._ (1999); Bernabei  _et al._ (2001); Li  _et al._
(2015) where full-volume coherence was assumed and there is no dependence on
the attenuation length222Notice the factor of $(\hbar c)^{3}$ rather than
$\hbar c$ as written in ref. Cebrian  _et al._ (1999) for dimensional
consistency.; the event rate in an energy window $[E_{1},E_{2}]$ is

$\dfrac{dN}{dt}=\pi g_{a\gamma}^{2}(\hbar
c)^{3}\dfrac{V}{v_{\textrm{cell}}^{2}}\sum_{\vec{G}}\bigg{[}\dfrac{d\Phi_{a}}{dE_{a}}\frac{|F_{j}(\vec{G})S_{j}(\vec{G})|^{2}}{|\vec{G}|^{2}}\sin^{2}(2\theta)\mathcal{W}\bigg{]}$
(22)

where $S_{j}$ is the crystal structure factor (see appendix), $F_{j}$ is the
atomic form factor for species $j$, and $d\Phi_{a}/dE_{a}$ is the solar axion
flux from Primakoff scattering and photon coalescence in the sun Redondo
(2013); Dicus  _et al._ (1978). For the solar axion flux, we take the
parameterized form appearing in ref. Di Lella  _et al._ (2000) which expands
upon the form originally given by CAST Andriamonje  _et al._ (2007) by
accounting for the axion mass; see Eq. 49. The event rate in Eq. 22 encodes
the effect of detector energy resolution $\Delta$ within the function
$\mathcal{W}$;

$\mathcal{W}(E_{a},E_{1},E_{2},\Delta)=\frac{1}{2}\bigg{(}\textrm{erf}\bigg{(}\frac{E_{a}-E_{1}}{\sqrt{2}\Delta}\bigg{)}-\textrm{erf}\bigg{(}\frac{E_{a}-E_{2}}{\sqrt{2}\Delta}\bigg{)}\bigg{)}$
(23)

The sum over the reciprocal lattice vectors $\vec{G}$ effectively counts the
contributions to the coherent scattering from each set of lattice planes,
illustrated in Fig. 1. The reader may refer to appendix A for a compact
description of the reciprocal lattice.

At this stage the effect of absorption will simply modify the event rate, as
seen in the previous section, by replacing the full coherent volume $V\to
V\times I(\vec{k},\vec{G})$ with
$\lambda=[\mu_{0}(1-\epsilon(\vec{G}))]^{-1}$, giving

$\displaystyle\dfrac{dN}{dt}=\pi g_{a\gamma}^{2}(\hbar
c)^{3}\dfrac{V}{v_{\textrm{cell}}^{2}}$
$\displaystyle\sum_{\vec{G}}\bigg{[}\dfrac{d\Phi_{a}}{dE_{a}}\cdot\frac{I(\vec{k},\vec{G})}{|\vec{G}|^{2}}$
$\displaystyle\times|F_{j}(\vec{G})S_{j}(\vec{G})|^{2}\sin^{2}(2\theta)\mathcal{W}\bigg{]}$
(24)

With $\sin^{2}(2\theta)$ simplifying to
$4(\hat{G}\cdot\hat{k})^{2}(1-(\hat{G}\cdot\hat{k})^{2})$ Bernabei  _et al._
(2001) where $\hat{k}$ is the unit vector pointing toward the Sun’s location,
we have

$\displaystyle\dfrac{dN}{dt}=\pi g_{a\gamma}^{2}(\hbar
c)^{3}\dfrac{V}{v_{\textrm{cell}}^{2}}$
$\displaystyle\sum_{\vec{G}}I(\vec{k},\vec{G})\bigg{[}\dfrac{d\Phi_{a}}{dE_{a}}|F_{j}(\vec{G})S_{j}(\vec{G})|^{2}$
$\displaystyle\times\frac{4(\hat{G}\cdot\hat{k})^{2}(1-(\hat{G}\cdot\hat{k})^{2})}{|\vec{G}|^{2}}\mathcal{W}\bigg{]}$
(25)

At this stage, we have also used the Bragg condition $E_{a}=\hbar
c|\vec{G}|^{2}/(2\hat{k}\cdot\vec{G})$. The time dependence is encoded in the
solar position, which we can express through
$\hat{k}=(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)$ for
$\theta=\theta(t)$ and $\phi=\phi(t)$. For the solar angle as a function of
time and geolocation, we use the NREL solar position algorithm Reda and
Andreas (2008).

In principle, the sum over reciprocal lattice vectors $\vec{G}$ is taken to
arbitrarily large combinations $(h,k,l)$, but due to the $1/|\vec{G}|^{2}$
suppression and the upper limit of the solar axion flux of around $\sim 20$
keV, we can safely truncate the sum at $max\\{h,k,l\\}=5$.

