# Improving Interface Physics Understanding in High-Frequency Cryogenic Normal
Conducting Cavities

Gerard E. Lawler
Department of Physics & Astronomy
University of California, Los Angeles
Los Angeles, CA 90045
<EMAIL_ADDRESS>
&Fabio Bosco
Department of Physics & Astronomy
University of California, Los Angeles
Los Angeles, CA 90045
&James B. Rosenzweig
Department of Physics & Astronomy
University of California, Los Angeles
Los Angeles, CA 90045
https://gelawler.github.io/

###### Abstract

As progress towards real implementations of cryogenic high gradient normal
conducting accelerating cavities continues, a more mature understanding of the
surface physics in this novel environment becomes increasingly necessary. To
this end, we here focus on developing a deeper understanding of one cavity
figure of merit, the radiofrequency (RF) surface resistivity, $R_{s}$. A
combination of experimental measurements and theory development form the basis
of this work. For many cases, existing theory is sufficient but there are
nuances leading to systemic errors in prediction which we address here. In
addition, for certain cases there exist unexpected local minimum in $R_{s}$
found at temperatures above 0K. We compare here several alternative models for
RF surface resistivity those which incorporate thin film like behavior which
we use to predict the location of the local minimum in surface resistivity.
Our experimental results focus on C-band frequencies for the benefit of
several future cryogenic linear accelerator concepts intended to operate in
this regime. To this end we have measured factor of $2.89\pm 0.05$
improvements in quality factor at $77$K and $4.61\pm 0.05$ at 45K. We further
describe the test setup and cooling capabilities to address systematic issues
associated with the measurements as well as a comparison of RF cavity
preparation and the significant effect on $R_{s}$. Some implications of our
measurements to linear accelerators combined with the theoretical
considerations are extended to a wider ranger of frequencies especially the
two additional aforementioned bands. Additional possible implications for
condensed matter physics studies are mentioned.

_Keywords_ Normal conducting RF cavities $\cdot$ C-band $\cdot$ Future
Accelerators

##  1 Introduction

An area of significant interest in particle accelerator research is the
development of accelerating cavities which can support very high field
gradients. Field gradients are usually limited by breakdown rates (BDR) so
significant experimental progress has been made in the advancement of
technologies which reduce BDR thus allowing for higher field gradient [1, 2].
In particular, empirical studies have observed ultra high peak fields in
excess of 500 MV/m at 45K which correspond to 250 MV/m accelerating fields
[3]. Significant consideration has been given to X-band $\left(8-12\text{
GHz}\right)$ and S-band $\left(2-4\text{ GHz}\right)$ for a number of reasons
[4, 5, 6, 7]. C-band $\left(4-8\text{ GHz}\right)$ however presents a number
of advantageous features especially from the stand point of a practical
achievable linear accelerator based on high gradient normal conducting
cryogenic cavities. Several future linear accelerator concepts have been
proposed which in this frequency range which utilize the high gradients made
possible from cryogenic operation [8, 9, 10].

Theoretical progress regarding surface behavior especially in the context of
BDR has been slow largely owing to complicated nature of the phenomena
involving extreme environments on varying time and length scales [11]. The gap
in understanding is in several ways insufficient for the needs of a growing
field of research into high gradient cavities. Indeed as progress towards
cryogenic operation of normal conducting cavities continues, a more mature
understanding of the surface physics is necessary. This is especially true at
cryogenic temperatures in an ultra-high gradient regime. Still there is a
significant body of work to which we can refer for our specific RF oscillating
field case, especially developed in the context of DC breakdown experiments
and theory [12, 13]. The theory here is complex and involved since it involves
multiple length and time scales.

To this end we want to develop improvements which are more modest in scope and
simple to compute but still notabley improve understanding especially for
experimental realization. Specifically, we here focus on one important figure
of merit (FoM), the radiofrequency (RF) surface resistivity $R_{s}$. We
attempt to iteratively improve predictions and understanding of this single
FoM from which we can derive more accurate experimentally relevant near term
useful predictions. The relevance of $R_{s}$ as a proxy for general cryogenic
cavity performance is multidimensional but can be explained primarily with
reference to the RF pulse heating which oscillating field cavities experience
per cycle. Through the affect on pulse heating, $R_{s}$ then becomes useful in
informing BDR behavior [14]. These numbers have been analytically computed in
the past to calculate for example temperature rise in room temperature
cavities [15]. Extension to cryogenic temperature then becomes possible as
long as one has an understanding of low temperature $R_{s}$.

