ON COHERENCE IN BRAGG-PRIMAKOFF AXION PHOTOCONVERSION

A Dissertation

by

ADRIAN RAPHAEL TRAQUINA THOMPSON

Submitted to the Office of Graduate and Professional Studies of

Texas A&M University

in partial fulfillment of the requirements for the degree of

DOCTORATE OF SCIENCE

Chair of Committee, Bhaskar Dutta Committee Members, Louis Strigari Teruki
Kamon Stephen A. Fulling Head of Department, Grigory Rogachev

August 2023

Major Subject: Physics

Copyright 2023 Adrian Raphael Traquina Thompson

## ABSTRACT

Axions and axion-like pseudoscalar particles with dimension-5 couplings to
photons exhibit coherent Bragg-Primakoff scattering with ordered crystals at
keV energy scales. This provides for a natural detection technique in searches
for axions produce in the Sun’s interior. I will motivate the utility of dark
matter direct detection experiments in searching for solar axions, emphasizing
the role crystal-based detector technologies. I present an updated theoretical
treatment of the Bragg-Primakoff photoconversion process for keV
pseudoscalars, and address simultaneously the effects of absorption of final
state photons in crystals on the loss of coherence, which can lead to large
suppressive corrections to the event rate sensitivity for this detection
technique. However, I also show that the Borrmann effect of anomalous
absorption significantly lifts the suppression. This phenomenon is studied 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, I investigate the future reach of multi-ton scale crystal
detectors and discuss strategies to maximize the discovery potential of
experimental efforts in this vein.

## DEDICATION

This body of work is the product of an arduous spirit that could not have been
kindled, stoked, and tended without the love and support of, first and
foremost, my mother Paula and father Kurt. And with the light of my friends,
bright and fluorescent even in the darkest trench, I always had a familiar
guide and tether to the things that were most important. For my advisor, Dr.
Bhaskar Dutta, I have such a happy gratitude for the delicate balance of
patience and ambition that was mentored. And without the kindness and
enthusiasm of Dr. Teruki Kamon, whom I first worked under, I would not have
started on this path. I want to deeply thank them, as well as my close
collaborators Dr. James Dent, Dr. Doojin Kim, and Dr. Louis Strigari for
engendering a creative environment where I was perfectly comfortable asking
stupid questions and letting my mind run free without fear of judgement.
Unfortunately that is a rare thing to find, sometimes, in our little
community, and so I count myself very lucky for that and for all the people in
my life. This work is dedicated to you all.

## ACKNOWLEDGMENTS

I am 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. I also 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.

I also thank Nicole Bell, Kiwoon Choi, and Sebastian Hoof for discussions
related to the work on inverse Primakoff scattering as it applied to dark
matter direct detection experiments.

I would also like to thank the Mitchell Institute for Fundamental Physics and
Astronomy for support. Portions of this research were conducted with the
advanced computing resources provided by Texas A&M High Performance Research
Computing.

## CONTRIBUTORS AND FUNDING SOURCES

#### Contributors

This work was supported by a thesis committee consisting of Professor Bhaskar
Dutta, Professor Louis Strigari, and Professor Teruki Kamon of the Department
of Physics and Astronomy, and Professor Stephen A. Fulling of the Department
of Mathematics.

Contributions to this work came from my close collaborators at Texas A&M, Los
Alamos National Laboratory, and abroad. The body of the thesis will include
work from the following publications that I have authored:

  1. 1.

J. B. Dent, B. Dutta, J. L. Newstead, A. Thompson, “Inverse Primakoff
Scattering as a Probe of Solar Axions at Liquid Xenon Direct Detection
Experiments” Phys. Rev. Lett. 125 (2020) 13, 131805 arXiv:2006.15118

  2. 2.

J. B. Dent, B. Dutta, A. Thompson, ‘‘Bragg-Primakoff Axion Photoconversion in
Crystal Detectors’’. To appear.

All other work conducted for this thesis was completed by the student
independently.

#### Funding Sources

This work has been supported in part by DOE grant DE-SC0010813 and the Los
Alamos National Laboratory LDRD program. I additionally thank the Mitchell
Institute for Fundamental Physics and Astronomy for support.

TABLE OF CONTENTS

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LIST OF FIGURES

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## 1\. INTRODUCTION

Axions and axion-like particles – 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 [7, 8], and to the dark matter (DM) problem [9, 10], and otherwise
appear ubiquitously in string theory [11, 12, 13] \- the so-called “axiverse”
- and the ultraviolet (UV) spectra of many other puzzle-solving models. In
this work I will dive into some of the mechanisms behind the aforementioned
axion and ALP scenarios in order to motivate a broad target parameter space to
search for ALPs.

While there has been a tremendous effort to probe high mass ALPs from the MeV
to GeV scales, there is a deep chasm of parameter space for ALPs below 1 keV
in mass that is neither constrained by laboratory probes nor astrophysics. In
this region the ALP can act as both a dark matter candidate and a solution to
the strong CP problem, among other roles in solutions to the current panorama
of anomalies and puzzles. There are a number of future and ongoing helioscope
experiments that use resonant cavities or large magnetic fields to search for
DM or solar ALPs, but the reach of these experiments terminates around masses
$\gtrsim 1$ eV. Dark matter direct detection experiments, on the other hand,
offer broadband sensitivity in the axion mass and may be powerful enough at
the tonne-scale to search for ALPs where the astrophysical constraints and
future helioscopes and haloscopes lose their sensitivity. In particular,
Bragg-Primakoff scattering – a form of ALP-to-photon conversion that is
coherent at the level of an ordered atomic lattice as well as coherent at the
level of the atomic charge distribution – can greatly enhance the event rates
for ALPs with keV energies. This makes for a natural detection technique in
the search for solar ALPs, and by leveraging crystal detectors we can get the
reach we need from direct detection experiments at smaller detector scales. In
this work I will thoroughly construct the methodology for this detection
technique, building off the existing literature but repairing several stark
inconsistencies along the way, in the end showing the importance and power of
crystallographic technologies to search for axion-like particles with the
aforementioned parameter space as a clear goal.

In Chapter 2 I discuss the theoretical motivations to search for axion-like
particles, and in Chapter 3 I will mark out the relevant amplitudes, cross
sections, and form factors relevant for Primakoff scattering, the key
phenomenological probe to look for axion-like particles coupling to photons.
In Chapter 4 I give a detailed treatment of Bragg-Primakoff conversion in
perfect crystals, working through the subtleties and salient phenomenological
impact of absorption effects (and the Borrmann effect) on the event rate for
Bragg-Primakoff conversion. In Chapter 5 I apply the results of the previous
two chapters to searches of axion-like particles produced in the Sun, and to
other potential discovery strategies to test QCD axion model parameter space.
Finally, in Chapter 6 I conclude and remark on future directions of work.

## 2\. THEORETICAL FOUNDATIONS FOR MOTIVATING THE EXISTENCE OF AXION-LIKE
PARTICLES

### 2.1 Axionic Solutions to the Smallness of Strong $CP$

We first begin with pseudoscalar particles that solve the strong CP problem,
which I will label simply as “axions” or “QCD axion” and I will call
pseudoscalars which do not necessarily solve strong CP, and span a much
broader parameter space of SM couplings, “axion-like particles” or ALPs. The
Peccei-Quinn mechanism [7, 8, 14, 15, 16, 17, 10, 9, 18], whereby a global
$U(1)_{PQ}$ symmetry is spontaneously broken, offers a dynamic mechanism to
minimize the QCD vacuum to a CP-conserving state.

The way this happens can be understood by looking at the properties of the QCD
vacuum. The QCD vacuum has two angles: $\theta$, the angle that appears in
front of the $G\tilde{G}$ topological term, and $\theta_{q}=\arg\det M_{q}$,
the phase of the quark mass matrix. These combine as a physical parameter
$\bar{\theta}=\theta+\theta_{q}$ which has the consequence of violating $CP$;

$\mathcal{L}\supset\frac{g^{2}}{32\pi^{2}}\bar{\theta}G\tilde{G}$ (2.1)

Here I have used the abbreviated notation
$G\tilde{G}=\epsilon_{\alpha\beta\mu\nu}G^{\alpha\beta}_{a}G^{\mu\nu}_{a}$.
The Vafa-Witten theorem states that if one considers two different
configurations of the QCD vacuum, one where $\theta=0$ and one where
$\theta\geq 0$, the one set to zero will always have lower energy density. In
a universe where $\theta$ is a fixed parameter, nothing would protect the
theory from having $\theta\geq 0$ and violating CP; we would just have to
throw up our hands and accept it. However, if $\bar{\theta}$ were promoted to
a dynamical field, then it would have the ability to roll down to the bottom
of its potential at $\theta=0$ where the Vafa-Witten theorem [19] tells us
that is the energetically favored configuration. This is precisely what the
axion field does for us, giving a mechanical explanation for why the observed
$CP$ violation is so small.

The next step in this line of thought is to construct a field $a$ that couples
to $G\tilde{G}$ and promotes $\bar{\theta}$ to a dynamical object. In Fig. 2.1
the common benchmark QCD axion models are shown schematically. Beginning with
the Kim-Shifman-Vainshtein-Zakharov (KSVZ) [20, 21] type models on the left,
which feature a set of heavy color-charged fermions $Q_{L}$ and $Q_{R}$ that
are also charged under $U(1)_{PQ}$, derive couplings to gluons and photons
through the operators $aG\tilde{G}$ and $aF\tilde{F}$ arising at loop level.

To see how this happens schematically, consider the PQ Yukawa terms like
$\Phi\bar{Q}_{L}Q_{R}$, which in the physical basis after $U(1)_{PQ}$ is
broken become

$\Phi\bar{Q}_{L}Q_{R}\to(v_{a}+\rho(x))e^{-ia/v_{a}}\bar{Q}_{L}Q_{R},$ (2.2)

giving $Q$ a mass term and a coupling to the goldstone and radial modes of
$\Phi$ in the broken phase. We can rotate away the goldstone angle
$e^{-ia/v_{a}}$ by performing a field redefinition,

$Q_{L,R}\to Q_{L,R}e^{-i\gamma^{5}a/2v_{a}}$ (2.3)

This brings a derivative coupling of the pseudoscalar axion field $a$ out of
the $Q$ kinetic pieces after integrating by parts and throwing away the total
derivative;

$i\bar{Q}_{L}\not{\partial}Q_{R}\to\frac{\partial_{\mu}a}{v_{a}}\bar{Q}_{L}\gamma^{\mu}\gamma^{5}Q_{R}$
(2.4)

The Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) type models [22, 23, 24, 17],
shown on the right, instead feature an extended Higgs sector which gives rise
to tree-level fermion couplings after a chiral rotation of the fields, while
couplings to gluons and photons are kept at loop level. Other variants exist,
for example the sub-types DFSZ-I, DFSZ-II, IIa, IIb, etc. [25] which
categorize different schema for the Higgs sector structure.

KSVZ-typeDFSZ-typeHeavy Quarks + Singlet: $Q_{L},Q_{R},\Phi$2HDM + Singlet:
$H_{u},H_{d},\Phi$$\mathcal{L}\supset-
m_{Q}\bar{Q}_{L}Q_{R}e^{-ia/v_{a}}$$\mathcal{L}\supset
aG\tilde{G},\,aF\tilde{F}$ via $Q$ loops$\mathcal{L}_{f=e,u,d}\supset-
m_{f}\bar{f}_{L}f_{R}e^{-i\chi
a/v_{a}}$$\delta\mathcal{L}_{f=e,u,d}=\frac{c_{af}}{f_{a}}(\partial_{\mu}a)\bar{f}\gamma^{\mu}\gamma_{5}f$$\mathcal{L}\supset
aG\tilde{G},\,aF\tilde{F}$ via $e,u,d$ loopsPQSB$f\to
fe^{-i\chi\gamma^{5}a/2v_{a}}$PQSB$Q\to Qe^{-i\gamma^{5}a/2v_{a}}$ Figure 2.1:
The KSVZ and DFSZ type QCD axion model mechanisms summarized graphically.

These different scenarios strongly motivate the existence of couplings to SM
fermions and to the photon through the dimension-5 operator $aF\tilde{F}$. The
treatment of this parameter space can be expanded in a model-independent way
by considering a simple effective field theory of ALP interactions;

$\mathcal{L}\supset\frac{1}{2}(\partial_{\mu}a)(\partial^{\mu}a)-m_{a}^{2}a^{2}-\dfrac{1}{4}g_{a\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu}-\dfrac{1}{4}g_{ag}a\textrm{Tr}[G_{\mu\nu}\tilde{G}^{\mu\nu}]-i\sum_{f}g_{af}a\bar{f}\gamma^{5}f$
(2.5)

The QCD axion parameter space will then be a subset of the space covered by
the axion mass and its couplings, as seen in Fig. 2.2.

