distribution (given by (42), plus the fully-coherent contribution analogous to
(40)) in terms of the logarithm of the scattering angle:

$\displaystyle\frac{dp}{d\ln\theta}$
$\displaystyle=\left(fnL\left(\frac{12\pi}{9+2(q_{T}R)^{4}}nR^{3}+1\right)+8\pi\left(fnL\Delta
r_{\perp}\right)^{2}e^{-\mathbf{q}_{T}^{2}\Delta
r_{\perp}^{2}/2}\right)\left|b(q_{T})\right|^{2}2\pi\theta\sin 2\theta$ (77)

where $f$ is the fraction of the target volume occupied by the grains, each
with atomic number density $n$ and average radius $R$, and $L$ is the
thickness of the target. We can rewrite this as the sum of three terms, which
dominate at different angles:

$\displaystyle\frac{dp}{d\ln\theta}$ $\displaystyle=A_{\rm inc}\theta\sin
2\theta+A_{\rm pc}\frac{(\theta/\theta_{\rm pc})^{2}}{1+(\theta/\theta_{\rm
pc})^{4}}+A_{\rm coh}\left(\frac{\theta}{\theta_{\rm
coh}}\right)^{2}e^{-(\theta/\theta_{\rm coh})^{2}}$ (78)

where $A_{\rm inc}$, $A_{\rm pc}$ and $A_{\rm coh}$ are constant prefactors
for incoherent, partially-coherent (i.e. coherent over grains but not over the
full target), and fully-coherent scattering, respectively, and $\theta_{\rm
pc}$ and $\theta_{\rm coh}$ are the angular scales of partial and full
coherence (i.e. the angle above which scattering from individual spheres
becomes incoherent and the angle above which the neutron size no longer
matters). Here we use $q_{T}\propto\theta$ rather than the exact form
$q_{T}\propto\sqrt{2-2\cos(2\theta)}=2\sin(\theta)$ for simplicity, since we
are interested only in small-angle effects in this section. The resulting
scattering distribution is illustrated schematically in Figure 7.

Figure 7: A schematic illustration of the scattering distribution (78). The
locations and values of local extrema indicated on the plot are approximate;
we have omitted various $\mathcal{O}(1)$ factors here for simplicity. Also
marked is the minimum angle $\theta_{X}$ accessible via X-ray scattering
(conservatively assumed to be larger than $\theta_{\rm min}$).

In a typical experiment, $\theta_{\rm pc}$ is likely to be comparable to the
smallest scattering angle at which neutrons can be observed, since the entire
purpose of employing structured targets is to maximize small-angle scattering.
The question, then, is whether neutrons observed to scatter by an angle around
$\theta_{\rm pc}$ are sourced by a single significant scattering of order that
angle, or by a series of smaller-angle scatterings that summed to it. In
particular, any experiment must ensure that the observed scattering
distribution is not significantly affected by scatterings at angles too small
to study using X-ray scattering, since there is no way to determine the
structure factor at such small momentum transfers.

To establish conditions for this to be the case, divide angles below
$\theta_{\rm pc}$ into two regimes: those large enough to be measured using
X-ray scattering ($\theta>\theta_{X}$), and those below that threshold. At
least some of the former will certainly affect the final observed scattering
distribution, but their effects can be numerically predicted once the
structure factor at these angles is determined with X-ray scattering. The
effects of angles below $\theta_{X}$, on the other hand, should be
insignificant as long as $\theta_{X}/\theta_{\rm pc}$ is sufficiently small,
as we show below.

Now, further subdivide the angles below $\theta_{X}$ further into two ranges:
those above the local minimum in the scattering distribution at $\theta_{\rm
min}$ (see Figure 7), and those below it. We begin with the former. The number
of scatterings per neutron from the $\theta_{\rm min}<\theta<\theta_{X}$ range
is upper bounded by a Poisson distribution with expected value
$\alpha=dp/d\ln\theta|_{\theta=\theta_{X}}\ln(\theta_{X}/\theta_{\rm min})$,
with each scattering by an angle upper bounded by $\theta_{X}$ by assumption;
we will conservatively use both of these upper bounds.

Under these assumptions, the probability of $N$ scatterings from this range is

$\displaystyle p(N)$ $\displaystyle=\frac{e^{-\alpha}\alpha^{N}}{N!}.$ (79)

The minimum number of scattering events from this angular range needed to
create an observed scattering by at least $\theta_{\rm pc}$ is $N_{\rm
min}=\lceil\theta_{\rm pc}/\theta_{X}\rceil$, though contributions from larger
numbers of scatterings may dominate when $N_{\rm min}\gg 1$ or $N_{\rm
min}-\theta_{\rm pc}/\theta_{X}\ll 1$ as it is unlikely for all of the small-
angle scatterings to be in the same direction. Even this weak lower bound is
likely sufficient for our purposes, however: the probability of at least
$N_{\rm min}$ scatterings is at most [113]

$\displaystyle p(N\geq N_{\rm min})\leq\frac{(e\alpha)^{N_{\rm
min}}e^{-\alpha}}{(N_{\rm min})^{N_{\rm min}}}$ (80)

and we have

$\displaystyle\frac{dp}{d\ln\theta}\propto\theta^{2}$ (81)

in this angle range, so this bound becomes

$\displaystyle p(N\geq N_{\rm min})\leq\frac{\left(eA_{\rm
pc}\frac{\theta_{X}^{2}}{\theta_{\rm pc}^{2}}\right)^{N_{\rm min}}e^{-A_{\rm
pc}\theta_{X}^{2}/\theta_{\rm pc}^{2}}}{(N_{\rm min})^{N_{\rm min}}}.$ (82)

Taking a typical value of $A_{\rm pc}\sim 0.1$ to balance maximizing
statistics with not being swamped by multiple scatterings at angles around the
peak, and assuming $\ln(\theta_{X}/\theta_{\rm min})\sim 1$, this gives

$\displaystyle p(N\geq 3)\lesssim 10^{-6}$ (83)

so we should only need to perform X-ray scattering down to an angle of around
one third (perhaps one fourth) of the minimal neutron scattering angle in
order to be able to accurately numerically predict the effects of multiple
scatterings in the $\theta_{\rm min}<\theta<\theta_{X}$ range. As we discuss
in Appendix H.3, this should not be particularly difficult.