The corresponding event rates for various energy windows are shown in Fig. 4
for Ge crystal, where we compare the relative enhancements with and without
the Borrmann effect to the case of full-volume coherence and to the case of
incoherent scattering on an amorphous lattice333Atomic Primakoff scattering is
still coherent here; we only turn off the coherence at the level of the
lattice for the sake of comparison with scattering on amorphous materials, in
this case, amorphous germanium.. The fluctuating features in the event rate
are the result of the sum over $\vec{G}$ which contributes to the Bragg peaks.
Here we have assumed a volume of 260 cm3 (corresponding roughly to the
volumetric size of a SuperCDMS germanium module), and so the relative
suppression for each $\vec{G}$ lattice plane goes like
$V^{1/3}/\lambda(\vec{k},\vec{G})$, giving a suppression on the order of
$10^{2}$ compared to the full-volume coherence assumption.

The time-dependence can be visualized further by viewing the event rates as a
function of incident angles integrated across the whole solar axion energy
window, as shown in Fig. 5. Depending on the time of year, different sets of
Bragg peaks will be traced over during the day, inducing an annual modulation
in addition to the intra-day modulation of the signal.

Figure 5: Event rates in germanium as a function of incident angles
$\theta,\phi$ for the integrated energy range $(1,10)$ keV and for reciprocal
lattice planes $(h,k,l)$ up to $max\\{h,k,l\\}=5$. The solid white lines trace
sample paths of the daily solar angle in January (solid), March (dashed), June
(dash-dotted), and September (dotted) at the latitude and longitude of the
Gran Sasso site.

Since the time of day fixes the solar zenith and azimuth $(\theta,\phi)$, we
can finally show the spectrum of the Primakoff signal as a function of energy
deposition and time of day; see Fig. 6.

Figure 6: Differential energy-time event rate with energy resolution
$\Delta=2.5$ keV. The time of year was taken to be January at the latitude and
longitude of the Gran Sasso site.

##  V Projected Sensitivities for Solar Axion Searches

Experiment |  Module Mass $\times$no. Modules |  Total Mass |  Energy Resolution | Threshold | Exposure (ton-years)
---|---|---|---|---|---
SuperCDMS (Ge) | 1.4 kg $\times$ 18 | 25.2 kg | 2.5 keV | 1 keV | 0.1
SuperCDMS (Si) | 0.6 kg $\times$ 6 | 3.6 kg | 2.5 keV | 1 keV | 0.0144
LEGEND-200 (Ge) | 2.6 kg $\times$ 75 | 195 kg | 2.5 keV | 1 keV | 0.78
LEGEND-1000 (Ge) | 2.6 kg $\times$ 400 | 1 tonne | 2.5 keV | 1 keV | 5
SABRE (NaI) | 2 kg $\times$ 25 | 50 kg | 1 keV | 1 keV | 0.15
10-tonne NaI | 2 kg $\times$ 2500 | 5 tonne | 1 keV | 1 keV | 50
10-tonne CsI | 2 kg $\times$ 2500 | 5 tonne | 1 keV | 1 keV | 50
Table 2: Assumed detector parameters for the SuperCDMS Albakry  _et al._
(2022), LEGEND Abgrall  _et al._ (2021), and SABRE Antonello  _et al._ (2019)
configurations.

We forecast the event rates for SuperCDMS Albakry  _et al._ (2022),
LEGEND-200, LEGEND-1000, SABRE, in addition to envisioned multi-tonne setups,
with detector specifications listed in Table 2. For the background-free
limits, we look for the Poisson 90% CL corresponding to $\simeq 3$ events
observed for a given exposure. The projected reach over the
$(g_{a\gamma}-m_{a})$ parameter space for these detector benchmarks is shown
in Fig. 7, where we show projections including the effects of absorption and
the Borrmann enhancement to the absorption length, in addition to the
projected limits assuming full volume coherence (FVC), i.e.
$I(\vec{k},\vec{G})\to 1$, indicated by the arrows and dotted lines.

The QCD axion parameter space is shown (yellow band) for the Kim-Shifman-
Vainshtein-Zakharov (KSVZ) type Kim (1979); Shifman  _et al._ (1980) and Dine-
Fischler-Srednicki-Zhitnitsky (DFSZ) type benchmark models Zhitnitsky (1980);
Dine  _et al._ (1981a, b); Dine and Fischler (1983), where the range is
defined by taking the anomaly ratios of $E/N=44/3$ to $E/N=2$ Di Luzio  _et
al._ (2020), although the space of heavier masses is also possible in high-
quality axion models and other scenarios Kivel  _et al._ (2022); Valenti  _et
al._ (2022). To probe this model parameter space beyond the existing bounds
from CAST and horizontal branch (HB) stars, when FVC is maintained, multi-
tonne scale experiments are needed. Additionally, the stellar cooling hints
that could be explained by ALPs with $g_{a\gamma}\lesssim 10^{-11}$ GeV-1 (and
for non-vanishing $g_{ae}\simeq 10^{-13}$), are also shown in Fig. 7,
indicated by the gray band (1$\sigma$) and down to vanishing $g_{a\gamma}$
Giannotti  _et al._ (2016); Hoof  _et al._ (2019); Ayala  _et al._ (2014).
These hints, though mild, could be tested by the multi-tonne setups with FVC
restored.

