#  ROS 2 on a Chip, Achieving Brain-Like Speeds and Efficiency in Robotic
Networking

Víctor Mayoral-Vilches  Juan Manuel Reina-Muñoz  Martiño Crespo Álvarez and
David Mayoral-Vilches
Acceleration Robotics
<EMAIL_ADDRESS>

###### Abstract

The Robot Operating System (ROS) pubsub model played a pivotal role in
developing sophisticated robotic applications. However, the complexities and
real-time demands of modern robotics necessitate more efficient communication
solutions that are deterministic and isochronous. This article introduces a
groundbreaking approach: embedding ROS 2 message-passing infrastructure
directly onto a specialized hardware chip, significantly enhancing speed and
efficiency in robotic communications. Our FPGA prototypes of the chip design
can send or receive packages in less than 2.5 microseconds, accelerating
networking communications by more than 62$\times$ on average and improving
energy consumption by more than 500$\times$ when compared to traditional ROS 2
software implementations on modern CPUs. Additionally, it dramatically reduces
maximum latency in ROS 2 networking communication by more than 30,000$\times$.
In situations of peak latency, our design guarantees an isochronous response
within 11 microseconds, a stark improvement over the potential hundreds of
milliseconds reported by modern CPU systems under similar conditions.

##  1 Introduction

As robots permeate various sectors - from healthcare to manufacturing - the
need for swift and energy-efficient communications becomes paramount.
Traditional computational methods, including CPUs and GPUs, are increasingly
inadequate for real-time robotics networking communications due to their
unbounded intra- and inter-network communication constraints. Robot-specific
circuitry can lead to thousands-fold faster communications and better energy-
efficiency, growing closer to the realm of human brain-like speeds and
efficiency.

In the fascinating realm of neuroscience, a single neuron firing an action
potential in the human brain lasts approximately 2 to 5 milliseconds,
consuming an incredibly minute amount of energy — roughly 0.03 to 0.3
microjoules. These latencies and energy-efficiency set a benchmark that
current message-passing infrastructures in robotics strive to match. The most
popular one is the Robot Operating System (ROS)[1]. The most modern
implementations of ROS (ROS 2), which build upon the Data Distribution Service
(DDS) [2] operate within similar latency margins on average, however these
systems often fall short in maintaining isochronous communications and
adhering to comparable energy profiles. The prevalent approach in the industry
to meet real-time demands involves an intensive, empirical, and piecemeal
tuning of systems. This CPU ’ _whack-a-mole_ ’ strategy, while common in
widely-used robotic frameworks such as ROS 2, proves unsustainable and
challenging to scale in industry, especially given the real-time nature of
robotic systems.

In this paper, we introduce a pioneering approach to addressing the
complexities and real-time demands of modern robotics communication by
embedding ROS 2 message-passing infrastructure functionalities onto
specialized hardware chips. This method significantly enhances the speed and
efficiency of robotic communications, aligning closer to the rapid and energy-
efficient communication seen in the human brain. Our chip design prototyped
with an FPGA achieve exceptional improvements in networking communications,
offering over 62$\times$ faster message transmission and over 500$\times$
better energy efficiency compared to best performing conventional CPU-based
ROS 2 implementations. Moreover, our approach dramatically reduces the maximum
latency in ROS 2 networking communication by up to more than 30,000$\times$,
ensuring isochronous responses within 11 microseconds even in peak latency
situations. This advancement marks a significant leap forward in robotic
communications, moving towards more efficient, reliable, and scalable robotic
systems.

##  2 Background

In the dynamic world of robotics, ROS emerges as a cornerstone for crafting
complex robotic systems, yet faces challenges in meeting the real-time and
deterministic communication demands of contemporary robotics. This backdrop
fosters a shift towards FPGA-based solutions, aiming to enhance compute and
communication speed, efficiency and ensure time-sensitive operations. Research
contributions, spanning from foundational tools facilitating FPGA integration
within ROS ecosystems [3, 4, 5, 6], to endeavors optimizing the networking
stack for improved node interactions [7], delineate a multifaceted approach
towards surmounting communication bottlenecks [8]. Further exploration reveals
practical applications [9, 10, 11, 12, 13, 14], where ROS’s computational
graph benefits from FPGA-based acceleration, underpinning the pivotal role of
hardware acceleration. This narrative underscores the necessity for a
comprehensive strategy that not only accelerates user-level applications and
ROS libraries but also optimizes the interplay among ROS Nodes across various
levels of interaction: inter-process, intra-process, intra-network and inter-
network. Such a holistic approach promises substantial latency reductions
across ROS and ROS 2’s computational landscapes, heralding a new era of
efficiency in robotic computation and communication. Tables 1 covers past
research efforts around FPGA-based computing in in ROS and ROS 2.

[startyear=2013,stopyear=2024,height=1.0ex]

[textwidth=3.5cm]2015(ROS) Proposal of ROS-compliant fpga component for low-
power robotic systems [15]

[textwidth=5cm,markdepth=60pt]09/2016(ROS) cReComp: Automated Design Toolfor
ROS-Compliant FPGA Component [3]

[textwidth=2.5cm,markdepth=-70pt]10/2016(ROS) Architecture Exploration of
Intelligent Robot System using ROS-compliant FPGA Component [9]

[textwidth=5.5cm,markdepth=-20pt]2017(ROS) Acceleration of publish/subscribe
messaging in ROS-compliant FPGA component [7]

[textwidth=2.5cm,markdepth=-70pt]05/2019(ROS) High Level Synthesis of ROS
Protocol Interpretation and Communication Circuit for FPGA [16]

[textwidth=4cm]12/2019(ROS) FPGA-ROS: Methodology to Augment the Robot
Operating System with FPGA Designs [17]

[textwidth=4cm,markdepth=120pt]12/2020(ROS 2) Automated Integration of High-
Level Synthesis FPGA Modules with ROS2 Systems [4]

[textwidth=4cm,markdepth=70pt]12/2020(ROS 2) ReconROS: ReconROS: Flexible
Hardware Acceleration for ROS2 Applications [5]

[textwidth=3cm,markdepth=-20pt]02/2021(ROS) ReconfROS: Running ROS on
Reconfigurable SoCs [6]

[textwidth=3cm,markdepth=-80pt]5/2022(ROS 2) RobotCore: An Open Architecture
for Hardware Acceleration in ROS 2 [18]

[textwidth=3cm,markdepth=100pt]5/2024(ROS 2) ROS 2 on a Chip (this
contribution)

[textwidth=3cm,markdepth=20pt]10/2023(ROS 2) fpgadds: An Intra-FPGA Data
Distribution Service for ROS2 Robotics Applications [19]