We can similarly bound the effects of scattering at angles below $\theta_{\rm
min}$, which are enhanced by total coherence over the neutron’s transverse
extent. In this case, the minimum number of scatterings to reach the first
angular bin outside of the beam is $\lceil\theta_{\rm pc}/\theta_{\rm
coh}\rceil$, which should be at least of order 10 for realistic experiments.
The fully-coherent scattering peak height $A_{\rm coh}$ can be estimated using
(40); relative to the partially-coherent peak, it is

$\displaystyle\frac{A_{\rm coh}}{A_{\rm pc}}\sim\frac{fL}{\sqrt{2}\pi R}.$
(84)

This is likely to give $A_{\rm coh}\gg 1$, so we can instead estimate the
result of these fully coherent scatters by the sum of $A_{\rm coh}$ random
2-dimensional vectors of length $\theta_{\rm coh}$. This, in turn, is well
approximated by a 2-dimensional Gaussian, which gives a probability for the
fully coherent scatters to sum to an observable angle of

$\displaystyle\begin{split}p(\theta_{\rm coh,total}\geq\theta_{\rm
pc})&\sim\exp\left(-\frac{3(nb\Delta
r_{\perp}\lambda_{0})^{2}}{4\sqrt{2}A_{\rm pc}^{2}}\right)\\\
&\sim\exp\left(-53\left(\frac{n}{10\text{
nm}^{-3}}\right)^{2}\left(\frac{b}{10\text{ fm}}\right)^{2}\left(\frac{\Delta
r_{\perp}}{10\text{ }\mu\text{m}}\right)^{2}\left(\frac{\lambda_{0}}{10\text{
\AA}}\right)^{2}\left(\frac{0.1}{A_{\rm pc}}\right)^{2}\right)\end{split}$
(85)

where $\lambda_{0}=2\pi/q_{0}$ is the incident neutron wavelength. This
probability needs to be smaller than $10^{-6}$ in order to reach our desired
control over systematic errors (see Section I.1). Assuming near-liquid
densities, this is always satisfied for argon-36 (see Table 1), but may not be
the case for xenon-136 if the neutron transverse size is less than
approximately $7~{}\mu$m (for a wavelength of $0.6$ nm); previous experiments
have measured values from roughly $1-100~{}\mu$m [114, 115, 116, 117].

Our discussion to this point has assumed scattering from a single material.
However, as we discuss in the main text, it is likely that a realistic
experiment would combine two materials: a solid lattice to form the granular
structure, and a noble liquid or gas within it whose scattering distribution
has simpler electromagnetic contributions. This leads to a suppression of the
partially-coherent scattering peak by the fractional difference between the
two materials’ scattering length densities since partial coherence depends on
the SLD contrast (see (44)) and an enhancement of the fully-coherent and
incoherent scattering peaks since both materials contribute to them. While
this does not affect the multiple scattering contribution of the intermediate
angular range, $\theta_{\rm min}<\theta<\theta_{X}$, it does enhance the
effect of small angles $\theta<\theta_{\rm min}$. This may prohibit certain
material combinations and porosities, depending on the value of $\Delta
r_{\perp}$. In particular, for two-material targets, the argument of the
exponent in (85) acquires a factor $f(\mathcal{S}_{\rm gas}-\mathcal{S}_{\rm
solid})/(f\mathcal{S}_{\rm gas}+(1-f)\mathcal{S}_{\rm solid})$, accounting for
the different scattering length densities relevant for fully- and partially-
coherent scattering.

Finally, we note that the frequency of multiple scattering events can be
further increased by short-range order, which tends to enhance scattering at
momentum transfers comparable to the inverse length scale of that order. This
is unlikely to be significant for short-range order within amorphous solids,
as the expected length scales should be shorter than those of the target
grains, leading to additional peaks at larger rather than smaller angles. Any
correlations in grain positions, however, may be more consequential, as the
necessarily large associated length scales would enhance scattering at angles
below $\theta_{\rm pc}$. The magnitude of this effect will depend on the
detailed structure of a particular target, but it appears unlikely that it
would significantly affect the discussion above except in highly ordered
targets. We will therefore not consider this effect further in this work.

##  Appendix F Thermal Effects

The observed neutron scattering distribution depends on the temperature of the
target in several ways. At high temperatures, the velocity of atoms in the
target leads to a significant difference between the center-of-mass velocity
of the scattering event and the lab-frame velocity of the neutron, creating an
apparent enhancement of the cross-section at low neutron energies [118, 119].
Additionally, higher temperatures lead to larger populations of excited states
of atoms, which can have different neutron scattering lengths due to
electromagnetic effects.

As discussed in Appendix A, nuclear scattering of neutrons is angle-
independent in the center-of-mass frame of the scattering event. When
considering scattering from a bulk target, however, it is more convenient to
work in the laboratory frame, in which individual atoms within the target have
a Maxwellian velocity distribution. Since the target atoms are no lighter (and
typically much heavier) than the incident neutron, this leads to large
differences in the center-of-mass velocity of the neutron. In particular, the
apparent scattering cross-section of neutrons slower than the atoms in the
target is enhanced because the scattering becomes the result of atoms striking
an essentially stationary neutron. This leads to an enhancement of the low-
energy cross section as $\sigma_{\rm lab}\propto 1/v$, with $v$ the neutron
velocity, whenever this velocity is lower than the thermal velocity of atoms
in the target, as well as a modification of the angular distribution of
neutrons after scattering in the lab frame.

