# European Workshop on Photocathodes for Particle Accelerator Applications
2022:
Summary Report

L. Monaco  D. Sertore  INFN Milano – LASA, Via Fratelli Cervi 201, Segrate
(MI), Italy  M. Baylac  Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3,
38000 Grenoble, France  L.B Jones  T.C.Q. Noakes  ASTeC, STFC Daresbury
Laboratory, Warrington, Cheshire, WA4 4AD, UK  Cockcroft Institute of
Accelerator Science & Technology, WA4 4AD, UK  J. Kühn  HZB, Helmholtz–Zentrum
Berlin für Materialien und Energie GmbH, 14109 Berlin, Germany  R. Xiang
HZDR, SRF–Gun Group, ELBE Department, Institute of Radiation Physics,
Helmholtz–Zentrum Dresden–Rossendorf, 01328 Dresden, Germany

Delegates at the EWPAA 2022 Workshop during their visit to the LASA facility

The European Workshop on Photocathodes for (particle) Accelerator Applications
(EWPAA) brings together experts in the field of photocathode based electron
sources for use in particle accelerators, with the aim of sharing their
knowledge and latest research and development progress in this crucial field
of particle accelerator science.

The workshop is convened every other year, and is thus complementary to the P3
workshop (Photocathode Physics for Particle accelerators) run in the USA.
Consequently, there is a workshop focusing on photocathodes for particle
accelerator applications convened every year, either in Europe or the USA.

The EWPAA 2022 is the 4th meeting in this workshop series. The event was
hosted by the INFN LASA Institute in Milan in collaboration with University of
Milano between September $20^{\mathrm{th}}$ and $22^{\mathrm{nd}}$. Details of
the event and the scientific programme can be found at the website:

https://agenda.infn.it/event/ewpaa2022

The programme was organised with 7 working groups, with each oral contribution
assigned to the most appropriate group. This report presents, summarised by
the work group conveners, the innovative ideas, the challenges and the main
points raised by each of the speakers.

###### Contents

  1. I WG1:Overview of Photocathode Research
    1. I.1 High Brightness Beams
    2. I.2 Recent Advancements in Photocathodes
    3. I.3 Experience on Polarized Photocathodes
  2. II WG2: Photocathode Performance in Accelerator Applications
    1. II.1 JLAB DC Gun Developments
    2. II.2 CsK2Sb Photocathodes for Application in an Industrial Accelerator
    3. II.3 Status of Precise Measurements of Electron Beam Polarization Changes During Long Term Operation
    4. II.4 Photocathode Activities at DESY
    5. II.5 Performance of Bialkali Photocathode in DC–SRF Photoinjector
    6. II.6 Photocathode Performance at LCLS–II
  3. III WG3: New Photocathode Ideas
    1. III.1 Ultra–thin MgO Films on Metal Photocathodes to Enhance QE
    2. III.2 Plasma Photocathode
    3. III.3 CERN Photocathode Activities
    4. III.4 Protective Layers on Bi–alkali Cathodes
    5. III.5 AI/ML–selection of Air–stable Photocathodes
  4. IV WG4: Metallic Photocathodes
    1. IV.1 Studies on the Evolution of MTE for Photocathodes Subjected to Controlled Degradation by Gas Exposure
    2. IV.2 Copper Photocathodes for CLARA
    3. IV.3 Copper photocathode experience at FERMI
    4. IV.4 Technology Aspects in Preparation at NCBJ of Nb–Pb Superconducting Photocathodes for XFEL–Type RF Electron Guns
  5. V WG5: Semiconductor Photocathodes
    1. V.1 Multi–alkali Antimonide Photocathodes for Highly Brilliant Electron Beams
    2. V.2 Characterisation at Daresbury Laboratory of CsTe Photocathodes Grown at CERN
    3. V.3 Cs2Te\KCsSb in Gun Operation at PITZ
    4. V.4 Operation of Cs2Te in SRF gun for THz User Shifts
  6. VI ERL Roadmap for Europe
  7. VII WG6: Photocathode Theory
    1. VII.1 Monte Carlo Transverse Emittance and Quantum Efficiency Study on Cs2Te
    2. VII.2 Monte Carlo Simulations of Electron Photoemission from Plasmon–enhanced Bialkali Photocathode
    3. VII.3 Exploring Cesium–-Tellurium Phase Space via High–throughput Calculations beyond Semi–local Density–functional Theory
    4. VII.4 Ultrafast Sub–threshold One-photon Photoemission
  8. VIII WG7: Advanced Photocathode Characterization
    1. VIII.1 The ASU photocathode research lab
    2. VIII.2 Photocathode Response Time Measurement
    3. VIII.3 Resolving Surface Chemical States of p-GaN:Cs Photocathodes by XPS Analysis
    4. VIII.4 High Crystalline Cesium Telluride Photocathode on Atomically–thin Graphene with Co–deposition
    5. VIII.5 Molecular Beam Epitaxy of Cs–Sb Thin Films: Structure–oriented Growth of High Efficiency Photocathodes
    6. VIII.6 Stoichiometry Control and Automated Growth of Alkali Antimonide Photocathode Films by Molecular Beam Deposition
  9. IX Conclusions

##  I WG1:Overview of Photocathode Research

Working Group 1 was focused on general overviews of key areas of photocathode
R&D. It was convened by Daniele Sertore.

###  I.1 High Brightness Beams

Alberto Bacci (INFN LASA, Italy) presented a view to photocathodes seen from
beam dynamics applications with particular attention to the achievement of
high brightness. Within the photocathode desiderata, high Quantum Efficiency
(QE) and fast response time were the most important to achieve final requested
beam performances. Linked to the photocathode requirements, laser beam
transverse shaping plays a key role. It was shown that a truncated Gaussian
delivers a smaller final emittance w.r.t. a uniform distribution, as usually
thought. The concepts before presented were applied to three different
scenarios, namely BRiXSinO ERL project based on a DC gun, the MariX project
based on a RF gun "Apex–like" and finally the EuPRAXIA configuration with a
1.6 cells S–band RF Gun. BriXSinO is based on a novel scheme where the beam
pass is accelerated through a Superconducting Linac, is bent in a compressing
arc and then re–injected into the linac from the exit side. The SC linac can
be then used to decelerate the beam in an ERL configuration, or to accelerate
the beam in the usual way in the so call Two–Pass Two–Way operation mode. The
MariX project was developed as a possible accelerator facility to be installed
in the new Milan university campus called MIND. A full start–to–end simulation
of the accelerator is presented with emphasis on the optimization of the
injector with a self–developed genetic algorithm GIOTTO to achieve optimal
compression in the MariX Loop and minimal transverse emittance for FEL
operation. For the last example, EuPRAXIA, Alberto presented preliminary
results on the optimization of the injector in order to achieve the best
performance from the beam driven plasma acceleration experiment.