Target |  Article |  Year |  Group
---|---|---|---
ROS |  Proposal of ROS-compliant FPGA component for low-power robotic systems [15]: First proposal of FPGA ROS-compliant component to simplify FPGA integration into robots. Demonstrated using image processing on Xilinx Zynq 7000, achieving 70% faster results than running on ARM cores. |  2015 |  Utsunomiya University
ROS |  cReComp: Automated Design Tool for ROS-Compliant FPGA Component [3]: Proposes an automated tool, cReComp, for creating ROS-compliant FPGA components, enhancing productivity with a code generator and componentization framework. |  2016 |  Utsunomiya University
ROS |  Architecture Exploration of Intelligent Robot System using ROS-compliant FPGA Component [9]: Explores a Visual SLAM study case using ROS-compliant FPGA components and the cReComp design tool. |  2016 |  Utsunomiya University
ROS |  Acceleration of publish/subscribe messaging in ROS-compliant FPGA component [7]: Reduces communication latency between ROS components by using hardware accelerated networking stack (SiTCP) and improving ROS API communications. |  2017 |  Utsunomiya University
ROS |  High Level Synthesis of ROS Protocol Interpretation and Communication Circuit for FPGA [16]: Enhances the ROS-compliant component concept with detailed component descriptions, HLS design flow, and feasibility study. |  2019 |  Utsunomiya University, Tokai University, Tokyo City University
ROS |  FPGA-ROS: Methodology to Augment the Robot Operating System with FPGA Designs [17]: Introduces a methodology for designing ROS-Compatible FPGA systems, featuring automatic generation of PL-equivalents to ROS messages and full hardware implementation of ROS entities. |  2019 |  Technische Universität Dresden
ROS 2 |  ReconROS: Flexible Hardware Acceleration for ROS2 Applications [5]: Presents a framework that integrates ROS with ReconOS for hardware acceleration of ROS applications, enabling ROS nodes fully mapped to hardware. |  2020 |  Paderborn University, University of Klangenfurt
ROS 2 |  Automated Integration of High-Level Synthesis FPGA Modules with ROS2 Systems [4]: Introduces the concept of ROS2-FPGA Nodes and the FOrEST tool for seamless integration of HLS-generated FPGA logic into ROS 2 systems, targeting PYNQ 2.5 and achieving significant speed-ups. |  2020 |  University of Toronto, Shibaura Institute of Technology, Keio University, Tokai University
ROS |  ReconfROS: Running ROS on Reconfigurable SoCs [6]: Proposes a method to integrate ROS into adaptive SOCs, focusing on general algorithmic acceleration and easy integration of FPGA-based hardware accelerators into ROS nodes. |  2021 |  Osnabrück University
ROS 2 |  RobotCore: An Open Architecture for Hardware Acceleration in ROS 2 [18]: Introduces RobotCore, a hardware acceleration robotics framework that demonstrates an intra-FPGA ROS 2 node communication queue to enable faster dataflows, and use it in conjunction with FPGA-accelerated nodes to achieve a 24.42% speedup over a CPU. |  2022 |  Acceleration Robotics, Harvard Unversity and Columbia University
ROS 2 |  fpgadds: An Intra-FPGA Data Distribution Service for ROS2 Robotics Applications [19]: Demonstrates fpgaDDS, a customized streaming communication architecture for low latency and high bandwidth between hardware-mapped ROS nodes, showing a 13x performance improvement over software DDS. |  2023 |  Paderborn University

Table 1: Past research efforts to bring ROS into FPGAs.

Given the emerging interest and demonstrated advantages, the exploration of
specialized circuitry for accelerating ROS 2’s message-passing infrastructure,
particularly through the Data Distribution Service (DDS), marks a pivotal
advancement towards ultra-efficient robotic communications. This initiative
aims to enable robotic systems to exchange information in just a few
microseconds, significantly outperforming the latency benchmarks set by
current software implementations of ROS 2. The subsequent section introduces
our work to bring _"ROS 2 on a Chip"_ and will elaborate on the integration of
ROS 2 functionalities into specialized hardware, focusing on the potential of
FPGA-based accelerations to significantly enhance real-time capabilities and
computational efficiency in robotic systems.

##  3 ROS 2 on a Chip

The architecture of robot communication stands as a critical foundation for
advanced computation and operational functionality. At the heart of this
architecture lies the computational graph, a model pioneered by the Robot
Operating System (ROS), which has become a standard for developing intricate
robotic systems. This graph structures the flow of data from an array of
sensors, weaving through a network of computational nodes that process,
analyze, and act upon this information in a coordinated fashion. As such, the
dataflow within and across robotic systems encapsulates the essence of robot
cognition, making intra-robot communications indispensable to the entirety of
robotic computations. Recognizing the significance of this communication
infrastructure, our investigation delves into the advancement of _ROS 2 on a
Chip_ — an initiative aimed at embedding ROS 2 functionalities into
specialized hardware.

|  ROBOTCORE for ROS 2 + RTPS + UDP/IP (FPGA@156MHz) |  ROS 2 + DDS1 + Linux net. stack (CPU) |  ROS 2 + DDS2 + Linux net. stack (CPU) |  ROS 2 + DDS3 + Linux net. stack (CPU)
---|---|---|---|---
ROBOTCORE UDP/IP (hardware, FPGA@156MHz) |  0.7 |  |  |
ROBOTCORE RTPS (hardware, FPGA@156MHz) |  2.3 |  |  |
ROBOTCORE for ROS 2 (hardware, FPGA@156MHz) |  2 |  |  |
Linux UDP/IP net. stack (software, CPU) |  |  70 |  73 |  71
DDS1 (software, CPU) |  |  835 |  |
DDS2 (software, CPU) |  |  |  127 |
DDS3 (software, CPU) |  |  |  |  141
ROS 2 (software, CPU) |  |  139 |  114 |  155
Total in us (speedup) |  5 (1$\times$) |  1044 (208$\times$) |  314 (62$\times$) |  369 (73$\times$)

Figure 1: Mean Round-Trip Network Latency in microseconds (us) breakdown
across various combination of hardware and software implementations. Round-
Trip Time (RTT) mean latencies measured after 1M samples and while sending
small ROS messages in a ping-pong format. ROBOTCORE for ROS 2, ROBOTCORE RTPS
and ROBOTCORE UDP/IP running in an FPGA@156MHz. Software implementations
running on an AMD Ryzen 5 PRO 4650G.

