FERMILAB-PUB-22-705-PPD-QIS-T Measuring the Migdal eect in semiconductors for dark matter detection Duncan Adams1Daniel Baxter2yHannah Day3zRouven Essig1xand Yonatan Kahn3 4

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FERMILAB-PUB-22-705-PPD-QIS-T

Measuring the Migdal eﬀect in semiconductors for dark matter detection

Duncan Adams,1, ∗Daniel Baxter,2, †Hannah Day,3, ‡Rouven Essig,1, §and Yonatan Kahn3, 4, ¶

1C.N. Yang Institute for Theoretical Physics, Stony Brook University, NY 11794, USA

2Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

3Department of Physics, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA

4Illinois Center for Advanced Studies of the Universe,

University of Illinois Urbana-Champaign, Urbana, IL 61801, USA

(Dated: March 24, 2023)

The Migdal eﬀect has received much attention from the dark matter direct detection community,

in particular due to its power in setting leading limits on sub-GeV particle dark matter. However, it

is crucial to obtain experimental conﬁrmation of the Migdal eﬀect through nuclear scattering using

Standard Model probes. In this work, we extend existing calculations of the Migdal eﬀect to the

case of neutron-nucleus scattering, with a particular focus on neutron scattering angle distributions

in silicon. We identify kinematic regimes wherein the assumptions present in current calculations

of the Migdal eﬀect hold for neutron scattering, and demonstrate that these include viable neutron

calibration schemes. We then apply this framework to propose an experimental strategy to measure

the Migdal eﬀect in cryogenic silicon detectors using an upgrade to the NEXUS facility at Fermilab.

A proliferation of direct detection experiments search-

ing for sub-GeV dark matter (DM) has been matched by

a suite of theoretical work to better understand the kine-

matics of low-energy scattering in the regime where parti-

cle physics and condensed matter intersect [1]. This kine-

matic regime primarily diﬀers from traditional WIMP

scattering in that the energy and momentum transfers

involved are comparable to the fundamental scales of the

target (set by the gap energy and inverse atomic size,

respectively), meaning that standard elastic scattering

approximations [2] no longer hold. Indeed, the primary

scattering channel of interest for sub-GeV DM searches

has long been DM-electron scattering [3], which must

account for both the inherent binding energy of the scat-

tered electron and the band structure of the target. More

recently, several theoretical advancements have uncov-

ered yet another inelastic scattering channel of interest

for sub-GeV DM, nuclear recoils that directly ionize the

scattered atom, a process denoted the “Migdal eﬀect”

(ME).

The theoretical underpinnings of the ME go back to

the early work of Arkady Migdal [4, 5], who calculated

the probability that a radioactive decay would directly

ionize the daughter nucleus. Such ionization has been

measured in radioactive decay, and is more commonly

referred to as “electron shake-oﬀ” [6–8]. Though a hand-

ful of papers [9–11] pointed out the likely relevance of this

eﬀect for DM-nucleus scattering, progress on the ME ac-

celerated after Ref. [12] derived the necessary electronic

excitation probabilities relevant for DM experiments. A

ﬂurry of theoretical activity [13–27] followed, expanding

the theory of the ME for galactic DM scattering in iso-

∗duncan.adams@stonybrook.edu

†dbaxter9@fnal.gov

‡hjday2@illinois.edu

§rouven.essig@stonybrook.edu

¶yfkahn@illinois.edu

lated atom [18], molecular [27], and solid-state [20, 21, 26]

targets, as well as for solar coherent elastic neutrino-

nucleus scattering [14]. The ME was also shown to dom-

inate over another important inelastic channel, namely

the bremsstrahlung process [14, 28]. Several experimen-

tal collaborations have since used these theoretical re-

sults to set what are currently the strongest limits on

DM-nuclear scattering below ∼1 GeV [29–35]. Out of all

of this work from the DM community, only Refs. [36–40]

have so far explicitly considered the ME for neutron scat-

tering. A parallel eﬀort in chemistry focused on neutron

scattering in isolated atoms and molecules [41, 42], in

part to explain anomalously large neutron cross sections

on hydrides [43–45]. Our work diﬀers from these propos-

als by focusing on the angular distribution of neutrons

scattered from solid-state targets.1While the existence

of the ME is well founded, the magnitude of the eﬀect

must be measured to understand the expected DM sig-

nal in a direct detection experiment.