###  I.2 Recent Advancements in Photocathodes

Laura Monaco (INFN LASA, Italy) gave an overview of the recent photocathode
advancements. In the field of metallic photocathodes, good results have been
obtained at HZDR in the SRF gun with Magnesium. Studies are also on–going to
lower the metal work function by depositing MgO film or exploring single vs
polycrystalline Ag or even creating structure on top of a flat metallic film
to enhance photoemission. Turning to Cs2Te semiconductor photocathodes, good
performance levels were reported with operating lifetime more than 1,400 days
while conserving good QE, and with response time in the
$200\text{\,}\mathrm{fs}$ range. The cesium telluride photocathode has been
also used extensively in the HZRD SRF Gun, even if degradation of the film is
still a not solved problem. Co–deposition of Cs2Te film has been studied by a
joint experiment CERN–STFC as well as at STFC directly for the CLARA project.
Alkali antimonide photocathodes are, nowadays, the more extensive studied
photocathodes due to their sensitivity to visible light and high QE. INFN–DESY
PITZ collaboration has tested a set of KCsSb photocathodes, deposited at INFN
LASA, in the PITZ RF gun. This experiment showed a larger field emission of
antimonide w.r.t. telluride but also a very fast response time (below
$100\text{\,}\mathrm{fs}$) and a small thermal emittance. At PITZ, there was a
confirmation of the sensitivity of these kind of photocathodes to vacuum
conditions as well as exposure to high accelerating gradients (above
$40\text{\,}\mathrm{MV}\text{/}\mathrm{m}$). A similar experiment has been
done in a joint collaboration between BNL and UCLA using a NaKCs photocathode.
In this case, the QE degraded consistently during the transportation.
Nevertheless, the team was able to measure QE, MTE and non-linear
photoemission effects. Also at HZB, NaKSb is studied and characterized in
preparation to its usage in the HZB $1.3\text{\,}\mathrm{GHz}$ SRF Gun. There
is significant on–going R&D activity in China on a broad set of photocathodes
and growth techniques. Moving to Cs3Sb material, the growth of an epitaxial
film of Cs3Sb(100) on a 3C Si(100) has allowed, for the first time, ARPES
measurement of the band structure of this material. The choice of a lattice
match material like strontium titanate has improved the smoothness of the film
and hence the contribution to MTE from roughness. On the way to improved
alkali antimonide robustness, Laura presented the results obtained at LANL on
graphene as substrate or graphene as protective layer for the Sb film. Both of
them show interesting results that need further studies. To improve QE, the
idea was presented of using Surface Plasmon Polaritons to enhance the laser
absorption by minimizing the reflection at a specific wavelength. The last
part of the presentation focused on new photocathode test facilities,
important tools for developing and testing new ideas and processes. Finally,
the Machine Learning and Artificial Intelligence approach to photocathode
material selection and photocathode growth closed the presentation.

###  I.3 Experience on Polarized Photocathodes

Kurt Aulenbacher (University of Mainz, Germany) presented on ‘polarized’
photocathodes. After a short introduction of Negative Electron Affinity
Photocathode (NEA) characteristics, Kurt introduced the well know GaAs
photocathodes, largely used for generation of spin polarized electrons, in the
strained and superlattice configuration, necessary for enhancing the electron
beam polarization. A new entrance in this family is the "resonant" strained
superlattice cathode that make use of an enhanced absorption achieved by a
DBR–Reflector that causes the active region to act as a cavity with enhanced
absorption at resonance. This type of photocathodes showed high QE and high
polarization in the $770\text{\,}\mathrm{nm}$ region. Moreover they should be
fast, not affected by ‘photovoltage’ effect and able to deliver
$\mathrm{A}\text{/}{\mathrm{cm}}^{2}$ while having small thermal emittance.
Polarized beams are important for nuclear an particle physics experiment. Two
machines are nowaday running with polarized electrons, namely MAMI in Germany
and CEBAF in USA. MESA (ERL), that is in construction, will also operate with
these beams. Future machines that aim to use polarized beams are, among
others, LHeC, CERL but they might be challenged by technical and physical
limitations of these material, because they require extraction of
$10\text{\,}\mathrm{kC}100\text{\,}\mathrm{kC}$. Changes in surface layer will
strongly affect the photocathode performance. Chemical reactions with gases of
the environment, ion back bombardment and thermal decomposition are the most
important players in this context. To mitigate them, UHV or XHV vacuum, very
limited beam losses in the source region and cathode temperature below
$50\text{\,}\mathrm{\SIUnitSymbolCelsius}$ are required. A different approach
is to stabilize the surface or protect it. Different labs have explore
different solutions: TU Darmstand & JLAb use Cs:F:Li while Cornell has
explored Cs:O:Sb and Cs2Te. These are still in the R&D phase bu the results
are encouraging. To control the cathode temperature, in the "BETHE" project a
new approach is presented where the puck is pressed against a thermal
conducting insulator like Boron Nitride (BN). This operation is done before
moving the cathode into the insulator. Given the properties of BN, this
solution should guarantee reasonable operation of the cathode also with large
incident laser power. Besides the technical limitation, a physical limitation
is the surface photovoltage, i.e. modified NEA–state reduce the QE due to near
band-gap excitation. This has significant impact of photocathode performances.
The last point touched in this presentation is the lack of a commercial
supplier for the lattice strained photocathode. This requires a common effort
between laboratories and institutions to support the manufacturing of these
fundamental photocathodes needed for all modern applications requiring
polarized electron beams.

##  II WG2: Photocathode Performance in Accelerator Applications

Working Group 2 was convened by Rong Xiang and focused on photocathode
performance in accelerator applications. It remains a goal in photocathode R&D
to improve the cathode performance in accelerator applications. On one hand,
we need to further push the study of mature photocathodes; on the other hand,
we are searching new materials with high QE, long time, low dark current, fast
response time, especially low Mean Transverse Energy (MTE) for special users
(hard X–ray FEL and microscope).

###  II.1 JLAB DC Gun Developments

Carlos Hernandez–Garcia (Thomas Jefferson National Accelerator Facility, USA)
discussed the successful application of photocathodes in different type of
electron sources (DC guns, NC RF guns and SRF guns). JLab’s experience of DC
guns with GaAs photocathodes represents a good example. With the now mature
GaAs photocathode, the CEBAF DC gun can provide highly polarized electron beam
at $130\text{\,}\mathrm{keV}$ and $0.2\text{\,}\mathrm{mA}$ CW for nuclear
physics experiments for over a decade. Recently JLab team has demonstrated up
to $300\text{\,}\mathrm{keV}$ and $5\text{\,}\mathrm{m}\mathrm{A}$ CW
un–polarized beam for ERL.

