# Endowing Interpretability for Neural Cognitive Diagnosis by Efficient
Kolmogorov-Arnold Networks

Shangshang Yang1 &Linrui Qin2 &Xiaoshan Yu1
1School of Artificial Intelligence, Anhui University,
2School of Computer Science and Technology, Anhui University
Hefei 230601, China
{yangshang0308, stonewallqr<EMAIL_ADDRESS>
###### Abstract

In the realm of intelligent education, cognitive diagnosis plays a crucial
role in subsequent recommendation tasks attributed to the revealed students’
proficiency in knowledge concepts. Although neural network-based neural
cognitive diagnosis models (CDMs) have exhibited significantly better
performance than traditional models, neural cognitive diagnosis is criticized
for the poor model interpretability due to the multi-layer perception (MLP)
employed, even with the monotonicity assumption. Therefore, this paper
proposes to empower the interpretability of neural cognitive diagnosis models
through efficient kolmogorov-arnold networks (KANs), named KAN2CD, where KANs
are designed to enhance interpretability in two manners. Specifically, in the
first manner, KANs are directly used to replace the used MLPs in existing
neural CDMs; while in the second manner, the student embedding, exercise
embedding, and concept embedding are directly processed by several KANs, and
then their outputs are further combined and learned in a unified KAN to get
final predictions. To overcome the problem of training KANs slowly, we modify
the implementation of original KANs to accelerate the training. Experiments on
four real-world datasets show that the proposed KA2NCD exhibits better
performance than traditional CDMs, and the proposed KA2NCD still has a bit of
performance leading even over the existing neural CDMs. More importantly, the
learned structures of KANs enable the proposed KA2NCD to hold as good
interpretability as traditional CDMs, which is superior to existing neural
CDMs. Besides, the training cost of the proposed KA2NCD is competitive to
existing models.

##  1 Introduction

As a crucial technique in intelligent education, cognitive diagnosis (CD) [2,
5] is responsible for revealing students’ proficiency in knowledge concepts by
mining their historical records of answering exercises. For better
understanding, an illustrative example of the CD process is given in Fig. 1.
In this case, two students’ response records are given ((i.e.,
$\\{e_{1},e_{3},e_{4}\\}$ and $\\{e_{1},e_{2},e_{3}\\}$)) and the
relationships between exercises and knowledge concepts are also depicted
through an exercise-concept relational matrix ($Q$-matrix for short). As can
be seen, the CD process models the student answering prediction to obtain the
two students’ knowledge mastery states by mining their response records and
utilizing the $Q$-matrix. With the diagnosed students’ knowledge proficiency,
many subsequent tasks [30, 3] on online education platforms can provide more
accurate and higher quality services, such as remedial instruction, learning
path recommendations, targeted training, and exercise/course assembly.

Figure 1: Illustration of cognitive diagnosis process of two students.

To develop convincing cognitive diagnosis models (CDMs) for meeting the
demands of fast-growing education platforms (e.g., ASSISTments [23], PTA
[14]), massive efforts have been devoted by researchers in different domains,
mainly from two research perspectives. The first perspective is to design
completely interpretable CDMs whose components and operations are drawn from
educational psychology, so that the users (including students, teachers, and
parents) are able to understand how the diagnosis results are obtained,
trusting the results. The representatives include IRT [10], DINA [26], MIRT
[24], and MF [18]. The second perspective is to leverage neural networks (NNs)
to model the student’s response prediction process, aiming to improve the
prediction accuracy and provide more accurate diagnosis results for making
subsequent tasks like recommendations more convincing. This type of CDMs is
called neural cognitive diagnosis models due to the NNs used, and the
representatives contain NCD [28], KaNCD [29], (RCD) [13], and so on [21].

Compared to traditional CDMs in educational psychology, neural CDMs indeed
exhibit significantly better performance [28], which can make subsequent tasks
benefit more. Despite that, the model interpretability of neural CDMs is not
as good as that of traditional CDMs because these models more or less employ
the multi-layer perceptions (MLPs), i.e., multiple fully connected (FC) layers
[11]. It is difficult to interpret what roles the employed MLPs or FC layers
play and how they process the inputs to get the output prediction [31] even
equipped with the monotonicity assumption [25]. This monotonicity property can
only ensure that their weights are positive and their outputs are
monotonically increasing with the inputs [28], which can not make users
understand how the diagnosis results are obtained. As a result, neural CDMs’
limited interpretability hampers their ability to engage users convincingly to
some extent.

To this end, this paper aims to build more convincing CDMs by enhancing the
interpretability of neural cognitive diagnosis models without sacrificing the
accuracy of the diagnosis results. Thus, this paper proposes leveraging
efficient Kolmogorov-Arnold networks for neural cognitive diagnosis models
(KA2NCD) to empower the model’s interpretability and maintain the accuracy of
diagnosis. Specifically, our main contributions are as follows:

  * •

This paper is the first work to leverage Kolmogorov-Arnold networks (KANs) to
empower the interpretability of neural CDMs while maintaining high diagnosis
accuracy. To achieve this, we propose two manners of using KANS for neural CD.
The first is replacing MLPs in neural CDMs by KANs to enhance their
interpretability directly, and the second is designing a completely new
aggregation framework consisting of multiple KANs without the help of any
neural CDMs.

  * •

In the new aggregation framework, there exist two levels of KANs to handle
input and output the prediction. Several KANs at the lower level are used to
directly process the student embedding, exercise embedding, and concept
embedding, respectively. Then, a unified two-layer KAN at the upper level is
used to further combine and learn the outputs of lower-level KANs to get final
predictions. Besides, considering high runtime in the original implementation
of KANs, we modified the implementation of KANs to accelerate the training.

  * •

To validate the effectiveness of the proposed KA2NCD, we compare it with some
representative CDMs on four popular education datasets. Experimental results
show that the proposed KA2NCD achieves better performance than traditional
CDMs and neural CDMs. Besides, the learned structures of KANs in the proposed
KA2NCD also demonstrate its higher interpretability than neural CDMs, as good
as traditional CDMs. Moreover, the modified implementation of KANs makes the
training cost of the proposed KA2NCD competitive to existing models.

##  2 Preliminaries and Related Work

###  2.1 Preliminaries of Cognitive Diagnosis Task

In the cognitive diagnosis task of intelligent education scenarios, we
consider $N$ students, $M$ exercises, and $K$ knowledge concepts, represented
by the sets $S=\\{s_{1},s_{2},\dots,s_{N}\\}$,
$E=\\{e_{1},e_{2},\dots,e_{M}\\}$, and $C=\\{c_{1},c_{2},\dots,c_{K}\\}$
respectively. The platform employs an exercise-concept relation matrix
provided by domain experts, referred to as the $Q$-matrix, denoted by
$Q=(Q_{jk}\in{0,1})^{M\times K}$. Here, $Q_{jk}=1$ implies that exercise
$e_{j}$ involves knowledge concept $c_{k}$, and $Q_{jk}=0$ indicates no
relation between them. Additionally, the platform maintains a log of students’
responses to exercises, recorded in $R_{log}$. This log comprises triplets
$(s_{i},e_{j},r_{ij})$, where $s_{i}\in S$, $e_{j}\in E$, and
$r_{ij}\in\\{0,1\\}$. In this context, $r_{ij}=1$ denotes a correct response
by student $s_{i}$ to exercise $e_{j}$, whereas $r_{ij}=0$ denotes an
incorrect response.

By utilizing the students’ response logs $R_{log}$ and the $Q$-matrix, the
cognitive diagnosis task involves mining students’ proficiency in knowledge
concepts by developing a model, $\mathcal{F}$, to predict students’ scores on
exercises. The model $\mathcal{F}$ utilizes three types of input features to
predict the score of student $s_{i}$ on exercise $e_{j}$: the student-related
feature vector $\mathbf{h}_{S}\in\mathbb{R}^{1\times D}$, the exercise-related
feature vector $\mathbf{h}_{E}\in\mathbb{R}^{1\times D}$, and the knowledge
concept-related feature vector $\mathbf{h}_{C}\in\mathbb{R}^{1\times K}$.
Specifically, the embedding for students, exercises, and knowledge concepts
can be obtained as follows:

$\small\left.\begin{aligned} \mathbf{h}_{S}=\mathbf{x}_{i}^{S}\times
W_{S},W_{S}\in\mathbb{R}^{N\times D},\mathbf{h}_{E}=\mathbf{x}_{j}^{E}\times
W_{E},W_{E}\in\mathbb{R}^{M\times D},\\\
\mathbf{h}_{C}=\mathbf{x}_{j}^{C}\times W_{Q},W_{Q}\in\mathbb{R}^{K\times
D},\mathbf{x}_{j}^{C}=\mathbf{x}_{j}^{E}\times
Q=(Q_{j1},Q_{j2},\cdots,Q_{jK})\\\ \end{aligned}\right.,$ (1)

where $D$ is the embedding dimension (usually equal to $K$ for consistency),
$\mathbf{x}_{i}^{S}\in\\{0,1\\}^{1\times N}$ is the one-hot vector for student
$s_{i}$, $\mathbf{x}_{j}^{E}\in\\{0,1\\}^{1\times M}$ is the one-hot vector
for exercise $e_{j}$, and $W_{S}$ and $W_{E}$ are trainable matrices in the
embedding layers. Then, the model $\mathcal{F}$ outputs the predicted response
$\hat{r}_{ij}$ as

$\hat{r}_{ij}=\mathcal{F}(\mathbf{h}_{S},\mathbf{h}_{E},\mathbf{h}_{C}),$ (2)

where $\mathcal{F}(\cdot)$ is the diagnostic function to combine three types
of inputs in different manners. Generally speaking, after training the model
$\mathcal{F}$ based on students’ response logs, each bit value of
$\mathbf{h}_{S}$ represents the student’s proficiency in the corresponding
knowledge concept.

###  2.2 Related Work on Cognitive Diagnosis

In the following, some representatives of traditional CDMs and neural CDMs
will be reviewed, which focuses on introducing their prediction process.

####  2.2.1 Traditional CDMs

As the most typical CDM, DINA [26] outputs the prediction $\hat{r}_{ij}$ as

$\hat{r}_{ij}=g^{1-nt}(1-sl)^{nt},\textrm{where}\
nt=\textstyle\prod_{k}\boldsymbol{\theta}_{k}^{\boldsymbol{\beta}_{k}},\
\boldsymbol{\beta}=\mathbf{h}_{C}.$ (3)

$\boldsymbol{\theta}\in\\{0,1\\}^{1\times K}$ and
$\boldsymbol{\beta}\in\\{0,1\\}^{1\times K}$ are two binary latent features to
indicate which concepts the student mastered and the exercise contains.
$\boldsymbol{\theta}$ can be obtained from $\mathbf{h}_{S}$ through a FC layer
and Gumbel-Softmax [15]. The guessing factor $g\in\mathbb{R}^{1}$ and slipping
factor $sl\in\mathbb{R}^{1}$ refer to the probability of correctly answering
the exercise only with guessing and the probability of mistakenly answering
even with the corresponding concept mastered, and they are transformed from
$\mathbf{h}_{E}$ by FC layers. As can be seen, the prediction process of DINA
is interpretable, but DINA suffers from poor prediction performance on large-
scale data.

As another typical CDM, the prediction process of IRT [10] can be denoted as
follows:

$\begin{aligned} \hat{r}_{ij}=Sigmoid(a(\theta-\beta)),\
\theta\in\mathbb{R}^{1}=FC(\mathbf{h}_{S}),\
&\beta\in\mathbb{R}^{1}=FC(\mathbf{h}_{E}),\
a\in\mathbb{R}^{1}=FC(\mathbf{h}_{E})\\\ \end{aligned},$ (4)

where $\theta$ is obtained from $\mathbf{h}_{S}$ by an FC layer, denoting the
student ability feature. $\beta$ and $a$ are transformed from $\mathbf{h}_{S}$
by two different FC layers, denoting the exercise difficulty and distinction
features. As can be seen, the prediction of IRT can be easily understood and
interpreted. However, IRT also does not perform well on some complex datasets.

