# Analysis on the Derivation of the Schr dinger Equation with Analogy to
Electromagnetic Wave Equation

Xuefeng Bao

##  1 Introduction

The Schr dinger equation is universally accepted due to its excellent
predictions aligning with observed results within its defined conditions.
Nevertheless, it does not seem to possess the simplicity of fundamental laws,
such as Newton’s laws of motion. Various insightful attempts have been made to
elucidate the rationale behind the Schr dinger equation [Derbes 1996,
Efthimiades 2006, Field 2010, Ward and Volkmer 2006]. This paper seeks to
review existing explanations and propose some prospectives on the derivation
of the Schr dinger equation.

##  2 Derivation

###  2.1 General Solution

Consider a complex wave which propagates in $x$ direction:

$\psi\left(x,t\right)=Ae^{i\left(kx-\omega t\right)}$ (1)

where $\psi$ denotes the wave function, $A$ the amplitude, $k$ the wave
number, $\omega$ the angular frequency, $x$ the position, and $t$ the time.

We can see that the real part of (1), i.e., $Re\left[Ae^{i\left(kx-\omega
t\right)}\right]=Acos\left(kx-\omega t\right)$, is a cosine function that can
represent a real wave and is independent from a wave equation, i.e., we don’t
need a wave equation to get this cosine form because this form itself
represents a wave that exists naturally. But, for mathematical convenience, we
would prefer to use its complex version.

Taking twice partial derivation on $t$ for (1) yields

$\frac{\partial^{2}}{\partial t^{2}}\psi=-\omega^{2}Ae^{i\left(kx-\omega
t\right)}$ (2)

and taking twice partial derivation on $x$ we can obtain

$\frac{\partial^{2}}{\partial x^{2}}\psi=-k^{2}Ae^{i\left(kx-\omega
t\right)}.$ (3)

Combining (2) and (3), and considering $k=\frac{2\pi}{\lambda}=\frac{2\pi
f}{v}=\frac{\omega}{v}$ (here $\lambda$ is wavelength, $f$ is frequency, and
$v$ is velocity), we have

$\frac{\partial^{2}}{\partial
x^{2}}\psi=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}\psi$ (4)

which is a typical wave equation. We can also verify that (1) is solution of
(4).

We may notice that (1) is represented by a pair $\left(k,\omega\right)$, and
(4) fits this representation. What if we change $\left(k,\omega\right)$ to
$\left(E,p\right)$ (here $E$ is energy and $p$ is momentum)? What wave
equation would fit the wave function represented by the new pair
$\left(E,p\right)$?

###  2.2 Electromagnetic Wave

According to the Photoelectric Effect, each photon in an electromagnetic wave
carries an energy in the form of

$E=\hbar\omega$ (5)

and, according to the basic properties of Matter Waves

$p=\hbar k$ (6)

and therefore, (1) can be rewritten as

$\psi\left(x,t\right)=Ae^{\frac{i}{\hbar}\left(px-Et\right)}.$ (7)

Taking $\frac{\partial^{2}}{\partial t^{2}}$ and $\frac{\partial^{2}}{\partial
x^{2}}$ on (7), we can get

$\frac{\partial^{2}}{\partial
t^{2}}\psi=-\frac{E^{2}}{\hbar^{2}}Ae^{i\left(px-Et\right)}$ (8)

and

$\frac{\partial^{2}}{\partial
x^{2}}\psi=-\frac{p^{2}}{\hbar^{2}}Ae^{i\left(px-Et\right)}.$ (9)

Relativistic total energy is defined as

$E^{2}=p^{2}c^{2}+m^{2}c^{4}$ (10)

where $c$ denotes the speed of light. As the moving photon does not carry mass
(i.e., $m=0$), (10) can be rewritten as

$E^{2}=p^{2}c^{2}.$ (11)

Combine (8), (9) and (11), we can get

$\frac{\partial^{2}}{\partial
x^{2}}\psi=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\psi$ (12)

which is a (plane) wave equation for (monochromatic) electromagnetic wave.

###  2.3 Wave Equation of Particles

Relativistic total energy of a photon purely equals its kinetic energy, i.e.,
(11). However, this is not true for a particle with nonzero mass, i.e, (10).
If we mechanically put the momentum and energy of a particle into (7) and want
to establish a wave equation, what type of energy we should use? The
relativistic total energy or the kinetic energy of the particle?

####  2.3.1 Derivation with relativistic total energy

Assuming the relationship between momentum and energy is governed by (10), and
substitute it in (8), we can get

$\frac{\partial^{2}}{\partial
t^{2}}\psi=-\frac{p^{2}c^{2}}{\hbar^{2}}\psi-\frac{m^{2}c^{4}}{\hbar^{2}}\psi$

which can be reformed to

$-\frac{p^{2}}{\hbar^{2}}\psi=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\psi+\frac{m^{2}c^{2}}{\hbar^{2}}\psi$

the left hand side exactly equals the right hand side of (9). Therefore, we
have

$\frac{\partial^{2}}{\partial
x^{2}}\psi-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\psi-\frac{m^{2}c^{2}}{\hbar^{2}}\psi=0$ (13)

which is the Klein-Gordon equation.

Then, we may approximate (10) in a way like

$E=mc^{2}\left(1+\frac{p^{2}}{m^{2}c^{2}}\right)^{\frac{1}{2}}\approx
mc^{2}\left(1+\frac{p^{2}}{2m^{2}c^{2}}\right)=mc^{2}+\frac{p^{2}}{2m}$

which is a combination of the rest energy and the classical kinetic energy.
Let’s define the classical kinetic energy as:

$\mathcal{T}=\frac{p^{2}}{2m}.$ (14)

Therefore, (7) can be written as

$\psi=Ae^{\frac{i}{\hbar}\left(px-mc^{2}t-\mathcal{T}t\right)}$

or

$\psi=e^{\frac{i}{\hbar}-mc^{2}t}\psi_{c}$ (15)

where

$\psi_{c}=Ae^{\frac{i}{\hbar}\left(px-\mathcal{T}t\right)}$ (16)

which is equivalent to (7) when the energy $E$ is purely kinetic energy
$\mathcal{T}$.

Taking $\frac{\partial^{2}}{\partial t^{2}}$ to (15), we obtain

$\frac{\partial^{2}}{\partial
t^{2}}\psi=-e^{-\frac{i}{\hbar}mc^{2}t}\left(\frac{m^{2}c^{4}}{\hbar^{2}}\psi_{c}+\frac{2i}{\hbar}mc^{2}\frac{\partial\psi_{c}}{\partial
t}\right)+e^{-\frac{i}{\hbar}mc^{2}t}\frac{\partial^{2}}{\partial
t^{2}}\psi_{c}.$ (17)

In [ward2006derive], the following condition seems to be assumed to hold:

$\left|-e^{-\frac{i}{\hbar}mc^{2}t}\left(\frac{m^{2}c^{4}}{\hbar^{2}}\psi_{c}+\frac{2i}{\hbar}mc^{2}\frac{\partial\psi_{c}}{\partial
t}\right)\right|\gg\left|e^{-\frac{i}{\hbar}mc^{2}t}\frac{\partial^{2}}{\partial
t^{2}}\psi_{c}\right|$ (18)

so that (17) can be approximated as

$\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\psi=-e^{-\frac{i}{\hbar}mc^{2}t}\frac{m^{2}c^{2}}{\hbar^{2}}\psi_{c}-\frac{2mi}{\hbar}e^{-\frac{i}{\hbar}mc^{2}t}\frac{\partial\psi_{c}}{\partial
t}.$ (19)

Combine (19) and (13), and cancel $\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\psi$, we can obtain

$-e^{-\frac{i}{\hbar}mc^{2}t}\frac{m^{2}c^{2}}{\hbar^{2}}\psi_{c}-\frac{2mi}{\hbar}e^{-\frac{i}{\hbar}mc^{2}t}\frac{\partial\psi_{c}}{\partial
t}=\frac{\partial^{2}}{\partial x^{2}}\psi-\frac{m^{2}c^{2}}{\hbar^{2}}\psi$

and use (15) again, we obtain

$i\hbar\frac{\partial\psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}$ (20)

which is exactly the form of Schr dinger Equation. But, we need to bear in our
mind that the solution of (20), i.e., (7) carries relativistic total energy,
i.e., (10) not $E=\frac{p^{2}}{2m}$. Also, several approximations are made to
get the final conclusion. Dogmatically, the Klein-Gordon equation is only
valid for spin-0 particles. And the speed of light is a large number but not
infinity. Nevertheless, the derivation above provides a great heuristic
inspiration.

