# What does BERT learn about prosody?

Sofoklis Kakourosa and Johannah O’Mahonyb

###### Abstract

Language models have become nearly ubiquitous in natural language processing
applications achieving state-of-the-art results in many tasks including
prosody. As the model design does not define predetermined linguistic targets
during training but rather aims at learning generalized representations of the
language, analyzing and interpreting the representations that models
implicitly capture is important in bridging the gap between interpretability
and model performance. Several studies have explored the linguistic
information that models capture providing some insights on their
representational capacity. However, the current studies have not explored
whether prosody is part of the structural information of the language that
models learn. In this work, we perform a series of experiments on BERT probing
the representations captured at different layers. Our results show that
information about prosodic prominence spans across many layers but is mostly
focused in middle layers suggesting that BERT relies mostly on syntactic and
semantic information.

Keywords: language model, BERT, prosody, prominence, part-of-speech

## 1\. Introduction

Pre-trained language models (LMs) have achieved high performance on several
natural language processing (NLP) tasks such as constituency parsing [1],
semantic role labeling, and coreference resolution [2]. Recently, prosody and
prosodic phenomena have witnessed increasing attention in natural language
applications enabled by pre-trained LMs leading to improved performance [3,
4]. Despite the gains, we do not yet fully understand what enables these
models to perform at this level. Their performance suggests that their
learning objectives potentially teach the models details about the structure
of the language. It remains, however, unclear, what specific prosodic
information these models are able to implicitly capture during their pre-
training. In this work, we focus on analyzing the prosodic information in one
of the most widespread models, BERT (Bidirectional Encoder Representations
from Transformers) [5].

Several recent works have investigated the representations learned at
different layers of BERT in an attempt to interpret and understand the
linguistic information captured by the model. These studies have uncovered
that BERT indeed learns various aspects of the language. For instance, [6]
showed that BERT layers capture a rich hierarchy of linguistic information
spanning from surface features in the lower layers to syntactic in middle
layers and semantic in the higher layers. These findings are further supported
by a number of other works with the general observation indicating that
learned representations vary with increasing network depth, with greater depth
typically involving linguistic functions that require larger contextual
relationships across the word tokens [2, 7, 8]. However, to the best of our
knowledge, BERT has not been examined with respect to its prosodic
information.

In general, prosody can be viewed as the characteristics in an utterance that
extend individual phonetic segments and encapsulate phonetic and phonological
properties that are not due to the choice of individual lexical items, but
depend on factors such as their semantic and syntactic relations [9, 10].
These characteristics convey information about the meaning and structure of an
utterance. Although prosody is a characteristic of spoken language, the
prosodic patterns in speech are connected and interact with their associated
sequences of syllables, words, and phrases. Therefore, it is meaningful to
assume that some aspects of the prosodic variation can be captured from text
alone [11, 3].

In this work, we investigate how prosodic information is linguistically
encoded by probing BERT with respect to the prosodic phenomenon of prominence.
Prosodic prominence is defined as the subjective impression of a linguistic
unit standing out of its context [12, 13, 14]. Given the recent success of
BERT in predicting prosodic prominence [3], we attempt to answer the question
of what BERT learns about prosody during its pre-training. Does the model rely
on general linguistic and syntactic knowledge for its prosodic predictions or
does it have a different pattern of weight allocations across its layers
suggesting that the model can capture prosodic information?

We use three datasets with prominence annotations and examine the weights at
different layers and compare them with existing findings from the literature
on other tasks. To further validate our approach with the literature we also
extract part-of-speech tags for our data and examine the learned layer
weights. The code to reproduce the results is publicly available at
github.com/skakouros/bert-prosody. In the next we describe the BERT model
architecture, related work, experimental methodology and results.

## 2\. BERT

BERT [5] is a language model based on the Transformer architecture [15] that
enables the bidirectional pre-training of representations by jointly
conditioning on the left and right context in all layers. This allows the
model to learn the entire surrounding context of a word which also means that
the same word in different context will have a distinct representation. In
contrast, earlier approaches looked at text sequences from left to right or by
combining left to right and right to left training. The BERT representations
are optimized based on two training objectives: (i) predicting randomly masked
words in the input, and (ii) predicting whether the next sentence is the
subsequent sentence in the input.

Our experiments are based on the bert-base-uncased variant of BERT. The model
consists of 12 layers, each with an embedding size of 768, and 12 attention
heads.

## 3\. Related Work

Probing network layers to investigate the structural knowledge of the language
that a model has captured is an active research area that falls under the
topic of neural network interpretability. In recent years there has been an
increasing number of studies examining the representations that language
models learn. Some works use probing tasks to unveil the linguistic features
encoded in neural models [6, 16], others use attribution methods such as
Integrated Gradients [17, 18] and some analyze Transformers’ attention heads
for evidence of linguistic and syntactic phenomena [19]. In this work, we
identify each layer’s contribution to a specific prosodic task by attaching
one trainable weight on each layer of BERT and training a light-weight
classification head on top of a frozen pre-trained BERT. This enables us to
observe the weight allocations across the layers and compare them with
existing findings.

## 4\. Layer-wise Analysis

### 4.1. Layer weights

We obtain the contribution of BERT layers to the prediction task by
introducing learnable scalar weights attached to each transformer layer of the
model. An overview of the experimental setup can be seen in Fig. 1. We take
representations from all transformer layers in the model and collapse them to
one via a weighted average. There is one weight for each layer (a total of
$L+1$) and all weights are trained jointly with the classification network.
The weights are implemented with a learnable vector of size $L+1$, followed by
the Softmax function.

### 4.2. Layer embeddings

To further probe the contribution of individual BERT layers, we extract the
embeddings from each layer separately and use them to train a classification
head consisting of a single dense layer followed by the Softmax function. For
each embedding layer, we then obtain the classification accuracy for the task.

Figure 1: Overview of the experimental setup.

## 5\. Experiments

In our experiments we use three datasets: two consisting of read speech and
one of spontaneous dialogue speech. These are presented next followed by a
description of the experimental setup.

### 5.1. Data

#### 5.1.1. BURNC

The Boston University Radio News Corpus (BURNC) is a corpus of professionally
read news data in American English [20]. The corpus consists of speech from
seven speakers (three female). The corpus also contains phonetic alignments,
orthographic transcriptions, part-of-speech tags, and prosodic labels. In this
work we use the text prompts and prosodic labels from the manually labelled
part of the corpus (six speakers; approximately 3 h of data). The prosodic
labeling system in BURNC is based on the Tones and Breaks Indices (ToBI)
labeling convention and includes prosodic phrasing, phrasal prominence, and
boundary tones. To obtain a single prominence label, all ToBI pitch accent
types (e.g., H*,L*,L*+H) were marked as prominent while the rest as non-
prominent.

#### 5.1.2. NXT Switchboard

The NXT Switchboard corpus is a dataset that includes the original Switchboard
corpus annotations [21] into one coherent integrated format (NITE XML; NXT)
enriched with annotations of prosody and contrast as well as syllable and
phone information [22]. The prosody annotations are available for a subset of
the data that includes 76 conversations labelled with the ToBI transcribing
convention. Similar to BURNC dataset, we use a binary prominence distinction
marking words with ToBI accents as prominent and the rest as non-prominent.

Figure 2: BERT layer weights for prominence (top) and POS (bottom) prediction.

#### 5.1.3. LibriTTS

The LibriTTS corpus [23] is a processed (automatically aligned, segmented, and
filtered) subset of the original audio and text data of the LibriSpeech corpus
[24] that is based on English audiobooks of the LibriVox project. From the
corpus we use the clean subset that consists of 262.5 hours of read speech
from 1230 speakers that were subsequently automatically labelled for
prominence in the Helsinki Prosody Corpus (HPC) [3]. From the HPC data, we use
the binary prominence tags.

### 5.2. Experimental Setup

To obtain the layer weights, we train the network (only the classification
head and layer weights) with frozen pre-trained BERT for prominence and POS
prediction for each dataset separately. For the training we use a 80-15-5
split for train, validation, and test. We run the training for 20 epochs with
a batch size of 4 and we repeat each experiment five times. The results are
averaged across the independent runs for each task. The network is configured
with a learning rate of $5e-5$ and a different learning rate $1e-2$ for the
layers weights. This was done in order to allow the model to adjust the
weights for the different layers more rapidly. We extract the weights from the
model checkpoint with the best development accuracy and average them over the
five runs. We repeat the same procedure and setup for POS prediction.

In addition to the frozen pre-trained BERT we also fine-tune the entire model
and report the results in Table 2 to compare the overall model performance
with both fine-tuned and frozen BERT models. We use the same learning rates
and epochs as the previous experiment with frozen BERT.

For POS prediction we used Spacy [25] to extract the part of speech
categories. This resulted into 17 discrete classes. POS classes include, for
example, adjectives, adpositions, adverbs (see [25] for a complete list of the
coarse POS categories).