The correlations between the QCD axion mass and its effective couplings are
given below, taken from ref. [26]. I will reiterate those correlations here
for convenience of the reader. The relation between the Peccei-Quinn breaking
scale $f_{a}$ and the axion mass is

$f_{a}=\bigg{(}\frac{5.691\times
10^{6}\textrm{eV}}{m_{a}}\bigg{)}\textrm{GeV}$ (2.6)

To find the correlations between the axion mass and its effective couplings to
photons in the KSVZ benchmark model is then given by Eq. 2.7;

$\displaystyle g_{a\gamma}$
$\displaystyle=\frac{m_{a}}{\textrm{GeV}}\bigg{(}0.203\frac{E}{N}-0.39\bigg{)}$
(2.7)

I will then take a region of model parameter space defined by considering
anomaly number ratios from $E/N=44/3$ to $E/N=2$. This defines a band in the
$(m_{a},g_{a\gamma}$ parameter space of ALP couplings and masses.

For the DFSZ benchmark model, for which couplings to electrons would be
dominant relative to the photon couplings, I take

$g_{ae}=\dfrac{m_{e}C_{ae}(m_{a},\tan\beta)}{f_{a}}$ (2.8)

where the coefficient $C_{ae}$ is dependent on the rotation angle $\beta$ for
the vacuum expectation values of the extended Higgs sector in DFSZI and DFSZII
models;

$\displaystyle\textrm{DFSZ(I):\, \, }C_{ae}$
$\displaystyle=-\frac{1}{3}\sin^{2}\beta+\frac{3\alpha^{2}}{4\pi^{2}}\bigg{[}\frac{E}{N}\log(f_{a}/m_{a})-1.92\log(1/m_{e})\bigg{]},\,\,\,\,\frac{E}{N}=8/3$
(2.9) $\displaystyle\textrm{DFSZ(II):\, \, }C_{ae}$
$\displaystyle=\frac{1}{3}\sin^{2}\beta+\frac{3\alpha^{2}}{4\pi^{2}}\bigg{[}\frac{E}{N}\log(f_{a}/m_{a})-1.92\log(1/m_{e})\bigg{]},\,\,\,\,\frac{E}{N}=2/3$
(2.10)

Here I allow $\tan\beta$ values between 0.25 and 120, allowing for a wide
range of coupling values for a particular axion mass [27].

As a brief aside on couplings to fermions: one can either have the Yukawa, or
manifestly-pseudoscalar operator;

$\mathcal{L}_{Yuk}\supset ig_{ae}a\bar{\psi}\gamma^{5}\psi$ (2.11)

or the gradient/derivative type operator associated with pNGBs with a shift
symmetry;

$\mathcal{L}_{Grad}\supset\frac{\partial_{\mu}a}{f_{a}}\bar{\psi}\gamma^{\mu}\gamma^{5}\psi$
(2.12)

These two operators can be transformed into each other since they are related
by a total derivative, but only if one assumes that the fermion $\psi$
satisfies its free equation of motion, i.e. $i\not{\partial}\psi=m\psi$.
Therefore, in general the two operators do not give equivalent amplitudes
except when fermions can be approximated as on-shell.

While searches for QCD axions, that is, ALPs which solve the strong CP problem
as part of a Peccei-Quinn mechanism, are well motivated, it is important to
also consider alternative solutions. For example, left-right symmetric models
have also been able to solve strong CP without needing an explicit axion. This
only offers more motivation to search for them, since if a discovery is not
made, and instead the viable parameter space for QCD axions is ruled out by
experiment, then it will give us the invaluable guidance to look elsewhere for
non-axionic solutions which we already know are possible.

Besides the scope of QCD axions mentioned here so far, there is a much broader
landscape of axion-like particle solutions to modern physics puzzles. String
axiverse pseudoscalars arise from string theory compactification scenarios,
which may number in the tens to hundreds from string theory compactification
scenarios [12, 13]. More generally, light pseudoscalars with weak couplings to
the SM could arise as pNGBs from other broken symmetries such as $B-L$ or
$T3R$, and may be introduced to address other important problems in the SM
(e.g. Majorons, Familons). We should keep this in mind when motivating tests
of ALP parameter space, having the understanding that the target space is
often much larger and open ended than that of the traditional QCD models.

### 2.2 Laboratory Probes

Figure 2.2: Existing ALP limits and constraints for ALPs coupling to photons
through $aF\tilde{F}$. Laboratory-derived constraints are shaded in solid
colors while astrophysical constraints appear in grayscale. Bands of constant
decay length are highlighted, whose thicknesses indicate a range of non-
relativistic to relativistic ALPs. Traditional QCD axion model parameter space
is shaded in yellow, corresponding to the KSVZ and DFSZ benchmark models.
Additional constraints from cosmological considerations and assuming a dark
matter ALP are not shown.

In Fig. 2.2 I show many of the existing limits that have been set on the
$g_{a\gamma}-m_{a}$ parameter space. Some of the most stringent constraints
come from astrophysics, shown in gray, like those from the axion-induced
stellar cooling of horizontal branch (HB) stars [28], supernova SN1987a energy
loss [29, 30, 31, 32, 33] and decaying axion signals [34, 33, 35], and low
energy supernovae [31]. Pure laboratory constraints – that is, those from
searches for axions that would be produced and detected terrestrially – are
shown with solid colors [36]. They include collider searches like those from
LEP [37], ATLAS [38] and CMS from $pp$ collisions as well as Pb-Pb collisions
[39], decaying particle lifetime measurements like those from PrimEx [40],
Belle II [41, 42], BESIII [43]. Beam dump experiments have also played a
significant role in testing parameter space of axions which decay on the scale
of the experimental baseline, namely E137 [41]. Myself and collaborators
exploited the intense electromagnetic and baryonic particle cascades inside
beam dump targets and reactor cores could be used in searches that would be
sensitive to high-mass ALPs in this parameter space [44, 45, 46, 47, 48].

To keep a reference of scale, I also show bands of constant decay length to
photon pairs, via the decay width

$\Gamma(a\to\gamma\gamma)=\dfrac{g_{a\gamma}^{2}m_{a}^{3}}{64\pi}$ (2.13)

The bands range from the size of the observable universe to nanometer scale,
where the width of each band sweeps from non-relativistic ($\beta\sim 0.01$)
to relativistic ($\beta>0.9$) ALPs. The model parameter space for traditional
QCD axions (KSVZ, DFSZ) is shown by the yellow band, however a much broader
parameter space is motivated by non-traditional QCD axion models or by non-QCD
ALP models. For example, many non-traditional QCD axion models are motivated
to ensure that higher-order operators in the ultraviolet scales, which break
$PQ$ symmetry, do not misalign the axion potential, spoiling the dynamical
minimization of CP violation in the QCD vacuum. This has been known as the
axion quality problem [49, 50, 51, 52, 53]. Many such models have been
realized which can shift the axion mass parametrically lighter [54] or heavier
[55, 56, 57]. These are just some examples, and of course the model parameter
space of generic pNGBs or axiverse pseudoscalars is much larger. The richness
of possibilities makes the physics targets for searches of ALPs also a broad
eneavor.

While the frontier for axion searches in the high mass regime is busy with
activity, the low mass regime is also very active but has its own challenges.
Helioscope experiments, like the International Axion Observatory (IAXO) [58],
aim to probe sub-eV masses of ALPs associated with solar axions as the next
generation successor to CAST [59]. At the same time, a multitude of resonant
cavity haloscopes and other coherent electromagnetic-based technologies search
for axions that might make up the local dark matter halo. These experiments
cover a wide swath of parameter space for light axions below the eV scale.
However, there is still a critical region of interest just at the eV scale
which hosts several theoretical targets. Looking again at Fig. 2.2, we have
(i) traditional QCD axion models, which are unconstrained from HB stars just
below $m_{a}=1$ eV, (ii) a number of anomalies in the cooling and growth rates
of stars, which can be explained by ALPs with photon couplings just below
$g_{a\gamma}\lesssim 10^{-11}$ GeV-1 [60, 61, 62] (not pictured), and (iii)
the lifetimes of ALPs in this same region of parameter space have lifetimes
beyond that of the age of the universe, making them excellent dark matter
candidates. It is this unique region of parameter space that is the prime
target of this thesis, and in the following chapters, I will motivate the
phenomenology of crystal detector technology that leverages coherence to be
sensitive to ALP detection in this context.

## 3\. ELECTROMAGNETIC-AXION INTERACTIONS

In this section I will review the production and detection processes that are
relevant to laboratory searches, in particular the Primakoff and inverse
Primakoff photoconversion processes relevant for axions coupled to photons, in
addition to a discussion on other processes relevant for the production of
axions in the Sun.

The form of the vertex operator for the dimension-5 photon coupling is
interesting, as the derivatives contained the electromagnetic field strength
tensors give rise to momentum dependence in the Feynman rules. Here I provide
a quick derivation. The operator concerning the $a\gamma\gamma$ vertex is

$\mathcal{L}_{a\gamma}=\dfrac{1}{4}g_{a\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu}$
(3.1)

where $\tilde{F}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$.
Expanding out the operator and moving to momentum space for the vertex
$a(k)\to\gamma(p)\gamma(q)$ yields

$\displaystyle aF_{\mu\nu}\tilde{F}^{\mu\nu}$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}a(k)\bigg{(}\partial_{\mu}A_{\nu}(p)-\partial_{\nu}A_{\mu}(p)\bigg{)}\bigg{(}\partial_{\alpha}A_{\beta}(q)-\partial_{\beta}A_{\alpha}(q)\bigg{)}$
$\displaystyle=-\epsilon^{\mu\nu\alpha\beta}a(k)\bigg{(}p_{\mu}A_{\nu}-p_{\nu}A_{\mu}\bigg{)}\bigg{(}q_{\alpha}A_{\beta}-q_{\beta}A_{\alpha}\bigg{)}$
$\displaystyle=-a(k)\bigg{(}\epsilon^{\mu\nu\alpha\beta}(p_{\mu}A_{\nu}q_{\alpha}A_{\beta}-p_{\mu}A_{\nu}q_{\beta}A_{\alpha})-\epsilon^{\mu\nu\alpha\beta}p_{\nu}A_{\mu}q_{\alpha}A_{\beta}+\epsilon^{\mu\nu\alpha\beta}p_{\nu}A_{\mu}q_{\beta}A_{\alpha}\bigg{)}$

Now relabel $\mu\Longleftrightarrow\nu$ and commute the Levi-Civita indices on
the last two terms:

$\displaystyle=-2\epsilon^{\mu\nu\alpha\beta}a(k)p_{\mu}A_{\nu}\bigg{(}q_{\alpha}A_{\beta}-q_{\beta}A_{\alpha}\bigg{)}$
$\displaystyle=-4\epsilon^{\mu\nu\alpha\beta}a(k)p_{\mu}A_{\nu}(p)q_{\alpha}A_{\beta}(q)$
(3.2)

To get the vertex function from here, we take

$\dfrac{(-i)^{3}\delta^{3}}{\delta a(k)\delta A^{\nu}(p)\delta A^{\beta}(q)},$

This gives us the final expression (and after multiplying back in the factor
of $\frac{1}{4}g_{a\gamma}$), which is

$ig_{a\gamma}\epsilon^{\mu\nu\alpha\beta}q_{\alpha}p_{\beta}$ (3.3)

This is the vertex rule for the ALP-photon coupling, just as expected by
analogy to the $\pi^{0}\gamma\gamma$ interaction. I will use this vertex rule
to derive amplitudes for several key processes involving the photon coupling
in the next few sections.

First we discuss ALP decays $a\to\gamma\gamma$ through the $g_{a\gamma}$
coupling (Fig. 3.1).

$a$$\gamma$$\gamma$ Figure 3.1: ALP decays through $a\to\gamma\gamma$ via the
$g_{a\gamma}$ coupling.

The decay width is given by Eq. 2.13 which has a very strong cubic dependence
on the ALP mass. This decay is kinematically accessible at all masses, but
becomes only relevant for masses $m_{a}\gtrsim 10$ keV, otherwise the lifetime
will be greater than about the age of the universe and becomes to rare to be
observed easily.