####  2.3.2 Derivation with classical kinetic energy

Assuming that in (7), the energy is purely the classical kinetic energy, i.e.,
$E$ in (7) equals $\mathcal{T}$ in (14), or explicitly:

$E=\frac{p^{2}}{2m}.$ (21)

We can rewrite (9) as

$-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial
x^{2}}=\frac{p^{2}}{2m}\psi$ (22)

and $\frac{\partial}{\partial t}$ on (7) and rearrange we can get

$-\frac{\hbar}{i}\frac{\partial\psi}{\partial t}=E\psi.$ (23)

Combining (22), (23), and (21), we have

$-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial
x^{2}}=-\frac{\hbar}{i}\frac{\partial\psi}{\partial t}$

which is

$-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial
x^{2}}=i\hbar\frac{\partial\psi}{\partial t}$ (24)

which is identical to (20), but the $E$ in (24) is the classical kinetic
energy, i,e., $E=\frac{p^{2}}{2m}$. If we want to add the potential energy
,$V$, into (24), we simply get

$\left[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+V\right]\psi=i\hbar\frac{\partial\psi}{\partial t}.$ (25)

During this derivation, $E=\hbar\omega$ (from (6) to (7)), which is for
photons, is applied to the particle with mass. If we assume
$E=\frac{1}{2}mv^{2}$, where $v$ is the velocity of the particle, we will get
exactly (4) through (2) and (3). But this is not the Schr dinger equation.

##  3 Interpretation

In the above analysis, we encounter a noteworthy problem. You may notice that
we transformed $\left(k,\omega\right)$ representation into $\left(E,p\right)$
using (5) (photoelectric effect) and (6) (matter waves), and thus, ultimately
arriving at (7). While the matter wave property holds true for both photons
and particles, the same cannot be assumed for the photoelectric effect. In the
above derivations, we directly started from (7), where (5) is implied to be
true. In other words, we imposed the photoelectric effect on particles by
assuming $E=\hbar\omega$ for a particle with mass. Is that a correction
operation? If so, why we can apply the photoelectric effect to particles? If
not, how can we derive the Schr dinger equation, which is widely considered to
be true?

###  3.1 Harmonic Oscillation

When an electron is performing harmonic oscillation, the ground state energy
is $E_{0}=\frac{\hbar\omega_{c}}{2}$, where $\omega_{c}$ is a classical
frequency that represents the frequency of the oscillation. Indeed, it is not
precisely $E_{0}=\hbar\omega_{c}$, and it is disturbing to assume that an
eletron is always in a harmonic oscillation. But at least, we have
$E_{0}\propto\hbar\omega_{c}$, Also, we can get a $\hbar\omega_{c}$-like
energy before we assume that he the Schr dinger equation itself is true. We
can start with the Hamiltonian,

$\hat{H}=\frac{1}{2m}\hat{p}^{2}+\frac{m\omega_{c}^{2}}{2}\hat{x}^{2}$ (26)

which is classical, where $\hat{\cdot}$ denotes operator and $\omega_{c}$ the
classical frequency. Applying substitution
$\hat{p}\rightarrow\frac{\hbar}{2\bigtriangleup x}$ and
$\hat{x}\rightarrow\bigtriangleup x$, where $\bigtriangleup x\bigtriangleup
p\geq\frac{\hbar}{2}$ (i.e., Heisenberg uncertainty relation), to (26), we
obtain [Bes 2007]:

$E\geq\frac{\hbar^{2}}{8m\left(\bigtriangleup
x\right)^{2}}+\frac{m\omega_{c}^{2}\left(\bigtriangleup x\right)^{2}}{2}$ (27)

which means that when $\bigtriangleup
x=\left(\frac{\hbar}{2m\omega_{c}}\right)^{\frac{1}{2}}$ we have:

$E_{0}=\frac{\hbar\omega_{c}}{2}.$

###  3.2 Feynman Integration

If we derive from the Feynman Integration, we still need an assumption like
$\psi=e^{\frac{i}{\hbar}S}$, where $S$ is the action, which is similar to (7),
and this is also the key to get the Schr dinger equation [Field 2010]. No one
explains why it can be applied to particles. Also, it’s hard to say if Feynman
path proves the Schr dinger equation or it’s derived to fit the Schr dinger
equation. Actually, we don’t need to be bothered with that, because the
photoelectric effect is more likely to be used to describe a path (maybe a
trajectory of a small part of an electron). Therefore, can we assume that (5)
is good for a small part of an electron? And all the small parts form the
electron? It sounds deviant as the electron is not separable according to the
Standard Model.

###  3.3 Virtual Photon

In fact, a closer examination of the interpretation of the wave function (1)
as a solution to an electromagnetic wave equation, such as (12), can offer
valuable insights. Suppose the amplitude of the electromagnetic wave is
directly related to the number of photons propagating in that direction
[Feynman 1998]. In that case, it is conceivable that $\psi$ in (4) is
associated with the number of photons, which form a ’beam.’ Hence, when we
explore $\psi$ in the context of the Schr dinger Equation for a particle, it
might similarly represent a ’beam,’ analogous to the electromagnetic wave. For
the electromagnetic wave, the constituents of the beam are photons. If we
consider a particle to be a ’beam,’ the question arises: what comprises this
beam? Are the constituents also photons? If so, why do we get the Schr dinger
equation instead of an electromagnetic wave equation? A closer examination
reveals that the relationship between energy and momentum for a particle is
defined by $E=\frac{p^{2}}{2m}$ (not $E=pc$), indicating that the mass of the
beam (or each constituent) is non-zero. This is the key reason that we observe
a second-order time derivative in the electromagnetic wave equation, as
opposed to the first-order time derivative in the Schr dinger equation. The
question that naturally follows is: how should we interpret it?

Virtual photons are widely recognized as a mathematical tool rather than a
physical reality. However, considering electrons as composed of virtual
photons becomes a useful framework for explaining various issues we have
discussed previously. Virtual photons exhibit photon-like behavior, allowing
us to conceptualize an electron as a stream of virtual photons. Just as the
amplitude of an electromagnetic wave depends on the number of photons, the
amplitude in the wave function of the Schr dinger equation also relies on the
number of virtual photons. Therefore, the $\psi$ in (12) and (24) share the
same interpretation. If electrons interact by exchanging virtual photons, it
may be reasonable to think that virtual photons are integral components of an
electron. Furthermore, virtual photons can interact with the Higgs field and
acquire mass. This could explain why an electron can be modeled as a beam of
virtual light and obeys $E=\frac{p^{2}}{2m}$. The enigma of why, when
measuring an electron, virtual photons tend to get gathered in locations with
the highest concentration is still mysterious. Nevertheless, this stochastic
process can be understood from a classical perspective [Nelson 1966], and the
gathering (collapse) process is at the speed of light. Moreover, it can easily
explains the double-slit experiments. Also, photon are like oscillations
according to QFT, and each virtual photon can travel along a path described by
Feynman.

##  4 Conclusion

In this work, some derivations of the Schr dinger equation were reviewed. One
interesting thing is that in many existing works, $E=\hbar\omega$ is assumed
to be true for a particle under the Schr dinger equation. In fact, if this is
true, and combine with the solution for an electromagnetic wave equation, the
Schr dinger equation can be naturally obtained. Therefore, instead of stating
that the Schr dinger equation is correct, we can rather say that
$E=\hbar\omega$ is true for the wave function of a particle. In this paper, an
effort is made to explain how $E=\hbar\omega$ can be true for a particle with
mass.