We implement the complete ROS 2 robotics communication framework stack split
into three hardware designs including all lower abstraction layers in hardware
to deliver unprecedented speed and efficiency in ROS 2 interactions over the
network111Note that in this work we focus on the intra-network ROS 2
interactions.: ROBOTCORE® for ROS 2, ROBOTCORE® RTPS and ROBOTCORE® UDP/IP.
Our chip design which includes all three is prototyped in an FPGA and
significantly enhances the networking architecture of robotic systems and
tackles one the widely criticized issues of ROS 2: its latency overhead over
the DDS communication middleware. It does so by building a hardware
implementation of the core ROS 2 abstraction layers (RCL, RMW) and by
establishing direct hardware datapaths with the underlying DDS middleware
implementation (3(a)). We then hardware-implement the Real-Time Publish
Subscribe protocol (RTPS) [20] with some minimal DDS capabilities for
interoperability (also in hardware) (3(b)). This allows to remove the latency
overhead of ROS 2 over DDS and to deliver unmatched speed and absolute
determinism. Finally, we connect the optimized datapaths of our minimal
hardware DDS implementation with an ultra low-latency hardware UDP/IP stack
that sits on top of a 10G PHY (3(c)).

Results comparing our ROS 2 circuitry with software implementations are
depicted in Figure 1, with all software benchmarks conducted on a modern AMD
Ryzen 5 PRO 4650G CPU equipped with 64 GB of RAM. Our comprehensive evaluation
showcases the exceptional performance of the ROS 2 hardware architecture,
significantly surpassing traditional software-based approaches. Notably, our
FPGA-based designs—ROBOTCORE® for ROS 2, ROBOTCORE® RTPS, and ROBOTCORE®
UDP/IP—demonstrate groundbreaking efficiency, achieving mean latency figures
as low as 0.7 microseconds for UDP/IP, 2.3 microseconds for RTPS, and an
aggregate of 2 microseconds for ROS 2 functionalities, all at a clock rate of
156 MHz.

In stark contrast, software implementations operating on the CPU exhibit
substantially higher latencies, with Linux UDP/IP networking stack alone
contributing up to a mean of 73 microseconds. When considering DDS
implementations, mean latency further escalates, with DDS1 reaching 835
microseconds, DDS2 at 127 microseconds, and DDS3 at 141 microseconds. Even the
aggregated ROS 2 software mean latency ranges from 114 to 155 microseconds,
dependent on the DDS version used. Consequently, the total latency for our
FPGA-based solution stands at an impressive 5 microseconds of mean latency,
marking a speedup factor ranging from 62$\times$ to 208$\times$ when compared
to various software configurations.

It’s crucial to note that all comparative experiments were conducted under
optimal conditions for the CPU-based setups: with an idle CPU processor and an
idle network. These conditions artificially inflate the performance of the CPU
implementations, which, unlike our FPGA-based solution, are not designed for
hard real-time or isochronous operations (further covered in subsection 3.2).
As such, the demonstrated latency figures for software configurations, while
already significantly higher than our hardware solution, represent a best-case
scenario222In real-world robotic applications, CPU processors and networks are
rarely idle and often experience varying levels of saturation. As demonstrated
by previous work at [8, 21], the latency and performance inconsistencies of
software-based ROS 2 communications would become more pronounced.. This
remarkable achievement not only underscores the superior performance of
hardware acceleration in ROS 2 communications but also highlights the
transformative potential of integrating ROS 2 functionalities directly into
specialized hardware for robotics applications, setting a new benchmark for
real-time robotic communication and computational efficiency.

###  3.1 Processor Design

Figure 2: Resource Utilization Breakdown for the "_ROS 2 on a Chip_ " Design
on FPGA Platforms: Detailing the allocations for integrated submodules.
ROBOTCORE® for ROS 2 facilitates core ROS 2 abstractions; ROBOTCORE® RTPS
provides core RTPS message-passing infrastructure and DDS interoperability;
and ROBOTCORE® UDP/IP enables high-speed, fixed low-latency networking stack
interactions. This composition highlights the comprehensive approach to
hardware-accelerated robotic communication

(a) Complete design resource allocation on AMD Zynq™ UltraScale+™ MPSoC.

(b) IP and UDP resource allocation on AMD Zynq™ UltraScale+™ MPSoC.

(c) PHY, IP, and UDP resource allocation on AMD Zynq™ UltraScale+™ MPSoC.

(d) RTPS and ROS 2 resource allocation on AMD Zynq™ UltraScale+™ MPSoC.

(e) PubSub resource allocation on AMD Zynq™ UltraScale+™ MPSoC.

|  LUT (%) |  FF (%) |  DSP (%) |  BRAM (%)
---|---|---|---|---
AMD Zynq™ UltraScale+™ MPSoC EV (XCK26) |  56,440 (48.19%) |  20,285 (8.66%) |  0 (0%) |  26 (18.06%)

|  ALUT (%) |  REG (%) |  DSP (%) |  M20K (%)
---|---|---|---|---
Intel Agilex® 7 FPGA F-Series, 1400 KLE, 2486A |  187,962 (19.29%) |  163,115 (8.37%) |  0 (0%) |  138 (1.94%)

The architecture of _ROS 2 on a Chip_ composed by ROBOTCORE® for ROS 2, RTPS
and UDP/IP submodules is designed to be highly scalable and flexible, enabling
it to seamlessly integrate into a variety of FPGA and FPGA SoC technologies.
Figure 2’s detailed portrayal of resource allocation for the _ROS 2 on a Chip_
processor design on AMD Zynq™ UltraScale+™ MPSoC and Intel Agilex® 7 FPGA
F-Series provides keen insights into the efficient use of FPGA resources. It
is noted that the resources required for implementing the chip design consume
less than 50% of the total available on the AMD platform and under 20% on the
Intel platform. This efficient resource usage underscores the design’s
scalability and potential for integration into a wide range of applications
without nearing resource capacity limits.

A significant observation from the resource allocation breakdown is that the
bulk of resources are dedicated to the RTPS and DDS logic, which are pivotal
for the communication capabilities of the chip. These components demand the
most from the FPGA’s resources due to their complexity and the need for high-
speed, deterministic communication. In contrast, a smaller portion of the
resources is allocated to the parallel UDP/IP networking stack, and even less
to the ROS 2 abstractions in hardware, which highlights the lightweight nature
of these abstractions and their minimal impact on resource consumption. The
resource sharing for RTPS and DDS logic further benefits the scalability of
the chip design, especially as ROS 2 requires minimal additional resources to
be implemented in hardware. This means that adding more ROS 2 abstractions
such as publishers, actions, or services would require only a marginal
increase in hardware space, thereby enhancing the chip’s parallelization
capabilities and its suitability for more complex ROS 2 applications. This
efficient and scalable use of resources not only emphasizes the advanced
design of the _ROS 2 on a Chip_ but also showcases its potential to impact
robotics applications by facilitating more complex computations and
interactions within a compact hardware footprint.