### 3.1 Atomic Primakoff Scattering

The Primakoff scattering process has been made famous in the search for axions
and ALPs. Originally named after $\pi^{0}\to\gamma$ photoconversion process
[63, 64], ALP Primakoff scattering is identical at the operator level and was
investigated in the context of $a\to\gamma$ conversion through the coherent
scattering with the strong atomic electric field of the nucleus and the
electron cloud [65]. The coherent scattering with the entire atomic and
nuclear charge density via a $t$-channel photon means that the cross section
enjoys a proportionality to the square of the atomic number, $\propto Z^{2}$.
But this is not the only form of coherence! Primakoff conversion may also
sometimes refer to the same process taking place in the presence of a
macroscopic electric or magnetic field, where in each case the coherency is
still manifest but on different length scales [66]. These make for the design
principles behind resonant cavity haloscopes and helioscopes look for axions
in the dark matter halo or those that might be free-streaming from the sun
converting into electromagnetic signals on the Earth. In this thesis, though,
I will contrast these search methods with a focus on atomic coherence, and
later, Bragg or Laue coherence at the level of ordered crystals.

$a$$\gamma$$Z$$Z$$\gamma$$a$$Z$$Z$

Figure 3.2: Primakoff scattering (left) and inverse-Primakoff scattering
(right) from an atom of atomic number $Z$.

The matrix element for Primakoff scattering with a free, heavy fermion
$a(k)N(p)\to\gamma(k^{\prime})N(p^{\prime})$ with momentum transfer
$q=k-k^{\prime}$, can be written down using the Feynman rules;

$\mathcal{M}_{\textrm{free}}=\bar{u}(p^{\prime})(-ie\gamma^{\mu})u(p)\bigg{(}\dfrac{-ig_{\mu\nu}}{q^{2}}\bigg{)}(ig_{a\gamma}\epsilon^{\nu\rho\alpha\beta}q_{\alpha}k^{\prime}_{\beta})\varepsilon^{*}_{\rho}(k^{\prime})$
(3.4)

Squaring and evaluating the trace in terms of the usual Mandelstam variables
$t=-q^{2}$ and $s=(k+p)^{2}$, and averaging over initial state spins, yields

$\braket{}{\mathcal{M}_{\textrm{free}}}{{}^{2}}=\frac{8\pi\alpha
g_{a\gamma}^{2}}{t^{2}}\bigg{[}m_{a}^{2}t\left(M^{2}+s\right)-t\left(\left(s-M^{2}\right)^{2}+st\right)-M^{2}m_{a}^{4}-\frac{1}{2}t\left(t-m_{a}^{2}\right)^{2}\bigg{]}$
(3.5)

To apply this to a real atomic target, we can factorize the free matrix
element and the atomic form factor separately to compute the interaction with
the nuclear and electron cloud charge density in a straightforward way, as

$\braket{\mid\mathcal{M}\mid^{2}}=\braket{\mid\mathcal{M}\mid^{2}}_{\textrm{free}}\times
F^{2}(q)$ (3.6)

For a hydrogenic potential, we use Tsai’s parameterization [65] of the atomic
form factor $F^{2}(t)$,

$F_{A}^{2}(q)=\dfrac{Z^{2}a^{4}q^{4}}{(1+a^{2}q^{2})^{2}}$ (3.7)

with $a=184.15e^{-1/2}Z^{-1/3}/m_{e}$. For more context about how this form
factor is constructed, I’ve added some notes in Appendix A.

In the forward limit, for an axion of momentum $k_{a}$, the inverse Primakoff
cross section is given by [67, 68, 69]

$\sigma(k_{a})=\dfrac{Z^{2}\alpha
g_{a\gamma}^{2}}{2}\bigg{(}\dfrac{2r_{0}^{2}k_{a}^{2}+1}{4r_{0}^{2}k_{a}^{2}}\ln\Big{(}1+4r_{0}^{2}k_{a}^{2}\Big{)}-1\bigg{)}$
(3.8)

Primakoff scattering is a very forward process; since the momentum transfer is
very small, the typical scattering angles of the outgoing photon or ALP
relative to the incoming momentum are typically small. This changes, however,
as the ALP becomes non-relativistic, as shown in Fig. 3.3.

Figure 3.3: Primakoff scattering ($\gamma A\to aA$) differential cross section
$d\sigma/d\theta$ as a function of the scattering angle $\theta$ with
$g_{a\gamma}=1$ MeV-1 and $Z=1$, on a log-linear graph (top) and log-log
(bottom).

## 4\. AXION-CRYSTAL INTERACTIONS

Axion-like particles in the keV to sub-eV mass range produced in the sun are
well motivated [70, 28, 27]. Searches were carried out by several experimental
collaborations by looking for $a\to\gamma$ Primakoff conversion in solid
crystal detectors, including DAMA [71] (NaI), CUORE [72, 73] (TeO2),
Edelweiss-II [74], SOLAX [75], COSME [76], CDMS [77], and Majorana [78] (Ge).
Other upcoming experiments like SuperCDMS [4], LEGEND [5], and SABRE [6] 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 coherency 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 Buchmuller & Hoogeveen [79] using the Darwin theory of
classical x-ray diffraction under the Bragg condition [80]. 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
[81] 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 [82],
theoretically explained by Zachariasen [83, 84] and others later (Battermann
[85, 86], Hirsch [87]). A quantum mechanical treatment was offered by Biagini
[88, 89] in which the Borrmann effect was explained by means of a statistical
treatment of the $\ket{\alpha}$ and $\ket{\beta}$ Bloch waves and their mutual
interference at the occupation number level. More recently, another treatment
of the Borrmann effect in the quantum limit was applied to study photon-photon
dissipation on Bragg-spaced arrays of superconduncting qubits [90].

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. [91, 92, 73] 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 [71, 72, 74, 75, 76, 77, 78]. However, absorption
effects in Bragg and Laue case diffraction were not considered in refs. [91,
92, 73]; 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. [79] and more recently ref.
[81]), 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 coherency. 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 this section, I will re-derive the event rate formula for solar axion
Primakoff scattering under the Bragg condition with full volume coherence. I
will then motivate the inclusion of absorption effects and discuss the
anomalous enhancement to the absorption length under the Borrmann effect. I
will then 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 to set up the subsequent chapter

### 4.1 Crystal structure

For crystals with diamond structure (Ge, Si, C), we may have the diamond
cubic, also seen as the inter-penetrating face-centered cubic (FCC), as the
unit cell;

Figure 4.1: Unit cell for a diamond cubic or interfaced-FCC lattice, where the
Bravais basis vectors $\vec{a}_{i}$ and primitive basis vectors
$\vec{\alpha}_{j}$ are shown. In the case of FCC lattices like those for NaI
or CsI crystals, the Na/Cs atoms are shown in blue and the I atoms shown in
red, while for diamond cubics like C, Ge, or Si, each scattering center is an
identical atomic species.

The primitive basis vectors just describe the first two atoms on the bottom-
left of Figure 4.1, left; these are

$\displaystyle\vec{\alpha}_{0}$ $\displaystyle=(0,0,0)$
$\displaystyle\vec{\alpha}_{1}$ $\displaystyle=\frac{a}{4}(1,1,1)$ (4.1)

while the basis vectors of the Bravais lattice are described by $\vec{a}_{1}$,
$\vec{a}_{2}$, and $\vec{a}_{3}$ in Figure 4.1, right:

$\displaystyle\vec{a}_{1}$ $\displaystyle=\frac{a}{2}(0,1,1)$
$\displaystyle\vec{a}_{2}$ $\displaystyle=\frac{a}{2}(1,0,1)$
$\displaystyle\vec{a}_{3}$ $\displaystyle=\frac{a}{2}(1,1,0)$ (4.2)

we can translate anywhere on the lattice by stepping in integer multiples of
these basis vectors;

$\vec{r}=n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3}$ (4.3)

We can then introduce the reciprocal lattice, which is like the Fourier
transform of the real space lattice basis vectors. They reciprocal lattice
basis vectors $\vec{b}_{i}$ satisfy
$\vec{b}_{i}\cdot\vec{a}_{j}=2\pi\delta_{ij}$. In general the transformations
give

$\displaystyle\vec{b}_{1}$
$\displaystyle=2\pi\dfrac{\vec{a}_{2}\times\vec{a}_{3}}{|\vec{a}_{1}\cdot(\vec{a}_{2}\times\vec{a}_{3})|}$
$\displaystyle\vec{b}_{2}$
$\displaystyle=2\pi\dfrac{\vec{a}_{3}\times\vec{a}_{1}}{|\vec{a}_{1}\cdot(\vec{a}_{2}\times\vec{a}_{3})|}$
$\displaystyle\vec{b}_{3}$
$\displaystyle=2\pi\dfrac{\vec{a}_{1}\times\vec{a}_{2}}{|\vec{a}_{1}\cdot(\vec{a}_{2}\times\vec{a}_{3})|}$
(4.4)

for our diamond cubic, we have

$\displaystyle\vec{b}_{1}$ $\displaystyle=\dfrac{2\pi}{a}(-1,1,1)$
$\displaystyle\vec{b}_{2}$ $\displaystyle=\dfrac{2\pi}{a}(1,-1,1)$
$\displaystyle\vec{b}_{3}$ $\displaystyle=\dfrac{2\pi}{a}(1,1,-1)$ (4.5)

One can check again in the figure that in real space, the reciprocal basis
vectors correspond to momentum (Bragg condition) that point along the surface
normals of the surfaces defined by the lattice basis vectors; the reciprocal
vectors are momenta that point normal to the lattice planes. Now use these
basis vectors to write any reciprocal lattice vector in terms of Miller
indices $h,k,l\in\mathbb{Z}$;

$\vec{G}=h\vec{b}_{1}+k\vec{b}_{2}+l\vec{b}_{3}$ (4.6)

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

$\vec{G}(hkl)=\dfrac{2\pi}{a}(h,k,l)$ (4.7)

The lattice constants, cell volumes, and basis vectors for a few examples (Ge,
Si, CsI, and NaI) are listed in Table 4.1.

Material |  Lattice Constant $a$ (Å) |  Cell Volume $v_{\textrm{cell}}$ (Å3) |  Primitive Basis |  Bravais Basis | $\vec{G}\times a/2\pi$
---|---|---|---|---|---
Ge (Diamond Cubic) | 5.657 | 181.0 |  $\vec{\alpha}_{0}=(0,0,0)$ $\vec{\alpha}_{1}=\frac{a}{4}(1,1,1)$ |  $\vec{a}_{1}=\frac{a}{2}(1,0,1)$ $\vec{a}_{2}=\frac{a}{2}(0,1,1)$ $\vec{a}_{3}=\frac{a}{2}(1,1,0)$ |  $(m_{1}-m_{2}+m_{3},$ $-m_{1}+m_{2}+m_{3},$ $m_{1}+m_{2}-m_{3})$
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 4.1: Lattice information for typical crystal detector technologies, FCC and diamond cubic. $hkl$ | $\sin\theta/\lambda$ | $|S(hkl)|$ | $1/|G|^{2}$ | $(|S(hkl)|/|\vec{G}|)^{2}$
---|---|---|---|---
111 | 0.1530891645 | 5.7 | 0.2702037463 | 8.778919716
022 | 0.3535442814 | 8 | 0.05066320242 | 3.242444955
202 | 0.3535442814 | 8 | 0.05066320242 | 3.242444955
220 | 0.3535442814 | 8 | 0.05066320242 | 3.242444955
113 | 0.3852659487 | 5.7 | 0.04266374941 | 1.386145218
131 | 0.3852659487 | 5.7 | 0.04266374941 | 1.386145218
311 | 0.3852659487 | 5.7 | 0.04266374941 | 1.386145218
133 | 0.4592674936 | 5.7 | 0.03002263847 | 0.975435524
313 | 0.4592674936 | 5.7 | 0.03002263847 | 0.975435524
331 | 0.4592674936 | 5.7 | 0.03002263847 | 0.975435524
333 | 0.4592674936 | 5.7 | 0.03002263847 | 0.975435524
224 | 0.4999871177 | 8 | 0.02533160121 | 1.621222478
242 | 0.4999871177 | 8 | 0.02533160121 | 1.621222478
422 | 0.4999871177 | 8 | 0.02533160121 | 1.621222478
004 | 0.6123566581 | 8 | 0.01688773414 | 1.080814985
040 | 0.6123566581 | 8 | 0.01688773414 | 1.080814985
400 | 0.6123566581 | 8 | 0.01688773414 | 1.080814985
444 | 0.6123566581 | 8 | 0.01688773414 | 1.080814985
335 | 0.6312027955 | 5.7 | 0.01589433802 | 0.5164070421
353 | 0.6312027955 | 5.7 | 0.01589433802 | 0.5164070421
533 | 0.6312027955 | 5.7 | 0.01589433802 | 0.5164070421
115 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
135 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
151 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
153 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
315 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
351 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
511 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
513 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
531 | 0.6789062885 | 5.7 | 0.01373917354 | 0.4463857483
044 | 0.7070885628 | 8 | 0.01266580061 | 0.8106112388
404 | 0.7070885628 | 8 | 0.01266580061 | 0.8106112388
440 | 0.7070885628 | 8 | 0.01266580061 | 0.8106112388
355 | 0.723471166 | 5.7 | 0.01209867521 | 0.3930859574
535 | 0.723471166 | 5.7 | 0.01209867521 | 0.3930859574
553 | 0.723471166 | 5.7 | 0.01209867521 | 0.3930859574
555 | 0.7654458227 | 5.7 | 0.01080814985 | 0.3511567886
Table 4.2: Example values of $\sin\theta/\lambda$ for a diamond cubic lattice
(Ge) in terms of Miller indices $hkl$ and the structure factor that appears in
the event rate.