With the effects of absorption included, we project SuperCDMS, LEGEND, and
SABRE to test parameter space unexplored by laboratory-based probes beyond the
CAST and XENONnT constraints for $m_{a}\gtrsim 1$ eV, but already excluded by
HB stars constraints. However, multi-tonne CsI and NaI setups would extend
this to nearly cover the HB stars exclusion. Similar reach could in principle
be found when considering the joint parameter space of multiple ALP couplings
to photons, electrons, and nucleons Dent  _et al._ (2020). For instance, by
considering the 57Fe solar axion flux, one could look for 14.4 keV energy
signatures and their Bragg-Primakoff peaks, although the sensitivity would
likely contend with astrophysics constraints as well Hardy and Lasenby (2017).

The existing bounds from DAMA Bernabei  _et al._ (2004), CUORE Li  _et al._
(2016), Edelweiss-II Armengaud  _et al._ (2013), SOLAX Avignone  _et al._
(1998), COSME Morales  _et al._ (2002), CDMS Ahmed  _et al._ (2009), and
Majorana Arnquist  _et al._ (2022) are not shown here, but their exclusions
would necessarily shift to larger coupling values to account for absorption
effects in the Bragg-Primakoff rates, depending on the detector volume and
material. Note that the relative reach between NaI and CsI crystals is
relatively suppressed when absorption is included here, due to the behavior of
the imaginary form factor for CsI giving more modest Borrmann enhancements at
the lower reciprocal lattice planes; see Fig. 9. In order to push the
sensitivity envelope beyond the current bounds by CAST and HB stars, even with
multi-tonne setups, the absorption effects need to be mitigated. Some
possibilities are discussed in the next section.

Figure 7: We show sensitivity projections for Ge experiments (SuperCDMS,
LEGEND-200, and LEGEND-1000), NaI (SABRE) and multi-tonne CsI and NaI setups
with absorption effects included. Existing bounds from solar axion searches at
XENONnT Aprile  _et al._ (2022) and CAST Andriamonje  _et al._ (2007) as well
as the constraint from HB star cooling Ayala  _et al._ (2014) are shown. The
complementary reach of axion helioscope BabyIAXO Armengaud  _et al._ (2019) is
shown for comparison. Changes to the projected reach for crystal detectors if
full volume coherence (FVC) is restored are indicated by the dotted lines.

##  VI Restoring Coherence

There may be ways to recover the sensitivity initially projected in the case
of full-volume coherence by mitigating the loss of coherence due to
absorption. These are of course speculative routes. Some of these routes for
future work are enumerated below;

  1. 1.

Since the attenuation of the coherent volume is direction-dependent, as shown
in Fig. 3, one could imagine optimizing a detector geometry such that the size
and orientation relative to the incoming flux of axions is ideal, maximizing
use of the Laue-type scattering and Borrmann effect to minimize the
absorption. This would require precise knowledge of the crystal purity and
plane orientation obtained from X-ray measurements.

  2. 2.

Along a similar vein, since the effects of absorption are minimized when the
detector scale $V^{1/3}$ becomes comparable to the photon mean free path
$\lambda$, one could instead prefer to use smaller detector volumes but with a
large total mass partitioned into many individual modules. As long as each
module is optically insulated from the others, the loss of coherence due to
absorption will be contained within each module and the suppression to the
event rate can be mitigated.

  3. 3.

It might be possible to apply the principles in this work to radioisotope
experiments like those proposed in ref. Benato  _et al._ (2019); Dent  _et
al._ (2022), where a keV-scale nuclear transition line (e.g. the 14.4 keV line
of 57Fe) could source ALPs through a coupling to nucleons. Subsequent
detection by an array of crystals encasing the radioactive source searching
for transition photons of known energy Primakoff-converting in the crystal
would leave a missing energy signature in the detector. By looking for
disappearing keV-scale transitions the signal rate would enjoy the coherent
enhancement relative to the incoherent scattering considered in ref. Dent  _et
al._ (2022).

  4. 4.

A dedicated keV photon source that would impinge on a crystal detector could
fire at a fixed angle of incidence such that the event rate enhancement from
the Borrmann effect and Laue effects are optimized and full volume coherence
is restored as best as possible. One might achieve this with a keV laser
Heeter  _et al._ (2021) or synchrotron sources in a similar fashion to LSW
experiments Yamaji  _et al._ (2018); Battesti  _et al._ (2010); Inada  _et
al._ (2017); Beyer  _et al._ (2022); Bähre  _et al._ (2013). By performing a
similar “missing” photon search as the one discussed above, the event rate for
the detection of missing energy will be proportional to $g_{a\gamma}^{2}$,
rather than $g_{a\gamma}^{4}$ as in solar axion searches, greatly enhancing
the sensitivity.