(a) ROS 2 hardware accelerator data flow.

(b) RTPS/DDS hardware accelerator data flow.

(c) UDP/IP hardware accelerator data flow.

Figure 3: Hardware accelerator data flow diagrams for UDP/IP, RTPS/DDS, and
ROS 2.

At its core, _ROS 2 on a Chip_ implements a streamlined and optimized hardware
processor for ROS 2 networking parallelizing computing across various hardware
sub-cores, each of which can have multiple instances and is responsible for a
specific ROS 2 operation streaming data in parallel. Figure 3 depicts the
architecture of the various accelerator designs that compose the _ROS 2 on a
Chip_ circuitry. Subfigure 3(a) corresponds with ROS 2 abstractions
implemented in hardware. Data-paths are fed from the underlying communication
middleware (depicted in subfigure 3(b)) and into the robotic application.
Control-paths exist similarly, incoming from the underlying communication
middleware and also between blocks of the hardware implementation of this
layer. ROS 2-level abstractions implemented in hardware (such as PUBLISHERS,
SUBSCRIBERS, ACTION_SERVERS, etc) can get hardware-instantiated multiple
times, which leads to higher throughput at an additional energy budget.
Determinism is guaranteed and maximum latency is established by the hardware
implementation datapaths, which is limited by the maximum frequency (fmax) of
operation, which in the case of the AMD Zynq™ UltraScale+™ MPSoC turned 156
MHz and for the Intel Agilex® 7 FPGA F-Series 161 MHz. Subfigure 3(b) depicts
an implementation of a hardware DDS communication middleware. Control- and
data-paths lie between the relationships across blocks. Readers and writers
across RTPS (the interoperability protocol) and DDS (the middleware on top)
can be instantiated multiple times for further parallelism while interfacing
with upstream robotic applications. Downstream data paths such as RTPS_OUT are
built in hardware, connected to underlying hardware implementation and can
also be set redundant and parallel if the application demands it. Isochronous
behavior is guaranteed by the hardware datapaths and maximum latency is
established by the hardware implementation, which is also limited by the
maximum frequency (fmax) of operation. Finally, 3(c) depicts an architecture
diagram of the UDP/IP stack implementated in hardware with dedicated data
paths per each networking interaction that run concurrently and without
interference. While they run in parallel (interleaving) with all the upstream
robotics stack layers implemented also in hardware.

###  3.2 Thousands-Fold faster, deterministic and isochronous

|  ROBOTCORE for ROS 2 + RTPS + UDP/IP (FPGA@156MHz) |  ROS 2 + DDS1 + Linux net. stack (CPU) |  ROS 2 + DDS2 + Linux net. stack (CPU) |  ROS 2 + DDS3 + Linux net. stack (CPU) |  ROS 2 + eCAL + Linux net. stack (CPU)
---|---|---|---|---|---
Mean latency (slowdown) |  5 (1$\times$) |  1044 (208$\times$) |  314 (62$\times$) |  369 (73$\times$) |  376 (75$\times$)
Max. latency (slowdown) |  11 (1$\times$) |  336750 (30613$\times$) |  22769 (2069$\times$) |  109776 (9979$\times$) |  5532 (502$\times$)

Figure 4: Mean and Maximum Round-Trip Network Latency in microseconds (us)
breakdown across various combination of hardware and software implementations.
Logarithmic scale. Figure depicts how a hardware implementation delivers an
isochronous and faster interaction.

The quantitative analysis presented in our study emphasizes not only the
remarkable speedup achieved by our _ROS 2 on a Chip_ FPGA prototype
implementation in terms of mean and maximum latency but also brings to light
its isochronous capabilities, which are essential for ensuring time-
deterministic communications in robotics. The data, as depicted in Figure 4,
offers a stark comparison between the hardware-accelerated and software-based
solutions, highlighting the intrinsic advantages of our FPGA-prototype
approach in achieving isochronous behavior.

In detail, our chip implementation prototyped in an FPGA maintains a mean
latency of 5 microseconds and a remarkably low maximum latency of 11
microseconds, demonstrating the system’s capability to deliver consistent
performance under varying conditions. This isochronous nature of communication
is crucial for real-time robotic applications, where even minor deviations in
message timing can lead to significant operational discrepancies. In contrast,
software implementations on a CPU, across various DDS configurations and eCAL,
exhibit significantly higher and more variable latencies, with maximum latency
figures soaring to as high as 336,750 microseconds for DDS1. This variability
undermines the predictability and reliability of software-based
communications, rendering them unsuitable for scenarios requiring stringent
real-time guarantees. Furthermore, the comparison reveals that our FPGA
implementation achieves up to a 30,613$\times$ improvement in maximum latency
over the software configurations, underscoring the substantial enhancement in
determinism. This deterministic performance, characterized by minimal latency
variations, is absent in the software implementations, where latency can
fluctuate dramatically depending on system and network load, further
exacerbating the challenge of achieving isochronous communications.

In essence, the FPGA-based "ROS 2 on a Chip" design not only accelerates
communication speeds but also solidifies the foundation for isochronous and
deterministic robotic communications. This advancement paves the way for more
sophisticated and reliable robotic systems, capable of performing synchronized
tasks with unparalleled precision and safety. Through the adoption of this
hardware-accelerated approach, we address and overcome a critical bottleneck
in robotic communications, marking a significant leap towards realizing the
full potential of real-time and autonomous robotic systems.

###  3.3 100-40000x better than other IP cores

|  ROBOTCORE for ROS 2 + RTPS + UDP/IP (FPGA@156MHz) |  Safe DDS (ARM Cortex-R5) |  Safe DDS (AMD MicroBlaze)
---|---|---|---
Mean Round-Trip Network Latency (speedup) |  5 (1$\times$) |  900 (180$\times$) |  200,000 (40,000$\times$)

Figure 5: Mean Round-Trip Network Latency in microseconds (us) across various
combination of hard and soft-processors providing ROS 2 interoperability
capabilities. Logarithmic scale.