In the high temperature limit, computing $R_{s}$ is as simple as using
Maxwell’s equations with appropriate boundary conditions to compute the real
component of the complex impedance as shown in Equ. 1.

$R_{s}\left(T\rightarrow\infty\right)=Re\left(Z_{s}\right)=\sqrt{\frac{2\pi
f\mu_{0}\rho}{2}}$ (1)

From this expression we observe the implicit dependence on temperature via the
bulk electrical resistivity $\rho$ and to a lesser extant the eigenmode
frequency $f$. The simplest theory for temperature dependence of $\rho$ which
we have comes via the work of Bloch and Gruneisen [16, 17]

$\rho\left(T\right)=A\left(\frac{T}{\Theta_{R}}\right)^{n}\int_{0}^{\Theta_{R}/T}\frac{t^{n}}{\left(e^{t}-1\right)\left(1-e^{-t}\right)}dt+C$
(2)

Where $A$ is a scaling constant, $\Theta_{R}$ is the Fermi temperature and $C$
is an constant left as a free parameter to adjust for the residual resistivity
ratio (RRR) defined as the ratio of room temperature resistivity to that at
4K. The remaining paremeter $n$ is dimensionless and established by the
dominant scattering mechanism at play for the situation. The value used in the
past has been $n=5$ which is true for ideal metals [17, 18]. We can compute
this temperature dependent resistivity and plot for the case of a number of
different RRR values. These are showed in Fig. 1. There are several features
here to note, namely that for $C=0$, the resistivity drops to 0 at 0K such
that the $RRR\rightarrow\infty$.

Figure 1: (a) Bloch-Gruneisen temperature dependent bulk resistivity from Equ.
2 with $n=5$ (b) Surface resistivity for Cband cavity using Reuter-Sondheimer-
Chambers theory to compute limits and using patching formula.

In the low temperature limit one common theory used for calculation of $R_{s}$
is derived from the work of Reuter and Sondheimer [19] with further
significant contribution from Chambers [20]. The theory was formulated to
explain the systematic underestimate of $R_{s}$ as the surface approaches 0K,
a phenomena termed the anomalous skin effect (ASE). The primary effect here is
the differing functional dependence on temperature of the electron mean free
path length (MFPL) and RF skin depth, $\delta$. MFPL reduces much faster than
$\delta$ and cryogenic temperatures such that at cryogenic temperatures MFPL
becomes large compared to $\delta$, more scattering events occur and $R_{s}$
is reduced. It is common to compute the asymptotic value of surface
resistivity at $0$ K. As shown in Equ. 3. It is worth noting for future
discussion that there is no explicit dependence on bulk properties of the
metal in this theory.

$R_{s}\left(T\rightarrow 0\right)=Z_{0}\left[\frac{\sqrt{3}v_{f}}{16\pi
c}\left(\frac{\omega}{\omega_{p}}\right)^{2}\right]^{\frac{1}{3}}$ (3)

A patching function is then used to interpolate between the two regimes with
dimensional parameters $a$ and $b$ given by the relative proportion of
diffusive to specular scattering of electrons in the surface material as shown
in Equ. 4.

$R_{s}\left(T\rightarrow
0\right)=R_{\infty}\left(1+a\alpha^{-b}\right)\qquad\text{for }\alpha\geq 3$
(4)

The dimensionless parameters range of order unity and different values are
used by different analysis to reflect once again the relavent scattering
behavior in the metal [21]. In the case of Reuter and Sondheimer’s original
formulation these values correspond to $a=1.004$ and $b=0.3333$; values which
were are circumstances experimentally established by Chambers. Of note then is
that predictions are useful but not sufficient for assessing $R_{s}$ from
first principles in the regime of practical consideration to normal conducting
cryogenic RF cavities. Intended temperature ranges for our normal conducting
cavities are primarily in the range of $35-77$ K, unfortunately placed
directly in the interpolation regime.