### 4.2 Laue and Bragg Conditions

For a real space coordinate $\vec{x}$, the Laue condition for diffraction
reads

$\exp(i\vec{x}\cdot\vec{G})=1$ (4.8)

or that $\vec{x}\cdot\vec{G}=2\pi N$ for $N\in\mathbb{Z}$. This implies that
$\vec{q}=\vec{G}$.

The Laue Condition: $\displaystyle\,\,\,\vec{q}=\vec{G}$ (4.9)

A special case of the Laue condition is the Bragg condition, and it comes from
imposing the elastic scattering condition. Rewrite Eq. 4.9 as follows;

$\displaystyle\vec{G}$ $\displaystyle=\vec{q}=\vec{k}_{f}-\vec{k}_{i}$
$\displaystyle\vec{k}_{i}$ $\displaystyle=\vec{k}_{f}-\vec{G}$
$\displaystyle|\vec{k}_{i}|^{2}$ $\displaystyle=|\vec{k}_{f}-\vec{G}|^{2}$
$\displaystyle|\vec{k}_{i}|^{2}$
$\displaystyle=|\vec{k}_{f}|^{2}+|\vec{G}|^{2}-2\vec{k}_{f}\cdot\vec{G}$
(4.10)

The Bragg condition is a statement that we are in the elastic scattering
regime; $|\vec{k}_{i}|=|\vec{k}_{f}|$. From Eq. 4.2 this implies

The Bragg Condition (Version I)
$\displaystyle\,\,\,2\vec{k}_{f}\cdot\vec{G}=|\vec{G}|^{2}$ (4.11)

This can be recast in its more familiar form, taking $\sin\theta$ as the sine
of the angle between $\vec{G}$ and $\vec{k}_{f}$,
$|\vec{k}_{f}|=2\pi/\lambda$, and $|\vec{G}|=2\pi n/d$:

The Bragg Condition (Version II) $\displaystyle\,\,\,2d\sin\theta=n\lambda$
(4.12)

Lastly, there is one more way to represent the Bragg condition which has shown
up in various ALP-crystal scattering papers, and it follows easily from Eq.
4.11:

The Bragg Condition (Version III)
$\displaystyle\,\,\,E_{i}=\dfrac{|\vec{G}|^{2}}{2\hat{u}\cdot\vec{G}}$ (4.13)

It should be noted that the Bragg and Laue conditions are different than
Bragg-case and Laue-case diffraction. The terms “Bragg-case" and “Laue-case"
diffraction (or conversion in the case of ALP-photon interactions) have to do
with the orientation of the crystal geometry with respect to the incident
wave.

Figure 4.2: Crystallographic planes cut by the $hkl=220$ (left), 224 (middle),
and 333 (right) reciprocal lattice vectors in CsI.

### 4.3 Event Rates for Coherent Bragg-Primakoff Scattering from Solar Axions

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 $\vec{k}^{\prime}$;

$f=\braket{\mathcal{M}_{\textrm{free}}}F_{A}(q)$ (4.14)

where $F_{A}$ is the atomic form factor given in Eq. 3.7 and
$\mathcal{M}_{\textrm{free}}$ is the single-atomic scattering amplitude, Eq.
3.5 taken in the relativistic limit

$\braket{\mathcal{M}_{\textrm{free}}}=\dfrac{4e^{2}g_{a\gamma}^{2}}{t^{2}}E_{\gamma}^{2}m_{N}^{2}k^{2}\sin^{2}\theta$
(4.15)

Similar to the approach illustrated by Bednyakov and Naumov to get the total
coherent amplitude [93] for neutrinos scattering over $N$ scattering centers
in a nucleus, we sum over the $N$ scattering centers in a crystal;

$\mathcal{M}(\vec{k},\vec{k}^{\prime})=\sum_{j=1}^{N}f_{j}(\vec{k},\vec{k}^{\prime})e^{i(\vec{k}^{\prime}-\vec{k})\cdot\vec{r}_{j}}$
(4.16)

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 $\alpha_{j}$ while the Bravais lattice vectors are described by
$\vec{a}_{1}$, $\vec{a}_{2}$, and $\vec{a}_{3}$. We can then represent any
scattering site as a linear combination of the $a$’s and either the first or
second primitive;

$\displaystyle\vec{r}_{i,0}$
$\displaystyle=\vec{R}_{i}+\vec{\alpha}_{0}=n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3}+\vec{\alpha}_{0}$
(4.17) $\displaystyle\vec{r}_{i,1}$
$\displaystyle=\vec{R}_{i}+\vec{\alpha}_{1}=n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3}+\vec{\alpha}_{1}$
(4.18)

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

$\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}=\sum_{i=1}^{N}\mid
f_{i}\mid^{2}+\sum_{j\neq
i}^{N}\sum_{i=1}^{N}f_{j}^{\dagger}f_{i}e^{-i\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})}$
(4.19)

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

$\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}=\sum_{j\neq
i}^{N_{c}}\sum_{i=1}^{N_{c}}\sum_{\mu=0}^{1}\sum_{\nu=0}^{1}f_{j}^{\dagger}f_{i}e^{-i\vec{q}\cdot(\vec{R}_{i}-\vec{R}_{j}+\vec{\alpha}_{\mu}-\vec{\alpha}_{\nu})}$
(4.20)

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;

$|\mathcal{M}|^{2}\equiv\sum_{j\neq
i}^{N_{c}}\sum_{i=1}^{N_{c}}\sum_{\mu,\nu=0}^{1}f_{j}^{\dagger}f_{i}e^{-i\vec{G}\cdot(\vec{\alpha}_{\mu}-\vec{\alpha}_{\nu})}$
(4.21)

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;

$|\mathcal{M}|^{2}=N_{c}^{2}f^{\dagger}f\sum_{\mu,\nu=0}^{1}e^{-i\vec{G}\cdot(\vec{\alpha}_{\mu}-\vec{\alpha}_{\nu})}$
(4.22)

Now the structure function is nothing but the sum over primtives;

$S(\vec{G})=\sum_{\mu}e^{i\vec{G}\cdot\alpha_{\mu}}$ (4.23)

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

$\displaystyle|\mathcal{M}|^{2}$
$\displaystyle=N_{c}^{2}|\mathcal{M}_{\textrm{free}}|^{2}|F_{A}(\vec{G})S(\vec{G})|^{2}$
(4.24)

Now let’s write down the cross section. From the Lorentz-invariant phase space
element we have

$d\sigma=\dfrac{1}{4E_{a}m_{N}v_{a}}|\mathcal{M}|^{2}\dfrac{d^{3}k^{\prime}}{(2\pi)^{3}2E_{\gamma}}\dfrac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}(2\pi)^{4}\delta^{4}(k+p-k^{\prime}-p^{\prime})$
(4.25)

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

$d\sigma=\dfrac{1}{64\pi^{2}E_{a}E_{\gamma}m_{N}^{2}}|\mathcal{M}|^{2}d^{3}k^{\prime}\delta(E_{a}-E_{\gamma})$
(4.26)

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;

$\int d^{3}q\to\frac{(2\pi)^{3}}{V}\sum_{\vec{G}}$ (4.27)

The event rate formula is constructed from a convolution of the detector
response (taken as a gaussian smearing function between true energy
$E_{\gamma}$ and electron-equivalent energy $E_{ee}$, with width
$\Delta\sim\mathcal{O}(1)$ keV), axion flux $\Phi_{a}$, and cross section;

$R=\int_{E_{1}}^{E_{2}}dE_{ee}\int_{0}^{\infty}dE_{a}\frac{(2\pi)^{3}}{V}\sum_{\vec{G}}\dfrac{d\Phi_{a}}{dE_{a}}\dfrac{1}{64\pi^{2}E_{a}E_{\gamma}m_{N}^{2}}|\mathcal{M}|^{2}\delta(E_{a}-E_{\gamma})\cdot\bigg{(}\dfrac{1}{\Delta\sqrt{2\pi}}e^{-(E_{ee}-E_{\gamma})^{2}/2\Delta^{2}}\bigg{)}$
(4.28)

Putting in the definition of $|\mathcal{M}|^{2}$ that we worked out and
substituting the free Primakoff cross section in the relativistic limit (Eq.
4.15), integrating over the energy delta function (and identifying
$E_{a}=E_{\gamma}=k$ for simplicity), and integrating over $dE_{ee}$ we arrive
finally at the solar axion event rate formula under full volume coherence,

$\boxed{R=\dfrac{(2\pi)^{3}e^{2}g_{a\gamma}^{2}}{8\pi^{2}}\dfrac{V}{v_{c}^{2}}\sum_{\vec{G}}\dfrac{d\Phi_{a}}{dE}\dfrac{k^{2}\sin^{2}(2\theta)}{|\vec{G}|^{4}}|F_{A}(\vec{G})S(\vec{G})|^{2}\mathcal{W}(E_{1},E_{2},E)}$
(4.29)

The event rate in Eq. 4.29 encodes the effect of detector energy resolution
$\Delta$ within the function $\mathcal{W}$ resulting from the $dE_{ee}$
integral;

$\mathcal{W}(E_{a},E_{1},E_{2},\Delta)=\frac{1}{2}\bigg{(}\textrm{erf}\bigg{(}\frac{E_{a}-E_{1}}{\sqrt{2}\Delta}\bigg{)}-\textrm{erf}\bigg{(}\frac{E_{a}-E_{2}}{\sqrt{2}\Delta}\bigg{)}\bigg{)}$
(4.30)

One can see the tabulated values of the structure factor and corresponding
miller indices $h,k,l$ in Table 4.2, as well as the decreasing values of
$(S(h,k,l)/|\vec{G}|)^{2}$ as they appear in the $\vec{G}$ sum, shown there
for a Ge diamond cubic lattice for reference.

Eq. 4.29 is almost identical to the rate in ref. [91], 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. 4.29 is still different than that
given in ref. [91] up to a factor of $4\sin^{2}(\theta)\sim\mathcal{O}(1)$,
although the event rate formula derived here is consistent with the
calculation performed in refs. [73, 72].

However, the most important aspect that has been neglected up until now is the
attenuation of the final state photon in dielectric that will alter and
suppress the coherence that we have relied on so far. In this next section I
will address this issue.

### 4.4 Coherence and Absorption

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 $\vec{k}^{\prime}$. Similar to the approach
illustrated by Bednyakov and Naumov to get the total coherent amplitude [93]
for neutrinos scattering over $N$ scattering centers in a nucleus, we sum over
the $N$ scattering centers in a crystal;

$\mathcal{M}(\vec{k},\vec{k}^{\prime})=\sum_{j=1}^{N}f_{j}(\vec{k},\vec{k}^{\prime})e^{i(\vec{k}^{\prime}-\vec{k})\cdot\vec{r}_{j}}$
(4.31)

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. If we
square this, we get

$\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}=\sum_{i=1}^{N}\mid
f_{i}\mid^{2}+\sum_{j\neq
i}^{N}\sum_{i=1}^{N}f_{j}^{\dagger}f_{j}e^{-i\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})}$
(4.32)

taking $\vec{q}\equiv\vec{k}-\vec{k}^{\prime}$. Demanding the Laue diffraction
condition, $\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})=2\pi n$ for
$n\in\mathbb{Z}$, then the phase factor in the exponential goes to one. In
this limit, the diagonal (first) term is subdominant and the final matrix
element squared tends to $\mathcal{M}^{2}\to N^{2}f^{2}$.

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,

$\vec{k}^{\prime}\to\bar{n}\vec{k}^{\prime},\,\,\,\bar{n}=n-i\kappa,$ (4.33)

where $\bar{n}$ is the complex index of refraction. Making this modification,
we have

$\displaystyle e^{i\bar{n}\vec{k}^{\prime}\cdot(\vec{r}_{i}-\vec{r}_{j})}$
$\displaystyle\to
e^{in\vec{k}^{\prime}\cdot(\vec{r}_{i}-\vec{r}_{j})}e^{-\frac{\mu}{2}|\hat{k}^{\prime}\cdot(\vec{r}_{j}-\vec{r}_{i})|}$
(4.34)

In the last line above, the absorption factor $\mu\equiv 2\kappa|\vec{k}|$
makes its appearance. Equivalently, we can write the standard absorption
coefficient as the product of the total photon absorption cross section
multiplied by number density of the material in transport, $\mu=n\sigma$, and
relates to the mean free path of the photon $\lambda\equiv 1/\mu$.
Conceptually, this factor encodes the effect of a reduced coherent
interference amplitude between any two scattering centers, since a photon
plane wave originated at one scattering center will have been attenuated after
reaching another scattering center.