Finally, to balance the data and enable better comparison, each dataset is
post-processed to include one full sentence per sample. As BURNC may include
an entire paragraph and Switchboard several sentences per dialogue turn within
a sample, we explicitly set sample size to be one sentence. Thus, a batch size
of four includes four sentences.

## 6\. Results and Discussion

We report overall model performance for POS and prominence prediction in Table
2, layer-specific performance in Table 1, and illustrate how layer weights
vary with respect to different tasks in Fig. 2. We are interested in examining
how different layers contribute to prominence and POS prediction with respect
to findings on other tasks that have indicated different linguistic functions
associated with different layers. Overall, for prominence we observe
widespread distribution of the layer weights while POS appears more focused in
the earlier BERT layers. These are presented in more detail in the next
sections.

### 6.1. Prosodic Prominence

Table 1: Layer-wise accuracy for the test set runs for prominence and POS prediction with frozen BERT. Numbers in bold denote the two top results in each task. Layer | BURNC | SWBD | LibriTTS
---|---|---|---
Prom | POS | Prom | POS | Prom | POS
0 | 81.51 | 93.84 | 75.32 | 94.62 | 80.18 | 89.50
1 | 83.52 | 93.90 | 76.14 | 94.67 | 80.32 | 90.08
2 | 85.20 | 94.33 | 76.01 | 94.02 | 80.29 | 89.53
3 | 85.20 | 93.84 | 75.47 | 93.50 | 81.08 | 89.00
4 | 85.26 | 93.47 | 75.25 | 92.52 | 80.28 | 88.40
5 | 85.26 | 93.17 | 75.82 | 91.87 | 80.00 | 87.29
6 | 84.13 | 92.98 | 75.71 | 90.95 | 79.71 | 86.12
7 | 84.46 | 91.76 | 75.82 | 90.11 | 79.40 | 84.86
8 | 83.79 | 91.03 | 75.45 | 88.37 | 79.05 | 83.48
9 | 82.92 | 89.20 | 74.71 | 87.20 | 78.63 | 82.36
10 | 82.59 | 88.47 | 74.29 | 86.64 | 78.32 | 81.14
11 | 83.26 | 87.19 | 73.08 | 82.09 | 77.27 | 76.67

For prosodic prominence, the three datasets tested have shown differences in
their overall performance. For example, for frozen BERT, prominence prediction
accuracy was 87.14% for BURNC, 78.10% for Switchboard, and 82.67% for
LibriTTS. These differences are likely due to the different speaking styles
involved in the data, with dialogue speech having the lowest performance.
Another interesting observation in the results is that the model performance
is degraded for the fine-tuned models. It seems that when fine-tuning the
model, generalizability decreases due to overfitting on the idiosyncrasies of
the training data. With the frozen model, performance is high and on a par
with results reported in the literature. For instance, for LibriTTS [3] report
83.20% and in our setup we obtained 82.67%.

When it comes to layer weights, prosodic prominence has a widespread pattern
of weights across BERT layers. Weights span most layers and are primarily
focused within layers 2-8 with a peak appearing at layer 3. Interestingly, the
pattern of weights appears to be quite similar across the three datasets
tested. The spread of the weights suggests that different types of linguistic
information are used in the prediction of prominent tokens. One interpretation
of the results is that the model relies greatly on surface linguistic features
such as POS but also involves syntactic and semantic information with
increasing layers (see also [6]). For BURNC, we also observe that the third
layer has a high weight for both prominence and POS prediction. It is possible
that prominence in BURNC (professionaly read speech) correlated more with POS
than audiobooks (LibriTTS) and dialogue speech (SWBD). Another possibility for
these differences could be also attributed to the different prominence coding
schemes in the three datasets.

### 6.2. POS

Part-of-speech information seems to be encoded in the early BERT layers with
model accuracy being high for both frozen and fine-tuned runs of the
experiments and across all datasets tested. Fine-tuning the model leads to
improved performance, where for BURNC we get an increase in accuracy from
95.97% with the frozen model to 97.56% with fine-tuning. Switcboard and
LibriTTS perform similarly with an increase in performance when the model is
fine-tuned.

Layer weights for POS demonstrate a very different pattern when compared to
prominence. POS information is found mainly in the lower layers of BERT with
weights across the datasets varying but being focused in the early layers of
the model. Most of the information seems to come from layers 0-4 which have
been shown to encode surface features [6]. This finding is also in agreement
with other work that shows maximum POS tagging performance in the lower layers
of models with accuracies ranging from 97.2% to 97.4% [7, 2].

Table 2: Accuracy for the test set runs for prominence and POS prediction with frozen and fine-tuned (ft) BERT. Prominence | BURNC | SWBD | LibriTTS
---|---|---|---
frozen | 87.14 | 78.10 | 82.67
ft | 85.53 | 75.95 | 80.32
POS |  |  |
frozen | 95.97 | 97.94 | 98.49
ft | 97.56 | 98.54 | 98.96

## 7\. Conclusions

In this work, we performed a series of experiments on prosodic prominence to
investigate whether prosody is part of the structural information of the
language that BERT learns. Our results show that BERT captures information
about prosodic prominence through a widespread allocation of weights across
its layers reaching high performance. The weight allocations suggest that BERT
relies on a variety of linguistic information for its predictions including
surface linguistic features such as POS but also involving syntactic and
semantic information. In future work, we will explore the same tasks with an
extended set of datasets and methodological approaches. In addition to layer
weights, we want to include layer integrated gradients as an attribution
method in the experiments. We also aim to examine differences between the
styles in the datasets, that is, read versus dialogue speech.

## 8\. Acknowledgements

This work was supported by the Academy of Finland project no. 340125
"Computational Modeling of Prosody in Speech" and Horizon 2020 Marie
Sklodowska-Curie grant agreement No 859588. The authors wish to acknowledge
CSC - IT Center for Science, Finland, for providing the computational
resources.

## References

  * [1] N. Kitaev and D. Klein, “Constituency parsing with a self-attentive encoder,” _arXiv preprint arXiv:1805.01052_ , 2018.
  * [2] M. E. Peters, M. Neumann, M. Iyyer, M. Gardner, C. Clark, K. Lee, and L. Zettlemoyer, “Deep contextualized word representations,” in _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)_. New Orleans, Louisiana: Association for Computational Linguistics, Jun. 2018, pp. 2227–2237. [Online]. Available: https://aclanthology.org/N18-1202
  * [3] A. Talman, A. Suni, H. Celikkanat, S. Kakouros, J. Tiedemann, and M. Vainio, “Predicting prosodic prominence from text with pre-trained contextualized word representations,” in _Proceedings of NoDaLiDa_ , 2019.
  * [4] B. Stephenson, L. Besacier, L. Girin, and T. Hueber, “Bert, can he predict contrastive focus? predicting and controlling prominence in neural tts using a language model,” _arXiv preprint arXiv:2207.01718_ , 2022.
  * [5] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” _arXiv preprint arXiv:1810.04805_ , 2018.
  * [6] G. Jawahar, B. Sagot, and D. Seddah, “What does bert learn about the structure of language?” in _ACL 2019-57th Annual Meeting of the Association for Computational Linguistics_ , 2019.
  * [7] M. E. Peters, M. Neumann, L. Zettlemoyer, and W.-t. Yih, “Dissecting contextual word embeddings: Architecture and representation,” _arXiv preprint arXiv:1808.08949_ , 2018.
  * [8] I. Tenney, D. Das, and E. Pavlick, “Bert rediscovers the classical nlp pipeline,” _arXiv preprint arXiv:1905.05950_ , 2019.
  * [9] M. Wagner and D. G. Watson, “Experimental and theoretical advances in prosody: A review,” _Language and cognitive processes_ , vol. 25, no. 7-9, pp. 905–945, 2010.
  * [10] S. Kakouros and O. Räsänen, “Perception of sentence stress in speech correlates with the temporal unpredictability of prosodic features,” _Cognitive science_ , vol. 40, no. 7, pp. 1739–1774, 2016.
  * [11] S. Kakouros, J. Pelemans, L. Verwimp, P. Wambacq, and O. Räsänen, “Analyzing the contribution of top-down lexical and bottom-up acoustic cues in the detection of sentence prominence,” _Proceedings Interspeech 2016_ , vol. 8, pp. 1074–1078, 2016.
  * [12] S. Kakouros and O. Räsänen, “3pro–an unsupervised method for the automatic detection of sentence prominence in speech,” _Speech Communication_ , vol. 82, pp. 67–84, 2016.
  * [13] J. Terken and D. Hermes, “The perception of prosodic prominence,” in _Prosody: Theory and experiment_. Springer, 2000, pp. 89–127.
  * [14] S. Kakouros and O. Räsänen, “Automatic detection of sentence prominence in speech using predictability of word-level acoustic features,” in _Sixteenth Annual Conference of the International Speech Communication Association_ , 2015.
  * [15] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [16] I. Tenney, P. Xia, B. Chen, A. Wang, A. Poliak, R. T. McCoy, N. Kim, B. Van Durme, S. R. Bowman, D. Das _et al._ , “What do you learn from context? probing for sentence structure in contextualized word representations,” _arXiv preprint arXiv:1905.06316_ , 2019.
  * [17] S. Ramnath, P. Nema, D. Sahni, and M. M. Khapra, “Towards interpreting bert for reading comprehension based qa,” _arXiv preprint arXiv:2010.08983_ , 2020\.
  * [18] A. Nayak and H. P. Timmapathini, “Using integrated gradients and constituency parse trees to explain linguistic acceptability learnt by bert,” in _Proceedings of the 18th International Conference on Natural Language Processing (ICON)_ , 2021, pp. 80–85.
  * [19] K. Clark, U. Khandelwal, O. Levy, and C. D. Manning, “What does bert look at? an analysis of bert’s attention,” _arXiv preprint arXiv:1906.04341_ , 2019\.
  * [20] M. Ostendorf, P. J. Price, and S. Shattuck-Hufnagel, “The boston university radio news corpus,” _Linguistic Data Consortium_ , pp. 1–19, 1995.
  * [21] J. J. Godfrey, E. C. Holliman, and J. McDaniel, “Switchboard: Telephone speech corpus for research and development,” in _Acoustics, Speech, and Signal Processing, IEEE International Conference on_ , vol. 1. IEEE Computer Society, 1992, pp. 517–520.
  * [22] S. Calhoun, J. Carletta, J. M. Brenier, N. Mayo, D. Jurafsky, M. Steedman, and D. Beaver, “The nxt-format switchboard corpus: a rich resource for investigating the syntax, semantics, pragmatics and prosody of dialogue,” _Language resources and evaluation_ , vol. 44, no. 4, pp. 387–419, 2010.
  * [23] H. Zen, V. Dang, R. Clark, Y. Zhang, R. J. Weiss, Y. Jia, Z. Chen, and Y. Wu, “Libritts: A corpus derived from librispeech for text-to-speech,” _arXiv preprint arXiv:1904.02882_ , 2019.
  * [24] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, “Librispeech: an asr corpus based on public domain audio books,” in _2015 IEEE international conference on acoustics, speech and signal processing (ICASSP)_. IEEE, 2015, pp. 5206–5210.
  * [25] M. Honnibal and I. Montani, “spaCy 3: Natural language understanding with Bloom embeddings, convolutional neural networks and incremental parsing,” 2017, to appear.