In addition, more advanced specially designed _mushroom dome_ RF cavity
measurements deviate from the aformentioned model in that there exists a local
minimum in $R_{s}$ around $30$ K [3]. Furthermore, subtle bulk material
property dependence was observed [11]. Neither of these features are predicted
at all with the existing framework of calculation. Observing that the complex
nature of the theory of BDR would be difficult to incorporate into
multiphysics simulations of the type found in CST or HFSS, we are thus we are
justified in seeking simple improvements of $R_{s}$ predictions.

In the field of RF cavities, $R_{s}$ is measured often measured via the
unloaded quality factor $Q_{0}$. $R_{s}$ for these simple cavities are then
derived from $Q_{0}$ as an inverse quantity scaled by some geometric factor
$G$. More information on the systemics this introduces into our measurements
and simulations will be covered in the next section but here we note that this
introduces the necessity to precisely consider another bulk property in
addition to the bulk resistivity, the coefficient of thermal expansion. Now
there is the additional implicit temperature dependence via the changing
eigenmode $\omega$ at which we measure $S11$ and the changing cavity coupling
via the RF antenna. More on the temperature dependent COE values will be
explained in the methodology section.

##  2 Methodology

Our intention is thus to compute with full transparency, the RSC predictions
for a number of different scattering parameters make adjustments along the way
where necessary. We will further introduce more subtle thin film physics
effects in RF cavities and propose simple alternative models. An experimental
setup and measurement suite was designed for measuring $R_{s}$ in Cband RF
cavities in order to validate the models down to at least application relevant
temperatures.

###  2.1 Theory

Our first modification beyond the introduced theory $R_{s}$ comes from a
nuance regarding the BG equation itself. We note that the value of $n=5$ is
only strictly speaking true in certain circumstances special circumstances
[18]. For transition metals, like copper, a more accurate value is $n=3$. The
calculations with this improvement are shown in Fig. 2. The notable
significance here is that the asymptotic high temperature and low temperature
values are agnostic to this change but the intermediary values decidedely are
not. For a not unreasonable number of $RRR=500$, the 77K $R_{s}$ is increased
by a factor of $\approx 1.3$ and nearly $2$ at $45$K.

Figure 2: Bloch-Gruneisen temperature dependent bulk resistivity with $n=3$
with this exception of $C_{BG}=0$ and a RRR500 curve which still use $n=5$ for
comparison. The feature to note here is the significant deviation at
intermediate values.

####  2.1.1 Thin Film Physics

To address the existence of experimental local minimum at intermediate
temperatures we want to explore more in depth the space of interface physics
and in doing so indeed find that there are situations where $R_{s}$ does not
monotonically decrease with temperature. Indeed for certain circumstances it
increases with decreasing temperatures within the intermediate range of
temperatures with which we are concerned. This is called the Gurzhi effect and
is well established for thin film physics representing a novel regime where
electrons in the material take on an effective viscosity [22, 23, 24]. The
justification of considering one skin depth as an effective thin film with
thickness $\delta$ has been made and we reiterate it here [25]. To this end
two alternative models we will be our focus here: the qualitative components
of the viscous electron model developed by Gurzhi and a more simple
application of thin film resistivity developed by Fuchs & Sondheimer [26].
Both predict local minimum in $R_{s}$. Based on these improved models for
surface resistivity we find a nonzero optimal temperature for normal
conducting copper cavities as a function of RRR.

As an application of Gurzhi theory, we first must consider some of the length
scales involved in computation of $R_{s}$ for our C-band cavities. We plot
these in Fig. 3. Specifically we have the same asymptotic values as as
existing theory with exception of an intermediary regime where as temperature
increases, the resistance increase. This regime corresponds to the situation
where the effective mean free path length is approximately equivalent to the
following expression.

