[inst1]organization=Deutsches Elektronen-Synchrotron DESY,
addressline=Notkestr. 85, city=22607 Hamburg, country=Germany

# Neural Networks for ID Gap Orbit Distortion Compensation in PETRA III

B. Veglia  I. Agapov  J. Keil

###### Abstract

Undulators are used in storage rings to produce extremely brilliant
synchrotron radiation. In the ideal case, a perfectly tuned undulator always
has a first and second field integrals equal to zero. But, in practice, field
integral changes during gap movements can never be avoided for real-life
devices. As they significantly impact the circulating electron beam, there is
the need to routinely compensate such effects. Deep Neural Networks can be
used to predict the distortion in the closed orbit induced by the undulator
gap variations on the circulating electron beam. In this contribution several
current state-of-the-art deep learning algorithms were trained on measurements
from PETRA III. The different architecture performances are then compared to
identify the best model for the gap-induced distortion compensation.

##  1 INTRODUCTION

††footnotetext: Work supported by the ACCLAIM Innovationspool Project (HGF)

The storage ring PETRA III at DESY [1] is being operated since 2009 and is one
of the brightest synchrotron radiation sources worldwide. It serves a broad
international multidisciplinary user community at currently 25 specialised
beamlines. With a storage ring energy of $6\text{\,}\mathrm{GeV}$, it delivers
mainly hard to high-energy X-rays for versatile experiments in a very broad
range of scientific fields. In this way, PETRA III provides the environment
for specialised experiments to address many of the pressing grand challenges
of the twenty-first century in energy, life and health, earth and environment,
mobility, and information technology. The circumference of
$2304\text{\,}\mathrm{m}$ makes it the largest storage-ring-based source
worldwide. Table 1 summarizes the main operational parameters of the PETRA III
storage ring.

Parameter | Value
---|---
Energy /GeV | 6
Circumference /m | 2304
Total current /mA | 120 | 100
Number of bunches | 480 | 40
Bunch Population /$10^{10}$ | 1.2 | 12
Emittance (horz.) /nm rad | 1.3
(vert.) /nm rad | 0.01
Beam Size at 5m Undulator (high $\beta$ Section) (horz.) /$\mu$m | 141.5
(vert.) /$\mu$m | 4.9
Beam Size at 5m Undulator (low $\beta$ Section) (horz.) /$\mu$m | 34.6
(vert.) /$\mu$m | 6.3
Beam Size at 10m Undulator (horz.) /$\mu$m | 141.6
(vert.) /$\mu$m | 6.6
Table 1: PETRA III Parameters.

Undulators are the most powerful generators of synchrotron radiation at
storage rings, they are a type of insertion devices (or IDs). The angle-
dependent emitted wavelength can be calculated as:

$\lambda(\theta)\approx\frac{\lambda_{U}}{2\gamma^{2}}\left(1+\frac{K^{2}}{2}+\gamma^{2}\theta^{2}\right)$
(1)

Where $\lambda_{U}$ is the period length of the undulator, $\gamma$ is the
Lorentz factor, $\theta$ is the emission angle and dimensionless quality K is
the deflection (or undulator) parameter that can expressed as:

$K=\frac{e}{2\pi m_{e}c}B_{0}\lambda_{U}=0.9336*B_{0}[T]\lambda_{U}[cm]$ (2)

with $e$ the electron charge and $m_{e}$ its mass, $c$ the speed of light and
$B_{0}$ the peak magnetic field on the undulator axis.

Radiation wavelength is adjusted by changing the undulator gap $g$, that is,
the distance of the opposing magnets, as shown in the following relation

$K\approx 0.168B_{r}\lambda_{U}\exp^{\left(-\pi\text{g}/\lambda_{U}\right)}$
(3)

where $B_{r}$ is the on-axis magnetic field. Figure 1 shows that the changes
applied by users are very frequent, do not follow specific patterns and, as
seen in Fig. 2 they are in general statistically uncorrelated (except for P01a
and P01b, where the two 5m long undulators are synchronized and considered as
one 10m long device). During operations normally most of the IDs would be
moving to small gaps and very rarely be fully open at the same time. Table 2
lists the beamlines operating with undulator radiation. The number in the
undulator type indicates the period length $\lambda_{U}$. Some experiments
perform wide scans of wavelengths while others apply small fast variations.
The beamline users have hence the control over the undulator movements, while
the accelerator operation is in charge of compensating any effect that those
might induce on the circulating electron beam.

Beamline | Length | Period length $\lambda_{U}$ | Gap range | Energy range
---|---|---|---|---
| (m) | (mm) | (mm) | (keV)
P01 a and b | 2 x 5 | 31.4 | 12.7 - 220 | 2.5 - 80
P02.1 P02.2 | 2 | 23.0 | 9.9 - 220 |  60 25.7, 42.9, 60
P03 | 2 | 29.0 | 10.2 - 217 | 9 - 23
P04 | 4.9 | 65.6 (APPLE*) | 11.5 - 148 | 0.25 – 3.0
P05 | 2 | 29.0 | 9.9 - 220 | 5 – 50
P06 | 2 | 31.4 | 9.8 - 220 | 5 - 45
P07 | 4 | 21.2 (In-Vacuum) | 7.0 - 40 | 30 – 200
P08 | 2 | 29.0 | 9.9 - 218 | 5.4 – 29.4
P09 | 2 | 31.4 | 9.8 - 220 | 2.7 -31
P10 | 5 | 31.4 | 9.7 - 220 | 5 -10
P11 | 2 | 31.4 | 9.9 - 220 | 5.5 – 30
P12 | 2 | 21.2 | 9.9 - 220 | 4 – 20
P13 | 2 | 29.0 | 9.8 - 220 | 4.5 – 17.5
P14 | 2 | 21.2 | 9.8 - 220 | 7 – 26.7
P21a | 2 | 29.0 | 9.9 - 220 | 52, 85, 100
P21b | 4 | 21.2 (In-Vacuum) | 7.0 - 40 | 40 - 150
P22 | 2 | 32.8 | 9.6 - 220 | 2.4 - 30
P23 | 2 | 31.4 | 9.9 - 220 | 5 - 35
P24 | 2 | 21.2 | 9.8 - 220 | 8, 17 - 44
P25 |  |  | in preparation |
P62 | 2 | 31.4 | 9.9 - 220 | 3.5 - 35
P63 |  |  | in preparation |
P64 | 2 | 32.8 | 10.4 - 220 | 4 - 44
P65 | 0.4 | 32.8 | 10.5 - 220 | 4 - 44
Table 2: PETRA III beamlines [2]. *APPLE is an abbreviation for Advanced
Planar Polarized Light Emitting undulator, a device made of four independent
arrays of permanent magnets which can be adjusted to control the polarisation
of the radiation emitted.

In 3rd generation light sources the orbit must be kept stable to within a few
percent of the electron beam size (see Tab.1). It is important to keep the
orbit constant during these field changes to not disrupt other users. With the
advent of 4th generation storage rings delivering high-brightness x-ray beams
with high coherent flux such as the upcoming upgrade PETRA IV [3], electron
beam sizes will become smaller and the number of IDs increase, making the
orbit stability even more challenging and important. The magnetic fields of
IDs introduce perturbations to the circulating electron beam and hence affect
the linear and nonlinear beam dynamics of the electron beam in the storage
ring.

The primary effect of the undulator on the electron beam trajectory is
determined by the field integrals [4]. Let $x$ be the horizontal position of
an electron, $y$ the vertical position, and $z$ the position along the
undulator (having $z_{0}$ as the initial longitudinal coordinate):

$\begin{array}[]{lcl}x^{\prime}(z)=-\frac{e}{\gamma
m_{e}v_{z}}\int\limits_{z_{0}}^{z}B_{y}(z_{1})dz_{1}\\\
y^{\prime}(z)=\frac{e}{\gamma
m_{e}v_{z}}\int\limits_{z_{0}}^{z}B_{x}(z_{1})dz_{1}\\\ x(z)=-\frac{e}{\gamma
m_{e}v_{z}}\int\limits_{z_{0}}^{z}\int\limits_{z_{0}}^{z_{2}}B_{y}(z_{1})dz_{1}dz_{2}\\\
y(z)=\frac{e}{\gamma
m_{e}v_{z}}\int\limits_{z_{0}}^{z}\int\limits_{z_{0}}^{z_{2}}B_{x}(z_{1})dz_{1}dz_{2}\\\
\end{array}$ (4)

Here, $e$ is the electron charge, $\gamma$ is the Lorentz factor, $m_{e}$ is
the electron rest mass, $v_{z}$ is the velocity along the undulator, and
$B_{x}$ and $B_{y}$ are the horizontal and vertical magnetic field components,
respectively.

Figure 1: Recorded gap sizes of PETRA III IDs over a year period between 2021
and 2022. Figure 2: Heatmap of the correlation matrix of gap movements over
the year period.

In the ideal case, a perfectly tuned undulator with an anti-symmetric magnet
structure always has a first field integral equal to zero. But due to
imperfections, magnetic field degradation caused by radiation damage [5, 6] as
well as concentration of ambient magnetic fields by undulator poles, field
integral changes during gap movements can never be avoided for real-life
devices. The traditional approach to compensate this effect is based on
measurements of the response to gap movements on the machine orbit (performed
only once during commissioning) to calculate look-up table for the undulator
control system in order to power the compensation coils (illustrated in Fig.
3) mounted symmetrically at each end of the insertion device according to the
current gap value [7].