## References

[1] Ward, D.W. and Volkmer, S.M., 2006. How to derive the schrodinger
equation. arXiv preprint physics/0610121.

[2] Efthimiades, S., 2006. Physical meaning and derivation of Schrodinger and
Dirac equations. arXiv preprint quant-ph/0607001.

[3] Derbes, D., 1996. Feynman’s derivation of the Schr dinger equation.
American Journal of Physics, 64(7), pp.881-884.

[4] Field, J.H., 2010. Derivation of the Schr dinger equation from the
Hamilton–Jacobi equation in Feynman’s path integral formulation of quantum
mechanics. European journal of physics, 32(1), p.63.

[5] Bes, D.R., 2007. Quantum mechanics: a modern and concise introductory
course (p. 43). Berlin Heidelberg: Springer.

[6] Nelson, E., 1966. Derivation of the Schr dinger equation from Newtonian
mechanics. Physical review, 150(4), p.1079.


# Asymmetric jet clustering in deep-inelastic scattering

M. Arratia<EMAIL_ADDRESS>Physics Department, University of
California, Riverside, CA 92521, USA  Thomas Jefferson National Accelerator
Facility, Newport News, VA 23606, USA  Y. Makris<EMAIL_ADDRESS>INFN Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy  D. Neill
<EMAIL_ADDRESS>Theoretical Division, MS B283, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  F. Ringer<EMAIL_ADDRESS>Nuclear
Science Division, Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA  N. Sato<EMAIL_ADDRESS>Theory Center, Jefferson Laboratory,
Newport News, Virginia 23606, USA

(August 27, 2024)

###### Abstract

We propose a new jet algorithm for deep-inelastic scattering (DIS) that
accounts for the forward-backward asymmetry in the Breit frame. The Centauro
algorithm is longitudinally invariant and can cluster jets with Born
kinematics, which enables novel studies of transverse-momentum-dependent
observables. Furthermore, we show that spherically-invariant algorithms in the
Breit frame give access to low-energy jets from current fragmentation. We
propose novel studies in unpolarized, polarized, and nuclear DIS at the future
Electron-Ion Collider.

††preprint: JLAB-THY-20-3209

Introduction. Understanding the structure of nucleons and nuclei in terms of
quark and gluons remains an open goal. Jet production in deep inelastic
scattering (DIS) provides an excellent tool for this endeavor. The future
Electron-Ion Collider (EIC) Accardi:2012qut  will produce the first jets in
polarized and nuclear DIS, which will enable a rich jet program Guzey:2019kik
; Boughezal:2018azh ; Klasen:2018gtb ; Klasen:2017kwb ; Chu:2017mnm ;
Abelof:2016pby ; Dumitru:2018kuw ; Hinderer:2017ntk ; Hinderer:2015hra ;
Liu:2018trl ; Kishore:2019fzb ; DAlesio:2019qpk ; Zheng:2018awe ; Boer:2016fqd
; Dumitru:2015gaa ; Pisano:2013cya ; Boer:2010zf ; Kang:2011jw ; Hatta:2019ixj
; Mantysaari:2019csc ; Roy:2019hwr ; Salazar:2019ncp ; Dumitru:2016jku ;
Hatta:2016dxp ; Altinoluk:2015dpi ; Liu:2018trl ; Sievert:2018imd ;
Kang:2013nha ; Kang:2012zr ; Aschenauer:2019uex ; Zhang:2019toi ; Li:2020sru ;
Kang:2020xyq .

The HERA jet measurements in DIS targeted gluon-initiated processes by
requiring large transverse momentum in the Breit frame Newman:2013ada . This
suppresses the Born configuration, $\gamma^{*}q\to q$, which has recently been
postulated as key to probe transverse-momentum dependent (TMD) PDFs Gutierrez-
Reyes:2018qez ; Gutierrez-Reyes:2019vbx ; Gutierrez-Reyes:2019msa .
Complementary to semi-inclusive DIS observables, jets avoid nonperturbative
TMD fragmentation functions. Moreover, modern jet substructure techniques
Larkoski:2017jix  offer new methods for precise QCD calculations and to
control nonperturbative effects, e.g grooming or a recoil-free axis can be
used to minimize hadronization effects or study TMD evolution Makris:2017arq .
These techniques also provide new ways to connect to lattice QCD calculations,
e.g. of the nonperturbative Collins-Soper kernel Ebert:2018gzl .

The Breit frame plays a central role in jet clustering for DIS Webber:1993bm ,
and it allows for a factorized TMD cross section in terms of the same soft and
un-subtracted TMD functions as in Drell-Yan and $e^{+}e^{-}\to$ dihadron/dijet
processes Gutierrez-Reyes:2018qez ; Gutierrez-Reyes:2019vbx ; Gutierrez-
Reyes:2019msa . However, the longitudinally-invariant (LI) algorithms commonly
used in DIS cannot cluster jets that enclose the beam axis given by the
proton/photon direction (see Fig. 1).

In this letter, we introduce a new jet algorithm that is longitudinally
invariant but can capture jets close to the Born configuration in the Breit
frame. In addition, we use spherically-invariant (SI) algorithms to study the
jet energy spectrum and find that they can separate the current and target
fragmentation region even for soft jets. Finally, we suggest novel studies of
jet energy and TMD observables.

Figure 1: DIS Born kinematics in the Breit frame.

Figure 2: Jet clustering in the Breit frame using the longitudinally-invariant
anti-$k_{T}(\text{LI})$, Centauro, and spherically-invariant
anti-$k_{T}(\text{SI})$ algorithms in a DIS event simulated with Pythia 8\.
Each particle is illustrated as a disk with area proportional to its energy
and the position corresponds to the direction of its momentum projected onto
the unfolded sphere about the hard-scattering vertex. The vertical dashed
lines correspond to constant $\theta$ and curved lines to constant $\phi$. All
the particles clustered into a given jet are colored the same.

Notation and DIS kinematics. In the Breit frame, the virtual photon momentum
is given by:

$q^{\mu}=\frac{Q}{2}(\bar{n}^{\mu}-n^{\mu})=Q(0,0,0,-1)\,,$ (1)

where $n^{\mu}\equiv(1,0,0,+1)$ and $\bar{n}^{\mu}\equiv(1,0,0,-1)$. The
proton momentum (up to mass corrections) is:

$P^{\mu}\simeq Q/(2x_{B})n^{\mu}=Q/(2x_{B})(1,0,0,+1)\,,$ (2)

with Bjorken $x_{B}\equiv Q^{2}/(2\,q\cdot P)$. At Born level, the struck
quark back-scatters against the proton and has momentum ($x\simeq x_{B}$):

$p_{q}^{\mu}=xP^{\mu}+q^{\mu}\simeq(Q/2)\bar{n}^{\mu}\,.$ (3)

The fragmentation of the struck-quark yields a jet that points to the beam
direction. The algorithms we introduce below are designed to capture this jet.
We define the scaling variable:

$z_{\text{jet}}=\frac{P\cdot p_{\text{jet}}}{P\cdot q}\simeq\bar{n}\cdot
p_{\rm jet}/Q=p^{-}_{\text{jet}}/Q\,.$ (4)

At leading-logarithmic accuracy, $z_{\text{jet}}$ is the fraction of the
struck-quark momentum carried by the jet.

New jet algorithms for DIS. The longitudinally-invariant $k_{T}$-type jet
algorithms Ellis:1993tq ; Dokshitzer:1997in ; Wobisch:1998wt ; Cacciari:2008gp
; Cacciari:2011ma  use the following distance measure:

$\displaystyle d_{ij}$ $\displaystyle=\min(p_{Ti}^{2p},p_{Tj}^{2p})\Delta
R^{2}_{ij}/R^{2}\;,$ $\displaystyle d_{iB}=p_{Ti}^{2p}\,,$ (5)

where $\Delta R_{ij}=(y_{i}-y_{j})^{2}+(\phi_{i}-\phi_{j})^{2}$. Here $d_{ij}$
is the distance between two particles in the event and $d_{iB}$ is the beam
distance. Since they cluster particles in the rapidity-azimuth ($y-\phi$)
plane, they cannot form a jet enclosing the $\bar{n}^{\mu}$ direction
$(y=-\infty)$.