# Resonant Ion Confinement Fusion Concept

Jose Luis Rosales 1 and Francisco Castejón 2 1Centre for Computational
Simulation
DLSIIS ETS Ingenieros Informáticos, Universidad Politécnica de Madrid,
Campus Montegancedo, E28660 Madrid, Spain.
2 Laboratorio Nacional de Fusión, CIEMAT, Avenida Complutense 40 E28040 Madrid
Spain<EMAIL_ADDRESS>
###### Abstract

Based on the theorized possibilities of resonant ion confinement, for a
Deuteron cloud in a Penning-Malmberg trap with a specially configured rotating
wall, the opportunity to design a new type of fusion device is prospected. It
is proven that, for some trap configurations, nuclear fusion reactions should
take place and, in that case, Lawson’s criterion for an efficient fusion
reactor is met. Furthermore, the reactor could have a compact design and,
since it should not require a large facility, it can function as a fusion cell
with a pure ion thermal gas.

Keywords: Charged plasma confinement, Penning traps, Fusion reactors.

##  1 Introduction

For an efficient fusion reactor, the pressure $p\;$of the confined plasma
times the energy confinement time $\tau_{E}\;$should be greater than a given
amount $p\tau_{E}>L;\;$ which was a criterium due to Lawson (Ref. [1]). For
Deuterium - Deuterium reactions this value is of the order of one hundred
$atm\times s.\;$This reaction happens in the stars, where plasma is confined
by gravity. The advantage of considering Deuterium-Deuterium reactions is that
this is a stable and abundant isotope of Hydrogen on Earth, although the cross
section of these reactions is smaller than that of Deuterium-Tritium. Up to
date, the magnetic confinement of neutral plasmas has been thought as the most
promising concept to build a fusion reactor and two main concepts of devices
have been developed: tokamaks and stellarators. In tokamaks, the magnetic
field confines the particles and energy long enough for ignition to occur and
it is expected that such a reactor will generate enough energy to achieve
$Q>10,\;$ which will probably be shown through the ITER experiment in the near
future (see Ref. [2], also Ref. [3]). Nevertheless, tokamaks still have to
overcome some challenges, namely the possibility of suffering disruptions, the
existence of edge localised modes (ELMs) and the intrinsic pulsed working, on
top of the possible impurity accumulation, (as from Ref. [4]). Turbulence will
be also playing a major role degrading the Tokamak plasma confinement. These
problems made it necessary to explore other possible magnetic confinement
fusion concepts like the stellarator one (see e.g. Ref. [5]), characterised by
enabling a continuous working and by the absence of large ELMs and
disruptions, since there plasmas are almost currentless. Stellarator
confinement is one generation after the tokamak one, due to the fact that they
have to reduce their neoclassical transport by optimizing the magnetic
configuration (see e.g. Ref. [6] and references therein),as well as to solve
the confinement of fast particles and to demonstrate power exhaust by a
suitable divertor concept (see again Ref. [5]). Therefore, it is still
necessary to overcome difficulties in both concepts to make it economically
feasible for the production of electricity by fusion. Other topic to discuss
in the future would be the size of the reactors, since all the scalings show
an improvement of confinement with the size of the device (see e.g. Ref. [7]
and references therein). Thus, to address the problems encountered in the case
of magnetically confined neutral plasmas and, at the same time, to explore
alternative confinement methods that minimise the resources involved, another
possibility to achieve nuclear fusion is described in this work; the new
method is based on the concept of resonant ion confinement (see Ref. [8],
which can be produced inside a cylindrical Penning-Malmberg ion trap with a
Deuteron plasma in perfect thermal equilibrium). A remarkable feature of this
fusion concept is the possibility of building small reactors, which brings
with it completely different engineering challenges from those encountered in
the design of magnetic confinement reactors. Indeed, as we shall see, on the
grounds of these resonant ionic confinement fundamentals, a kind of compact
reactor, say a fusion cell, can be conceived with, predictably, energy yields
which, interestingly, are proportionally lower than those expected in ITER.
These fusion cell have, therefore, complementary purposes and scopes.

In this device ions are confined by applying an electric potential
difference,$V_{0},\;$and a magnetic axial field $\mathbf{B}.\;$The ion trap,
additionally, has a quadrupole electric field of a frequency, $\omega,\;$such
that it allows the Deuterium nuclei to rotate also with that same angular
velocity. In this system, a fraction of ions performs trajectories that should
bring them close enough to produce fusion reactions by tunnelling. The
confined ion cloud takes the shape of a highly flattened spheroid. Although
these characteristics are common to any Penning-Malmberg configuration, there
is a very precise set of values for the trap parameters that determines the
intensity of the quadrupole field for which resonant, coalescing paths, arise
among all the Deuterium ion trajectories, and, in this way, nuclear fusion
reactions will likely take place. Thus, the precise dependence of this
quadrupole electric field as a function of the electric field potential,
$V_{0},\;$ and of the magnetic field intensity, $\mathbf{B},\;$as well as the
radius of the trap cylinder, $R_{0},\;$will determine the resonance conditions
required for the equipment to produce a finite number of fusion reactions.
Finally, the pressure and confinement time necessary for ignition in the
centre of the ionic trap should be reached only for special configurations of
the confinement system parameters. What propose here, contrarily to other
possible candidates for fusion devices, is to build a device that confines
non-neutral plasmas in thermal equilibrium where, also, the ergodic theorem
for the average trajectories of the confined ions is satisfied.