Figure 3: Image showing the air coils correctors (copper windings) at the end
of the undulator structure.

PU08

(a)

(b)

PU14

(c)

(d)

Figure 4: Measurement of two IDs varying gap vs. the horizontal (a and c) and
vertical (b and d) orbit. Each coloured curve corresponds to a different BPM.

Measurements of the individual impact of the IDs gap variations on the closed
orbit are shown in Fig. 4. Each plotted line represents the detected position
at the corresponding BPM. The ability of neural networks to learn non-linear
patterns [8], approximating any complex function through an ensemble of latent
variables, makes them ideal candidates to capture and model the effect of
undulator movements on the circulating electron beam, which presents complex
nonlinear behaviour. This contribution shows how the application of a model-
independent neural network could allow for such closed orbit distortion
corrections.

##  2 NEURAL NETWORKS

Neural networks (NNs) have proved to be most effective for nonlinear function
fitting, both theoretically and empirically [9, 10]. Recently, improved
techniques from the fields of machine learning (ML) and artificial
intelligence (AI) have been incorporated into the design of control systems
for particle accelerators. In particular, techniques based on neural networks
(NNs) are well-suited to modeling, control, and diagnostic analysis of
complex, time-varying systems, and systems with large parameter spaces [11,
12].

###  2.1 ARCHITECTURES CONSIDERED

Here, we propose a NN approach which, taking arbitrary ID gap configurations
as input, predicts the corresponding impact on the electron beam orbit giving
the transverse position at the BPMs as output. The NN are capable to learn
complex nonlinear relationships between the ID settings and transverse
position using training data. Figure 5 illustrates a schematic representation
of a generalized NN architecture mapping the ID gap sizes to the closed orbit.

Figure 5: Schematic layout of a Neural Network mapping the ID gaps to the
closed orbit.

Each training input is loaded into the neural network in a process called
forward propagation. Once the model has produced an output, this predicted
output is compared against the given target output in a process called back-
propagation. Stochastic gradient descent is used to minimize a loss function,
quantifying the error produced by the model. The parameters of the model are
updated at each iteration (or epoch) so that its output better fits the target
output.

The NNs are implemented using Keras with Tensorflow backend [13, 14], using
mean squared error as the loss function. The models are trained using the
back-propagation method employing the Adam optimizer [15] for 200 or less
epochs. The training takes between 10 and 50 minutes on a single desktop-class
CPU depending on the model. A variety of NN architecture features was screened
including regularization methods and activation functions in order to optimise
the model structures. Weights&Biases [16] was used to automate hyperparameters
tuning, exploring the space of possible models and allowing to identify the
best suited ones for the problem. In the following subsections the model
architectures considered are described.

####  2.1.1 FFNN

The simplest architecture is a NN where the output from the ”neurons” of one
layer is used as input to the next layer. Such networks are called feed-
forward neural networks (FFNN) or Multilayer Perceptron (MLP) and can use
linear or non-linear activation functions. The information is always
propagated forward within the network and there are no loops. In this study we
considered a FFNN with only one hidden layer (shallow) and a deeper one with
three hidden layers. We expect that the deep FFNN achieves higher accuracy
than the shallow one (at the same computational power) thanks to its ability
to better extract features.

####  2.1.2 RNN

Recurring neural networks, or RNNs, relie on a simple principle, feeding the
output of a layer back to itself, thus creating a feedback loop between
different layers of the network, allowing output from some neurons to affect
subsequent input to the same neurons. This enables it to exhibit temporal
dynamic behavior acting like a neural memory. Working with sequential data,
the order in which information appears is important and by feeding back the
previous input to the current layer, it allows to apply updates in the context
of the sequence seen so far. The recurrent network keeps a record of the input
signal finding patterns between different parts of the input. The specific
architecture studied in this contribution consisted of two hidden layers, each
composed of recurrent units, followed by the fully connected output layer.

Figure 6: Schematic representation of a single recurrent layer with two inputs
and four hidden units.

####  2.1.3 CNN

Convolutional layers represent major building blocks in many deep neural
networks. The design was inspired by the visual cortex, where individual
neurons respond to a restricted region of the visual field known as the
receptive field. Convolutional neural networks (CNNs) [17] get their name
because their layers use convolution to learn a feature map from the input
data. Convolution is a mathematical function that is used in place of matrix
multiplication in the layers of CNN. There are four main components in CNNs:
(a) convolution filters, (b) nonlinear activations, (c) spatial coarsening
(via pooling or strided convolution), (d) a prediction module, often
consisting of fully connected layers that operate on a global instance
representation. The same concept can be successfully applied to one-
dimensional sequences of data, such as in the gap-to-orbit case discussed
here. The network learns to extract features from sequences of observations
and how to map the internal features to predict the ID-induced orbit
distortion. The NN used for the closed orbit prediction consist of a 1D
convolutional layer, followed by the application of a 1D pooling and two fully
connected layers [18].

###  2.2 Hyperparameter Sweeps

Hyperparameters, such as the learning rate and the number of hidden layers,
play a crucial role in determining a machine learning model’s performance. A
small difference in a single hyperparameter can lead to a large performance
variation. Therefore, a central component of machine learning research is to
find the correct set of hyperparameters for a given task. It can be
challenging to identify optimal hyperparameters for a data distribution
sequentially since the parameter search space is large and has very few
optimal values. Systematic hyperparameter sweeps were performed with
Weights&Biases in order to find the best working combination for each of the
NN structures considered, allowing to fairly compare their performance. The
following hyperparameters were investigated:

  * •

Activation function determines how to compute the input values of a layer into
output values.

  * •

Batch size defines the number of samples propagated through the network.

  * •

Hidden layer size is the number of neurons in a layer which controls the
representational capacity of the network, at least at that point in the
topology.

  * •

Learning rate controls the step size for a model to reach the minimum loss
function [19].

  * •

Dropout is a regularization technique to avoid overfitting by randomly
dropping out neurons during training [20].

  * •

Filters indicates the amount of filters applied to the input in the
convolutional layer.

  * •

Kernel size specifies the amplitude of the convolution window in convolutional
layers.

Figure 7: Representation of the optimization of various hyperparameters for
the shallow FFNN. Figure 8: Graphical representation of the hyperparameter
tuning for the deep FFNN. Figure 9: Graphical representation of the
hyperparameter tuning for the RNN.

In Figures 7 to 10 the impact of the hyperparameters relevant for each
architecture on the training loss and validation loss is displayed. Each
vertical axis represents one hyperparameter while the colorbar indicates the
loss relative to each hyperparameter combination tested.

Figure 10: Hyperparameter tuning for the CNN.

##  3 MEASUREMENTS

###  3.1 Training data

The data used for training the neural networks was acquired in July 2022
during a dedicated machine study shift by scanning 18 of the IDs operational
gap size range and recording the orbit at each of the functioning 239 BPMs.
The values for the remaining undulators were taken from previous measurement
campaigns performed following the same procedures. For observing the raw
impact of the ID gap variation on the circulating electron beam, the fast
orbit feedback system was disabled. The previously mentioned Figure 4 shows
the measurements of the orbit varying with the gap opening in one of the IDs
while the other were fully open. For each measurement the IDs were moved
towards the minimum gap at small steps and then reopened again and kept to the
maximum gap during the movement of the other IDs. The maximum and minimum gap
size depends on the type of undulator, the so-called ’in-vacuum’ ones, for
example, reach very small gaps but are constrained by the vacuum chamber when
opening. To train the network the collected dataset was augmented by linear
interpolation.

Models for each of the four different architectures (Shallow and Deep FFNN,
RNN and CNN) were trained on 80% of the training measurements. The remaining
20% of the data was used for validation in order to gauge the models’ ability
to predict the electron beam orbit. The training results were compared for
each architecture once the hyperparameters were optimized. Figure 11 shows the
final training loss and the epoch at which the training stopped. The loss
value gives an indication on how well the model fits the data, while the
number of epochs specify how many training iterations were required.

Figure 11: Comparison of the performance of the four different model
architectures.

The convolutional and recurrent structures outperform the fully connected NNs
reaching better accuracy in a reasonable amount of epochs. This is due to the
format of the measurements used for training, where the gap size was
recursively increased and decreased for each undulators. The input data
exhibited a serial structure that is better modelled by NNs containing
recurrent or convolutional features. As seen in Fig. 12 the tuned NNs,
although giving slightly different outputs, are generally able to accurately
model the training data.

Figure 12: Horizontal (top) and vertical (bottom) beam orbit predictions by
the trained NNs (dotted coloured lines) vs validation values of the training
dataset (black solid line).

###  3.2 Operational data

Since the training set was measured moving only one undulator at the time, to
test the trained network predictions with realistic operative gap values orbit
data was collected. In this set multiple undulators were closed at the same
time, observing their combined impact on the orbit. The undulators were moved
to randomly chosen gaps as shown in Fig. 13.

Figure 13: Random gap variations used for the operational data measurements.
Each colour represents a PETRA III Undulator (PU).

Constrains were in place to avoid losing the beam by inducing a too large
transverse kick. At each step a random portion of the insertion devices is
moved by an amount of millimetres sampled from a univariate normal
distribution of mean 0 and variance 10.

Figure 14: Predicted orbit distortion induced by multiple IDs closing compared
to the prediction obtained with the RNN model trained on training data only.