As a multidimensional variant of IRT, MIRT [24] applies the same logistic
function to the linear transformation of the student ability vector
$\boldsymbol{\theta}\in\mathbb{R}^{1\times K}$, the exercise difficulty
feature ${\beta}\in\mathbb{R}^{1}$, and the knowledge concept latent vector
$\boldsymbol{\alpha}\in\mathbb{R}^{1\times K}$. That can be denoted as

$\begin{aligned}
\hat{r}_{ij}=Sigmoid({\sum\boldsymbol{\alpha}\odot\boldsymbol{\theta}-\beta}),\
\boldsymbol{\theta}=\mathbf{h}_{S},\ &\beta=FC(\mathbf{h}_{E}),\
\boldsymbol{\alpha}=FC(\mathbf{h}_{C})\\\ \end{aligned},$ (5)

where student ability feature $\boldsymbol{\theta}$ and knowledge concept
latent feature $\boldsymbol{\alpha}$ are multidimensional and can handle
multidimensional data [7]. Therefore, MIRT exhibits better performance than
IRT without losing interpretability.

Different from the above traditional CDMs, MF [18] is originally devised for
recommender systems but can be used for CD. MF holds very high
interpretability because its prediction process [28] is very easy as
$\hat{r}_{ij}=\sum\mathbf{h}_{S}\odot\mathbf{h}_{E}$. MF directly applies the
inner product to student embedding $\mathbf{h}_{S}$ and exercise embedding
$\mathbf{h}_{E}$ to compute the similarity. Larger similarity represents a
higher probability of correctly the student answering the exercise. MF is
quite simple yet effective compared to above CDMs.

####  2.2.2 Neural CDMs

To improve the diagnosis accuracy, Wang _et al._ tried to incorporate NNs with
high-interpretability traditional CDMs like IRT and thus proposed a neural
cognitive diagnosis framework (NCD) [28]. NCD’s prediction is as

$\begin{aligned} \mathbf{f}_{S}=Sigmoid(\mathbf{h}_{S}),\
&\mathbf{f}_{diff}=Sigmoid(\mathbf{h}_{E}),\
f_{disc}=Sigmoid(FC(\mathbf{h}_{E}))\\\
&\hat{r}_{ij}=FC_{3}(FC_{2}(FC_{1}(\mathbf{y}))),\
\mathbf{y}=\mathbf{h}_{C}\odot(\mathbf{f}_{S}-\mathbf{f}_{diff})\times
f_{disc}\\\ \end{aligned}.$ (6)

$\mathbf{f}_{S}\in\mathbb{R}^{1\times K}$ represents student ability vector,
$\mathbf{f}_{diff}\in\mathbb{R}^{1\times K}$ and $f_{disc}\in\mathbb{R}^{1}$
represent exercise difficulty vector and distinction feature. The computation
process of $\mathbf{y}$ is similar to IRT, and $\hat{r}_{ij}$ is obtained by
applying three FC layers to $\mathbf{y}$. It can be seen the three-FC-layer is
difficult to interpret in NCD. Although NCD attempts to utilize the
monotonicity assumption to enhance its interpretability, the assumption can
not make users understand how the diagnosis results are obtained. In summary,
NCD indeed achieves promising improvements in diagnosis accuracy yet
sacrifices its model interpretability to some extent.

As the follow-up work of NCD, KSCD [21] adjusts the positions of FC layers in
NCD to obtain more meaningful student and exercise latent vectors by
incorporating with knowledge concept vector first. Its prediction process can
be denoted as

$\begin{aligned}
\hat{\mathbf{h}_{S}}=Sigmoid(&FC_{1}([\mathbf{h}_{S},\mathbf{h}_{C}])),\hat{\mathbf{h}_{E}}=Sigmoid(FC_{2}([\mathbf{h}_{E},\mathbf{h}_{C}]))\\\
&\hat{r}_{ij}=(\sum{\mathbf{h}_{C}\times(\hat{\mathbf{h}_{S}}-\hat{\mathbf{h}_{E}})})/D\\\
\end{aligned},$ (7)

where $\hat{\mathbf{h}_{S}}$ and $\hat{\mathbf{h}_{E}}$ are two latent vector
with length of $D$. Despite the promising performance of KSCD, we can see the
second equation can be interpreted yet the first one containing two FC layers
holds poor interpretability.

As another representative of neural CDMs, the diagnostic function of RCD [13]
gets the prediction $\hat{r}_{ij}$ as follows:

$\begin{aligned}
\hat{\mathbf{h}_{S}}=Sigmoid(&FC_{1}([\mathbf{h}_{S},\mathbf{h}_{C}])),\hat{\mathbf{h}_{E}}=Sigmoid(FC_{2}([\mathbf{h}_{E},\mathbf{h}_{C}]))\\\
&\hat{r}_{ij}=Sigmod(\frac{1}{D}\sum
FC_{3}(\hat{\mathbf{h}_{S}}-\hat{\mathbf{h}_{E}}))\end{aligned}.$ (8)

Although previous papers [22] show RCD holds dominant performance over other
neural CDMs, it can be seen that RCD’s interpretability is worse than KSCD
because the second equation further contains an FC layer.

###  2.3 Kolmogorov-Arnold Networks (KANs)

To address the lack of interpretability in existing neural CDMs, which can be
mainly attribute to the opaque nature of the MLP. This paper aims to
incorporate KANs into neural CDMs or directly leverage KANs for cognitive
diagnosis because high model interpretability of KANs as shown in [19]. In the
following, KANs will be introduced briefly.

A $L$-layer MLPs can be written as interleaving of transformations ${W}$ and
activations $\sigma$:

$\operatorname{MLP}(\mathbf{x})=\left({W}_{L-1}\circ\sigma\circ{W}_{L-2}\circ\sigma\circ\cdots\circ{W}_{1}\circ\sigma\circ{W}_{0}\right)\mathbf{x},$
(9)

which approximates complex functional mappings through multiple layers of
nonlinear transformations. However, its deeply opaque nature constrains the
model’s interpretability, posing challenges to intuitively understanding the
internal decision-making process.

To address the issues of low parameter efficiency and poor interpretability in
MLPs, Liu _et al._ [19] introduced the Kolmogorov-Arnold Network (KAN) that is
inspired by Kolmogorov-Arnold representation theorem [17] [4]. Similar to MLP,
a $L$-layer KAN can be described as a nesting of multiple KAN layers:

$\operatorname{KAN}(\mathbf{x})=\left(\boldsymbol{\Phi}_{L-1}\circ\boldsymbol{\Phi}_{L-2}\circ\cdots\circ\boldsymbol{\Phi}_{1}\circ\boldsymbol{\Phi}_{0}\right)\mathbf{x},$
(10)

where $\boldsymbol{\Phi}_{i}$ represents the $i$-th layer of the whole KAN
network. For each KAN layer with $n_{in}$ -dimensional input and $n_{out}$
-dimensional output, $\boldsymbol{\Phi}$ consist of $n_{in}*n_{out}$ 1-D
learnable activation functions $\phi$:

$\boldsymbol{\Phi}=\left\\{\phi_{q,p}\right\\},\quad p=1,2,\cdots,n_{\text{in
}},\quad q=1,2\cdots,n_{\text{out }}.$ (11)

When computing the result of the KAN network from layer $l$ to layer $l+1$, it
can be represented in matrix form as follows:

$\mathbf{x}_{l+1}=\underbrace{\left(\begin{array}[]{cccc}\phi_{l,1,1}(\cdot)&\phi_{l,1,2}(\cdot)&\cdots&\phi_{l,1,n_{l}}(\cdot)\\\
\phi_{l,2,1}(\cdot)&\phi_{l,2,2}(\cdot)&\cdots&\phi_{l,2,n_{l}}(\cdot)\\\
\vdots&\vdots&&\vdots\\\
\phi_{l,n_{l+1},1}(\cdot)&\phi_{l,n_{l+1},2}(\cdot)&\cdots&\phi_{l,n_{l+1},n_{l}}(\cdot)\end{array}\right)}_{\boldsymbol{\Phi}_{l}}\mathbf{x}_{l}.$
(12)

In conclusion, KANs differentiate themselves from traditional MLPs by using
learnable activation functions on the edges and parametrized activation
functions as weights, eliminating the need for linear weight matrices. This
design allows KANs to achieve comparable or superior performance with smaller
model sizes. Moreover, their structure enhances model interpretability without
compromising performance, making them suitable for applications like
scientific discovery. In cognitive diagnostic tasks, KANs may offer precise
diagnosis and analysis of learners’ knowledge structures, aiding personalized
teaching and precision education with intuitive data interpretation.

Figure 2: The Overview of the proposed KA2NCD, containing two implementation
manners.

##  3 The Proposed KA2NCD

Overview. Figure 2 presents the overview of the proposed KA2NCD containing two
implementation manners. In the first manner (termed KA2NCD-native), all FC
layers (MLPs) in the utilized CDM (NCD or KaNCD or KSCD or RCD) will be
replaced by KANs with the same settings. In the second manner, there are two
levels of KANs in the main module and two alternative embedding modules. In
the main module, KANs at the lower level process the received student,
exercise, and concept embeddings, while a KAN at the upper level combines and
learns the lower KANs’ outputs to get the prediction. For the second manner,
KAN2CD-e adopts the common embedding layers as its embedding module, while
KAN2CD-kan adopts KANs as its embedding module.

###  3.1 Manner 1: KA2NCD-native

KA2NCD-native directly replaces all FC layers in the utilized neural CDM. For
example, when adopting the NCD as the main module, KA2NCD-native (can be
denoted as NCD+) outputs the prediction $\hat{r}_{ij}$ as

$\begin{aligned} \mathbf{f}_{S}=&Sigmoid(\mathbf{h}_{S}),\
\mathbf{f}_{diff}=Sigmoid(\mathbf{h}_{E}),\
f_{disc}=Sigmoid(KAN_{1}(\mathbf{h}_{E}))\\\
&\hat{r}_{ij}=KAN_{2}(\mathbf{y}),\
\mathbf{y}=\mathbf{h}_{C}\odot(\mathbf{f}_{S}-\mathbf{f}_{diff})\times
f_{disc}\\\ \end{aligned}.$ (13)

$KAN_{1}(\cdot)$ is used to get the exercise difficulty scalar, and
$KAN_{2}(\cdot)$ is used to get $\hat{r}_{ij}$ from $\mathbf{y}$. Due to the
page limit, more KA2NCD-native variants (taking KaNCD, KSCD, and RCD as the
main modules) can be found in Appendix, denoted as KaNCD+, KSCD+, and RCD+.

###  3.2 Manner 2: KA2NCD-e and KA2NCD-kan

Different from manner 1 under existing CDMs, in manner 2, a novel aggregation
framework is designed for cognitive diagnosis based on KANs, which consists of
two modules, i.e., the embedding module to get the input embedding and the
main module to process input embedding and get the prediction.

####  3.2.1 The embedding module

This paper designs two alternative embedding modules: the vanilla embedding
module using the embedding layers and the KAN embedding module using KANs to
get the embedding. For whichever type of embedding module,there are $k$ sub-
embedding modules to get $k$ different initial embeddings
$\\{\mathbf{h}_{Si},\mathbf{h}_{Ei},\mathbf{h}_{Ci}|1\leq I\leq k\\}$.
Multiple embeddings will provide diverse representations and may cause better
learning results, which is similar to multiple heads in Transformer [27]

For the vanilla embedding module, each sub-embedding module’s forward process
is the same as Eq.(1), and the process of the vanilla embedding module can be
denoted as

$\begin{aligned} \mathbf{h}_{Si}=\mathbf{x}_{i}^{S}&\times
W_{Si},W_{Si}\in\mathbb{R}^{N\times
D},\mathbf{h}_{Ei}=\mathbf{x}_{j}^{E}\times
W_{Ei},W_{Ei}\in\mathbb{R}^{M\times D}\\\
&\mathbf{h}_{Ci}=\mathbf{x}_{j}^{C}\times W_{Qi},W_{Qi}\in\mathbb{R}^{K\times
D}\\\ \end{aligned},1\leq i\leq k.$ (14)

For the KAN embedding module, each sub-embedding module’s forward process is
as

$\begin{aligned} \mathbf{h}_{Si}=KAN_{1*k}(\mathbf{x}_{i}^{S}|\Phi_{1*k}),&\
\mathbf{h}_{Ei}=KAN_{2*k}(\mathbf{x}_{i}^{E}|\Phi_{2*k}),\Phi_{2},\\\
&\mathbf{h}_{Ci}=KAN_{3*k}(\mathbf{x}_{i}^{C}|\Phi_{3}),\Phi_{3},\end{aligned},1\leq
i\leq k.$ (15)

$\mathbf{h}_{Si}$ has a length of $D$, and thus $\Phi_{1*k}$ in
$KAN_{1*k}(\cdot)$ contains $D\times N$ learnable functions to learn the
embedding $\mathbf{h}_{Si}$. $\Phi_{2*k}$ and $\Phi_{3*k}$ contains $D\times
M$ and $D\times K$ learnable functions.