##  2 Methods

### Resonant Ion Confinement.

Let us consider $N\;$Deuteron nuclei of mass $m\;$confined in a cylindrical
Penning-Malmberg trap of radius $R_{0};\;$we will assume perfect thermodynamic
equilibrium. Let us denote the applied electrostatic potential as $V_{0}\;$and
the axial magnetic intensity as $\mathbf{B}.\;$A rotating wall electric
quadrupolar field of intensity $\lambda=1/2+\delta\;$and angular frequency
$\omega\;$is then added which prompts the plasma cloud to rotate collectively
with the same rotating wall angular frequency $\small\omega.\;$The latter is
the guiding orbit magnetron frequency of the ion cloud. The charged plasma
remains confined in the central plane of the trap and the form of the confined
ion cloud is a spheroid of semi-major axis $R_{C}\;$and semi-minor axis
$z.\;$Using the fact that ions should be in equilibrium inside the confined
cloud, it is easy to see that the electric axial oscillations of ions should
be related to the volume density number,$n,\;$of this cloud as
$\small\omega_{z}=(ne^{2}/m\epsilon_{0})^{1/2}.\;$Moreover, the oscillator
axial frequency also depends on the electric potential $V_{0}\;$and the radius
of the trap cylinder, $R_{0},\;$as $\omega_{z}=(2/R_{0})\sqrt{eV_{0}/m}.\;$In
addition, ought to the applied axial magnetic field $\mathbf{B},\;$the ions
inside the confined spheroid have an internal fast cyclotron oscillation whose
frequency is defined in terms of $\mathbf{\Omega}=e\mathbf{B}/m.\;$Due to the
diamagnetic behaviour of charge currents inside the plasma cloud, the
achievable cyclotron frequency of the Deuterons is actually smaller:
$\Omega^{\prime}=\Omega-2\omega,\;$which this is called the vortex frequency.
On the other hand, dynamic equilibrium of the plasma imposes that
$\omega(\Omega-\omega)\rightarrow\omega_{z}^{2}/2.\;$This means that there
exists a key parameter $\vartheta\;$to define confinement quality of the
Penning-Malmberg trap such that
$\omega=\Omega\sin^{2}\frac{\vartheta}{2}\;$and
$\omega_{z}=\Omega\sin\vartheta/\sqrt{2}.\;$When $\omega_{z}\ll\Omega\;$the
aspect ratio satisfies the condition $\ell=z/R_{C}\ll 1\;$(see Ref. [9]). Ion
confinement is expected to be perfectly stable free of the magnetohydrodynamic
instabilities that happen in magnetic neutral plasma confinement. This is the
confinement theorem of charged plasmas. In special situations, there are other
kind of periodic solutions of the equations of motion (see Appendix). These
solutions represent the trajectories of two correlated confined nuclei that
follow coalescing paths to the centre of the trap, in the end this will allow
for their respectively trajectories to cross where they collide, and so D-D
fusion reactions ought to happen with high probability through quantum
tunnelling the Coulomb barrier. Yet, this effect will hold only at some given
rate, since most of the ions in the cloud will still follow circular magnetron
guiding orbits. Thus, the correlation only occurs for the special condition
between the quadrupole field strength parameter $\delta\;$and the trap
parameter $\vartheta\;$takes place, which is the ion resonant confinement
condition. To see this, recall that, in the rotating centre of mass frame of
every two ions, the trap field interaction Lagrangian is given in terms of the
relative distance $\varrho\;$between the correlated nuclei as
$L=\frac{1}{2}\mu\dot{\varrho}^{2}-U(\varrho,t),\;$the potential energy being
$U(\varrho,t)=\frac{1}{2}\mu\omega_{z}^{2}\varrho^{2}(\lambda\cos 2\omega
t-\frac{1}{2}),\;$where $\mu=m/2\;$stands for the reduced mass of the ions. It
is seen that the fast cyclotron motion frequency does not appears in
$U(\varrho,t).\;$The solution of the equation of motion of the two-correlated-
nuclei complex is given in terms of Mathieu functions. If $R_{C}\;$is the
maximum radius of the orbit, denoting,
$\chi=\cot^{2}(\vartheta/2)\rightarrow\frac{eB^{2}R_{0}^{2}}{4\mu
V_{0}}-2\;$(for $\vartheta\ll 1$ ), one gets,
$\varrho=\frac{2R_{C}}{q_{0}}\bi{C}_{e}[-\tiny\chi,-\lambda\chi,\omega
t\;],\;$ where $q_{0}\;$is a normalization constant. In general, these orbits
are exponentially unstable, yet numerical calculations show that there exists
a special, resonant,$\pi/\omega\;$periodic orbit whenever the quadrupolar
field strength $\delta$ takes the value, (see Appendix)

$\displaystyle\delta\equiv\lambda-\frac{1}{2}\rightarrow\frac{1}{\sqrt{2\chi}}.$
(1)

The minimum distance between the nuclei in these resonant orbits will
eventually be given by the following numerical solution

$\varrho_{0}=2R_{C}\kappa,\;\normalsize
with\small\;\kappa\simeq\exp\\{1/2-1.35\sqrt{\chi}\\},$ (2)

which can reach arbitrarily small values at any temperature of the confined
plasma in the limit $\vartheta\rightarrow 0.\;$Moreover, the kinetic energy
per Deuteron in this case becomes $E\rightarrow W/2$ where
$W=\frac{e^{2}}{4\pi\epsilon_{0}\varrho_{0}},\;$ which provides
$\varrho_{0}.\;$On the other hand, a charged plasma in thermodynamic
equilibrium at temperature $T\;$should behave as a solid rotor in a Penning-
Malmberg trap with a magnetron frequency oscillatory stroboscopic rotating
wall quadrupolar field (see Appendix). The plasma global angular velocity
$\omega\;$, coinciding with that of the individual ions guiding orbits,
satisfies the condition $\omega=1/R_{C}\sqrt{(k_{B}T/\mu)}\;$. Then, since
$\omega=\Omega(\chi+1)^{-1},\;$we get $\chi+1=\Omega
R_{C}\sqrt{k_{B}T/\mu}\;$which, owing to the dynamic internal equilibrium of
the plasma, it is equivalent to saying that $\chi\simeq
eV_{0}/(k_{B}T)\times(R_{C}/R_{0})^{2}.\;$Finally, using this thermodynamic
constraint and given that the axial degree of freedom is thermal, i.e.,
$k_{B}T=1/2\mu\omega_{z}^{2}z_{max}^{2},\;$we conclude that the aspect ratio
of the confined ion cloud minimal height spheroid is $z_{max}/R_{C}\simeq
1/\sqrt{\chi},\;$which is an important relation that will be used later.
Moreover the probability that two correlated ions follow the resonant orbits
is (see Appendix) $\wp(\chi)=\chi^{-1}.\;$Recall that in the resonant case,
all the macroscopic features of the Deuteron plasma can be written only in
terms of the microscopic parameter $W$ and of the trap confinement value for
$\chi.\;$Then, $R_{C}=e^{2}/(8\pi\epsilon_{0})W^{-1}\kappa^{-1}\;$and the
confined volume of the ion cloud becomes
$V\rightarrow\frac{4}{3}\pi(R_{C}/\sqrt{\chi})^{3}\times\chi,\;$ which is
coincident with the volume of $\chi$ ion clump spheres of radius
$R_{C}/\sqrt{\chi}.\;$The density number becomes
$n=\frac{\epsilon_{0}B^{2}}{4\mu}\sin^{2}\vartheta\rightarrow
4\omega^{2}\epsilon_{0}\mu\chi/e^{2}\;$ and the total number of confined
Deuterons $N\rightarrow V\times n.\;$

##  3 Results

### Fusion Conditions.

For Deuterium fusion, the occurring reactions are
${}^{2}H+^{2}H\to^{3}He+n;\;and\;\;^{2}H+^{2}H\to\;\;^{3}H\;+^{1}H\;$ while
their corresponding fusion energies are $E_{f1}[^{3}He+n]=3.27\;MeV$ and
$E_{f2}[^{3}H\;+^{1}H]=4.04\;MeV$. Each reaction takes place with
approximately $50\%\;$probability. Thus, together with the necessary Lawson
criterium, in a self-sustaining energy balanced fusion Deuterium device the
only heating terms shall be that from the kinetic energy of the confined
charged fusion reactions products. The kinetic energy contribution of the
neutrons, which cannot be confined in the trap, must be subtracted from the
nuclear energy balance, a fraction $\eta_{n}=E_{n}/(E_{f1}+E_{f2}).\;$The
neutron energy escapes directly to hit the reactor walls, where its energy can
be extracted by a breeding blanket. On top of that, other energy losses have
to be taken into account in the power balance analysis, namely the
Bremsstrahlung term due to accelerated charges in the plasma, as well as the
transport term, which is assumed to be due to ion-ion Coulomb collisions.
Positive power balance of the Deuteron plasma, then, requires

$P_{f}(1-\eta_{n})>P_{B}+P_{L},$ (3)

where $P_{f}\equiv J_{f}V\;$is the power due to fusion reactions that heats
the whole plasma volume $V,\;$$P_{B}\;$is the Bremsstrahlung term and
$P_{L}\;$is the power loss caused by transport, assumed collisional. Again,
let $E\;$be the energy of the Deuteron nuclei in the resonant orbit (this
correlates with the effective temperature of the nuclei inside the density
clusters) and let $n^{\prime}\;$be the density number of the plasma in the
resonant clusters, then the collision frequency is, (see Ref. [10])
$\nu(\chi,E)=n^{\prime}E^{-3/2}\frac{3e^{4}}{16\pi^{2}\epsilon_{0}^{2}m^{1/2}}\ln\Lambda(\chi,E).\;$
The Coulomb logarithm is
$\ln\Lambda(\chi,E)=\ln\big{\\{}12\pi\epsilon_{0}^{3/2}n^{\prime-1/2}e^{-3}E^{3/2}\big{\\}}.\;$Now,
we write $N^{\prime}=\wp(\chi)N\;$as the number of resonant nuclei in the
density clusters; then $V^{\prime}=\kappa V\;$is the volume of the density
clump because only one of the two major axis of the spheroid is compressed due
to the resonant orbits of the nuclei in it. Then every ion swirls around the
centre in a vortex from $r=R_{C}\;$to $r=\varrho_{0}/2\;$and, therefore, the
effective volume that affects all the resonant orbits can be approximated as
$V^{\prime}=4/3\pi R_{C}\big{\\{}\kappa R_{C}\big{\\}}z,\;$which will be the
volume available for fusion reactions in the centre of mass of the resonant
complex. Then, $n^{\prime}=\wp(\chi)/\kappa\times n.\;$Now, from these
definitions, the reactor effective power per unit of volume becomes