Additionally, the maximum gap size the undulators were moved to was limited to
80 mm, so that the impact on the circulating beam wasn’t negligible. Comparing
the expected orbit distortions obtained with any of the trained and optimized
model, it was found that they significantly differ from the measurements. Fig.
14 shows the comparison of the closed orbit measured with an operational IDs
configuration and the prediction calculated with the RNN model.

The RNN, which achieved the smallest loss on training data was thus retrained
on a subset of operational data (randomly distributed between training and
validation sets). The updated model was still not able to predict the orbit
distortion. It was the deep fully connected architecture that once retrained
including the new operational measurements could effectively model the IDs
effects as shown in Fig. 15, where the predictions are made on a different
subset of operational data.

Figure 15: Horizontal (top) and vertical (bottom) beam orbit predictions by
the re-trained deep FFNN (dotted yellow lines) vs. the measured value (black
solid line). The IDs gap configuration was randomly selected within a test
subset of the operational data.

##  4 DISCUSSION

The NNs take as input a vector containing the gap size of each ID for any
operational configuration and give as output the predicted transverse position
at the BPM locations.

With the final model it is possible to scan 23 IDs through their entire
operational parameter space and evaluate the expected transverse position at
the BPMs. The RNN was successfully trained to make accurate predictions of the
beam transverse position. The orbit distortion predicted by the trained model
can then be used to calculate the required strength in the corrector magnets.
The global correction scheme that we suggest uses a pyAT [21] calculated orbit
response matrix (ORM) with singular value decomposition [22]. Through the ORM
it is possible to compute the kick at each of the correctors along the ring
necessary to counteract the existing dipolar field errors such that the orbit
distortion measured with the BPMs is minimized.

A diagram depicting the general process towards the orbit distortion
correction is shown in Fig. 16.

Figure 16: Description of the steps followed in this study to develop a
reliable model to predict the undulator induced orbit distortion.

In order to verify whether linear combinations of the explanatory variables
(ID gaps) can be used to predict the response variable (closed orbit
distortion) we created a linearity test that is defined by the following.

Having $F_{j}$ as the orbit at BPM $j$ and

$g_{l}=(0,0,0,0,0,x_{l},0,0)$

$g_{m}=(0,x_{m},0,0,0,0,0,0)$

(where $l\neq m$)

as single-entry vectors (or scaled standard-basis vectors) representing a
specific configuration of undulator gaps where all the IDs are open to the
maximum and only the one identified with the location of the non-zero value
moved to the gap size $x_{l}$ and $x_{m}$ respectively. The system shows
linearity when

$F_{j}\left(\sum_{i=1}^{N<N_{Gaps}}g_{i}\right)=\sum_{i=1}^{N<N_{gaps}}F_{j}\left(g_{i}\right)$

where with $F_{j}\left(\sum_{i=1}^{N<N_{Gaps}}g_{i}\right)$ we indicate the
orbit distortion induced by the undulator gaps configuration obtained by
summing the single-entry vectors. This should hold for all possible gap
combinations $\forall\\{g_{i}\\}$.

Hence the criterion of non-linearity is

$\Delta
Prediction=||F_{j}\left(\sum_{1}^{N_{gaps}}g_{i}\right)-\sum_{1}^{N_{gaps}}F_{j}\left(g_{i}\right)||\neq
0$

We performed the test by randomly assigning a gap value $g_{0}$ to PU01a and
varying the gap in PU01b, while having all the other undulators open, and
using the model to calculate the expected impact on the orbit. The operation
was then repeated for all the other IDs. Figure 17 shows the difference
between the predicted orbit distortions at one point along the ring for
combinations of PU01a and any other undulator gap closing. The difference
decreases to 0 when the undulator is open to the maximum gap as expected. The
process was repeated for all the other combinations of two undulator closing
obtaining similar curves, indicating that the non-linearity criterion is
satisfied.

Figure 17: Difference of the predicted orbit (measured at one BPM) as
calculated with the NN with two closed IDs (one of which is always PU01a) and
as the sum of the predictions obtained for closing only one ID at a time. The
gap of the undulators is linearly re-scaled to the 0-1 interval instead of the
ID dependent min-max gap size.

##  5 CONCLUSIONS AND OUTLOOK

Four NNs models were developed and trained with PETRA III measurements to
successfully predict the beam orbit at any given operational ID gaps
configuration. The models were optimised through hyperparameter sweeps to
obtain high predictability and comparisons indicated the 1D recurrent neural
network as the best fitting. The model initially failed to agree with
measurements when fed with operational ID gap configurations, as the impact of
multiple undulators closing at the same time couldn’t be modelled from the
training data. The best performing architecture, the RNN, was then trained
again using a sample of 302 orbit measurements with multiple IDs closing. The
updated RNN model reached a training loss of
$0.0501\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a validation loss of
$0.0588\text{\,}\mathrm{\SIUnitSymbolMicro m}$ after 200 epochs and showed
good agreement with test data. To make sure the model was reliable we compared
the predictions with measurements taken on different days.

A similar approach could also be considered to counteract the perturbation
introduced by ID gap variations to the betatron coupling and the vertical
dispersion, as demonstrated at ALS [23]. The effect is currently not
observable for PETRA III specifications but could impact the extremely low
emittances that will be reached in PETRA IV [3]. In such case the impact (on
the beam size) could be measured even with the fast orbit feedback activated
thus collecting the necessary data during operations would be possible. In
this way, the NN would be constantly updated, keeping up with drifts in the
machine as well as changes in IDs applied by users without requiring
additional dedicated measurement shifts.

##  6 ACKNOWLEDGEMENTS

The authors thank Andreas Schoeps and his colleagues from the Undulator Group
for the support for the measurements in PETRA III.

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# A physically cryptographic warhead verification system
using neutron induced nuclear resonances

Ezra M. Engel  Areg Danagoulian∗

Department of Nuclear Science and Engineering, Massachusetts Institute of
Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, USA

∗Correspondence and requests for materials should be addressed to A.D. (email:
<EMAIL_ADDRESS>
###### Abstract

Arms control treaties are necessary to reduce the large stockpiles of the
nuclear weapons that constitute one of the biggest dangers to the world.
However, an impactful treaty hinges on effective inspection exercises to
verify the participants’ compliance to the treaty terms. Such procedures would
require verification of the authenticity of a warhead undergoing
dismantlement. Previously proposed solutions lacked the combination of
isotopic sensitivity and information security. Here we present the
experimental feasibility proof of a technique that uses neutron induced
nuclear resonances and is sensitive to the combination of isotopics and
geometry. The information is physically encrypted to prevent the leakage of
sensitive information. Our approach can significantly increase the
trustworthiness of future arms control treaties while expanding their scope to
include the verified dismantlement of nuclear warheads themselves.

## Abstract

## Introduction

As of 2019 there are estimated 13000 nuclear weapons that make up the nuclear
arsenals of the United States and Russia[1, 2]. Such large arsenals may be one
of the greatest threat to our civilization. While high, these numbers are a
significant reduction from the Cold War era, as a result of series of arms
control treaties. The past treaties between United States of America and
Soviet Union / Russia, however, primarily focused on the verified
dismantlement of the delivery systems, such as ballistic missiles and
strategic bomber aircraft. This circumstance in part was driven by the notion
that the delivery systems can be a good proxy for the states’ strike
capability during a nuclear war. It was also driven by the difficulty of
verifiably dismantling the warheads themselves without leaking sensitive
information about the weapon designs in the process[3]. This approach has left
behind large stockpiles of surplus nuclear weapons, exposing them to the risk
of theft and unauthorized or accidental use, as well as transfer to third
countries. This state of the affairs elevates the risk of nuclear terrorism
and nuclear proliferation. The need for more effective arms control treaties
is recognized by the 2018 US Nuclear Posture Review, which noted ”The United
States will continue its efforts to seek arms control agreements that enhance
security, and are verifiable and enforceable.”[4] Furthermore, there are
increased worries that unless new treaties are implemented the current arsenal
sizes will stagnate at the current numbers[2, 5]. New treaties involving the
verified dismantlement of nuclear warheads will significantly improve global
security. However new types of technologies are necessary to enable such
treaties. These technologies will have to detect hoaxing attempts, clear
honest warheads as such, while simultaneously protecting sensitive information
about the weapon designs.

A variety of paradigms of verification have been proposed by a number of
scholars and scientists. The most recent of these, the template verification
paradigm, is based on the expectation that the inspection party will be able
to acquire an authentic device (henceforth referred to as the genuine
reference object), and then use relevant encrypted data (known as a template)
from this device to compare with equivalent data acquired from candidate
devices of the same design. The key to this verification process is a proof
system that can guarantee that no treaty accountable item (TAI) undergoing
verified dismantlement is secretly modified. A number of works have been
published on the philosophy behind this template verification - in the US
national laboratories, think tanks, and academia[6, 7, 8] \- and a variety of
concepts have been proposed [9, 10, 11, 12, 13, 14, 8, 15, 16]. Common to all
these studies is the concept of the genuine reference object, which is
acquired based on a situational context and brought to the testing facility
via a chain of custody. For a more detailed delineation of the procedure
describing the acquisition of the genuine reference object (sometimes referred
to as golden copy in the literature) and its use in the verification exercises
see Results section in our previous work, Hecla and Danagoulian[17]. That work
as well as the one presented in this study leverage Neutron Resonance
Transmission Analysis (NRTA). This technique uses neutron induced nuclear
resonances in the electron-Volt (eV) energy range, common in many high Z
elements including molybdenum, tungsten, uranium, and plutonium, to achieve
high resolution imaging of the individual isotopes of those elements[18]. The
strength of NRTA stems from the uniqueness of the resonance energies and
amplitudes to individual isotopes, thus making their observation a unique tag
of a particular isotope’s abundance in the sample. NRTA has been previously
used to image and study various nuclear fuel samples and archaeological
objects - for an in depth discussions see Losko et al. and Bourke et al.[19,
20].

