∎

11institutetext:  H. Jelodar 22institutetext<EMAIL_ADDRESS>

R. Orji
<EMAIL_ADDRESS>

S. Matwin
<EMAIL_ADDRESS>

S. Weerasinghe
<EMAIL_ADDRESS>

O. Oyebode
<EMAIL_ADDRESS>

Y. Wang
<EMAIL_ADDRESS>
33institutetext:
1School of Computer Science and Technology, Nanjing University of Science and
Technology, Nanjing 210094, China

2Faculty of Computer Science, Dalhousie University, Halifax, NS, Canada

3 Faculty of Medicine, Dalhousie University, Halifax, NS, Canada

4 Institute of Computer Science Polish Academy of Sciences, Warsaw, Poland

# Artificial Intelligence for Emotion-Semantic Trending and People Emotion
Detection During COVID-19 Social Isolation

Hamed Jelodar 1 2 Rita Orji 2 Stan Matwin 2 4 Swarna Weerasinghe 3 Oladapo
Oyebode 2 Yongli Wang 1

(Received: date / Accepted: date)

###### Abstract

Taking advantage of social media platforms, such as Twitter, this paper
provides an effective framework for emotion detection among those who are
quarantined. Early detection of emotional feelings and their trends help
implement timely intervention strategies. Given the limitations of medical
diagnosis of early emotional change signs during the quarantine period,
artificial intelligence models provide effective mechanisms in uncovering
early signs, symptoms and escalating trends. Novelty of the approach presented
herein is a multitask methodological framework of text data processing,
implemented as a pipeline for meaningful emotion detection and analysis, based
on the Plutchik/Ekman approach to emotion detection and trend detection. We
present an evaluation of the framework and a pilot system. Results of confirm
the effectiveness of the proposed framework for topic trends and emotion
detection of COVID-19 tweets. Our findings revealed Stay-At-Home restrictions
result in people expressing on twitter both negative and positive emotional
semantics (feelings), where negatives are “Anger” (8.5% of tweets), followed
by “Fear” (5.2%), “Anticipation” (53.6%) and positive emotional semantics are
“Joy” (14.7%) and “Trust” (11.7%). Semantic trends of safety issues related to
staying at home rapidly decreased within the 28 days and also negative
feelings related to friends dying and quarantined life increased in some days.
These findings have potential to impact public health policy decisions through
monitoring trends of emotional feelings of those who are quarantined. The
framework presented here has potential to assist in such monitoring by using
as an online emotion detection tool kit.

###### Keywords:

Twitter, NLP, Deep Learning, COVID-19, Emontion

##  1 Introduction

Over 73 million people have been affected by COVID-19 across the globe [3].
This more than a yearlong outbreak is likely to have a significant impact on
mental health of many individuals who lost loved ones, who lost personal
contacts with others due to strictly enforced public health guidelines of
mandatory social segregation. Complex psychological reactions to COVID-19
regulatory mechanisms of mandatory quarantine and related emotional reactions
has been recognized as hard to disentangle [1] – [4]. A study conducted in
Belgium found social media being positively associated with constructive
coping for adolescents with anxious feelings during the quarantine period of
COVID-19 [4]. Another study conducted among social media users during COVID-19
pandemic in Spain was able to capture added stress placed on people’s
emotional health during the pandemic period [5]. However, social media
providing a platform of risk communication and exchange of feelings and
emotions to curb social isolation, this text data provides a wealth of
information on the natural flow of people’s emotional feelings and expressions
[6]. This rich source of data can be utilized to curb the data collection
barriers during the pandemic. The goal of this research was to use AI to
uncover the hidden, implicit signal related to emotional health of people
subject to mandatory quarantine, embedded in a latent manner in their twitter
messages.

Within the context of this paper, an NLP-based emotion detection system aims
to provide useful information by examining unstructured text data used in
social media. The purpose of the NLP system used herein is to show the meaning
and emotions of users’ expressions related to a particular topic, which can be
used to understand their psychological health and emotional wellbeing. In this
regard, use of NLP-based approach for emotion detection from complex textual
structures such as social media (e.g., Twitter) remains a challenge in
biomedical applications of AI.

The goal of this paper is to contribute an AI based methodological framework
that can uncover emotion semantic trends that can better understand the impact
and the design of quarantine regulations. The two-fold objectives of this
paper are: (a) to develop and AI framework based on machine learning models
with for emotion detection and (b) to pilot this model on unstructured tweets
that followed quarantine regulation using stay at home messaging during the
first wave of COVID-19. We investigate emotions and semantic structures to
discover knowledge from general public tweeter exchanges. We analyze the
structure of vocabulary patterns used on Twitter with a specific focus on the
impact of the Stay-At-Home public health order during the first wave of the
COVID-19. The AI framework is described and its implemented pipeline pilot
herein can be used in the emotion detection of social media information
exchange during the second wave of COVID-19 and beyond to investigate the
impact on any future public health guideline.

We aim to demonstrate the effectiveness of deep learning models for detecting
emotions from COVID-19 tweets. In addition, our findings will provide
directions public health decision making on emotion trend detection over a
four weeks period in relation to “Stay at Home” order. The contributions of
this paper can be summarized as follows:

  * •

A cleaned and standardized tweet dataset of COVID-19 issues is built in this
research, and a new database of emotion-annotated COVID-19 tweets is
presented, and this could be used for future comparisons and implementations
of detection-systems based on machine learning models.

  * •

We design a triple-task framework to investigate the emotions in eight
standard positions (explained in section II B) via Plutchik’s model using the
COVID-19 tweets in which all three different tasks are complementary to each
other towards a common goal.

  * •

We discover semantic-word trends via various models such as latent Dirichlet
allocation (LDA) and probabilistic latent semantic analysis (PLSA). We aim to
have a semantic knowledge discovery based on topic trends and semantic
structures during the first wave of the pandemic, which provides an effective
mechanism for managing future waves.

  * •

A deep learning model based convolutional neural network (CNN)) is presented
for emotion detection from the COVID-19 tweets. To the best of our knowledge,
this is the first attempt that detects emotion automatically for people’s
reaction to stay at home during the pandemic based on the online comments,
especially for #StayAtHome.

This paper is organized into sections: (a) a review of literature on existing
models on emotion detection for social media pertaining to health online
communities (section 3) (b) introduce a multi-tasks framework to COVID-19
emotion detection (section 4), (c)describe data collection of twitter and
research experiment (section 5), (d) discuss the effectiveness of the
presented AI framework and future research directions (section 6) with final
section on the conclusions on findings of emotion detection during stay at
home(section 7).

##  2 NOVELTY OF THE PROPOSED FRAMEWORK

Although machine learning based emotion detection approaches have been
proposed within social media text analysis with the context of COVID-19, there
are still many challenges remained to be addressed. In this regard, most of
the existing studies related to COVID-19, on Twitter, and other social media
platforms were performed on a general public opinion, no research have
specifically investigated emotions related to quarantine “stay at home” order,
public health policy of social segregation, that is widely used across the
globe. Novelty of the methods used in this paper consists of a multi-task
framework that can be directly applied to COVID-19 related mood discovery,
using eight types of emotional reaction and designing a deep learning model to
uncover emotions based on the first wave of the pandemic public health
restriction of mandatory social segregation. We argue that the framework can
discover semantic trends of COVID-19 tweets during the first wave of the
pandemic to predict new concerns that may be associated with furthering into
the new waves of COVID-19 quarantine orders and other related public health
regulations. Our novel approach presented herein can help future public health
crisis management in the new waves of the Coronavirus pandemic.

Moreover, public health decision makers need to understand the temporal
patterns of emotional reactions on the population when these public health
regulatory measures are continued. To fulfill this need, we investigate the
semantic topic and emotion trends to better understand the people’s reactions
from the initial wave of the pandemic.

##  3 RELATED WORK

NLP and Machine Learning has been used within the context of identifying the
type of emotions in twitter texts. In this section, we provide a review of
literature on recent emotion detection studies with focus on; Emotion
detection in online health communities, Emotion-based Lexical models, Deep
learning and machine learning, and Directions for Public health decision
making using social media during COVID-19 related text analytics.
Emotion detection analytics through information retrieval and NLP as a
mechanism have been used to explore large text corpora of online health
community communications in psychiatry, dentistry, cancer and health and
fitness. For example, a communication tool was introduced for mental health
care to understand counseling content based on emotion detection and natural
language processing using chat assistants [7] - [12]. Similar to the proposed
approach in our work, a research analyzed messages in online health
communities (OHCs) to understand the most prominent emotions in health-related
posts and proposed a computational model that can exploit the semantic
information from the text data [9]. They presented a dataset from a cancer
forum with the six common emotions based on the Ekman model and investigated
the most prominent emotions in OHCs. We proposed to use broader types of
emotions using Plutchik’s model that contains eight emotions.

In our previous work [13], sentiment and latent-topics techniques application
to COVID-19 comments in social media shed light on the usefulness of NLP
methods in uncovering issues directly related to COVID-19 public health
decision making. We expect to extend the methodology, in this study, within
our goal of extracting meaningful knowledge of emotional expression words from
people’s reactions during mandatory quarantine using the StayAtHome hashtag on
Twitter. This knowledge is essential as it can help decision makers to take
necessary actions to control the adverse emotional effects of various public
health policies, especially during the emerging waves of the pandemic.
Clearly, negative emotional effects such as anger and fear can lead to
negative social reactions. To the best of the authors’ knowledge, little
research have been done to understand the emotional expression during
mandatory quarantine, partly due to difficulties in collecting such personal
level data during the pandemic. The authors of a study in India analyzed real-
time posts on Twitter during COVID-19, and they were able to identify the
expression of mood of the nation through their analysis [14]. Also, they
developed a platform for viewing the trends in emotional change across the
country during a specific interval. As a result, their system allowed users to
view the mood of India on specific events happening in the country during
COVID-19. Development of such a platform for Canada may be a far reaching goal
of this study. This study presents the first step towards development of such
online tools to monitor the moods and emotions to inform public health
decision makers.

##  4 METHODS

This paper’s methods provide step-by-step approach to text data processing,
emotion detection and intensity scoring, emotion semantic trends calculation
and finally evaluation of the deep learning algorithm using training and
testing data.

###  4.1 Multi-Task Framework

In this section, we present a multi-task framework based on Plutchik’s Emotion
model [15] and deep learning techniques to address aforementioned research
objectives of this paper. Our approach includes three main tasks. Plutchik’s
is an operationalization of Ekman [16]. The first task is to create models to
investigate emotional reaction to mandatory restrictions of Stay-At-Home using
tweets. The second task, is to show how to discover semantic and emotional
trends to obtain patterns depicted in the first wave of the pandemic for the
30 days period from April 28th to June 1st, 2020. Finally, the third task, a
machine learning deep neural network is built as an emotion detection system
that can be used for social media exchange data analysis during the quarantine
period. Our framework, including these three tasks, is presented in Fig. 1.

1) #StayAtHome-Related Tweets and Data-Dimension Reduction:
Our inclusion criteria include only tweets related to the COVID-19 and Stay-
At-Home order. Application of this inclusion criteria is a critical step
because the quality of the input data directly affects the output.Lexical text
analysis, Data-Dimension Reduction, and NLP Preprocessing of data are
necessary to clean the data by removing the noisy and inconsistent tweets and
also analyzing the relevant data to identify relevant and appropriate
information related to the topic of interest. For this purpose, four NLP
techniques are used: sentence splitting and word tokenization, removing stop-
words, HTML cleaning to remove unnecessary contents, removing of stop-words
and hashtags, and stemming to remove prefixes and suffixes hence returning to
the root. The main purpose of the text data cleaning process is to eliminate
all tweets unrelated to the subject of our stay at home public health order.
Each tweet passed through a set of filters that are created based on the
points described above. As stated in the objectives of this paper we want to
detect the emotions expressed in the tweets that are in English language, non-
English tweets are eliminated from the dataset and the classifier is then
trained using English tweets only.