$\displaystyle J=J_{f}-\eta\bigg{\\{}n^{\prime
2}\alpha_{B}E^{\frac{1}{2}}+\frac{N^{\prime}}{V}E\nu(\chi,E)\bigg{\\}}$ (4)

where the fusion power density is
$J_{f}=\bigg{\\{}\frac{1}{2}E_{f_{1}}\langle\sigma
v\rangle_{1}+\frac{1}{2}E_{f_{2}}\langle\sigma
v\rangle_{2}\bigg{\\}}\frac{n^{\prime 2}}{2}.\;$ Notice that in Eq. (4), we
consider the total volume of the plasma in the collision term, instead of that
of the resonant cluster of ions; the reason is that we consider the most
unfavourable case in which the collisional transport is large enough to drive
the local inhomogeneities that appear in the resonant zone to the entire
plasma volume. In Eq. (4) $\eta\approx 1$ because, neglecting the ${}^{3}H$,
${}^{1}p\;$ and ${}^{3}He_{2}\;$concentrations, i.e., those of the nuclear
reaction products, there is only a single particle species in the confined
plasma. For long working period of the reactor, the concentrations of these
species could not be negligible and should be taken into account to develop a
possible exhaust device. The trap disconnection can be used for exhauteing
these particles in any case. In Eq. (4),
$\alpha_{B}=1.4\;10^{-40}\;(m_{e}/m_{p})^{3/2}\;\;[W/m^{3}K^{-1/2}]\;$ and,
for $E(keV)<300$, we can make the following analytical approximation for the
cross-section of the reaction: $\langle\sigma
v\rangle_{i}=b_{i}{E}^{\beta_{i}}\exp\\{c_{i}{E}^{\chi_{i}}\\}$
$b_{1}=1.198\times 10^{-18}$,$b_{2}=3.5501\times
10^{-19}$,$\beta_{1}=-1.0759$, $\beta_{2}=-0.9462$, $c_{1}=-23.511$,
$c_{2}=-22.04$, $\chi_{1}=0.29221$, $\chi_{2}=0.2922$. Using these values in
Eq. (4) one sees that the Bremsstrahlung and the collisional transport terms
have small effects on the power balance.

### Fusion Cells.

In order for the reactor to be energetically self-sustained, the generated
energy must be greater than the sum of the emitted heat plus the magnetic back
reaction pressure loss (as required from Brillouin’s theorem). Then, the
actual thermodynamically available power density is $J_{C}=J-\dot{W},\;$ where
$\dot{W}=\frac{b^{2}}{2\mu_{0}}\times\frac{\omega}{\pi}\;$ is the power loss
per unit of volume due to the diamagnetic currents of the confined ions
calculated for the resonance frequency. Recall that the contribution to the
pressure losses is written in terms of the plasma back reaction magnetic
field, namely,
$\mathbf{b}=-4(\mu/e)\mathbf{\omega}=-2\mathbf{B}(1+\chi)^{-1}.\;$Now, the
energy confinement time $\tau_{E}$ may be calculated in terms of the
temperature $T$ and the density number $n^{\prime}$ for the nuclei in the
resonance $\tau_{E}=n^{\prime}E\times J_{C}^{-1}.\;$Since the pressure is
$p=2/3n^{\prime}E,\;$then (see Ref. [10])

$\displaystyle p\tau_{E}=\frac{2}{3}n^{\prime 2}E^{2}/J_{C}.$ (5)

The function $p\tau_{E}$ should have a minimum value for some $\chi\;$and
$E=W/2.\;$As seen in Fig. 1, Lawson’s minimum is actually reached for
$\vartheta\leq 0.105,\;$and $W=30.7\;keV.\;$Indeed, this minimum is obtained
for approximately the same confinement parameters, independently of the actual
temperature of the confined plasma cloud. Thus $p\tau_{E}\sim 90\;atm\cdot
s\;$, which is expected to be experimentally achievable. Moreover, the value
of the Coulomb barrier energy for the Lawson minimum computed is remarkably
close to the expected experimental value (see Ref. ([10]), a fact that
supports the Resonant Ion Confinement reactor concept. Therefore, in order to
obtain the actual power efficiency of the reactor, recall that, since the
species ${}^{3}He\;$, $p\;$and ${}^{3}H\;$are positively charged, they will be
retained in the trap but, contrarily to that, the neutrons will escape from
the trap cavity and will be absorbed by the surrounding walls of the reactor
chamber, where a separate equipment to absorb their energy should be
installed. Their kinetic energy might be, then, transformed into heat (with an
efficiency $\eta_{h}\sim 1/3\;$). This heat can used to produce electricity by
means of high efficient thermoelectric materials.

Figure 1: Lawson triple product $p\tau_{E}\;$as a function of the Coulomb
barrier energy achievable for D-D collisions $W\;$in keV and the parameter
$\vartheta=\arcsin(\frac{\sqrt{2}\omega_{z}}{\Omega}).\;$As seen here, an
stable minimum does exist, $p\tau_{E}\sim 90\;atm\times s,\;$for
$\vartheta<0.105\;$and $W\sim 30.7\;keV.$

Moreover, we must also take into account that every $D-D\;$collision takes
place by quantum tunnelling the barrier $W.\;$The Gamow-Sommerfeld probability
of this transition needed to undergo the nuclear reaction is
$\eta_{f}(W)\sim\exp\\{-\pi\alpha c\sqrt{2\mu/W}\\},\;$again, the Deuteron
reduced mass is $\mu=m/2.\;$It gives, for the Lawson minimum $W\simeq
30.7\;keV,\;$$\eta_{f}\simeq 1/300.\;$Having this in mind, the following
estimate for the electric power of the reactor can be provided
$P_{f}\sim\eta_{n}\eta_{h}J_{C}\times
V\rightarrow\eta_{n}\eta_{h}J_{C}\times\\{\mathcal{Q}_{t}V^{\prime}\\},\;$
where we have used the fact that, for the reactants in the cell, their actual
available volume, $V,\;$must be calculated considering that the number of
reactions in it must coincide, in average, with the sum of the ”Quantum
tunneling” configurations, $\mathcal{Q}_{t}$, all over the time, at some given
space point; if this particular point is selected as the centre of the cell,
we assume that $V^{\prime}=\kappa V$. Moreover, the quantity
$\mathcal{Q}_{t}\;$can be estimated as
$\mathcal{Q}_{t}=\eta_{f}(W)\times\wp(\chi)^{2}\times N^{2}/2,\;$in terms of
the total number of available resonant ion pairs configurations times the
tunnelling through the barrier probability. This taken into account, the
following equation determines the key constraints for the reactor confinement
configurations

$\mathcal{Q}_{t}\times\kappa=1,$ (6)

for $R_{C}\leq R_{0},\;$ where the radius of the Penning-Malmberg trap should
be $R_{0}=2(\chi+1)(eB)^{-1}\times(eV_{0}\mu/\chi)^{1/2}\;$and the maximum
radius of the confined plasma cloud
$R_{C}=\frac{1}{2}\varrho_{0}\kappa^{-1}\;$. The total thermoelectric power
becomes

$\mathbf{P}=P_{f}-(1-\eta_{h})aT^{4}[\pi/\omega]V.$ (7)

Here, to obtain the realistic model of the fusion device, we have subtracted
the black body radiation term ($a$ is the radiation constant), recall also
that $k_{B}T\simeq eV_{0}/\chi\times(R_{C}/R_{0})^{2}.\;$In addition to that,
the maximum variation of the number of Deuterons per unit of time in the
reactor is $\dot{N}_{D}=\frac{1}{\eta_{n}}P_{f}/\langle E_{f}\rangle\;$ and,
therefore, the thermonuclear reaction frequency is $\nu_{f}=\dot{N}_{D}/N$. On
the other hand, the D-D collision frequency should be of the order of the
cyclotron one, $\Omega.\;$Thus, for the actual resonant ion confinement
device, we propose that the Deuteron fuel will enter the resonant confinement
device cavity in a pulsed way. To this point, the fuel is introduced in the
reactor chamber through very high frequency pulses and the achievable electric
power of the ion cell will depend on the ultra high frequency and high voltage
circuit relays to control the rapid pulsed fuel refilling of the reactor
chamber. Then, if we denote the frequency of these circuit relays as
$\nu_{CB}\;$, the achievable power would be:
$2\pi\nu_{CB}/\Omega\times\mathbf{P}.\;$A solution of the ergodic constraint
in Eq. 6 does exist and is $\chi=477.102,\;$$B=3.73\;T,\;$and
$W=30.74\;keV.\;$Recall that, remarkably, the solution of the ergodic
condition for the barrier energy $W\;$coincides with that provided by the
stable Lawson point from the nuclear model alone. The resulting maximum
achievable electric power would be $\mathbf{P}\sim 11\times
2\pi\nu_{CB}/\Omega\;(MW)$ for Deuteron pulses of $N=2.2\times 10^{10}\;$ions,
if the electric potential is within the range $5.9\;kV<V_{0}<10.5\;kV.\;$In
order to estimate the actual achievable power of the cell a possible
approximation should be to provide $2\pi\nu_{CB}<O(\omega),\;$i.e, that, in
order to allow for the pulses to reach the thermal equilibrium during
confinement, and to the aim to achieve the resonance, the refueling frequency
should be slightly smaller than the resonant one. In that case, a solution of
Eq. 6 is represented in Fig.2 for the attainable thermoelectric power in Eq.
7. This configuration gets a maximum of approximately