In the case where we assume full volume coherence, shown in Fig. 7, dotted
lines, ton-scale setups like LEGEND-200 and LEGEND-1000 can reach
significantly smaller couplings, probing values of $g_{a\gamma}$ beyond the
existing bounds fom HB Stars Ayala  _et al._ (2014); Giannotti  _et al._
(2016) and CAST Anastassopoulos  _et al._ (2017) for masses $m_{a}\lesssim 10$
keV, losing sensitivity for higher masses for which the axion production rates
from photon coalescence and Primakoff scattering are diminished (see also Fig.
8). These reach more than an order of magnitude lower in the coupling than
previous Bragg-Primakoff solar axion searches.

##  VII Conclusions

In this work, we have taken into account a more proper estimate of the effects
of anomalous absorption into the event rate, i.e. via the Borrmann effect on
the coherence condition of Bragg-Primakoff photoconversion of solar axions.
The sensitivity of crystal technologies used in the SuperCDMS, LEGEND, and
SABRE setups has been demonstrated, and we find that the inclusion of
absorption effects even with Borrmann-enhanced signal rates still would
require multi-tonne scale detectors to surpass the existing astrophysical
constraints in sensitivity to ALPs. However, a dedicated study with a thorough
and careful treatment of the absorption suppression and Borrmann effects is
definitely needed to better understand its impact on experiments that utilize
Bragg-Primakoff conversion. In particular, the evaluation of the imaginary
form factor in other crystals (namely, PbWO4 may be an interesting option)
would help determine potential enhancements to the anomalous absorption effect
in other detector materials.

Crystal detector technologies are also necessary tools to discriminate axion-
like particle signals from other types of BSM and neutrino signatures, with
high sensitivity to time modulation from the directional sensitivity of Bragg-
Primakoff scattering. This is a powerful tool for background rejection as
well, and ideally a joint analysis of multiple detectors situated at different
latitudes and longitudes would benefit greatly from leveraging the time
modulation of the signal. They are also complimentary to future helioscope
experiments like IAXO; while the projected reach for IAXO over the axion-
photon coupling parameter space is vast, the sensitivity to solar axions with
masses $m_{a}\gtrsim 1$ eV becomes weaker to coherent Primakoff conversion in
magnetic field helioscopes. Sensitivity to this region of parameter space is
necessary in order to test QCD axions, especially in non-traditional models of
high quality axions and the like, which have parametrically larger masses
Gaillard  _et al._ (2018); Kivel  _et al._ (2022); Hook  _et al._ (2020). It
was shown in ref. Dent  _et al._ (2020) that future liquid noble gas detectors
for dark matter direct detection at kiloton-year scales could begin to probe
couplings beyond the astrophysics constraints for axion-like particles, while
in this work we find that equivalent reach is possible at ton-year exposures
with crystal detector technology, if utilized to its fullest potential. The
presence of complimentary searches at these mass scales is essential for a
complete test of the axion solution to the strong CP problem and the broader
space of ALPs.

## Acknowledgements

We are very grateful to Imran Alkhatib, Miriam Diamond, Amirata Sattari Javid,
and John Sipe for the vigorous discussions and studies on the theoretical
treatment of coherent Primakoff scattering in crystals and the comparison of
numerical computations. We graciously thank Tomohiro Yamaji for the insight on
Laue-type diffraction, Timon Emken for the technical correspondence on the
DarkART package, and Alexander Poddubny for the useful comments on Biagini’s
theory of anomalous absorption. The work of BD and AT is supported by the DOE
Grant No. DE-SC0010813. JBD acknowledges support from the National Science
Foundation under grant no. PHY-2112799. Portions of this research were
conducted with the advanced computing resources provided by Texas A&M High
Performance Research Computing. We also thank the Center for Theoretical
Underground Physics and Related Areas (CETUP*) and SURF for facilitating
portions of this research.

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##  Appendix A Crystal structure

For convenience of the reader we repeat the standard discussion on the
description of the lattice vector space for the crystals we have considered,
much of which can be found in Warren (1969) and other canonical literature.
The $\vec{\alpha}_{j}$ describe the positions of each atom within the cell,
while the basis vectors $\vec{a}_{i}$ describe the Bravais lattice. The linear
combination of the two is used to translate anywhere on the lattice by
stepping in integer multiples of these basis vectors;

$\vec{r}_{i}=n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3}+\vec{\alpha}_{i}$
(26)

We can then introduce the reciprocal lattice, giving reciprocal lattice basis
vectors $\vec{b}_{i}$ which satisfy
$\vec{b}_{i}\cdot\vec{a}_{j}=2\pi\delta_{ij}$. In general the transformations
give