####  3.2.2 The main module

After getting the input embedding set
$H=\\{H_{1},\dots,H_{3*k}\\}=\\{\mathbf{h}_{Si},\mathbf{h}_{Ei},\mathbf{h}_{Ci}|1\leq
I\leq k\\}$, the main module is used to porocess these embeddings to get the
final prediction $\hat{r}_{ij}$ by two levels of KANs.

In the lower level, there are $3\times k$ KANs used for handling the input
embedding set and obtaining the latent vector
$\mathbf{v}=\\{v_{1},v_{2},\dots,v_{3*k}\\}$, where the forward pass is as
follows:

$v_{i}=KAN_{i}^{low}(H_{i}|\Phi_{i}^{low}),\ 1\leq i\leq 3*k.$ (16)

Afthewards, in the upper level, a unified $2$-layer KAN $KAN^{up}$ is further
used to process to the latent vector $\mathbf{v}$ as

$\hat{r}_{ij}=KAN^{up}(\mathbf{v}|\Phi^{up}_{1},\Phi^{up}_{2})=\Phi^{up}_{2}\circ\mathbf{ls}=\Phi^{up}_{2}\circ\Phi^{up}_{1}\circ\mathbf{v},$
(17)

where $\Phi^{up}_{1}$ contains $D\times 3*k$ learnable functions and
$\Phi^{up}_{2}$ contains $1\times K$ learnable functions. Note that there is a
latent vector $\mathbf{ls}\in\mathbb{R}^{1\times K}$ with length of $K$, which
can be used to represent the student’s knowledge mastery vector.

While in manner 1, the student’s knowledge mastery vector (i.e., the student
knowledge proficiency vector) is still represented by the latent student
ability vector. For example, $\mathbf{f}_{S}$ represents the student knowledge
proficiency in NCD+ and KaNCD+, while $\hat{\mathbf{h}_{S}}$ represents the
student knowledge proficiency in KSCD+ and RCD+.

###  3.3 Model Training and Implementation Modification

Model Training. To train the proposed KA2NCD model, the Adam optimizer [16] is
used to mine the following cross entropy loss [8] between model’s output
$\hat{r}_{ij}$ and true response $r_{ij}$:

$\mathcal{L}=-\textstyle\sum_{\left(s_{i},e_{j},r_{ij}\right)\in\mathcal{R}_{log}}\left(r_{ij}\log
y_{ij}\right)+\left(1-r_{ij}\right)\log\left(1-y_{ij}\right).$ (18)

Implementation Modification. The original implementation of KANs is not
efficient because all intermediate variables $X$ need to be expanded to
perform different pre-given activation functions, which will require more
memory and cost more to train. Considering most activation functions can be
linear combinations [9] of a fixed set of B-splines basis functions
$B=\\{B_{1}(\cdot),\dots,B_{L}(\cdot)\\}$, the process of one activation
function can be rewritten as $[B_{1}(X),\dots,B_{L}(X)]\times
W_{linear},W_{linear}\in\mathbb{R}^{L\times 1}$, i.e., activation with
multiple basis functions and combine them linearly.

##  4 Experiments

The following experiments aim to answer the following researcher questions:

  * •

RQ1: How about the effectiveness of manner 1 of KA2NCD?

  * •

RQ2: How about the effectiveness of manner 2 of KA2NCD, i.e., KA2NCD-e(-kan)?

  * •

RQ3: How about the interpretability (visualization) of KA2NCD?

  * •

RQ4: How about the efficiency and complexity of the proposed KA2NCD?

Table 1: Statistics of four popular education datasets. Dataset |  ASSISTments [12] ① |  SLP [20] ② |  JunYi [6] ③ |  FrcSub [1] ④
---|---|---|---|---
# Students/Concepts/Exercises | 4,163/123/17,746 | 1,499/34/907 | 1000/39/712 | 536/8/20
# Response logs | 324,572 | 57,244 | 203,945 | 10,720

###  4.1 Experimental Settings

Table 2: Performance comparison on four datasets. The best, second-best, and third-best results are in bold, underlined, and underwaved: KA2NCD-native’s results are not involved. Dataset | Assistments | SLP | JunYi | FrcSub
---|---|---|---|---
Method | AUC$\uparrow$ | ACC$\uparrow$ | AUC$\uparrow$ | ACC$\uparrow$ | AUC$\uparrow$ | ACC$\uparrow$ | AUC$\uparrow$ | ACC$\uparrow$
IRT | 72.02% | 70.25% | 80.91% | 74.29% | 74.80% | 72.74% | 80.63% | 57.14%
MIRT | 65.84% | 63.90% | 72.78% | 71.90% | 69.59% | 69.50% | 81.93% | 69.12%
DINA | 72.15% | 68.06% | 77.24% | 71.43% | 75.81% | 68.18% | 80.66% | 83.12%
MF | 70.55% | 68.26% | 79.22% | 72.80% | 79.48% | 74.15% | 84.10% | 84.10%
NCD | 74.84% | 72.15% | 84.76% | 80.72% | 80.70% | 76.73% | 90.12% | 70.15%
NCD+ | 75.71% | 71.91% | 84.72% | 80.38% | 80.38% | 76.12% | 90.66% | 74.30%
KaNCD |  76.44% |  73.33% | 85.21% |  81.61% | 80.80% | 76.15% | 90.11% | 76.68%
KaNCD+ | 76.99% | 73.54% | 85.25% | 81.91% | 82.06% | 77.23% | 91.44% | 78.36%
RCD | 75.91% | 72.99% |  85.57% | 79.37% |  83.25% |  78.67% | 89.39% | 73.83%
RCD+ | 77.10% | 73.78% | 86.38% | 80.12% | 83.33% | 78.76% | 89.46% | 74.53%
KSCD |  76.55% |  73.04% |  85.90% |  81.02% |  82.17% |  77.83% | 90.49% | 80.27%
KSCD+ | 76.72% | 73.01% | 86.06% | 80.87% | 83.43% | 78.67% | 90.66% | 82.93%
KA2NCD-e |  76.64% |  72.96% |  86.08% |  82.66% |  83.18% |  78.39% |  91.27% |  84.58%
KA2NCD-kan | 72.89% | 71.55% | 85.91% | 81.91% | 83.14% | 78.41% | 90.38% | 83.30%
Table 3: Model parameter and training runtime comparisons between neural CDMs
and KA2NCD on SLP & Junyi and ASSISTments & Junyi. (’K’ means kilo, ’h’
represents hours).

Param. | NCD | NCD+ | KaNCD | KaNCD+ | RCD | RCD+ | KSCD | KSCD+ | KA2NCD-kan | KA2NCD-e
---|---|---|---|---|---|---|---|---|---|---
SLP | 232K | 83K | 91K | 50K | 279K | 115K | 94K | 81K | 4143K | 471K
Junyi | 219K | 68K | 79K | 36K | 266K | 97K | 74K | 67401 | 3420K | 416K

Runtime | NCD | NCD+ | KaNCD | KaNCD+ | RCD | RCD+ | KSCD | KSCD+ | KA2NCD-kan | KA2NCD-e
---|---|---|---|---|---|---|---|---|---|---
ASSIST | 0.12h | 0.17h | 0.15h | 0.2h | 0.16h | 0.28h | 0.46h | 1.12 | 15h | 1.2h
JunYi | 0.02h | 0.02h | 0.06h | 0.08h | 0.03h | 0.03h | 0.08h | 0.13h | 1.3h | 0.33h

Datasets. To verify the proposed KA2NCD, we conducted experiments on four
real-world education datasets, including ASSISTments [12], SLP [20] JunYi [6]
and FrcSub [1]. Their statistics are presented in Table 1, and more details
can be found in Appendix.

Compared Models. Four traditional CDMs (including IRT, MIRT, DINA, and MF)
were taken as baselines. Besides, four neural CDMs (including NCD, KaNCD,
KSCD, and RCD) are taken as baselines to validate the effectiveness of KA2NCD-
native.

Model Settings. For all models, their embedding dimension $D$ is set to the
number of concepts $K$. The hyperparameters of comparison models follow the
original papers. For KA2NCD, the batch size and learning rate are set to 128
and 0.002, the training epoch number is set to 20. All models were implemented
in PyTorch and executed under an Intel 13700k CPU. Accuracy (ACC) and Area
Under the ROC Curve (AUC) were used as evaluation metrics. Our source code has
been uploaded as a supplementary file.

###  4.2 Effectiveness of KA2NCD-native, KA2NCD-e, and KA2NCD-kan (RQ1 & RQ2)

To answer RQ1 and RQ2, Table 2 summarizes the performance of all compared CDMs
on four datasets in terms of AUC and ACC values, and we can get the following
observations.

Firstly, for KA2NCD-native variants, NCD+, KaNCD+, KSCD+, and RCD+ hold
significantly better AUC and ACC performance than four traditional CDMs on
three datasets (ASSISTments, SLP, and JunYi). On the FrcSub dataset, their AUC
values are better than four traditional CDMs but their ACC values are worse
than MF, whereas neural CDMs are also worse than MF regarding ACC values.
Secondly, on ASSISTments and FrcSub datasets, KA2NCD-native variants hold
better AUC values than NCD, KaNCD, KSCD, and RCD, where most of their ACC
values are slightly better than neural CDMs. While for results on SLP and
JunYi, most of KA2NCD-native variants’ results are similar to NCD, KaNCD,
KSCD, and RCD, but some of their results are still superior (such as RCD+ on
both datasets, KaNCD+ on JunYi, and KSCD+ on Junyi). Thirdly, compared to
existing CDMs, KA2NCD-e nearly achieves the top-best performance on all
datasets, while KA2NCD-kan’s performance is worse than KA2NCD-e and not very
promising, especially on ASSIStments. However, the performance of KA2NCD-kan
on the other three datasets is still promising, which is slightly worse than
KSCD but better than other CDMs.

The first two observations show the proposed KA2NCD in manner 1 (i.e., KA2NCD-
native ) is effective. The third one demonstrates the superiority of KA2NCD-e
and KA2NCD-kan’s effectiveness, further indicating leveraging KANs for CD is
promising and worth exploring.

Figure 3: Visualization of NCD+ (upper)and KaNCD+ (lower) on JunYi datasets.

###  4.3 Visualization of KA2NCD (RQ3)

To answer RQ3, Figure 3 plots the structures of NCD+ and KaNCD+ only on the
JunYi dataset because its $K$ is relatively small. The structure visualization
of NCD+ and KaNCD+ on SLP can be found in Appendix. In Figure 3, only the last
KAN in NCD+ and KaNCD+ is plotted because they replace MLPs to get the
prediction. As can be seen, firstly, the learned structures of their KANs are
very close, which indicates the learned KAN has good transferability between
two models; secondly, only about 8 of 39 connections are kept in both models,
which represents the learned models are easy to interpret how they work.
Similar observations and conclusions can be drawn from the Figures in
Appendix, which shows the proposed KA2NCD can indeed enhance the
interpretability of existing CDMs.

###  4.4 Complexity and Efficiency of KA2NCD (RQ4)

To show the model complexity of the proposed KA2NCD, Table 3 presents the
comparison of neural CDMs, KA2NCD-native variants, KA2NCD-e, and KA2NCD-kan
regarding model parameter number and runtime of training models. As can be
seen, the parameters of KA2NCD-native are less than half of the corresponding
neural CDM, while parameters of KA2NCD-e and KA2NCD-kan are much more than
neural CDMs, especially for KA2NCD-kan. The high parameters of KA2NCD-kan lead
to its extremely high runtime. Despite that, the runtimes of KA2NCD-native are
close to those of neural CDMs, which indicates the efficiency of KA2NCD is
competitive and acceptable. The good efficiency is attributed to our modified
implementations of KANs, because the original implementation of KANs is very
slow: on ASSISTments and JunYi, NCD+ took about 0.9 and 0.3 hours, and KaNCD+
took about 1.1 and 0.35 hours, which is four times as our modified KANs.