$\mathbf{P}\sim 20\;kW\;$

at $V_{0}\simeq 8\;kV\;$for a fuel rate of $\dot{N}_{D}\simeq 2.1\times
10^{17}\;Deuteron/s.\;$ The confinement radius of the ion cloud would be
$R_{C}\simeq 91\;mm\;$ and the trap radius is $R_{0}\simeq 92.56\;mm.\;$The
actual fusion device requires a pulsed refilling of the reactor cavity and
that the resonant configuration would be attainable only adiabatically. This
kind of technological requirements will be the subject of the experimental
research for real fusion facilities based on the grounds of the resonant ion
confinement concept. Yet, in order to double check that the above estimates
are correct, we can do a kind of Fermi analysis. As said, in the resonant cell
(all over the time), the number of possible configurations between two
Deuterons, leading to quantum tunneling reactions, is approximately
$\mathcal{Q}_{t}\sim\eta_{f}(W)\wp(\chi)^{2}\times N^{2}/2.\;$Since the
natural colliding frequency is $\Omega/\pi,\;$the nuclear reaction rate should
be $\mathcal{Q}_{t}\times\Omega/\pi,\;$and the number of emitted neutrons per
second is half this value. Then, roughly, the actual achievable electric power
must be
$\mathbf{P}\sim\eta_{h}2.44\;MeV\times[\frac{1}{2}\mathcal{Q}_{t}\times\Omega/\pi]\times(\omega/\Omega)\;$(the
last term takes into consideration that we refuel the cell at the typical
resonant frequency rates). For the numbers above, i.e.,
$B=3.7\;T,\;$$N=2\times 10^{10}\;Deuterons,\;$$\chi\sim 477,\;$and $R_{C}\sim
91\;mm\;$(which, for the resonant condition in Eqs. 2, is equivalent to saying
that $\varrho_{0}\simeq 47\;fm$ or $W\simeq 30.7\;keV\;$), we get
$\mathbf{P}\sim 26\;kW,\;$which is of the order of the figure that we did
obtain from nuclear theory alone. This fact makes us confident in the
correctness of the derived result. We can call this optimum and small size
configuration of the confinement device parameters attaining the maximum power
as a fusion cell.

Figure 2:  Optimal achievable power of the standard fusion device as a
function of the applied confinement electric potential
$V_{0},\;$$B=3.73\;T,\;$$W=30.7keV\;$and $\vartheta=0.0915.\;$For the
estimation, the frequency of the Deuteron refueling pulse was taken to be
exactly coincident with the resonant one.

##  4 Discussion

For a compact reactor of small size, a new type of fusion technology can be
developed according to the basic conditions described in this article,
although some practical considerations must be taken into account. This
possibility arose from imposing the parametric relations in Eqs. 1 and 2 to
the standard Penning-Malmberg ion trap, which lead to the necessary resonant
kind of ion confinement. On the other hand, the trap fusion conditions are the
key configurations that satisfy the Eq. 6 and, very promisingly, both the
resonant and the operative conditions for Deuteron confinement are found
within the range of the current state of the art of technological
capabilities. In other words, due to the resonance, the Coulomb barrier can be
overcome due to the movement of resonant ions. Additionally, one must expect
that a pure Helium-3 resonant ion confinement device will work as another
fusion possibility in exactly the same principles as the Deuterium one exposed
in this article. Recall that such Helium-3 Cell will be a neutronic one and,
as a safe reactor, it opens the possibility to implement a sort of
thermonuclear battery in the event that it could power an autonomous device
provided with some finite reservoir of Helium-3 gas. Even though the physics
of both types of cells should be very alike, it does not necessarily imply
that their engineering designs, which have to be associated with solving the
problems of each type of reactors, should be also, in turn, similar. Therefore
the Helium-3 cell must be studied separately and the results will be presented
elsewhere.

##  5 Conclusion

In this work, the theory of resonant ion confinement has been developed for a
charged gas in thermal and dynamic equilibrium. Precise relationships between
the trap parameters and Deuteron-Deuteron collision probabilities have been
developed for ions following coalescent trajectories, leading, this way, to
fusion reactions. The analysis takes into account the statistical ergodicity
criterion of the system, as well as the assumption of thermal equilibrium of
the ion gas, which should be necessary, we claim, for any sustainable and
stable fusion reactor. In these circumstances, the Lawson criterion is
fulfilled. This device is capable of producing $20$ kW of electrical power
with a small reactor design that we have termed a fusion cell. Despite of the
final energy production efficiency, the experimental test of this concept must
be based on the eventual emitted neutron measurements.

Acknowledgements. Jose Luis Rosales has been supported by an internal research
contract owned by the Universidad Politécnica de Madrid, he is also indebted
to previous collaborations with Professor Vicente Martín leading to the
concept of Resonant Ion Confinement in Penning-Malmberg traps.

## Appendix

A cylindric Penning Malmberg trap is characterised by the introduction of an
axial magnetic field $\mathbf{B}\;$and a electrostatic potential $V_{0},\;$in
a cylindrical cavity where the ions are introduced. Additionally the full
stability can be obtained with the help of a rotating electric field with
angular frequency $\omega$ (see Ref. [9]) and field strength $\lambda;\;$in
this case, the motion of the ions in the trap is decomposed into separated
rotationally confined mode and an axial oscillation. Let $q\;$be the charge of
the ion, then, the Lagrangian is given in terms of the electrostatic
quadrupole and the magnetic field frequencies of the trap, i.e., denoting the
cyclotron frequency $\Omega=qB/m$

$L=\sum_{i}\frac{1}{2}m\\{\dot{x}_{i}^{2}+\dot{y}_{i}^{2}+\dot{z}_{i}^{2}+\Omega(x_{i}\dot{y}_{i}-y_{i}\dot{x}_{i})\\}-U_{t}(\mathbf{r}_{i}),$
(8)

$U_{t}(\mathbf{r})=qV_{0}+\frac{1}{2}m\\{\omega_{z}^{2}[z^{2}-\frac{1}{2}\rho^{2}]+\lambda\omega_{z}^{2}[(x^{2}-y^{2})\cos
2\omega t-2xy\sin 2\omega t]\\},\;$ and $\rho^{2}=x^{2}+y^{2}.\;$This rotating
quadrupolar electrical potential wall of intensity $\lambda$ might be required
for the adiabatic stability of the ions in the trap as it will be confirmed
down from. Let us consider the rotating coordinate frame which rotates with
angular frequency $\omega$: $(x,y)\rightarrow(\xi,\zeta).\;$Then the rotating
wall quadrupole perturbation becomes time independent,
$(x^{2}-y^{2})\cos(2\omega t)-2xy\sin(2\omega
t)\rightarrow\xi^{2}-\zeta^{2},\;$

$L=\sum_{i}\frac{1}{2}m\\{\dot{\xi}_{i}^{2}+\dot{\zeta}_{i}^{2}+\dot{z}_{i}^{2}+(\Omega-2\omega)(\xi_{i}\dot{\zeta}_{i}-\zeta_{i}\dot{\xi}_{i})\\}-U^{\prime}(\mathbf{r}_{i}),$
(9)

$U^{\prime}(\mathbf{r})=qV_{0}+\frac{1}{2}m\omega_{z}^{2}z^{2}-\frac{m\rho^{2}}{4}\omega_{z}^{2}+\frac{m\rho^{2}}{2}\Omega\omega-\frac{m\rho^{2}}{2}\omega^{2}+\frac{m}{2}\lambda\omega_{z}^{2}(\xi^{2}-\zeta^{2}).\;$Hereafter
the constant energy potential term $qV_{0}\;$will be omitted. The
$z\;$coordinate motion is an harmonic oscillator of frequency $\omega_{z}$.
The restoring force gives the effective axial component of the electric field
$eE_{z}=-m\omega_{z}^{2}z$. Along these lines, if the density of the plasma
cloud is $n$, $E_{z}\;$can be thought as that coming from the displacement of
a positive charge in the plasma cloud (effectively a relative negative charge)
$\Delta q=-qz\oint dSn$. Then, $\oint E_{z}dS=\Delta q/\epsilon_{0}\;$and from
this the charge density becomes
$n=\frac{m\omega_{z}^{2}\epsilon_{0}}{q^{2}}.\;$ On the other hand, in order
to compensate the axial displacement of every ion, this motion should be
correlated with that of some other ion in the opposite $z\;$coordinate;
moreover this z axis degree of freedom can be assumed to be thermal and,
therefore, for these two correlated ions