$\ell_{eff}\sim\frac{\delta^{2}}{\ell_{ee}}$

That is to say the effective electron mean free path becomes approximately
equal to the square of the sample size divided by the electron electron mean
free path length. For our effective mean free path length we can use NIST
polylog fit curves published from measured data which relate the thermal
conductivity to temperature dependence. We can then relate this to the
$\ell_{eff}$ with which it should scale linearly. We further consider
Mathiessen’s rule such that we can say the following [27]

$\ell_{eff}=\left(\frac{1}{\ell_{phonon}}+\frac{1}{\ell_{impurities}}+\cdots\right)^{-1}$

So for the case where $\ell_{ee}$ is largely a material dependent property and
thus independent of temperature we can compute the regime where this behavior
should occur for the case of $RRR450$ where if $\ell_{ee}\approx 40$ nm [28]
we would have the Gurzhi effect regime kicking in around $35-40$K. In
addition, for a low purity material like $RRR050$ we would not expect the
regime to be present at all. We can see this in Fig. 3. Our intention thus is
to cool to below this value for our existing pillbox structures to determine
this negative slope $dR_{s}/dT$ in experimental data.

(a) Ratio of RF skin depth (calculated classically) to $\ell_{eff}$ calculated
via NIST log functions

(b) $\delta$ times the ratio from plot (a) showing an idea of where the Gurzhi
effect may occur in cavities relevant to Cband cryogenic operation

Figure 3: Length scales involved in theoretical computations of $R_{s}$
relevant to our C-band cavity dimensions

A fully analytic application of Gurzhi theory is too involved for the time
being so considering the need for a more computationally tractable theory. For
simplicity and the sake of computational feasability, we also propose a model
where since the RF skin depth is treated as a thin film sample with thickness
equal to the skin depth we calculate directly the resistivity of a thin film.
We refer here to the formulation of the Fuchs and Sondheimer theory to compute
an thin film resistance[26]. With constants of proportionality composed of
elementary constants and material properties not explicitly dependent on
temperature. So we have the following where the value $p$ relates the
proportion of specular to diffusive scattering at the boundary

$\frac{\rho_{film}}{\rho_{bulk}}\approx\left[1.0-\frac{C_{0}}{\rho^{3/2}}\left(1-p\right)\right]^{-1}$

We again make the assumption that the impurity mean free path length dominates
at low temperature and is independent of temperature. For this model we assume
the case of ideal metals, not transition metals, so we will limit our
calculations in these cases to $n=5$. Also since $p$ is free parameter here we
can make a semi-empirical model where this is scaled to previous data. This
can then be said to be a near total specular scattering case since such that
$1-p$ is vanishingly small. We can then use this thin film resistivity in
place of the bulk resistivity in our existing formalism for computing $R_{s}$.
Plotting these results as a function of temperature gives is shown in Fig. 7

Figure 4: E-field profile in Cband cavity with antenna port

###  2.2 Experimental Methods

As mentioned in the introduction, experimentally the easiest method to measure
the $R_{s}$ in RF cavities is via unloaded quality factors, $Q_{0}$. In order
to do this we insert microwave antennas into the cavities and measure a the
reflection coefficient called $S11$ produced from a low level input signal.
$Q_{0}$ can then be computed from the shorted position as the central
eigenmode frequency divided by the $f3db$ bandwidth both of which are measured
on a vector network analyzer (VNA). We are then left with the need to
calculate the geometric factor $G$ which related $Q_{0}$ to $R_{s}$ in the
following manner.

$R_{s}=\frac{G}{Q_{0}}$ (5)

The resonant cavities that we use are as close to ideal cylindrical resonant
cavity as possible, a geometry commonly called a pillbox. Such a cavity is
ideal since it has analytic solutions for the eigenmodes in terms of
cylindrical Bessel functions from which we can quickly compute analytic curves
based on direct material properties. These offer useful checks of measured
data during experiment in addition to full multiphysics simulations which we
later perform in CST. The idealized Bessel function fields are depicted in
Fig. 4.

The RF geometry of the cavities are machined as close to the idealization as
possible. There are of course perturbations from the placement of four
features in particular: an RF antenna, a vacuum port, a fillet for machining
considerations, and the possibility of excess braze material in the edge where
the cavity cap is added. These features are replicated in the CST multiphysics
model for simulation. The consideration is now especially notable due to our
attempt to remove as many systematics as possible in a measurements since high
precision is needed.