##  5 Conclusion

In this paper, we proposed to enhance the model interpretability of neural
CDMs by leveraging KANs, and thus we proposed KA2NCD. There are two manners to
implement KA2NCD, where the first manner directly replaces MLPs in existing
neural CDM by KANs and the second manner builds a novel CDM completely with
KANs. Experimental results demonstrate the effectiveness of the proposed
KA2NCD and its high interpretability.

## References

  * [1] AICFE.  Frcsub dataset.  http://staff.ustc.edu.cn/%7Eqiliuql/data/math2015.rar, 2018.
  * [2] Ashton Anderson, Daniel Huttenlocher, Jon Kleinberg, and Jure Leskovec.  Engaging with massive online courses.  In Proceedings of the 23rd International Conference on World Wide Web, pages 687–698, 2014.
  * [3] Joseph Beck.  Difficulties in inferring student knowledge from observations (and why you should care).  In Proceedings of the 13rd International Conference of Artificial Intelligence in Education, pages 21–30, 2007.
  * [4] Jürgen Braun and Michael Griebel.  On a constructive proof of kolmogorov’s superposition theorem.  Constructive approximation, 30:653–675, 2009.
  * [5] Hugh Burns, Carol A Luckhardt, James W Parlett, and Carol L Redfield.  Intelligent Tutoring Systems: Evolutions in design.  Psychology Press, 2014.
  * [6] Haw-Shiuan Chang, Hwai-Jung Hsu, Kuan-Ta Chen, et al.  Modeling exercise relationships in e-learning: A unified approach.  In Proceedings of the 2015 International Educational Data Mining Society (EDM, pages 532–535, 2015.
  * [7] Song Cheng, Qi Liu, Enhong Chen, Zai Huang, Zhenya Huang, Yiying Chen, Haiping Ma, and Guoping Hu.  Dirt: Deep learning enhanced item response theory for cognitive diagnosis.  In Proceedings of the 28th ACM International Conference on Information and Knowledge Management, pages 2397–2400, 2019.
  * [8] Pieter-Tjerk De Boer, Dirk P Kroese, Shie Mannor, and Reuven Y Rubinstein.  A tutorial on the cross-entropy method.  Annals of operations research, 134:19–67, 2005.
  * [9] Carl De Boor and Carl De Boor.  A practical guide to splines, volume 27.  springer-verlag New York, 1978.
  * [10] Susan E Embretson and Steven P Reise.  Item Response Theory.  Psychology Press, 2013.
  * [11] Feng-Lei Fan, Jinjun Xiong, Mengzhou Li, and Ge Wang.  On interpretability of artificial neural networks: A survey.  IEEE Transactions on Radiation and Plasma Medical Sciences, 5(6):741–760, 2021.
  * [12] Mingyu Feng, Neil Heffernan, and Kenneth Koedinger.  Addressing the assessment challenge with an online system that tutors as it assesses.  User Modeling and User-adapted Interaction, 19(3):243–266, 2009\.
  * [13] Weibo Gao, Qi Liu, Zhenya Huang, Yu Yin, Haoyang Bi, Mu-Chun Wang, Jianhui Ma, Shijin Wang, and Yu Su.  Rcd: Relation map driven cognitive diagnosis for intelligent education systems.  In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 501–510, 2021.
  * [14] Liya Hu, Zhiang Dong, Jingyuan Chen, Guifeng Wang, Zhihua Wang, Zhou Zhao, and Fei Wu.  Ptadisc: A cross-course dataset supporting personalized learning in cold-start scenarios.  In Proceedings of the 37th Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2023.
  * [15] Eric Jang, Shixiang Gu, and Ben Poole.  Categorical reparameterization with gumbel-softmax.  arXiv preprint arXiv:1611.01144, 2016.
  * [16] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  arXiv preprint arXiv:1412.6980, 2014.
  * [17] Andreĭ Kolmogorov.  On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables.
  * [18] Yehuda Koren, Robert Bell, and Chris Volinsky.  Matrix factorization techniques for recommender systems.  Computer, 42(8):30–37, 2009.
  * [19] Ziming Liu, Yixuan Wang, Sachin Vaidya, Fabian Ruehle, James Halverson, Marin Soljačić, Thomas Y Hou, and Max Tegmark.  Kan: Kolmogorov-arnold networks.  arXiv preprint arXiv:2404.19756, 2024.
  * [20] Yu Lu, Yang Pian, Ziding Shen, Penghe Chen, and Xiaoqing Li.  SLP: A multi-dimensional and consecutive dataset from K-12 education.  In Proceedings of the 29th International Conference on Computers in Education Conference, volume 1, pages 261–266, 2021.
  * [21] Haiping Ma, Manwei Li, Le Wu, Haifeng Zhang, Yunbo Cao, Xingyi Zhang, and Xuemin Zhao.  Knowledge-sensed cognitive diagnosis for intelligent education platforms.  In Proceedings of the 31st ACM International Conference on Information & Knowledge Management, pages 1451–1460, 2022.
  * [22] Haiping Ma, Jinwei Zhu, Shangshang Yang, Qi Liu, Haifeng Zhang, Xingyi Zhang, Yunbo Cao, and Xuemin Zhao.  A prerequisite attention model for knowledge proficiency diagnosis of students.  In Proceedings of the 31st ACM International Conference on Information & Knowledge Management, pages 4304–4308, 2022.
  * [23] Thanaporn Patikorn, Neil T Heffernan, and Ryan S Baker.  Assistments longitudinal data mining competition 2017: A preface.  In Proceedings of the 2018 International Conference on Educational Data Mining, 2018.
  * [24] Mark D Reckase.  Multidimensional Item Response Theory Models.  Springer, 2009.
  * [25] Wojciech Samek, Alexander Binder, Grégoire Montavon, Sebastian Lapuschkin, and Klaus-Robert Müller.  Evaluating the visualization of what a deep neural network has learned.  IEEE Transactions on Neural Networks and Learning Systems, 28(11):2660–2673, 2016.
  * [26] D. L. Torre and J.  Dina model and parameter estimation: A didactic.  Journal of Educational and Behavioral Statistics, 34(1):115–130, 2009.
  * [27] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  Advances in neural information processing systems, 30, 2017.
  * [28] Fei Wang, Qi Liu, Enhong Chen, Zhenya Huang, Yuying Chen, Yu Yin, Zai Huang, and Shijin Wang.  Neural cognitive diagnosis for intelligent education systems.  In Proceedings of the 2020 AAAI Conference on Artificial Intelligence, volume 34, pages 6153–6161, 2020.
  * [29] Fei Wang, Qi Liu, Enhong Chen, Zhenya Huang, Yu Yin, Shijin Wang, and Yu Su.  Neuralcd: a general framework for cognitive diagnosis.  IEEE Transactions on Knowledge and Data Engineering, 2022.
  * [30] Shangshang Yang, Haoyu Wei, Haiping Ma, Ye Tian, Xingyi Zhang, Yunbo Cao, and Yaochu Jin.  Cognitive diagnosis-based personalized exercise group assembly via a multi-objective evolutionary algorithm.  IEEE Transactions on Emerging Topics in Computational Intelligence, 7(3):829–844, 2023.
  * [31] Yu Zhang, Peter Tiňo, Aleš Leonardis, and Ke Tang.  A survey on neural network interpretability.  IEEE Transactions on Emerging Topics in Computational Intelligence, 5(5):726–742, 2021.

##  Appendix A More Details

###  A.1 KA2NCD-native Variants

KSCD+. When adopting the KSCD as the main module, the prediction process of
KA2NCD-native (further denoted as KSCD+) is as

$\begin{aligned}
\hat{\mathbf{h}_{S}}=Sigmoid(&KAN_{1}([\mathbf{h}_{S},\mathbf{h}_{C}])),\hat{\mathbf{h}_{E}}=Sigmoid(KAN_{2}([\mathbf{h}_{E},\mathbf{h}_{C}]))\\\
&\hat{r}_{ij}=(\sum{\mathbf{h}_{C}\times(\hat{\mathbf{h}_{S}}-\hat{\mathbf{h}_{E}})})/D\\\
\end{aligned},$ (19)

where the first KAN $KAN_{1}(\cdot)$ is used to combine student embedding and
concept embedding to get the richer student ability vector and the second KAN
$KAN_{2}(\cdot)$ is used to get the richer exercise difficulty vector.

RCD+. When adopting the RCD as the main module, the prediction of KA2NCD-
native (further denoted as RCD+) can be obtained according to the following
process:

$\begin{aligned}
\hat{\mathbf{h}_{S}}=Sigmoid(&KAN_{1}([\mathbf{h}_{S},\mathbf{h}_{C}])),\hat{\mathbf{h}_{E}}=Sigmoid(KAN_{2}([\mathbf{h}_{E},\mathbf{h}_{C}]))\\\
&\hat{r}_{ij}=Sigmod(\frac{1}{D}\sum
KAN_{3}(\hat{\mathbf{h}_{S}}-\hat{\mathbf{h}_{E}}))\end{aligned}.$ (20)

where the first two KANs are the same as those of KSCD+, which is used to get
a richer student ability vector and a richer exercise difficulty vector. The
third KAN $KAN_{3}(\cdot)$ is used to extract the response prediction.

KaNCD+. When adopting the KaNCD as the main module, we can follow the
following equation to get the prediction of KA2NCD-native (further denoted as
KaNCD+):

$\begin{aligned} \mathbf{f}_{S}=&Sigmoid(KAN_{1}(\mathbf{h}_{S})),\
\mathbf{f}_{diff}=Sigmoid(KAN_{2}(\mathbf{h}_{E})),\
f_{disc}=Sigmoid(KAN_{3}(\mathbf{h}_{E}))\\\
&\hat{r}_{ij}=KAN_{4}(\mathbf{y}),\
\mathbf{y}=\mathbf{h}_{C}\odot(\mathbf{f}_{S}-\mathbf{f}_{diff})\times
f_{disc}\\\ \end{aligned}.$ (21)

It can be seen that the last two KANs have the same effect as that of NCD+,
and the first and second KANs are used to get the enhanced student ability
vector and exercise difficulty vector.

###  A.2 Details about datasets

The detailed descriptions of four datasets are as follows:

  * •

ASSISTments [12] is an openly available dataset created in 2009 by the
ASSISTments online tutoring service system. Here we adopted the public
corrected version that does not contain duplicate data. The dataset contains
more than 4,000 students, nearly 18,500 exercises, and over 300,000 response
logs.

  * •

SLP [20] is another public education dataset published in 2021. SLP collects
learners’ regularly captured academic performance data during their three-year
studies in eight subjects: Chinese, mathematics, English, physics, chemistry,
biology, history, and geography.

  * •

JunYi [6] JunYi is a public dataset collected by the JunYi Education Platform
in Taiwan, containing nearly $20M$ responses from 1,000 students. E-Math is a
private dataset collected by a well-known electronic educational product,
mainly containing primary and secondary school students’ math exercises and
test logs.

  * •

FrcSub 111http://staff.ustc.edu.cn/%7Eqiliuql/data/math2015.rar The FrcSub
dataset, commonly utilized for evaluating cognitive diagnosis models, focuses
on fraction addition and subtraction problems. It encompasses performance data
from 536 learners across 20 objective questions, with scores represented as
binary values (1 for correct, 0 for incorrect). Additionally, it includes a
knowledge point correlation matrix that details the relationship between the
20 questions and 8 distinct knowledge points, where a value of 1 indicates a
related knowledge point, and 0 indicates no relation. This dataset serves as a
valuable resource for cognitive modeling research.

For the four datasets, we filtered out students with fewer than 15 response
logs to ensure sufficient data for model learning. Each student’s logs are
randomly split into training and testing datasets at the splitting ratio of
7/3.

##  Appendix B More Visualization

Figure 4: Overview-Visualization of KANs in NCD+ (upper)and KaNCD+ (lower) on
the SLP datasets.

Figure 5:  Visualization of pruned KANs in NCD+ (upper)and KaNCD+ (lower) on
the SLP datasets. Figure 6: Visualization of the last KAN in KA2NCD-e ) on the
JunYi datasets. Figure 7: Visualization of KA2NCD-e (upper) on the SLP
datasets.