###  II.2 CsK2Sb Photocathodes for Application in an Industrial Accelerator

Recent years have seen several new/emerging electron gun projects. In
addition, the stability and reliability of photocathodes in laboratories has
pushed their application in industrial accelerator projects. Veronica Kuemper
(Research Instruments, Germany) spoke about work by Research Instruments as
part of the Lighthouse project, highlighting the promising application of two
$350\text{\,}\mathrm{kV}$ DC guns with a CsK2Sb preparation system has the
goal to provide $40\text{\,}\mathrm{mA}$ in CW mode for continuous operation
23h/7 days a week.

###  II.3 Status of Precise Measurements of Electron Beam Polarization Changes
During Long Term Operation

Currently the mature semiconductor cathodes ready for various applications
include Cs2Te, Cs2KSb and GaAs. Jennifer Groth (University of Mainz) shared
data from MESA on their use of GaAs photocathodes in DC guns realize high
polarization during long–term operation, highlighting their observations thtat
preparation with NF3 allows more activation cycles with stable QE.

###  II.4 Photocathode Activities at DESY

Groups operating photoinjectors have accumulated rich experience during their
beam operation. Firstly, extremely high vacuum (XHV) environment is the key
for long operation lifetime in every electron gun. Furthermore, it is clear
that the ion back–bombardment is responsible for the cathode surface damaging
during DC gun operation. However, further studies are needed to confirm that
ion back–bombardment is detrimental in RF guns. David Pavel Juarez–Lopez
(DESY, Germany) presented recent results on the successful application is the
Cs2Te cathode for $1.3\text{\,}\mathrm{GHz}$ normal conducting RF guns at
FLASH and European XFEL, which work reliably at DESY for years already and fit
well for FLASH and European XFEL as a user facility.

###  II.5 Performance of Bialkali Photocathode in DC–SRF Photoinjector

Haumu Xie (Peking University, China) shared their experience of DC–SRF gun
operation, noting that the cryogenic temperature has strong influence on the
QE of Cs2KSb, which is lower at cryogenic temperature than that at room
temperature. The bialkali photocathode in the DC–SRF–II gun has already
produced the first beam, with typical QE around $1\text{\,}\mathrm{\char
37\relax}$ at $36\text{\,}\mathrm{K}$.

For the first goal, the analysis of cathode after gun operation without
explosion to air will help us to understand the cathode surface changing or QE
distribution evolution during gun operation. And for the second goal, a real
test platform (test gun) is important to prove the cathode properties suitable
for the gun operation.

###  II.6 Photocathode Performance at LCLS–II

INFN style Cs2Te photocathodes have been used in many large facilities for
decade. In last years, Cs2Te cathodes were successful prepared and operated in
VHF gun for LCLS-II project too. Theo Vecchione (SLAC, USA) gave an update on
progress in commissioning the LCLS–II VHF gun, and their use of Cs2Te
cathodes. He drew attention to their work to reduce dark current, highlighted
the importance of post–mortem cathode evaluation, and the particularly the
preservation of the used cathode state through the use of in–vacuum transfer
to the post–mortem analysis facility. He also gave an overview of SLAC’s work
to construct a cryogenically–cooled momentatron system to measure MTE.

##  III WG3: New Photocathode Ideas

Working Group 3 was focused on new ideas and concepts within the field of
photocathode technology. It was convened by Laura Monaco.

###  III.1 Ultra–thin MgO Films on
Metal Photocathodes to Enhance QE

Chris Benjamin (STFC Daresbury Laboratory, UK) presented work on his
application of ultra–thin oxide films to reduce work function and so increase
the photocathode QE. Metal photocathodes are used for their ease of use and
relative robustness to less than ideal vacuum conditions. However, their high
work function and consequent low quantum efficiency is a hindrance in their
utilisation as high beam currents require high power levels of UV light. The
application of an ultra–thin MgO film enhances the quantum efficiency by
nearly an order of magnitude on both Ag(100) and Cu(100) single crystals by
reducing the work function of the metal cathode. At the same time, the
robustness of the photocathode to residual vacuum gases is improved. The
chemical nature of this modified surface was investigated through X–ray
photoelectron spectroscopy (XPS), low energy electron diffraction (LEED) and
quantum efficiency measurements which confirmed an increase in electron yield.
Measurements of the mean transverse energy (MTE) using the TESS system at
Daresbury show that the photoemission threshold is extended through the
deposition of an ultra–thin oxide layer, and also that the energy spread of
the photoemitted electrons agrees well with the Dowell–Schmerge approximation.

###  III.2 Plasma Photocathode

Carlo Benedetti (LBNL, USA) presented an update on progress in the field of
plasma photocathodes. Plasma–based accelerators have received significant
interest in recent years owing to their ability to sustain large acceleration
gradients several orders of magnitude larger than in conventional RF
accelerators ($10100\text{\,}\mathrm{GV}\text{/}\mathrm{m}$ demonstrated
experimentally), enabling compact accelerating structures for future
high–energy applications (e.g., particle colliders) and radiation production
(e.g., X–ray FELs). In a plasma accelerator a drive beam with suitable
duration and intensity (either a laser pulse in a laser wakefield accelerator,
or a particle bunch in a plasma wakefield accelerator) propagating in a plasma
excites a plasma wave (or wakefield). The plasma wave has a relativistic phase
velocity, a characteristic size of
$\sim$$10100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (i.e., on the order of the
plasma wavelength, which depends on the background plasma density), and can
accelerate and focus a properly delayed particle beam. For instance, laser
wakefield accelerators have demonstrated the production of high charge
($10100\text{\,}\mathrm{pC}$), short ($110\text{\,}\mathrm{fs}$), and
high–quality (< 1 $\mu$m normalized emittance with $\sim 1$ % relative energy
spread) electron bunches, resulting in a 4D brightness of $\sim 10^{18}$
A/mrad2. These results were obtained in regimes where bunch production and
acceleration was done by means of a single laser driver.

In the plasma photocathode scheme the production and acceleration of the
electron bunch are decoupled. The (laser or particle) driver is responsible
for the creation of the plasma wave while bunch generation is achieved by
means of a short–wavelength, low–intensity, trailing ionization pulse which
liberates tunnel–ionized electrons from a high–Z gas (not fully ionized by the
driver) directly into the plasma wave. Modeling shows that the plasma
photocathode concept allows, in principle, the production of electron bunches
with a normalized emittance of $\sim$$10\text{\,}\mathrm{nm}$ which results in
a significant increase of the 4D brightness up to $\sim 10^{20}$ A/mrad2.
Experiments to demonstrate the plasma photocathode concept are underway at
FACET (SLAC) and the BELLA Center (LBNL).