Our _ROS 2 on a Chip_ design, when evaluated against other IP core solutions
offering DDS and RTPS functionalities on different processing architectures,
showcases distinct performance metrics, particularly in terms of mean round-
trip network latency. As presented in Figure 5, our circuitry implemented on
an FPGA operating at a frequency of 156MHz, records a mean round-trip network
latency of 5 microseconds. This result is juxtaposed with 900 microseconds
observed for Safe DDS running on an ARM Cortex-R5 and 200,000 microseconds on
an AMD MicroBlaze, indicating latency reductions by factors of 180 and 40,000,
respectively[22]. Such disparities in performance are critical in the context
of robotic communications, where latency impacts the system’s responsiveness
and operational efficiency. Our _ROS 2 on a Chip_ FPGA-prototype provides a
stark contrast in network latency compared to implementations on both hard-
core (ARM Cortex-R5) and soft-core (AMD MicroBlaze) processors. This
comparative analysis highlights the significant differences in how each
architecture handles ROS 2 interoperability requirements, with our FPGA-based
prototype demonstrating the lowest latency figures.

Furthermore, the comparison sheds light on the inherent limitations and
performance bottlenecks associated with soft-core and hard-core processor-
based deployments for DDS and RTPS functionalities within ROS 2 frameworks. By
detailing these quantitative results, the study underscores the potential
advantages of a custom compute architecture in reducing network latency for
robotic communication systems, thereby enhancing real-time data exchange and
processing capabilities essential for autonomous robotic operations.

###  3.4 500$\times$ more energy-efficient

Sending a ROS 2 message with _ROS 2 on a Chip_ takes about 1.775 microjoules
(uJ). Our chip design stands out remarkably, outperforming contemporary
workstation CPU and GPU solutions when it comes to power performance. _ROS 2
on a Chip_ exhibits an exceptional efficiency, being 500$\times$ more power-
efficient compared to modern workstation CPUs and GPUs as depicted in figure
6.

|  FPGA@156MHz
(ROBOTCORE for ROS 2 + RTPS + UDP/IP) |  Workstation CPU
---|---|---
Mean communication frequency-per-Watt (energy efficiency) |  281,690 (543$\times$) |  518 (1$\times$)

Figure 6: Energy efficiency measured in terms of mean communication frequency-
per-Watt. Logarithmic scale.

The comparison extends to the domain of communication frequency-per-Watt, a
metric that encapsulates the balance between communication throughput and
energy consumption. Our hardware configuration achieves an impressive mean
communication frequency-per-Watt of 281,690, signifying an efficiency that is
543$\times$ greater than that of conventional workstation CPUs, which register
a mean of merely 518. Such metrics not only highlight the hardware’s adeptness
at minimizing energy expenditure but also its capability to sustain high-
frequency communications with minimal power input.

These findings, depicted in Figure 6, resonate with the broader objective of
advancing robotic systems towards not only achieving real-time performance but
doing so in an energy-conscious manner. The ROBOTCORE suite’s ability to
deliver such efficiency gains illustrates the tangible benefits of integrating
specialized hardware for ROS 2 functionalities, paving the way for the
development of more sustainable, energy-efficient robotic systems.

##  4 Discussion and future work

###  4.1 Achieving Brain-Like Speeds and Efficiency in Robotic Networking

Sending a ROS 2 message with _ROS 2 on a Chip_ Design takes about 1.78
microjoules ($\mu$J) and about 2.5 microseconds ($\mu$s). For comparison, a
single neuron of a human brain firing one action potential takes roughly 0.03
to 0.3 microjoules ($\mu$J) and lasts approximately 2 to 5 milliseconds (ms)
[23, 24], while the energy required to Blink an Eye is roughly estimated to be
around 1 millijoule (mJ) lasting about 300 to 400 milliseconds (ms) [25]. The
remarkable efficiency and speed of the _ROS 2 on a Chip_ Design in exchanging
data between robots not only brings robotic networking closer to biological
processing speeds but also highlights the potential for robotics to emulate
more closely the energy efficiency found in natural systems. This comparison
not only underscores the advanced technological achievements of _ROS 2 on a
Chip_ but also opens avenues for further exploration into making robotic
systems that can operate within the energy budgets of biological counterparts.
By reducing the energy consumption and latency in robotic communication, _ROS
2 on a Chip_ Design paves the way for the development of more responsive,
autonomous systems that can perform complex tasks with a minimal energy
footprint, mirroring the efficiency of biological organisms.

###  4.2 Future work

Our circuitry presented herein as _ROS 2 on a Chip_ served within the
ROBOTCORE® suite of IP operates with an energy budget comparable to the
fundamental biological processes in the human brain and with much faster
speeds. This is relevant for future research in both robotics and
neuroscience, as _"communication consumes 35 times more energy than
computation in the human cortex"_ [24]. In upcoming research, we’ll focus on
leveraging our specialized hardware chip design and benchmark it against
industry use cases wherein these accelerated communication and computation
capabilities can lead to faster robots in real scenarios.

Beyond industry, these advancement holds promise for enabling new applications
in robotics where energy efficiency and rapid response times are critical,
such as those requiring compute-demading levels of encryption, authentication
and authorization. Further research could explore the integration of this
technology into broader networks of robots, exploring the inter-network level
of interactions, creating collectives capable of complex, coordinated actions
with efficiency and communication speeds that empower new applications.

## References

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# Coulomb explosion imaging of carbon monoxide dimers

A. Méry  CIMAP, CEA-CNRS-ENSICAEN-UNICAEN, Normandie Université,
BP5133, F-14050 Caen Cedex 04, France  V. Kumar  CIMAP, CEA-CNRS-ENSICAEN-
UNICAEN, Normandie Université,
BP5133, F-14050 Caen Cedex 04, France  X. Fléchard  Normandie Univ, ENSICAEN,
UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France  B. Gervais  CIMAP, CEA-
CNRS-ENSICAEN-UNICAEN, Normandie Université,
BP5133, F-14050 Caen Cedex 04, France  S. Guillous  CIMAP, CEA-CNRS-ENSICAEN-
UNICAEN, Normandie Université,
BP5133, F-14050 Caen Cedex 04, France  M. Lalande  CIMAP, CEA-CNRS-ENSICAEN-
UNICAEN, Normandie Université,
BP5133, F-14050 Caen Cedex 04, France  J. Rangama  CIMAP, CEA-CNRS-ENSICAEN-
UNICAEN, Normandie Université,
BP5133, F-14050 Caen Cedex 04, France  W. Wolff  Instituto de Física -
Universidade Federal do Rio de Janeiro, Cidade Universitária, Rio de Janeiro,
Brazil  A. Cassimi  CIMAP, CEA-CNRS-ENSICAEN-UNICAEN, Normandie Université,
BP5133, F-14050 Caen Cedex 04, France

###### Abstract

We report on experimental results obtained from collisions of slow highly
charged $\rm Ar^{9+}$ ions with a carbon monoxide dimer $\rm(CO)_{2}$ target.
A COLTRIMS setup and a Coulomb explosion imaging approach are used to
reconstruct the structure of the $\rm CO$ dimers. The three dimensional
structure is deduced from the 2-body and 3-body dissociation channels from
which both the intermolecular bond length and the relative orientation of the
two molecules are determined. For the 3-body channels, the experimental data
are interpreted with the help of a classical model in which the trajectories
of the three emitted fragments are numerically integrated. We measured the
equilibrium intermolecular distance to be $\rm R_{e}=4.2~{}\text{\AA}$. The
orientation of both CO molecules with respect to the dimer axis is found to be
quasi-isotropic due to the large vibrational temperature of the gas jet.