Some of the previously proposed concepts made use of physical cryptography.
This has the advantage of avoiding computational cryptography, which can be
prone to hacks and backdoor exploits and as such shifts the burden of
verification to the computer code and electronic components. Work by Philippe
et al.[11] had the advantage of achieving the highest level of information
security via a zero-knowledge measurement, yet it is not fully sensitive to
isotopic hoaxes due to the similarity of fast neutron processes between
various nuclei. Additional work is currently underway to mitigate this
circumstance, e.g. by observing fast neutrons from neutron-induced
fission[21]. Others, such as those by Vavrek et al.[16] leverage nuclear
resonance phenomena to achieve strong isotopic sensitivity to hoaxes, but do
not have zero-knowledge information security and involve complex systems that
may be unpractical in a verification setting. Our previous study, Hecla and
Danagoulian[17], was based on Monte Carlo (MC) simulations, where we combined
the advantages of a neutron-based radiographic system with the isotopic-
sensitivity of a nuclear resonance sensitive measurement to achieve an
effectively zero knowledge proof system.

## Results

###  0.1 Template verification

This work builds upon the general concepts described previously by Hecla and
Danagoulian[17], with important differences and modifications. The study by
Hecla and Danagoulian was based on Monte Carlo simulations alone and
envisioned an imaging system where the physical encryption is achieved by a
geometrically complex reciprocal mask. The concept proposed in this work
meanwhile achieves an experimental feasibility proof of concept involving a
simple encrypting filter, and uses a single pixel neutron detector. The
detector measures the combined flux through the object and can determine the
energy of individual neutrons. While a single pixel measurement along one axis
of measurement does not have geometric uniqueness, such a uniqueness can be
achieved via multiple measurements with object rotations, not dissimilar to
the principles of a tomographic system. This requires a new procedure for the
comparisons between template data from the genuine reference object and the
equivalent data from the candidate items. This study’s advantage is in the
simplicity of the measurement as it avoids actual imaging and thus enables
higher degree of information security. It is also a direct experimental proof
of feasibility which is sensitive to the backgrounds, noise, and system
instabilities not captured in a numerical simulation.

The overall high level verification exercise is described in detail in Results
section of Hecla and Danagoulian[17]. In brief, in an unannounced visit to an
ICBM base the joint team of the inspectors and the hosts acquires a randomly
selected nuclear weapon, whose authenticity is thus established by its
situational context. The weapon is then transported under joint custody, i.e.
in the presence of both parties and via application of tamper-proof seals and
checks[22], to a facility where it is disassembled in a controlled environment
such that no new objects are introduced or removed. The fissile hollow sphere,
from here on referred to as the pit, is removed and used as a genuine
reference object. This study focuses on the stage involving the actual
measurements test, where the genuine reference object is being compared to the
TAI candidate objects. The test needs to detect both isotopic hoaxes as well
as geometric hoaxes, described later in this section. The experimental
procedure consists of the following steps:

  1. 1.

The genuine object, an encrypting filter, and the neutron detector are aligned
along the beam z-axis. The encrypting filter is produced by the host, and its
composition is unknown to the inspectors. The system schematic can be seen in
Figure 1. The filter can be made of any material, with the assumption but not
the requirement that the hosts will add elements identical to those in the
TAI, with the only requirement being that the filter allow a significant flux
of neutrons to reach the detector for an agreed upon measurement time. A
spectroscopic measurement is taken, providing the template data (from here on
referred to as template) that the next measurements will be compared to.

  2. 2.

Keeping the encrypting filter and the detector in their places the genuine
object is replaced with the candidate object at the same orientation. A
measurement under experimental conditions identical to step (1) is performed.

  3. 3.

The data from the measurement of the candidate object is compared with the
template in a statistical test. If the test identifies a statistically
significant difference then the candidate is declared a hoax. If the new
measurement passes the statistical test then the procedure moves on to the
next stage, where measurements under random rotations are performed.

  4. 4.

Measurements described in the steps (1) and (2) are now repeated with both
genuine and candidate objects rotated at a random angle. To keep the number of
unknowns larger than the number of measurements a new encrypting filter is
provided by the hosts. Like before, the filter is kept in place during the two
(genuine, candidate) rotated measurements. Similar to step (3) the two
measurements are compared in a statistical test. If the measurements pass the
statistical comparison then the candidate is declared isotopically and
geometrically identical to the genuine, and is thus accepted as genuine as
well.

This work focuses on a sensitivity study, experimentally demonstrating the
feasibility of the comparison. The technique is shown capable of clearing
objects identical to the genuine reference object and rejecting objects which
are isotopically or geometrically different. The verification exercise focuses
on the weapons grade plutonium (WGPu) pit, primarily due to the weapon’s
hydrogenous components’ near-opacity to epithermal neutrons. HMX explosive,
for example, has a mean free path of $\sim$1.8 cm for epithermal neutrons.
Public domain data on estimates of a Soviet thermonuclear design[23] indicate
that the explosive’s thickness can be $\sim$6.5 cm, translating to an
attenuation factor of about 1000. While not impossible, a transmission
measurement of weapons at a lesser degree of disassembly would require either
more intense neutron sources, or longer measurement times. This may become
possible in the future, as the laser driven and other high intensity neutron
sources currently under development become more applicable for field use[24,
25]. It should be added that the plutonium pit may also contain other
materials, such as coatings to prevent the air’s corrosive effects on
plutonium, as well as the beryllium neutron reflector shell which may be
bonded to the plutonium shell. The later has a mean free path of 1.3 cm for eV
neutrons, and depending on its thickness its attenuation effects may translate
to measurement times somewhat longer than those estimated in this work. This
study thus focuses on an inspection scenario where the plutonium pit has been
extracted in a controlled environment and undergoes verification, as described
in prior work[17].

Figure 1: General schematic of the experiment (not to scale). The combination
of isotopic and geometric sensitivity is achieved by neutron spectroscopy via
time-of-flight (TOF) technique and comparisons between the candidate and the
genuine reference under random projections. For simplicity, in this experiment
the angles of the projections were chosen to be 0∘, 45∘, and 90∘. Figure 2:
Cross sections. Total interaction cross sections for neutron induced
interactions for a subset of molybdenum and tungsten isotopes with significant
resonances in the 1-200 eV range. Figure 3: Histograms of neutron count
energies for two isotopic hoaxes. a The template data from an authentic
reference of 90/10 Mo/W composition (blue) is compared to the data from a
50/50 Mo/W isotopic hoax (red). b The data from the authentic reference is
compared to that from a 10/90 Mo/W isotopic hoax (red). The legend lists the
$\chi^{2}$ value, the number of degrees of freedom, and the corresponding
probability (p-value) for the $\chi^{2}$ test. Both hoaxes are rejected. The
solid and dashed vertical lines denote the locations of some of the known
tungsten and molybdenum resonances, respectively. Data collection lasted
approximately five minutes per object.

###  0.2 The experiment

The overall diagram of the measurement setup can be seen in Figure 1, and is
discussed in more detail in Methods section. The neutrons are produced in a
pulsed mode from a pulsed electron linear accelerator, via a tantalum
bremsstrahlung-photoneutron converter, which produces neutrons via the
$(\gamma,n)$ reaction. A 2.54 cm polyethylene moderator degrades the neutrons’
energy from the $\sim$MeV scale to the $\sim$eV scale. As the epithermal
neutrons traverse the object and the encrypting filter, their spectrum is
modulated in accordance to the cross sections and areal densities of the
relevant isotopes, after which they are detected by a 6Li glass scintillator
detector. The detector produces a timing pulse, which is compared to the
timing pulse of the linear accelerator. This comparison allows to determine
$t_{\text{tof}}=t-t_{0}$ of the neutrons. The energy of individual neutrons
can then be reconstructed via $E=m(d/t_{\text{tof}})^{2}/2$, where $m$ and $d$
are the neutron mass and the flight path length, respectively. The
transmission spectra are normalized to the incident neutron count, determined
by a fission chamber (not shown) upstream of the target object.

Figure 4: Histograms of neutron count energies for a geometric hoax. The
geometric hoax has been constructed as to have the same areal density along
z-axis as the genuine reference. The template data from an genuine reference
(blue) is compared to the data from the geometric hoax (red) at
$\theta=0^{\circ}$ (a), $\theta=45^{\circ}$ (b), and $\theta=90^{\circ}$ (c).
As expected per design the $\theta=0^{\circ}$ yields a perfect agreement, as
shown by the p-value. The rotations readily expose the hoax. Figure 5:
Spectral comparisons between a genuine reference and an honest candidate. The
measurements were performed at rotation angles $\theta=0^{\circ}$ (a),
$\theta=45^{\circ}$ (b), and $\theta=90^{\circ}$ (c). All comparisons yield
p-values of 50-95%, indicating an agreement, and thus clearing the candidates
as genuine objects.