There is an additional subtlety associated with $G$ also since it has an
implicit dependence on temperature via the coefficient of thermal expansion we
can consider it separately in simulation. We can analytically solve for the
$G$ for the case of the ideal pillbox cavity to obtain the following
expression relating $Q_{0}$ and $R_{s}$

$R_{s}=\frac{\chi_{01}\eta}{2}\frac{1}{1+a/h}\frac{1}{Q_{0}}$ (6)

Where $a$ and $h$ are the radius and height of the cylinder. Thankfully then
for an isotropic material the effect contraction cancels out and $G$ remains
temperature independent. For completeness we simulate the situation with the
real geometry with the expectation that the additional effect of antenna
cooling and contraction should be of vanishing significance if any measurable
at all.

Figure 5: (a) CYBORG beamline cryostat which can be configured for LLRF tests.
Including measurements presented here. (b) COMEB manufactured Cband cavity
measured within CYBORG cryostat to measure data here presented.

With respect to the cryogenics we have commissioned at UCLA multiple new
cryocoolers for novel RF testing including a single stage low temperatuer high
power cryocooler used for a new CrYogenic Brightness Optimized Radiofrequency
Photogun (CYBORG) [29]. In parallel to development of the photogun we have the
versatility to configure our cryostat for pillbox cavity testing by direct
coupling to the cooler heat exchanger. The CYBORG cryosat openned for access
is shown in Figure 5 along with the cavity under testing. The cooler relevant
to these measurements is a Sumitomo Heavy Industries CH-110LT. Because of the
high cooling capacity of this device our setup simply consists of the pillbox
cavities mounted to the cold head via a simple polished C101 copper coupling
plate. The composite system of cooler, coupler, and cavity are then wrapped in
14 layers of multi-layer insulation (MLI) with small penetration for
temperature sensors and the RF antenna cabler. For cooling, the device under
test is then placed within the CYBORG vacuum cryostat and pumped to around
$10^{-5}$ torr.

##  3 Results

We first address the results of our examination of improved simple models. For
the Fuchs and Sondheimer case we note now they because of the additional
dependence of bulk resistivity there is now additional dependence on RRR, a
bulk material property. The theory then also predicts a local minimum around
25K for the high purity case. For a full space of solutions over many values
of RRR we calculate a sweep for the contour plot shown in Fig. 7.

(a) Cavity detuning as a function of temperature with uncertainty derived from
published NIST curved for coefficient of thermal expansion as function of
temperature.

(b) Q factor enhancement measured in-situ via S11 curve fitting algorithm

Figure 6: Data from measured COMEB pillbox cavity during cryogenic testing.
Curves represent warming of cavities after cool down to minimum value of 38K

Contained within Fig. 6 are some of the results of our measurements of $Q_{0}$
enhancement factor as a function of temperature. The values of note from the
plot are the 45K and 77K values which are most relevant for the UCXFEL and C3
use cases. These and select other values are shown in Table 1. We also from
the $Q_{0}$ enhancement compute the surface resistivity values. In addition,
to measuring the temperature dependence of the cavity quality factor we also
measure the detuning of the cavity itself by tracking the minimum of the
cavity S11 value. The detuning when paired with NIST log fitting for cryogenic
copper coefficients of thermal expansion can then be used as a secondary
verification measurement of the temperature of the cavity.

To address some of the systematics, the cavity data was recorded on warming of
the cavity in order to better ensure thermalization at each temperature. As a
note on the temperature dependence of the geometric, our CST simulations show
a relatively minute impact on the low level RF figures of merit that were
measured and reported here. There is more significant impact on the coupling
coefficient which were here note are necessary for data analysis consideration
but are omitted simply for brevity.

##  4 Discussion

(a) Single curve from contour plot in (b) shown compared to existing Reuter-
Sondheimer-Chambers formulation

(b) Solution space of predictions for various RRR values as a function of
temperature.

Figure 7: Results of simplified Fuchs-Sondheimer style toy model used to
compute $R_{s}$(T) in our relevant regime of Cband frequency.