One way to bypass this problem is to use spherically-invariant algorithms.
Catani et al. first proposed to adapt spherically-invariant algorithms to DIS
in ref. Catani:1992zp . In this study we consider the $k_{T}$-type algorithms
for $e^{+}e^{-}$ collisions Catani:1991hj ; Cacciari:2011ma , which have the
following distance measure:

$\displaystyle d_{ij}$
$\displaystyle=\min(E_{i}^{2p},E_{j}^{2p})\frac{1-c_{ij}}{1-c_{R}}\;,$
$\displaystyle d_{iB}=E_{i}^{2p}\,,$ (6)

where $c_{ij}=\cos\theta_{ij}$ and $c_{R}=\cos R$. However, these algorithms
lack the longitudinal invariance that connects the class of frames related to
the Breit frame by $\hat{z}$ boosts, which is a crucial feature of jet
clustering Webber:1993bm . For example, it is important for multijet events
where the parton kinematics is not constrained by $x_{B}$ and $Q^{2}$, and to
identify photo-production or separate the beam remnant from forward jets
Catani:1993hr .

To solve this issue, we introduce a new jet algorithm that is longitudinally
invariant along the Breit frame beam axis but yet captures the struck-quark
jet. Recently, Boronat et al. Boronat:2014hva  proposed a hybrid algorithm
that suppresses $\gamma\gamma$ background in $e^{+}e^{-}$ colliders. In
contrast, we suggest a jet algorithm that is asymmetric in the backward and
forward directions, and suggest novel studies for spherically-invariant
algorithms in DIS.

Figure 3: The distribution of jets in the Breit frame in terms of their
pseudorapidity $\eta_{\text{jet}}$ and momentum fraction $z_{\text{jet}}$. The
left, center, and right panels correspond to jets identified with the
anti-$k_{T}(\text{LI})$, Centauro, and anti-$k_{T}(\text{SI})$ algorithms,
respectively. The dashed lines indicate the separation of jets in the current
and target fragmentation region.

Starting with the distance measure of the Cambridge/Aachen (C/A) algorithm for
$e^{+}e^{-}$ (i.e Eq. (6) for $p=0$), we write the numerator in Eq. (6) in
terms of the unit vectors along the directions of particles $i$ and $j$,

$\displaystyle 1-c_{ij}$
$\displaystyle=1-\hat{n}_{i}\cdot\hat{n}_{j}=1-s_{i}s_{j}\cos\Delta\phi_{ij}-c_{i}c_{j}\,.$
(7)

with $c_{i}=\cos\theta_{i}$ and $s_{i}=\sin\theta_{i}$. Expanding in the very
backward limit (i.e. $\bar{\theta}_{i}\equiv\pi-\theta_{i}\ll 1$) we find:

$1-c_{ij}\simeq\frac{1}{2}(\bar{\theta}_{i}-\bar{\theta}_{j})^{2}+\bar{\theta}_{i}\bar{\theta}_{j}(1-\cos\Delta\phi_{ij}).$
(8)

We then introduce the replacements:

$\displaystyle\bar{\theta}_{i}$ $\displaystyle\to f_{i}=f(\bar{\eta}_{i})\,,$
$\displaystyle\bar{\eta}_{i}$ $\displaystyle\equiv-\frac{2Q}{\bar{n}\cdot
q}\frac{p_{i}^{\perp}}{n\cdot p_{i}}\,,$ (9)

where the function $f$ must satisfy: $f(x)=x+\mathcal{O}(x^{2})$, and
$p^{\perp}_{i}$ is the transverse momentum in the Breit frame. The term
$\bar{\eta}_{i}$ (which in the Breit frame is $2p^{\perp}_{i}/(n\cdot p_{i})$)
introduces an asymmetry: in the backward region the distance between particles
is given by their separation in $\bar{\eta}$, which decreases as particles
become closer in angle. In contrast, in the forward region $\bar{\eta}$
diverges and thus prevents jets from enclosing the proton beam direction, like
the anti-$k_{T}\text{(LI)}$ algorithm. We thus introduce the following
distance measure:

$\displaystyle d_{ij}$ $\displaystyle=\Big{[}(\Delta
f_{ij})^{2}+2f_{i}f_{j}(1-\cos\Delta\phi_{ij})\Big{]}/R^{2}\;,$ $\displaystyle
d_{iB}=1$ (10)

which defines a new class of algorithms, which we call Centauro algorithms.
Two relevant choices 111Different choices can be optimized to match jets found
by other algorithms, for example we found that jets clustered in the forward
region are better matched to the jets clustered with the anti$-k_{T}$ (LI)
algorithm. for the function $f$ are:

$\displaystyle f(x)$ $\displaystyle=x\,,$ $\displaystyle f(x)$
$\displaystyle=\sinh^{-1}(x)\,.$ (11)

The Centauro algorithm is invariant along the $\hat{z}$ direction, but in the
backward hemisphere it matches the spherically-invariant algorithms (see Eq.
(6)). This feature is largely independent of the choice of $f$ 222For the
analysis in this section we started with the choice $p=0$ in Eq.(6). Analogous
algorithms can be obtained for the choices $p=+1$ and $p=-1$..

Simulation results and applications. Throughout this letter we analyze DIS
events with $Q>10$ GeV simulated in Pythia 8 Sjostrand:2007gs  with 10 and 100
GeV electron and proton beam energies respectively 333In addition, we verified
that our conclusions are the same when using the DIRE shower of ref.
Hoche:2015sya .. We exclude neutrinos and particles with $|\eta|>4$ or
$p_{T}<200$ MeV in the laboratory frame.

Fig. 2 illustrates the anti-$k_{T}(\text{LI})$, Centauro, and
anti-$k_{T}(\text{SI})$ jet clustering for an exemplary Pythia 8 event. The
anti-$k_{T}(\text{LI})$ algorithm clusters the particles from the
fragmentation of the struck-quark into four different jets 444Note that this
cannot be corrected for by choosing a larger jet radius due to the exponential
increase of the rapidity in the forward and backward region.. In contrast, the
anti-$k_{T}(\text{SI})$ and Centauro algorithms cluster all of these particles
into a single jet with $z_{\text{jet}}\sim 1$. The Centauro algorithm mimics
the features of the anti-$k_{T}(\text{SI})$ in the backward direction and the
anti-$k_{T}(\text{LI})$ in the forward direction.

Furthermore, with the use of Centauro and anti-$k_{T}(\text{SI})$ jets it is
also possible to suppress the target fragmentation with a cut on
$z_{\text{jet}}\sim 0.2-0.7$, as shown in Fig. 3 (center panel). This allows
for direct studies of quark TMD observables. For the anti-$k_{T}(\text{SI})$
555Note that we include a factor of $\pi/2$ for the jet radius of the
anti-$k_{T}(\text{SI})$ algorithm to account for the difference in the
denominator of the clustering metric in eq. (6) compared to eq. (5).
algorithm, a cut on $\eta_{\text{jet}}<1$ separates current and target regions
(right panel of Fig. 3). This reveals the full $z_{\text{jet}}$ spectrum,
which cannot be accessed with hadron measurements due to the contamination
from the target fragmentation region Boglione:2016bph ; Aschenauer:2019kzf .
For comparison we also show the result for the anti-$k_{T}(\text{LI})$
algorithm in the left panel of Fig. 3.

Figure 4: Pseudorapidity (top panel) and momentum fraction $z_{\text{jet}}$
(bottom panel) of jets clustered with anti-$k_{T}(\text{LI})$,
anti-$k_{T}(\text{LI})$ and Centauro algorithms in the Breit frame.