We note that Eq. 4.33 is a heuristic choice, since the attenuated plane wave
solution is not a true eigenstate of the interaction Hamiltonian, but rather a
simple ansatz made to capture 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
$\lambda=1/\mu$, taking the real part of the index of refraction $\simeq 1$.
We then have

$\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}=\sum_{i=1}^{N}\mid
f_{i}\mid^{2}+\sum_{j\neq
i}^{N}\sum_{i=1}^{N}f_{j}^{\dagger}f_{j}e^{-i\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})}e^{-z_{ij}/(2\lambda)}$
(4.35)

At this stage evaluating the sum is tricky, so I will first give a rough
estimation for the suppression factor of the modified coherent sum. Again, if
we apply the Laue diffraction condition
$\vec{q}\cdot(\vec{r}_{i}-\vec{r}_{j})=2\pi n$ and if the scattering centers
are all identical ($f_{j}=f_{i}$), then

$\displaystyle\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}$
$\displaystyle\simeq f^{\dagger}f\sum_{j\neq
i}^{N}\sum_{i=1}^{N}e^{-z_{ij}/(2\lambda)}$
$\displaystyle=f^{\dagger}f\sum_{j\neq
i}^{N}\int_{\textrm{Crystal}}d^{3}r\,e^{-z_{j}/(2\lambda)}\sum_{i=1}^{N}\delta^{3}(\vec{r}-\vec{r}_{i})$
(4.36)

Then going to the continuum limit, we have

$\displaystyle\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}$
$\displaystyle\simeq f^{\dagger}f\sum_{j\neq i}^{N}\dfrac{N}{V}\int
d^{3}r\,e^{-z_{j}/(2\lambda)}$ $\displaystyle\simeq f^{\dagger}f\sum_{j\neq
i}^{N}\dfrac{N}{V}\int_{0}^{L_{x}}\int_{0}^{L_{y}}\int_{0}^{L_{z}}dxdydz\,e^{-|z-\hat{z}\cdot\vec{r}_{j}|/(2\lambda)}$
$\displaystyle\simeq f^{\dagger}f\dfrac{L_{x}L_{y}N}{V}\sum_{j\neq
i}^{N}\int_{0}^{L_{z}}dz\,e^{-|z-\hat{z}\cdot\vec{r}_{j}|/(2\lambda)}$ (4.37)

Above, we used the fact that for Primakoff forward scattering,
$\hat{k}\simeq\hat{k}^{\prime}$, so
$z_{ij}\simeq\mid\hat{z}\cdot(\vec{r}_{i}-\vec{r}_{j})\mid$, choosing the
direction of the incoming ALP momentum $\hat{k}=\hat{z}$ without loss of
generality.

Let us use the definition $z_{0}\equiv\hat{z}\cdot\vec{r}_{j}$, then we see

$\displaystyle\int_{0}^{L_{z}}e^{-|z-\hat{z}\cdot\vec{r}_{j}|/(2\lambda)}dz=\int_{0}^{L_{z}}e^{-|z-z_{0}|/(2\lambda)}$
(4.38) $\displaystyle=$
$\displaystyle\int_{0}^{z_{0}}e^{(z-z_{0})/(2\lambda)}dz+\int_{z_{0}}^{L_{z}}e^{-(z-z_{0})/(2\lambda)}dz$
(4.39) $\displaystyle=$
$\displaystyle\lambda-\frac{\lambda}{2}\left(e^{-z_{0}/(2\lambda)}+e^{-(L_{z}-z_{0})/(2\lambda)}\right)$
(4.40)

For $L_{z}\gg\lambda$, this result varies from $\lambda$ at $z_{0}=0$ to a max
of $2\lambda$ at $z_{0}=L_{z}/2$, and back to $\lambda$ at $z_{0}=L_{z}$.
Taking the sum $\sum_{j\neq i}^{N}$ to the continuum limit, one finds that the
term in parentheses in Eq. 4.40 is $\mathcal{O}(\lambda^{2})<<L_{z}$.
Therefore, we have

$\displaystyle\mid\mathcal{M}(\vec{k},\vec{k}^{\prime})\mid^{2}$
$\displaystyle\gtrsim f^{\dagger}f\sum_{j\neq i}^{N}\dfrac{\lambda
L_{x}L_{y}N}{V}$ $\displaystyle\gtrsim
f^{\dagger}fN^{2}\dfrac{\lambda}{L_{z}}$ (4.41)

Comparing Eq. 4.37 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 $\lambda/L_{z}$.

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

$I\equiv\sum_{j\neq i}^{N}\sum_{i=1}^{N}e^{-z_{ij}/(2\lambda)},$ (4.42)

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. 4.42 explicitly. After rederiving the coherent
sum using the replacements in Eqns. 4.33, the event rate becomes

$\boxed{\dfrac{dN}{dt}=\dfrac{(2\pi)^{3}e^{2}g_{a\gamma}^{2}}{8\pi^{2}}\dfrac{V}{v_{\text{cell}}^{2}}\sum_{\vec{G}}I(\vec{k},\vec{G})\dfrac{d\Phi_{a}}{dE}\dfrac{k^{2}\sin^{2}(2\theta)}{|\vec{G}|^{4}}|F_{A}(\vec{G})S(\vec{G})|^{2}\mathcal{W}(E_{1},E_{2},E)}$
(4.43)

### 4.5 The Borrmann Effect

Yamaji et. al. [81] has found that for the Laue-case conversion of ALPs, the
attenuation length is modified as

$L_{\text{att}}\to L_{\alpha/\beta}\equiv
2L_{\text{att},\alpha/\beta}\bigg{(}1-\exp\bigg{(}-\dfrac{L}{2L_{\text{att},\alpha/\beta}}\bigg{)}\bigg{)}$
(4.44)

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:

$\epsilon\equiv\dfrac{\text{Im}\\{F(\vec{G})\\}}{\text{Im}\\{F(\vec{0})\\}}$
(4.45)

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. The notes below
will attempt to explain this further.

The total scattering form factor can be decomposed into the real and imaginary
parts [94];

$f=f^{0}+\Delta f^{\prime}+i\Delta f^{\prime\prime}$ (4.46)

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

$f^{0}(q)\equiv\int d^{3}\vec{x}\rho(\vec{x})e^{i\vec{q}\cdot\vec{x}}$ (4.47)

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. There is a useful
relationship between the imaginary part and the photoelectric absorption cross
section;

$\Delta f^{\prime\prime}(E)=\dfrac{E\sigma_{\text{PE}}(E)}{2\hbar cr_{e}}$
(4.48)

The anomalous absorption due to the Borrmann effect modifies the absorption
coefficient $\mu_{0}=1/\lambda$ as

$\frac{1}{\lambda_{\text{eff}}}=\mu_{\text{eff}}=\mu_{0}\bigg{[}1-\dfrac{F^{\prime\prime}(hkl)}{F^{\prime\prime}(000)}\bigg{]}$
(4.49)

Here $F^{\prime\prime}(hkl)$ is the combination of structure function and
imaginary form factor, $F^{\prime\prime}(hkl)=S(hkl)\Delta f^{\prime\prime}$
[85, 86]. The ratio in the second term of the expression is usually denoted as
$\epsilon$111In Yamaji et al [81], they use $\kappa$.

$\displaystyle\epsilon$
$\displaystyle\equiv\dfrac{F^{\prime\prime}(hkl)}{F^{\prime\prime}(000)}$
$\displaystyle=\dfrac{e^{-M}\sum_{j}Z_{j}\eta_{j}f_{j}^{0}(hkl)}{\sum_{j}Z_{j}\eta_{j}}\cdot\dfrac{S(h,k,l)}{S(0,0,0)}$
$\displaystyle=\dfrac{e^{-M}\Delta
f^{\prime\prime}(hkl)}{\sum_{j}Z_{j}\eta_{j}}\cdot\dfrac{S(h,k,l)}{S(0,0,0)}$
(4.50)

The exponential $e^{-M}$ is a Debye-Waller factor to account for thermal
effects and is usually close to 1.

Alternatively, we can calculate the imaginary form factor using the relation
in Wagenfield’s paper. 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 [95]

$\Delta
f^{\prime\prime}=\dfrac{\pi\hbar^{2}}{m_{e}}\bigg{(}\int\psi_{f}^{*}(\boldsymbol{r})\boldsymbol{\varepsilon}_{0}\cdot\boldsymbol{\nabla}e^{i\boldsymbol{k}\cdot\boldsymbol{r}}\psi_{i}(\boldsymbol{r})d^{3}r\bigg{)}\bigg{(}\int\psi_{f}(\boldsymbol{r})\boldsymbol{\varepsilon}^{\prime}_{0}\cdot\boldsymbol{\nabla}e^{-i\boldsymbol{k}^{\prime}\cdot\boldsymbol{r}}\psi_{i}^{*}(\boldsymbol{r})d^{3}r\bigg{)}$
(4.51)

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

$\Delta
f^{\prime\prime}=\dfrac{\pi\hbar^{2}}{m_{e}}\bigg{(}\boldsymbol{\varepsilon}_{0}\cdot\int\psi_{f}^{*}(\boldsymbol{r})e^{i\boldsymbol{k}\cdot\boldsymbol{r}}\boldsymbol{\nabla}\psi_{i}(\boldsymbol{r})d^{3}r\bigg{)}\bigg{(}\boldsymbol{\varepsilon}^{\prime}_{0}\cdot\int\psi_{f}(\boldsymbol{r})e^{-i\boldsymbol{k}^{\prime}\cdot\boldsymbol{r}}\boldsymbol{\nabla}\psi_{i}^{*}(\boldsymbol{r})d^{3}r\bigg{)}$
(4.52)

Referring to Catena et al [96], we can then apply the definition of the
vectorial form factor (eq B18, but with some changes made to keep the notation
more consistent),

$\boldsymbol{f}_{1\to 2}(\boldsymbol{q})=\int
d^{3}r\psi^{*}_{f}(\boldsymbol{r})e^{i\boldsymbol{q}\cdot\boldsymbol{r}}\frac{i\boldsymbol{\nabla}}{m_{e}}\psi_{i}(\boldsymbol{r}).$
(4.53)

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

$\displaystyle\Delta f^{\prime\prime}$
$\displaystyle=\dfrac{\pi\hbar^{2}}{m_{e}}\bigg{(}\boldsymbol{\varepsilon}_{0}\cdot(-im_{e})\boldsymbol{f}_{1\to
2}(\boldsymbol{k})\bigg{)}\bigg{(}\boldsymbol{\varepsilon}^{\prime}_{0}\cdot(im_{e})\boldsymbol{f}^{*}_{1\to
2}(\boldsymbol{k}^{\prime})\bigg{)}$
$\displaystyle=\pi\hbar^{2}m_{e}\bigg{(}\boldsymbol{\varepsilon}_{0}\cdot\boldsymbol{f}_{1\to
2}(\boldsymbol{k})\bigg{)}\bigg{(}\boldsymbol{\varepsilon}_{0}^{\prime}\cdot\boldsymbol{f}^{*}_{1\to
2}(\boldsymbol{k}^{\prime})\bigg{)}$ (4.54)

We can rewrite this expression with vector indices $i,j$ and take an average
over polarization vectors using $\sum\epsilon_{i}\epsilon_{j}=\delta_{ij}$;

$\displaystyle\Delta f^{\prime\prime}$
$\displaystyle=\pi\hbar^{2}m_{e}(\boldsymbol{\varepsilon}_{0})_{i}(\boldsymbol{f}_{1\to
2}(\boldsymbol{k}))_{i}(\boldsymbol{\varepsilon}_{0}^{\prime})_{j}(\boldsymbol{f}^{*}_{1\to
2}(\boldsymbol{k}^{\prime}))_{j}$
$\displaystyle=\pi\hbar^{2}m_{e}\boldsymbol{f}_{1\to
2}(\boldsymbol{k})\cdot\boldsymbol{f}^{*}_{1\to 2}(\boldsymbol{k}^{\prime})$
(4.55)

Then, in the Bragg limit $|\vec{k}|=|\vec{k}^{\prime}|$ and for spherically
symmetric wavefunctions we may take

$\Delta f^{\prime\prime}(k)=\pi\hbar^{2}m_{e}|\boldsymbol{f}_{1\to 2}(k)|^{2}$
(4.56)

The incoherent form factors were also calculated and reported by Freeman [97].