For deep insight into the learned KANs’ structures to answer RQ3, Figures 4
and 5 plot the structures of NCD+ and KaNCD+ learned on the SLP dataset,
Figure 6 plots the structures of KA2NCD-e learned on the JunYi dataset, while
Figure 7 plots the structures of KA2NCD-e learned on the SLP dataset. For easy
observation, for Figure 4, only the last KAN in NCD+ and KaNCD+ is plotted
because they replace MLPs to get the prediction; while for Figures 6 and 7,
only the structures of the upper-level KANs are presented.

As can be seen from Figures 4 and 5, the learned structures of KANs are very
close, which indicates the learned KAN has good transferability between two
models; about 15 of 34 connections are kept in both models, which makes the
learned models easy to be interpreted. As can be seen from Figures 6 and 7,
which latent features are used and how they are combined can be obviously
observed, which is very important for users to understand.

In conclusion, the proposed KA2NCD can indeed enhance the interpretability of
existing CDMs, and the manner 2 of the proposed KA2NCD is indeed effective,
where the learned structures are really easy to interpret.


# Classification of vacuum arc breakdowns in a Pulsed DC System

Anton Saressalo<EMAIL_ADDRESS>Andreas Kyritsakis  Flyura
Djurabekova  Helsinki Institute of Physics and Department of Physics,
University of Helsinki, PO Box 43 (Pietari Kalmin katu 2), 00014 Helsingin
yliopisto, Finland  Iaroslava Profatilova  Jan Paszkiewicz  Sergio Calatroni
Walter Wuensch  CERN, European Organization for Nuclear Research, 1211 Geneva,
Switzerland

###### Abstract

Understanding the microscopic phenomena behind vacuum arc ignition and
generation is crucial for being able to control the breakdown rate, thus
improving the effectiveness of many high-voltage applications where frequent
breakdowns limit the operation. In this work, statistical properties of
various aspects of breakdown, such as the number of pulses between breakdowns,
breakdown locations and crater sizes are studied independently with almost
identical Pulsed DC Systems at the University of Helsinki and in CERN. In
high-gradient experiments, copper electrodes with parallel plate capacitor
geometry, undergo thousands of breakdowns. The results support the
classification of the events into primary and secondary breakdowns, based on
the distance and number of pulses between two breakdowns. Primary events
follow a power law on the log–log scale with the slope $\alpha\approx$1.30$$,
while the secondaries are highly dependent on the pulsing parameters.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Grasping the underlying physical processes leading to electrical vacuum arc
outbursts – breakdowns (BDs) – is important for many applications across
various fields in modern science and technology. The phenomenon occurs in
devices that operate in (ultra) high vacuum and which are subject to high
electric fields. Applications include vacuum switches and interrupters, vacuum
arc metal processing, ion beam and pulsed sources, fusion reactors, satellites
and radio-frequency (RF) particle accelerators Boxman  _et al._ (1996); Latham
(1995); Falkingham and Montillet (2004); McCracken (1980).

Investigation on the origin of BDs has been underway for more than a century
Boxman  _et al._ (1996). Numerous experimental, theoretical and, more
recently, computational studies have been performed over the decades in order
to understand the phenomenon Dyke  _et al._ (1953); Charbonnier  _et al._
(1967); Schade and Shmelev (2003). Several different processes have been
suggested to explain the arc formation, but none of them have provided
adequate analytic explanation Zhou  _et al._ (2019).

It is clear that surface electric fields below
$\mathrm{G}\mathrm{V}\mathrm{/}\mathrm{m}$ range are not strong enough to
break a metal surface in order to initiate the plasma of the vacuum arc. The
most common explanation is that micro and nanoscale protrusions on the surface
locally increase the surface electric field and this enhanced field results in
electron field emission induced evaporation of neutral atoms as well. The
emitted electrons accelerated under the electric field ionize some of the
atoms which in turn are accelerated back towards the cathode, sputtering more
neutrals into the vacuum and starting an avalanche process Barengolts  _et
al._ (2018); Kyritsakis  _et al._ (2018). Leading to an exponential growth in
the number of charged particles in the vacuum and practically short circuiting
the anode and cathode, this whole process is seen as a vacuum discharge –
vacuum arc – also known as a breakdown. Once the system is short circuited,
large currents flow through even with relatively small voltages. Origin and
experimental observation of these protrusions is unclear. Some hypotheses link
them to near-surface dislocations causing deformations on the surface Nordlund
and Djurabekova (2012); Engelberg  _et al._ (2018).

The Compact Linear Collider (CLIC) is an example of an application in which
breakdowns play an important role in limiting the design. The project is a
high-energy physics facility proposed to be built at CERN in order to
accelerate and collide electrons and positrons Burrows  _et al._ (2018). The
accelerating structures for the main beam of CLIC operate at room temperature
and use $50\text{\,}\mathrm{MW}$ X-band RF pulses to accelerate electrons and
positrons in an ultra-high vacuum environment. In order to minimize the length
and construction costs of the facility, electric fields up to
$100\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ are used to accelerate
the particles for the highest collision energy stage. These high accelerating
fields correspond to surface electric fields in excess of
$200\text{\,}\mathrm{MV}\text{/}\mathrm{m}$, a value limited by vacuum
electrical breakdowns. If a breakdown occurs during the operation of the CLIC
accelerator, the particle beam is kicked and no $e^{+}e^{-}$ collisions can
occur for that pulse. Thus, the accelerator’s luminosity is reduced, which is
why the breakdown rate (BDR) is required to be kept below
$3\text{\times}{10}^{-7}$ per pulse per meter for the accelerator to operate
efficiently Aicheler  _et al._ (2012).

Copper has been chosen as the material of these accelerating structures
Zennaro  _et al._ (2008). Despite its relatively low average breakdown field
after conditioning, $170\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$,
the other properties of copper, such as good conductivity, machinability,
ductility and availability made it the best choice for the material
Descoeudres  _et al._ (2009).

In order to optimize the accelerating structure design and operation, the
structures are experimentally tested in klystron-based X-band test facilities
in CERN Catalan-Lasheras  _et al._ (2014); Catalan Lasheras  _et al._ (2016);
Volpi  _et al._ (2018). These test stands allow $200\text{\,}\mathrm{ns}$
pulses with output up to $50\text{\,}\mathrm{MW}$ and repetition rate up to
$400\text{\,}\mathrm{Hz}$ with breakdown behaviour being one of the most
important parameters investigated Shipman (2014); Woolley (2015).

Since building and operating these RF test facilities is both expensive and
time-consuming, DC pulse experiments have been designed specifically to study
the breakdowns with much higher repetition rates and simpler setup. In spite
of the differences between the RF and DC systems, the DC pulsing is made as
close as possible to the RF case. This way the DC experiments are not only
useful in studying the BD resistance during CLIC-like pulsing, but also in
understanding the basic physics behind the BD initiation general.

##  II Experimental setup

###  II.1 Equipment

The experiments were conducted using a Pulsed DC System aimed to emulate the
RF pulsing, but with a higher repetition rate. The system includes two
parallel copper electrodes inside a vacuum chamber (Large Electrode System,
LES), connected to a high-voltage power supply and a pulse generator along
with an oscilloscope and a measurement computer. Almost identical Pulsed DC
Systems in CERN and at the University of Helsinki were used in this work. In
CERN, there are also continuous experiments done with the actual CLIC
accelerating structures, using $12\text{\,}\mathrm{GHz}$ RF pulses with
repetition rates up to $400\text{\,}\mathrm{Hz}$.

LES is a compact vacuum chamber designed at CERN for high-gradient studies
(Figure 1). Inside the chamber, there are two diamond-machined cylindrical
electrodes separated by an aluminum oxide spacer which maintains a desired gap
between the electrode surfaces. In these studies, spacers resulting in a
$60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ gap were used, but there are also
other options for a gap from $20\text{\,}\mathrm{\SIUnitSymbolMicro
m}100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The surface roughness and
dimensional precision of the spacers as well as the electrodes is below
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by design. Together, these
electrodes act as a parallel plate capacitor. One of the electrodes is charged
to a positive voltage (anode) while the other one is grounded (cathode). The
small gap allows generation of electric fields of even
$100\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ with DC voltages of
only a few kilovolts. During measurements, the chamber is pumped down to high-
vacuum below 10-7 $\mathrm{mbar}$ using a turbo pump in series with a roughing
pump.

Figure 1:  A cross-section from a 3D model of the LES vacuum chamber. The
illustration shows the electrodes in the middle, connected to vacuum
feedthroughs and otherwise insulated by an Al2O3 spacer. Four viewports on the
sides of the chamber allow also visual detection of a breakdown.

High-voltage microsecond-pulses with a repetition rate of typically $12$
$\mathrm{kHz}$ are generated using a Marx Generator EPULSUS®-FPM1-10 by Energy
Pulse Systems 201 (2015). The generator utilizes SiC MOSFET technology for
amplifying the voltage from a power supply by a factor of at least $10$,
depending on the model, and enabling pulses with lengths from
$200\text{\,}\mathrm{ns}$ upwards. During a pulse, the effective capacitor
inside the LES is charged with a current spike, and a similar spike in the
opposite direction discharges the electrodes after a specified up-time (pulse
length), provided that no breakdown occurred. The rise and fall time of the
pulses are in the order of $100\text{\,}\mathrm{ns}$ Redondo  _et al._ (2017).
Pulse length and repetition rate can be varied programmatically.

The generator is also used for detecting the electrical breakdowns between the
electrodes by monitoring the current during pulsing. When a breakdown occurs,
there is a rapid current peak as the anode and cathode are briefly short
circuited. This current peak is typically at least by a factor of $2$ higher
than the charging peak and can thus be distinguished by the generator. After
the breakdown peak, the short circuit stays open and there will be a constant
$20\text{\,}\mathrm{V}$ burning voltage of across the gap for around $250400$
$\mathrm{ns}$ Jûttner (2001); Shipman  _et al._ (2012). Examples of waveforms
without and with a breakdown in Figures 2 and 3. Sampling rate for these
waveforms in the oscilloscope is $1\text{\,}\mathrm{GHz}$.

Figure 2:  A typical waveform of a $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$
pulse generated by Marx generator without a breakdown. The figure shows the
charging and discharging current peaks at the start and at the end the pulse,
with the voltage staying constant in between. Figure 3:  A typical waveform of
a pulse with a breakdown, showing the extra current peak followed by roughly
$600\text{\,}\mathrm{ns}$ of burning voltage. During the breakdown, the
voltage drop occurs within approximately five nanoseconds.

###  II.2 Pulsing & conditioning algorithm

The ultimate aim of the experiments is to increase the breakdown resistance of
the electrodes during pulsing with as high electric field as possible while
keeping the breakdown rate within predefined limits. The increase in the
breakdown resistance is achieved by conditioning the copper electrodes with
electric pulses Descoeudres  _et al._ (2009). For copper, this conditioning
typically requires a great number of pulses (typically more than $10^{7}$
Volpi  _et al._ (2018)), during which numerous breakdowns occur (typically
more than $100$) Cox (1974).

The conditioning starts with relatively low electric fields, for example,
$10\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ and the voltage is
gradually increased in small steps after each pulsing period, of typically
$100\,000$ pulses. If a breakdown occurs, the pulsing for that period is
terminated and the electric field is either slightly decreased or not changed
at all, depending on the number of pulses which took place until the
breakdown. The algorithm is similar to that used in the RF experiments and is
explained in details in Korsbäck  _et al._ (2019).

During the first pulses after a breakdown, there is a ramping period, where
the field is briefly decreased to one fifth of the value before the BD and
then it is asymptotically increased back to the target value during a course
of $20$ voltage steps with $100$ pulses in each step as demonstrated in Figure
4. The objective of the ramping is to reduce the possibility of cascades of
secondary breakdowns.

After the electrode has reached a conditioned state – i.e. the frequency of
breakdowns is such that the algorithm keeps the electric field at a steady
level, thus the field has saturated – the measurements are continued in so-
called flat mode runs. In these runs the target voltage is set to a constant
value where the breakdown rate is relatively stable. The different pulsing
modes are visualized in Figure 5.