###  III.3 CERN Photocathode Activities

Miguel Martinez–Calderon (CERN) gave an update on photocathode activities at
CERN, focusing on the fabrication and performance ofCs2Te cathodes for use in
the CERN Electron Accelerator for Research (CLEAR) and the Advanced
proton–driven plasma Wakefield acceleration Experiment (AWAKE). For the CLEAR
facility, CERN has implemented an in–situ solution utilising two evaporators
within the RF photoelectron gun in an isolated small preparation chamber. This
allows the cathode to be rejuvenated without opening the gun or exchanging the
cathode plug through the sequential evaporation of a Tellurium layer and a
Cesium layer, resulting in initial quantum efficiency (QE) values of up to
$23\text{\,}\mathrm{\char 37\relax}$ and sustained QE values over
$0.5\text{\,}\mathrm{\char 37\relax}$ during year–long runs. Cs2Te
photocathodes for the AWAKE experiment are fabricated and tested at CERN’s
photoemission laboratory using the co–evaporation technique, achieving
measured post–fabrication QE levels up to $24\text{\,}\mathrm{\char 37\relax}$
in the same laboratory in a test $65\text{\,}\mathrm{kV}$ DC electron gun.
These photocathodes are then transferred to the AWAKE RF photoelectron gun
using a UHV transport carrier, and during AWAKE 2021 – 2022 runs, the QE
values have been shown to reach a maximum of $25\text{\,}\mathrm{\char
37\relax}$ and a consistent performance.

In addition, Miguel presented CERN’s latest research studies on ultrafast
laser material surface nanopatterning as an alternative to improve the
photoemissive properties of metallic photocathodes. By tailoring the physical
dimensions of these surface nanostructures, one can localize the optical field
intensity and exploit plasmonic effects occurring in such nanostructures. As a
result, this surface nanopatterning technique can become a great tool for
improving metallic photocathodes photoemission behavior enabling their use for
next generation high brightness electron sources. Miguel presented the
performance of two different femtosecond laser nanopatterned plasmonic copper
photocathodes, analyzed by measuring the quantum yield with a
$65\text{\,}\mathrm{kV}$ DC electron gun utilizing $266\text{\,}\mathrm{nm}$
laser excitation generated by a nanosecond laser with $5\text{\,}\mathrm{ns}$
pulse duration and $10\text{\,}\mathrm{Hz}$ repetition rate. By comparing the
electron emission of the copper surface nanostructured areas with that of a
flat area, their results suggest quantum yield enhancements of up to 5 times.

###  III.4 Protective Layers on Bi–alkali Cathodes

Nathan Moody (LANL, USA) presented work on the use of 2D protective coatings
for photocathodes. The aim of this project is to address a recognised
accelerator R&D priority which is the development of a photocathode technology
which simultaneously provides long operational lifetime, high efficiency and
high brightness. The solution under investigation involves the use of a 2D
material which acts as a membrane to protect the photoemissive surface from
exposure to pollutants. Work published in 2017 demonstrated the concept for
metals. It showed that that a graphene layer on a copper photocathode
conferred an improvement of 8 orders of magnitude in operating pressure whilst
maintaining QE. The next phase of this project focused on using a 2D material
to protect a bi–alkali photocathode. XRD was used to to demonstrate a high
level of K2CsSb crystallinity following deposition onto graphene, nickel and
stainless steel substrates, and XRF to confirm near–ideal stoichiometry
(K1.85Cs1.08Sb). High resolution QE maps showed that graphene was the best
protective layer with the QE of the deposited film being uniform and typically
$1517\text{\,}\mathrm{\char 37\relax}$. The latest stage of this project has
shifted attention to studying the effect of the thickness of the 2D graphene
layer on the optical performance and QE of the photocathode structure.
Measured transmission through 2, 3 and 5 layers of graphene were consistently
lower than that predicted by theory: transmission of $5\text{\,}\mathrm{\char
37\relax}$ measured at $5\text{\,}\mathrm{eV}$ through 2 layers of graphene
compared to $50\text{\,}\mathrm{\char 37\relax}$ predicted by theory. It has
emerged that graphene is the most resilient protective layer in that the
difference between the QE for a ‘new’ and a ‘used’ graphene surface is very
small, while that for silicon and molybdenum is much larger. QE is therefore
more reproducible and resilient on graphene when reused. It was also noted
that the deposition of a bi–alkali layer on a graphene–coated metal substrate
serves to increase the QE by enhancing the ‘mirroring’ effect whereby incident
photons not initially absorbed are reflected from the metal surface can still
drive photoemission on their second pass through the graphene/bi–alkali
photoemissive layer. Future work will focus on the use of hexagonal boron
nitride instead of graphene as the protective layer as theory predicts higher
levels of QE with the same level of protection.

###  III.5 AI/ML–selection of Air–stable Photocathodes

Evan Antoniuk (Stanford University & LLNL, USA) gave an overview of progress
in the application of machine learning to screen potential photocathode source
candidate materials. Despite considerable research effort, the discovery of
new photocathode materials has historically proven to be challenging.
Photocathodes must meet a vast array of material design requirements and the
experimental characterization of photocathode materials is time intensive.
Recent advances in computational modelling, screening methodology, and machine
learning offer a unique opportunity to overcome these challenges by allowing
for hundreds of thousands of materials to be simultaneously screening against
multiple selection criteria.

The team have carried out the first such data–driven screening to identify
novel photocathode materials from a list of more than 74,000 candidates.
Notably, the search space of this effort is over 3 orders of magnitude larger
than what has been experimentally explored to date and encompasses a diverse
spectrum of candidate materials. In addition, we utilize machine learning
predictions of work functions of 90 ,698 surfaces to solve the problem of
identifying low work function ($<3$ eV) photocathode materials. Two distinct
screenings are performed to identify photocathodes which meet two sets of
selection criteria. First they identify novel high brightness photocathodes,
resulting in the discovery of candidates that exhibit intrinsic emittances
that are up to 4x lower than currently used photocathodes. Then they identify
air–stable photocathode materials to overcome the extreme air–sensitivity of
current photocathodes. This screening identifies the air–resistant
photocathodes (M2O, M=Na,K,Rb) that are also visible–light photosensitive.
Finally, they extensively validate the performance of these materials by
utilizing previous reports in the literature and state–of–the–art simulation
capabilities.

The high brightness and air–resistant photocathode materials discovered in
this work should pave the way for the experimental realization of higher
brightness and longer lifetime photonic devices. Their vast photoemission
dataset created in this work also provides statistical insights into
engineering better photocathode materials.

##  IV WG4: Metallic Photocathodes

Working Group 4 looked at the performance of metal photocathodes in
accelerator environments, and also at work to improve their performance. It
was convened by Lee Jones.