2) Task I: Emotion-detection of #StayAtHome tweets:
To achieve our research goal Task 1 is the most important process for
automatic detection of emotions from #StackAtHome tweets. It is the first step
towards the initial determination of the type of COVID-19 emotions, which also
has a direct influence on Tasks 2 and 3. We take advantage of the NRC-Word-
Emotion [17] lexicon based on Plutchik’s Wheel of Emotions to perform this
task, which is beneficial for standard scoring of the word-emotion
association. In this research, three sub processing steps are carried out on
every annotated tweet: (a) identifying type of emotion using Plutchik-theory
and hence reproducibility is warranted, (b) assigning the emotion score
obtained from the National Research Council Canada (NRC)/ NRC Emotion Lexicon
[17] and (c) identifying the emotion and the maximum association score based
on the scores computed according to the following rule. In part (b) for
assigning the score, we calculate the total emotion association score as the
sum of scores of the terms depicting higher scores for higher intensity of the
emotion of the tweets (See Table I, example). The scores denote the intensity
of the emotion for the COVID-19 tweets. By default, every emotion in the tweet
will receive a value of 1. This value will be increased or decreased by the
intensifier words used in the tweets. However, if there are multiple mentions
of emotion then the intensity will have a higher score, as shown in Table I.
In part (c) the maximum association score of a tweet represents the maximum
score noted for any of the eight emotions. A tweet that does not associate
with any emotion receives a score of zero (0), as showed in Table 1. This AI
based emotion detection task uncovers emotion semantics with emotion valuation
(strength) attached to each emotion lexicon in each of the tweets.

Figure 1: Research framework and pipeline for the COVID-19 tweet emotion
capture and analysis Figure 2: Example of the process of determining the score
for a pure tweet that is related to #StayAtHome

Since the labelling is done automatically and no human tagging is used, we
will have consistent data annotation. This model defines eight basic emotions
and makes it possible to provide consistent classification of texts to uncover
the trend in the data to reach the objectives of this research. Fig. 2
provides an example of selecting the score for COVID-19 tweets. For example,
from the tweet after text processing showed “Sad man friend whos livin skin
cant stand company” and will have the emotion SAD associated with FEAR and
this emotional expression provided the highest score from NRC based Lexicon
when the process is detecting the predominant emotion.

Table 1: EXAMPLE OF THE TWEETS WITH VARIOUS EMOTIONS

Tweet ID. | Tweet without stop-words | Label
---|---|---
A | Tweet | Today has been a challenging day, here’s to tomorrow | Anticipation
Score | Anger=0$\sim$Anticipation=1$\sim$Disgust=0$\sim$Fear=0 $\sim$Joy=0 $\sim$Sadness=0$\sim$Surprise=0$\sim$Trust=0
B | Tweet | A day is a long time in the coronavirus pandemic. | Anticipation
Score | Anger=0$\sim$Anticipation=2$\sim$Disgust=0$\sim$Fear=0 $\sim$Joy=0 $\sim$Sadness=0$\sim$Surprise=0$\sim$Trust=0
C | Tweet | Looking forward to those summer days when I can enjoy the beach and the ocean breeze again????. Stay positive and healthy everyone. | Joy
Score | Anger=0$\sim$Anticipation=1$\sim$Disgust=0$\sim$Fear=0$\sim$Joy=3$\sim$Sadness=0$\sim$Surprise=0$\sim$Trust=1

3) Task II: Emotion/Semantic-Trends of #StayAtHome
Researchers have identified timing of the emotional progression and noted that
positive emotions arose significantly earlier and the negative emotions took
longer [18] – [20]. Identification of the emotion and semantic trends over
time can be helpful and effective to understand temporal changes of the
opinions related to the human behavior. In fact, understanding the mood
changes or awareness of the emotion trends can have a practical application
for public health decision making. We use semantic topics [21] discovered in
the entire dataset to detect and describe semantic trends.

In order to obtain semantic topics we need to design a topic model. Infact, we
consider two popular methods for evaluating and determining an optimal
approach to obtain semantic-trends from #StayAtHome tweets of the online
community during the stay-at-home. For applying this task, the PLSA [21] and
LDA [22] models are employed to obtain the best semantic related-words and
discovering semantic structures of the COVID-19 tweets, as described below:
\- The probabilistic latent semantic analysis (PLSA) model is used as an NLP
technique that can display topical similarities between words.
\- The latent Dirichlet allocation (LDA) model has proven very useful for
semantic extraction and generating trends over time. LDA has been successfully
applied in several applications such as topic discovery, temporal semantic
trends, document classification, and finding relations between documents.
Another advantage of this method is the identification of semantic-trends over
time, which we consider in this research as means to discover unusual semantic
trends based on the first wave of the pandemic of #StayAtHome tweets. Overall,
The main aim of the task 2 is to capture two kinds of trends based on emotion
and semantic aspects of the COVID-19 tweets. However, we know that LDA model
discover ‘semantic related-words’ from the semantic structure of the text.
Then by investigating the distributions of these semantic-topics in various
days, we obtain semantic trends.

As a second part of this task, we compute the types of each emotion to
identify the trends among different emotions based on task I. By considering
the length and strength of the staying at home public health order in the
first wave, we believe that it is necessary to examine the changes in people’s
emotions by monitoring the time trends and fluctuations of directions using
twitter data.

4) Task III: Modeling Sentence and COVID-19 Emotion-Detection
Machine learning offers the advantage of automatic emotion detection beyond
using existing lexicial dictionaries for emotion analysis. In particular, deep
learning models have proven successful in many NLP applications for emotion
detection from health and medical text data [23] – [25]. We focused use a
convolutional neural network (CNN) [26] – [27] to implement an emotion
detection system based on emotion vectors of #StayAtHome tweets. Our network
layers involve embedding layers, convolution layers, drop out layers, and Max-
Pooling and filter layers.

In the following, we discuss the details of the designed deep-learning model,
the number of convolutional layers, and dense layers to build our COVID-19
emotion detection framework; since the input layer is the sentence
representation, a convolutional layer is then deployed to obtain the sequence
level feature from the sentence sequences. Moreover, the convolutional layer
is considered as the core functional block and includes a collection of
filters. These filters serve as a learner in the network act when they find
some certain type of feature at a determined input. Overall, we consider three
flatten layers for the designed method. Finally, we concatenate the output of
all three learned features by considering dense layers to generate scores and
recognize the type of emotion of the COVID-19 tweets.

##  5 EXPERIMENTAL EVALUATION

This section describes the dataset, generates emotion/semantic trends and the
informative results with various experiments to evaluate the performance of
the research model. In fact, to show the value of the framework, we conduct
experiments on collected COVID-19 tweets and generate informative
emotion/semantic trens. We provide a comparison with a standard base-line to
demonstrate the superiority of our CNN model for emotion detection using F1
score and accuracy as standard metrics. In this research, we use 90% of the
data for training and 10% for testing for all experiments. We focused on the
tweets from the first waves of the COVID-19 pandemic based on #StayAtHome.
However, NLP pre-processing methods (such as stop-words and stemming) are used
to reduce the noises and improve the quality of the output of the task. Table
II shows various times along with the number of tweets based on pure content
and status of data after removing duplicate contents.

Table 2: TABLE II. DETAILS OF THE # STAYATHOME DATASET

#StatAtHome Days | Pure Tweets | Obviate Duplicate Tweets | After Data-Reduction
---|---|---|---
28-4 To 6-5 | 409,000 | 145,000 | 92,000
7-5 To 15-5 | 289,000 | 112,000 | 12,000
16-5 To 24-5 | 226,000 | 88,000 | 10,000
25-5 To 6-2 | 122,000 | 54,000 | 6,000
Total | 1,046,000 | - | -

###  5.1 Informative Trends of the first wave: Emotion and Semantic

Trending topics, to a certain extent, describe the opinion of a community and
provide the means to analyze it, knowing where public attention is at a
certain time point and this has become a matter of interest for researchers
and health professionals. Regarding task II, we need to predict the trend of
topics and give some explanations for the important variation of trends about
COVID-19 trends. To test our machine learning approach with respect to this
task, we randomly split our dataset into 90% for training and the remaining
10% for testing.

###  5.2 Relationship between semantic trends and StayAtHome Tweets

It is difficult to identify the key concepts discussed by users from a million
tweets in traditional ways, so we examine NLP methods (LDA and PLSA) to
extract topics based on semantic aspects to better understand behaviors and
People’s reactions, during stay at home. Then, we investigate the distribution
of generated topics on different days of the initial wave of the outbreak,
which as a result of this process can be helpful in managing public health in
the community. First, we investigate PLSA and LDA models to analyze and
validate the relationship between semantic-topics extracted from COVID-19
tweets and related issues of the pandemic. For this purpose, we use a Mallet
package. Then, we generate 100 topics and only focused on 5 top topics of all
COVID-19 tweets as a result of topic modeling for discovering semantic
related-words. Fig 3, compares the performance of the two topic modeling
methods we have considered to identify emotional trends in our twitter data.
And showed, LDA can capture semantic topics better than the PLSA model for
extracting semantic-topics of #StayAtHome tweets. Therefore, we consider an
LDA model for performing Task II.

Figure 3: The precision metric of word clustering for semantic-topic trending
based on LDA and PLSA

To implement our analytic framework’s detection of semantic trends shown in
the topics during the initial wave of the pandemic, we investigate five top
topics (i.e., S1, S2, S3, S4, and S5, Fig.4) to better understand the online
community reactions change over time. These topics are distributed over
different days and we were able to isolate time varying nature of the semantic
trends of #StayAtHome tweets labeled by an automatic process described in Task
1. According to Fig. 4, the highest ranked (most frequent) topic is
characterized by the words Home, Staysafe, Lockdown, Love, and Family. These
correspond to the safety issues related to staying at home. We label this
topic as S1. It rapidly decreases over time at the rate of 0.11 (p=0.04)
within the 28 days and the decline was greater within the last 14 days 0.28
(p=0.001) (Fig. 4) with some day to day fluctuations shown in the graph. Topic
S2( words Live, Free) shows a decline within the first 14 days and then an
increasing trend is detected within the last 14 days. We need to confirm these
with data related to quarantine and it is possible that after the 14 day
quarantine period the individuals feel free to live. It is important to notice
that negative feelings grow over time, eg S4 (words Friends, Die, Virus)
increases at a rate of 0.14 (p=0.0001) and S5 (Home, remote, quarantine,
health life) increases at a rate of 0.06 (p=0.0005) over the course of the 28
days of tweet text data collection period. Herein we show dynamic behaviour of
statistically significant trends of topics from April 27th to June 1st.

Figure 4: Semantic trends of the initial waves of COVID-19 pandemic by
#StayAtHome

###  5.3 Relationship Between Emotion Trends and StayAtHome Tweets

In Tasks 1 and 2 of the framework, we take the advantage of NRC emotional
lexical, which is supported by Plutchik’s theory based on 14,000 words for
finding the eight primary emotions: anger, anticipation, joy, surprise,
sadness, disgust, trust, and fear. The NRC dictionary has been widely
validated for emotion analysis on social media such as Twitter emotion;
therefore, we consider the advantage of this lexicon for #StayAtHome tweets in
this research. In the application of task II analysis in this stage, we have
accomplished the identification of the relationship between emotion trends in
COVID-19 tweets based on Plutchik theory of classification.

Figure 5: Distribution of emotion trends in #StayAtHome tweets over time in
the initial wave of COVID-19 pandemic. Figure 6: Emotion Trends of the
COVID-19 tweets in line graphs.