$2\times\frac{1}{2}k_{B}T=\frac{1}{2}\mu\omega_{z}^{2}z_{max}^{2}.$ (10)

Then, in order to determine the orbits of what may represent those
complementary bundles of ions, let us first recall that in the rotating frame
the net quadrupole radial force can be averaged out to zero in every cycle
($\langle F_{q}\rangle=-\lambda m\omega_{z}^{2}\langle\rho(t)\cos 2\omega
t\rangle\to 0,\;$ for $0\leq\omega t\leq 2\pi\;$). Indeed, three forces are
acting on each of the ions, namely, the centrifugal force
$+m\omega^{2}\rho\;,$ the radial electric force
$+\frac{1}{2}m\omega_{z}^{2}\rho\;$and that one associated with the radially
electric field induced by the rotation $\omega\;$through the direction of the
axial magnetic field whose value is $-m\omega\Omega\rho.\;$It is this field
that will provide the radial confinement. To see how, recall that, from the
symmetry of a cylindrical trap, the total angular momentum (that from the ions
plus that from the magnetic field),
$\sum_{i}\\{m_{i}v_{\theta\;i}\rho_{i}+qB\rho_{i}^{2}/2\\},\;$ is preserved
and, therefore, for instance for large $B$, $\sum_{i}\rho_{i}^{2}\;$must reach
some constant, which, as said, implies radial confinement; this argument is
due to O’Neil (Ref. [11], Ref. [12]). With that in mind, the simplest case
where the perturbed quadrupole frequency should be taken as that case where
the net radial force acting on each of the charges is zero. Then,
$\omega_{z}^{2}=2\omega(\Omega-\omega).\;$ It means that a ”trap angle” can be
defined such that
$\small\omega_{z}^{2}=\frac{1}{2}\Omega^{2}\sin^{2}\vartheta,\;\;\omega=\Omega\sin^{2}(\frac{\vartheta}{2}).\;$
This condition corresponds to the confinement situation of a thin, rigidly
rotating, spheroid of plasma at angular velocity $\omega$ (Ref. [13] Ref.
[14]). Recall also that for that confined cloud of ions of mass $m\;$ and
charge $q=Ze\;$a relation between the axial frequency $\omega_{z}\;$, the
cylindric radius $R_{0}\;$and the applied electric potential $V_{0}\;$ can be
found as the solution of $V(R_{0})=0,\;$where
$V(R)=V_{0}-\frac{m}{4}\omega_{z}^{2}R^{2}/q.\;$This gives,
$\omega_{z}^{2}=\frac{4qV_{0}}{mR_{0}^{2}},\;$ Lagrange’s equations read
(denote $\tau(t)=\Omega^{\prime}t\;$and
$\varepsilon=\lambda\omega_{z}^{2}/\Omega^{\prime 2}$ )

$\displaystyle\frac{d^{2}\xi}{d\tau^{2}}-\frac{d\zeta}{d\tau}+\xi\varepsilon=0,\;\;\;\frac{d^{2}\zeta}{d\tau^{2}}+\frac{d\xi}{d\tau}-\zeta\varepsilon=0.$
(11)

Elseways, since the Lagrangian is not time dependant, the energy is preserved
and a first integral of motion is obtained
$E=\textstyle\sum_{i}\\{\dot{\xi_{i}}\partial_{\dot{\xi_{i}}}L+\dot{\zeta_{i}}\partial_{\dot{\zeta_{i}}}L\\}-L=\textstyle\sum_{i}\frac{m}{2}\\{\dot{\xi_{i}}^{2}+\dot{\zeta_{i}}^{2}+\lambda\omega_{z}^{2}(\xi^{2}-\zeta^{2})\\}.\;$This
means that the motion is bounded and that a time period $t_{0}$ can be found
such that $2E/(\Omega^{\prime
2}m)=\varepsilon\textstyle\sum_{i}\\{\xi_{i}(t_{0})^{2}-\zeta_{i}(t_{0})^{2}\\}.\;$As
said, the orbits of the ions are combinations of rapid bare cyclotron
$\Omega^{\prime}\;$oscillations plus a slow guiding centre magnetron
trajectory of angular velocity $\omega\;$which becomes stabilised by the
rotating quadrupole force. Notwithstanding with this, other solutions can be
seen as representing binary collisions at the centre of the trap. Now,
define,$\;\eta=\sqrt{1+4\varepsilon^{2}},$
$\;\sigma=\sqrt{1/2(\eta-1)}\rightarrow\varepsilon\;$,
$\gamma=\frac{\varepsilon+\sigma^{2}}{\sigma}\rightarrow 1\;$,
$\beta=\sqrt{1/2(\eta+1)}\rightarrow 1\;$and
$\gamma^{\prime}=-\frac{\beta}{\varepsilon+\beta^{2}}\rightarrow-1\;,$ for
$\varepsilon\ll 1.\;$With this notation the exact solutions are

$\displaystyle\xi_{h}=a\cosh\sigma\tau,\;\;\zeta_{h}=a\gamma\sinh\sigma\tau,\;\;\;\;\;\;\;\;\;$
$\displaystyle\xi_{C}=b\cos\beta\tau,\;\;\;\;\;\zeta_{C}=b\gamma^{\prime}\sin\beta\tau.\;\;\;\;\;\;\;\;\;$
(12)

The $\xi_{C}$, $\zeta_{C}$ should correspond to the mentioned bare cyclotron
orbits (see e.g., Hasegawa et al. Ref. [9] for the solutions of the cyclotron
orbits in the rotating wall trap configuration). Thus, for instance, if the
general solution is $\vec{r}=\vec{r}_{h}+\vec{r}_{C},\;$ the limit
$\varepsilon\rightarrow 0\;$is just the usual Penning trap constant radius
magnetron orbit $|\vec{r}|=a$ provided with a rapid cyclotron oscillation
around this guiding orbit. Incidentally, in spite of the fact that $\xi_{h}$,
$\zeta_{h}$ can be seen as spurious solutions (they are not bounded hyperbolae
that do not meet the required confinement conditions), recall that,
admissibly, some of the actual orbits could also be represented by linear
combinations of these hyperbolic plus the bare cyclotron solutions:
$\xi^{(i)}=\xi_{h}^{(i)}+\xi_{C}^{(i)},\;\;\zeta^{(i)}=\zeta_{h}^{(i)}+\zeta_{C}^{(i)}.\;$
These trajectories would exist during some period of time $t_{0},\;$say,
$-t_{0}\leq t\leq t_{0},\;$. Numerical simulation shows that, in the rotating
frame, the trajectory is just an hyperbola provided with rapid cyclotron
oscillations having two turning points (at which
$\dot{\xi}(t_{0})=\dot{\zeta}(t_{0})=0.)\;$Is obvious that the physical
situation corresponds to the orbits of two long range coupled ion density
fluctuations that collide (and repel each other) at the centre of the trap
with $\xi^{(1)}=-\xi^{(2)},\;\zeta^{(1)}=-\zeta^{(2)}.\;$This is obviously
correct because, otherwise, single ions would not preserve linear momentum
individually. The only relevant degree of freedom ought be the relative
distance between the two charged aggregates,
$\varrho=2(\xi^{2}+\zeta^{2})^{1/2}\;$. For this coordinate, the collision is
described from the Lagrangian Eq. 9 replacing $\xi=\varrho\cos\omega
t,\;\;\zeta=\rho\cos\omega t,\;$and $m\rightarrow\mu$

$\mathcal{L}=\frac{1}{2}\mu\\{\dot{\varrho}^{2}+\frac{1}{2}\omega_{z}^{2}\varrho^{2}-\omega_{z}^{2}\varrho^{2}\lambda\cos
2\omega t\\}.$ (13)

Denoting $\varphi=\omega t\;$and
$\mathbf{\chi}=\frac{\omega_{z}^{2}}{2\omega^{2}}=\cot^{2}(\frac{\vartheta}{2})\;$
the equation of motion reads

$\frac{d^{2}}{d\varphi^{2}}\varrho-\\{\mathbf{\chi}-2\lambda\mathbf{\chi}\cos
2\varphi\\}\varrho=0.$ (14)

Eq. 14 is Mathieu’s equation whose time symmetric solution is obtained in
terms of the Mathieu Cosine function:
$\varrho(\varphi)=\frac{2R_{C}}{c}C_{e}(-\mathbf{\chi},-\lambda\mathbf{\chi},\varphi),\;\;\;\;\;c=C_{e}(-\mathbf{\chi},-\lambda\mathbf{\chi},0).\;$
Thus, as in the case of the inverse pendulum, there are periodic stable bound
solutions only within a very narrow parametric region
$\lambda(\mathbf{\chi})\;$(see Ref. [15]), they have period $\pi$ for the
variable $\varphi.\;$For $\omega_{z}\gg\omega,\;$numerically, this parametric
stability constraint corresponds to a dependency between the quadrupolar
electric force intensity and the parameters of the Penning trap