In this Letter, we highlight many of the subtle diﬀer-

ences in the ME between the cases of sub-GeV DM and

traditional neutron probes. These diﬀerences arise be-

cause sub-GeV DM is lighter than the neutron and thus

carries less momentum than a neutron of the same kinetic

energy. We carefully delineate the theoretical approxi-

mations made when calculating ME rates to deﬁne the

regime where they continue to hold for neutron scatter-

ing, which diﬀers considerably from the regime of validity

for DM scattering depending on the neutron energy. We

expand the framework of Ref. [38] to include the angular

dependence of neutron scattering, as is used in standard

neutron calibration experiments involving neutron detec-

1See, however, Ref. [46] which employed a similar setup to perform

the ﬁrst inelastic scattering measurements of eV-energy neutrons

from liquid targets and which motivated the ﬁrst derivation of

the ME in molecules [41], though the neutron energies were too

low to observe the Migdal signal.

arXiv:2210.04917v2 [hep-ph] 22 Mar 2023

tor backing arrays. A key ﬁnding of this study is that,

for an isotropic target in the limit of small momentum

transferred to the electronic system (the “soft limit”),

the angular distribution of the scattered neutron factor-

izes from the electronic matrix element, allowing for a

direct calibration of this matrix element for DM scatter-

ing. We demonstrate that the electronic matrix element

for the ME can be measured with a greatly reduced or

even completely absent elastic scattering background by

a judicious choice of the neutron beam energy and the

neutron scattering angle. However, the expected rates we

ﬁnd for such a measurement are at the boundary of what

is currently feasible with existing setups and techniques.

We conclude that, in order to measure the ME with neu-

trons in semiconductors, a dedicated low-energy neutron

calibration setup is required, and propose one such ex-

periment using modiﬁcations to the existing NEXUS fa-

cility [47] at Fermi National Accelerator Laboratory (Fer-

milab).

The ME is deﬁned as the ionization or excitation of an

atomic electron accompanying the recoil of the atom’s

nucleus [5]. For sub-GeV DM, the ME greatly enhances

the sensitivity of direct detection experiments to the DM-

nucleon cross section [15, 16, 48] because the electronic

excitations are observable even when the nuclear recoil is

below threshold. For both isolated atoms and semicon-

ductors, under various sets of assumptions (which will

be discussed further below and delineated in detail in

Appendices B and C), the ME rate spectrum RMfac-

torizes into a quasi-elastic nuclear recoil rate Rel and an

electronic excitation probability de

Pe/dω, such that

d2RM

dErdω =dRel

dEr×q2de

dω .(1)

Here, Eris the nuclear recoil energy, ωis the total energy

deposited in the electronic system (excitation or ioniza-

tion), and ~q =qˆqis the momentum transfer from the

neutron probe to the target. For both classes of targets,

the electronic spectrum scales as q2, and we have explic-

itly factored out this scaling. The goal of this Letter is to

devise a scheme to measure de

Pe/dω in semiconductors.

For isolated atoms, the electronic ionization spectrum

is [12]

dω !atom

=me

mN21

2πX

i,f |hψ(ω)

f|ˆq·~re|ψii|2,(2)

where meand mNare the electron and nucleus mass, re-

spectively, ~reis the electron position operator, the sum

runs over initial and ﬁnal single-electron orbital quan-

tum numbers, and the ﬁnal state is a spherical wave

with wavenumber k=p2me(ω− |Eb|) where Ebis the

binding energy of the initial state. Eq. (2) was de-

rived within the context of the Born-Oppenheimer ap-

proximation [12], but in fact does not require this as-

sumption and is correct to O(me/mN)2[1, 41]. On the

other hand, Eq. (2) does assume qmN/(mea0)'

200 MeV mN

26 GeV where a0is the Bohr radius, since it

was derived from the dipole approximation to the expo-

nential exp ime

mN~q ·~re. Note that the dependence on

ˆqin Eq. (2) drops out when summing over spherically-

symmetric ﬁlled electron shells, such that de

Pe/dω is

isotropic.

In a solid-state system, the ME derivation must be

modiﬁed because the constituent atoms are no longer

free, which gives a characteristic energy scale ωph for op-

tical phonon excitation (≈10 − −100 meV in typical

solid-state systems [1]) and also removes the constraint

of exact momentum conservation for the nucleus because

the atoms are no longer in momentum eigenstates. How-

ever, the form of Eq. (1) can be recovered under the fol-

lowing three assumptions:

•Impulse approximation: q&p2mNωph

•Free-ion approximation: initial nucleus state is a

zero-momentum plane wave

•Soft limit: kqand ~q ·~

kmNω, where ~

kis the

momentum transferred to the electronic system.

The result for a solid-state system is [20]

dω !sol.

=4α

ω4m2

NZd3~

(2π)3Z2

ion(k)(ˆq·ˆ

k)2W(~

k, ω),

(3)

where Zion(k) is an eﬀective momentum-dependent

charge of the nucleus plus inner-shell electrons, α'

1/137 is the ﬁne structure constant, and W(~

k, ω) is the

energy loss function (ELF) of the target which mea-

sures its response to charge perturbations. If the ELF

is isotropic and only depends on the magnitude of ~

k, the

dependence on ˆqdrops out, as in the atomic case. We

have veriﬁed that for all of the kinematic conﬁgurations

we will consider, the impulse approximation is valid.