According to Fig. 5 the most of the emotions depict in tweets across time are
“Anticipation”. As shown in Fig. 6 the mean percentage of “anticipation”
detected per day across 30 day period is 53.6 (95% CI: 52.4-55.0) with the
least shown by “Surprise” (mean 1.5% , CI: 1.4-1.6%), and “Sadness” (mean
2.2,CI:2.1-2.3%). Among negative emotions “Anger” is shown with the highest
(mean=8.5,CI:8.2-8.7%), followed by “Fear” (mean=5.2, CI:5.0-5.5%), “disgust”
(mean=2.3, CI:2.2-2.4) and “Sadness” (mean=2.2, CI:2.1-2.3). Among positive
feelings “joy” (mean=14.7,CI:14.4-15.4) is the second highest emotion.

According to psychology literature [36], Anticipation and Surprise can be
related to positive or negative health emotional outcomes. Nevertheless, in
this study anticipation stemmed out of the hashtag “stay-at-home”, a
restriction on a socially undesirable action and therefore, one can assume
anticipation is mostly directed towards a negative emotional feeling, of
perceived susceptibility. It is important to consider that these tweets were
exchanged during the early pandemic period of April 28th to June 1st, 2020 of
the first wave and North America reported the peak in May, 2020. Anger may be
directed towards missing summer outdoor activities due to stay-at-home
restrictions, whereas fear may be expressed by those living in high risk
clusters of elderly and those with chronic conditions. It is important to note
that the negative feeling of disgust was minimal and people may be aware of
the importance of quarantine regulations. These emotion expressions and trend
detection over time provide important messages where public health decision
makers can be aware of in the future public health regulation ordering. Though
people trust the public health measures, their anticipation towards negative
emotions need to be considered in the way the public health regulations are
ordered and imposed. Negative connotation of anticipation and fear can be
overcome with public education using social media.

###  5.4 Deep learning model configurations and Training details

The objective of task III of this work is to automatically detect emotions
from #StayAtHome tweets by enabling Multi-Channel CNN methodology as a
computational model for the emotion detection of the COVID-19 tweets. The mode
trained on tweets based on the created dataset of COVID-19 emotion (task II).
First we get the input data of the Task I, we leverage our COVID-19 tweet data
to train our own word embedding with Word2Vec technique [13] which provides a
much richer text representation than the classical, word-based approaches. The
corpus for training word2vec is generated by selecting all the necessary
words, followed by preprocessing the data and removing all stop words. The
output of word2vec are real values are vectors (we used 100 as the dimension).
For our implementation, we used to Keras library. Then the dense layers are
used to output the result of the deep learning model for COVID-19 emotion
detection. As already discussed, every training tweet is automatically
annotated (labelked) using one of eight different emotions including fear,
joy, trust, sadness, anger, surprise, and anticipation. To the best of our
knowledge, this is one of the few studies of automatic emotion detection from
COVID-19 tweets by focusing on Stay-At-Home issues.

###  5.5 Validation and comparison of deep learning model

As Long short-term memory (LSTM) [28] – [29] is a standard base-line for this
research area, we also use that deep model in our study. We compare the CNN
model discussed in sec. IV with the LSTM-Softmax for COVID-19 emotion
detection of the #StayAtHome dataset. The LSTM network used in this research
consists of 64 units. Here, we consider various parameters to train our model
with the different number of epochs such as 10, 20, 30, 40 and 50 to ensure
the significance of the obtained results. However, for each COVID-19 tweet, we
have 8 labels that are features for our detection. Therefore, the output of
the deep-learning model can determine the type of COVID-19 tweets with the
labels.

Figure 7: The F1-score for COVID-19 emotion detection by comparing the CNN
model and the LSTM model

###  5.6 Evaluation of the effectiveness of the framework

We evaluated the performance of the research model with emotions classes. Fig.
7 provides a clear view of variation with different parameters using word
embedding trained. In particular, our preliminary results indicate that the
multi-channel CNN out performs the LSTM-Softmax in terms of various epochs
based on a multi-class F-score as a standard metric. The advantage of the CNN
model comes for detection of the type of emotions in COVID-19 tweets, which
enables to avoid overfitting and still be able to find complex patterns to
emotion detection in the introduced data.

##  6 DISCUSSION, LIMITATION AND FUTURE WORK

Our results, in general, suggested that the machine learning methods we use
are appropriate for the emotion detection of COVID-19 tweets. The study
results clearly demonstrated anticipation as a prominent emotional semantic.
Among various definitions for this emotion semantic analysis, anticipation is
considered as one of “the mature ways of dealing with real stress”[30].
Regarding this definition, people can lower their stress during the COVID-19
pandemic by anticipating and preparing how they are going to deal with it.
Anticipation can be interpreted as either future positive and negative events,
according to [31], and are aligned with hope and fear which are the typical
anticipatory feelings that arise in response to possibilities of future such
events. A study that included multiple unigrams and bi-grams related to
COVID-19 twitter feeds were analyzed using machine learning approaches and
their findings were similar to ours in that the dominant theme identified was
anticipation with a mixed of feelings of trust, anger and fear [32].

To develop a framework that can understand the type of standard emotions
contained in COVID-19 sentences in social media is among the challenging
topics of NLP for the public health and mental healthcare delivery [33] –
[35]. Therefore, in this paper, a multi-task framework is presented to make a
smart emotion detection system based on the Stay-At-Home aspects of COVID-19
tweets. All the experiments are performed using different parameter settings.
The results suggest that the CNN model with two convolutional layers with
filter sizes of 3 and 4 can achieve good performance with various metrics for
emotion detection and classification.

Use of online social network text data to understand user health behaviors and
emotions has become an emerging research area in NLP and health informatics
[36] – [49]. COVID-19 introduced an unprecedented global threat that public
health planning and policy making community are still struggling to find best
practices to curb the pandemic. As the pandemic evolved public health
guidelines became strict measures imposed on the general public. This one
track minded approach of combating the spread or better known as flattening
the curve, neglected emotional and mental health of the individuals who were
subject to those strict public health ordering. This study findings showed a
mechanism of how the emotions and semantic trends of people’s reactions to
COVID-19 public health restrictions can be obtained for knowledge discovery
and can inform related decision making. The advantage of such an approach is
that identifying these online trends provide easy and helpful information
about public reactions to particular issues and thus it has recently attracted
the attention of medical and computer researchers. The framework proposed in
this research covers three practical tasks that are related to each other with
a common goal to develop a deep-learning system for emotion detection and
analysis of informative trends from COVID-19 tweets of people’s reaction
during the stay-at-home. Our final results uncovered important directions for
public health policy makers [40] and decision makers [41] to pay attention to
emotional issues that stemmed from those strict public health restrictions.

This research has some limitations, e.g the size of dataset, data inclusion
limited to emotions based on texts of COVID-19 issues. Currently, our data
consists of 1,047,968 tweets based on #StatAtHome tweets from 28-April to
2-June of 2020. Although more tweets can be extracted based on #StayAtHome, we
believe that the number of current tweets is sufficient to draw reasonable
conclusions to direct possibilities of uncovering importance of consequences
of public health orders and restrictions. We acknowledge that the imbalanced
dataset representing different emotions is also a limitation of our work. We
did not consider slang or emoticons to compute emotions in the tweet contents,
it would be useful to build a new emotional lexical to cover slang words
related to COVID-19 issues. A significantly longer temporal horizon
longitudinal dataset would allow us using LSTM on sequences of tweets, as well
as replacing the cross-validation. train/test approach with one based time-
stamp rather than the cross-validation approach. Further advantage of a
temporally larger dataset is an opportunity of a longitudinal study combining
geographically-based tweeter -detected emotions with COVID-19 incidences and
expanded public health regulations to enable geographic-area targeted public
health decision making. The framework we developed showed potential to
accurately uncover emotional responses and temporal trend detection of mood
changes due to quarantine related public health orders.

##  7 CONCLUSION

This paper presented a novel framework for emotion detection using COVID-19
tweets in relation to the “stay-at-home” public health guidelines. For this
framework, a multi-task framework of COVID-19 emotions detection via a CNN
model was presented. The research further shows that the framework is
effective in capturing the emotions and semantics trends in social media
messages during the pandemic. Moreover, it presents a more insightful
understanding of COVID-19 tweets by automatically identifying the type of
emotions including both negative and positive reaction and the magnitude of
their presentation. The framework can be applied to uncover reactions to
similar public health policies that affect people’s well being. We identified
ways to improve the findings in future research. We discuss potentially
significant, realistic future work, such as extending the longitudinal
character of the results, inclusion of geography-based public health orders
and spatially-annotated COVID-19 case loads.

## Ethical Approval

All procedures performed in studies involving human participants were in
accordance with the ethical standards of the institutional and/or national
research committee and with the 1964 Helsinki declaration and its later
amendments or comparable ethical standards.

Declaration of Conflict of Interest : All authors declare no conflict of
interest directly related to the submitted work.