$\lambda\rightarrow\frac{1}{2}+1/\sqrt{2\mathbf{\chi}}$ (15)

Additionally, the closest distance between the two ion bundles defines the
squeezing factor, $\kappa,\;$of the coalescing orbits, i.e., numerically

$\displaystyle\ln\\{\varrho_{0}/2R_{C}\\}\rightarrow\ln\kappa\rightarrow\frac{1}{2}-1.35\sqrt{\mathbf{\chi}}$
(16)

here $\varrho_{0}\equiv\varrho(\pi/2)\;.$The implication is that, if the
resonant condition in Eq. 15 is satisfied, any closeness, even small, between
the two positive ion clumps can be reached near the centre of the trap when
$\mathbf{\chi}\gg 1.\;$Eqs. 15 and 16 constitute the grounds of the Resonant
Ionic Confinement Method. Owing to the existence of the quadrupole, the
magnetron degree of freedom is stabilised and the plasma is macroscopically
described as a rigid rotor of angular velocity $\omega.\;$Notwithstanding with
this, microscopically, each individual ion radial velocity should be
Maxwellian and the stability of the plasma requires that the solid rotor
energy be thermal. Incidentally, the actual inertia momentum of every ”rigidly
rotating” stabilised thin disk of plasma becomes $I_{Disk}=\sum_{i\in
Disk}m_{i}r_{i}^{2}=\frac{1}{2}N_{Disk}mR_{C}^{2},\;$providing a rotational
energy giving by $E_{Disk}=\frac{1}{2}I_{Disk}\omega^{2};\;$yet, to this
rotational degree of freedom of each individual ion, a thermal energy
$k_{B}T/2\;$must be allocated. Along these lines, the condition of thermal
equilibrium of the stabilised, rigidly rotating, plasma is compelled to be
$N_{Disk}\times\frac{k_{B}T}{2}=\frac{1}{2}I_{Disk}\omega^{2}.\;$ It imposes
$\omega R_{C}=\sqrt{k_{B}T/\mu}\;$ which, since
$\omega\simeq\Omega/\chi,\;$also implies, in the approximation
$\omega_{z}\gg\omega,\;$that
$\chi\sim\frac{qV_{0}}{k_{B}T}(\frac{R_{C}}{R_{0}})^{2}.\;$ Additionally, the
density clumps in the coalescing orbits are practically confined in the centre
of the trap most of the time and they move away to reach some maximum
confinement radius $R_{C}\leq R_{0}\;$ where they bounce back again to the
centre. In this case, the ions inside those large aggregates, may interact
individually when, owing to the resonance, the relative distance between the
clumps is reduced to a tiny minimum. It is straightforward to derive, the
following relations for $\chi\gg
1:\;$$\;B\simeq\frac{2\chi}{qR_{C}}\sqrt{k_{B}T\mu},\;n\simeq\frac{\epsilon_{0}}{\mu\chi}B^{2},\;z_{max}\simeq
R_{C}/\sqrt{\chi},\;N\simeq n\chi\times V_{c},\;$ where
$V_{c}=V/\chi=\frac{4}{3}\pi(R_{C}/\sqrt{\chi})^{3}\;$is the clump volume.
Therefore, we see, the confined plasma volume can be calculated as if there
were $\chi\;$granular spheres of ions of radius $R_{C}/\sqrt{\chi}.\;$From
these equation it is possible to derive that

$n=\\{(N/\chi)\;2\times\frac{1}{2}k_{B}T/E_{C}\\}\times{(N/\chi)}/\frac{4}{3}\pi
R^{\prime 3}\\}$

where $R^{\prime}=R_{C}/\sqrt{\chi},\;$whereas,
$E_{C}=(qN/\chi)^{2}/(8\pi\epsilon_{o}R^{\prime}),\;$corresponds to the
Coulomb energy of a bubble of $N/\chi\;$ions on the surface of a sphere whose
radius is precisely $R^{\prime}.\;$Yet, recall that the plasma is fully
thermal, imposing that the two dimensional surface energy satisfies
$2\times\frac{1}{2}(N/\chi)k_{B}T=E_{C}.\;$These thermal conditions of the
plasma are, indeed, fully compatible with the existence of small fluctuations
in the statistics that, according to our interpretation of the two coalescing
ion density clumps, in the resonant case, will orbit the cloud at periodic
Mathieu trajectories. For some clump pair in the plasma, the resonant
situation will hold with some probability, say
$\mathbf{\wp}[i\in\\{coalescing\\}].\;$To estimate this recall that, owing to
the periodic radial displacement of the density bundles, analogously to the
axially periodic degree of freedom, dynamical equilibrium imposes that there
should be some effective radial restoring force $q\delta
E_{\varrho}=-\mu(2\omega)^{2}\delta\varrho.\;$Again, this corresponds to a
negative effective displaced charge (a hole in the positively charged ion
cloud) of $-q\delta\varrho n^{\prime}\oint dS,\;$where $n^{\prime}\;$is the
density of the displaced density clumps inside the plasma. Now, Gauss theorem
applied to a thin disk surface of the plasma states that
$n^{\prime}=2m\epsilon_{0}\omega^{2}/q^{2},\;$which, from its statistical
definition $n^{\prime}\equiv n\mathbf{\wp},\;$gives

$\mathbf{\wp}=\frac{\omega}{\Omega-\omega}=\mathbf{\chi}^{-1},$ (17)

which, given that $\omega<\Omega/2,\;$it is always lower than $1\;$as it
should be. Recall that near to the centre of the trap, the minimum distance
between the colliding ions can be approximated by
$\varrho(t)\simeq|\varrho_{0}+2R_{C}\varepsilon\cos\Omega^{\prime}t|,\;$ it
means that the distance obtains its minimum when $\Omega^{\prime}\Delta
t=\pi.\;$

### References

## References

  * [1] J.D. Lawson ($1957$),“Some criteria for power producing thermonuclear reactor”, Proc. Phys. Soc. B, $70$, $7$.
  * [2] A. J. H. Donné ($2019$), “The European roadmap towards fusion electricity”, Phil. Trans. R. Soc. A $377$: $20170432$
  * [3] K. Lackner, N. Gottardi ($1990$), “Tokamak confinement in relation to plateau scaling”, Nucl. Fusion $30$, $767$.
  * [4] A. Hassanein, V. Sizyuk. Hassanein, A., Sizyuk, V. Potential design problems for ITER fusion device”. Sci. Rep. 11, 2069 (2021). https://doi.org/10.1038/s41598-021-81510-2
  * [5] A. Dinklage et al. ($2018$), “ Magnetic configuration effects on the Wendelstein 7-X stellarator”, Nature Phys. $14$, $855-860$.
  * [6] F. Castejón et al. “Stellarator optimization under several criteria using metaheuristics”. Plasma Physics and Controlled Fusion 55 (2013) 014003. Doi: http://iopscience.iop.org/0741-3335/55/1/014003
  * [7] F. Castejón, A. J. Rubio-Montero, A. López-Fraguas, E. Ascasíbar, and R. Mayo-García, ($2016$) “Fusion Science and Technology”, $70$ pp. $406–416$.
  * [8] J.L. Rosales, and V. Martin, ($2021$) “A resonant ion confinement fusion reactor”. Spanish Patent Application No. 202131221, Universidad Politécnica de Madrid, (December $2021$).
  * [9] T. Hasegawa, M. J. Jensen and J. J. Bollinger, ($2005$) “Stability of a Penning trap with a quadrupole rotating electric field”, Phys. Rev A $71$, $023406$.
  * [10] J. Freidberg ($2007$), “Plasma Physics and Fusion Energy”, Ed. Cambridge Uni. Press, ISBN $978-0-521-85107-7$. pp. $65$, $111$.
  * [11] T.M. O’ Neil, “Plasma with a single sign of charge (an overview)”, ($1995$), Phys. Scripta T$59$ pp $341$-$351$.
  * [12] O’ Neil, T.,($1980$),“A confinement theorem for nonneutral plasmas”, Phys. Fluids, $23$, $2216$
  * [13] D. H. E. Dubin and T. M. O’Neil ($1999$) “Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states)”, Rev. Mod. Phys. $71$, p$87$.
  * [14] J.J. Bollinger, D.J. Wineland and D.H.E. Dubin, ($1994$), Phys. Plasmas,$1$, “Non-neutral ion plasmas and crystals”, laser cooling, and atomic clocks, $1403$
  * [15] L. Meirovitch, “Methods of Analytical Mechanics”, ($1970$). McGraw-Hill, reprinted by Dover Books, NY, ($2003$) ISBN 0-486-43239-4, pp $282-288$.