To demonstrate the main kinematic features of the

ME, we model the valence shell of silicon with Eq. (3), us-

ing the isotropic GPAW ELF from DarkELF [49], which has

a regime of validity ω.75 eV, k.22 keV. For larger

ω, we model the inner-shell electrons with Eq. (2) [12].

This division is somewhat artiﬁcial, and we discuss its

limitations in Appendix C. As we show in Appendices A

and B, in the soft limit, we can convert the nuclear recoil

spectrum into an angular spectrum:

d2PM

dcos θndω =de

dcos θnEn, ω, cos θnde

dω ω,(4)

where Enis the kinetic energy of the incident neutron,

θnis the lab-frame angle of the scattered neutron, ˜

is a kinematic prefactor containing all angular depen-

dence, and we have expressed the spectrum as a diﬀer-

ential probability PMof Migdal scattering per incident

neutron (rather than a ﬂux-dependent rate). Fig. 1 illus-

trates these kinematics and the experimental setup.

FIG. 1. A diagram of an ideal neutron scattering experiment with a backing array, which consists of a series of active (red) and

passive (grayscale) elements. Neutrons are generated isotropically from a source placed inside of a shield with a small opening

to collimate the beam. These neutrons then enter the vacuum chamber (often a dilution refrigerator) with energy En, and

scatter with lab-frame angle θninto a circular backing array element after transferring Erof energy to the nuclear recoil and

energy ωto electrons, which together are detected as ionization energy Eion. Unscattered neutrons, meanwhile, pass through

a capture detector (e.g. 3He counter) to help normalize the simulated beam ﬂux before arresting in a beamstop.

In Eq. (4), we have explicitly noted the separation of

the ionization (ME) probability and the kinematic pref-

actor containing the angular dependence, and absorbed

the q2scaling from Eq. (1) into e

Psuch that it now has

units of (events/neutron)×[eV]2. This unconventional

choice of normalization allows us to group all the terms

depending on the experimentally-controllable variables

Enand cos θntogether in the explicit expression

dcos θn

=N0ρTLσel

µ2mNEn

βm2

nmn

cos θn+β2

×(1−µ2

nmn

cos θn+β2

−ω

En),

(5)

where mnis the neutron mass, µ≡mnmN/(mn+mN) is

the reduced mass, σel is the elastic neutron cross section

on a target material with density ρTand thickness L

in the beam direction, ANis the target’s atomic mass

number, N0is Avogadro’s number, and

β≡s1−m2

(1 −cos2θn)−mnω

µEn

.(6)

In Appendix C, we demonstrate that the factorization

of Eq. (4) that allows this separation does not hold out-

side of the soft limit for a semiconductor. We therefore

note that calibrating the semiconductor ME outside of

the soft limit, for example with high-energy (MeV-scale)

neutrons, is fundamentally no longer probing the same

regime as sub-GeV DM-nucleus scattering, where the soft

limit approximations always hold. For the proposed cal-

ibrations discussed in the rest of this Letter, we will fall

safely within the soft limit (see Appendix B), and thus

the results of any such calibration are eﬀectively a mea-

surement of de

Pe/dω that can be directly translated to

DM-nucleus scattering.

Since we do not measure ωdirectly, we change variables

again to the observable Eion, the total amount of energy

available as ionization, deﬁned as

Eion ≡ω+fn(Er)Er,(7)

where fn(Er) denotes the ionization eﬃciency for elastic

nuclear recoils as a function of the nuclear recoil energy

Er(ω, θn). For the purposes of this work we consider

the Sarkis model [50] as a best theoretical approxima-

tion for the ionization eﬃciency in the mostly uncali-

brated regime of small Er[51]. In general, calibrations

of the ME will be dependent on this ionization eﬃciency

(quenching) model; however, we propose two speciﬁc cal-

ibration schemes in this Letter designed to minimize this

dependence. A more complete treatment would also con-

sider the systematic or theoretical ﬂuctuations in fn(Er),

but this is outside the scope of this work.

To predict the number of electron-hole pairs neas a

function of Eion, we use the charge production model

presented in Ref. [52]. This is a data-driven model of

impact ionization in silicon that more accurately models

the response for low nethan a model of Fano statistics

alone (i.e. Ref. [53]). Ref. [52] provides a set of functions

pne(Eion) for the probability of producing nepairs for

energy deposit Eion. Thus, to compute measured ion-

ization rates as a function of angle, we integrate Eq. (4)

against pneto ﬁnd the diﬀerential angular probability of

Migdal events binned in ne,

dPne

dcos θn

=ZdEion pne(Eion)d2PM

dcos θndEion

.(8)

The inherent widths of the pneleads to a smearing eﬀect

that can aﬀect our signal (even before considering ex-

perimental factors, such as non-monochromaticity of the

beam). We show two examples of observable spectra in

Fig. 2 for diﬀerent choices of Enand θn, where we decom-

pose the spectral contributions to the rate from elastic,

valence band ME, and inner shell ME scatters.