††preprint: CEI_CO_AM

##  I INTRODUCTION

Carbon monoxide dimers $\rm(CO)_{2}$ have been studied since the 90′s using
infrared and millimeter-wave spectroscopy [Brookes, 1999,Surin, 2003,Rezaei,
2013]. However, the interpretation of the measured rovibrational spectra has
been puzzling for many years because of the possible contribution of two
distinct isomers of the dimer with very little binding energy difference. The
lowest in energy has been assigned to a planar C-bonded geometry in which the
carbon atoms of each CO molecule are located in the inner region of the dimer.
In contrast, the other isomer would correspond to a planar O-bonded
configuration with two inner oxygen atoms. Moreover, these two isomers have
been found to have different equilibrium intermolecular distance $\rm R_{e}$.
The corresponding three-dimensional structures of the dimer can be described
using four independent parameters: the intermolecular bond length $\rm R_{e}$,
the relative orientation $\rm\uptheta_{1}$ and $\rm\uptheta_{2}$ of each
molecule with respect to the dimer axis and the torsional angle $\rm\Phi$ (see
Fig. 1). Table 1 shows a summarized comparison of the geometry of the C-bonded
and O-bonded structures.

Figure 1: Definition of the dimer geometry using internal coordinates. The CO
molecular bond length is fixed to $\rm R_{CO}=1.13~{}\text{\AA}$. Upper panel:
complete three-dimensional geometry ($\rm\uptheta_{1}$, $\rm\uptheta_{2}$,
$\rm\Phi$, $\rm R_{e}$). Lower panel: Partial geometry accessible through the
2-body and 3-body dissociation of the dimer ($\rm\uptheta$, $\rm R_{e}$). The
non-dissociating $\rm CO^{+}$ molecular ion is considered as a point-like
particle located at its center of mass and the measurement is not sensitive to
the $\rm\Phi$ coordinate (see details in the text).

Conjointly, theoretical efforts were dedicated to the detailed investigation
of the potential energy surface of the neutral CO dimer [Han, 1997,Surin,
2007,Dawes, 2013]. Han et al. [Han, 1997] reported about the possible
existence of T-shaped structures (corresponding to $\rm\uptheta_{1}\approx
0^{\circ}$ and $\rm\uptheta_{2}\approx 90^{\circ}$) as well as slipped anti-
parallel structures. The likely presence of planar anti-parallel geometries
has been confirmed more recently through extensive efforts in the
determination of the potential energy surface (PES) of CO dimers [Surin,
2007,Dawes, 2013]. The PES shows a global minimum corresponding to the
C-bonded geometry and a local minimum corresponding to the O-bonded geometry
(see table 1). These two stable structures differ both by their relative
orientation $\rm\uptheta_{1}$ and $\rm\uptheta_{2}$ but also by their
intermolecular distance $\rm R_{e}$ and thus may be identified thanks to two
distinct experimental observables. The energies of these two minima have been
calculated to be about -135 $\rm cm^{-1}$ and -120 $\rm cm^{-1}$ respectively.
The two potential wells are separated by low barriers following a dis-rotatory
path along the $\rm X=\frac{\uptheta_{1}+\uptheta_{2}}{2}$ coordinate while
passing through some intermediate parallel conformations. The low barrier
(about 18 $\rm cm^{-1}\approx 2~{}meV$) easily enables vibrational excitation
along this dis-rotatory coordinate [Dawes, 2013] even for relatively low
excitation energies of the dimer.

The Coulomb explosion imaging (CEI) technique is known to be a powerful tool
to investigate the fragmentation dynamics of covalent molecules [Légaré,
2005,Matsuda, 2007,Neumann, 2010,Wu, 2015]. The method has also recently
allowed the experimental determination of the structure of van der Waals
complexes such as rare gas trimers [Ulrich, 2011] but also mixed $\rm ArN_{2}$
[Wu, 2014] or $\rm ArCO$ [Gong, 2013] dimers. The CEI technique is
particularly well suited for van der Waals clusters where large intermolecular
distances lead to dissociation following a pure Coulomb potential energy curve
[Wu, 2013]. This experimental method also requires a rapid ionization of the
target molecules in such a way that the nuclear motion can be considered as
frozen during the ionization process.

The CEI technique gives access to several experimental observables, such as
the kinetic energy released (KER) in the fragmentation and the relative angles
of emission of the fragments, which provide unique information on the 3D
molecular structure. It is worth noting that the direct measurement of the
complete structure of a carbon monoxide dimer using the CEI technique would
require the Coulomb explosion of the two constitutive CO molecules and the
coincident detection of the four emitted atomic ions. Although, the cross
sections for creation of 4-fold ionized dimers leading to such 4-body
fragmentation channels are very small in the present collision system.
Moreover, the data analysis procedure is complex because of the strong
ambiguity for pairing two sets of carbon and oxygen atomic ions coming from
two identical molecules. Thus, only the 2-body and 3-body dissociation
channels will be presented in the forthcoming discussion. As described later,
the 2-body channels (where only intermolecular fragmentation occurs) give a
direct measurement of the intermolecular distance while the molecular
orientation can be deduced from the 3-body channels. In the following we will
refer to the molecular orientation using a single angle $\rm\uptheta$ because
the fragmentation in three bodies does not enable measuring both
$\rm\uptheta_{1}$ and $\rm\uptheta_{2}$ and neither the torsional angle
$\rm\Phi$ (Fig. 1, lower panel). Note that the $\rm\uptheta$ angle is defined
with respect to the van der Waals bond and the oxygen atom. As a result, the
O-bonded structure corresponds to $\rm\uptheta=45^{\circ}$ while the C-bonded
structure is associated to $\rm\uptheta=135^{\circ}$.