First of all, the LLRF measurements have immediate implications for the UCXFEL
[8] and more long term implications for the Cool Copper Collider [30]. As
shown in the Table 1, the two considered operation points are $45$ and $77$K
corresponding to the cryocooled UCXFEL photoinjector and the liquid nitrogen
cooled linac sections to be used in both concepts. Quality factor enhancement
approximately $4.61\pm 0.05$ and $2.89\pm 0.05$ respectively are necessary
numbers for RF design in the the two cases. In the Cband regime to our
knowledge these are reported here for the first term for a cavity manufactured
to accelerating cavity specifications.

In addition to the Q factor enhancement, the most significant observable
implication is the impact on the RF pulse heating. Our model predicts that
there is an optimum working temperature for a cryogenic normal conducting
cavity definitively based on minimizing pulse heating. During the initial
stages of the commissioning of CYBORG tests RF pulse heating as a a function
of temperature are currently underway and will provide an additional
measurement in addition to the S11 reflection coefficient [31].

Now with respect to the bulk material properties, we should clarify the
physical significance of what RRR means. In our consideration we use this as a
free parameter derived ex post facto from experiment and incorporated into the
Bloch-Gruneisen model. Strictly speaking this FoM is only a proxy for the
quantity of interest which is the material purity in terms of defects, grain
boundaries etc. We can then refer to high RRR as high purity metal and lower
RR as less pure. Less pure here need not refer to a failure of metallurgy,
rather it can also refer to intentional alloying. The main implication of the
models presented here is that there may exist local surface resistivity minima
for purity. More specifically, if one is willing to make the voyage to single
digit kelvin, an alloy with purity corresponding to below RRR250 is
preferable.

If higher temperatures are the only option then a more pure metal is required
with RRR450 matching the performance at 25K. So hardening the material with
alloying CuAg alloys then presents an appealing option to reduce the thermal
loads required. In addition, arbitrary purity is known to lead to unmanageable
soft copper from the machining perspective [32] so hardening at room
temperature first before cooling may be further be a necessity. We have begun
to consider this case with alloys, some of the measurements reported here
[33]. We reiterate this here for the sake of saying that the future testing of
the pillbox tests here are not only intending to iteratively reduce
temperature to below 30K but also repeat the $R_{s}$ measurements for varying
alloys, especially CuAg, using this pillbox template.

5.718 GHz cavity | 295K | 77K | 45K
---|---|---|---
$\Delta f$ [MHz] | 0 | 17.8 | 18.7
$Q_{0}$ enhancement | 1 | $2.89\pm 0.05$ | $4.61\pm 0.05$
$R_{s}$ [$\Omega$ m] | $3\times 10^{-2}$ | $1.05\times 10^{-2}$ | $6.5\times 10^{-3}$
Table 1: Measured properties for the two relevant use cases: the 45K operation
point for UCXFEL and the 77K operation point for $C^{3}$

Finally we can further the discussion of the implications beyond the scope of
cryogenic cavity development by considering the new thin film model as a
possible novel physical environment in which to study thin film physics. We
consider the possibility of going beyond the scope of linear accelerator
development and instead using cryogenic RF cavities as a new test environment
for the study of viscous electron flow. This is an area of intense study for
condensed matter and new examples of viscous electron are of certain interest
[24, 23].

##  5 Conclusion

We are here convinced of the validity of our surface physics understanding for
a Cband resonant cavity down to at least 38K. With a Q factor enhancement of
$4.61\pm 0.05$ at 45K and $2.89\pm 0.05$ at 77K this has immediate impacts on
the development of cryogenic copper linear accelerator development, especially
UCXFEL and $C^{3}$. Preparation and machining become the important knobs for
precision study but insofar as these are established processes in accelerator
physics, we can envision here low level RF cavities as test beds for material
physics properties relevant to high gradient cavities in parallel to the
CYBORG beamline at UCLA. Future studies will include in order of chronology, a
more in depth analysis of $R_{s}$ in the CYBORG photogun from the standpoint
of RF heating, cooling of the cavity to the limits of our cryocooler $<30K$,
and the manufacture of equivalent cavities with hard CuAg alloys. We also
begin to consider cryogenic RF cavities as possible novel test beds for
studying thin film and surface physics in a novel environment.