###  IV.1 Studies on the Evolution of MTE for Photocathodes Subjected to
Controlled Degradation by Gas Exposure

Liam Soomary (University of Liverpool, UK) presented work on the degradation
of polycrystalline copper photocathodes, and the performance of
caesium–implanted copper photocathodes. The degradation work involved use of
the Transverse Energy Spread Spectrometer (TESS) to monitor the evolution of
the photoemission footprint from a copper photocathode under illumination at
$256\text{\,}\mathrm{nm}$ which was subjected to continuous exposure to a
number of typical gas species found in the residual atmosphere in a vacuum
system. The work showed that following an initial decline in QE of 6
percentage points over the first 10 Langmuirs of oxygen exposure, the QE
remained largely stable at a level of $94\text{\,}\mathrm{\char 37\relax}$ of
the initial QE over an exposure of 100 Langmuirs. Work has also been done to
manufacture and characterise polycrystalline copper photocathodes implanted
with caesium ions. The implantation was carried out in two stages using a
Hiden IG5C Cs surface ion source. The first stage involved implantation at an
energy of $500\text{\,}\mathrm{eV}$, and the second at
$300\text{\,}\mathrm{eV}$. When characterised using the TESS over a range of
wavelengths, it was found that the caesium implanted copper photoemitted over
broad range of wavelengths from $236\text{\,}\mathrm{nm}$ to more than
$500\text{\,}\mathrm{nm}$. Energy spread measurements with the TESS show that
the $25\text{\,}\mathrm{meV}$ thermal floor was reached for the
$500\text{\,}\mathrm{eV}$ implanted cathode at an emission threshold around
$495\text{\,}\mathrm{nm}$, increasing to around $525\text{\,}\mathrm{nm}$
after the second implantation at $300\text{\,}\mathrm{eV}$. Both data sets fit
well with the Dowell–Schmerge model with workfunctions of 2.2 and 2.3 eV
respectively. The photoemission images form TESS indicated the presence of two
different photoemission sources when illuminated at short wavelengths, the
dominant low–MTE source being attributed to copper and the weaker high-MTE
source driven by the presence of caesium. Degradation studies showed that this
composite photocathode is more robust on exposure to oxygen than GaAs by a
factor of around 10.

###  IV.2 Copper Photocathodes for CLARA

In this presentation, Tim Noakes (STFC Daresbury Laboratory, UK) reviewed the
current status of the CLARA accelerator. He also updated on current phase 2
development work, the Full Energy Beam Exploitation (FEBE) and aspirations for
phase 3. He also reviewed the performance of the Alpha X
$10\text{\,}\mathrm{Hz}$ RF gun and the history of the solid copper backplate
photocathodes used therein, before continuing to discuss progress with the
newly–designed 2.6 cell S–band gun. The new gun uses a molybdenum INFN–style
photocathode puck as its baseline. The gun was conditioned using such a
cathode, but a high level of dark current was seen which was attributed to the
high levels of measured surface roughness of this cathode (c.
$113\text{\,}\mathrm{nm}$). The gun was also operated using thin film copper
was deposited onto a molybdenum cathode puck by magnetron sputtering. Though
initial QE measurements were favourably high, the lifetime was very poor. A
hybrid photocathode assembly with a bulk copper tip mounted in a molybdenum
body was also investigated. Initial tests were not promising, but following a
series of in–accelerator tests a hybrid photocathode and preparation process
were reached. This involved the use of BPS172 to clean the copper, on–axis and
off-axis diamond turning with a notable reduction in surface roughness of the
copper tip, dimensional adjustment to move the position of the photocathode
surface in the RF cavity (affecting the surface field), and finally argon
plasma treatment. The resulting hybrid photocathode demonstrated an
operational 1/e lifetime of 270 days, with an acceptable level of dark current
around $250\text{\,}\mathrm{pC}$. This cathode operated for 6 months during
the last exploitation period. The next steps will be the commissioning of a
new high repetition rate 1.5 cell $400\text{\,}\mathrm{Hz}$ S–band gun. The
photocathode choice for this gun will benefit from the previous experience
with the hybrid cathode for the 10 Hz gun, and also from the use of CsTe
photocathode currently under development at Daresbury.

###  IV.3 Copper photocathode experience at FERMI

Mauro Trovò (Elettra Sincrotrone Trieste, Italy) opened this talk by
highlighting the demands placed on the FERMI gun through 6,000 hours per year
of FEL operations. The FERMI @ Elettra facility uses a 1.6 cell RF gun. It
incorporates a cooled backplate but does not have a cathode exchange
mechanism. Radiabeam Technologies manufactured the gun, which has been
operational since May 2013. Gun chamber pressure logging has demonstrated that
the pressure rose progressively through 2020 from
$4\text{\times}{10}^{-10}\text{\,}\mathrm{mbar}$ to
$4\text{\times}{10}^{-9}\text{\,}\mathrm{mbar}$. QE data for 2021 shows
several steps in QE, with a general downward trend which is most likely linked
to the gradually increasing pressure. The cathode was replaced in August 2021,
and although this new cathode exhibited a lower QE than the previous one, the
QE level has remained fairly consistent during operations through to July
2022. However, differences have been noted between the virtual cathode laser
image and the extracted beam image which had very different Twiss parameters
and necessitated an adjustment of the machine optics. Differences in
photocathode QE may in general impact machine optics and prompt retuning. Dark
current levels up to 1 nC during the RF pulse without the presence of a drive
laser beam were noted, and while only a small proportion of these electrons
are transported any distance, they present a problem to the machine protection
system. ICT measurements suggest that the integrated dark current over the
course of 1 RF pulse is approximately equal in terms of total charge to the
actual electron beam pulse, emphasising the need to avoid dark current
generation in the first instance, and the requirement for a robust and stable
photocathode electron source.

###  IV.4 Technology Aspects in Preparation at NCBJ of Nb–Pb Superconducting
Photocathodes for XFEL–Type RF Electron Guns

Jerzy Lorkiewicz (Narodowe Central Badań Jądrowych, Poland) first summarised
the design parameters for the $1.3\text{\,}\mathrm{GHz}$ RF gun for an XFEL
and the PolFEL. A Pb/Nb thin film has been selected to achieve this due to
compatibility with the superconducting gun, durability to surface
contamination and laser-induced damage, its long lifetime, and its short
Cooper pair recombination time which facilitates repetition rates up to
$100\text{\,}\mathrm{kHz}$. The photocathode preparation process was outlined,
and the choice of cathodic arc Pb deposition justified through data showing
that the required QE was achieved in this way. The components of intrinsic
emittance were identified as those arising from thermal emittance, surface
roughness and surface field. This emphasised the need to minimise surface
roughness. Two cathodic arc deposition methods were presented: Filtered Arc
and Two-Step coating. Filtered arc produces Pb layers
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ in thickness, but with the presence of spherical protrusions up to
$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter. The Two-Step process
generated thicker films up to 18 microns, with surface features up to
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter reduced significantly
following exposure to the Ar plasma in the second step. XRD data implies a
higher level of crystallinity from the Two-Step process, predominantly
Pb(111), Pb(222) and Pb(333).