FIG. 2. Diﬀerential probability spectra dPne/dθn(in units of events/neutron/degree of angular coverage) are shown per

detectable charge quanta nein the left (right) plot for an ideal 1 cm thick silicon detector in a En= 24 (2) keV monoenergetic

neutron beam at a ﬁxed scattering angle of θn= 72 (10) degrees, assuming the Sarkis ionization eﬃciency (quenching) model [50]

and Ramanathan charge production model [52]. In both cases, we assume perfect backing detector with full azimuthal coverage.

Left: for higher neutron energies and wide angles, the contribution from the inner shell [12] is distinct above the elastic peak.

Right: for low neutron energies and shallow angles, the contribution from the valence band [49] separates from the elastic

peak.

Fig. 2 illustrates two possible strategies for calibrating

the ME in the correct kinematic regime. The q2scaling

of the Migdal probabilities translates to an enhancement

approximately proportional to En(1 −cos θn). Larger

momentum transfers (which lead to larger nuclear recoil

energies), achieved either by raising the neutron beam

energy or by looking at a larger scattering angle, will

therefore give a higher rate of Migdal events. However,

to avoid the aforementioned diﬃculties of the inherent

smearing due to the Fano statistics, it is important to

keep the nuclear recoil energy scale small enough that

the elastic spectrum does not smear too much into the

Migdal tail. Thus, the ﬁrst experimental strategy is to

target setups that balance the q2rate enhancement with

low recoil energy, in order to clearly isolate the high-

side Migdal rate tail. As can be seen in the left plot of

Fig. 2, this strategy is particularly useful for calibrating

Eq. (2), the ME contribution for inner shell electrons, but

care must be taken not to increase the neutron energy

and scattering angle outside of the kinematic regime of

interest (see Appendix B).

The second strategy is to employ low-energy neutrons

scattering at low angles such that the quenched nuclear

recoils are too small to produce any secondary ionization,

eﬀectively eliminating the observable elastic contribution

(the second term of Eq. (7)). This strategy is challeng-

ing in that it involves novel neutron source development,

but is able to calibrate Eq. (3), the ME contribution from

valence electrons independent of other contributions, as

shown in the right plot of Fig. 2. Since sub-GeV DM

will typically only produce single- to few-electron events,

this setup more closely mimics what a sub-GeV DM sig-

nal would look like in a single-electron threshold charge

detector.

In a real experiment, no neutron beam will be perfectly

monochromatic, such that contamination from higher-

energy neutrons and gamma backgrounds must be taken

carefully into account through validated simulations.

There are a number of common neutron sources and

methods that are implemented in the lab, each of which

can be turned into a fairly monochromatic beam with

careful application. These include deuterium-deuterium

(D-D) and deuterium-tritium (D-T) generators [54], pro-

ton accelerators incident on a 7Li [55] or 51V [56] target,

and photoneutron sources that exploit the 9Be disinte-

gration threshold of 1.67 MeV [57, 58]. Each of these op-

tions comes with its own advantages and disadvantages,

so we will emphasize that the fairly rare probability of

a Migdal scatter, even in an ideal setup, necessitates (a)

a low-background environment, as is typically achieved

with signiﬁcant overburden, thus complicating the use

of proton accelerators, and (b) a high ﬂux of low-energy

neutrons, which can be achieved with either photoneu-

tron sources or moderated D-D (or D-T) generators. Col-

limation in both of these cases is achieved through robust

4πshielding minus a small beam hole, typically around

1 cm in diameter (dependent on detector size).

Because of logistical challenges associated with hav-

ing a suﬃciently high-activity gamma source to produce

a high ﬂux of neutrons from a photoneutron source, we

will focus the rest of the Letter on using a D-D neutron

generator, as is employed by NEXUS. A D-D generator

leverages fusion reactions to generate isotropic 2.5 MeV

neutrons without any primary gamma backgrounds [54]

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FERMILAB-PUB-22-705-PPD-QIS-TMeasuringtheMigdaleectinsemiconductorsfordarkmatterdetectionDuncanAdams,1,DanielBaxter,2,yHannahDay,3,zRouvenEssig,1,xandYonatanKahn3,4,{1C.N.YangInstituteforTheoreticalPhysics,StonyBrookUniversity,NY11794,USA2FermiNationalAcceleratorLaboratory,Batavia,IL60510,USA3Depa...

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