$\displaystyle\vec{b}_{1}$
$\displaystyle=2\pi\dfrac{\vec{a}_{2}\times\vec{a}_{3}}{|\vec{a}_{1}\cdot(\vec{a}_{2}\times\vec{a}_{3})|}$
$\displaystyle\vec{b}_{2}$
$\displaystyle=2\pi\dfrac{\vec{a}_{3}\times\vec{a}_{1}}{|\vec{a}_{1}\cdot(\vec{a}_{2}\times\vec{a}_{3})|}$
$\displaystyle\vec{b}_{3}$
$\displaystyle=2\pi\dfrac{\vec{a}_{1}\times\vec{a}_{2}}{|\vec{a}_{1}\cdot(\vec{a}_{2}\times\vec{a}_{3})|}$
(27)

The reciprocal lattice basis vectors are used to construct the reciprocal
lattice vector $\vec{G}$ that point along the surface normals of the
scattering planes. In terms of integers $m_{1}$, $m_{2}$, and $m_{3}$, each
scattering plane is defined;

$\vec{G}=m_{1}\vec{b}_{1}+m_{2}\vec{b}_{2}+m_{3}\vec{b}_{3}$ (28)

Sometimes the integers $h,k,l$ are used instead, and in some contexts one can
use this basis to express $\vec{G}$ as

$\vec{G}(hkl)=\dfrac{2\pi}{a}(h,k,l)$ (29)

The lattice constants, cell volumes, and basis vectors for a few examples (Ge,
Si, CsI, and NaI) are listed in Table 3.

Material |  Lattice Constant $a$ (Å) |  Cell Volume $v_{\textrm{cell}}$ (Å3) | Primitive Basis | Bravais Basis | $\vec{G}(m_{1},m_{2},m_{3})\times a/2\pi$
---|---|---|---|---|---
Ge (Diamond Cubic) | 5.657 | 181.0 |  $\vec{\alpha}_{0}=(0,0,0)$ $\vec{\alpha}_{1}=\frac{a}{4}(1,1,1)$ |  $\vec{a}_{0}=\frac{a}{2}(0,0,0)$ $\vec{a}_{1}=\frac{a}{2}(1,0,1)$ $\vec{a}_{2}=\frac{a}{2}(0,1,1)$ $\vec{a}_{3}=\frac{a}{2}(1,1,0)$ |  $(m_{1}-m_{2}+m_{3},$ $-m_{1}+m_{2}+m_{3},$ $m_{1}+m_{2}-m_{3})$
Si (Diamond Cubic) | 5.429 | 160.0
CsI (FCC) | 4.503 | 91.3 |  $\vec{\alpha}_{0}=(0,0,0)$ $\vec{\alpha}_{1}=\frac{a}{2}(1,1,1)$
NaI (FCC) | 6.462 | 67.71
Table 3: Lattice information for typical crystal detector technologies.

##  Appendix B Derivation of the Event Rate

Let $f(\vec{k},\vec{k}^{\prime})$ be the Primakoff scattering matrix element
for a single atomic target, for an incoming ALP 3-momentum $\vec{k}$ and
outgoing $\gamma$ 3-momentum $\vec{k}^{\prime}$;

$f=\mathcal{M}_{\textrm{free}}F_{A}(q)$ (30)

where $\mathcal{M}_{\textrm{free}}$ is the single-atomic scattering amplitude
with the angle of scattering defined by
$\vec{k}_{a}\cdot\vec{k}_{\gamma}=E_{\gamma}k\cos 2\theta$, averaged over
spins and taken in the limit $k\gg m_{a}$, $m_{N}\gg k,E_{\gamma}$;

$\braket{\mathcal{M}_{\textrm{free}}}=\dfrac{8e^{2}g_{a\gamma}^{2}}{q^{4}}E_{\gamma}^{2}m_{N}^{2}k^{2}\sin^{2}2\theta$
(31)

We sum over the $N$ scattering centers in a crystal;

$\mathcal{M}(\vec{k},\vec{k}^{\prime})=\sum_{j=1}^{N}f_{j}(\vec{k},\vec{k}^{\prime})e^{i(\vec{k}^{\prime}-\vec{k})\cdot\vec{r}_{j}}$
(32)

where $e^{i(\vec{k}^{\prime}-\vec{k})\cdot\vec{r}_{j}}$ is a phase factor that
comes from assuming plane wave solutions for the in and out states. The
position vector $\vec{r}_{j}$ can be expressed in terms of the Bravais lattice
basis vectors and the primitive basis vectors for each unit cell of the
crystal. For germanium crystal with lattice constant $a$, we have primitive
basis vectors

$\displaystyle\vec{\alpha}_{0}$ $\displaystyle=(0,0,0)$
$\displaystyle\vec{\alpha}_{1}$ $\displaystyle=\frac{a}{4}(1,1,1)$ (33)

while the basis vectors of the Bravais lattice are described by $\vec{a}_{1}$,
$\vec{a}_{2}$, and $\vec{a}_{3}$;