###  0.3 Targets

As the research was performed in academic setting, plutonium and uranium
targets were not available. Thus, targets were built of more commonly
available materials which have significant resonances in the relevant region
of $1\text{eV}\leq E\leq 200\text{eV}$ and can thus act as proxies for most
uranium or plutonium isotopes. Natural molybdenum and tungsten were chosen as
element-to-isotope proxies for 239Pu and 240Pu, respectively. Instead of
hollow spheres, which are typical to nuclear weapon pits, simpler cylindrical
geometries were chosen, such that the genuine proxy object is a cylinder of
50.8 mm diameter and 30 mm length, with the first 27 mm consisting of
molybdenum and 3 mm consisting of tungsten. Such a combination was chosen to
mimic the $\sim$90% enrichment of WGPu. Plots of total neutron interaction
cross sections for a subset of molybdenum and tungsten isotopes can be seen in
Figure 2. Unlike molybdenum or tungsten the actinide targets would produce
additional neutrons via the (n,fission) reaction. However the resulting
neutrons would be of the $\sim$MeV energies, making the 6Li detector
insensitive to them. The encrypting filters were assembled from 3 mm and 6.35
mm thick tungsten and molybdenum plates, respectively, that had 76.2 mm
$\times$ 76.2 mm outer dimensions. The measurements at 0∘ rotation, described
in Figures 3, 4(a), and 5(a), used a filter consisting of three tungsten and
three molybdenum plates aligned with the beam. The measurements at 45∘ used a
filter consisting of two tungsten and one molybdenum plates, and at 90∘ the
filter consisted of one tungsten and one molybdenum plates. See Methods for
additional detail on experimental settings.

###  0.4 Epithermal Neutrons

In this experiment the neutron energies at the source spanned from the
$~{}\sim$MeV scale, known as fast neutrons, to $\sim$meV, referred to as
thermal neutrons. However the use of a thin cadmium filter near the source
filtered out most neutrons below 0.5 eV. The detector used to register the
neutron hits was based on 6Li isotope, which interacts with neutrons via the
6Li$(n,t)\alpha$ reaction, with the triton and the alpha causing scintillation
in the glass and thus triggering detection. When combined with the TOF
technique this allows to determine the neutron’s kinetic energy with $\ll$eV
precision, allowing to resolve narrow absorption lines such as the 0.13 eV
wide line at 70.9 eV from 97Mo. The spectra in Figures 3, 4, and 5 were
produced via this method.

The results of isotope-specific resonant absorption are most readily
observable in Figure 5. The solid vertical gray lines indicate the expected
positions of the tungsten resonances, while the dotted lines indicate the
positions of the molybdenum resonances. Depending on an isotope, the
resonances are due to the (n,$\gamma$) and (n,n′) reactions, which selectively
remove neutrons from the beam at the resonant energies. The relative depth of
the absorption lines primarily depend on the resonance width, with broader
widths translating to higher integrated cross sections and thus more
absorption. The shape of the absorption line in part depends on the type of
the interaction. These combined absorption features yield a unique spectral
fingerprint of a particular configuration of isotopic, geometric, and density
distribution. Future research however should focus on providing a systematic
proof of this uniqueness, i.e. that no combination of other elements can
reproduce uranium or plutonium signatures. See Supplementary Note 4 for a
detailed discussion.

###  0.5 Isotopic Hoax resistance

Isotopic hoax resistance refers to a system’s ability to detect hoaxing
attempts which involve modifications of the isotopic ratios in an otherwise
geometrically identical object. For a real warhead this may involve replacing
the WGPu pit, which has enrichment levels of more than 90%, with a pit made
from the more available Reactor Grade Plutonium (RGPu), which has a lowered
239Pu enrichment of approximately 60%, with the remaining $\sim$40% in the
form of 240Pu.

To test the sensitivity of the technique to isotopic variations the 90%:10%
Mo:W genuine reference object’s transmission spectrum was compared to the
transmission spectra of 50%:50% and 10%:90% Mo:W objects of the same
cylindrical shape and overall dimensions. Figure 3 shows the plots and
comparisons of the spectra. The comparisons show that as the proportion of
tungsten is increased the tungsten absorption lines show increased absorption,
while the molybdenum lines exhibit reduced absorption, as expected. A simple
$\chi^{2}$ test can be applied, determining the values of $\chi^{2}$ and the
corresponding probability $p$ that the difference is merely due to random
fluctuations. For these comparisons the $p$-value is consistent with zero to
ten significant digits, and thus indicates that a systematic difference is
present. The $\chi^{2}$ test thus rejects the comparison, and indicates a
hoaxing attempt. All measurements lasted approximately 5 minutes.

###  0.6 Geometric Hoax resistance

A single measurement along the beam’s z-axis is only sensitive to the
projected isotopics along that axis. The transmission $T$ for a an energy bin
$E$ and a spherically symmetric geometry can be approximated as

$\ln
T\simeq-\sum_{i}\frac{\sigma_{i}(E)N_{\text{Av}}\rho_{i}}{A_{i}}2\pi\int_{1}^{\cos\theta^{\prime}}\int_{0}^{X}f_{i}(r,\cos\theta)drd\cos\theta,$

where $\sigma_{i}(E)$, $\rho_{i}$, $X$, $\theta^{\prime}$ and $A_{i}$ are the
cross section, density, total thickness, polar angle, and atomic number for
isotope layer $i$ of the object, respectively. $f_{i}(r,\cos\theta)$ is the
fractional abundance of isotope $i$ at a particular radial coordinate, and
$N_{\text{Av}}$ is Avogadro’s number. The summation takes place over all the
isotopes, and the integration is along the radius and the polar angle. Thus,
the single pixel camera measurement effectively collapses the three dimensions
into a zero-dimensional energy-dependent-only measurement. This limitation can
be readily overcome by making multiple measurements at various angles, similar
to tomographic measurements where a single pixel can ultimately produce a
three dimensional sensitivity. Unlike a tomography where the goal is to
produce a three dimensional image, here the goal is to simply verify that two
objects are identical to within some level of geometric complexity. Prior work
has shown that two to three measurements at randomly selected angles are
sufficient to achieve this goal[12]. The random selection of the angles at the
time of the verification exercise renders the hosts unable to optimize a
possible geometric hoax to the measurement. The problem is thus reduced to
proving that the system is sensitive to geometric differences under rotations.

To show this sensitivity we perform measurements at 45∘ and 90∘ relative to
the beam’s z-axis. For this purpose the geometric hoax object was chosen to be
a rectangular parallelepiped, i.e. a box, of the same composition along the
z-axis as the genuine reference object. It consists of a 50.8 x 50.8 x 27 mm3
box of molybdenum followed by a 50.8 x 50.8 x 3 $mm^{3}$ plate of tungsten.
Since the beam diameter is only 44.5 mm, the spectral measurement results from
two objects should look identical at 0∘. However at 45∘ and 90∘ the geometric
differences should amount to different values of $\ln T$.

To test the above hypothesis and the sensitivity of the system under rotations
the measurements were performed under identical beam conditions. Furthermore,
for every rotation the encrypting mask was modified but kept the same between
genuine-candidate comparisons. This is a necessary step, as to prevent any
differential analysis of data from multiple angles which may otherwise reveal
geometric information. In analogy to an under-defined system, every new
measurement needs an additional unknown, in the form of a different encrypting
filter. Figure 4 shows the spectral comparisons. It can be seen that at 0∘ the
spectra are essentially identical, when compared with a $\chi^{2}$ test.
However, when rotated the small but significant differences in geometries give
rise to detectable deviations. While not as large as in the isotopic hoax, the
differences are nevertheless statistically significant. This result shows that
for these experimental settings, object types, and measurement times the
system can detect these geometric hoaxes with rotations by $\theta\geq
45^{\circ}$.

In this experiment all the rotations were centered on the geometric center of
the object. However, hollow spheres of the same thickness but different radii
would produce identical signals under such rotations, due to their spherical
symmetry. This would result in an effective geometric hoax. To break this
symmetry the rotations should either involve a randomly chosen center, or be
followed by random translations. Overall this study focuses on a limited set
of hoaxing scenarios, and future studies need to focus on further extending
these tests. As with most cryptographic protocols, the concept presented in
this work should undergo future tests and checks. The use of the K-transforms
in particular may prove promising in providing a proof of geometric
uniqueness[12].

###  0.7 Completeness

Finally, the test procedure needs to also show that it is able to accept
honest candidates. As with the geometric hoax sensitivity test, here the test
needs to clear the honest candidates at 0∘, $45^{\circ}$ and $90^{\circ}$
rotations, as a way of confirming that the objects are identical both
isotopically and geometrically. We thus repeat the procedure described in the
previous subsection. This is necessary in order to rule out the possibility
that the previously observed differences were merely the results of system
instabilities, such as beam energy variations or detector gain shifts. Figure
5 shows comparisons where the genuine reference object measurements are
compared to those from identical, honest objects. Between these measurements
the honest objects as well as the encrypting filters were replaced with other
targets and filters and then were placed back for the comparison measurements.
A marker on the target holder allowed for the precise setting of the
rotational angle. In one case the accelerator had to be turned off and
restarted due to beam instabilities. As expected, and despite technical
disruptions, in all cases the $\chi^{2}$ test indicates that the differences
are merely statistical, as the $p$-value of the test indicates an agreement.
It is then concluded that the system is capable of clearing honest candidate
objects. A summary of the test results are listed in Table 1. Per procedure
described at the beginning of the section, the test immediately rejects the
attempts at isotopic hoaxing. The geometric hoax shows an identical signal to
the genuine reference object at measurements along the z-axis, however
rotations by 45∘ or larger immediately results in a rejection. Finally, the
honest candidate measurements pass the test for all angles of rotation, and
are thus accepted as honest objects.