Structure | C-bonded | O-bonded
---|---|---
| expt. | calc. | expt. | calc.
| [Brookes, 1999] | [Surin, 2007,Dawes, 2013] | [Brookes, 1999] | [Surin, 2007,Dawes, 2013]
$\rm{\uptheta(^{\circ})}$ | — | 134.2/134.79 | — | 61/65.18
$\rm{R_{e}(\text{\AA})}$ | 4.4 | 4.33/4.331 | 4.0 | 3.67/3.620
Table 1:  Structural parameters for the planar C-bonded and O-bonded
geometries of the neutral CO dimer. The relative orientation angle
$\rm\uptheta$ with respect to the dimer axis and the intermolecular distance
are defined on Fig. 1.

In the present study, multiply charged dimers are produced following multiple
electron capture by slow highly charged ions on neutral dimer targets. The
velocity of the $\rm Ar^{9+}$ projectiles is about $\rm 8~{}\text{\AA}/fs$
(0.37 atomic units) and the geometrical size of the dimer target is of the
order of $\rm 4.5~{}\text{\AA}$. For such projectile ions, electron capture is
by far the dominant process and occurs in the femtosecond timescale. The
subsequent concerted fragmentation of the ionized target also occurs on the
femtosecond timescale and ensures the validity of the axial recoil
approximation on which relies our Coulomb explosion approach. We focus here on
collisions where at least one electron is captured on each molecular site of
the CO dimer thus leading to the cleavage of the intermolecular van der Waals
bond. Covalent bond breakup also occurs when two or more electrons are
captured on one of the CO molecules. The relevant fragmentation channels can
be classified into 2-body or 3-body channels depending on either bound or
dissociative states of the multi-ionized CO molecule are formed during the
collision process. The following 2-body channels have been experimentally
identified and are referred to as (m,n) where m and n are the charge states of
the emitted CO molecular ions:

(1,1) : $\rm(CO)_{2}^{2+}\rightarrow CO^{+}+CO^{+}$

(2,1) : $\rm(CO)_{2}^{3+}\rightarrow CO^{2+}+CO^{+}$

The simultaneous break-up of both the van der Waals bond and of one covalent
bond leads to the following 3-body channels:

(1,[1,1]) : $\rm(CO)_{2}^{3+}\rightarrow CO^{+}+C^{+}+O^{+}$

(1,[2,1]) : $\rm(CO)_{2}^{4+}\rightarrow CO^{+}+C^{2+}+O^{+}$

(1,[1,2]) : $\rm(CO)_{2}^{4+}\rightarrow CO^{+}+C^{+}+O^{2+}$

(1,[2,2]) : $\rm(CO)_{2}^{5+}\rightarrow CO^{+}+C^{2+}+O^{2+}$

These channels are referred to as (m,[p,q]) where m=1 is the charge state of
the non-dissociating $\rm CO^{+}$ cation and where p and q are respectively
the charge states of the carbon and oxygen atomic ions. The simultaneous
measurement of several 2-body and 3-body fragmentation channels allows an
unambiguous determination of the $\rm R_{e}$ and $\rm\uptheta$ coordinates of
the CO dimer from the present data set.

##  II EXPERIMENT

The experiment uses the COLd Target Recoil Ion Momentum Spectroscopy technique
to perform the coincident detection of the positively charged fragments
emitted from collisions of CO dimers with $\rm Ar^{9+}$ projectile ions. Both
target and projectile beams cross at right angle at the center of the
spectrometer. A homogeneous perpendicular electric field of about 40 V/cm is
oriented along the vertical X axis and allows the collection of the cations
resulting from fragmentation on a micro-channel plate detector coupled to
delay line anodes. The $\rm 4\pi$ collection efficiency is restricted by the
80 mm diameter of the recoil ion detector located 100 mm from the collision
point resulting in a limit of 16 eV (resp. 32 eV) for the detection of singly
(resp. doubly) charged ions. Before reaching the detector the ions are further
post-accelerated to about 3 keV per charge to ensure an optimal detection
efficiency of the MCP. Finally, the three dimensional coordinates of the
momentum vector of each fragment are computed from their time-of-flight and
position on the detector. The highly charged $\rm Ar^{9+}$ projectile ions are
extracted from an electron cyclotron resonant ion source at a kinetic energy
of 135 keV on the ARIBE/GANIL beam line. The ion beam then passes through a
$\rm 600~{}\mu m$ collimation aperture before colliding with the gas jet. At
the exit of the spectrometer, the final charge state of the projectile ions is
analyzed thanks to a parallel plates deflector before reaching a dedicated
position sensitive detector. Argon projectiles with charge state ranging from
5+ to 8+ give the start for the time-of-flight measurement while the intense
primary $\rm Ar^{9+}$ beam is deflected out of the detector. No selection is
imposed on the final charge state of the projectile ion because it shows no
influence on the fragmentation dynamics of the ionized target.

The neutral CO dimer targets are produced from a supersonic gas jet with a
stagnation pressure of 15 bars, expending through a $\rm 30~{}\mu m$ nozzle. A
skimmer of 0.5 mm diameter is located 6 mm away from the nozzle and a
collimator of 0.5 mm diameter is placed before the collision chamber. The
resulting diameter of the gas jet at the collision point is about 1.5 mm. The
translational and rotational temperatures of the gas jet are estimated to be
below 10 K. However, the vibrational degrees of freedom are only partially
cooled down in the supersonic expansion and the vibrational temperature of the
jet is expected to be close to the temperature of the nozzle which is here 300
K [McClelland, 1979,Wall, 2016]. The proportion of CO dimers within the gas
jet is estimated to be about 1 % of the dominant monomer targets and the yield
of larger cluster $\rm(CO)_{n\geq 3}$ targets is at least one order of
magnitude lower than the dimers. As a consequence, an accurate data filtering
method is mandatory to extract the relatively small portion of events
originating from the relevant dimer break-ups. For this purpose, the
coincident measurement of all emitted cationic fragments and the restriction
imposed by the momentum conservation law enable to efficiently select the
desired fragmentation channels.

Figure 2: KER spectra of the monomer dication dissociation $\rm
CO^{2+}\rightarrow C^{+}+O^{+}$. Upper panel: KER spectrum from the present
measurement. Lower panel: KER spectrum from electron impact ionization
measurement [Lundqvist, 1995] and identification of the corresponding
electronic states of the $\rm CO^{2+}$ dication [Lundqvist, 1995]. The
relative deviation from the reference data is below 1 % over the 7 to 10 eV
energy range.