$\displaystyle\vec{a}_{1}$ $\displaystyle=\frac{a}{2}(0,1,1)$
$\displaystyle\vec{a}_{2}$ $\displaystyle=\frac{a}{2}(1,0,1)$
$\displaystyle\vec{a}_{3}$ $\displaystyle=\frac{a}{2}(1,1,0)$ (34)

we can represent any scattering site as a linear combination of the $a$’s and
either the first or second primitive;

$\displaystyle\vec{r}_{i,0}$
$\displaystyle=\vec{R}_{i}+\vec{\alpha}_{0}=n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3}+\vec{\alpha}_{0}$
(35) $\displaystyle\vec{r}_{i,1}$
$\displaystyle=\vec{R}_{i}+\vec{\alpha}_{1}=n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3}+\vec{\alpha}_{1}$
(36)

where the index $i$ maps to a unique combination ($n_{1},n_{2},n_{3}$). If we
square this, we get

$\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}=\sum_{i=1}^{N}\mid
f_{i}\mid^{2}+\sum_{j\neq
i}^{N}\sum_{i=1}^{N}f_{j}^{\dagger}f_{i}e^{-i\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})}$
(37)

taking $\vec{q}\equiv\vec{k}-\vec{k}^{\prime}$. Rewriting in terms of a sum
over $N_{c}$ cells and the cell primitives, the coherent part (second term) is

$\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}=\sum_{j\neq
i}^{N_{c}}\sum_{i=1}^{N_{c}}\sum_{\mu=0}^{1}\sum_{\nu=0}^{1}f_{j}^{\dagger}f_{i}e^{-i\vec{q}\cdot(\vec{R}_{i}-\vec{R}_{j}+\vec{\alpha}_{\mu}-\vec{\alpha}_{\nu})}$
(38)

When the Laue condition is met, we have $\vec{q}=\vec{G}$ and
$\vec{G}\cdot\vec{R}_{i}$ is a $2\pi$ integer multiple;

$|\mathcal{M}|^{2}\equiv\sum_{j\neq
i}^{N_{c}}\sum_{i=1}^{N_{c}}\sum_{\mu,\nu=0}^{1}f_{j}^{\dagger}f_{i}e^{-i\vec{G}\cdot(\vec{\alpha}_{\mu}-\vec{\alpha}_{\nu})}$
(39)

Now we can factorize the sum over primitives, and since we are considering a
monoatomic crystal we can also take the $f_{i}=f_{j}$, simplifying things;

$|\mathcal{M}|^{2}=N_{c}^{2}f^{\dagger}f\sum_{\mu,\nu=0}^{1}e^{-i\vec{G}\cdot(\vec{\alpha}_{\mu}-\vec{\alpha}_{\nu})}$
(40)

In Eq. 40 the structure function can be substituted, which is nothing but the
sum over primtives;

$S(\vec{G})=\sum_{\mu}e^{i\vec{G}\cdot\alpha_{\mu}}$ (41)

and we have no need for a species index $j$ on $S_{j}(\vec{G})$ since we only
have one atomic species, but it is trivial to extend this derivation to
include it - we just need to add another index to the primitive basis vectors
and sum over it. With this identification and also taking
$f^{\dagger}f=|\mathcal{M}_{\textrm{free}}|^{2}F^{2}_{A}(\vec{G})$, we have

$\displaystyle|\mathcal{M}|^{2}$
$\displaystyle=N_{c}^{2}|\mathcal{M}_{\textrm{free}}|^{2}|F_{A}(\vec{G})S(\vec{G})|^{2}$
(42)

Now let’s write down the cross section.

$d\sigma=\dfrac{1}{4E_{a}m_{N}v_{a}}|\mathcal{M}|^{2}\dfrac{d^{3}k^{\prime}}{(2\pi)^{3}2E_{\gamma}}\dfrac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}(2\pi)^{4}\delta^{4}(k+p-k^{\prime}-p^{\prime})$
(43)

Taking the ALP velocity $v_{a}=1$, momentum transfer minimal such that
$E_{p^{\prime}}=m_{N}$, and integrating out the $\delta^{3}$ we get

$d\sigma=\dfrac{1}{64\pi^{2}E_{a}E_{\gamma}m_{N}^{2}}|\mathcal{M}|^{2}d^{3}k^{\prime}\delta(E_{a}-E_{\gamma})$
(44)

Performing a change of variables to $d^{3}k^{\prime}\to d^{3}q$ (since
$q=k-k^{\prime}$ and $k$ is fixed), we would integrate this over $\vec{q}$.
Since we have $\vec{q}=\vec{G}$ at this stage, we should replace the integral
with a sum;

$\int d^{3}q\to\frac{(2\pi)^{3}}{V}\sum_{\vec{G}}$ (45)

The event rate formula is constructed from a convolution of the detector
response, axion flux $\Phi_{a}$, and cross section;