As we discuss in Section H.2, typical neutron wavelengths at beamlines that
may be appropriate for our purposes are around 0.4-0.8 nm. Targets with atomic
weight $A$ then have thermal velocities equal to the neutron speed at
temperature

$\displaystyle T\sim A\left(\frac{\lambda}{0.4\text{ nm}}\right)^{-2}\times
10\text{ K}.$ (86)

Most of the targets we consider in this work have $A\gg 1$, and can therefore
likely be cooled below this temperature with little difficulty. In this case,
frame differences will only lead to small corrections in the observed
scattering distribution, which should not meaningfully affect final
sensitivities. If this is not the case—for example if using helium, or for
argon at larger neutron wavelengths–low velocity enhancement may become more
significant. While this should not prevent measurements of the sort we
describe in this work from being performed, it may have a more significant
effect on the final sensitivity, which we do not attempt to estimate.

The other effect of high target temperatures on cold neutron scattering is the
enhancement of populations of excited atomic states. In particular, our
modeling of neutron scattering from noble elements throughout this work
assumes their ground state electron configuration, with zero total spin or
orbital angular momentum. These assumptions no longer hold for excited states
of the atoms.

Fortunately, the excitation energies of noble elements are too high for this
to be a significant effect: the lowest excited state of xenon is at an energy
of $1.3$ eV [120, 121, 122], corresponding to an excited state population
fraction of order $10^{-22}$ even at room temperature; the excited states of
other noble elements are even more suppressed. This is far below the $10^{-6}$
benchmark that we use throughout this work, so any effects of excited
electronic states in the noble gas should be negligible. Any thermal effects
in the solid component of a target should be accounted for when separating
scattering contributions, so they do not affect our analysis further; see
Appendix D.

##  Appendix G Atomic Interaction Effects

In this appendix, we discuss the effects of interactions between atoms on
structured targets’ scattering distributions. We consider both interactions
between atoms within the noble gas (or liquid; while we have ignored the
distinction in most of this work, the high density of the noble fluid will be
relevant here) and between those noble atoms and the atoms in the solid.
Interactions among atoms within the solid are not relevant for our purposes
since we are generally impartial to the total scattering length of the solid.

###  G.1 Interactions at the Grain Surface

There are two ways in which interactions at the surface between the solid
component of the target and the liquid or gas within it can affect the
observed neutron scattering distribution. First, they can modify the
distribution of the noble atoms within their grain, modifying the structure
factor. For example, an attractive potential near the surface could increase
the density of the noble gas near the surface, whereas our estimates in
Appendix C assumed a uniform distribution of the gas within each grain.
Second, they can modify the electron orbitals of the noble atoms, potentially
changing the electromagnetic scattering length. In particular, we have assumed
throughout this work that the noble atoms have zero net electron spin and
angular momentum, but this may cease to be the case in the presence of
electromagnetic fields induced by the solid.

The former effect does not significantly affect us, so long as it does not
lead to an order-unity change in the structure factor, since it is independent
of the scattered particle and thus will be accounted for when combining
measurements, as described in Appendix D. This condition should certainly be
satisfied for typical materials, considering both the weakness of Van der
Waals interactions and that optimal sensitivity is generally achieved for
near-liquid densities of the noble element, in which case there is little room
for atoms to deviate from uniform packing.

Significant modification of the noble atoms’ electronic structure could be
more difficult to handle, however. In the presence of an inhomogeneous
magnetic field, the atomic Hamiltonian should acquire off-diagonal terms of
order $|\boldsymbol{\mu}||\Delta\mathbf{B}|$, with $\mu\sim\mu_{B}$ the
characteristic magnetic moment of atomic electrons and $\Delta\mathbf{B}$ the
variation in the magnetic field over the extent of the atom; we ignore any
numerical factors here, given the considerable uncertainties in the discussion
below. This perturbation then leads to a mixing of the ground state by a
fraction

$\displaystyle\frac{\Delta H}{\Delta
E}\sim\frac{\mu_{B}|\Delta\mathbf{B}|}{\Delta E}$ (87)

of the excited state, with $\Delta E$ the energy of that state, and thus gives
a correction to the neutron scattering length of order

$\displaystyle|\Delta b|\sim\frac{|g_{n}|e^{2}}{8\pi
m_{e}}\frac{\mu_{B}|\Delta\mathbf{B}|}{\Delta E};$ (88)

see Appendix A.

The maximum magnetic field resulting from a single atom on the solid surface
is of order

$\displaystyle|\mathbf{B}|_{\rm max}\sim\frac{\mu_{B}}{R_{\rm atom}^{3}},$
(89)

with $R_{\rm atom}\sim 0.1$ nm the characteristic size of the atoms. The
neutron scattering length of such an atom is then modified by approximately

$\displaystyle|\Delta b|\sim\frac{|g_{n}|e^{2}\mu_{B}^{2}}{8\pi m_{e}R_{\rm
atom}^{3}\Delta E}\lesssim 3\times 10^{-3}\text{ fm},$ (90)

where we use the lowest excited state energy of a noble element ($1.3$ eV for
xenon [120, 121, 122]) for this upper bound. This should be compared to
nuclear scattering lengths of order 10 fm; see Table 1.

Such a correction to every atom would be well above our systematic error
target of one part in $10^{6}$, if it were not further suppressed by two
additional effects. First, the rapid decay of dipole magnetic fields with
distance from the dipole means that only an order-unity number of atoms near
the surface dipole will be affected; only a fraction of order $R_{\rm
atom}/R_{\rm grain}$ of the atoms in a grain of radius $R_{\rm grain}$ are
therefore affected. Second, since we are not considering ferromagnetic
targets, we expect the magnetic field directions, and thus the signs of the
neutron scattering length changes for a given momentum transfer, to vary
within (as well as among) grains. Assuming that any correlations in the
alignment of surface dipoles occur on length scales $\xi<R_{\rm grain}$, we
expect a total of approximately $(R_{\rm grain}/\xi)^{2}$ independently chosen
directions over the extent of the grain surface.