##  V WG5: Semiconductor Photocathodes

Working Group 5 focused on the development and application of
alkali–antimonide photocathodes and their use in accelerators. It was convened
by Maud Baylac.

###  V.1 Multi–alkali Antimonide Photocathodes for Highly Brilliant Electron
Beams

The first presentation of the semiconductor session was given by Chen WANG
(HZB, Germany) on multi–alkali antimonide photocathodes for highly brilliant
electron beams. The Sealab/bERLinPro project aims to build a SRF gun with
multi–alkali antimonides to reach high QE and long operation lifetime. NaKSb
photocathodes on a Mo substrate were produced in a UHV preparation chamber at
HZB. The influence of the deposition parameters was studied with XPS and QE
measurements. The correlation between chemical composition and QE values are
discussed. A hybrid deposition (combination of sequential and co–deposition)
proved to produce the best performance.

###  V.2 Characterisation at Daresbury Laboratory of CsTe Photocathodes Grown
at CERN

Lee Jones (STFC Daresbury Laboratory, UK) talked on the performance
characterization at Daresbury Laboratory of CsTe photocathodes manufactured at
CERN. In the framework of a future UK X–FEL, high–performance photocathodes
are studied: surface characteristics are analyzed as they limit the achievable
electron beam quality (emittance, brightness, energy spread). Detailed
analysis of CsTe samples produced at CERN were performed with XPS, STM and the
TESS spectrometer to evaluate the MTE. Multiple chemical species were found to
be acting as photoemitters (CsxTey, Cs and CsO), and correlations between the
measured surface characteristics and the MTE values for different wavelengths
were shown.

###  V.3 Cs2Te\KCsSb in Gun Operation at PITZ

The third talk, dedicated to development of multi–alkali antimonides
photocathodes for high brightness photoinjectors, was given by Sandeep MOHANTY
(DESY, Germany). For the KSbCs photocathodes synthesized at INFN–LASA,
different factors affecting the growth procedure, such as substrate
temperature and deposition rate, were detailed. The photocathodes tested in
the PITZ RF gun at DESY under high gradients showed promising results (QE,
thermal emittance, and response time) despite high dark current and short
operational lifetime. Density–Functional Theory calculations were done with
Quantum ESPRESSO. NaKSb(Cs) exhibited several–year lifetime and resistance up
to $120\text{\,}\mathrm{\SIUnitSymbolCelsius}$ without degradation.

###  V.4 Operation of Cs2Te in SRF gun
for THz User Shifts

The session concluded with Rong Xiang (HZDR, Germany) reporting on the
operation of Cs2Te in SRF gun for ELBE. For THz applications at the ELBE
facility, the SRF gun–II operates with Cs2Te photocathodes. After deposition
on a Cu substrate, photocathodes were loaded and tested in the SRF gun. The
evolution of QE during operation was analyzed and XPS measurement were done.
It was shown that Cs2Te photocathodes perform well in the SRF gun in terms of
QE and charge lifetime (> $10\text{\,}\mathrm{C}$) with an acceptable level of
dark current. It was underlined that a dedicated RF start up procedure is
essential to avoid multipacting and thus preserve the cathode surface.

##  VI ERL Roadmap for Europe

Boris Militsyn (STFC Daresbury Laboratory, UK) gave a review of the
accelerator R&D roadmap on Energy Recovery Linacs (ERL) and the significance
of photocathode research in this field. Within the 5 pillars of the European
strategy for particle physics (https://europeanstrategy.cern/), ERLs aim to
push the energy frontier for collider applications: they are foreseen for
FCC–eh and LHeC. On the pathway to the demonstration of LHeC at the GW level,
the PERLE project pursues multi–turn, multi–MW operation. For these ERL
applications, electron injectors must generate high average current (10s to
100s mA). The presentation included a review of the state–of–the–art in the
fields of DC, NCRF and SRF guns, extending to the major R&D topics to be
pursued such as green photocathodes, laser systems and novel gun technologies.

##  VII WG6: Photocathode Theory

Working Group 6 focused on the construction, development and testing of
theoretical models supporting photoemission from photocathodes. It was
convened by Tim Noakes.

###  VII.1 Monte Carlo Transverse Emittance and
Quantum Efficiency Study on Cs2Te

The first talk of the theory session was given by Gowri Adhikari (SLAC, USA)
and concerned the theoretical modelling of Caesium Telluride photocathodes
using Monte Carlo techniques. The model focused on the ideal orthorhombic
stoichiometric Cs2Te material, with the band structure and projected density
of states determined for the bulk structure using density functional theory
(DFT). This was then used in the three step photoemission model. The initial
energy distribution was taken from the valence band distribution with the
depth dependence calculated for the first $15\text{\,}\mathrm{nm}$. Particular
attention was paid to the various scattering phenomena during electron
transport to determine the distribution in the conduction band. The quantum
efficiency and thermal emittance could then be calculated and both agreed well
with experimental data if an assumed electron effective mass of 0.25 was used.
Unusually for this type of calculation, a response time was also extracted
from the simulations which indicated a value around $200\text{\,}\mathrm{fs}$.

###  VII.2 Monte Carlo Simulations of
Electron Photoemission from
Plasmon–enhanced Bialkali Photocathode

The second talk of the session was given by Zenggong Jiang (Shanghai Advanced
Research Institute, China). This talk concerned Monte Carlo simulations of
photoemission from plasmon–enhanced bi–alkali photocathodes. The motivation of
this study is to increase the adsorption of light in the surface region of the
photocathode, both increasing QE and improving the response time. Surface
plasmon adsorption can be enhanced by introducing a regular array of features
into the surface with appropriate dimensions to match the wavelength. Three
step photoemission was simulated. For a series of feature sizes, QE was
enhanced by between 2 – 3 times as a result of increased adsorption without
any increase in the calculated emittance. Future work is planned to fabricate
test structures to assess this enhancement technique experimentally.

###  VII.3 Exploring Cesium–-Tellurium Phase Space
via High–throughput Calculations beyond
Semi–local Density–functional Theory

The third talk was presented by Holger–Dietrich Sassnik (University of
Oldenburg, Germany) where they have an active collaboration with
experimentalists at HZB. The talk was about density functional theory
calculations of binary Caesium based compounds (Cs3Sb and Cs2Te) using high
throughput methods. By using these methods the group were able to look at a
whole range of different compositional and structural compounds in the two
binary systems and establish the most stable ones. For the Cs-Te system they
also carried out simulations taking into account the surface and evaluating
the stability of different terminations ad determined their electronic
properties.