$\dfrac{dN}{dt}=\int_{E_{1}}^{E_{2}}dE_{ee}\int_{0}^{\infty}dE_{a}\frac{(2\pi)^{3}}{V}\sum_{\vec{G}}\dfrac{d\Phi_{a}}{dE_{a}}\dfrac{1}{64\pi^{2}E_{a}E_{\gamma}m_{N}^{2}}|\mathcal{M}|^{2}\delta(E_{a}-E_{\gamma})\cdot\bigg{(}\dfrac{1}{\Delta\sqrt{2\pi}}e^{-(E_{ee}-E_{\gamma})^{2}/2\Delta^{2}}\bigg{)}$
(46)

Putting in the definition of $|\mathcal{M}|^{2}$ that we worked out and
substituting the free Primakoff cross section, integrating over the energy
delta function (and identifying $E_{a}=E_{\gamma}=E$ for simplicity), and
integrating over $dE_{ee}$ we get

$\dfrac{dN}{dt}=\dfrac{(2\pi)^{3}e^{2}g_{a\gamma}^{2}}{8\pi^{2}}\dfrac{V}{v_{\text{cell}}^{2}}\sum_{\vec{G}}\dfrac{d\Phi_{a}}{dE}\dfrac{k^{2}\sin^{2}(2\theta)}{|\vec{G}|^{4}}|F_{A}(\vec{G})S(\vec{G})|^{2}\mathcal{W}(E_{1},E_{2},E)$
(47)

This is almost identical to the rate in ref. Cebrian  _et al._ (1999), which
uses a different definition of the atomic form factor up to a factor of
$\frac{q^{2}}{ek^{2}}$. After some algebra, the event rate in Eq. 47 is still
different than that given in ref. Cebrian  _et al._ (1999) up to a factor of
$4\sin^{2}(\theta)$. However, the event rate formula derived here is
consistent with the calculation performed in refs. Li  _et al._ (2015, 2016).
After rederiving the coherent sum using the replacements in Eqns. 6-10, the
event rate becomes

$\dfrac{dN}{dt}=\dfrac{(2\pi)^{3}e^{2}g_{a\gamma}^{2}}{8\pi^{2}}\dfrac{V}{v_{\text{cell}}^{2}}\sum_{\vec{G}}I(\vec{k},\vec{G})\dfrac{d\Phi_{a}}{dE}\dfrac{k^{2}\sin^{2}(2\theta)}{|\vec{G}|^{4}}|F_{A}(\vec{G})S(\vec{G})|^{2}\mathcal{W}(E_{1},E_{2},E)$
(48)

##  Appendix C Solar Axion Flux

We use the parameterization appearing in ref. Di Lella  _et al._ (2000) for
massive axion production in the sun; the flux parameterizations are repeated
here for convenience

$\displaystyle\dfrac{d\Phi_{\gamma\to a}}{dE_{a}}$
$\displaystyle=\dfrac{4.20\cdot
10^{10}}{\textrm{cm}^{-2}\textrm{s}^{-1}\textrm{keV}^{-1}}\bigg{(}\dfrac{g_{a\gamma}}{10^{-10}\textrm{GeV}^{-1}}\bigg{)}^{2}\dfrac{E_{a}p_{a}^{2}}{e^{E_{a}/1.1}-0.7}(1+0.02m_{a})$
(49) $\displaystyle\dfrac{d\Phi_{\gamma\gamma\to a}}{dE_{a}}$
$\displaystyle=\dfrac{1.68\cdot
10^{9}}{\textrm{cm}^{-2}\textrm{s}^{-1}\textrm{keV}^{-1}}\bigg{(}\dfrac{g_{a\gamma}}{10^{-10}\textrm{GeV}^{-1}}\bigg{)}^{2}m_{a}^{4}p_{a}\bigg{(}1+0.0006E_{a}^{3}+\frac{10}{E_{a}^{2}+0.2}\bigg{)}e^{-E_{a}}$
(50)

where $\Phi_{\gamma\to a}$ is the Primakoff solar flux and
$\Phi_{\gamma\gamma\to a}$ is the flux resulting from resonant photon
coalescence, both in units of
$\textrm{cm}^{-2}\textrm{s}^{-1}\textrm{keV}^{-1}$, given for axion energy and
momentum $E_{a}$ and $p_{a}$ in keV, and for the coupling $g_{a\gamma}$ in
GeV-1. The solar axion flux from photon coalescence and Primakoff conversion
is shown in Fig. 8 for several benchmark axion masses.

Figure 8: Solar Axion fluxes produced from Primakoff scattering ($\gamma Z\to
aZ$) and coalescence ($\gamma\gamma\to a$) in the sun.

##  Appendix D Utilizing DarkARC/DarkART for Calculation of the Absorptive
Form Factor

Figure 9: Borrmann parameter $\epsilon$ as a function of the momentum transfer
$\sin\theta/\lambda=|\vec{G}|/4\pi$ for several crystal materials.