The average change in scattering length is then

$\displaystyle\left|\overline{\Delta
b}\right|\sim\frac{|g_{n}|e^{2}\mu_{B}^{2}}{8\pi m_{e}R_{\rm atom}^{3}\Delta
E}\left(\frac{R_{\rm atom}}{R_{\rm grain}}\right)\frac{1}{\sqrt{(R_{\rm
grain}/\xi)^{2}}}\lesssim\left(\frac{R_{\rm atom}\xi}{R_{\rm
grain}^{2}}\right)3\times 10^{-3}\text{ fm},$ (91)

which should be below our target of order $10^{-5}$ fm for all targets we
consider.

Note that this average change in the scattering length is the correct quantity
to consider (as opposed to, for example, its root mean square change) even for
incoherent scattering: since the maximal change in the scattering length of
one noble atom is much smaller than its nuclear scattering length, its non-
negligible effect comes from the interference between nuclear and
electromagnetic scattering (i.e. from the product of the nuclear and
electromagnetic scattering lengths). This term is linear in the
electromagnetic scattering length, and thus the change in the total scattering
probability depends precisely on $\left|\overline{\Delta b}\right|$.

###  G.2 Interactions Among Noble Atoms

We now turn to the effects of interactions within the noble liquid or gas
itself. Similarly to the surface interactions described above, the effects of
interactions between the atoms in a fluid can affect the observed scattering
distribution both by modifying the target’s structure factor and by inducing
changes in the electromagnetic scattering length of the atoms.

The high density of liquids leads to significant correlations between the
positions of atoms and their neighbors, i.e. a non-trivial pair distribution
function [123, 124, 125, 126, 127, 128, 129]. These correlations will then
lead to a modification of the structure factor at inverse momentum transfers
comparable to the atomic spacing in the liquid. Fortunately, these are not an
issue for the measurements we propose: since this effect is independent of the
scattered particle, it will automatically be accounted for when combining
measurements (see Section III.2), and it should have little effect on
achievable sensitivity since it occurs at much larger momentum transfers than
the focus of this work.

Interactions between the atoms within the liquid may, in principle, also lead
to changes in their electronic states, which would then modify the
electromagnetic scattering lengths of the target atoms. Since we are
considering noble elements, whose ground states have zero magnetic moment,
there are no apparent mechanisms to induce significant changes in the atoms’
electronic states. Thus this should not be a significant effect for liquids or
gases. It is less clear whether spontaneous magnetization could appear in
noble solids, but we leave consideration of this to future work.

##  Appendix H Instrument Parameters

In this appendix, we summarize the key features of neutron and X-ray
scattering instruments and describe the parameters of such instruments that we
assume for our projections in the main text. We begin with an outline of
small-angle neutron scattering instruments as a whole, before focusing on the
properties of neutron sources and beams in particular, which are the limiting
parameters for our proposal’s reach. We then consider the analogous properties
of X-ray beams, justifying our assumption in the main text that these
contribute subdominant uncertainties.

###  H.1 Target Geometry

A sketch of a simplified neutron scattering experiment layout is shown in
Figure 1. In this section, we consider a few aspects of this layout and their
impact on the sensitivity of such an experiment to the new forces we consider.

The importance of the distance between the neutron source and the scattering
target for collimation is discussed in the next section of this appendix. We
begin instead with the distance beyond the target, between it and the neutron
detectors, which is significant in comparison to three other length scales:
the pixel size of the neutron detector, and the transverse and longitudinal
target dimensions.

Modern neutron detectors can achieve resolutions approaching 1 $\mu$m [130,
131, 132, 133, 134, 135], though this enormously exceeds our needs; as we
discuss below, there is little benefit to resolutions significantly below the
target’s transverse size. The ratio of the pixel size to the distance between
the target and the neutron detector sets the maximum angular resolution of the
experiment. This angular resolution should certainly be no greater than the
minimum scattering angle we wish to measure, and there is likely to be some
benefit to an angular resolution a few times better, in order to better
resolve the new force peak. Our projections below assume a minimum scattering
angle of 3 mrad, requiring a detector distance of at least 300 times the pixel
size.

The minimum useful pixel size, in turn, is set by the target’s transverse
length scale. Below this length scale, scattering events at the same angle can
appear in different pixels if they occurred at different locations within the
target. While this is not intrinsically problematic, there is little purpose
to using much smaller pixels, since the angular distribution of the neutrons
will be washed out by the target size at these scales. Note as well that, when
the target transverse size is larger than the pixel size, it is the ratio of
the detector distance to the former that sets the maximum resolution. Thus,
the 10 cm2 targets that we assume in our projections require a target to
detector distance of approximately 10 m.

There is a similar effect from the target’s longitudinal size (i.e. its
depth), though it is likely to be sub-dominant for our purposes due to our
focus on small-angle scattering. There are three other constraints on this
depth, however. First, for a given set of target materials, the target depth
sets the fraction of neutrons that are scattered into observable angles; as we
discuss in Appendix E, we assume that this is set to $0.1$, in order to
maintain control over multiple scattering events. Second, as we noted in
Section IV.1, excessively thin targets may be difficult to work with; combined
with the maximum depth from multiple scattering, this may preclude certain
material combinations.

The third effect of finite target depth is a suppression of fully coherent
scattering at large angles. Our calculation of fully coherent scattering in
Section C assumed that the direction of momentum transfer was perpendicular to
the neutron’s incident direction, but this is only true in the limit of zero
scattering angle. At larger angles, neutrons scattered from one end of the
target become incoherent with those scattered from the other end, reducing the
fully-coherent scattering contribution at these angles. This is completely
insignificant for all parameters we consider, however, since full coherence is
only significant at very small angles to begin with.