Figure 4:  Asymptotic ramping period after a BD at
$V=$5000\text{\,}\mathrm{V}$$ (which is
$E=$83\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$$, based on $E=V/d$).
Each dot represents the start of a new ramping step of $100$ pulses. The
ramping starts from one fifth of the target voltage. A BD may also occur at
any time during the ramping, after which the ramping starts over again, though
the target voltage stays the same as before. Figure 5:  Conditioning and first
flat mode runs of Soft Cu Helsinki visualized. The graph shows evolution of
the electric field and number of breakdowns as a function on pulses. The
conditioning part, as well as the different flat mode runs are separated by
red dashed lines.

###  II.3 Electrodes

The LES electrodes are of a cylindrical shape with the contact area of
$62\text{\,}\mathrm{mm}$ in diameter, while the diameter of the bottom of the
electrodes, i.e. including the areas below the spacer, is
$80\text{\,}\mathrm{mm}$. More technical detail on the shape of electrodes is
available elsewhere Gudkov (2014). Electrode thickness is greatest below the
$62\text{\,}\mathrm{mm}$ contact area, $30\text{\,}\mathrm{mm}$. The contact
surface of the electrodes is diamond machined to have roughness below one
micron, which is also the accuracy of the shoulder height which maintains the
electrode separation via an aluminum oxide spacer.

As discussed previously, the material of interest is copper. The two main
types of copper being tested are hard copper and soft copper. Hard copper is
high-purity, oxygen-free electronic copper that is machined into the required
shape. The grain diameter on the surface of hard copper is between
$10\text{\,}\mathrm{\SIUnitSymbolMicro
m}100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (at least $4$ on the ASTM E112
standard ASTM International (2013)). Soft copper additionally undergoes a
treatment first at $1040\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in hydrogen
atmosphere and usually afterwards at
$650\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in vacuum to breath out the
hydrogen. However, the electrodes of the measurement Soft Cu CERN did not
undergo the breath-out treatment. The average grain diameter of the Soft Cu
CERN was estimated to be $1.3\pm 0.2\text{\,}\mathrm{mm}$ based on the Heyn
Linear Intercept Procedure described in the ASTM E112 Standard.

###  II.4 Breakdown localization

The Pulsed DC System at CERN is additionally equipped with cameras for
localization of the breakdowns. They are installed close to two viewports of
the vacuum chamber, perpendicular to each other. Positions of the breakdowns
are determined by the light emitted during each breakdown. If both cameras
record a single line, the positioning of breakdown on the electrode surface
can be determined. This technique allows to see real-time spatial distribution
of breakdowns without the necessity of disassembling the vacuum chamber in
order to see the BD spots which would cause several days’ halt with the
measurements. In these experiments, smaller electrodes with contact disk
diameter of 40 mm were used. The localization algorithm is described in detail
in Reference Profatilova  _et al._ (2019).

Collecting the data from the cameras, the high voltage generator, the
oscilloscope and post-mortem microscopy, a wide range of parameters such as
electric field, number of pulses between previous breakdown, distance between
subsequent breakdowns and crater sizes on the anode and cathode surfaces can
be analyzed for better understanding of each breakdown event.

Electrodes are also imaged both before breakdown experiments and post-mortem
with optical microscopes. In these images, breakdowns can be seen as dark
craters on the surface, though it is practically impossible to connect them to
the pulse that caused the BD without the cameras for localization. Examples of
breakdowns on a cathode surface of Soft Cu CERN are shown in Figure 6. Machine
vision algorithm was used to detect the BD craters from the image and estimate
their sizes and locations. The detection algorithm used two-stage circular
Hough transform to identify the circular objects in an image.

Figure 6:  Optical microscope image of Soft Cu CERN surface with several BD
craters visible and detected by machine vision.

###  II.5 Other parts of the measurements and analysis

During the pulsing and breakdown experiments, also secondary parameters of the
system were observed. These include monitoring the vacuum pressure as well as
voltage and current waveforms. In some runs, we also used a mass spectrometer
in order to detect residual particles in the vacuum.

The waveform and pulsing data are saved for each breakdown. This allows
investigation of the breakdown current, voltage, timing within the pulse,
short circuit width, pulse number and timing. Reference values are also saved
for some non-breakdown pulses.

Also, simple simulations were used to understand the results. Particularly,
the distribution of distances between breakdowns was simulated with Monte
Carlo methods. That is, by repeatedly scattering series of $1000$ BDs within
$3\text{\,}\mathrm{mm}$ from the edge of the electrode and calculating the
distribution of distances between the subsequent spots in order to compare the
distribution with the experimentally measured ones. The
$3\text{\,}\mathrm{mm}$ annulus was selected due to the fact that generally,
especially for hard copper, more than $90\text{\,}\mathrm{\char 37\relax}$ of
the breakdowns lie within the $3\text{\,}\mathrm{mm}$ from the edge. We will
refer to these breakdowns as edge BDs. The increased BD density near the edges
is currently explained by locally enhanced electric fields in the region
Parker (2002); Catalán Izquierdo  _et al._ (2017).

In addition, post-mortem analysis is performed to the surfaces for example to
measure the electrode tilt and roughness with a profilometer or to investigate
the BD craters with a Scanning White Light Interferometry microscope
Kassamakov  _et al._ (2017).

A breakdown event concentrates tens millijoules of energy within a small area
Latham (1995); Kyritsakis  _et al._ (2018), which results in creation of the
BD craters. It has been observed that a cathode crater, generated by a BD in
LES with a $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ gap, usually contains a
pit that has a typical depth of $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
a radius of $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$. At the edges of the
pit, there is typically a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick
annulus of molten and recrystallized material, which has numerous sharp edges
and protrusions that serve as sites for additional BDs Wang and Loew (1989).

##  III Results

All the data presented below are the results of the flat mode runs with
conditioned electrodes. The conditions of the experiment were kept constant as
long as possible in order to collect enough data for statistical analysis.
Most importantly, the pulsing voltage was kept constant except immediately
after each breakdown, when the asymptotic ramping described earlier was used
to ramp up the electric field from one fifth to the target voltage in $2000$
pulses ($2100$ for the setup in CERN).

###  III.1 Pulses between breakdowns

Number of pulses between two consecutive breakdowns were analyzed for four
flat mode runs with different electrodes – two sets in CERN and two in
Helsinki. The key numbers of the runs are shown in Table 1. Hard Cu CERN and
Soft Cu CERN had a contact surface diameter of 40 mm in order to enable BD
localization. For reference, the same analysis was also conducted for an RF
run conducted with the X-band test stand at CERN Zennaro  _et al._ (2017).

Table 1: The four DC data sets and one RF run used for comparing the statistics on pulses between breakdowns. Name | Pulse length | Electric field | RepRate
---|---|---|---
Hard Cu CERN | $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ | $83\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ | $2000\text{\,}\mathrm{Hz}$
Soft Cu CERN | $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ | $85\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ | $2000\text{\,}\mathrm{Hz}$
Hard Cu Helsinki | $3\text{\,}\mathrm{\SIUnitSymbolMicro s}$ | $83\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ | $2000\text{\,}\mathrm{Hz}$
Soft Cu Helsinki | $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ | $83\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ | $4000\text{\,}\mathrm{Hz}$
RF Run CERN | $83\text{\,}\mathrm{ps}$ | $108\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ | $50\text{\,}\mathrm{Hz}$
Name | BDs | Pulses | BDR
Hard Cu CERN | $2383$ | $3.15\text{\times}{10}^{7}$ | $7.55\text{\times}{10}^{-5}$
Soft Cu CERN | $1489$ | $2.03\text{\times}{10}^{7}$ | $7.34\text{\times}{10}^{-5}$
Hard Cu Helsinki | $414$ | $9.69\text{\times}{10}^{7}$ | $4.27\text{\times}{10}^{-6}$
Soft Cu Helsinki | $1830$ | $1.98\text{\times}{10}^{8}$ | $9.24\text{\times}{10}^{-6}$
RF run CERN | $93$ | $3.44\text{\times}{10}^{7}$ | $2.70\text{\times}{10}^{-6}$
Figure 7:  Probability distribution of pulses between breakdowns on doubly
logarithmic scale. The black dotted line shows probability distribution
function of a Poisson distribution for the comparison. A semilogarithmic inset
shows a zoom-in of the distribution during the ramping when the voltage is
between $50\text{\,}\mathrm{\char 37\relax}90\text{\,}\mathrm{\char 37\relax}$
($2001000$ pulses) of the target voltage. The red dashed line indicates the
end of the ramping phase. The uncertainties are obtained from the standard
deviations of the slope values.

A probability distribution function of the number of pulses between two
subsequent breakdowns was generated by collecting the events of a given number
of pulses into logarithmically spaced bins in the pulse range. The results are
shown in Figure 7. The graphs show that the probability for a breakdown to
occur within the ramping period is remarkably higher than after the ramping,
but the values vary significantly within this period, correlating with the
ramping steps. Comparison shows that the points within the each step follow
the linear part of Poisson distribution, except for the latest points of each
step.

After the ramping has ended, the probability decreases linearly on the log–log
scale. There are no big differences in the trends between the PDFs for
different runs, except for the Hard Cu Helsinki, which has a jump in the
probability at around $10^{5}$ pulses, but continues with almost the same
slope even after the jump. The linear decay of the probabilities follow the
power law $P(S)=kS^{-\alpha}$, with $\alpha\approx$1.30$$, which was similar
for all the runs. The RF run does not have comparable steps in the ramping
algorithm, though the results still nicely follow the power law with only
slightly smaller slope.

The inset of Figure 7 reveals that the probability of BDs shows a sawtooth-
pattern with peaks at every 100 pulses – exactly at the beginning of each
ramping step. The sawtooth-pattern confirms an earlier qualitative observation
that the breakdown probability increases whenever the pulsing is paused and
the conditions are changed. In this case, the pause was a few seconds, which
was required for the power supply to adjust to the new voltage for each
ramping step. Due to the asymptotic ramping, the relative changes in the
electric field are the largest in the first ramping steps (below $800$
pulses), where the relative increase per step is more than
$3\text{\,}\mathrm{\char 37\relax}$. Above $800$ pulses in the ramping, the
voltage is already above $90\text{\,}\mathrm{\char 37\relax}$ of the target
value. This explains why the clearest sawtooth-pattern is seen between
$200800$ pulses, where both the relative change and the absolute voltage are
large enough. It is important to note that by bypassing the ramping and
starting directly at the target voltage, there would be even more breakdowns
during the first pulses after the previous BD.

###  III.2 Distances between breakdowns

Figure 8:  Distribution of distances between breakdowns for Hard and Soft Cu,
grouped by pulses between breakdowns with error bars showing the standard
deviation. a) and c) show the distributions for ramping pulses where the only
features in the distributions are visible within the first
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$. b) and d) show the distributions
for all the BDs at all distances for BDs after the ramping period. In
addition, b) shows the distribution distances between simulated uncorrelated
edge BDs. The integral of each distribution equals to the fraction of
breakdowns that fall within each pulse and distance range, so that both ranges
combined equal $100\text{\,}\mathrm{\char 37\relax}$.

Distances between consecutive breakdowns were measured using the localization
technique described in Section II. Figure 8 shows the probability for a
breakdown to occur at a certain distance from the previous one (center to
center) for Hard and Soft Cu. The left panel (Figures 8a and 8c) in the graph
shows the distributions for breakdowns that occurred during ramping. They are
mainly localized within the distance of
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$, especially on Hard Cu surface,
while almost no breakdowns are seen at the larger distances. The right panels
(Figures 8b and 8d) show the distributions of distances between the breakdowns
registered after the ramping mode was completed. There we see that the
probabilities are almost equal for all the distances. especially on Hard Cu,
where the distribution is very similar to simulated, uncorrelated BD locations
within $3\text{\,}\mathrm{mm}$ from the edge of the electrode.

In the subfigures 8a) and 8c), we see that there is an increased probability
to have a breakdown at around $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
which happens to be close to the average radius of a BD crater on cathode, as
seen in Figure. 9. This behaviour is very similar for both Hard and Soft Cu.

Figure 9:  Distributions of distances between ramping breakdowns near the
previous breakdown spot (same as Figure 8a)) and the radii of cathode
breakdown craters compared, both from Hard Cu CERN.