Now we can express the Borrmann parameter with the quadrupole component of the
imaginary form factor [95, 98, 99];

$\epsilon\equiv D\bigg{(}1-2\sin^{2}\theta_{B}\frac{\Delta
f^{\prime\prime}_{Q}}{\Delta
f^{\prime\prime}}\bigg{)}\frac{|S(h,k,l)|}{|S(0,0,0)|}$ (4.57)

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. [100] as well as fits to $\Delta
f^{\prime\prime}$ for several pure materials of interest.

Figure 4.3: The absorption factor $I(\vec{k},\vec{G},\lambda)$ as a function
of the mean free path $\lambda=1/\mu$ for a crystal of cubic volume with side
length 5 cm. Figure 4.4: Borrmann parameter $\epsilon$ as a function of the
momentum transfer $\sin\theta/\lambda=|\vec{G}|/4\pi$ for several crystal
materials.

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

$I(\vec{k},\vec{G},\lambda)\equiv\sum_{j\neq
i}^{N}\sum_{i=1}^{N}e^{-\frac{(\vec{k}-\vec{G})}{|\vec{k}-\vec{G}|}\cdot(\vec{r}_{i}-\vec{r}_{j}))/(2\lambda)}$
(4.58)

Taking the Ge lattice as an example, with lattice constant $d=5.657$ Å, we
evaluate $I(\vec{k},\vec{G},\lambda)$ 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 geoometry is enough to obtain a
convergent error of around 5%. Some evaluations of
$I(\vec{k},\vec{G},\lambda)$ as a function of varying mean free path $\lambda$
are shown in Fig. 4.3 for several choices of scattering planes $\vec{G}$ and
incoming wavevectors $\vec{k}$.

One interesting phenomenon that can be seen in Fig. 4.3 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;

$\displaystyle\hat{k}^{\prime}\cdot(\vec{r}_{i}-\vec{r}_{j})$
$\displaystyle=\bigg{(}\frac{\vec{G}}{2\vec{k}\cdot\hat{G}}-\frac{\vec{G}}{k}\bigg{)}\cdot(\vec{r}_{i}-\vec{r}_{j})=0$
(4.59)

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

$\theta=\cot^{-1}\left(\frac{n_{x}\cos(\phi)-n_{y}\sin(\phi)}{n_{z}}\right)+\pi
c_{1}$ (4.60)

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. 4.3. 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. [81].

We employ a numerical cutoff on the sum over reciprocal lattice planes
$\vec{G}(hkl)$ at $max(h,k,l)=6$, where a test for convergence of the event
rate shows that the corrections beyond add corrections less than 5% (see Fig.
4.5). Additionally, the values of the structure factor for a Ge diamond cubic
lattice are tabulated in Table 4.2, ordered by decreasing
$|S(hkl)|^{2}/|\vec{G}|^{2}$ as they appear in the event rate sum.

Figure 4.5: Left: convergence rate for the sum over $\vec{G}$ in the event
rate $R$ as a function of the cutoff $max(h,k,l)$. Right: convergence of the
event rate with the absorption sum $I(\vec{k},\vec{G})$ for a given simulated
lattice size.

In Fig. 4.6 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.
4.49. 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 4.6: 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 the 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.

## 5\. LABORATORY SEARCHES FOR SOLAR AXIONS AT DIRECT DETECTION EXPERIMENTS
*††footnotetext: * Parts of this chapter are adapted from ref. [1] with
permission from Adrian Thompson and co-authors.

### 5.1 Dark Matter Direct Detection Experiments

Dark matter direct detection experiments, initially designed to search for
WIMP-like dark matter, have been adapted more broadly as detectors of Beyond
Standard Model (BSM) physics. Notable among the wide class of BSM physics
searches at direct detection facilities is the extraordinary sensitivity to
possible axion or axion-like particles coupling to Standard Model particles
(SM) [101, 102, 103, 104, 105, 106, 107]. By examining electronic recoils
produced by a solar axion flux through the detector, these searches have
probed a variety of $a$-SM couplings including axion-electron, axion-photon,
and axion-nucleon interactions.

In ref. [1], we investigated inverse Primakoff scattering as a new detection
channel at liquid xenon based direct detection experiments. We showed that
sole use of the coupling $g_{a\gamma}$ can fit the recent XENON1T excess of
electron recoils in their low energy (1-30 keV) data, with a rise above the
background-only model occurring below 7 keV [108]. Although a null result was
later found with 1.16 tonne-years of exposure [109], we found at the time that
the fitting of the excess is free of the leading helioscope CAST constraint
for $m_{a}\gtrsim 0.03$ eV. However, even then the tension associated with the
astrophysical constraints, HB star cooling from $R$ parameter measurements,
was still 8$\sigma$ [110].

The more interesting prospect that emerged out of this analysis was that next-
generation xenon experiments were projected to overcome the HB stars limit,
and for $g_{ae}=10^{-13}$, the 2.4$\sigma$ hint region of stellar cooling can
be probed within $1\sigma$; see Fig. 5.3. In addition, these future bounds
would be applicable for masses $m_{a}<1$ keV, covering complementary regions
of parameter space (including that of KSVZ axions) for which future
helioscopes, such as IAXO, start to lose sensitivity near $m_{a}\gtrsim 0.01$
eV. A similar region of the $g_{a\gamma}-m_{a}$ space will also be
investigated at LZ [111] and SuperCDMS SNOLAB [112], where the reach for
$g_{a\gamma}$ needs to be scaled for the new detector type roughly by
$\sqrt{M_{D}Z^{2}_{D}/M_{Xe}Z^{2}_{Xe}}$ (where $M_{D}$ is the detector mass
and $Z_{D}$ is the atomic number of the detector nucleus) for the same
exposure.

There are three prominant sources of solar axion flux that we considered with
keV-scale energy spectra, each with a dependence on a different axion coupling
parameter. First is the the “ABC" flux, driving axion production from Atomic
de-excitation and recombination, Bremmstrahlung, and Compton scattering
processes [113], dependent on the $g_{ae}$ coupling. Next, the Primakoff
production process, $\gamma Z\to aZ$, occurs via the $g_{a\gamma}$ coupling
through the $t$-channel exchange of a virtual photon scattering with electrons
or ions in the solar interior [114, 115]. Finally, de-excitation of 57Fe in
the sun can produce a monoenergetic axion population at 14.4 keV [116, 117,
118]. This flux would arise from an effective axion-nucleon coupling $g_{\rm
an}^{\rm eff}=-1.19g_{\rm an}^{0}+g_{\rm an}^{3}$, where $g_{\rm an}^{0(3)}$
are the isoscalar (isovector) coupling constants for the nucleons [119, 118].
Each of these flux components are shown in Fig. 5.1.

Figure 5.1: Solar axion fluxes at the Earth’s surface are shown for the ABC,
Primakoff, and 57Fe components. The bulk shape of the ABC component is due to
axion-bremsstrahlung and Compton scattering, $e^{-}I\to e^{-}I\,a$ and
$e^{-}\gamma\to e^{-}a$, while the numerous peaks are due to atomic
transitions in $I^{*}\to Ia$ and $e^{-}I\to I^{-}a$. The Primakoff flux
exhibits a smooth thermal distribution from the coherent conversion of photons
into axions, while the 57Fe flux is monoenergetic and is expected to broaden
out from detector energy response effects. Reprinted from ref. [1] with
permission from Adrian Thompson and co-authors.

We considered inverse Primakoff scattering in addition to the axioelectric
absorption process outlined in the analysis performed by XENON1T (see also
[120, 121, 122, 118]). Also, it is possible that axions undergo inverse
Compton scattering off electrons at rest in LXe, $ae^{-}\rightarrow\gamma
e^{-}$ [67], but this is a subdominant process ($\propto Z$) in comparison to
axioelectric scattering ($\propto Z^{5}$). If both axion-photon and axion-
electron couplings are present, there are interference terms present in the
total matrix element of the combined processes, which are also subdominant,
but we included them as a matter of completeness.

To predict the event spectra from axions produced through ABC, Primakoff, and
57Fe, we convolved the fluxes in each case with the total cross sections, for
inverse Primakoff scattering or axioelectric absorption, and multiply by the
detector efficiency [108]. In addition, we approximated the detector response
for the energy resolution by convolving the simulated differential event
distribution with an energy-dependent Gaussian smearing function [108, 123,
124]. The event distribution for Primakoff-produced axions that undergo
inverse Primakoff scattering in the LXe fiducial volume over a tonne-year
exposure is shown in Figure 5.2.

Figure 5.2: The event rate distributions for inverse Primakoff scattering in
LXe for a tonne-year exposure from Primakoff-produced, ABC-produced, and 57Fe-
produced axions are shown for select choices of axion couplings, and added to
the “$B_{0}$" background model. Reprinted from ref. [1] with permission from
Adrian Thompson and co-authors.

In Fig. 5.3 (bottom), I show the next-generation xenon (G3 Xe) constraint
(with a 1 kilotonne-year exposure [125]) where we found that the 95% CL can
overcome even the HB stars constraint and start exploring the mild hint
(2.4$\sigma$) region of stellar cooling within 1$\sigma$. Interestingly, this
is only possible with the inclusion of the inverse Primakoff channel since
without this channel the constraint could be worse by a few orders of
magnitude. We also found that our projected sensitivity for a 1
kton$\cdot$year exposure at a G3 LXe experiment is competitive with future
helioscope experiments. The proposed DARWIN detector would achieve a 200
tonme-year exposure [126], thereby covering the current HB Stars constraint.
Comparing the 1 kt$\cdot$year projection against the projected sensitivities
for IAXO+ with masses $m_{a}>0.1$ eV, one can see where the sensitivity begins
to diminish for larger masses [58] and the direct detection experiments play
an important role as a broadband search over ALP masses. Additionally, future
direct detection experiments with directional sensitivity would be able to use
the directional information to reduce backgrounds and further increase their
sensitivity to solar axions. This is especially useful in the Primakoff
channel, where the axion’s incoming direction is approximately preserved due
to the low momentum transfer / forward scattering by the photon in the
relativistic limit.

Figure 5.3: Projections of the 95% CL future exclusions (using the Wigner-
Seitz FF in light gray and RHF form factor in dark grayWe have changed this
value of the screening radius from 2.45 Å [2] to 0.685 Å in accordance with
the more common expression for the Wigner-Seitz radius [3], which accounts for
at most a factor of $10$ reduction in the cross section at 1 keV down to a
factor of 3 reduction at 5 keV with respect to the old value.) set by G3 Xe
over a 1 kton$\cdot$year exposure given background-only observations. The
exclusion line for 1 kton$\cdot$year without inverse Primakoff (I.P.)
scattering is shown for comparison (dotted red). We also show the IAXO+
projection (blue) which begins to lose sensitivity for $m_{a}\gtrsim 0.01$ eV.
Reprinted from ref. [1] with permission from Adrian Thompson and co-authors.

### 5.2 Pushing the Sensitivity Envelope with Bragg-Primakoff Conversion

Using the event rate formula for Bragg-Primakoff coherent scattering with a
perfect crystal worked out in § 4, the event rate in an energy window
$[E_{1},E_{2}]$ is

$\displaystyle\dfrac{dN}{dt}=\pi g_{a\gamma}^{2}(\hbar
c)^{3}\dfrac{V}{v_{\textrm{cell}}^{2}}$
$\displaystyle\sum_{\vec{G}}I(\vec{k},\vec{G})\bigg{[}\dfrac{d\Phi_{a}}{dE_{a}}|F_{j}(\vec{G})S_{j}(\vec{G})|^{2}\frac{4(\hat{G}\cdot\hat{k})^{2}(1-(\hat{G}\cdot\hat{k})^{2})}{|\vec{G}|^{2}}\mathcal{W}\bigg{]}$
(5.1)

where $S_{j}$ is the crystal structure factor (see Ch. 4), $F_{j}$ 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 [28, 127,
128]. 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. 4.2.