Finally, we consider the effects of the cell containing the target
material(s), which we have heretofore ignored. The additional scattering
contribution from the neutron beam passing through the walls of the cell must
be separated in order to isolate the scattering distribution of the ideal gas
within. However, the separation of contributions procedure discussed in
Appendix D is not immediately applicable to this issue because scattering from
the ideal gas cannot be performed without the cell. Note, however, that this
is only a concern if there is coherent scattering from the cell walls combined
with the cell contents (i.e. the cross-term discussed in Appendix D);
otherwise the cell’s scattering distribution can simply be measured separately
and then subtracted out of the final distribution. Our projections assume that
the cell walls lack any structure at length scales comparable to the inverse
momentum transfer, such that this should be the case, leaving characterization
of cell wall roughness to future work.

###  H.2 Neutron Beam Parameters

Neutrons used in small-angle neutron scattering (SANS) experiments are
typically produced in nuclear reactors (e.g. [29, 30, 31, 32, 33, 34]) or
neutron generators (e.g. [35, 36]). Since such sources produce neutrons at
much higher energies than desirable for SANS experiments, they are then cooled
by passing through one or more cold moderators (e.g. water and liquid
hydrogen), resulting in an uncollimated collection of neutrons with a broad
(though not necessarily thermal) distribution of energies.

The simplest approach to forming a neutron beam is simply to reject all
neutrons that do not pass through two small, widely separated apertures.
Neutrons can also be transported via waveguides and focused with optics,
though we will not review such devices here (see e.g. [136, 137]). The key
feature of neutron collimation for our purposes is simply that it is
statistically costly, with neutron count proportional to accepted phase space.

Neutron energy distributions are, similarly, narrowed primarily through
rejection of velocities other than those desired. A typical design for a
neutron velocity selector is described in [37]: a rotating cylinder of
absorbing material with helical channels, such that neutrons are absorbed
unless they have the right velocity to pass through in a straight line without
striking any surfaces. An alternative approach to velocity selection using
slotted disks is described in [38]. The statistical cost of these procedures
depends somewhat on the particular neutron source used and the corresponding
energy distribution after moderation.

A variety of existing neutron scattering instruments may be able to
accommodate our proposal: the NIST Center for Neutron Research’s VSANS
instrument [138], Oak Ridge National Laboratory’s EQ-SANS diffractometer
[139], the Institut Laue-Langevin’s D22 diffractometer [140], and various
instruments at J-PARC’s Materials and Life Science Experimental Facility
[141], among others. Since this work is not specific to any particular source,
our projections assume a set of parameters approximately representative of
these optimal sources: a flux of $10^{8}$ cm-2 s-1 neutrons of $0.6$ nm
wavelength over a target area of $10$ cm2, with a minimum resolvable angle
(including collimation, detector pixel size, and detector distance to target
size ratio) of $3$ mrad, corresponding to a momentum transfer of approximately
$(30\text{ nm})^{-1}$.

Other parameters of the neutron beam should be less important for our
sensitivity projections. In particular, energy spread acts only to “wash out”
the low-angle peak that would indicate the presence of a new force; while this
increases the uncertainty on a detected new force’s mediator mass $\mu$, it
has little effect on the total number of new force scattering events within
that peak and therefore has minimal impact on our ability to detect a new
force’s presence (see Appendix I). Similarly, our proposal should not require
particularly good angular resolution, so long as sufficiently small angles can
be observed.

###  H.3 X-Ray Beam Parameters

Small-angle X-ray scattering (SAXS) sources are typically based on undulation
of an electron beam by a series of magnets of alternating polarity [142]. SAXS
instruments are available over a wide range of X-ray energies, including well
above the keV momentum scales of the neutron sources we consider. Using such
high-energy sources is likely to be necessary for most targets of interest,
due to X-ray absorption: as discussed in Appendix B, low-energy X-rays are
rapidly attenuated within dense materials, especially at large atomic weights.
For the xenon targets that are the focus of this work, energies of at least 40
keV are likely desirable.

A variety of SAXS facilities could potentially meet the requirements of our
proposal; see, for example, [143, 144]. Just as for neutron sources, we do not
choose a particular source in this work, and in fact we are generally agnostic
to the parameters of the source used so long as they are sufficient for the
dominant uncertainties in the final measurement to arise from neutron
scattering; we assume that this holds for all of our projections. The precise
X-ray photon counts required for this to hold were discussed in Section III.2:
an X-ray count of $10^{5}$ times the neutron count should always be
sufficient, with as little as a few times the neutron count sufficient at the
smaller mediator masses we focus on. Even the former condition can be
satisfied by instruments such as the European Synchrotron Radiation Facility’s
ID15A [144], and an enormous variety of X-ray sources can meet the weaker flux
requirement.

There is one other property of X-ray sources that complicates our proposal
somewhat: their beamsize. Typical X-ray scattering instrument beams are far
narrower than the centimeter-scale targets that are necessary to maximize
neutron count. Since targets may not be spatially uniform over their
transverse extent, it is necessary to scan the X-ray scattering measurement
over the target in order to obtain a spatially-average structure factor
applicable to the neutron scattering measurement. Note that the neutron beam
may not be spatially uniform either, so the spatial distribution of neutrons
must be measured as well. This has a negligible impact on the total neutron
count needed for our experiment, however, as the absence of a scattering
target for this measurement eliminates the usual factor of 10 loss in flux
from most of the beam passing through a target without scattering (see
Appendix E).

It is similarly critical that the X-ray and neutron beamlines be precisely
coaxial in order to see the same effective target thickness and structures.
This may pose challenges for the use of recently developed techniques for
simultaneous X-ray and neutron scattering measurements [51].