##  6 Acknowledgements

This work was supported by the Center for Bright Beams, National Science
Foundation Grant No. PHY-1549132 and DOE HEP Grant DE-SC0009914

## References

  * [1] Evgenya I. Simakov, Valery A. Dolgashev and Sami G. Tantawi  “Advances in high gradient normal conducting accelerator structures” Advances in Instrumentation and Experimental Methods (Special Issue in Honour of Kai Siegbahn)  In _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 907, 2018, pp. 221–230  DOI: https://doi.org/10.1016/j.nima.2018.02.085
  * [2] Valery A. Dolgashev  “Design Criteria for High-Gradient Radio-Frequency Linacs”  In _Applied Sciences_ 13.19, 2023  DOI: 10.3390/app131910849
  * [3] A.D. Cahill et al.  “Ultra High Gradient Breakdown Rates in X-Band Cryogenic Normal Conducting Rf Accelerating Cavities”  In _Proc. of International Particle Accelerator Conference (IPAC’17), Copenhagen, Denmark, 14-19 May, 2017_ , International Particle Accelerator Conference 8  Geneva, Switzerland: JACoW, 2017, pp. 4395–4398  DOI: https://doi.org/10.18429/JACoW-IPAC2017-THPIK125
  * [4] V.. Dolgashev et al.  “High-gradient rf tests of welded $X$-band accelerating cavities”  In _Phys. Rev. Accel. Beams_ 24 American Physical Society, 2021, pp. 081002  DOI: 10.1103/PhysRevAccelBeams.24.081002
  * [5] A.D. Cahill  “Measurements of Copper RF Surface Resistance at Cryogenic Temperatures for Applications to X-Band and S-Band Accelerators”  In _Proc. of International Particle Accelerator Conference (IPAC’16), Busan, Korea, May 8-13, 2016_ , International Particle Accelerator Conference 7  Geneva, Switzerland: JACoW, 2016, pp. 487–490  DOI: doi:10.18429/JACoW-IPAC2016-MOPMW038
  * [6] J.B. Rosenzweig et al.  “Ultra-high brightness electron beams from very-high field cryogenic radiofrequency photocathode sources”  In _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 909, 2018, pp. 224–228  DOI: 10.1016/j.nima.2018.01.061
  * [7] J.. Rosenzweig et al.  “Next generation high brightness electron beams from ultrahigh field cryogenic rf photocathode sources”  In _Phys. Rev. Accel. Beams_ 22 American Physical Society, 2019, pp. 023403  DOI: 10.1103/PhysRevAccelBeams.22.023403
  * [8] J B Rosenzweig et al.  “An ultra-compact x-ray free-electron laser”  In _New J. Phys._ 22 IOP Publishing, 2020, pp. 093067  DOI: 10.1088/1367-2630/abb16c
  * [9] Mei Bai et al.  In _arXiv preprint arXiv:2110.15800_ , 2021  DOI: 10.48550/arXiv.2110.15800
  * [10] G.E. Lawler et al.  “RF Testbed for Cryogenic Photoemission Studies”  In _Proc. IPAC’21_ , International Particle Accelerator Conference 12  JACoW Publishing, Geneva, Switzerland, 2021, pp. 2810–2813  DOI: 10.18429/JACoW-IPAC2021-WEPAB096
  * [11] Alexander David Cahill  “Ultra-High Accelerating Gradients in Radio-Frequency Cryogenic Copper Structures”  University of California, Los Angeles, 2017
  * [12] A. Grudiev, S. Calatroni and W. Wuensch  “New local field quantity describing the high gradient limit of accelerating structures”  In _Phys. Rev. ST Accel. Beams_ 12 American Physical Society, 2009, pp. 102001  DOI: 10.1103/PhysRevSTAB.12.102001
  * [13] K. Nordlund and F. Djurabekova  “Defect model for the dependence of breakdown rate on external electric fields”  In _Phys. Rev. ST Accel. Beams_ 15 American Physical Society, 2012, pp. 071002  DOI: 10.1103/PhysRevSTAB.15.071002