Fig. 4 shows the $z_{\text{jet}}$ and $\eta_{\text{jet}}$ distributions of
inclusive jets as described above. While in the backward region
($\eta_{\text{jet}}<0$), the Centauro and anti-$k_{T}(\text{SI})$ algorithms
result in a peak at large $z_{\text{jet}}\sim 1$, the anti-$k_{T}(\text{LI})$
algorithm separates that jet into several and yields a peak at
small-$z_{\text{jet}}$. The two peaks at $z_{\text{jet}}\sim 1$ and
$z_{\text{jet}}\sim 0$ correspond to backward and mid rapidity jets. The
intermediate $z_{\text{jet}}$ region is described in terms of jet functions
and DGLAP evolution Dasgupta:2014yra ; Kaufmann:2015hma ; Kang:2016mcy ;
Dai:2016hzf . The large-$z_{\text{jet}}$ jets probe the threshold region
Dai:2017dpc , whereas the small-$z_{\text{jet}}$ region is related to soft
fragmentation in $e^{+}e^{-}$ collisions Mueller:1981ex ; Bassetto:1982ma ;
Dokshitzer:2005bf ; Kom:2012hd ; Anderle:2016czy  and small-$x$ physics
Basso:2006nk ; Hatta:2008st ; Neill:2020bwv .

We propose a measurement of $z_{\text{jet}}$ at HERA, which has not been done
before, and the future EIC. The high-$z_{\text{jet}}$ region corresponds to
jets with high-$p_{\mathrm{T}}$ in the laboratory frame that can be measured
with high precision and with an accuracy limited by the jet energy scale
uncertainty, which reached 1$\%$ at HERA Newman:2013ada . The measurement of
the small-$z_{\text{jet}}$ region will be challenging because these jets
correspond to jet $p_{\mathrm{T}}$ up to a few GeV in the laboratory frame
666This depends on $Q$, and a high $Q$ is preferred to reach the lowest
$z_{\text{jet}}$, a region that can be limited by calorimeter noise and
resolution. These issues could be bypassed by defining jets with charged
particles only, which would require the inclusion of track-based jet functions
on the theory side Chang:2013iba ; Chang:2013rca .

Figure 5: The $q_{T}=p_{\text{jet}}^{\perp}/z_{\text{jet}}$ spectrum for
Centauro and anti-$k_{T}(\text{SI})$ jets with $\eta_{\text{jet}}<1.0$.

We also propose to use Centauro jets to study quark TMDs by measuring
$q_{T}=p_{\text{jet}}^{\perp}/z_{\text{jet}}$. Fig. 5 shows that the $q_{T}$
spectrum for $z_{\text{jet}}>0.5$ peaks at $q_{T}<Q_{\text{min}}/4$, which is
ideal for TMD phenomenology. With the polarized proton beams available at the
EIC, this observable would provide clean access to the Sivers PDFs. Fig. 5
also shows the $q_{T}$ distribution for anti-$k_{T}(\text{SI})$ jets for
$z_{\text{jet}}<0.5$ and $0<z_{\text{jet}}<1$. Note the latter is only
possible since we can suppress the target fragmentation by requiring
$\eta_{\text{jet}}<1$. While for $z_{\text{jet}}>0.5$ we find similar result
as the Centauro jets, for $z_{\text{jet}}<0.5$ the spectrum peaks at
$q_{T}\sim Q$. Novel theoretical techniques are necessary to describe this
kinematic $q_{T}$ region of mid-rapidity jets.

In addition, the longitudinal invariance and ability to measure jets close to
Born kinematics makes the Centauro algorithm an attractive option to: i)
extract the strong-coupling constant from the rates of $n$-jets Newman:2013ada
; ii) enable “tag-and-probe” studies of nuclei Arratia:2019vju ; iii) identify
the background for gluon helicity and Sivers PDF studies Page:2019gbf ;
Zheng:2018ssm , iv) study jet substructure event shape observables in DIS, v)
probe TMD evolution observables that can be related to lattice QCD. We leave
those studies for future work.

Conclusions. We have proposed a new jet clustering approach tailored to the
study of energetic jets with low transverse momentum in DIS that relies on
spherically-invariant algorithms and a new longitudinally-invariant algorithm
that is asymmetric in the backward and forward directions, which we call
Centauro. The Centauro algorithm enables novel studies of transverse-momentum-
dependent observables in the Breit frame. Furthermore, we find that
spherically symmetric $k_{T}$-type algorithms yield clean access to the soft
jet fragmentation region, which also reveals a new $q_{T}$ regime where
$q_{T}\sim Q$. The new jet algorithms introduced here are relevant for the
studies of jet energy spectra, quark TMDs and spin physics, and cold-nuclear
matter effects. All these studies will be central for the jet physics program
of the future Electron-Ion Collider.

###### Acknowledgements.

Acknowledgements. We thank Anselm Vossen, Renee Fatemi, and Fernando Torales-
Acosta and Benjamin Nachman for insightful comments on our manuscript. M.A and
N.S are supported through DOE Contract No. DE-AC05-06OR23177 under which JSA
operates the Thomas Jefferson National Accelerator Facility. Y.M is supported
by the European Union’s Horizon 2020 research and innovation programme under
the Marie Skłodowska-Curie grant agreement No. 754496 - FELLINI. D.N. was
supported by the U.S. DOE under Contract DE-AC52-06NA25396 at LANL and through
the LANL/LDRD Program. F.R. was supported by the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231 and the National Science Foundation under
Grant No. ACI-1550228.

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# High-entropy ceramics: propelling applications through disorder

Cormac Toher  Dept. Mechanical Engineering and Materials Science and Center
for Autonomous Materials Design, Duke University, Durham, NC 27708, USA  Corey
Oses  Dept. Mechanical Engineering and Materials Science and Center for
Autonomous Materials Design, Duke University, Durham, NC 27708, USA  Marco
Esters  Dept. Mechanical Engineering and Materials Science and Center for
Autonomous Materials Design, Duke University, Durham, NC 27708, USA  David
Hicks  Dept. Mechanical Engineering and Materials Science and Center for
Autonomous Materials Design, Duke University, Durham, NC 27708, USA  George N.
Kotsonis  Dept. Materials Science and Engineering, The Pennsylvania State
University, University Park, PA 16802, USA  Christina M. Rost  Dept. Physics
and Astronomy, James Madison University, Harrisonburg, VA 22807, USA  Donald
W. Brenner  Dept. Materials Science and Engineering, North Carolina State
University, Raleigh, NC 27695, USA  Jon-Paul Maria  Dept. Materials Science
and Engineering, The Pennsylvania State University, University Park, PA 16802,
USA  Stefano Curtarolo<EMAIL_ADDRESS>Dept. Mechanical Engineering and
Materials Science and Center for Autonomous Materials Design, Duke University,
Durham, NC 27708, USA

###### Abstract

Disorder enhances desired properties, as well as creating new avenues for
synthesizing materials. For instance, hardness and yield stress are improved
by solid-solution strengthening, a result of distortions and atomic size
mismatches. Thermo-chemical stability is increased by the preference of
chemically disordered mixtures for high-symmetry super-lattices. Vibrational
thermal conductivity is decreased by force-constant disorder without
sacrificing mechanical strength and stiffness. Thus, high-entropy ceramics
propel a wide range of applications: from wear resistant coatings and thermal
and environmental barriers to catalysts, batteries, thermoelectrics and
nuclear energy management. Here, we discuss recent progress of the field, with
a particular emphasis on disorder-enhanced properties and applications.

## Introduction

Entropy is expected to dominate the formation of single-phase multicomponent
ceramics [1], opening up new synthesis routes and facilitating property
optimization without the need for toxic or expensive elements. High-entropy
ceramics [2] typically consist of a disordered multi-cation sublattice and an
ordered single-anion sublattice [3], with multi-anion ceramics now also being
developed [4, 5, 6]. The earliest work on high-entropy ceramics involved thin
films synthesized by sputtering high-entropy alloys in a N2 or O2 atmosphere
[7, 8, 9]. The original attempts typically resulted in amorphous or multi-
phase films due to the presence of non-nitride forming metals such as Cu [7,
8]; while the first single-phase high-entropy ceramic thin film reported was
(AlCrTaTiZr)N, displaying an fcc lattice based on the rocksalt structure [9].
In 2015, (MgCoNiCuZn)O became the first bulk single-phase high-entropy ceramic
to be synthesized — it also displayed a rocksalt-type fcc lattice [3], and was
demonstrated to be entropy-stabilized: when annealed at high temperatures, the
non-cubic CuO and ZnO secondary phases merged with the rocksalt phase [3].
High-entropy ceramics have now expanded to include carbides [10], borides
[11], silicides [12], niobates [13, 14], zirconates [15, 16], as well as
multi-anion boro-carbides [4] and carbo-nitrides [5, 6].