Other parameters of X-ray sources are generally not a concern for our
proposal. As discussed in Appendix E, the X-ray collimation requirement is a
factor of a few stronger (in terms of transverse momentum) to the neutron
requirement, due to the need to measure the structure factor at smaller
momentum transfers in order to predict the impact of multiple scattering
events; this is easily satisfied by many X-ray instruments. Similarly, the
distribution of incident X-ray energies is generally far narrower than that of
neutrons, so energy spread should not be a meaningful constraint for our
purposes.

##  Appendix I Statistics

In this appendix, we describe our approach to estimating the potential
sensitivity of the various target materials considered in the main text. We
first tabulate the various systematic errors faced by any implementation of
our proposal. We then describe our approach to calculating the statistical
reach of single-material targets, before explaining our approximation of the
statistical error for two-material targets.

###  I.1 Systematic Errors

We begin by summarizing the systematic errors that limit the achievable
sensitivity of new force searches implementing our proposed strategy. Most of
the effects we consider in this section have been discussed elsewhere in this
work; our goal here is to tabulate their respective magnitudes before we
calculate realistically achievable experimental sensitivities.

A relatively fundamental limit on searches for new neutron-atom interactions
is our limited ability to predict strong nuclear interactions. In most of this
work, we have described nuclear scattering as entirely angle-independent, but
this is an approximation: significant deviations from angle-independence occur
at momentum transfers comparable to the inverse strong force range, i.e.
inverse femtometers [96, 97, 98, 99, 100]; see Appendix A. Corrections to the
angle-independence of nuclear scattering are therefore suppressed by
$\mathcal{O}((q_{T}b_{\rm nuc})^{2})\lesssim 10^{-8}$ even at the largest
momentum transfers we consider, leaving them far below our systematic target.
In fact, a more significant systematic error may arise from the contribution
of the nuclear charge form factor to the neutron’s electric polarizability
scattering length, which is suppressed by $\mathcal{O}(q_{T}\sqrt{\left\langle
r_{n}^{2}\right\rangle}b_{P}/b_{\rm nuc})\sim 10^{-9}$ (see Appendix A.2).
This error may be reducible using knowledge of this form factor, but we will
not explore this here, given that we do not expect this to be a limiting error
for our proposal.

Errors related to modeling of electromagnetic scattering may be more
significant, though these depend considerably on the exact target used. As we
discuss in Section III.3, contributions from the new force and from
electromagnetic scattering become difficult to distinguish once
$q_{o}\sim\mu$; we avoid this issue in our projections by conservatively
restricting to $\lambda>10^{-1}$ nm. Moreover, even for ideal noble gases, our
description of electromagnetic scattering (see Appendix A) ignored various
terms suppressed by $m_{e}/m_{n}$, $r_{N}/r_{A}$, or $q_{T}r_{N}$, with
$r_{N}$ the nuclear radius and $r_{A}$ the atomic radius. All three of these
factors are of order $10^{-6}$, but may be enhanced sufficiently to be
relevant by the large atomic numbers of the target atoms we consider. We note,
however, that these higher-order terms can likely be worked out more precisely
if this is useful for future experiments, since, unlike the nuclear
corrections described above, they are the result of well-understood
electromagnetism.

Scattering from realistic targets leads to a number of other electromagnetic
corrections, however. Non-noble elements generally have non-zero magnetic
dipole moments, leading to additional neutron scattering (see Appendix A), and
even noble atoms may have induced magnetic moments due to interatomic
interactions (see Appendix G). The separation of scattering contributions
explained in Appendix D allows non-noble elements’ contributions to be
removed, so they do not lead to a systematic error (up to small caveats
discussed below). Magnetic moments induced in the target noble atoms, however,
cannot be separated out. We show in Appendix G that magnetic moments induced
by surface interactions should lead to scattering length corrections of no
more than roughly $3\times 10^{-3}(R_{\rm atom}\xi/R_{\rm grain}^{2})\text{
fm}$ when correlations in the magnetic dipole moments within the solid have
length scale $\xi$; effects of interactions within noble liquids (or dense
noble gases) should be negligible by comparison.

Separation of contributions may fail to entirely remove systematic
electromagnetic backgrounds from non-noble elements if there are any changes
to the target between neutron and X-ray scattering or between different
measurements of one type will inhibit the accuracy of this separation,
however. One potential cause of such changes is material degradation from the
scattering processes themselves, though this is likely to be minimal for the
low energies we consider. In the case of materials consisting of piles of
grains, the structure may also change simply due to motion of the target
between measurements (though this may be circumventable by, for example,
rotating such targets continuously during measurements and using the resulting
average structure, as one would need to do for aerosols or boiling liquids).
Xenon snow may be so unstable that it degrades even if not moved. Since all of
these effects are strongly dependent on the specific materials used, we will
not include them in our sensitivity projections below, but we note that they
may be significant in some circumstances.

A related systematic effect arises due to the differing energies of
appropriate neutron and X-ray sources, which change their respective
correspondences between momentum transfer and scattering angle. As a result,
neutrons and X-rays at a fixed momentum transfer take different paths through
the target and do not, in fact, see identical target structures, potentially
changing their respective structure factors. While we expect this effect to be
small, given the generally thin targets and small angles that we are most
interested in, it will likely need to be simulated numerically or tested with
additional measurements given our desired precision. We leave a more detailed
treatment of this effect to future work.

Systematic errors may also arise if materials’ compositions change between
measurements. This could occur due to finite noble gas purity, or due to
imperfect separation of materials, for example from adsorption of atoms by the
solid components of a target. Again, these depend heavily on details beyond
the general principles outlined in this work, so we will not include these
effects below.