## References

  * Leek et al. (2016) [1] Becker, M., et al., Natural language processing of German clinical colorectal cancer notes for guideline-based treatment evaluation. International journal of medical informatics, 2019. 127: p. 141-146.
  * Leek et al. (2016) [2] Ong, D., et al., Modeling emotion in complex stories: the Stanford Emotional Narratives Dataset. IEEE Transactions on Affective Computing, 2019.
  * Leek et al. (2016) [3] Organization, W.H., Coronavirus disease (COVID-19) pandemic. 2020: p. https://www.who.int/emergencies/diseases/novel-coronavirus-2019.
  * Leek et al. (2016) [4] Cauberghe, V., et al., How Adolescents Use Social Media to Cope with Feelings of Loneliness and Anxiety During COVID-19 Lockdown. Cyberpsychology, Behavior, and Social Networking, 2020.
  * Leek et al. (2016) [5] de Las Heras-Pedrosa, C., P. Sánchez-Núñez, and J.I. Peláez, Sentiment analysis and emotion understanding during the covid-19 pandemic in spain and its impact on digital ecosystems. IJERPH, 2020. 17(15): p. 5542.
  * Leek et al. (2016) [6] Li, Q., et al., Tracking and Analyzing Public Emotion Evolutions During COVID-19: A Case Study from the Event-Driven Perspective on Microblogs. IJERPH, 2020. 17(18): p. 6888.
  * Leek et al. (2016) [7] Yu, S., et al. Emoticon analysis for chinese health and fitness topics. in International Conference on Smart Health. 2014. Springer.
  * Leek et al. (2016) [8] Johnsen, J.-A.K., et al., Differences in Emotional and Pain-Related Language in Tweets About Dentists and Medical Doctors: Text Analysis of Twitter Content. JMIR public health and surveillance, 2019. 5(1): p. e10432.
  * Leek et al. (2016) [9] Khanpour, H. and C. Caragea. Fine-grained emotion detection in health-related online posts. Empirical Methods in Natural Language Processing. 2018.
  * Leek et al. (2016) [10] Plaza-del-Arco, F.M., et al., Improved emotion recognition in Spanish social media through incorporation of lexical knowledge. Future Generation Computer Systems, 2020. 110: p. 1000-1008.
  * Leek et al. (2016) [11] asan, M., E. Rundensteiner, and E. Agu, Automatic emotion detection in text streams by analyzing twitter data. International Journal of Data Science and Analytics, 2019. 7(1): p. 35-51.
  * Leek et al. (2016) [12] Jelodar, H., et al., Deep sentiment classification and topic discovery on novel coronavirus or covid-19 online discussions: Nlp using lstm recurrent neural network approach. IEEE JBHI , 2020.
  * Leek et al. (2016) [13] Aslam, F., et al., Sentiments and emotions evoked by news headlines of coronavirus disease (COVID-19) outbreak. Humanities and Social Sciences Communications, 2020. 7(1): p. 1-9.
  * Leek et al. (2016) [14] Venigalla, A.S.M., S. Chimalakonda, and D. Vagavolu. Mood of India During Covid-19-An Interactive Web Portal Based on Emotion Analysis of Twitter Data. in Conference Companion Publication of the 2020 on Computer Supported Cooperative Work and Social Computing. 2020.
  * Leek et al. (2016) [15] Plutchik, R., A general psychoevolutionary theory of emotion, in Theories of emotion. 1980, Elsevier. p. 3-33.
  * Leek et al. (2016) [16] Ekman, P., Basic emotions. Handbook of cognition and emotion, 1999. 98(45-60): p. 16.
  * Leek et al. (2016) [17] Mohammad, S., S. Kiritchenko, and X. Zhu. NRC-Canada: Building the State-of-the-Art in Sentiment Analysis of Tweets. Proceedings of the Seventh International Workshop on Semantic Evaluation . 2013.
  * Leek et al. (2016) [18] Kwon, S., MLT-DNet: Speech emotion recognition using 1D dilated CNN based on multi-learning trick approach. Expert Systems with Applications, 2020: p. 114177.
  * Leek et al. (2016) [19] Sun, X., Y. Song, and M. Wang, Towards sensing emotion with deep visual analysis: A long-term psychological modeling approach. IEEE MultiMedia, 2020.
  * Leek et al. (2016) [20] Gautam, R. and M. Sharma, Prevalence and Diagnosis of Neurological Disorders Using Different Deep Learning Techniques: A Meta-Analysis. Journal of Medical Systems, 2020. 44(2): p. 49.
  * Leek et al. (2016) [21] Hofmann, T., Unsupervised learning by probabilistic latent semantic analysis. Machine learning, 2001. 42(1-2): p. 177-196.
  * Leek et al. (2016) [22] Blei, D.M., A.Y. Ng, and M.I. Jordan, Latent dirichlet allocation. Journal of machine Learning research, 2003. 3(Jan): p. 993-1022.
  * Leek et al. (2016) [23] Li, H. and H. Xu, Deep reinforcement learning for robust emotional classification in facial expression recognition. Knowledge-Based Systems, 2020. 204: p. 106172.
  * Leek et al. (2016) [24] Lee, R.Y., et al., Identifying Goals of Care Conversations in the Electronic Health Record Using Natural Language Processing and Machine Learning. Journal of Pain and Symptom Management, 2020.
  * Leek et al. (2016) [25] Wu, P., et al., Social media opinion summarization using emotion cognition and convolutional neural networks. International Journal of Information Management, 2020. 51: p. 101978.
  * Leek et al. (2016) [26] Kim, Y. Convolutional Neural Networks for Sentence Classification.Empirical Methods in Natural Language Processing (EMNLP). 2014.
  * Leek et al. (2016) [27] Kim, J.-C. and K. Chung, Discovery of knowledge of associative relations using opinion mining based on a health platform. Personal and Ubiquitous Computing, 2019: p. 1-11.
  * Leek et al. (2016) [28] Hochreiter, S. and J. Schmidhuber, Long short-term memory. Neural computation, 1997. 9(8): p. 1735-1780.
  * Leek et al. (2016) [29] Uddin, Md Zia, and Erik G. Nilsson. ”Emotion recognition using speech and neural structured learning to facilitate edge intelligence.” Engineering Applications of Artificial Intelligence 94 (2020): 103775.
  * Leek et al. (2016) [30] Conte, H.R. and R. Plutchik, Ego defenses: Theory and measurement. 1995: John Wiley & Sons.
  * Leek et al. (2016) [31] MacLeod, A., Prospection, well-being, and mental health. 2017: Oxford University Press.
  * Leek et al. (2016) [32] Xue, J., et al., Twitter Discussions and Emotions About the COVID-19 Pandemic: Machine Learning Approach. Journal of medical Internet research, 2020. 22(11): p. e20550.
  * Leek et al. (2016) [33] Ta, N., et al., Evaluating Public Anxiety for Topic-based Communities in Social Networks. IEEE Transactions on Knowledge and Data Engineering, 2020.
  * Leek et al. (2016) [34] Yang, X., et al., A big data analytics framework for detecting user-level depression from social networks. IJIM, 2020. 54: p. 102141.
  * Leek et al. (2016) [35] Xue, J., et al., Twitter Discussions and Emotions About the COVID-19 Pandemic: Machine Learning Approach. JMIR, 2020. 22(11): p. e20550.
  * Leek et al. (2016) [36] Pérez-Rodríguez, Gael, Martín Pérez-Pérez, Florentino Fdez-Riverola, and Anália Lourenço. ”Mining the sociome for Health Informatics: Analysis of therapeutic lifestyle adherence of diabetic patients in Twitter.” Future Generation Computer Systems (2020).
  * Leek et al. (2016) [37] Puppala, Mamta, Tiancheng He, Shenyi Chen, Richard Ogunti, Xiaohui Yu, Fuhai Li, Robert Jackson, and Stephen TC Wong. ”METEOR: an enterprise health informatics environment to support evidence-based medicine.” IEEE Transactions on Biomedical Engineering 62, no. 12 (2015): 2776-2786.
  * Leek et al. (2016) [38] Fang, R., Pouyanfar, S., Yang, Y., Chen, S. C., & Iyengar, S. S. (2016). Computational health informatics in the big data age: a survey. ACM Computing Surveys (CSUR), 49(1), 1-36.T4
  * Leek et al. (2016) [39] Roberts, K., Boland, M. R., Pruinelli, L., Dcruz, J., Berry, A., Georgsson, M., … & Jiang, Y. (2017). Biomedical informatics advancing the national health agenda: the AMIA 2015 year-in-review in clinical and consumer informatics. Journal of the American Medical Informatics Association, 24(e1), e185-e190.
  * Leek et al. (2016) [40] Zon, Hilaire, Milena Pavlova, and Wim Groot. ”Exploring decision makers’ knowledge, attitudes and practices about decentralisation and health resources transfer to local governments in Burkina Faso.” Global Public Health (2020): 1-1432. Karmegam, D., T. Ramamoorthy, and B. Mappillairajan, A systematic review of techniques employed for determining mental health using social media in psychological surveillance during disasters. Disaster medicine and public health preparedness, 2020. 14(2): p. 265-272.
  * Leek et al. (2016) [41] Stead, William W., and Nancy M. Lorenzi. ”Health informatics: linking investment to value.” Journal of the American Medical Informatics Association 6, no. 5 (1999): 341-348.


††thanks: Corr.author

# Observation of the diocotron instability in a diode with split cathode

Y. Bliokh  Physics Department, Technion, Israel Institute of Technology, Haifa
320003, Israel  Ya. E. Krasik<EMAIL_ADDRESS>Physics
Department, Technion, Israel Institute of Technology, Haifa 320003, Israel  J.
G. Leopold  Physics Department, Technion, Israel Institute of Technology,
Haifa 320003, Israel  E. Schamiloglu  Department of Electrical and Computer
Engineering, University of New Mexico, Albuquerque, New Mexico 87131-0001, USA

###### Abstract

Diocotron instability has been observed in the pure electron plasma formed in
a split cathode coaxial diode. This plasma consists of electrons, trapped in
the longitudinal potential well between the two parts of the cathode. The
mathematical model of the electron squeezed state, which allows calculation of
the equilibrium plasma density, is presented. The model is applied in a
comprehensive analysis of experimental data and the presence of the diocotron
instability is unambiguously confirmed.

##  I Introduction

Single-charged non-neutral, pure electron plasmas [1, 2] are of interest both
for basic fundamental [3] plasma physics [4], including astrophysics [5],
atomic physics [6], and for various practical applications [7]. Examples of
such plasmas are present in the electron cloud in the Malmberg-Penning trap
[8, 9], the squeezed state of electrons [10, 11], and electron beams. However,
the properties of each of these plasmas are very different. Electrons in the
Malmberg-Penning trap are isolated from the environment, their density is
quite low, $\leq 10^{8}$ cm-3, and their energy is very low as well (several
eV) [4]. The electron squeezed state is in permanent contact with the electron
source and is characterized by high density, ($\leq 10^{12}$ cm-3), and high
energy electrons (10s keV) [12]. The density and the energy of electron beams
can vary over a very broad range.

In the majority of cases, the geometry of a single-charged plasma is a solid
cylinder or an annular column. The diocotron instability [13, 1] development
results in the radial expansion of the electrons and azimuthal modulation of
the electron density. Sometimes this instability is undesirable, but sometimes
it is useful. There is some evidence [14] that the diocotron instability can
enhance formation of spokes in magnetrons.

In this article, the development of the diocotron instability in a pure
electron plasma trapped in the potential well existing between the two parts
of a split cathode geometry (see Fig. 1), is studied experimentally and
analytically. Dynamical equilibrium of this plasma is known as a squeezed
electron state. Recently, the split cathode was successfully used as a source
of electrons in a relativistic magnetron [15], which explains our interest in
this configuration. Diocotron instability in a squeezed state was observed,
but only in numerical simulations [16].

##  II  EXPERIMENTAL SETUP

Experiments were carried out using the same experimental setup as that
described in detail in [17] and sketched in Fig. 1. The experiment was
performed in a 124 mm diameter and 600 mm long stainless-steel tube at the
center of which a 40 mm diameter, 62 mm long stainless-steel anode insert was
placed. An axial magnetic field, $H_{0}$, of up to 1.2 T and half-period of 15
ms is produced by a solenoid. The high voltage pulse, formed by a Marx
generator ($\sim 200$ kV, $\sim 150$ ns with a resistive load of $\sim
84\Omega$) is applied to 20 carbon-epoxy capillary tubes (each 5 mm long and
1.5 mm/0.75 mm outer/inner diameter) placed along a 16 mm-diameter circle on a
25 mm diameter aluminium cathode. The axial distance between the edge of the
capillaries and the anode was 20 mm. The reflector consists of a 2 mm-thick,
40 mm-diameter aluminium disk, and a 6 mm diameter ring at its periphery. The
reflector is connected to the cathode by a 6 mm diameter aluminium axial rod.
The cathode, reflector and the rod were hard coated with an 80 $\mu$m-thick
Al2O3 ceramic layer to avoid explosive emission plasma formation. The distance
between the reflector and the anode downstream edge was 40 mm.

The diode voltage and current waveforms were measured using a resistive
voltage divider and a self-integrating Rogowski coil, respectively. These
probes were placed inside transformer oil tank at the output of the Marx
generator. The beam potential was measured using an anode capacitive voltage
divider calibrated in shots with a 16 mm diameter rod traversing the anode
axis. This probe was placed at the inner surface of the anode insert. The
diocotron instability appears as rotating azimuthal modulations of the
electron density. These modulations were measured by two probes inserted
through holes drilled at the longitudinal center of the anode and separated
$180^{\circ}$ from each other. These probes, consist of 80$\mu$m-diameter
copper wires, placed inside ceramic capillaries leaving 4 mm bare intended to
contact with the electron plasma. The voltage, current, and probe waveforms
were acquired by a DSO80604B digitizing oscilloscope.

The temporally-and-spatially-resolved transverse distribution of the electron-
beam current density was obtained by fast x-ray imaging of the beam. These
experiments were carried out either with a split cathode, with the plate of
the reflector replaced by a 40 mm 90% transparent tungsten wire grid, or
without the reflector and the rod. The images were obtained at a distance of
100 mm from the anode where a 2 mm thick fast EJ-200 plastic polyvinyl toluene
(PVT) thick-scintillator was placed, attached to the back of a grounded 80
$\mu$m-thick Ta foil. The mean free path of the electrons (electron energy
<250 keV) in the Ta foil is less than the foil thickness. However, the mean
free path of the x-rays produced by the electrons inside the foil is larger
than the foil thickness. The interaction of these x-rays with the scintillator
produces a temporally-and-spatially-resolved image, which was recorded using a
4QuikE camera (of 1.2 ns frame duration) synchronized with the high voltage
generator.