In addition to contributing to synthesizability, disorder directly alters and
improves the properties of materials such as high-entropy ceramics [2]. Solid-
solution strengthening combined with reduced grain growth enhances mechanical
properties including strength, hardness and toughness [10, 17, 18, 19] — vital
for structural applications and wear resistant coatings [20]. The strong
thermodynamic preference for high-symmetry lattices increases the thermo-
chemical stability, suppressing structural defect formation and phase
transitions — crucial for thin-film diffusion barriers in nanoelectronics [21]
and for improving cycling ability in battery materials [22, 23] — while
entropic stabilization of unusual oxidation states opens up catalyst design
rules [24]. Interatomic force constant and mass disorder reduce lattice
thermal conductivity, often reaching the amorphous limit [25] — useful for
increasing the efficiency of thermoelectric devices [26, 27, 28]. The
combination of low thermal conductivity, mechanical strength and corrosion
resistance make high-entropy ceramics particularly suitable as thermal and
environmental barriers (e.g., for hypersonics) [13, 14, 15, 16]. In magnetic
systems, disorder introduces localization, reducing the orbital overlap and
increasing the importance of double and superexchange [29], while disrupting
long-range order promotes the formation of relaxor ferroelectrics [30, 31,
32].

## Synthesis and modeling techniques

Synthesis approaches. Synthesis techniques for high-entropy ceramics include
sputtering [7, 8, 9]; pulsed laser deposition [33]; (reactive) spark plasma
sintering [10, 34]; boro-, carbo- and boro-carbothermal synthesis [35]; sol-
gel synthesis [24]; and solution combustion synthesis [36]. One current
research direction is developing techniques to avoid grain-coarsening to
enhance mechanical properties, such as new spark-plasma sintering protocols
with lower temperatures or shorter annealing times [37], or carbothermal
synthesis using finely ground metal or metal oxide precursors [35, 38]. Growth
kinetics (e.g., substrate temperature for pulsed laser deposition) and
epitaxial strain can be used to engineer electrical and magnetic properties
[33]. Rapid high-temperature Joule heating was used to synthesize a Pd-
containing denary nanoparticle catalyst for methane combustion [39], where
short, high-temperature heat treatments overcame the kinetic barriers for
mixing multiple elements while avoiding agglomeration and collapse of pore
structure.

Computational modeling. First-principles modeling of disordered materials
often uses special quasi-random structures (SQS) [40], in which the lattice is
decorated so as to maximize the similarity of the radial correlation functions
to that of the completely disordered material, in combination with the ideal
approximation for the configurational entropy. SQS models the fully
disordered, infinite-temperature limit: useful for understanding how high
levels of disorder can affect properties, but neglecting the effect of short-
and long-range ordering at finite temperatures.

The AFLOW partial occupation module (AFLOW-POCC) enables the modeling of
disorder as a function of temperature [41]. The disordered material is
represented by an ensemble of small ordered cells, generated by enumerating
all possible decorations of the parent structure that produce the target
compositions up to a cell-size cut-off. A descriptor, the entropy forming
ability (EFA), was formulated based on the energy distribution obtained from
first-principles calculations of these configurations [10]. A narrow
distribution implies that it is easy to introduce new configurations and
disorder, while a broad distribution indicates a thermodynamic preference for
certain ordered states. The EFA descriptor has guided the synthesis of high-
entropy carbides, successfully predicting whether compositions would form a
single-phase or undergo phase separation [10]. Cr-containing compositions
forming a single phase were also identified with machine-learning models
trained on EFA data [42].

The properties of high-entropy ceramics have also been modeled computationally
[10, 43, 44]. AFLOW-POCC was used to calculate the thermodynamically weighted
average of the elastic moduli of the different configurations to predict the
mechanical properties of disordered carbides [10]. The temperature-dependent
elastic properties of (HfTaTiWZr)C and (MoNbTaVW)C and their component binary
carbides were investigated using ab initio molecular dynamics [43]. Molecular
dynamics simulations with deep-learning potentials were used to predict the
thermal and elastic properties of (HfNbTaTiZr)B2 as a function of temperature
[44].

## Disorder-enhanced properties

Figure 1: Thermal and mechanical properties. a) Disorder simultaneously
enhances hardness and fracture toughness of high-entropy nitrides (panel
adapted from Ref. 19). b) Flexural strength as a function of grain size (panel
adapted from Ref. 45). c) Microstructures of (HfNbTaTiZr)C annealed at 1800∘C
(top) and 1900∘C (bottom), showing the increase in grain size with temperature
(panel adapted from Ref. 38). d) Crack deflection by SiC phase (panel adapted
from Ref. 46). e) Thermal conductivity versus elastic modulus: disorder lowers
thermal conductivity while maintaining stiffness (panel adapted from Ref. 25).

Mechanical properties. The mechanical properties of high-entropy ceramics are
enhanced by a combination of grain refinement and solid-solution strengthening
— lattice distortions due to differing atomic radii and Jahn-Teller effects
[47, 48] create barriers that disrupt the propagation of dislocations,
hindering plastic deformation and increasing hardness and yield strength. The
reduction in grain coarsening has been attributed to increased crystalline
energy due to lattice distortions, which reduces the free energy gained by
shrinking the grain surface area [49, 50], as well as to slow diffusion [50].
Finer grains lead to improved toughness, and to Hall-Petch strengthening due
to the nanograin microstructure impeding the motion of dislocations.

Bulk (HfNbTaTiZr)C, prepared by spark-plasma sintering, shows a hardness 50%
higher than what would be expected from the rule-of-mixtures of the binary
carbides [10, 17, 51], while carbo-nitride versions of this composition are
slightly harder than the carbides [5, 6], and an 8-cation carbide is even
harder [52]. High-entropy nitrides show even greater improvements, doubling
both hardness and fracture toughness compared to their components (Figure 1a)
[19]. Thin films deposited using magnetron sputtering had even higher hardness
[53, 54, 55], likely due to internal stresses induced by ion bombardment [53,
54, 55]: the highest reported hardness was 70 GPa for a (HfNbTaTiV)N film
[56]. The highest fracture toughness values of 8.4 MPa m1/2 were reported for
(MoNbTaTiVW)C(N) [57], whereas the highest flexural strengths tend to be for
compositions with fewer elements such as (NbTaTiZr)C (544 to 560 MPa [45]) or
(TaTiZr)C at 700 MPa [58].

The enhanced mechanical properties of high-entropy ceramics are generally
maintained to relatively high temperatures [58, 35, 59]. (TaTiZr)C retained
half its flexural strength up to 1800∘C, while its fracture toughness
increased almost 40% when heated to 1800∘C [58]. The flexural strength of a
fine-grained (HfNbTaTiZr)C sample remained almost constant up to
${\sim}1800^{\circ}$C, before decreasing significantly between 2000∘C and
2300∘C [35]. The steady state creep rates of (HfNbTaZr)C are about 10 times
lower than the rates for the component binary carbides [59], attributed to
lattice distortion and sluggish diffusion.

Synthesis procedures, defects and grain size all have a major impact on
mechanical properties of high-entropy materials (Figure 1b for dependence of
strength on grain size) [18, 60, 37, 50]. The fracture strength of (HfNbTaZr)C
was significantly reduced by defects in the form of pores or secondary phases
[18]. Reducing the number of carbon vacancies in high-entropy carbides
increases the hardness up to carbon saturation [60], while excess carbon
precipitates at the grain boundaries, first reducing hardness and then
increasing it again due to the formation of a diamond-like carbon matrix.
Average grain size in (HfMoNbTaTiZr)C increased ${\sim}4$-fold with increase
in sintering temperature up to 2500∘C [37], reducing the Vickers hardness.
Sub-micron grains of (HfNbTaTiZr)C were prepared using a two-step sintering
process [50]: the small grain size was thermally stable, and the sample showed
improved strength, hardness and toughness. Fine-grained high-entropy carbide
samples can also be prepared by carbothermal synthesis of oxide precursors,
with grain size controlled by the hot-pressing temperature (Figure 1c) [35].