For this analysis we may be also interested in axion masses that are heavier,
around $\sim$1-10 keV. For this we use the parameterization appearing in ref.
[128] for massive axion production in the sun; the flux parameterizations are
repeated here for convenience

$\displaystyle\dfrac{d\Phi_{\gamma\to a}}{dE_{a}}$
$\displaystyle=\dfrac{4.20\cdot
10^{10}}{\textrm{cm}^{-2}\textrm{s}^{-1}\textrm{keV}^{-1}}\bigg{(}\dfrac{g_{a\gamma}}{10^{-10}\textrm{GeV}^{-1}}\bigg{)}^{2}\dfrac{E_{a}p_{a}^{2}}{e^{E_{a}/1.1}-0.7}(1+0.02m_{a})$
(5.2) $\displaystyle\dfrac{d\Phi_{\gamma\gamma\to a}}{dE_{a}}$
$\displaystyle=\dfrac{1.68\cdot
10^{9}}{\textrm{cm}^{-2}\textrm{s}^{-1}\textrm{keV}^{-1}}\bigg{(}\dfrac{g_{a\gamma}}{10^{-10}\textrm{GeV}^{-1}}\bigg{)}^{2}m_{a}^{4}p_{a}\bigg{(}1+0.0006E_{a}^{3}+\frac{10}{E_{a}^{2}+0.2}\bigg{)}e^{-E_{a}}$
(5.3)

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. 5.4.

Figure 5.4: Solar axion fluxes from Primakoff conversion of photons and photon
coalescence are shown for several mass benchmarks.

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 [129].

The corresponding event rates for various energy windows are shown in Fig. 5.5
for Ge crystal, where we compare the relative enhancements with and without
the Borrmann effect to the case of full-volume coherency and to the case of
incoherent scattering on an amorphous lattice111Atomic Primakoff scattering is
still coherent here; only the coherency at the level of the lattice is lost
for the sake of comparison with scattering on amorphous materials.. 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 coherency assumption.

Figure 5.5: Solar ALP scattering rates in a 250 cm3 Ge crystal, comparing the
rates with full volume coherency 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 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. 5.6 (left). Depending on the time of year, different
sets of Bragg peaks will be traced over during the day, inducing a annual
modulation in addition to the intra-day modulation of the signal. 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. 5.6 (right).

Figure 5.6: Right: 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. Left: 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.

We forecast the event rates for SuperCDMS [4], LEGEND-200, LEGEND-1000, SABRE,
in addition to envisioned multi-ton setups, with detector specifications
listed in Table 5.1. For the background-free limits, we look for the Poisson
90% CL corresponding to $\simeq 3$ events observed for a given exposure. The
limits on the $(g_{a\gamma}-m_{a})$ parameter space are shown in Fig. 5.7,
where we show projections assuming full volume coherency without absorption
effects (top) as well as with absorption effects (bottom). In the case where
we assume full volume coherency, we forecast the multi-kg-scale SuperCDMS
setup to set the first laboratory based limits for $m_{a}>1$ eV beyond the
bounds set by XENONnT [130]. Ton-scale setups like LEGEND-200 and LEGEND-1000
can reach further, probing couplings up to the existing bounds fom HB Stars
[62, 60] and CAST [59] for masses $m_{a}\lesssim 10$ keV, losing sensitivity
for higher masses where the axion production rates from photon coalescence and
Primakoff scattering diminish (see also Fig. 5.4). These reach more than an
order of magnitude lower in the coupling than previous Bragg-Primakoff solar
axion searches, also shown here for DAMA [71], CUORE [72], Edelweiss-II [74],
SOLAX [75], COSME [76], CDMS [77], and Majorana [78]. The QCD axion parameter
space is shown here for the Kim-Shifman-Vainshtein-Zakharov (KSVZ) and Dine-
Fischler-Srednicki-Zhitnitsky (DFSZ) type benchmark models, where the range is
defined by taking the anomaly ratios of $E/N=44/3$ to $E/N=2$ [26], although
the space of heavier masses is also possible in high-quality axion models and
other scenarios, e.g. refs. [56, 131]. To probe this model parameter space
beyond the existing bounds from CAST and HB stars, multi-ton scale experiments
are needed.

However, with the effects of absorption included (Fig. 5.7, bottom), the
suppression of the event rate brings our projections for LEGEND, SuperCDMS and
SABRE to test parameter space already excluded by HB stars constraints.
However, the LEGEND-1000 background-free scenario is projected to have the
leading laboratory-based sensitivity beyond the existing limits from CAST and
XENONnT, but multi-ton CsI and NaI setups would extend this to nearly cover
the HB stars exclusion. The existing bounds from Edelweiss-II, COSME, SOLAX,
CDMS, DAMA, and Majorana 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 volume and detector material.
For the future projections, 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 only modest Borrmann
enhancements at the lower reciprocal lattice planes; see Fig. 4.4. In order to
push the sensitivity envelope beyond the current bounds by CAST and HB stars,
even with multi-ton setups, the absorption effects need to be mitigated in
order to recover the event rate sensitivity in full-volume coherence. Some
possibilities are discussed in the following section.

Experiment |  Module Mass $\times$no. Modules |  Total Mass |  Energy Resolution | Threshold | Exposure (tonne-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 ton | 2.5 keV | 1 keV | 4.16
SABRE (NaI) | 2 kg $\times$ 25 | 50 kg | 1 keV | 1 keV | 0.15
Ton-scale NaI | 2 kg $\times$ 2500 | 5 ton | 1 keV | 1 keV | 50
Ton-scale CsI | 2 kg $\times$ 2500 | 5 ton | 1 keV | 1 keV | 50
Table 5.1: Assumed detector parameters for the SuperCDMS [4], LEGEND [5], and
SABRE [6] configurations, as well as those taken for the NaI and CsI multi-
tonne benchmarks. Exposures are based on 4-5 years of active run time.

We forecast the event rates for SuperCDMS [4], LEGEND-200, LEGEND-1000, SABRE,
in addition to envisioned multi-tonne setups, with detector specifications
listed in Table 5.1. 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. 5.7, 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) and Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) type
benchmark models, where the range is defined by taking the anomaly ratios of
$E/N=44/3$ to $E/N=2$ [26], although the space of heavier masses is also
possible in high-quality axion models and other scenarios, e.g. refs. [56,
131]. To probe this model parameter space beyond the existing bounds from CAST
and HB stars, when full volume coherence is maintained, multi-tonne scale
experiments are needed.

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 [1]. 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 [132].

The existing bounds from DAMA [71], CUORE [72], Edelweiss-II [74], SOLAX [75],
COSME [76], CDMS [77], and Majorana [78] 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. Please see appendix C for the projected sensitivity and
existing constraints assuming full volume coherence. In fig. 5.7, 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. 4.4. 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 5.7: Sensitivity projections using the exposures listed in Table 5.1
for SuperCDMS, LEGEND-200, LEGEND-1000, SABRE, and multi-tonne setups with
absorption effects. The extension of these sensitivities to full volume
coherency are indicated with the arrows.

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. 4.6, 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. [133], 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. [133].

  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 [134]
or synchrotron sources in a similar fashion to LSW experiments [135, 136, 137,
138]. 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. 5.7, 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 [62, 60] and CAST [59] 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. 5.4). These reach more than an order of magnitude lower in the
coupling than previous Bragg-Primakoff solar axion searches.

Lastly, I will make some comments about extending this work to other important
phenomenological probes. First, one can additionally consider non-relativistic
ALPs, which may come from the dark matter halo, or from stellar basins like
those considered in ref. [139, 140]. In these cases one can think about a
modified Bragg-Primakoff condition, or instead a photoionization-like process
which can actually dominate the detection channels in the non-relativistic
regime [141]. For this second option, we investigated this possibility in an
alternative method to ref. [141] that again makes use of the DarkARC code to
compute the ionization amplitudes in inverse Primakoff photoionization; see
appendix B. Secondly, the anomalies observed in white dwarf (WD), red giant
(RG), and horizontal branch (HB) cooling data, known as the WD stellar cooling
hints, have shown to be explained by ALP-assisted stellar cooling that would
correspond to parameter space near the $g_{a\gamma}\lesssim 10^{-11}$ GeV-1
level and below, and for non-vanishing $g_{ae}\simeq 10^{-13}$ [60, 61, 62].
These are also shown in Fig. 5.7 by the gray band for the RG and HB cooling
hints explained at the $1\sigma$ level in $g_{a\gamma}$, which extends down at
the $2\sigma$ level to vanishing photon coupling, but can be explained at the
2$\sigma$ level by a non-vanishing electron coupling. Testing the ALP
explanation is a further motivator of probing this parameter space, and an
analysis that combines both electron and photon couplings in this context for
future crystal detectors would be able to directly test this anomaly.

## 6\. CONCLUSION

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 coherency 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-ton 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 factors to determine the anomalous absorption effect in materials other
than Ge is of particular interest for future work.

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. It was shown in ref. [1] that future liquid noble
gas detectors for dark matter direct detection at kilotonne-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
tonne-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 QCD axion parameter space and the
parameter space for heavier axion-like particle models.

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##  APPENDIX A COHERENT ATOMIC FORM FACTOR

The atomic form factor, in the original sense, is given by the Fourier
transform of the spatial parts of the initial and final state atomic wave
functions;

$F(q)=\int\psi^{*}_{\rm f}\psi_{\rm i}e^{i\vec{q}\cdot\vec{r}}d^{3}r$ (A.1)

This term can usually factor out of the entire matrix element, as long as
there are no spin-dependent pieces in the spatial wave functions, such that
the matrix element can be factorized as $(\text{Lorentz
structures})\times(\text{Form Factor})$. The exception to this rule is seen in
nucleon scattering which has separate form factors for the electric and
magnetic dipole moments, due to the different spin rules for each interaction.

For atomic scattering, typically one considers transitions from the ground
state wave function $\psi_{0}$ to an excited state $\psi_{E}$, or more simply,
in the case that there is no change to the spatial in- and out-states, only a
momentum transfer to the whole target. In that case, we can take

$\displaystyle F(q)$
$\displaystyle=\int|\psi|^{2}e^{i\vec{q}\cdot\vec{r}}d^{3}r$
$\displaystyle\propto\int\rho(\vec{r})e^{i\vec{q}\cdot\vec{r}}d^{3}r$ (A.2)

where $\rho$ is the mass or charge distribution that is proportional to the
wave function amplitude. There are two form factors used in ref. [81] which I
will discuss later;

$f_{\gamma}(\vec{q})\equiv\frac{1}{e}\int
d^{3}\vec{r}\rho(\vec{r})e^{i\vec{q}\cdot\vec{r}}$ (A.3) $f_{a}(\vec{q})\equiv
k_{a}^{2}\int d^{3}\vec{r}\phi(\vec{r})e^{i\vec{q}\cdot\vec{r}}$ (A.4)

The atomic form factor has the property

$f_{\gamma}(0)=Z$ (A.5)

which normalizes $\rho(r)$ to the total atomic charge, $eZ$. The axion form
factor, on the other hand, should be treated carefully. We can derive an
analog of the Mott-Bethe relation, which ordinarily relates the X-ray and
electron scattering form factors, to relate the X-ray and axion form factors
$f_{\gamma}$ and $f_{a}$, respectively. I use Bethe’s approach of inserting
the Laplacian acting on $e^{-\vec{q}\cdot\vec{r}}$ equation and integrating by
parts;

$\displaystyle f_{a}(\vec{q})$ $\displaystyle=k_{a}^{2}\int
d^{3}\vec{r}\phi(\vec{r})e^{i\vec{q}\cdot\vec{r}}$ (A.6)
$\displaystyle=-\dfrac{1}{q^{2}}k_{a}^{2}\int
d^{3}\vec{r}\phi(\vec{r})\nabla^{2}e^{i\vec{q}\cdot\vec{r}}$ (A.7)
$\displaystyle=-\dfrac{1}{q^{2}}k_{a}^{2}\int
d^{3}\vec{r}\nabla^{2}\phi(\vec{r})e^{i\vec{q}\cdot\vec{r}}$ (A.8)
$\displaystyle=\dfrac{k_{a}^{2}}{q^{2}}\int
d^{3}\vec{r}\Big{[}eZ\delta(r)-\rho(r)]\Big{]}e^{i\vec{q}\cdot\vec{r}}$ (A.9)
$\displaystyle=\dfrac{ek_{a}^{2}}{q^{2}}\Big{[}Z-f_{\gamma}(\vec{q})\Big{]}$
(A.10)

where I have used Poisson’s equation
$\nabla^{2}\phi(\vec{r})=-(eZ\delta(r)-\rho(r))$ in natural units, which
includes the nucleus charge density $eZ\delta(r)$ and the electron charge
distribution with $\int\rho(\vec{r})d^{3}\vec{r}=eZ$. The final expression in
Eq. A.10 agrees with [79].