Figure 1: The experimental setup for (a) a wire grid reflector, (b) to record
the beam in the absence of the reflector and rod and (c) with the probes #1
and #2.

##  III Experimental results

Time-integrated patterns of the electron beam on the 0.4mm-thick Mylar or 1mm-
thick aluminum targets placed at the distance of 10 cm from the anode insert,
were obtained in experiments without the reflector and rod. In these
experiments, at a magnetic field of 6.4 kG, the electron beam leaves on the
targets a $\sim\leavevmode\nobreak\ 2.5$ mm thick circular pattern with an
average diameter of $\sim 18$ mm and without signs of discrete structures. In
Fig. 2(a), waveforms of the voltage, electron beam current, and the potential
of the electron beam are shown for the setup shown in Fig. 1(a) (no reflector
and rod). One can see that the potential of the electron beam is $\sim 25$ kV
when the beam current is $\sim 1$ kA. The $\sim 200$ ns long plateau of the
current, despite the significant decrease in the amplitude of the applied
voltage, is probably the result of the decrease in the effective gap between
the boundary of the cathode plasma expanding along the magnetic field upstream
from the anode towards it at a velocity of $\sim 10^{7}$ cm/s [18], In Fig.
2(b), waveforms of the voltage, electron beam current, and potential when the
reflector is absent and only the 6 mm central rod is attached to the cathode,
are shown. One can see that the beam potential increases to $\sim 75$ kV due
to the presence of the rod.

Figure 2: Waveforms of the voltage, beam current and beam potential (a)
without rod and reflector and (b) with the rod but no reflector. The magnetic
field here was 6.4 kG

In Fig. 3, the voltage, total current and beam potential in a split cathode
configuration for two values of the magnetic field are presented. The first
oscillations seen on the current waveform are the result of the displacement
current. Next, during $\sim 75$ ns, the amplitude of the total current is
$\leq 150$ A, which is $\sim 6.5$ times smaller than when the reflector is
absent. However, from $t\approx 80$ ns and on, the total current increases up
to $\sim 1$ kA at $t\approx 380$ ns. Inspection of the reflector and central
rod alumina coating after hundreds of shots showed only a few damage points,
indicating that explosive electron emission from these parts are not
responsible for the current increase. Thus, the gradual increase in the total
current at $t>80$ ns while the voltage decreases can be related to the
accumulation of the electrons in the split cathode potential well and the
electron space charge radial expansion above cathode radius of 12.5 mm to
become an upstream electron flow increasing the total current. For the larger
value of the magnetic field in Fig. 3(b) the current reaches lower values.

The smaller value of the potential for $H_{0}=10$ kG than that obtained for
$H_{0}=6.4$ kG (see Fig. 3) indicates that the space charge accumulated in the
potential well between the cathode and reflector increases with increasing
magnetic fields. However, one can reasonably consider that this space charge
is expanding radially faster for smaller magnetic fields accompanied by a
change in the anode voltage divider’s coefficient. Since all the quantities,
that is, the electron density, its spatial distribution, and the high voltage
change together, the signal from the voltage divider presented in Figs. 3(a)
and 3(b), cannot be used to determine the value of the electron cloud
potential. Nevertheless, a fast increase of the probe signal followed by an
abrupt decrease to zero allows one to extract useful information. This happens
when the electron cloud’s expansion leads to electron losses to the high
voltage electrode of the divider followed by breakdown along the insulator
surface. Thus, the breakdown time is the characteristic time of the electron
cloud’s radial expansion. The results obtained showed that this time grows
with increasing magnetic field.

Figure 3: Waveforms of the voltage, total current and electron beam potential
relative to the anode with a split cathode for an applied axial magnetic field
of (a) 6.4 kG and (b) 10 kG.

No x-ray electron beam images were obtained in experiments with a split
cathode when the aluminum reflector was replaced with a tungsten wire grid.
This is because the energy of the electrons near the reflector is strongly
reduced and too little of the beam is transmitted. Images were obtained only
in the absence of the reflector and registered at different times relative to
the onset of the voltage as seen in Fig. 4 for a magnetic field of $H_{0}=8$
kG. Similar images were obtained for other magnetic fields values as well,
independent of whether the rod was present or absent. One can see that the
electron beam pattern is not azimuthally uniform and bunching is discernible,
typical for a diocotron instability[1].

Figure 4: Framing images of the luminescence pattern appearing on the plastic
scintillator due to the interaction of the electron beam with a
127-$\mu$m-thick Ta foil placed in front of the scintillator. Frame duration
1.2 ns, magnetic field 8 kG.

The existence of the diocotron instability in a split cathode geometry is
confirmed by high frequency oscillations observed on the probe signals shown,
for example, in Fig. 5 together with the voltage and current waveforms.
Experiments with probes #1 and #2 were carried out in three configurations:
(a) the split cathode; (b) no reflector and rod; and (c) no reflector but rod
present. In all these cases, high frequency oscillations were observed in the
probe signals. Analysis of the waveforms of these probes is in Section IV.

Figure 5: The probe signal, voltage, and current waveforms. Split cathode
configuration, magnetic field 8 kG.

##  IV Mathematical model

###  IV.1 Steady-State Hollow Electron Cloud

Let us consider an electrons placed in the space between an anode tube of
radius $R_{ext}$ and a coaxial central rod of radius $R_{in}$ (Fig 6). It is
assumed that this cloud is bounded between cylindrical surfaces of the radii
$r_{a}$ and $r_{b}$ and has radially-uniform density. In the longitudinal
direction the electron motion is restricted by the cathode and reflector (see
Fig. 1).

Figure 6: The electron cloud bounded in the space between cylindrical surfaces
of the radii $r_{a}$ and $r_{b}$ between the radius $R_{in}$ central rod and
the radius $R_{ext}$ of the anode tube.

There is a potential difference, $\varphi_{0}$, between the tube and the rod,
and an axial magnetic field $H_{0}$ which keeps the cloud from expanding
radially.

The nonrelativistic electron equations of motion are:

$\displaystyle\frac{dv_{r}}{dt}=\frac{e}{m}E_{r}+\omega_{H}v_{\theta}+\frac{1}{r}v_{\theta}^{2},$
$\displaystyle\frac{dv_{\theta}}{dt}=\frac{e}{m}E_{\theta}-\omega_{H}v_{r}-\frac{1}{r}v_{\theta}v_{r}.$
(1)

Here $\omega_{H}=eH_{0}/mc$ is the electron gyrofrequency, $v_{r}$ and
$v_{\theta}$ are electron radial and azimuthal velocities, respectively.

The steady-state is characterized by zero radial velocity, $v_{r}=0$, and zero
azimuthal electric field, $E_{\theta}=0$. Then, it follows from Eq. (IV.1),
that at steady-state

$v_{\theta}(r)^{2}+r\omega_{H}v_{\theta}+\frac{e}{m}rE_{r}(r)=0$ (2)

Introducing the angular frequency $\omega_{d}(r)=v_{\theta}/r$ of the rotating
electrons, the solution of Eq. (2) can be written as

$\omega_{d}(r)=-\frac{1}{2}\omega_{H}\pm\sqrt{\frac{1}{4}\omega^{2}_{H}-\frac{eE(r)}{mr}},$
(3)

The lower frequency corresponds to the rotation with azimuthal velocity
$v_{\theta}$ close to the drift velocity $v_{d}=cE_{r}/H_{0}$,
$v_{\theta}\simeq v_{d}$ .

The solution of the Poisson equation in the region $r_{a}<r<r_{b}$, the
continuity of the potential $\varphi(r)$ and the radial electric field
$E_{r}=-\partial\varphi/\partial r$ at the interfaces $r=r_{a}$ and $r=r_{b}$,
and the boundary conditions $\varphi(R_{in})=0$ and
$\varphi(R_{ext})=\varphi_{0}$ allow one to determine the potential and
electric field distributions in the entire space $R_{in}\leq r\leq R_{ext}$,
as for example in [13]. Finally, the radial electric field allows one to
calculate the electrons rotation frequency, $\omega_{d}(r)$, using Eq. (3).
Note that the rotating electrons produce additional axial magnetic field so
that the gyrofrequency, $\omega_{H}$, in Eq. (3) depends, strictly speaking,
on the radius and should be calculated simultaneously with $\omega_{d}(r)$.
However, for the experimental conditions considered here, this additional
magnetic field is small and can be neglected.

###  IV.2 Diocotron instability. Dispersion relation

The diocotron instability, which can develop under certain conditions, breaks
the axial symmetry of the described above equilibrium configuration of the
electron cloud. The stability analysis, developed in [1, 13, 25], is
applicable to the system being studied. Note, that a system with a central
electrode was considered in [13], but the normalized variables used in this
paper are inconvenient when the electron rotation frequency $\omega_{d}(r)$ is
defined by the external electric field rather than the space-charge field.

The stability analysis is reduced to the eigenvalue problem for Poisson
equation

$\frac{1}{r}\frac{d}{dr}\left(r\frac{d\phi}{dr}\right)-\frac{\ell^{2}}{r^{2}}\phi=-\frac{\ell\,d\omega_{p}^{2}(r)/dr}{\omega_{H}r[\omega-\ell\omega_{d}(r)]}\phi$
(4)

with boundary conditions $\phi(R_{in})=\phi(R_{ext})=0$ for the potential
perturbation of the form $\delta\varphi(r,\theta,t)=\phi(r)\exp(-i\omega
t+i\ell\theta)$, $\ell=0,\pm 1,\pm 2,\ldots$.

The electron Langmuir frequency $\omega_{p}(r)$ is defined by the equilibrium
distribution of the electron density $n_{e}(r)$. For the model with a step-
function density profile, used in the previous section, the eigenfrequencies
$\omega$ are roots of a quadratic equation (dispersion equation), whose
derivation and particular form are presented in Appendix A. The general
expression for the eigenfrequencies is rather cumbersome. Here we present the
solution for a thin electron cloud, the thickness of which $h=r_{b}-r_{a}$ is
small compared to its mean radius $r_{e}=(r_{b}+r_{a})/2$, $h\ll r_{e}$.

Instead of frequencies $\omega_{d}(r_{a,b})$, it is convenient to introduce
$\bar{\omega}_{d}=[\omega_{d}(r_{a})+\omega_{d}(r_{b})]/2$ and
$\delta\omega_{d}=[\omega_{d}(r_{a})-\omega_{d}(r_{b})]/2$. The quantity
$\bar{\omega}_{d}$ is the mean electron rotation frequency, the quantity
$\delta\omega_{d}$ characterizes the difference between the electron rotation
frequencies at the inner and outer surfaces of the annular cloud.

Assuming that the rather weak conditions $(r_{e}/R_{ext})^{2\ell}\ll 1$ and
$(R_{in}/r_{e})^{2\ell}\ll 1$ with $\ell=2,3,..$ are satisfied, the solution
of the dispersion equation takes the form (see details in Appendix A)

$\omega\simeq\ell\bar{\omega}_{d}\pm\sqrt{\ell\delta\omega_{d}\left(\ell\delta\omega_{d}+\frac{\omega_{p}^{2}}{\omega_{H}}\right)-\frac{\omega_{p}^{4}\ell
h}{2\omega_{H}^{2}r_{e}}}.$ (5)

The electron cloud becomes unstable when the radical expression is negative.
It follows from Eq. (5) that in the instability region
$\Re\,\omega\simeq\ell\omega_{d}(r_{e})$, i.e., the azimuthal velocity
$\Re\omega r_{e}/\ell$ of the electron density modulation (electron bunches)
is close to the electron rotation velocity $v_{\theta}\simeq
cE_{r}(r_{e})/H_{0}\simeq\bar{\omega}_{d}r_{e}$. The numerical solution of the
dispersion equation (17), presented in Fig. 7, confirms this analysis. Note
that the dependence of the mean electron rotation frequency
$\bar{\omega}_{d}(n_{e})$ on the density is weak,
$\bar{\omega}_{d}(n_{e})\simeq\left.\bar{\omega}_{d}\right|_{n_{e}=0}$.