Mechanical properties of high-entropy ceramics can be further enhanced by
incorporating SiC secondary phases: SiC particles deflect cracks and increase
toughness (see Figure 1d) [46, 61]. In particular, 4-point bending strengths
increased by 40% to 60%, while fracture toughness improved by about 20%.

Thermal conductivity. Disorder suppresses lattice thermal conductivity in
high-entropy ceramics without compromising mechanical strength and stiffness
(Figure 1e) [25, 62, 63, 64]. (MgCoNiCuZn)O has a lattice thermal conductivity
of $2.95\pm 0.25$ W m-1 K-1, while six-cation systems formed by adding other
elements such as Sc, Sb, Sn, Cr or Ge have even lower values. The lowest
thermal conductivities for high-entropy carbides were for porous samples [65]
— when fully densified, thermal conductivities range from ${\sim}2$ W m-1 K-1
to 9.2 W m-1 K-1 [66, 63, 62]. Reducing the carbon content of films increased
the metallic bonding, raising the electronic contribution to the thermal
conductivity [62]. Very low thermal conductivities were reported for multi-
component chalcogenides [27, 28] — promising for thermoelectric applications.

The reduction in thermal conductivity has been attributed to disorder in the
interatomic force constants [25] and atomic radii [64], while mass disorder
appears to be less significant [25]. Extended x-ray absorption fine structure
measurements show that the oxygen sublattice distortion is much larger in
6-cation systems such as (MgNiCuCoZnCr)O than in (MgCoNiCuZn)O [25],
indicating the importance of lattice distortion and force constant disorder —
this is further supported by molecular dynamics simulations [25]. For a set of
22 oxides based on the pyrochlore structure, the thermal conductivity had a
good correlation ($r^{2}=0.73$) with the overall size disorder [64].

Figure 2: High-entropy ceramics for magnets, batteries, thermoelectrics and
radioactive waste immobilization. a) High-entropy oxide thin films showed
strong magnetic anisotropy when subjected to in-plane epitaxial strain (panel
adapted from Ref. 67). b) High-entropy battery materials maintain efficiency
under cycling (panel adapted from Ref. 22). c) Increasing the number of
components improves the thermoelectric efficiency (panel adapted from Ref.
27). d) High-entropy ceramics show reduced leaching rates when used as
immobilization matrices for radioactive waste (panel adapted from Ref. 68).

Electrical and magnetic properties. The low long-range order in high-entropy
ceramics strongly affects their electrical and magnetic properties.
Ferroelectric and magnetic domains tend to be smaller and the number of
microstates is increased. Frustrated dipolar and magnetic ordering can promote
the formation of “relaxor ferroelectrics” [30, 31, 32] and spin glasses [69].
In magnetic systems, disorder introduces localization, reducing orbital
overlap and raising the importance of double and superexchange [29]. Entropy
also enables the fabrication of thermodynamically stable single-phase magnetic
insulators with high quality surfaces such as (MgNiFeCoCu)Fe2O4 [67]:
promising for spintronic, spincaloritronic, nonvolatile memory and microwave
applications.

The magnetic and electrical properties of high-entropy ceramics can be tuned
by altering their composition [70, 69] or synthesis conditions [70, 33], or by
applying strain [29, 67]. (MgCoNiCuZn)O films grown at low temperature exhibit
enhanced Co3+ concentrations that result in a compressed lattice parameter,
reduced optical band gap, increased electrical conductivity, and more diluted
magnetic lattice [33, 70]. Tensile strain generates magnetic anisotropy by
increasing the out-of-plane magnetic hardness (Figure 2a) [29], apparently due
to the modification of the superexchange coupling between the transition metal
atoms. Increasing the Cu content of (MgCoNiCuZn)O creates structural disorder
since Cu2+ ions tend to undergo a tetragonal distortion from the octahedral
coordination (Jahn-Teller effect [48]), leading to increased spin frustration
and a reduction in the driving force for a preferred magnetization axis [69].
Increasing the non-magnetic Co3+ concentration in (MgCoNiCuZn)O results in
more uncompensated spins when interfaced with a ferromagnet, leading to
enhanced magnetic exchange bias [33].

Relaxor ferroelectrics have much smaller permanent electric dipole domains
than conventional ferroelectrics, forming “polar nano-regions” on the
nanoscale instead of the microscale. They take less energy to align, giving
them very high specific capacitance and making them useful for energy storage.
They also have high dielectric permittivity, high piezoelectric coefficient
and giant field induced strain: important for applications in dielectric
capacitors, piezoelectric sensors and actuators. In high-entropy ceramics such
as perovskite-structure (NaBiBaSrCa)TiO3 and (BiNaKBaCa)TiO3, or Aurivillius
phase (CaSrBaPb)Bi2Nb2O9 and (Ca0.2Sr0.2Ba0.2Pb0.2Nd0.1Na0.1)Bi2Nb2O9 (space
group $A2_{1}am$, #36), differences in the valence and ion size disrupt long-
range ferroelectric ordering and result in the formation of several different
types of electric dipoles. This promotes the formation of polar nano-regions,
reducing the saturation polarization, remanent polarization and coercive field
[30, 31, 32], and increasing the piezoelectric coefficients [32].
(NaBiBaSrCa)TiO3 also displayed an electrocaloric effect: the application of
an electric field changes the dipole arrangement from disordered to ordered,
resulting in an entropy-change that can be exploited for refrigeration [30].

Corrosion resistance. The high melting temperatures of carbides and borides
offer great potential for applications in extreme environments such as
combustion chambers, where resistance against corrosion is critical. Oxidation
behavior has been extensively studied in (HfNbTaTiZr)C [71, 72, 73, 74, 75].
Kinetic investigations showed that oxidation follows a parabolic rate law
between 1073 and 1773 K, indicating a diffusion-controlled process [71, 72,
73]. The rate constant increases until 1273 K while the passivating layer is
highly porous, then decreases with increasing temperature as more complex
quaternary and quinary oxides form and the passivating layer densifies [71,
72]. The oxide layer is heterogeneous: group IV oxides primarily form near the
surface (due to the readily oxidized corresponding carbides), while group V
oxides form near the interface with the ceramic [74, 75] — similar results
were found in the diboride analog [74, 75]. Experiments using water vapor
showed a denser protective layer, which increased corrosion resistance by
preventing inward diffusion of H2O [73]. The same study found improved
corrosion resistance compared to ZrC — attributed to sluggish diffusion in the
high-entropy material due to lattice distortions, suppressing outward
diffusion of the cations. This was supported by decreased oxidation rates in
(HfNbTaTiZr)C compared to (TiNbTaZr)C and (NbTiZr)C.

A separate study in air found improved oxidation resistance in (HfNbTaZr)C
compared to the five-metal carbide [76, 77]. The four-metal carbide also
showed improved performance compared to the binary carbide reactants, and the
oxide layer was more structurally stable than in (HfTa)C, even though (HfTa)C
had a lower oxidation onset temperature [76]. Mechanistic studies confirmed
that outward diffusion of TiO is the rate-determining step in the oxidation of
(HfNbTaTiZr)C and thus key to increased corrosion resistance of (HfNbTaZr)C
[78]. Performance was further improved by doping with SiC as it forms a dense
oxide that prevents inwards diffusion of oxygen. Additionally, SiO is an
effective diffusion barrier for Ta and Nb oxides, preventing outward diffusion
of these metals [78, 77]. (HfNbTaZr)C with 20 vol-% SiC showed enhanced
oxidation resistance compared to the ultra-high temperature ceramic ZrB2-SiC
[77]. Composition and porosity of the passivating oxide layer are therefore
crucial to the stability of high-entropy ceramics at high temperatures under
oxidizing conditions. The ability to combine many elements to tailor chemical
properties makes high-entropy carbides and borides strong candidates for
applications in extreme heat conditions.