In the subfigures 8b) and 8d), however, we see large differences between the
copper types. While the distribution for Hard Cu is more or less uniform, the
Soft Cu still has an increased amount of BDs close to the previous one. At
first, this seems to contradict the earlier observation reported in Korsbäck
_et al._ (2019), which showed intense spatial clustering of BDs on Hard Cu,
while on Soft Cu, the spatial BD distribution was much more uniform across the
whole surface. However, a closer look at the locations of consecutive BDs
reveals that in Soft Cu, it is common to have several consequent BDs within a
close distance (less than a millimeter) from each other, after which the next
breakdown can be anywhere. On Hard Cu, the BDs are more clustered, but it is
rare to have more than one consecutive BD in the same cluster – the next BD is
more likely to occur in another cluster anywhere on the surface. The
clustering effect is visualized in Figure 10, where we see that the BD density
on Hard Cu exceeds $500$ BDs per million pulses per ${\mathrm{mm}}^{2}$ in
several spots, whereas on Soft Cu there are only less prominent clusters and
the BDs are more widespread outside of clusters.

Figure 10:  Density graph of the breakdown positions on Hard Cu CERN and Soft
Cu CERN. The plots show that on Hard Cu, the BDs tend to appear in distinct
sites, whereas on Soft Cu, the clusters are not as strong and the distribution
is more widely spread (leading to lower BD density). It is also important to
note that on both electrodes, majority of the BDs lie on within a few
millimeters from the edge.

In Figure 9, the distances between consecutive ramping breakdowns are shown
again for the Hard Cu CERN. This time the distances are compared with the size
distribution of the cathode spots. The distributions show similar trends, with
peaks around $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Gaussian fits for
the distributions yield medians of $104\pm
4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $110\pm
1\text{\,}\mathrm{\SIUnitSymbolMicro m}$, for the distances and radii,
respectively.

Figure 11 shows that a large part ($32\text{\,}\mathrm{\char 37\relax}$) of
non-ramping breakdowns occur within the first $1.5\text{\,}\mathrm{mm}$ from
each other, after which the distribution is flat and close to zero for the
rest of the $38.5\text{\,}\mathrm{mm}$ of the surface. The
$1.5\text{\,}\mathrm{mm}$ mm cut-off value is close to the average grain
diameter of $1.3\pm 0.2\text{\,}\mathrm{mm}$

Figure 11:  Distribution of distances between non-ramping breakdowns within
$4\text{\,}\mathrm{mm}$ from the previous breakdown spot on Soft Cu CERN, i.e.
a zoom to the first three bars (distances $04.5$ $\mathrm{mm}$) of Figure 8d)
compared with the average grain diameter distribution (solid line) and its
standard deviation (dashed line).

##  IV Discussion

The results show that the probability for a breakdown to occur is the highest
within the next few hundred pulses after the previous one – and within
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the center of the previous BD
spot. That is why the ramping has been introduced to keep the breakdown rate
approximately stable. During the ramping period, the BD probability strongly
follows the ramping steps so that it is the highest at the beginning of each
step. The lowest probability within each ramping step is always the latest
data point of the step. This value is higher when the electric field is higher
at the corresponding ramping step. This is seen in overall ascending trend in
the sawtooth function shown in the inset of Figure 7.

This, linked to the high localization of the consequent ramping BDs, shows
that there is a correlation between the events. Since these breakdowns appear
as follow-up events, we call them secondary breakdowns. The breakdowns, which
take place after a large number of pulses and do not exhibit any spatial or
temporal correlation with the preceding one, are called primary BD.

After the ramping, the BD probability decays linearly on the log–log scale, as
a function of the number of pulses, with the slope being
$\alpha\approx$1.33$$. The value is really close to the slope obtained with
the previous system, reported in Wuensch  _et al._ (2017) and also relatively
close to the RF experiments in CERN. The observed jump at around 105 pulses in
Hard Cu Helsinki plot is most probably an artifact from the changing of
measurement period, which also happens every 105 pulses, granted that there
was no BD. This kind of power law behaviour is seen in various seemingly
unrelated phenomena, such as avalanche size distribution and earthquake
frequency – and also the behaviour of dislocations in metals Papanikolaou  _et
al._ (2011); Chrzan and Mills (1994). Observed $\alpha$ in the references is
typically $1.5$, so relatively close to the measured values. Important feature
of this kind of behaviour is the universality across several magnitudes of
scales.

These events that happen during the linear part of the Figure 7 can be seen as
primary breakdowns as they are mostly independent from one another and
typically followed by secondary, highly dependent, BDs. Their initiation
requires some local changes in the material, making that particular spot
”hot”. To understand the BD ignition, it is really important to understand
what makes this particular spot more favourable for a BD than any other on the
surface with an area larger than $10\text{\,}{\mathrm{cm}}^{2}$. Earlier work
have hypothesized linking this to dislocations piling up near the surface,
causing formation of protrusions, which, in turn, enhance the electric field
locally Engelberg  _et al._ (2018). The power law behaviour can be explained
by dislocation avalanches.The relation between localized pre-breakdown field
emission currents, hot spots and breakdown initiation was found in References
Davies (1973); Davies and Biondi (1966). The phenomenon of hot, breakdown-apt
regions have also been observed in the cells of RF structures Adolphsen  _et
al._ (2001); Dal Forno  _et al._ (2016); Degiovanni  _et al._ (2016) and been
previously connected to surface contamination Mesyats (2013).

The third main observation is the most probable distance between two
consecutive breakdowns. If the BD would not fully destroy the underlying
material, it would be natural that the next breakdown is likely to hit in the
same crater again. Even more interesting is the finding of increased
probability for a BD to occur at around
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the previous spot. This
happens to be very close to the average radius of a breakdown crater. This
suggests that the molten areas near the edges of the crater form ideal
conditions for next breakdowns to happen as already observed in Wang and Loew
(1989). The phenomenon can actually be seen in the post-mortem surface images,
showing chains of breakdowns exactly one crater radius apart from each other
as seen in Figures 6 and 12. The small difference in the mean values could be
explained by systematic uncertainties in the BD crater recognition algorithm
and its definition of BD crater ”edge”.

Figure 12:  Optical microscope image of a series of $13$ consecutive soft
copper BDs in a cluster on, bounded by a grain on a soft copper surface.
$\triangleright$ and $\Box$ indicate the first and last BDs, respectively. The
circle edges and centers were identified manually as the machine vision
algorithm was not usable due to lack of contrast. The order of the BD sequence
was obtained from the BD localization information. Figure 13:  Optical
microscope image of the full surface of the Hard Cu CERN cathode. On the
surface, a series of $19$ consecutive BDs is shown, each in a separate site
randomly distributed near the edge of the cathode. $\triangleright$ and $\Box$
indicate the first and last BDs, respectively.

The subfigures 8b) and 8d) show clear differences between Hard and Soft Cu.
The Hard distribution pretty much follows the shape of random, uncorrelated
edge BDs, as the simulation result shows in the same figure, suggesting that
BDs on hard Cu surfaces after the ramping period are mostly uncorrelated,
stochastic events. Soft Cu BDs however, show correlation even after the
ramping period, as most of the BDs fall within a few millimeters from the
previous one. This can also be observed when studying series of consecutive
non-ramping BDs that are located within $1\text{\,}\mathrm{mm}$ from each
other. In Hard Cu, less than $2\text{\,}\mathrm{\char 37\relax}$ of the BDs
fall into this category, while in Soft Cu, the fraction is around
$10\text{\,}\mathrm{\char 37\relax}$ of the events (counting the series with
at least two BDs).

Also previous research suggests that the breakdowns can be classified as
either primary or secondary events Wuensch  _et al._ (2017); Mesyats (2013).
These are defined based on the physical distance and pulses between the
events. The primary BDs initiate when some features, such as dislocations
congest near surface in a particular hot grain. With hard copper, the grain
diameter is in the order of $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, so one
breakdown does diminish the whole area of the grain and it becomes inactive
for a large number of pulses. This can be understood in terms of dislocation
activities. A new set of dislocations needs to be activated before the grain
becomes active again. Thus, it takes a large number of pulses before that spot
can recover and a new primary breakdown can occur there. During this recovery
time, it is more probable that the dislocations are mobilized in some other
grain, leading to a breakdown in this spot. Figure 13 shows an example of this
kind of a series of $19$ consecutive primary BDs randomly scattered on the
electrode surface (though nearly all of them are near the edge).

With soft copper, the grain diameter can be up to a few millimeters, allowing
multiple primary breakdowns to occur close to each other before the grain is
quenched. Secondary breakdowns are those that occur within the same breakdown
crater, also typically close in terms of pulses in between. Figure 12 shows an
example of a series of $13$ consecutive BDs (including some ramping BDs)
within one grain.

The different behaviour between surfaces with different grain sizes fits the
dislocation hypothesis suggested in References Nordlund and Djurabekova
(2012); Engelberg  _et al._ (2018), which is linked to the grain size: the
dislocations are known to get pinned or annihilated at the grain boundaries
Hull and Bacon (2011). With hard copper, presumably only some of the grains
have mobile dislocations at a time, and once a BD has occurred on one, the
source of protrusions – a hot spot – is at least temporarily deactivated.

The small size of grains allows for fewer mobile dislocations, since they have
higher probability to be stopped (pinned) at the grain boundaries. The large
grains of well-annealed Cu offer freer movement of dislocations, which can
arrive at the surface participating, for instance, in growth of surface
asperities Pohjonen  _et al._ (2012).

With soft copper again, most grains are so large that they are prone to having
at least some mobile dislocations. When a BD occurs in this kind of a hot
grain, one event is not enough to quench the whole surface of the grain and
some other BDs are likely to occur due to nearby dislocations. The dislocation
hypothesis is also supported by the temperature dependence of copper breakdown
susceptibility Cahill  _et al._ (2018).

##  V Conclusions

Vacuum arc breakdowns between two Cu electrodes were generated by DC pulses in
order to understand their generation processes and how to limit the frequency
of the BD events. In the analysis, the pulses between breakdowns, breakdown
locations and their correlation to cathode crater size were compared over
various flat mode measurement runs and two Cu types, both at the Accelerator
Laboratory of the University of Helsinki and in CERN.

The results support the previous observation of independent primary and
correlated secondary follow-up breakdowns. The BD locations show differences
between hard and soft copper, which can be explained by grain size difference
and dislocations. The dislocation hypothesis is also supported by the power
law trend on pulses between breakdowns of the primary events.

For future work, it will be important to analytically understand the
differences in the behaviour between Hard Cu and Soft Cu and to investigate
the ramping algorithm so that it minimizes the breakdown rate without
affecting the pulsing efficiency too much. Also, understanding the
contaminants on electrode surfaces can play important role in understanding
the full span of the breakdown processes Abe  _et al._ (2018).

##  VI Acknowledgements

The research was funded by the K-contract between Helsinki Institute of
Physics and CERN. The optical microscopy was performed by Timo Hilden and Essi
Kangasharju at the Detector Laboratory of the Helsinki Institute of Physics,
as well as by Enrique Rodriguez Castro at CERN. The electrode surface
profilometry measurements were performed by Jouni Heino, also at the Detector
Laboratory of the Helsinki Institute of Physics. The breakdown craters were
also studied with the Scanning White Light Microscope by Anton Nolvi and Ivan
Kassamakov at the Electronics Research Laboratory of the University of
Helsinki.