A final, more generic source of systematic error is low-angle multiple
scattering, discussed in Appendix E. As we note in that appendix, multiple
scattering should not be an issue for scattering from argon, but may or may
not be a significant constraint on xenon-based targets. The error introduced
by multiple scattering grows exponentially as a function of various
experimental parameters (see (85)) so it is generally either enormous or
irrelevant. In particular, this error depends on the transverse size of
individual neutrons, which is not currently known sufficiently well for any
apparatus to definitively determine which material combinations are compatible
with a scattering fraction of $0.1$; reducing the scattering fraction
suppressed this effect. We assume in most of this work that neutrons have
sufficient transverse sizes to use the materials we consider with scattering
fractions of $0.1$ without significant multiple scattering errors, but we note
that some solids discussed in Section IV.1 may in actuality require reduced
statistics.

###  I.2 Projecting Sensitivity for Single-Material Targets

We can now estimate the sensitivity of a neutron scattering experiment
implementing our proposal. Since we expect to have many neutrons observed in
each angular bin, we can approximate the exact maximum-likelihood analysis of
the data with an F-test [145], comparing the $\chi^{2}$ values obtained when
fitting the observed data including ($\chi^{2}_{\rm with}$) and not including
($\chi^{2}_{\rm without}$) a new force. More precisely, let

$\displaystyle F=\frac{\left(\chi^{2}_{\rm without}-\chi^{2}_{\rm
with}\right)/2}{\chi^{2}_{\rm with}/N^{\rm dof}_{\rm with}},$ (92)

for $N^{\rm dof}_{\rm with}$ degrees of freedom in the with-new force fit (and
assuming two degrees of freedom for the new force: $\mu$ and $g$). Then the
distribution of $F$ values will follow an $F$-distribution with $d_{1}=2$ and
$d_{2}=N^{\rm dof}_{\rm with}$ degrees of freedom. We can then constrain any
new force for which the resulting $F$ is expected to exceed some threshold
value.

In the case of single-material scattering, we can straightforwardly compute
the expected values of $\chi^{2}$ both with and without the new force included
in the fit as follows. We generically expect a $\chi^{2}$ contribution of $1$
for each degree of freedom in a fit, simply from Poisson statistics within
each angular bin, independent of which fit is used. In the presence of a new
force, we expect an additional number of scatterings into the bin at angle
$\theta$ given by $2\kappa_{\rm new}N_{\rm
expected}(\theta)/(1+q_{T}(\theta)^{2})$, with $N_{\rm expected}$ the expected
number of neutrons scattered into this bin by nuclear scattering. In the
absence of any fit corrections—that is, using the fit parameters that would be
optimal in the absence of a new force—this leads to an expected $\chi_{\rm
without}^{2}$ contribution given by $4\kappa_{\rm new}^{2}(\theta)N_{\rm
expected}(\theta)/(1+q_{T}(\theta)^{2})^{2}$, but no contribution to
$\chi_{\rm with}^{2}$. Note that this total expected contribution is
independent of the number of bins in the limit that the nuclear scattering
distribution (including coherence) is constant within each bin:

$\displaystyle\Delta\chi^{2}_{\rm no-fit}=\sum\limits_{\rm
bins}\frac{4\kappa_{\rm new}^{2}(\theta)N_{\rm
expected}(\theta)}{(1+q_{T}(\theta)^{2})^{2}}\to N_{\rm scattered}\int
d\theta\frac{4\kappa_{\rm
new}^{2}(\theta)}{(1+q_{T}(\theta)^{2})^{2}}\frac{dp}{d\theta}$ (93)

for $N_{\rm scattered}$ total scattered neutrons and (nuclear, coherence-
enhanced) scattering distribution $dp/d\theta$, normalized to integrate to
unity over angles above some minimum. This additional $\chi^{2}$ contribution
is reduced after fitting empirically-determined Standard Model scattering
parameters: fitting over the angle-independent nuclear scattering length, for
example, eliminates the average number of additional scatters per bin (of
constant solid angle), leaving only the variation in new force scattering over
different angles.

In terms of this additional post-fit $\chi^{2}$ contribution, (92) predicts

$\displaystyle F=1+\frac{1}{2}\Delta\chi^{2}_{\rm fit}.$ (94)

For $N^{\rm dof}_{\rm with}\gg 1$, the cumulative distribution function is
well approximated at $F$ of no more than a few by

$\displaystyle{\rm CDF}_{2,N^{\rm dof}_{\rm with}}(F)\approx 1-e^{-F},$ (95)

so we expect to be able to detect a new force at 95% confidence whenever
$1+\Delta\chi^{2}_{\rm fit}/2>\ln(20)\approx 3.0$, i.e. when
$\Delta\chi^{2}_{\rm fit}>4.0$. Since the finite true number of angular bins
weakens this somewhat, we conservatively require $\Delta\chi^{2}_{\rm
fit}>5.0$ for the projections in this work.

###  I.3 Projecting Sensitivity for Two-Material Targets

In the previous subsection, we explained our method for estimating the
sensitivity of a single-material neutron scattering experiment following the
approach discussed in this work. An exact analysis of the two-material case is
necessarily numerical. However, in this appendix, we describe an approximate,
analytic approach to estimating the two-material sensitivity, which avoids the
computational cost of the numerical analysis while offering additional insight
into the parameter dependence of the sensitivity.

Note that, throughout this appendix, we assume that the spin-dependent
scattering cross-term discussed in Appendix D is not significant, such that
measurements from two noble gases are sufficient. Including the spin-dependent
cross-term complicates the analysis further, but we will not consider it in
detail in this work; in any case it is unlikely to have more than an order-
unity effect on the final sensitivity.