It is necessary to acknowledge that a small rate of false positives may be
caused by difference in ages between honest pits, caused by slight differences
in 241Pu concentrations due its beta decay into 241Am, as well as machining
tolerances in pit production. As demonstrated in Supplementary Note 3 the
former is a minor effect that should not cause false positives for pits with
age difference of less than ten years. The later effect can be accurately
estimated only given classified information on pit production, and should be
addressed in the classified setting. Our current modeling indicates that
differences of less than 1 mm should not trigger false positives.

candidate object Mo/W composition and shape |  rotation |  $\chi^{2}/$NDF |  p value | decision
---|---|---|---|---
Isotopic Hoaxes |  |  |  |
50/50, cylinder | 0∘ | 5491 / 1198 | 0.00 | reject
10/90, cylinder | 0∘ | 10957 / 1175 | 0.00 | reject
Geometric Hoax |  |  |  |
| 0∘ | 1196 / 1216 | 0.42 | accept
90/10, cube | 45∘ | 1532 / 1225 | 0.00 | reject
| 90∘ | 1823 / 1245 | 0.00 | reject
Honest Candidate |  |  |  |
| 0∘ | 1140 / 1218 | 0.94 | accept
90/10, cylinder | 45∘ | 1124 / 1224 | 0.98 | accept
| 90∘ | 1234 / 1245 | 0.58 | accept
Table 1: Statistical comparisons between the candidates and the genuine
reference object. The composition here refers to the relative lengths along
z-axis of molybdenum/tungsten components of the candidate objects. The total
length of the objects is kept identical to that of the genuine reference. The
genuine reference is a cylinder of the length of 3cm, and Mo/W composition of
90/10. In order to pass inspection, the objects have to first pass the
$\chi^{2}$ test along z axis ($\theta=0$), and then along two rotations at 45∘
and 90∘. The procedure is shown to accept all honest candidates and rejects
both isotopic and geometric hoaxes.

###  0.8 Information Security

One of the important goals of the verification system is the information
security, that is the inspectors’ inability to learn significant information
about the TAI. In our prior work we showed via computational simulations that
while the inspectors can learn information about the combined content of the
TAI and the encrypting filter, they cannot learn anything specific to the TAI
itself – see Hecla and Danagoulian[17], subsection on Isotopic Information
Security. The primary focus of this study is the sensitivity analysis, with
the purpose of showing that the concept is capable of detecting hoaxing
attempts. Nevertheless, we use experimental data from the measurements in
combination with a Geant4[26, 27] computational model of the experiment to
simulate scenarios with a WGPu pit and an encrypting filter of WGPu weighting
1.8 kg. In these data-driven simulations we show that various opposite
combinations of the TAI and filter geometries produce statistically
indistinguishable signals. This makes it impossible for the inspectors to
learn information of value, similar (but not identical) to the concept of
zero-knowledge proof. At the same time, additional simulations show that pairs
of 5 minute long measurements at an experimental facility similar to the one
used in this study can readily detect cheating scenarios where the WGPu pit
has been replaced with RGPu, or where its size has been reduced by 2 mm or
less, depending on experimental conditions. See Supplementary Note 3 for a
detailed discussion. In Supplementary Note 1 it is demonstrated that using an
encrypting filter of just 1.8 kg will result in an inferred range of pit’s
possible mass that spans from zero to a value that is significantly larger
than the critical mass. At the same time the combined target thickness will be
small enough to allow for significant counting statistics and thus reasonable
measurement times. The impact of temperature dependent broadening of the
resonances is also estimated and described in Supplementary Note 2, showing
that the effect can be made negligible.

The presence of absorption lines associated with weapon materials whose
presence may be a priori unknown to the inspecting side is an important
consideration in the context of information security. If those materials were
to have resonances in the energy domain of the measurement, the inspector
would learn about their presence. Given that this study focuses on a scenario
where a bare pit undergoes verification, the list of the elements that would
be of significant quantities is limited to the actinides and gallium, which is
used to stabilize the plutonium in its $\delta$-phase. Nevertheless, to
address possible concerns, the treaty signatories can agree that the host will
put samples of all elements that contain resonances in the agreed-upon range
in their encrypting filters. In this manner the inspectors will not be able to
determine whether the absorption lines come from the weapon, or the filter.
The problem also has a technological solution. A neutron-opaque chopper
synchronized with the neutron pulse can be added to the beamline, with an
opening chosen such that it only lets through neutrons of a specific energy
range. For example, for this experiment the [1,50] eV range translates to TOF
of [150,1061] $\mu$s. With an accelerator pulse rate of 25 Hz, a chopper
rotating at 12.5 Hz would need two 41∘-wide openings centered at 27.25∘ and
207.25∘ phases relative to the pulse to pass only the [1,50] eV neutrons. This
opening angle and the rotation phase can be tuned to match the desired energy
range, including that of a single resonance. Any hoaxing attempts involving
shifting of phase or frequency would be readily exposed by the TOF data.

Finally, the chain of custody for the selection of the genuine object and its
tamper-proof transportation to the dismantlement and verification facility is
an important topic. Past literature has discussed various approaches for
achieving this goal[6, 28, 29]. Exercises by the US and European national
laboratories and organizations have taken place to test and characterize
series of techniques necessary for this goal. The Letterpress Exercise in
particular has focused on the high level procedure, as well as the various
simple but robust tamper-proof methods. Such techniques include electrical,
mechanical, as well as reflective particle tag seals. Many of these low-tech
methods use a physical randomness to ensure that a broken seal cannot be faked
or reproduced. For a detailed discussion see Smartt and Marleau[22] and Bunch
et al.[28]. In addition to the particularities of the chain of custody, the
arms control treaties stipulating verified dismantlement will have to
negotiate the various conditions under which a particular verification
technique is applied. For the method described in this study in particular,
the two sides should negotiate the following parameters: the measurement time,
beam current, and the resulting statistics, such that it is sufficient to
detect the likely hoaxes without raising any worries about excessive
information; the minimum count rate to be observed by the detector, derived
from broad considerations of the weapon material quantities, as a way of
ensuring that the hosts do not add neutron-opaque materials to the encrypting
filter and thus make the measurements impossible; the manner in which the
detector and source instrumentation will be validated before and after the
measurements, as away of ensuring that no hoaxing and no information leakage
has occurred. Generally, the use of physical cryptography makes the acquired
data fundamentally encrypted, thus significantly reducing information security
risks associated with the instrumentation that acquires the data. However
validation of the instrumentation can further strengthen confidence. To this
end, the inspection side could be asked to provide $n$ copies of the
instrumentation package, with the option for the hosts to perform invasive
validation of the $n-2$ copies, leaving the last two copies to be used
interchangeably in the verification exercises. In this case the probability of
the inspecting party to introduce two malicious packages undetected is
$p=1/{n\choose 2}=2/[n(n-1)]$. It should be clarified that we make no claims
that the concept proposed in this study is a zero-knowledge proof. Physical
cryptography implies a degree of information security less than zero
knowledge, with a goal of protecting most useful information, but not
necessarily all information. The considerations listed in this section are
only part of a larger set of possible circumstances to consider. As is the
case with most cryptographic schemes, future research should focus on the
pursuit of possible vulnerabilities and for solutions for those.

## Discussion

Past arms control treaties lacked a method for reliably verifying the
dismantlement of the nuclear weapons, and instead focused on the delivery
systems, such as ICBM and strategic bomber aircraft. This work experimentally
demonstrates the feasibility of using resonance phenomena via epithermal
neutrons transmission to compare two objects and verify that they are
isotopically and geometrically identical, as part of a template verification
exercise. Any differences in isotopics and geometry between the candidate and
the genuine reference object can be detected via isotope-sensitive
measurements by using multiple rotations. Similar to the requirements of zero-
knowledge proof, the inspectors cannot learn significant information about the
object itself, and at most can infer the combined mass and isotopics of the
object and the encrypting filter. These combined inferred quantities thus
constitute upper bounds for the object, corresponding to the scenario of an
empty encrypting filter. It is shown that by selecting thicker encrypting
filters of just 1-2 kg of WGPu these upper bounds can be made large enough to
make it impossible to infer any useful, new information. Unlike most
information barrier concepts the sensitive information is encrypted in the
physical domain, thus achieving a very high degree of information security.
For an in-depth discussion see the sections on information security in Hecla
and Danagoulian[17]. Using the neutron beam at an experimental facility
similar to the one described in this study the measurements can be performed
in the combined time of less than an hour.

Additional and continued research in security studies arena and by the weapons
laboratories is needed to address the various high level political and
technical aspects of a treaty that would employ the techniques described in
this study. In particular the joint chain of custody is paramount to the
validity of the genuine object which establishes the template. While no
treaties stipulating the verified dismantlement of the weapons have been
enacted, exercises by US and European national laboratories and institutes
have explored the various real world considerations, testing a variety of
protocols and techniques to insure a secure chain of custody[22]. This study
presents only a basic concept, which will need to be embodied in particular
designs adapted to the particular treaty conditions and the characteristics of
the available facilities and instrumentation.