## High-entropy ceramic applications

Batteries. The entropy-driven preference for high-symmetry lattices gives
materials such as (MgCoNiCuZn)O high thermo-chemical stability, suppressing
structural deformation and defect formation, giving them the potential to
maintain performance under charge cycling (Figure 2b) [22, 79]. This led to
investigations of the suitability of high-entropy ceramics for use as anodes,
cathodes and solid-state electrolytes in Li-ion batteries [22, 23]. Shortly
after its initial synthesis, room temperature superionic conductivity was
reported for Li-doped (MgCoNiCuZn)O [80], significantly exceeding that of the
currently-used lithium phosphorous oxy-nitride solid-state electrolyte.

High-entropy oxides based on (MgCoNiCuZn)O are also promising as anodes, with
high specific capacities and robust responses to charge cycling. Initial
capacities of 980 mAh g-1 [22] to 1585 mAh g-1 [79] have been reported, with a
reversible capacity of 920 mAh g-1 at 100 mA g-1 after 300 cycles [79].
(MgCuNiZn)0.65Li0.35O had an even higher initial discharge capacity of 1930
mAh g-1 [81]. These values compare very favorably with the current standard
graphite anodes, which have a capacity of ${\sim}360$ mAh g-1 [23].

Cation-disordered rocksalt-type oxyfluorides were investigated for use as Li-
ion cathodes [82], delivering specific energies of up to 307 mAh g-1. High-
entropy oxides based on the layered delafossite $\alpha$-NaFeO2 structures
were also examined, but their specific capacities were low compared to state-
of-the-art cathode materials [83].

X-ray absorption spectroscopy was used to investigate the lithiation
mechanisms in (MgCoNiCuZn)O [84, 85]. Cu(II) was observed to reduce first to
Cu(I) and then to metallic Cu, followed by reduction of Ni and Co to the
metallic state. The high-entropy oxide structure appeared to collapse for
charges greater than 400 mAh g-1, and at 800 mAh g-1, a significant portion of
Zn metal was present [85]. Removing Zn or Mg from (MgCoNiCuZn)O resulted in
significantly worse cycling performance, indicating that these cations are
important for maintaining the rocksalt structure.

Catalysts. High-entropy catalysts, particularly oxides, demonstrate several
advantages over ordered counterparts: compositional freedom translating to
substantial electronic structure and coordination environment tunability [86,
39], better performance than fewer-component variants [86, 39], and excellent
stability and durability [24, 86, 87, 88]. High entropy enables the design of
new catalysts that combine and even enhance beneficial features of their
components. In FeNiCoCrMn-glycerate, Fe-, Ni-, Co-, and Mn-based materials are
known to be highly active oxygen evolution reaction electrocatalysts [86],
while Cr induces strain in the structure that weakens active site
chemisorption. The layered glycerate structure allows rapid transport of the
reactants to the material and provides additional catalytic active sites. High
entropy can also stabilize exotic active sites: Cu in high-entropy rocksalt
phases can cycle between redox states Cu(II) and Cu(I) as it is treated with
CO and O2 at temperatures from $130^{\circ}$C to ${\sim 250}^{\circ}$C [24],
whereas CuO is irreversibly reduced to Cu(I) under the same conditions.

Poly-cation oxides like (FeMgCoNi)O1.2 have shown great potential for hydrogen
production through two-step thermochemical water splitting [89, 87].
Improvements have been reported by integrating microwave-absorbing SiC foam
and performing the reaction under short-term, low-energy microwave
irradiation: i. the thermal reduction process was expedited to 4 minutes
versus the usual 30 minutes or greater, ii. high H2 generation rates, and iii.
power consumption that is 3% that of conventional heat treatments. Microwave
irradiation significantly increases the oxygen vacancies in the structure and
can also generate plasma that enhances hydrogen production, but at the cost of
accelerated degradation of the material.

High-entropy structures were also shown to be a catalytic booster for aerobic
oxidative desulfurization of diesel [90]. Carbon and oxygen co-doped hexagonal
boron nitride catalysts with the highest density of grain boundaries
demonstrated the shortest induction period, achieving complete sulfur
conversion in six hours.

Thermoelectrics. Thermoelectric materials generate a voltage when subjected to
a thermal gradient, and are used for energy generation and refrigeration. They
are evaluated by the “figure of merit”: $zT=\sigma S^{2}/\kappa$, where
$\sigma$ is the electrical conductivity, $S$ is the Seebeck coefficient and
$\kappa$ is the thermal conductivity — the reduced lattice thermal
conductivity in disordered materials leads to increased $zT$. The thermal
conductivity of perovskite-structure (BaCaLaPbSr)TiO3 was ${\sim}5$ times
lower than that of SrTiO3 [26], with a maximum $zT$ over 0.2. The high-entropy
chalcogenide Pb0.89Sb0.012Sn0.1Se0.5Te0.25S0.25 had an ultralow thermal
conductivity of 0.3 W m-1 K-1 [27] with $zT$ of 1.8 compared to 0.8 for
Pb0.99Sb0.012Se (Figure 2c). Adding Sn enabled the stabilization of the
Pb(SeTeS) system due to increased configurational entropy.
Pb0.975-xCdxNa0.025Se0.5S0.25Te0.025, where $x\leq 0.05$, had a conversion
efficiency at $\Delta T=507$ K of 12% (among the highest reported values) with
a power output of 2.7 W [28].

Thermal and environmental barriers. The simultaneous suppression of the
thermal conductivity [25] and enhancement of the mechanical properties [10,
51] in high-entropy materials makes them highly attractive for thermal and
environmental protection applications, with recent work focusing on niobates
[13, 14] and zirconates [15, 16]. Pyrochlore-structure (LaNdSmEuGd)2Zr2O7
could withstand up to 5 times as much thermal cycling as La2Zr2O7 [15]. Rare
earth niobates had ultra-low thermal conductivities of $<1$ W m-1 K-1 [13] and
hardnesses up to $13.9$ GPa [14].

Nuclear energy applications. Their thermochemical stability and mechanical
behavior make high-entropy materials promising for nuclear energy applications
ranging from i. structural components in fission and fusion reactors, which
need resistance to neutron irradiation damage, corrosive coolants, thermal and
irradiation creep, helium embrittlement, etc. [91]; to ii. immobilizing
matrices for high-level waste, which require excellent radiation tolerance and
aqueous durability [68]. Radiation-induced defect clusters and dislocation
loops in (NbTaTiZr)C remained small — likely due to the lattice distortion
slowing their growth — and no void formation or radiation-induced segregation
near grain boundaries were observed [91]. For pyrochlore-structure
(Eu1-xGdx)2(TiZrHfNbCe)2O7, the normalized leaching rates for simulated
radionuclides Ce (surrogate for Pu) and Gd were two orders of magnitude lower
than that for Gd2Zr2O7 (Figure 2d). Electrical conductivity was much lower in
the high-entropy material, indicating significantly fewer oxygen vacancies —
leading to slower cation diffusion and higher chemical stability — while
lattice distortion also leads to a high lattice potential energy, reducing
cation mobility.

## Conclusions

Disorder is propelling the development of new ceramic materials.
Configurational entropy provides new thermodynamic pathways to stabilize
multi-component materials. Disorder directly enhances properties, facilitating
the design of materials with features that are traditionally considered
mutually exclusive, such as high stiffness and low thermal conductivity, or
improved hardness and fracture toughness. High-entropy ceramics will continue
to impact fields as diverse as batteries, catalysts, thermoelectrics, magnets,
ferroelectrics, nuclear energy, and structural materials, for the foreseeable
future.

Acknowledgments. The authors thank Xiomara Campilongo, Eva Zurek, William
Fahrenholtz, Douglas Wolfe and Darrell Schlom for valuable discussions.
Research sponsored by DOD-ONR (N00014-15-1-2863, N00014-21-1-2515) and NSF
(DMR-1921909, DGE-2022040).

Author contributions. The authors contributed equally to the article.

Competing interests. The authors declare no competing interests.

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