In the single-material case, the effects of a new force can be resolved from
those of similar-length scale structure simply by looking at ratios of neutron
to X-ray scattering. Unfortunately, this fails in the two-material case due to
interference between scattering from the solid and gas, which differs for
neutrons and X-rays. In particular, the total neutron scattering distribution
in this case can be written as (see (70))

$\displaystyle\left.\frac{dp}{d\Omega}\right|_{\rm 2-material}$
$\displaystyle=\left.\frac{dp}{d\Omega}\right|_{\rm
s,inc}+\left.\frac{dp}{d\Omega}\right|_{\rm g,inc}+\left|\left\langle
B_{0}(\mathbf{q}_{T})\right\rangle+b_{g}(\mathbf{q}_{T})\left\langle
W(\mathbf{q}_{T})\right\rangle\right|^{2}$ (96)

where the first two terms on the right-hand side are the incoherent scattering
distributions from the solid alone and the gas alone, while the last term is
the total coherent scattering length, including both the unknown solid
contribution $B_{0}(\mathbf{q}_{T})$ and the gas contribution, which is given
by the product of the gas’s single-atom scattering length
$b_{g}(\mathbf{q}_{T})$ and the target phase sum $W(\mathbf{q}_{T})$. Note
that we have taken the expectation values of $B_{0}(\mathbf{q}_{T})$ and
$W(\mathbf{q}_{T})$ above in order to separate the coherent and incoherent
scattering contributions to the scattering distribution.

We can eliminate all of the terms not enhanced by the gas’s structure by
making three measurements—the two materials together, the solid alone, and the
gas alone (which gives precisely the incoherent gas scattering distribution
above)—and taking the following linear combination:

$\displaystyle\begin{split}\left.\frac{dp}{d\Omega}\right|_{\rm
difference}&=\left.\frac{dp}{d\Omega}\right|_{\rm
2-material}-\left.\frac{dp}{d\Omega}\right|_{\rm s\
only}-\left.\frac{dp}{d\Omega}\right|_{\rm g\ only}\\\
&=2~{}\text{Re}\left(\left\langle B_{0}^{*}(\theta)\right\rangle
b_{g}(\theta)\left\langle
W(\theta)\right\rangle\right)+\left|b_{g}(\theta)\left\langle
W(\theta)\right\rangle\right|^{2}.\end{split}$ (97)

Now suppose that we have obtained $W(\mathbf{q}_{T})$ from X-ray scattering
measurements. This is not a precise description of the two-material analysis,
since the X-ray scattering distribution similarly suffers from interference,
but it should act as a reasonable approximation of the process, since the
purpose of the X-ray measurements is precisely to distinguish any new force
from the (shared) structure factor.

Then, if we ignore the electromagnetic and new force contributions to the last
term, we can predict it from a combination of the gas-only measurement of
$b_{g}(\theta)\approx b_{0}$ and the X-ray measurement of $W(\theta)$;
including an estimate of the electromagnetic contribution (from other
measurements, or from theoretical calculations) can make this prediction even
more precise. Similarly to the handling of the phase sum above, this is not
necessary in the true numerical analysis, but is useful for our simple
estimate here; note that this approximation is likely to be conservative, as
we are discarding part of the effect of the new force. Then let

$\displaystyle\left.\frac{dp}{d\Omega}\right|_{\rm cross}$
$\displaystyle=\left.\frac{dp}{d\Omega}\right|_{\rm
2-material}-\left.\frac{dp}{d\Omega}\right|_{\rm s\
only}-\left.\frac{dp}{d\Omega}\right|_{\rm g\
only}-\left|b_{g}(\theta)\left\langle W(\theta)\right\rangle\right|^{2}_{\rm
predicted}.$ (98)

Since the only complex component to the single-atom scattering length comes
from the irrelevant Schwinger scattering length (see Appendix D), we can also
write this measurement combination as

$\displaystyle\left.\frac{dp}{d\Omega}\right|_{\rm cross}$
$\displaystyle=2~{}\text{Re}\left(\left\langle
B_{0}^{*}(\theta)\right\rangle\left\langle
W(\theta)\right\rangle\right)b_{g}(\theta).$ (99)

Crucially, this is now fully factored into a gas-independent term
$2~{}\text{Re}\left(\left\langle B_{0}^{*}(\theta)\right\rangle\left\langle
W(\theta)\right\rangle\right)$ and a gas-specific scattering length
$b_{g}(\theta)$. Thus, if we perform this process for two different noble
elements, we can take the ratio

$\displaystyle\frac{\left.dp/d\Omega\right|_{\rm
cross,1}}{\left.dp/d\Omega\right|_{\rm cross,2}}$
$\displaystyle=\frac{b_{g,1}(\theta)}{b_{g,2}(\theta)},$ (100)

which is now independent of both the solid and the structure factor.

We can now use this to detect a new force by detecting a difference in the
angle-dependence of the two elements’ single-atom scattering lengths. In
particular, using (46), we have

$\displaystyle\frac{\left.dp/d\Omega\right|_{\rm
cross,1}}{\left.dp/d\Omega\right|_{\rm cross,2}}$ $\displaystyle=({\rm
const.})\left(1+\kappa_{\rm EM,1}f_{1}(q_{T})-\kappa_{\rm
EM,2}f_{2}(q_{T})+\frac{\Delta\kappa_{\rm
new}}{1+(q_{T}/\mu)^{2}}+\mathcal{O}(\kappa^{2})\right)$ (101)

where $\Delta\kappa_{\rm new}=\kappa_{\rm new,1}-\kappa_{\rm new,2}$. From
here, the fitting procedure is much the same as for the single-material case,
with seven free parameters (the constant prefactor, two electromagnetic
parameters for each noble element, and two new force parameters). The four
electromagnetic parameters were not included in the fits for the rough
projections of this work, as this significantly reduced computational expense.
We do not expect this to have a meaningful impact on the final sensitivity,
however, as electromagnetic scattering has a significantly different angular
dependence from new force scattering; this was also confirmed numerically for
the single-material case.

As we note in the main text, the dependence of this final ratio only on the
difference $\Delta\kappa_{\rm new}$ makes it advantageous to use noble gases
with very different atomic weights, in order to maximize this difference.
However, this must be balanced against the weakness of the gas’s scattering
contribution relative to that of the solid, given the low value of helium’s
SLD in particular–see Table 1–leading to the approximate parity between argon-
and helium-based scattering seen in Figure 3.

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