Wagenfield’s form factor for the anomalous dispersion of X-rays with incoming
and outgoing momenta and polarizations $\boldsymbol{k}$,
$\boldsymbol{\varepsilon}_{0}$, $\boldsymbol{k}^{\prime}$,
$\boldsymbol{\varepsilon}_{0}^{\prime}$ is Wagenfeld (1987)

$\displaystyle\Delta f^{\prime\prime}$
$\displaystyle=\dfrac{\pi\hbar^{2}}{m_{e}}\bigg{(}\int\psi_{f}^{*}(\boldsymbol{r})\boldsymbol{\varepsilon}_{0}\cdot\boldsymbol{\nabla}e^{i\boldsymbol{k}\cdot\boldsymbol{r}}\psi_{i}(\boldsymbol{r})d^{3}r\bigg{)}\bigg{(}\int\psi_{f}(\boldsymbol{r})\boldsymbol{\varepsilon}^{\prime}_{0}\cdot\boldsymbol{\nabla}e^{-i\boldsymbol{k}^{\prime}\cdot\boldsymbol{r}}\psi_{i}^{*}(\boldsymbol{r})d^{3}r\bigg{)}$
(51)

Applying the gradient and expanding, we get some terms proportional to
$\boldsymbol{\varepsilon}_{0}\cdot\boldsymbol{k}$ which vanish, leaving us
with

$\displaystyle\Delta f^{\prime\prime}$
$\displaystyle=\dfrac{\pi\hbar^{2}}{m_{e}}\bigg{(}\boldsymbol{\varepsilon}_{0}\cdot\int\psi_{f}^{*}(\boldsymbol{r})e^{i\boldsymbol{k}\cdot\boldsymbol{r}}\boldsymbol{\nabla}\psi_{i}(\boldsymbol{r})d^{3}r\bigg{)}\bigg{(}\boldsymbol{\varepsilon}^{\prime}_{0}\cdot\int\psi_{f}(\boldsymbol{r})e^{-i\boldsymbol{k}^{\prime}\cdot\boldsymbol{r}}\boldsymbol{\nabla}\psi_{i}^{*}(\boldsymbol{r})d^{3}r\bigg{)}$
(52)

Referring to Catena et al Catena  _et al._ (2020), we can then apply the
definition of the vectorial form factor (eq B18, but with some changes made to
keep the notation more consistent),

$\boldsymbol{f}_{1\to 2}(\boldsymbol{q})=\int
d^{3}r\psi^{*}_{f}(\boldsymbol{r})e^{i\boldsymbol{q}\cdot\boldsymbol{r}}\frac{i\boldsymbol{\nabla}}{m_{e}}\psi_{i}(\boldsymbol{r}).$
(53)

Here the final state and initial state wave functions have quantum numbers
$i=n,\ell,m$ and $f=p^{\prime},\ell^{\prime},m^{\prime}$ where $p^{\prime}$ is
the final state electron momentum, and
$\\{n,\ell,m\\},\\{\ell^{\prime},m^{\prime}\\}$ are the initial and final
quantum numbers, respectively. Applying this definition, we have

$\displaystyle\Delta f^{\prime\prime}$
$\displaystyle=\dfrac{\pi\hbar^{2}}{m_{e}}\bigg{(}\boldsymbol{\varepsilon}_{0}\cdot(-im_{e})\boldsymbol{f}_{1\to
2}(\boldsymbol{k})\bigg{)}\bigg{(}\boldsymbol{\varepsilon}^{\prime}_{0}\cdot(im_{e})\boldsymbol{f}^{*}_{1\to
2}(\boldsymbol{k}^{\prime})\bigg{)}$
$\displaystyle=\pi\hbar^{2}m_{e}\bigg{(}\boldsymbol{\varepsilon}_{0}\cdot\boldsymbol{f}_{1\to
2}(\boldsymbol{k})\bigg{)}\bigg{(}\boldsymbol{\varepsilon}_{0}^{\prime}\cdot\boldsymbol{f}^{*}_{1\to
2}(\boldsymbol{k}^{\prime})\bigg{)}$ (54)

If our photons are unpolarized, then we can take a sum over the helicity
states, giving the completeness relation
$\sum_{s}(\boldsymbol{\varepsilon}_{0}(s))_{i}(\boldsymbol{\varepsilon}_{0}^{\prime}(s))_{j}=\delta_{ij}$.
Taking $\boldsymbol{k}^{\prime}=\boldsymbol{k}-\boldsymbol{q}$, this reduces
the polarization-summed imaginary form factor to

$\Delta
f^{\prime\prime}(\boldsymbol{k},\boldsymbol{q})=\pi\hbar^{2}m_{e}\bigg{(}\boldsymbol{f}_{1\to
2}(\boldsymbol{k})\cdot\boldsymbol{f}^{*}_{1\to
2}(\boldsymbol{k}-\boldsymbol{q})\bigg{)}$ (55)