###  VII.4 Ultrafast Sub–threshold
One-photon Photoemission

The final talk of the session came from Andreas Schroeder (University of
Illinois at Chicago, USA). This was a comparison of a new theoretical model
for single photon photoemission with data collected using their QE and MTE
measurement equipment. This experimental set up was described where the laser
system has been upgraded to increase power in the UV region. In the model, the
temperature of hot electrons and the ensuing Boltzmann tail in the energy
distribution give rise to significant electrons that are able to access higher
energy states within the conduction band that are in excess of the vacuum
level. Hence there is no requirement for more than one photon to produce
photoemission even below the work function. A comparison of data (QE and MTE)
with theory for systems including Cu(111), GaSb(001), N–doped diamond(001) and
Ag(111) demonstrated the general applicability of the approach to both metal
and semiconductor photocathodes.

##  VIII WG7: Advanced Photocathode Characterization

The last session of the EWPAA workshop in 2022 was dedicated to advanced
photocathode characterization. It was convened by Julius Kühn.

###  VIII.1 The ASU photocathode research lab

Siddarth Karkare (Arizona State University, USA) talked about his
Photoemission and Bright Beams Laboratory at ASU. An ideal photocathode lab
would consider photon and (cryogenic)–temperature dependence, surface
preparation, photoemission characterization and tests in an electron gun.
Everything has to be carried out in–situ under excellent extreme UHV
conditions to avoid degradation of high QE Cs-based photocathodes. The system
at ASU will be equipped with to growth chambers, one for alkali–antimonide
thin film growth and one with RHEED and EDX capabilities to enable epitaxial
grown films. Surface characterization can be done by UHV–AFM, LEED or Auger
electron spectroscopy. For MTE-measurements and Electron Energy Analyzer with
sub–meV resolution is available. A Photoemission Electron Microscope will
allow to study nano–scale electron source, cathode emission uniformity and
photonics/plasmonics integrated photocathodes. The system will also be
equipped with an $200\text{\,}\mathrm{kV}$ DC electron gun with an
acceleration gradient of $8\text{\,}\mathrm{MV}\text{/}\mathrm{m}$ at
$200\text{\,}\mathrm{kV}$ and a beam line for 4D space mapping and a
$3.0\text{\,}\mathrm{GHz}$ deflection cavity for temporal response
measurements. The systems is supported by a tunable wavelength laser and
cryocooling capabilities.

###  VIII.2 Photocathode Response Time Measurement

The second talk of this session was given by Gregor Loisc (DESY Zeuthen,
Germany) on behalf of the PITZ and INFN Milano cathode groups. He presented
his results on direct response time measurements on semiconductor (and metal)
photocathodes by measuring the bunch lengthening during the photoemission
process. The measurement setup at the PITZ test facility is capable of
studying the bunch length at low charge and short bunches with high TDS, to
identify the RF–contribution to the bunch length via two–bunch probe and to
disentangle cathode response time from the laser shape. The shortest response
time for Cs2Te measured was $184\pm 42\text{\,}\mathrm{fs}$ and the longest
$257\pm 41\text{\,}\mathrm{fs}$. No aging effects and no dependence on the QE
related of the cathode response time was observed. For an Au cathode a
response time of $183\pm 10\text{\,}\mathrm{fs}$ was measured. The simulation
results of the emission process show reasonable agreement.

###  VIII.3 Resolving Surface Chemical States of p-GaN:Cs Photocathodes by XPS
Analysis

The third talk of this session was given by Jana Schaber (Helmholtz Zentrum
Dresden–Rossendorf, Germany) on resolving the surface chemical states of
p-GaN:Cs photocathodes by in–situ photoelectron spectroscopy. A preparation
chamber at low $10\text{\times}{10}^{-10}\text{\,}\mathrm{mbar}$ vacuum level
is equipped with a $310\text{\,}\mathrm{nm}$ UV–LED with a power of
$50\text{\,}\mathrm{\SIUnitSymbolMicro W}$ and an XPS system. P-GaN was
investigated as a potential photocathode material which can be used in several
cycles whereby one cycle is defined as thermal cleaning, Cs–activation and
QE–decay. It was found that Carbon an Oxygen are incorporated in the material
by XPS and that Argon ion sputtering causes surface damage. The Cs activation
step was also studied by XPS. QE values between $712\text{\,}\mathrm{\char
37\relax}$ were achieved in the range of
$400500\text{\,}\mathrm{\SIUnitSymbolCelsius}$. p-GaN will be considered to be
operated in the gun, but the cathode holder design still remains challenging.

###  VIII.4 High Crystalline Cesium Telluride Photocathode on Atomically–thin
Graphene with Co–deposition

The fourth talk was given by Mengjia Gaowei (Brookhaven National Laboratory,
USA) on high crystalline cesium telluride photocathode on atomically–thin
graphene via co–deposition. At BNL a preparation system equipped with
evaporators for alkali metals, Sb and Te, and with QCM, RHEED and QE-setup is
operated which can be used at the NSLS–II light source doing XRD, XRR, XRF
during the growth of the photocathode. Cesium telluride was grown via
co–deposition and characterized by XRD online monitoring and XRR after growth.
The growth recipe was optimized to achieve the composition of Cs2Te on Si. The
co–deposition incorporates more Cs than the sequential growth and leads to
better crystallization, with low surface roughness and higher QE at
$266\text{\,}\mathrm{nm}$. Furthermore the growth of Cs2Te on graphene/Si was
studied, showing that the nucleation on graphene is more oriented than that on
the Si substrate and the film is textured in the early stage of the growth.
The use of graphene as a substrate for cesium telluride is a promising route
to produce cathode film with better crystallinity and better cathode
performance.

###  VIII.5 Molecular Beam Epitaxy of Cs–Sb Thin Films: Structure–oriented
Growth of
High Efficiency Photocathodes

The fifth talk of the session was given by Alice Galdi (University of Salerno,
Italy and CLASSE, Cornell University, USA) on the molecular beam epitaxy of
Cs–Sb thin films to grow structure–oriented high efficiency photocathodes.
Cs3Sb photocathodes have been grown epitaxially to control roughness and the
surface potential by orientation control. As substrate for epitaxial growth
3C–SiC was identified. Reflection High Energy Electron Diffraction (RHEED)
assisted MBE has been carried out to follow the crystallization process during
thin film growth. RHEED can be operated in real time with sub–monolayer
sensitivity and gives information about the surface roughness and
crystallinity. The growth procedure has been optimized for single crystalline
Cs3Sb.