Figure 7: Solution of the dispersion equation (17) for the diocotron modes
$\ell=2$ (dashed lines) and $\ell=3$ (solid lines). The rotation frequencies
$\ell\bar{\omega}_{d}(n_{e})$ are shown by dotted lines. Geometrical
parameters, magnetic field value, and the voltage between internal and
external electrodes correspond to the experimental conditions. The vertical
line marks the electron density, calculated within the framework of the
squeezed state model.

###  IV.3 The Squeezed Electron State

The cathode and the reflector in a split cathode form a potential well which
traps the emitted electrons while their number increases gradually. To this
charge increase, the trapped electrons react by relaxing to a force-free state
(the so-called squeezed state) [11] characterized by reduced energy and high
density defined by the depth of the potential well restricted radially for
very strong magnetic field. For strong but finite magnitude magnetic fields,
the external radial electric field can extract electrons from the squeezed
state in radial direction. This is the case when the squeezed state can be
employed as an electron source with radial emission [15].

The steady-state electron density is defined by the balance between the
external force that keeps electrons in the potential well and the Coulomb
force of the space charge of accumulated electrons in the well. The Coulomb
electric field can be calculated using the Green function method in the same
manner as shown in [19, 20, 21] (for details see the Appendix B).

We now present the annular electron cloud as a set of charged rigid annular
rings $r_{a}\leq r\leq r_{b}$ of infinitesimal thickness $dz$. The expression
for the axial component of the electric field
$\bar{E}_{z}(z^{\prime},z^{\prime\prime})$ produced in the cross section
$z=z^{\prime}$ by the charged ring located at $z=z^{\prime\prime}$ can be
written as

$\bar{E}_{z}(z^{\prime},z^{\prime\prime})=2\pi\sigma{\rm
sign}(z^{\prime}-z^{\prime\prime})G(|z^{\prime}-z^{\prime\prime}|),$ (6)

where $\sigma=en_{e}dz$ is the surface charge density, ${\rm
sign}(z^{\prime}-z^{\prime\prime})=1\,{\rm or}\,-1$ for
$z^{\prime}>z^{\prime\prime}$ and $z^{\prime}<z^{\prime\prime}$, respectively.

It was shown in [20, 21, 22] that the function
$G(|z^{\prime}-z^{\prime\prime}|)$ for a solid cylindrical electron cloud can
be approximated by the exponential

$G(|z^{\prime}-z^{\prime\prime}|)\simeq\exp(-|z^{\prime}-z^{\prime\prime}|/\ell_{C}),$
(7)

where the characteristic scale of the Coulomb field $\ell_{C}\simeq r_{b}/2$.
An annular cloud is characterized by additional of geometrical parameters.
Nevertheless, the exponential approximation Eq. (7) is still applicable (see
Fig. 14). The dimensions of the conductors and the electron cloud in the
experiment described in Section III, are $r_{a}=0.7$cm, $r_{b}=0.9$cm,
$R_{in}=0.3$cm, and $R_{ext}=2.0$ cm. The characteristic scale of Coulomb
field for these parameters is $\ell_{C}\simeq 0.56$cm.

The equation of motion of the charged rings, comprising the electron cloud in
the external and the self-consistent space charge electric fields can be
written as [11]

$\frac{d^{2}z}{dt^{2}}=-\frac{e}{m}\frac{\partial\varphi}{\partial
z}+2\pi\frac{e^{2}}{m}\intop_{0}^{L}dz^{\prime}n_{e}(z^{\prime}){\rm
sign}(z-z^{\prime})G(|z-z^{\prime}|).$ (8)

Here $\varphi$ is the external potential and $L$ is the system length.
Assuming that the characteristic spatial scale of the variation of the
external potential $\varphi$ and the electron density $n_{e}$ is large
compared to $\ell_{C}$, one can evaluate the integral in Eq. (8) as

$\intop_{0}^{L}dz^{\prime}n_{e}(z^{\prime}){\rm
sign}(z-z^{\prime})G(|z-z^{\prime}|)\simeq\intop_{0}^{L}dz^{\prime}n_{e}(z^{\prime}){\rm
sign}(z-z^{\prime})e^{(|z-z^{\prime}|)/\ell_{C}}\simeq
2\frac{dn_{e}(z)}{dz}\ell_{C}^{2}.$ (9)

It follows from Eq. (9) that the steady-state solution exists for

$n_{e}(z)=\frac{\varphi(z)}{4\pi e\ell_{C}^{2}}\simeq 5.7\cdot
10^{8}\frac{\varphi(z)[kV]}{\ell_{C}^{2}[cm]}[cm^{-3}].$ (10)

Note, that $\varphi(z)$ is the potential profile along the electron
propagation path, i.e., $\varphi(z)=\varphi(z,r)_{r={\rm const}}$.

The depth of the potential well, $\varphi_{0}$, where the squeezed electron
state (cloud) forms, is defined by the potential profile along the line
$r=r_{e}\simeq(r_{b}+r_{a})/2$ of the electron trajectory. In the
configuration with a central rod,
$\varphi_{0}=U_{0}(|\ln\rho_{e}/\ln\rho_{in}|-1)\simeq 0.5U_{0}$, where
$\rho_{e}=r_{e}/R_{ext}$, $\rho_{in}=R_{in}/R_{ext}$ and $U_{0}$ is the
potential difference between the central electrode and the external tube
(anode). In the experiments (see Section V for details) $U_{0}=75$kV so that
$\varphi_{0}\simeq 37.5$kV. Using $\ell_{C}\simeq 0.56$cm and
$\varphi_{0}\simeq 37.5$kv, in Eq. (39), the electron density can be estimated
as $n_{e}\simeq 6.9\cdot 10^{10}$cm-3.

##  V Analysis of the Experimental Results

As mentioned in Section II, two probes were placed at two diametrically
opposite locations along the anode circumference. Typical waveforms of the
probe signals and the potential difference between the anode and central rod
are presented in Fig. 8.

Figure 8: (a) Potential difference between anode and the rod electrode and (b)
two probe signals.

###  V.1 Processing the Probe Signals

First, the fast oscillating component of the probe signals was separated from
the slowly varying basis as illustrates in Fig. 9.

Figure 9: (a) Raw probe signal (gray) and filtered base line (red). (b)
Normalized fast oscillating component.

Next, a wavelet analysis of the normalized filtered signals is performed. The
resulting spectrograms of two probe signals are seen in Fig. 10.

Figure 10: (a) Correlation functions of the signals within the time intervals
275 - 300 ns and 220 - 270 ns; (b) and (c) Spectrograms of the signals
registered by the two probes.

One can see a sharp change in the spectra in the time interval 275-300 ns. It
seems reasonable to analyze the correlation between the signals from two
probes in these two time domains, 220-270 ns and 275-300 ns, separately. The
resulting correlation functions are shown in Fig. 10(a). These will be
discussed in Section V.B.

###  V.2 Data Analysis

Periodic probe signals appear as the result of rotating electron density
modulation produced by the diocotron instability. The rotation frequency of
the density modulation, as shown in Eq. (5), is close to the electron rotation
frequency, defined by the drift velocity $v_{d}=cE/H$ in the crossed radial
electric, $E$, and axial magnetic, $H$, fields. In order to support this
statement, we check that the temporal variations of the voltage and the
spectrum correlate. To do this, let us superimpose a spectrogram and an
appropriately scaled voltage waveform as shown in Fig. 11.

Figure 11: Two different spectrograms and the corresponding scales voltages.
The voltage waveforms $U(t)$ are scaled so that they coincide with the
dominant frequency at some point marked as the reference point.

It can be clearly seen that the time dependence of the voltage waveform
coincidences with the variation in the spectrum over a long period of time.
This indicates that the diocotron instability can indeed be responsible for
the observed oscillating signal behavior. Note, that similar methods to detect
the diocotron instability using azimuthally separated probes or a segmented
anode cylinder, was described in [23, 24].

The raw data used in Fig. 10 demonstrates an additional property, which can
also be considered as manifestation of the diocotron instability. The
correlation functions, presented in Fig. 10(a), show that the oscillating
signals are in-phase in the time domain $220<t<270$ ns, and out of phase in
the time domain $275<t<300$ ns. This can be explained by the fast
transformation of the azimuthally rotating electron column from even to odd
number of the azimuthal high-density bunches of electrons. If this assumption
is correct, then the difference between the frequencies in the time domains is
the experimental determination of the rotation frequencies of the bunches, as
shown in Fig. 12.

Figure 12: Experimental determination of the rotation frequency of the
bunches. (a) Spectrogram. Vertical red line marks the point in time when the
frequency jump occurs. The horizontal white lines mark the values of the
frequencies. (b) The voltage waveform. Vertical red line marks the voltage
value at the time the frequency jump occurs. (c) The rotation frequency,
calculated using the theoretical model, described in Section IV.

Calculation of the rotation frequency, $f(r)=\omega(r)/2\pi$, was performed
using the model described in Section IV. The frequency jump occurs when the
voltage is $\simeq 75$kV. Following Section IV, the corresponding electron
density of the squeezed state at this moment is $\simeq 6.9\cdot 10^{10}$cm-3.
The calculated dependence $f(r)$ for these values of the voltage and the
electron density for the experimental system dimensions is shown in Fig. 12c.
The calculated frequency value agrees well with the defined experimentally
difference $\Delta f\simeq 0.25$GHz between frequencies, shown in Fig. 12(a).
The minor frequency from the considered is nearly twice as large as the
rotation frequency, while the higher frequency is approximately three times
larger. It means, that the lower frequency corresponds to the electron cloud
configuration with two (even!) azimuthal bunches, the higher frequency
corresponds to configuration with three (odd!) bunches. The latter agrees with
the result of correlation analysis, presented in Fig. 10(a).

These results – number of azimuthal bunches and modulation frequencies – agree
perfectly with numerical solution of the dispersion equation, presented in
Fig. 7 in Section IV B.

##  VI Conclusions

The split cathode is a promising source of electrons for high power microwave
devices. Electrons, emitted by an annular cathode, are trapped in the
longitudinal potential well forming electron cloud, the radial expansion of
which is limited by a strong axial magnetic field. First experimental studies
of a relativistic magnetron with a split cathode confirmed that the absence of
the cathode explosive emission plasma inside the anode interaction space
prevents microwave pulse shortening [15].

The azimuthal symmetry of an electron cloud in a magnetron can be violated by
two processes. First, the interaction of the electrons with electromagnetic
fields of the magnetron cavity eigenmodes. Second, by the diocotron
instability, which for the split cathode geometry, has not been studied so
far. In the present article, for the split cathode fed smooth bore anode, it
has been shown, by a comprehensive theoretical analysis of the experimental
data, that the diocotron instability indeed develops. Thus, we expect that for
a corresponding magnetron, the diocotron instability and the magnetron
eigenmodes could act simultaneously and influence each other. Matching between
the diocotron and magnetron eigenmodes can improve the magnetron
characteristics.

## Acknowledgments

The authors are grateful to two anonymous referees for valuable comments that
have lead to a substantial improvement in the manuscript. At the Technion,
this work was supported by Technion (Grant No. 2029541) and ONRG (Grant No.
N62909-21-1-2006). We are thankful for the assistance in experiments Y. Cao,
S. Gleizer and E. Flyat. At the University of New Mexico, this work was
supported by ONR (Grant No. N00014-19-1-2155).