## References

  * Boxman  _et al._ (1996) Raymond L Boxman, David M Sanders, and Philip J Martin,  _Handbook of vacuum arc science & technology: fundamentals and applications_ (William Andrew, 1996).
  * Latham (1995) R. V. Latham,  _High voltage vacuum insulation: Basic concepts and technological practice_ (Elsevier, 1995).
  * Falkingham and Montillet (2004) L T Falkingham and G F Montillet, “A history of fifty years of vacuum interrupter development-(the English connection),” in  _IEEE Power Engineering Society General Meeting, 2004._ (2004) pp. 706–711.
  * McCracken (1980) G M McCracken, “A review of the experimental evidence for arcing and sputtering in tokamaks,” Journal of Nuclear Materials 93, 3–16 (1980).
  * Dyke  _et al._ (1953) Walter P Dyke, J K Trolan, E E Martin, and J P Barbour, “The field emission initiated vacuum arc. I. Experiments on arc initiation,” Physical review 91, 1043 (1953).
  * Charbonnier  _et al._ (1967) Francis M Charbonnier, Carol J Bennette, and Lynwood W Swanson, “Electrical breakdown between metal electrodes in high vacuum. I. Theory,” Journal of Applied Physics 38, 627–633 (1967).
  * Schade and Shmelev (2003) Ekkehard Schade and Dmytry Leonidovich Shmelev, “Numerical simulation of high-current vacuum arcs with an external axial magnetic field,” IEEE Transactions on Plasma Science 31, 890–901 (2003).
  * Zhou  _et al._ (2019) Zhipeng Zhou, Andreas Kyritsakis, Zhenxing Wang, Yi Li, Yingsan Geng, and Flyura Djurabekova, “Direct observation of vacuum arc evolution with nanosecond resolution,”  (2019), 10.1038/s41598-019-44191-6.
  * Barengolts  _et al._ (2018) S. A. Barengolts, V. G. Mesyats, V. I. Oreshkin, E. V. Oreshkin, K. V. Khishchenko, I. V. Uimanov, and M. M. Tsventoukh, “Mechanism of vacuum breakdown in radio-frequency accelerating structures,” Physical Review Accelerators and Beams (2018), 10.1103/PhysRevAccelBeams.21.061004.
  * Kyritsakis  _et al._ (2018) Andreas Kyritsakis, Mihkel Veske, Kristjan Eimre, Vahur Zadin, and Flyura Djurabekova, “Thermal runaway of metal nano-tips during intense electron emission,” Journal of Physics D: Applied Physics 51, 225203 (2018).
  * Nordlund and Djurabekova (2012) Kai Nordlund and Flyura Djurabekova, “Defect model for the dependence of breakdown rate on external electric fields,” Physical Review Special Topics-Accelerators and Beams 15, 71002 (2012).
  * Engelberg  _et al._ (2018) Eliyahu Zvi Engelberg, Yinon Ashkenazy, and Michael Assaf, “Stochastic model of breakdown nucleation under intense electric fields,” Physical review letters 120, 124801 (2018).
  * Burrows  _et al._ (2018) P Burrows, Nuria Catalan-Lasheras, Lucie Linssen, M Petric, Aidan Robson, Daniel Schulte, Eva Sicking, and Steinar Stapnes, “The Compact Linear e+ e- Collider (CLIC) 2018 Summary Report,” CERN Yellow Reports: Monographs (2018).
  * Aicheler  _et al._ (2012) Markus Aicheler, P Burrows, M Draper, Terence Garvey, P Lebrun, K Peach, N Phinney, H Schmickler, Daniel Schulte, and N Toge,  _A Multi TeV Linear Collider based on CLIC technology: CLIC Conceptual Design Report_, Tech. Rep. (CERN, 2012).
  * Zennaro  _et al._ (2008) Riccardo Zennaro, Alexej Grudiev, G Riddone, A Samoshkin, Walter Wuensch, Sami G. Tantawi, J W Wang, and Toshiyasu Higo, “Design and Fabrication of CLIC Test Structures,” in  _Proceeding of LINAC 2008_ (Victoria, BC, Canada, 2008) pp. 533–535.
  * Descoeudres  _et al._ (2009) A Descoeudres, T Ramsvik, Sergio Calatroni, Mauro Taborelli, and Walter Wuensch, “DC breakdown conditioning and breakdown rate of metals and metallic alloys under ultrahigh vacuum,” Physical Review Special Topics-Accelerators and Beams 12, 32001 (2009).
  * Catalan-Lasheras  _et al._ (2014) Nuria Catalan-Lasheras, Alberto Degiovanni, S Doebert, Wilfrid Farabolini, Jan Kovermann, Gerard McMonagle, S Rey, Igor Syratchev, L Timeo, Walter Wuensch, Benjamin Woolley, and J Tagg, “Experience Operating an X-Band High-Power Test Stand At CERN,” in  _Proceeding of IPAC2014_ (2014) pp. 2288–2290.
  * Catalan Lasheras  _et al._ (2016) Nuria Catalan Lasheras, Anastasiya Solodko, Theodoros Argyropoulos, Benjamin Woolley, Walter Wuensch, Daniel Esperante Pereira, J Tagg, Gerard McMonagle, Cedric Eymin, and Igor Syratchev, “Commissioning of XBox-3: A very high capacity X-band test stand,” Proceedings of LINAC2016 (2016).
  * Volpi  _et al._ (2018) Matteo Volpi, Nuria Catalan-Lasheras, Alexej Grudiev, Thomas Geoffrey Lucas, Gerard McMonagle, Jan Paszkiewicz, V Del, Pozo Romano, Igor Syratchev, Anna Vnuchenko, Benjamin Woolley, Walter Wuensch, P J Giansiracusa, Roger Paul Rassool, Claudio Serpico, and Mark James Boland, “High Power and High Repetition Rate X-band Power Source Using Multiple Klystrons; High Power and High Repetition Rate X-band Power Source Using Multiple Klystrons,”  (2018), 10.18429/JACoW-IPAC2018-THPMK104.
  * Shipman (2014) Nicholas Shipman,  _Experimental study of DC vacuum breakdown and application to high-gradient accelerating structures for CLIC_ , Ph.D. thesis (2014).
  * Woolley (2015) Benjamin Woolley,  _High power X-band RF test stand development and high power testing of the CLIC crab cavity_ , Ph.D. thesis (2015).
  * 201 (2015) “Manual EPULSUS-FPM1-10,” (2015).
  * Redondo  _et al._ (2017) L. M. Redondo, Aleh Kandratsyeu, M. J. Barnes, Sergio Calatroni, and Walter Wuensch, “Solid-state Marx generator for the compact linear collider breakdown studies,” in  _2016 IEEE International Power Modulator and High Voltage Conference, IPMHVC 2016_ (2017).
  * Jûttner (2001) Burkhard Jûttner, “Cathode spots of electric arcs,” J. Phys. D: Appl. Phys 34, 103–123 (2001).
  * Shipman  _et al._ (2012) Nicholas Shipman, Sergio Calatroni, Walter Wuensch, and R. M. Jones, “Measurement of the Dynamic Response of the CERN DC Spark System and Preliminary Estimates of the Breakdown Turn-on Time,” in  _IPAC2012_ (IEEE, New Orleans, Louisiana, USA, 2012).
  * Cox (1974) B M Cox, “Variation of the critical breakdown field between copper electrodes in vacuo,” Journal of Physics D: Applied Physics 7, 143 (1974).
  * Korsbäck  _et al._ (2019) Anders Korsbäck, Laura Mercadé Morales, Iaroslava Profatilova, Enrique Rodriquez Castro, Walter Wuensch, Sergio Calatroni, and Tommy Ahlgren, “Vacuum electrical breakdown conditioning study in a parallel plate electrode pulsed DC system,” Arxiv 1905.03996 (2019).
  * Gudkov (2014) Dmitry Gudkov, “CDD: Drawings, Fixed Gap System Anode / Cathode Disk,”  (2014).
  * ASTM International (2013) ASTM International, “ASTM E112-13, Standard Test Methods for Determining Average Grain Size,”  (2013).
  * Profatilova  _et al._ (2019) Iaroslava Profatilova, Xavier Stragier, Sergio Calatroni, Aleh Kandratsyeu, Enrique Rodriquez Castro, and Walter Wuensch, “Breakdown localisation in a pulsed DC electrode system,” Nuclear Inst. and Methods in Physics Research, A , 163079 (2019).
  * Parker (2002) G. W. Parker, “Electric field outside a parallel plate capacitor,” American Journal of Physics 70, 502–507 (2002).
  * Catalán Izquierdo  _et al._ (2017) S. Catalán Izquierdo, José M. Bueno Barrachina, César S. Cañas Peñuelas, and Francisco Cavallé Sesé, “Capacitance evaluation on parallel-plate capacitors by means of finite element analysis,” Renewable Energy and Power Quality Journal 1, 613–616 (2017).
  * Kassamakov  _et al._ (2017) Ivan Kassamakov, Sylvain Lecler, Anton Nolvi, Audrey Leong-Hoï, Paul Montgomery, and Edward Hæggström, “3D super-resolution optical profiling using microsphere enhanced Mirau interferometry,” Scientific reports 7, 3683 (2017).
  * Wang and Loew (1989) J W Wang and G A Loew, “Rf breakdown studies in copper electron linac structures,” in  _Proceedings of the 1989 IEEE Particle Accelerator Conference,.’Accelerator Science and Technology_ (1989) pp. 1137–1139.
  * Zennaro  _et al._ (2017) Riccardo Zennaro, Terence Garvey, Theodoros Argyropoulos, Rolf Wegner, Alexej Grudiev, Walter Wuensch, Daniel Esperante Pereira, Gerard McMonagle, Benjamin Woolley, and Thomas Lucas, “High power tests of a prototype X-band accelerating structure for CLIC,” CLIC Note 1134 (2017).
  * Wuensch  _et al._ (2017) Walter Wuensch, Alberto Degiovanni, Sergio Calatroni, Anders Korsbäck, Flyura Djurabekova, Robin Rajamäki, and Jorge Giner Navarro, “Statistics of vacuum breakdown in the high-gradient and low-rate regime,” Physical Review Accelerators and Beams 20, 11007 (2017).
  * Papanikolaou  _et al._ (2011) Stefanos Papanikolaou, Felipe Bohn, Rubem Luis Sommer, Gianfranco Durin, Stefano Zapperi, and James P Sethna, “Universality beyond power laws and the average avalanche shape,” Nature Physics 7, 316 (2011).
  * Chrzan and Mills (1994) D C Chrzan and M J Mills, “Criticality in the plastic deformation of L 1 2 intermetallic compounds,” Physical Review B 50, 30 (1994).
  * Davies (1973) D. Kenneth Davies, “The initiation of electrical breakdown in vacuum—a review,” Journal of Vacuum Science and Technology 10, 115–121 (1973).
  * Davies and Biondi (1966) D. Kenneth Davies and Manfred A. Biondi, “Vacuum electrical breakdown between plane-parallel copper electrodes,” Journal of Applied Physics 37, 2969–2977 (1966).
  * Adolphsen  _et al._ (2001) C. Adolphsen, W Baumgartner, K Jobe, F Le Pimpec, R Loewen, Doug McCormick, M Ross, T Smith, J W Wang, and Toshiyasu Higo, “Processing studies of X-band accelerator structures at the NLCTA,” in  _Proceeding of the 2001 Particle Accelerator Conference_, 1 (Chicago, Illinois, 2001) pp. 478–480.
  * Dal Forno  _et al._ (2016) Massimo Dal Forno, Valery A. Dolgashev, Gordon Bowden, Christine Clarke, Mark Hogan, Doug McCormick, Alexander Novokhatski, Bruno Spataro, Stephen Weathersby, and Sami G. Tantawi, “rf breakdown tests of mm-wave metallic accelerating structures,” Physical Review Accelerators and Beams 19 (2016), 10.1103/PhysRevAccelBeams.19.011301.
  * Degiovanni  _et al._ (2016) Alberto Degiovanni, Walter Wuensch, and Jorge Giner Navarro, “Comparison of the conditioning of high gradient accelerating structures,” Physical Review Accelerators and Beams (2016), 10.1103/PhysRevAccelBeams.19.032001.
  * Mesyats (2013) Gennady A. Mesyats, “Ecton mechanism of the cathode spot phenomena in a vacuum Arc,” IEEE Transactions on Plasma Science 41, 676–694 (2013).
  * Hull and Bacon (2011) Derek Hull and David J Bacon,  _Introduction to dislocations_ (Butterworth-Heinemann, 2011).
  * Pohjonen  _et al._ (2012) Aarne S. Pohjonen, Flyura Djurabekova, Antti Kuronen, Steven P. Fitzgerald, and Kai Nordlund, “Analytical model of dislocation nucleation on a near-surface void under tensile surface stress,” Philosophical Magazine (2012), 10.1080/14786435.2012.700415.
  * Cahill  _et al._ (2018) A. D. Cahill, J. B. Rosenzweig, Valery A. Dolgashev, Sami G. Tantawi, and Stephen Weathersby, “High gradient experiments with X -band cryogenic copper accelerating cavities,” Physical Review Accelerators and Beams 21, 102002 (2018).
  * Abe  _et al._ (2018) Tetsuo Abe, Tatsuya Kageyama, Hiroshi Sakai, Yasunao Takeuchi, and Kazuo Yoshino, “Direct observation of breakdown trigger seeds in a normal-conducting rf accelerating cavity,” Physical Review Accelerators and Beams 21 (2018), 10.1103/PhysRevAccelBeams.21.122002.