  * [14] Gaoxue Wang, Evgenya I. Simakov and Danny Perez  “Ab initio Cu alloy design for high-gradient accelerating structures”  In _Applied Physics Letters_ 120.13, 2022, pp. 134101  DOI: 10.1063/5.0084266
  * [15] David Peace Pritzkau  “RF pulsed heating”, 2001
  * [16] Jack Bass, William P. Pratt and Peter A. Schroeder  “The temperature-dependent electrical resistivities of the alkali metals”  In _Rev. Mod. Phys._ 62 American Physical Society, 1990, pp. 645–744  DOI: 10.1103/RevModPhys.62.645
  * [17] Mamdouh Hassan Abbas Ahmed Nasr  “Distributed-Coupling Linear Particle Accelerators”  Stanford University, 2021
  * [18] R.. Matula  “Electrical resistivity of copper, gold, palladium, and silver”  In _Journal of Physical and Chemical Reference Data_ 8.4, 1979, pp. 1147–1298  DOI: 10.1063/1.555614
  * [19] GEH Reuter and EH Sondheimer  “The theory of the anomalous skin effect in metals”  In _Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences_ 195.1042 The Royal Society London, 1948, pp. 336–364
  * [20] RGj Chambers  “The anomalous skin effect”  In _Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences_ 215.1123 The Royal Society London, 1952, pp. 481–497
  * [21] Weiren Chou and Francesco Ruggiero  “Anomalous skin effect and resistive wall heating”, 1995
  * [22] RN Gurzhi  “Minimum of resistance in impurity-free conductors”  In _Sov. Phys. JETP_ 44, 1963, pp. 771
  * [23] Noboru Takimoto  “A Theory of the Anomalous Skin Effect in a Metal Film”  In _Journal of the Physical Society of Japan_ 25.2, 1968, pp. 390–403  DOI: 10.1143/JPSJ.25.390
  * [24] Marco Polini and Andre K. Geim  “Viscous electron fluids”  In _Physics Today_ 73.6 AIP Publishing, 2020, pp. 28–34  DOI: 10.1063/pt.3.4497
  * [25] RN Gurzhi  “Contribution to the theory of the skin effect in metals at low temperatures”  In _Sov. Phys. JETP_ 20, 1964, pp. 1228–1230
  * [26] CR Pichard et al.  “Alternative analytical forms of the Fuchs-Sondheimer function”  In _Journal of materials science_ 20 Springer, 1985, pp. 4185–4201
  * [27] Sergio Calatroni  “Materials and Properties: Thermal and Electrical Characteristics”, 2020  arXiv:2006.02842 [physics.acc-ph]
  * [28] Daniel Gall  “Electron mean free path in elemental metals”  In _Journal of Applied Physics_ 119.8, 2016, pp. 085101  DOI: 10.1063/1.4942216
  * [29] G.E. Lawler et al.  “CrYogenic Brightness-Optimized Radiofrequency Gun (CYBORG)”  In _Proc. IPAC’22_ , International Particle Accelerator Conference 13  JACoW Publishing, Geneva, Switzerland, 2022, pp. 2544–2547  DOI: 10.18429/JACoW-IPAC2022-THPOST046
  * [30] Emilio A Nanni et al.  “Status and future plans for C3 R&D”  In _Journal of Instrumentation_ 18.09 IOP Publishing, 2023, pp. P09040
  * [31] G. al.  “Temperature stability in CrYogenic Brightness-Optimized Radiofrequency Gun (CYBORG)”  In _Proc. IPAC’23_ , IPAC’23 - 14th International Particle Accelerator Conference 14  JACoW Publishing, Geneva, Switzerland, 2023, pp. 1425–1427  DOI: 10.18429/JACoW-IPAC2023-TUPA039
  * [32] Jack Ekin  “Experimental techniques for low-temperature measurements: cryostat design, material properties and superconductor critical-current testing”  Oxford university press, 2006
  * [33] G. al.  “Effects of bulk material properties on RF surface resistivity”  In _Proc. IPAC’23_ , IPAC’23 - 14th International Particle Accelerator Conference 14  JACoW Publishing, Geneva, Switzerland, 2023, pp. 1422–1424  DOI: 10.18429/JACoW-IPAC2023-TUPA038