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request

## Appendix A

In this Section we apply the stability analysis, developed in [13, 25] to the
system with step-like profile of the electron cloud, considered in Section IV
A.

Normalizing all frequencies to $c/R_{ext}$ ($\Omega=\omega R_{ext}/c$,
$\Omega_{p}=\omega_{p}R_{ext}/c$, etc.) and radii to $R_{ext}$
($\rho=r/R_{ext}$, $\rho_{a}=r_{a}/R_{ext}$, etc.), let us write Eq. (4) in
the dimensionless form:

$\frac{d}{d\rho}\left(\rho\frac{d\phi}{d\rho}\right)-\frac{\ell^{2}}{\rho}\phi=-\frac{\ell\,d\Omega_{p}^{2}(\rho)/d\rho}{\Omega_{H}[\Omega-\ell\Omega_{d}(\rho)]}\phi.$
(11)

For the step-like profile of the electron density, $n_{e}(\rho)=n_{0}$ when
$\rho\in(\rho_{a},\rho_{b})$ and $n_{e}=0$ elsewhere,

$d\Omega_{p}/d\rho=\Omega_{p0}[\delta(\rho-\rho_{a})-\delta(\rho-\rho_{b})],$
(12)

where $\delta(x)$ is the Dirac $\delta$-function.

General solution of Eq. (11) is

$\phi=\alpha\rho^{\ell}+\beta\rho^{-\ell},$ (13)

where $\alpha$ and $\beta$ are arbitrary constants. Using the boundary
conditions $\phi(\rho_{in})=\phi(1)=0$ and taking into account Eq. (12), the
solution of Eq. (11) can be written as

$\displaystyle\phi\equiv\phi_{\rm
I}=A\left(\rho^{\ell}-\rho^{-\ell}\rho_{in}^{2\ell}\right),\hskip
14.22636pt\rho\in(\rho_{in},\rho_{a}),$ $\displaystyle\phi\equiv\phi_{\rm
II}=C\rho^{\ell}+D\rho^{-\ell},\hskip 14.22636pt\rho\in(\rho_{a},\rho_{b}),$
(14) $\displaystyle\phi\equiv\phi_{\rm
III}=B\left(\rho^{\ell}-\rho^{-\ell}\right),\hskip
14.22636pt\rho\in(\rho_{b},1),$

Function $\phi$ is continuous everywhere over the interval $(\rho_{in},1)$

$\displaystyle\phi_{\rm I}(\rho_{a})=\phi_{\rm II}(\rho_{a}),$
$\displaystyle\phi_{\rm III}(\rho_{b})=\phi_{\rm III}(\rho_{b}),$ (15)

and its derivative $\phi^{\prime}=d\phi/d\rho$ experiences stepwise variations
at the interfaces between the regions:

$\displaystyle\phi^{\prime}_{\rm II}(\rho_{a})-\phi^{\prime}_{\rm
I}(\rho_{a})=-\frac{\ell\Omega_{p}^{2}\phi(\rho_{a})}{\rho_{a}\Omega_{h}\left[\Omega-\ell\Omega_{d}(\rho_{a})\right]},$
$\displaystyle\phi^{\prime}_{\rm III}(\rho_{b})-\phi^{\prime}_{\rm
II}(\rho_{b})=\frac{\ell\Omega_{p}^{2}\phi(\rho_{b})}{\rho_{b}\Omega_{h}\left[\Omega-\ell\Omega_{d}(\rho_{b})\right]}.$
(16)

The dispersion equation appears as a compatibility condition for the system
(15)–(16). Introducing
$\bar{\Omega}_{d}=[\Omega_{d}(\rho_{a})+\Omega_{d}(\rho_{b})]/2$,
$\delta\Omega=[\Omega_{d}(\rho_{a})-\Omega_{d}(\rho_{b})]/2$, and
$w=\Omega-\ell\bar{\Omega}_{d}$, one can write the dispersion equation in the
following form:

$\alpha w^{2}+\beta w+\gamma=0.$ (17)

Here

$\alpha=\Delta_{-}(1-g_{a}g_{b})-\Delta_{+}(g_{a}-g_{b}),\hskip
8.53581pt\beta=\mu\Delta_{-}(g_{a}+g_{b}),$
$\gamma=\Delta_{-}\mu^{2}+\ell\mu\delta\Omega[2\Delta_{+}-\Delta_{-}(g_{a}-g_{b})]-\alpha\ell^{2}\delta\Omega^{2},\hskip
8.53581pt\mu=\Omega_{p}^{2}/\Omega_{H},$
$\Delta_{-}=\rho_{a}^{\ell}\rho_{b}^{-\ell}-\rho_{a}^{-\ell}\rho_{b}^{\ell},\hskip
8.53581pt\Delta_{+}=\rho_{a}^{\ell}\rho_{b}^{-\ell}+\rho_{a}^{-\ell}\rho_{b}^{\ell},$
$g_{a}=[1+(\rho_{in}/\rho_{a})^{2\ell}]/[1-(\rho_{in}/\rho_{a})^{2\ell}],\hskip
8.53581ptg_{b}=(1+\rho_{b}^{2\ell})/(1-\rho_{b}^{2\ell}).$

For the qualitative analysis of the dispersion equation, it is convenient to
consider a thin annular cloud, assuming that
$\rho_{b}-\rho_{a}=d\ll\rho_{a}+\rho_{b}=2\rho_{e}$, where $d$ is the cloud
thickness and $\rho_{e}$ is the cloud mean radius. Assuming also that
$(\rho_{in}/\rho_{e})^{2\ell}\ll 1$ and $\rho_{e}^{2\ell}\ll 1$, where
$\ell=2,3,\ldots$, the dispersion equation (17) can be simplified to

$w^{2}=\ell\delta\Omega(\ell\delta\Omega+\mu)-\ell\mu^{2}\delta/2\rho_{e}.$
(18)

The diocotron instability develops when the right-hand side of Eq. (18) is
negative. For this case

${\rm Re}\,\Omega=\ell\bar{\Omega}_{d}.$ (19)

This result is confirmed by the solution of Eq. (17), presented in Fig. 13.
One can see that the dependence of the mean electron rotating frequency
$\bar{\Omega}_{d}$ on the electron density is weak, so that its value can be
calculated for a ‘‘cold’’ system, when the space charge field of the electron
cloud is small compared to the external electric field. Eq. (19) means that
the azimuthal modulation of the electron density rotates practically with the
same velocity as the electrons themselves.

Figure 13: The solution $\Omega$ of the dispersion equation (17) as function
of the parameter $\mu$. The dashed line is the value of $\ell\bar{\Omega}_{d}$
vs $\mu$. The geometrical parameters are the same as in the experiment. In the
instability region ${\rm Re}\,\Omega=\ell\bar{\Omega}_{d}$

## Appendix B

###  VI.1 The Space Charge Electric Field

Consider the electron cloud occupying the annular region $r_{a}\leq r\leq
r_{b}$ arranged coaxially in a conducting tube of radius $R_{ext}$, as shown
in Fig. 6. In general, the conducting tube can also contain a central rod. Let
us find the potential $\varphi({\bf r})$ induced by an annular disk $r_{a}\leq
r\leq r_{b}$ of infinitesimal thickness $dz$ in the $z$ direction.

The electrostatic potential, produced by an axially-symmetrical charged layer,
placed at $z=0$, in the half-space $z\geq 0$, can be written as

$\varphi({\bf r})=\sum_{n}A_{n}e^{-\mu_{n}z}F_{n}(r),$ (20)

where $F_{n}(r)$ are the eigenfunctions of the Laplace equation

$\frac{1}{r}\frac{d}{dr}\left(r\frac{dF_{n}}{dr}\right)+\mu_{n}F_{n}=0$

with zero boundary conditions either in the ring area $R_{in}\leq r\leq
R_{ext}$ or in the circular area $0\leq r\leq R_{ext}$ in the absence of the
central conducting rod. Here $\mu_{n}$ are the eigenvalues. Because of
symmetry, on the entire $z=0$ plane, $E_{z}|_{z=0}=-(\partial\varphi/\partial
z)_{z=0}=0$, except on the charged layer itself at $r\in(r_{a},r_{b})$.
Differentiating Eq. (20) and setting $z=0$, gives

$\left(\partial\varphi/\partial z\right)_{z=0}=-\sum_{n}A_{n}\mu_{n}F_{n}(r).$
(21)

The coefficients $A_{n}$ of the expansion Eq. (20) can be calculated as

$A_{n}=-\frac{1}{\mu_{n}\|F_{n}\|^{2}}\intop_{r_{a}}^{r_{b}}\left(\frac{\partial\varphi}{\partial
z}\right)_{z=0}rF_{n}(r)dr,$ (22)

where $\|F_{n}\|=\sqrt{\intop_{R_{in}}^{R_{ext}}F_{n}^{2}(r)rdr}$ is the norm
of the eigenfunction.

The electric field ${\bf E}$ at the surface of charged layer is directed along
the $z$-axis and is equal to $2\pi\sigma$, where $\sigma$ is the layer surface
charge density. Thus,

$E_{z}|_{z=0}=-\left(\partial\varphi/\partial z\right)_{z=0}=2\pi\sigma.$

Assuming, that the layer is charged homogeneously, one can rewrite Eq. (22) as

$A_{n}=\frac{2\pi\sigma}{\mu_{n}\|F_{n}\|^{2}}\intop_{r_{a}}^{r_{b}}rF_{n}(r)dr.$

Expression (20) acquires the form

$\varphi(r,z)=2\pi\sigma\sum_{n}\mu_{n}^{-1}\|F_{n}\|^{-2}e^{-\mu_{n}z}F_{n}(r)\intop_{r_{a}}^{r_{b}}rF_{n}(r)dr.$
(23)

Accordingly, the axial electric field reads as

$E_{z}(r,z)={\rm
sign}(z)2\pi\sigma\sum_{n}\|F_{n}\|^{-2}e^{-\mu_{n}|z|}F_{n}(r)\intop_{r_{a}}^{r_{b}}rF_{n}(r)dr,$
(24)

where ${\rm sign}(z)=1\,{\rm or}\,-1$ for $z>0$ and $z<0$, respectively.

Let the radial thickness $d$ of the electron cloud be small compared to the
external cylinder radius $R_{ext}$. For $r_{a}=0$ and no central rod, the
cloud cross section is a solid disk. Consider the disk radius $r_{b}$ be
small, $r_{b}\ll R_{ext}$. Then, assuming that the electron density $n_{e}$ is
homogeneous in the transverse plane, the radial dependence of the electric
field, $E_{z}$, which governs the axial motion of the electrons comprising the
layer $dz$, can be neglected and the problem reduces to the one-dimensional.

Following Tien et al.[20], let us replace $dz$-thick slices of the electron
cloud by charged rings (or disks) with the same total charge so that
$\sigma=en_{e}dz$. In the model of rigid slices, the acting electric field
$E_{z}(r,z)$ should be replaced by its average over the slice surface area

$\bar{E}_{z}(z)=\frac{2}{r_{b}^{2}-r_{a}^{2}}\intop_{r_{a}}^{r_{b}}E_{z}(r,z)rdr.$

Thus,

$\bar{E}_{z}(r,z)={\rm
sign}(z)2\pi\sigma\left\\{\frac{2}{r_{b}^{2}-r_{a}^{2}}\sum_{n}e^{-\mu_{n}|z|}\left[\intop_{r_{a}}^{r_{b}}F_{n}(r)rdr\right]^{2}\Biggm{/}\intop_{R_{in}}^{R_{ext}}F_{n}^{2}(r)rdr\right\\},$
(25)

Equation (25) is valid for all three configurations of the cloud: (i) annular
cloud between the tube and the central electrode, (ii) angular cloud in the
tube and no central rod, and (iii) solid cloud of radius $r_{b}$ in the
cylindrical tube. The only difference between these configurations is the
explicit form of the eigenfunctions and corresponding eigenvalues.