Finally, research needs to be performed to study the feasibility of this
technique in more compact, possibly relocatable systems. The work presented in
this study used a large facility due to its convenience and the high precision
neutron beam that it produced. However smaller, more compact neutron sources
may also enable such measurements. Such sources could be in the form of a
pulsed deuterium-tritium (DT) generator, or small, commercial, linear
accelerator driven sources which use bremsstrahlung beams to produce neutrons
via ($\gamma$,n) reactions. The later systems are advantageous due to the
large neutron flux that they would generate, and would be similar to the
facility used for this work, albeit at much smaller, laboratory-size scales.
While this study focused on the verification of plutonium components of a
disassembled weapon, the advent of higher intensity neutron sources, e.g.
using laser-driven methods[24, 25], may allow for the verification of fully
assembled weapons. The use of compact neutron source platforms may also allow
for implementations of the technique described in this study using non-
electronic detectors and preloads, similar to the concept by Philipe et
al[11], with the possibility of extending this physically cryptographic method
to a fully zero-knowledge proof system.

## Methods

###  0.9 Experimental Facility

The experiment was performed at the 15 m station of the Gaerttner LINAC
Research Center at the Rensselaer Polytechnic Institute. The neutron beam was
produced via a 60 MeV electron beam from the LINAC, which produced
1$\mu$s-long pulses with a repetition of 25 Hertz. An electronic signal from
the pulse sets the start time $t_{0}$ necessary for the TOF calculation. The
beam was incident upon a tantalum photoneutron target. Tantalum, due to its
high atomic number, is a powerful source of bremsstrahlung photons when
impinged by electrons. At 60 MeV most of the energy loss by the electrons is
radiative, i.e. via bremsstrahlung electromagnetic radiation. At the same time
the nucleus of 181Ta has a photoneutron production threshold of 7.6 MeV, via
the 181Ta($\gamma$,n)180Ta reaction. The cross section for this reaction peaks
to approximately 0.37 barns at 14.7 MeV. Since the energy distribution of the
bremsstrahlung photons is continuous, most of the photons overlap with the
energy domain at which the photodisintegration of 181Ta is at a maximum. The
neutrons have an energy that is $E_{\text{n}}=E_{\gamma}-7.6$ MeV, where
$E_{\gamma}$ is the energy of the bremsstrahlung photon. Thus most of the
neutrons are produced in the $\mathcal{O}$(MeV) range. To produce an
epithermal beam a 2.54 cm polyethylene target was placed near the tantalum
target, allowing a small fraction of the fast neutrons to undergo elastic
scattering on the hydrogen in polyethylene and thus lose energy in a process
referred to as moderation. The resulting neutron energy distribution spans
thirteen orders of magnitude from fast, MeV scale to the thermal, meV scale.
For a detailed description of the facility and the target design see Section I
in Danon et al.[30].

The resulting polyenergetic neutron beam was then transported via a beamline
to the 15 m station, named so because of the target to detector distance of
approximately 15 m. A 6Li glass detector detected the neutrons, and registered
their arrival times relative to the $t_{0}$ of the electron pulse. The neutron
detector consisted of 3 mm-thick lithium glass scintillator with 6.6% lithium
content, enriched to 95% in 6Li. The scintillator is also sensitive to
photons, however the photon arrival times would be $\mathcal{O}$(50ns), and
thus could be rejected during the TOF analysis. Additional photons are present
during the neutron arrival times, primarily due to (n,$\gamma$) capture of the
thermalized neutrons near the detector volume. Those events however can be
suppressed due to the relatively weak light output by photon interactions in
the scintillator. The only remaining background in the TOF analysis are the
epithermal and thermal neutrons that scattered around the beamline. This
effect can be seen for example as the counts at the bottom of the 15-30 eV
absorption lines in Figure 3. This background however is constant, doesn’t
change between multiple measurements, and thus doesn’t affect the comparisons
performed in this study. For a detailed description of the detector, the data
acquisition electronics, and the general experimental setup see Section II in
Danon et al.[31].

###  0.10 Energy reconstruction via TOF and statistical analysis

Due to a thermalization times of $\lesssim$ 1 $\mu$s and flight times of
$\sim$ms, the pulse $t_{0}$ amounted to an good initial measure of the
epithermal neutron emission time. The detection of the neutron in the
scintillator produces a second pulse with a time $t$, and the energy of the
neutron can be then inferred from $E=m(d/t_{\text{tof}})^{2}/2$, where
$t_{\text{tof}}=t-t_{0}$. During the analysis of the data however it became
apparent that the slight dependency of the moderation time on resulting
neutron’s energy significantly distorted the energy reconstruction. This
effect was particularly significant for the higher energy neutrons in the
spectrum. To quantify this effect simulations of this process were performed
for the known geometries of the moderator, using the Geant4 object oriented
simulation toolkit[32]. The energy-dependent corrections for thermalization
time, which varied in the 4-8 $\mu$s range, were applied to the data,
resulting in a precise reconstruction of the known molybdenum and tungsten
resonances up to 200 eV.

The energy dependence of the detected neutrons in the open beam varied
significantly with energy. This was caused in part by the intrinsic neutron
flux, which changes significantly due to the incident neutron spectrum, caused
by the underlying moderation process in the polyethylene moderator. It is also
caused by the decreasing sensitivity of the 6Li glass detector with increased
energy, due to the power-law dependence of the 6Li(n,t)4He reaction, which has
the form of $\sigma\propto E^{-1/2}$. The data acquisition of the facility
bins the counts in 0.1 $\mu$s wide time bins in the 1-200 eV range. When
converting to the energy via TOF, the TOF distribution, which can be described
as the time differential of the cumulative counting probability $dP/dt$, is
weighted by the Jacobian, $dE/dt=-2E/t_{\text{tof}}$, such that
$dP/dE=\frac{dP}{dt}/|\frac{dE}{dt}|$. All statistical errors are propagated
accordingly to the final energy spectrum. Finally, to even-out the statistics
throughout the broad 1-200 eV range, the uniform binning is changed to a non-
uniform binning, where increasingly larger bins are assigned with increasing
neutron energy.

The resulting spectral data from a pair of measurements were compared in a
$\chi^{2}$ statistical test. Assuming normally distributed errors, it is
defined as $\chi^{2}=\sum_{1}^{N}(c_{0,i}-c_{1,i})^{2}/(c_{0,i}+c_{1,i})$,
where $c_{0}$ and $c_{1}$ are the count arrays for the spectra undergoing
comparison, and $N$ is the total number of bins. Given a value of $\chi^{2}$
and $N$, the probability, also known as p-value, can be determined from the
$\chi^{2}$ distribution. p-value describes the probability that two spectra’s
differences are due to statistical fluctuations alone. A low p-value, e.g.
$\ll$1%, is indicative of statistical effects due to cheating by the hosts.

###  0.11 Data availability

All the data supporting the findings of this study are available from the
corresponding author upon reasonable request.

## References

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## Acknowledgments

This work is supported in part by by the Consortium for Verification
Technology under Department of Energy National Nuclear Security Administration
Award DE-NA0002534. The authors are grateful for the support and encouragement
from their colleagues within the Consortium of Verification Technologies. The
authors thank the staff at The Gaerttner LINAC Research Center and Yaron Danon
in particular for assistance with the experiment, as well as valuable
suggestions and advice. The authors are grateful to Igor Jovanovic for
valuable comments on the manuscript. The authors also thank Ethan A. Klein for
cataloging the nuclear data for (n,$\gamma$) and (n,n′) reactions on
molybdenum and tungsten. We express our gratitude to Jacopo Bongiorno for help
with equilibrium heat transfer calculations, and to the reviewers for their
insightful comments.

## Author Contributions

A.D. conceived the project. A.D. and E.M.E. performed the experiments and
analyzed the data. A.D. wrote the manuscript.

###  0.12 Competing Interests:

The authors declare no competing interests.


[a]Xuan Li

# Forward silicon vertex/tracking detector design and R$\&$D for the future
Electron-Ion Collider

Melynda Brooks  Matt Durham  Yasser Corrales Morales  Astrid Morreale
Christopher Prokop  Eric Renner  Walter Sondheim

###### Abstract

The proposed high-luminosity high-energy Electron-Ion Collider (EIC) will
provide a clean environment to precisely study several fundamental questions
in the fields of high-energy and nuclear physics . A low material budget and
high granularity silicon vertex/tracking detector is critical to carry out a
series of hadron and jet measurements at the future EIC especially for the
heavy flavor product reconstruction or tagging. The conceptual design of a
proposed forward silicon tracking detector with the pseudorapidity coverage
from 1.2 to 3.5 has been developed in integration with different magnet
options and the other EIC detector sub-systems. The tracking performance of
this detector enables precise heavy flavor hadron and jet measurements in the
hadron beam going direction. The detector R$\&$D for the proposed silicon
technology candidates: Low Gain Avalanche Diode (LGAD) and radiation hard
depleted Monolithic Active Pixel Sensor (MALTA), which can provide good
spatial and timing resolutions, is underway. Bench test results of the LGAD
and MALTA prototype sensors will be discussed.