Next, it was pointed out by [142] that while $f_{\gamma}(0)=Z$, $f_{a}$ does
not approach zero in the $q=0$ limit, as the Yamaji paper suggests. This is
because after expanding $f_{\gamma}(q)$ in powers of q, an even series
develops whose leading term is $Z$, cancelling the other $Z$ in the bracket,
and subleading term is $\propto q^{2}$ which gets cancelled by the $q^{2}$ in
the denominator of Eq. A.10. See below;

$\displaystyle f_{\gamma}(\vec{q})$ $\displaystyle=\frac{1}{e}\int
d^{3}\vec{r}\rho(\vec{r})e^{i\vec{q}\cdot\vec{r}}$ (A.11)
$\displaystyle=\frac{1}{e}\int\rho(\vec{r})\dfrac{\sin(qr)}{qr}r^{2}dr$ (A.12)
$\displaystyle=\frac{1}{e}\int\rho(\vec{r})\Big{(}1-\frac{(qr)^{2}}{3!}+\frac{(qr)^{4}}{5!}-\dots\Big{)}r^{2}dr$
(A.13)
$\displaystyle=Z\Big{(}1-\frac{q^{2}}{3!}\braket{r^{2}}+\frac{q^{4}}{5!}\braket{r^{4}}-\dots\Big{)}$
(A.14)

Therefore, inserting this expansion into Eq. A.10, we have

$f_{a}(q)=eZk_{a}^{2}\Big{(}\frac{1}{3!}\braket{r^{2}}-\frac{q^{2}}{5!}\braket{r^{4}}+\dots\Big{)}$
(A.15)

##  APPENDIX B PHOTOELECTRIC PRIMAKOFF IONIZATION

Consider the scattering of an ALP and an atomic system such that the energy
transfer of the incoming ALP ionizes the atom, ejecting one of the outer shell
electrons into a continuum final state. For the ALP-photon coupling
$aF\tilde{F}$, this can happen through a Primakoff-like process,

$a+A\to\gamma+A^{+}+e^{-}$ (B.1)

for initial and final 4-momenta $k_{a}$, $p_{e}$, $k_{\gamma}$,
$p_{e}^{\prime}$ of the ALP, initial state electron, final state photon, and
final state electron, respectively (see also Fig. B.1).

$\gamma$$a$$A^{+}$$A$$e^{-}$ Figure B.1: Tree-level ALP ionization through an
inverse Primakoff-like scattering with an atomic system.

Consider Primakoff scattering, but instead of an atomic coherence, we work in
the incoherent regime in which a target atom at rest is ionized by the
exchange photon. Define the free particle 4-momenta as follows.

$\displaystyle k_{a}^{\mu}\to(E_{a},\vec{k}_{a})$ (B.2) $\displaystyle
p_{e}^{\mu}\to(E_{e},\vec{p}_{e})$ $\displaystyle
k_{\gamma}^{\mu}\to(E_{\gamma},\vec{k}_{\gamma})$
$\displaystyle{p_{e}^{\prime}}^{\mu}\to(E_{e}^{\prime},\vec{p}_{e}^{\prime})$

We start with the $2\to 2$ differential cross section element and follow the
guidance of refs. [96, 143] for the following calculation.

$\displaystyle d\sigma=$
$\displaystyle\dfrac{1}{4E_{a}E_{e}v_{a}}\dfrac{d^{3}\vec{p}^{\prime}_{e}}{(2\pi)^{3}}\dfrac{1}{2E_{e}^{\prime}}\dfrac{d^{3}\vec{k}_{\gamma}}{(2\pi)^{3}}\dfrac{1}{2E_{\gamma}}|\mathcal{M}|^{2}(2\pi)^{4}\delta^{4}(k_{a}+p_{e}-p_{e}^{\prime}-k_{\gamma})$
(B.3)

The matrix element can be broken up as the point-wise matrix element combined
with the atomic electron matrix element. The electronic part will describe the
transition of an electron in the $(n,l)$ shell to a continuum state with
momentum $p_{e}^{\prime}$ and final state quantum numbers
$(n^{\prime},l^{\prime})$;

$|\mathcal{M}|^{2}=\int\dfrac{d^{3}p_{e}}{(2\pi)^{3}}\braket{\vec{e}_{2}}{{\vec{p}_{e}}^{\prime}}\braket{}{\mathcal{M}_{\text{free}}(k_{a},p_{e},k_{\gamma},p_{e}^{\prime})}{{}^{2}}\braket{\vec{p}_{e}}{\vec{e}_{1}}$
(B.4)

Here, $\braket{}{\mathcal{M}}{{}^{2}}$ is the matrix element for point-wise
Primakoff scattering:

$\braket{}{\mathcal{M}}{{}^{2}}=\dfrac{e^{2}g_{a\gamma}^{2}}{2t^{2}}[m_{a}^{2}t(m_{e}^{2}+s)-m_{a}^{4}m_{e}^{2}-t((s-m_{e}^{2})^{2}+st)-t(t-m_{a}^{2})/2],$
(B.5)

Next, we can remove the spatial part of the delta function by absorbing it
with the $d^{3}p_{e}^{\prime}$ integration, making use of the momentum
transfer $\vec{q}=\vec{p}_{e}-\vec{p}_{e}^{\prime}$;

$\displaystyle d\sigma=$
$\displaystyle\dfrac{V}{4E_{a}E_{e}v_{a}}(2\pi)\delta(E_{f}-E_{i})\dfrac{1}{2E_{e}^{\prime}}\dfrac{d^{3}\vec{k}_{\gamma}}{(2\pi)^{3}}\dfrac{1}{2E_{\gamma}}\int\dfrac{d^{3}p_{e}}{(2\pi)^{3}}\braket{\vec{e}_{2}}{\vec{q}+\vec{k}}\braket{}{\mathcal{M}_{\text{free}}}{{}^{2}}\braket{\vec{p}_{e}}{\vec{e}_{1}}$
(B.6) $\displaystyle=$
$\displaystyle\dfrac{V\delta(E_{f}-E_{i})}{16E_{a}E_{\gamma}E_{e}E_{e}^{\prime}v_{a}}\dfrac{d^{3}\vec{k}_{\gamma}}{(2\pi)^{2}}\int\dfrac{d^{3}p_{e}}{(2\pi)^{3}}\braket{\vec{e}_{2}}{\vec{q}+\vec{k}}\braket{}{\mathcal{M}_{\text{free}}}{{}^{2}}\braket{\vec{p}_{e}}{\vec{e}_{1}}$

Now we make a change of variables $d^{3}k_{\gamma}\to d^{3}q$ and express
$d^{3}q=2\pi q^{2}dqd(\cos\theta)$, where $\cos\theta$ is the cosine angle
between the momentum transfer and the ALP direction. Although the integration
over $d^{3}p_{e}^{\prime}$ was already performed, we need to put back the
phase space of the asymptotic free electron final state by acting on $d\sigma$
with the integral operator;

$\dfrac{V}{2}\sum_{\text{states}}\int\dfrac{{p_{e}^{\prime}}^{3}dT_{e}}{(2\pi)^{3}T_{e}}$
(B.7)

where $T_{e}=E_{e}^{\prime}-m_{e}\approx\frac{{p_{e}^{\prime}}^{2}}{2m_{e}}$.
At this stage we can also safely write $E_{e}E_{e}^{\prime}\approx m_{e}^{2}$.
This gives

$\displaystyle d\sigma=$
$\displaystyle\dfrac{\delta(E_{f}-E_{i})}{128E_{a}E_{\gamma}m_{e}^{2}v_{a}}\dfrac{2\pi
q^{2}dqd(\cos\theta)}{(2\pi)^{2}}\dfrac{dT_{e}}{T_{e}}\braket{}{\mathcal{M}_{\text{free}}}{{}^{2}}|f_{ion}^{nl}(p_{e}^{\prime},q)|^{2}$
(B.8)

taking a familiar definition for the ionization form factor [96, 143];

$|f_{ion}^{nl}(p_{e}^{\prime},q)|^{2}=\dfrac{V4{p_{e}^{\prime}}^{3}}{(2\pi)^{3}}\sum_{states}\bigg{|}\int
d^{3}x\psi_{p_{e}^{\prime},l^{\prime},m^{\prime}}^{*}(\vec{x})\psi_{n,l,m}(\vec{x})e^{i\vec{q}\cdot\vec{x}}\bigg{|}^{2}$
(B.9)

This ionization form factor has been calculated using a number of schemes, and
several codes have been made available by other authors. For this work, we
have used [NEEDS CITATION]. We perform the integration over $d(\cos\theta)$ by
making use of the relation

$\int d(\cos\theta)\delta(E_{f}-E_{i})=\dfrac{E_{\gamma}}{k_{a}q}$ (B.10)

after writing the energy delta function in terms of $\cos\theta$ and using the
identity $\delta(f(x))=|f^{\prime}(x=x_{0})|^{-1}\delta(x-x_{0})$ where
$f(x_{0})=0$. This reduces the cross section element to

$\displaystyle d\sigma=$
$\displaystyle\dfrac{E_{\gamma}}{k_{a}}\dfrac{1}{128E_{a}E_{\gamma}m_{e}^{2}v_{a}}\dfrac{2\pi
qdq}{(2\pi)^{2}}\dfrac{dT_{e}}{T_{e}}\braket{}{\mathcal{M}_{\text{free}}}{{}^{2}}|f_{ion}^{nl}(T_{e},q)|^{2}$
(B.11)

The energy prefactors can be written in a manifestly Lorentz-invariant way,
since it can be shown that

$\dfrac{E_{\gamma}}{k_{a}}\dfrac{1}{E_{\gamma}E_{a}m_{e}^{2}}=\dfrac{4}{(s-m_{e}^{2}-m_{a}^{2})(s-m_{e}^{2}+m_{a}^{2})}$
(B.12)

and, therefore,

$\displaystyle d\sigma=$
$\displaystyle[(s-m_{e}^{2}-m_{a}^{2})(s-m_{e}^{2}+m_{a}^{2})]^{-1}\dfrac{1}{64\pi
v_{a}}qdq\dfrac{dT_{e}}{T_{e}}\braket{}{\mathcal{M}_{\text{free}}}{{}^{2}}|f_{ion}^{nl}(T_{e},q)|^{2}$
(B.13)

Figure B.2: Right: Primakoff scattering cross sections for a variety of atomic
form factors (red) compared against the Primakoff photoionization cross
section (blue). Left: coherent Primakoff scattering compared with Primakoff
photoionization plotted on a log scale to visualize the behavior at low
energies where photoionization becomes dominant.

I can connect this expression to the point-wise inverse Primakoff differential
scattering cross section, which can be written as

$\bigg{(}\dfrac{d\sigma}{dq^{2}}\bigg{)}_{\text{point}}=\dfrac{1}{16\pi(s-(m_{e}-m_{a})^{2})(s-(m_{e}+m_{a})^{2})}\braket{}{\mathcal{M}_{\text{free}}}{{}^{2}}.$
(B.14)

Factoring out B.14 from B.13 gives us

$\displaystyle d\sigma=$
$\displaystyle\dfrac{1}{4v_{a}}qdq\dfrac{dT_{e}}{T_{e}}\xi(s,m_{e},m_{a})\bigg{(}\dfrac{d\sigma}{dq^{2}}\bigg{)}_{\text{point}}|f_{ion}^{nl}(T_{e},q)|^{2}$
(B.15)

where we have defined

$\xi(s,m_{e},m_{a})\equiv\dfrac{(s-(m_{e}-m_{a})^{2})(s-(m_{e}+m_{a})^{2})}{(s-m_{e}^{2}-m_{a}^{2})(s-m_{e}^{2}+m_{a}^{2})}.$
(B.16)

For $m_{a}^{2}<<m_{e}^{2}$ and $v_{a}\to 1$, this simplifies to

$\displaystyle\dfrac{\partial^{2}\sigma}{\partial T_{e}\partial q}$
$\displaystyle=\dfrac{q}{4T_{e}}\bigg{(}\dfrac{d\sigma}{dq^{2}}\bigg{)}_{\text{point}}|f_{ion}^{nl}(T_{e},q)|^{2}$
$\displaystyle=\dfrac{1}{8T_{e}}\bigg{(}\dfrac{d\sigma}{dq}\bigg{)}_{\text{point}}|f_{ion}^{nl}(T_{e},q)|^{2}$
(B.17)

This process was investigated further using a different theoretical formalism
in ref. [141].

Lastly, let’s compare the photoionization process to the coherent Primakoff
process. I show this comparison in Fig. B.2. We see that the usual process
without ionization has the dominant cross section in the relativistic / high
energy limit, while below 1 keV, ionization begins to dominate. Here I take
$m_{a}\ll 1$ keV.

##  APPENDIX C FURTHER ALP PARAMETER SPACE CONSTRAINTS

The existing bounds from DAMA [71], CUORE [72], Edelweiss-II [74], SOLAX [75],
COSME [76], CDMS [77], and Majorana [78] are shown in Fig. C.1 assuming full
volume coherence (FVC), as well as the projected limits for SuperCDMS,
LEGEND-200, LEGEND-1000, SABRE, and future CsI and NaI detectors with FVC.