####  VI.1.1 Annular Electron Cloud in the Presence of a Central Rod

As a set of the eigenfunction, one can choose

$F_{n}(r)=N_{0}(\nu_{n}\rho_{in})J_{0}(\nu_{n}\rho)-J_{0}(\nu_{n}\rho_{in})N_{0}(\nu_{n}\rho),$
(26)

where $\nu_{n}\equiv\mu_{n}R_{ext}$ are the roots of the equation

$N_{0}(\nu_{n}\rho_{in})J_{0}(\nu_{n})-J_{0}(\nu_{n}\rho_{in})N_{0}(\nu_{n})=0.$
(27)

Here $J_{0}(x)$ and $N_{0}(x)$ are zeroth order Bessel and Neumann functions,
$\rho=r/R_{ext}$, and $\rho_{in}=R_{in}/R_{ext}$. The integrals in the
numerator and denominator of the expression in Eq. (25) can be evaluated as

$\displaystyle\intop_{r_{a}}^{r_{b}}F_{n}(r)rdr=\left\\{N_{0}(\nu_{n}\rho_{in})\left[\rho_{b}J_{1}(\nu_{n}\rho_{b})-\rho_{a}J_{1}(\nu_{n}\rho_{a})\right]\right.$
$\displaystyle-\left.J_{0}(\nu_{n}\rho_{in})\left[\rho_{b}N_{1}(\nu_{n}\rho_{b})-\rho_{a}N_{1}(\nu_{n}\rho_{a})\right]\right\\}R_{ext}^{2}/\nu_{n}\equiv
g_{n}R_{ext}^{2}/\nu_{n},$ (28)
$\intop_{R_{in}}^{R_{ext}}F_{n}^{2}(r)rdr=\frac{2R_{ext}^{2}}{\pi^{2}\nu_{n}^{2}}\left[\frac{J_{0}^{2}(\nu_{n}\rho_{in})}{J_{0}^{2}(\nu_{n})}-1)\right]\equiv\frac{2R_{ext}^{2}}{\pi^{2}\nu_{n}^{2}}f_{n}.$
(29)

Here $\rho_{a,b}=r_{a,b}/R_{ext}$. Finally, the expression for the axial
component of the electric field $\bar{E}_{z}(z^{\prime},z^{\prime\prime})$
produced in the cross section $z=z^{\prime}$ by the charged ring located at
$z=z^{\prime\prime}$ can be written as

$\bar{E}_{z}(z^{\prime},z^{\prime\prime})=2\pi\sigma{\rm
sign}(z^{\prime}-z^{\prime\prime})G(|z^{\prime}-z^{\prime\prime}|),$ (30)

where

$G(z)=\frac{\pi^{2}}{\rho_{b}^{2}-\rho_{a}^{2}}\sum_{n}e^{-\nu_{n}|z|/R_{ext}}g_{n}^{2}/f_{n}$
(31)

####  VI.1.2 Annular Electron Cloud – No Central Rod

For this configuration the eigenfunctions are
$F_{n}(r)=J_{0}(\nu_{n}r/R_{ext})$, where $\nu_{n}$ are the zeros of the
Bessel function $J_{0}(x)$. The electric field
$\bar{E}_{z}(z^{\prime},z^{\prime\prime})$ can be written in the same form as
in Eq. (30) but with different functions $G$, $g$ and $f$, given by

$G(z)=\frac{4}{\rho_{b}^{2}-\rho_{a}^{2}}\sum_{n}e^{-\nu_{n}|z|/R_{ext}}\frac{g_{n}^{2}}{\nu_{n}^{2}f_{n}},$
(32)

where

$g_{n}=\rho_{b}J_{1}(\nu_{n}\rho_{b})-\rho_{a}J_{1}(\nu_{n}\rho_{a})\hskip
8.53581pt{\rm and}\hskip 8.53581ptf_{n}=J_{1}^{2}(\nu_{n}).$ (33)

####  VI.1.3 Solid Cylindrical Electron Cloud

The expression for the electric field of a solid electron cloud of radius
$r_{b}$ ($r_{a}=0$) follows from Eq. (33):

$G(z)=4\sum_{n}e^{-\nu_{n}|z|/R_{ext}}\frac{g_{n}^{2}}{\nu_{n}^{2}f_{n}},$
(34)

where

$g_{n}=J_{1}(\nu_{n}\rho_{b})\hskip 8.53581pt{\rm and}\hskip
8.53581ptf_{n}=J_{1}^{2}(\nu_{n}).$ (35)

####  VI.1.4 Approximation by Exponent

It was shown in [20, 21, 22] that the function $G(z)$ for a solid cylindrical
electron cloud can be approximated by the exponential

$G(z)\simeq\exp(-|z|/\ell_{C}),$ (36)

where the characteristic scale of the Coulomb field $\ell_{C}\simeq r_{b}/2$
and its value is practically independent of the ratio $r_{b}/R_{ext}=\rho_{b}$
for $\rho_{b}\leq 0.3$. Annular clouds, either in the presence of a central
rod or not, are characterized by additional of geometrical parameters.
Nevertheless, the exponential approximation Eq. (36) is still applicable.
Note, that this is valid far from the potential well’s edges. The Coulomb
field of the charged plane slice is not symmetrical with respect to the slice
position when the distance between the boundary electrodes and the slice is
compared with $\ell_{C}$. This asymmetry can be properly considered, but its
role is small when the distance between these electrodes is large as compared
with $\ell_{C}$.

###  VI.2 Equation of Motion and the Steady-State Electron Density

As mentioned above, the electron cloud can be represented as a set of charged
rigid rings/disks with surface charge density $\sigma=en_{e}dz$. The equation
of motion of the rings/disks in the external electric field and the self-
consistent Coulomb field of the electron space charge can be written as [11]

$\frac{d^{2}z}{dt^{2}}=-\frac{e}{m}\frac{\partial\varphi}{\partial
z}+2\pi\frac{e^{2}}{m}\intop_{0}^{L}dz^{\prime}n_{e}(z^{\prime}){\rm
sign}(z-z^{\prime})G(|z-z^{\prime}|).$ (37)

Here $\varphi$ is the external potential and $L$ is the system length.
Assuming, that the characteristic spatial scale of the variation of the
external potential $\varphi$ and the electron density $n_{e}$ is large
compared to $\ell_{C}$, one can evaluate the integral in Eq. (37) as

$\intop_{0}^{L}dz^{\prime}n_{e}(z^{\prime}){\rm
sign}(z-z^{\prime})G(|z-z^{\prime}|)\simeq\intop_{0}^{L}dz^{\prime}n_{e}(z^{\prime}){\rm
sign}(z-z^{\prime})e^{(|z-z^{\prime}|)/\ell_{C}}\simeq
2\frac{dn_{e}(z)}{dz}\ell_{C}^{2}.$ (38)

It follows from Eq. (38) that the steady-state solution exists for

$n_{e}(z)=\frac{\varphi(z)}{4\pi e\ell_{C}^{2}}\simeq 5.7\cdot
10^{8}\frac{\varphi(z)[kV]}{\ell_{C}^{2}[cm]}[cm^{-3}].$ (39)

Note, that $\varphi(z)$ is the potential profile along the electron
propagation path, i.e., $\varphi(z)=\varphi(z,r)_{r={\rm const}}$.

The steady-state density of a solid circular electron cloud of radius $r_{d}$
is

$n_{e}(z)\simeq 2.3\cdot 10^{9}\frac{\varphi(z)[kV]}{r_{d}^{2}[cm]}[cm^{-3}].$
(40)

The dimensions of the conductors and the electron cloud in the experiment
described in Section III, are $r_{a}=0.7$cm, $r_{b}=0.9$cm, $R_{in}=0.3$cm,
and $R_{ext}=2.0$ cm. The function $G$, defined by Eq. (31), and its
exponential fit are shown in Fig. 14 as functions of the dimensionless
distance $\xi=z/\sqrt{r_{b}^{2}-r_{a}^{2}}$. The characteristic scale of
Coulomb field for these parameters is $\ell_{C}\simeq 0.56$cm. Note that
$\ell_{C}$ remains the same when no central electrode is present.

Figure 14: Annular electron cloud including the central electrode. Exponential
fit of function $G(\xi)$, $\xi=z/\sqrt{r_{b}^{2}-r_{a}^{2}}$.

## References

  * [1] R.C. Davidson, Physics of Nonneutral Plasmas, 2nd Ed. (World Scientific Publishing, Singapore, 2001).
  * [2] T.M. O’Neil, Physics Today 52, 24 (1999).
  * [3] K. Blaum, Yu.N. Novikovb, and G. Werth, Contemp. Phys. 51, 149 (2010).
  * [4] D.H.E. Dubin and T.M. O’Neil, Rev. Mod. Phys. 71, 87 (1999).
  * [5] J. Pétri, Astronomy & Astrophysics 478, 31 (2008).
  * [6] S. Sturm, G. Werth, and K. Blaum, Ann. Phys. (Berlin) 525, 620 (2013).
  * [7] K. Katagiri, K. Kawamura, T. Wakui, M. Muramatsu, and N. Hayashizaki, Nuclear Inst. and Methods in Phys. Res. A 987, 164837 (2021).
  * [8] F.M. Penning, Physica 3, 873 (1936).
  * [9] J.S. deGrassie and J.H. Malmberg, Phys. Rev. Lett. 39, 1077 (1977).
  * [10] A.M. Ignatov and V.P. Tarakanov, Phys. Plasmas 1, 741 (1994).
  * [11] Y. Bliokh, J.G. Leopold, and Ya.E. Krasik, Phys. Plasmas 28, 072106 (2021).
  * [12] A.E. Dubinov and V.P. Tarakanov, Tech. Phys. 65, 1002 (2020).
  * [13] R.H. Levi, Phys. Fluids 8, 1288 (1965).
  * [14] S. Riyopoulos, Phys. Plasmas 6, 323 (1999).
  * [15] Ya.E. Krasik, J.G. Leopold, Y. Hadas, Y. Cao, S. Gleizer, E. Flyat, Y.P. Bliokh, D. Andreev, A. Kuskov, and E. Schamiloglu, J. Appl. Phys. 131, 023301 (2022).
  * [16] N.S. Frolov, A.A. Koronovskii, and A.E. Hramov, Bull. Rus. Acad. Sci.: Physics 81, 27 (2017).
  * [17] J.G. Leopold, Ya.E. Krasik, Y.P. Bliokh, and E. Schamiloglu, Phys. Plasmas 27, 103102 (2020).
  * [18] S.P. Bugaev, A.A. Kim, A.I. Klimov and V.I. Koshelev, Zh. Tech. Fiz., v. 50, 2463 (1980) (in Russian).
  * [19] W.R. Smythe, Static and Dynamic Electricity, (Taylor & Francis, Boca Raton, FL, 1989).
  * [20] P.K. Tien, L.R. Walker, and V.M. Wolontist, Proc. IRE 43, 260 (1955).
  * [21] J.E. Rowe, Nonlinear Electron Wave Interaction Phenomena (Academic Press, New York, NY, 1965).
  * [22] I.J. Morey and C.K. Birdsall, IEEE Trans. Plasma Sci. 18, 482 (1990).
  * [23] G. Rosental, G. Dimonte, and A.Y. Wong, Phys. Fluids 30, 3257 (1987).
  * [24] G. Bettega, F. Cavaliere, B. Paroli, R. Pozzoli, M. Romé, and M. Cavenago, Phys. Plasmas 15, 032102 (2008).
  * [25] R.C. Davidson and G.M. Felice, Phys. Plasmas 5, 3497 (1998).