##  1 Introduction

The proposed Electron-Ion Collider (EIC) aims to solve several fundamental
questions in the field of high-energy nuclear physics. An example is exploring
the hadronization process [1]. The EIC will utilize polarized and unpolarized
electron+proton ($e+p$) and electron+nucleus ($e+A$) collisions with a variety
of nuclear species (mass number A = 2-208) at the center of mass energy from
20 GeV to 141 GeV. The expected instantaneous luminosity is around
$10^{33-34}cm^{-2}sec^{-1}$, which is a factor of 1000 higher than the Hadron-
Electron Ring Accelerator (HERA) collider. The beam crossing rate varies from
1 $ns$ to 10 $ns$ depending on the collision system. A high granularity, low
material budget tracking detector, which can provide good spatial and momentum
resolutions, is required especially in the hadron beam going direction to
precisely measure charged particles produced by the asymmetric collisions at
the EIC. A Forward Silicon Tracker (FST) based on the Monolithic Active Pixel
Sensor (MAPS) technology has been developed to provide precise track
reconstruction in the pseudorapidity region of 1.2 to 3.5. A series of heavy
flavor hadron and jet observables have been enabled by this detector and will
provide enhanced sensitivities to explore the hadronization process in vacuum
and in a nuclear medium [2, 3]. The conceptual design of the proposed FST, its
tracking performance and progresses of the associated detector R$\&$D will be
discussed.

##  2 The Proposed Forward Silicon Tracker Conceptual Design and Its
Performance

The current design of the proposed FST consists of three MAPS [4] disks with
the pixel pitch of 10 $\mu$m and two Depleted MAPS (referred to as MALTA) [5]
disks with the pixel pitch of 36.4 $\mu$m. In addition to good momentum and
spatial resolutions which can be offered by general MAPS trackers [4], this
hybrid silicon tracker provides fast timing resolution at around 5 $ns$, which
can help separate events belonging to different beam bunch crossings at the
EIC and suppress backgrounds. The latest geometry parameters of the FST
conceptual design are listed in Table 1. This detector concept has higher
pixel densities compared to existing silicon detector at Relativistic Heavy
Ion Collider (RHIC) and the Large Hadron Collider (LHC). The detailed detector
layout, which includes the silicon sensor segmentation, cooling system,
readout distribution and mechanical structure has been implemented in the
GEANT4 simulation.

Disk index | technology | inner radius | outer radius | z location | pixel pitch |  sensor thickness |  No. of pixels
---|---|---|---|---|---|---|---
1 | MAPS | 3.5 cm | 15.5 cm | 30 cm |  10 $\mu$m |  50 $\mu$m |  $\approx$0.72B
2 | MAPS | 3.5 cm | 33.5 cm | 53.75 cm |  10 $\mu$m |  50 $\mu$m |  $\approx$3.49B
3 | MAPS | 5.1 cm | 41.1 cm | 77.5 cm |  10 $\mu$m |  50 $\mu$m |  $\approx$5.22B
4 | MALTA | 7.0 cm | 45.0 cm | 106.25 cm |  36.4 $\mu$m |  100 $\mu$m |  $\approx$0.39B
5 | MALTA | 8.0 cm | 46.0 cm | 125 cm |  36.4 $\mu$m |  100 $\mu$m |  $\approx$0.46B
Table 1: The FST conceptual design geometry [3].

The top panel of Figure 1 shows the geometry of the proposed FST integrated
with the other EIC detector subsystems inside the Babar magnet, which provides
the maximum magnetic field at 1.4 T. The track momentum dependent momentum
resolutions and transverse momentum dependent transverse Distance of Closet
Approach ($\text{DCA}_{\text{2D}}$) resolutions in seven different
pseudorapidity ranges from 0 to 3.5 are illustrated in the right panel of
Figure 1. Different FST designs and the corresponding tracking performances
from different integrated detector and magnet options have been studied [3].
Recent simulation studies have validated that this proposed FST can enable
precise heavy flavor hadron and jet measurements in the forward pseudorapidity
region of $1.2<\eta<3.5$ at the EIC [2, 3].

Figure 1: The FST conceptual design implemented in GEANT4, which is integrated
with the other EIC detector subsystems is illustrated in the left. The
tracking momentum dependent momentum resolutions and the transverse momentum
dependent transverse Distance of Closet Approach ($\text{DCA}_{\text{2D}}$)
resolutions in different pseudorapidity regions of $0\leq\eta\leq 3.5$ with
the Babar magnet are shown in the middle and right. Figure 2: Configuration of
the two-layer LGAD telescope ${}^{90}Sr$ source test. Each LGAD sensor has 15
pixels wire bonded to the readout output (left). The analog pulse from
individual pixel is shown in the right top panel. The LGAD digitized pulse
processed by the CAEN 1730s digitizer is shown in the right bottom panel.

Figure 3: The bias voltage magnitude dependent charge collections for four
different types of LGAD sensors. Sensor test result with a laser scan for one
set of four LGAD sensors is shown in the left and analysis results of the
${}^{90}Sr$ source test digitized data with an another set of four LGAD
sensors are illustrated in the right.

##  3 R$\&$D for Advanced Silicon Technology Candidates

In addition to the MAPS and Depleted MAPS technologies, other advanced silicon
sensor candidates such as Low Gain Avalanche Diode (LGAD) [6] and AC-Coupled
LGAD (AC-LGAD) [7] are under consideration for the generic EIC silicon
vertex/tracking detector. These technologies can provide either better than 5
$\mu$m single hit spatial resolution or less than 30 $ps$ timing resolution.
Test benches for both LGAD (AC-LGAD) and MALTA prototype sensors have been set
up at Los Alamos National Laboratory.

Figure 4: Bench test configuration for a single MALTA sensor is shown in the
left. Threshold and noise scan results for a partial region of a MALTA sensor
are illustrated in the right.

To characterize the LGAD (AC-LGAD) prototype sensor performance, ${}^{90}Sr$
source tests are used. The LGAD prototype sensor has 15 pixels wire bonded to
the output of the carrier board. The LGAD pixel has a dimension of 1.3 mm by
1.3 mm, which are further reduced for AC-LGAD sensors. The readout chain based
on the CAEN 1730s digitizers has been established to process the data from a
single LGAD (AC-LGAD) sensor or a telescope of two LGAD (AC-LGAD) sensors with
a ${}^{90}Sr$ source. The ${}^{90}Sr$ source bench test configuration for a
two-layer LGAD telescope is shown in Figure 2. The analog signal from
individual pixel has a pulse width of $\sim$500 ps and pulse amplitude of
$\sim$100 mV. LGAD sensors with different doping techniques have nonidentical
slopes in the bias voltage dependent charge collection. The charge collection
versus the bias voltage magnitude of four different types of LGAD sensors
characterized with both laser scans and ${}^{90}Sr$ source tests are shown in
Figure 3. Similar trends have been observed for two sets of four LGAD sensors
between the ${}^{90}Sr$ source digitized data and the analog outputs using
laser beams.

The test bench to study the MALTA sensor performance has been setup and the
test configuration is shown in the left panel of Figure 4. This test utilizes
the XILINX KC705 FPGA evaluation board to process the data. A router, which
can expand the test capability from a single sensor to a four-layer telescope,
is used for the communication between the FPGA board and the DAQ computer.
Each MALTA prototype sensor has a 512$\times$512 pixel matrix [5] that can be
further divided into eight different regions using different pixel spacing,
dopings and reset mechanisms. The pixel size of MALTA sensors is 36.4 $\mu$m
by 36.4 $\mu$m. MALTA sensors can inject charges into individual pixels to
evaluate their performance. Based on this feature, a series of threshold and
noise scans have been carried out for different regions in a MALTA sensor. The
right panel of Figure 4 presents the threshold and noise scan results for 1010
pixels in the 5th region of a MALTA sensor. The average threshold over noise
ratio is larger than 20, which indicates the MALTA sensor can significantly
suppress the background with this threshold configuration. The noise tail due
to the Random-Telegraph-Signal (RTS) has been mitigated by using a larger
transistor in the new MALTA prototype (Mini-MALTA) [8]. Further studies such
as the evaluation of hit/tracking spatial resolution and efficiency will be
performed on a multi-layer telescope using a ${}^{90}Sr$ source and cosmic
rays.

##  4 Summary and Outlook

The proposed Forward Silicon Tracker (FST) for the future EIC provides fine
tracking momentum and spatial resolutions in the forward pseudorapidity
region. Fast timing capability can be provided by this detector design as
well. Initial characterization results of the LGAD and MALTA prototype sensors
obtained from lab bench tests are consistent with their design parameters,
suggesting simulations of tracking momentum resolution and DCA can be
realized. Ongoing and future silicon detector R$\&$D and developments towards
finer pixel pitch and faster readout speed will pave the path for the EIC
detector construction and operation planned for starting in 2025.

## References

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  * [2] X. Li et al., _A New Heavy Flavor Program for the Future Electron-Ion Collider_ , EPJ Web of Conferences 235, 04002 (2020).
  * [3] C. P. Wong, X. Li, M. Brooks, M. J. Durham, M. Liu, A. Morreale, C. da Silva, W. Sondheim, _A Proposed Forward Silicon Tracker for the Future Electron-Ion Collider and Associated Physics Studies_ , arXiv: 2009.02888 (2020).
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  * [5] H. Pernegger et al., _First tests of a novel radiation hard CMOS sensor process for Depleted Monolithic Active Pixel Sensors_ , JINST 12 (2020) P06008.
  * [6] G. Pellegrini et al., _Technology developments and first measurements of Low Gain Avalanche Detectors (LGAD) for high energy physics applications_ , NIM A 765 (2014) 12-16.
  * [7] A. Apresyan et al., _Measurements of an AC-LGAD strip sensor with a 120 GeV proton beam_ , JINST 15 (2020) P09038.
  * [8] M. Dyndal et al., _Mini-MALTA: radiation hard pixel designs for small-electrode monolithic CMOS sensors for the High Luminosity LHC_ , JINST 15 (2020) P02005.