###  VIII.6 Stoichiometry Control and Automated Growth
of Alkali Antimonide Photocathode Films
by Molecular Beam Deposition

The last talk of the session was given by Vitaly Pavlenko (Los Alamos National
Laboratory, USA) and focussed on stoichiometry control and automated growth of
alkali antimonide photocathode films by molecular beam deposition. A method of
film growth with support of a feedback loop that controls stoichiometry in
real time is presented. Photoemissive properties exhibit a distinct dependence
on the stoichiometric composition that depends on the ratio of the incident
fluxes. The results were obtained on Cs3Sb but are expected to be relevant to
other alkali antimonides and tellurides.

##  IX Conclusions

This has been the first in-person Photocathode Workshop since the Covid-19
Pandemic. More than 40 participants attended the workshop, both from Europe
and from the USA.

We also greatly appreciated the willingness of colleagues from America and
China who have endured the many hours of time zone difference in order to be
present remotely and give their contribution to the success of the Workshop.

We thank all the Workshop participants for their contributions to very
fruitful and stimulating discussions.

The Local Organizer Committee thanks also the EWPAA Scientific Program Members
for their support and help in organizing and achieving such a successful
event.

The next workshop in the European Workshop On Photocathodes for Accelerator
Applications series will be hosted by the Helmholtz Zentrum Dresden (HZDR) in
the middle of 2024.


# The triangular numbers are finitely stable

###### Abstract.

This small note proves that the set of triangular numbers is a finitely stable
additive basis. This, together with a previous result by the author, shows
that triangular numbers and squares are, among all polygonal numbers, the only
ones that are finitely stable additive bases.

Luan Alberto Ferreira

##  1\. Introduction

If $A$, $B\subseteq\mathbb{N}=\\{0,1,2,3,4,5,\ldots\\}$, we define

$A+B=\\{a+b;\ a\in A,\ b\in B\\}.$

Also, if $h\in\mathbb{N}$, $h\neq 0$, then we write

$hA=\underbrace{A+\cdots+A}_{h\text{ times}}=\\{a_{1}+\cdots+a_{h};\
a_{1},\ldots,a_{h}\in A\\}.$

If there is $h\in\mathbb{N}$, $h\neq 0$, such that $hA=\mathbb{N}$, then $A$
is said to be an additive basis. The smallest $h\in\mathbb{N}$ satisfying this
equality is called the order of $A$, and is denoted by $h=o(A)$. For example,
Lagrange’s four-square theorem says that the set of the squares
$\mathbb{N}^{2}=\\{0,1,4,9,16,25,\ldots\\}$ is an additive basis whose order
is four, i.e., $o(\mathbb{N}^{2})=4$. Finally, if $A\subseteq\mathbb{N}$ and
$n\in\mathbb{N}$, then we write

$A(n)=\\#\\{a\in A;\ 1\leq a\leq n\\}.$

In [1], it was defined that an additive basis $A$ is _finitely stable_ if the
order of $A$ is equal the order of $A\cup F$, for all finite subset
$F\subseteq\mathbb{N}$, and it was proved the following theorem:

######  Theorem 1.1.

Let $A$ be an additive basis such that $o(A)=h\geq 3$. If

$\displaystyle\lim_{n\rightarrow\infty}\frac{((h-2)A)(n)}{n}=0$

and

$\limsup_{n\to\infty}\displaystyle\frac{((h-1)A)(n)}{n}<1,$

then $A$ is finitely stable.

Using this result, it was shown that the set of the squares $\mathbb{N}^{2}$
is a finitely stable additive basis. But theorem 1.1 also implies that the set
of triangular numbers $T=\\{0,1,3,6,10,\ldots\\}$ is finitely stable. The
purpose of this humble note is to prove this and to show that among all
polygonal numbers, the triangular numbers and the squares are the only ones
that are finitely stable additive bases.

##  2\. The result

For the main result we will need the following facts. For completeness, a
proof of the second will be given.

######  Theorem 2.1 (Gauss’ num $=\Delta+\Delta+\Delta$ theorem).

The set of the triangular numbers $T$ is an additive basis of order $3$.

###### Proof.

See [3], chapter $1$, section $1.7$. ∎

######  Proposition 2.2.

If $n\equiv 5$ or $8\pmod{9}$, then $n$ is not the sum of two triangular
numbers.

###### Proof.

Suppose by absurd that $n\equiv 5\pmod{9}$ is the sum of two triangular
numbers. Then there exists $m$, $x$, $y\in\mathbb{N}$ such that

$9m+5=\frac{x(x+1)}{2}+\frac{y(y+1)}{2}.$

This implies

$18m+10=x(x+1)+y(y+1).$

Taking this equation mod $9$,

$1\equiv x(x+1)+y(y+1)\pmod{9}.$

But this equation does not have a solution mod $9$, contradiction. Analogously
for the case $n\equiv 8\pmod{9}$. ∎

######  Corollary 2.3.

The triangular numbers $T$ is a finitely stable additive basis.

###### Proof.

Since $o(T)=3$, the previous proposition shows that the hypotheses of theorem
1.1 are fulfilled for $T$. Then $T$ is a finitely stable additive basis. ∎

This corollary, together with the next two results, show that the only
polygonal numbers that are finitely stable additive bases are the squares
$\mathbb{N}^{2}$ and triangular numbers $T$.

######  Theorem 2.4 (Fermat-Cauchy polygonal number theorem).

Let $k\in\mathbb{N}$, $k\geq 3$. If $\ \mathbb{N}_{k}$ denotes the set of the
$k$-gonal numbers, then $o(\mathbb{N}_{k})=k$.

###### Proof.

See [2]. ∎

######  Theorem 2.5 (Legendre polygonal number theorem).

Let $m\geq 3$ and $N\geq 28m^{3}$. If $m$ is odd, then $N$ is the sum of four
polygonal numbers of order $m+2$. If $m$ is even, then $N$ is the sum of five
polygonal numbers of order $m+2$, at least one of which is $0$ or $1$.

###### Proof.

See [3], chapter $1$, section $1.7$. ∎

######  Corollary 2.6.

The only polygonal numbers that are finitely stable additive bases are the
squares $\mathbb{N}^{2}$ and triangular numbers $T$.

###### Proof.

Fix $m\in\mathbb{N}$, $m\geq 3$. Now consider the set

$\mathbb{N}_{m+2}\cup\\{j\in\mathbb{N};\ j<28m^{3}\\}$

and compare the order of this set with the order of $\mathbb{N}_{m+2}$. ∎

## References

  * [1] Ferreira, L. A.: Finitely stable additive bases. _Bulletin of the Australian Mathematical Society_ (3) 97 (2018), 360–362.
  * [2] Nathanson, M. B.: A short proof of Cauchy’s polygonal number theorem. _Proceedings of the American Mathematical Society_ (1) 99 (1987), 22–24.
  * [3] Nathanson, M. B.: _Additive Number Theory: The Classical Bases_. Grad. Texts in Math. 164, Springer, New York, 1996.