In the framework of the Coulomb imaging technique, the intermolecular bond
length $\rm R_{e}$ is derived from the KER measurement of the 2-body
dissociation channels. Therefore, special attention has been devoted to the
absolute momentum calibration of the spectrometer. This calibration requires
both time-of-flight and position calibrations along the two transverse
directions Y and Z of the spectrometer. The time-of-flight (TOF) calibration
is performed by using the centroids of several well identified peaks in the
TOF spectrum resulting from the detection of different atomic or molecular
ions and allows a precise determination of the spectrometer characteristics
(flight length and electric field strength). Additionally, the measured TOF
are further corrected to first order from the short flight time of the recoil
ions in the post acceleration region. The position calibration along the Y and
Z directions of the delay line anodes was adjusted to get the optimal KER
resolution, which is only achieved when the three components of each fragment
momentum are properly reconstructed. We used the dominant $\rm
CO^{2+}\rightarrow C^{+}+O^{+}$ fragmentation of the monomer dication and the
reference KER spectrum obtained with a Doppler free instrument [Lundqvist,
1995] for tuning the absolute energy calibration of our spectrometer (Fig. 2).
Even if our energy resolution does not allow to separate the vibrational
levels, the KER peak associated to the $\rm{}^{3}\Sigma^{+}$ state (KER=7.833
eV [Lundqvist, 1995]) can be used to test the absolute calibration of our
apparatus as it contains mainly one vibrational level. Our experimental energy
resolution has also been estimated from the width of this KER peak and is
found to be $\rm\Delta E$=0.28 eV (FWHM) for a KER of 7.833 eV. Moreover, the
contributions of two vibrational levels associated to the $\rm
2^{1}\Sigma^{+}$ state (Fig. 2) have also been deconvoluted and fitted from
the high energy peak around 9.5 eV. The results show that our KER calibration
is consistent with the reference data from [Lundqvist, 1995] with a precision
better than 1 % over the 7 eV to 10 eV range. The resulting uncertainty on the
intermolecular distance $\rm R_{e}$ is thus way below 10 % and would allow
separating the two predicted C-bonded and O-bonded structures of the CO dimer.

##  III EXPERIMENTAL RESULTS

###  III.1  2-body dissociation channels

We first consider the Coulomb explosion of the dimer resulting from a single
electron capture on each molecule $\rm(CO)_{2}^{2+}\rightarrow CO^{+}+CO^{+}$.
Fig. 3.a.) shows the associated KER spectrum with a single peak centered at
$\rm KER^{(1,1)}=3.21\pm 0.04~{}eV$. This value agrees very well with the
former experimental measurement by Ding et al. [Ding, 2017] in which the
similar dissociation channel of the $\rm CO$ dimers was triggered by multiple
ionization with ultra-short intense laser pulses. The presence of a single
peak in the KER spectrum indicates no sizable contribution of indirect
relaxation processes such as ICD (Intermolecular Coulomb Decay) or RCT
(Radiative Charge Transfer). The Coulomb explosion approximation can thus be
used to infer the intermolecular distance from the KER spectrum. In this
approximation, the equilibrium distance between the centers of mass of the two
CO molecules is found to be $\rm R_{e}^{(1,1)}=4.49~{}\text{\AA}$.

However the $\rm CO^{+}$ ion has a quite large electrostatic dipole moment
($\rm\mu\approx 2.75D$) and a significant part of the initial potential energy
can thus be released as rotational energy. This rotational energy is not
accounted for in the KER spectrum deduced from the $\rm CO^{+}$ fragment
momenta and leads to an overestimation of the $\rm R_{e}$ value deduced from
the pure Coulomb model. The correction of the KER measurement due to the
undetected rotational energy will be detailed in section V.B and leads the
actual intermolecular distance to be $\rm R_{e}^{(1,1)}=4.2~{}\text{\AA}$.

Figure 3: KER spectra of the 2-body a. (1,1) and b. (2,1) dissociation
channels. For both spectra, the background contribution is due to random
coincidences and is subtracted using a linear function and the red line is a
Gaussian fit of the experimental data. The expected position of the KER peaks
corresponding to theoretical predictions $\rm R_{e}=4.3~{}\text{\AA}$
(C-bonded) and $\rm R_{e}=3.6~{}\text{\AA}$ (O-bonded) are represented by
dashed black lines.

A small contribution of the $\rm(CO)_{2}^{3+}\rightarrow CO^{2+}+CO^{+}$
dissociation channel has also been identified (Fig. 3.b). Its low relative
intensity results mainly from the low proportion of metastable states (with a
lifetime longer than $\rm 1\mu s$) of the $\rm CO^{2+}$ dication in comparison
with the dissociative states leading to the competing 3-body channel
$\rm(CO)_{2}^{3+}\rightarrow CO^{+}+C^{+}+O^{+}$. The KER peak is centered at
$\rm KER^{(2,1)}=6.37\pm 0.07eV$ which is two times the KER measured in the
$\rm CO^{+}+CO^{+}$ channel as expected within the approximation of the
Coulomb explosion. A correction from the rotational energy (see section V.B)
leads to an intermolecular distance of $\rm R_{e}\cong 4.3~{}\text{\AA}$ for
the $\rm CO^{2+}+CO^{+}$ channel.

The two intermolecular distances inferred from different dissociation channels
($\rm R_{e}\cong 4.2~{}\text{\AA}$) seems more likely to correspond to the
C-bonded structure predicted by theoretical calculations (see table 1). As
already discussed in a previous section, this geometry has been calculated to
be the global minimum of the PES of the neutral CO dimer and is associated to
$\rm\uptheta_{a}=\uptheta_{b}=135^{\circ}$. The measured KER distributions of
Fig. 3 do not show any visible peak corresponding to the intermolecular
distance of $3.6~{}\text{\AA}$ expected for the isomeric O-bonded structure.
Contrary to previous IR measurements where several subbands were interpreted
to arise from two different dimer bond lengths ($4.0$ and $4.4~{}\text{\AA}$),
we observe here only one intermolecular distance. Our value is close to the
calculated distance of the true ground state of the dimer with a C-bonded
orientation. Further interpretation of this bond length measurement will be
given in the next section especially in regards of the supersonic jet
temperature.

###  III.2  3-body dissociation channels

In order to look for a more direct evidence for such a geometrical structure,
the 3-body channels will now be investigated. We focus here on several
particular 3-body dissociation channels (m=1,[p,q]) of the dimer with the aim
to determine the $\rm\uptheta$ coordinate, i.e. the relative orientation of
the CO molecules in the dimer. These channels result from a single electron
capture on one molecule and a double, triple or quadruple electron capture on
the other partner of the dimer. The use of the CEI technique requires that the
3-body break-up occurs simultaneously or at least on a smaller time scale than
the relative motion of each fragment ion. Such a specific case will be later
referred to as concerted fragmentation. Conversely, the existence of
dissociative states of the multi-ionized CO molecule with lifetimes much
longer than 10 ps will lead to a so-called sequential fragmentation. Such
long-lived states do not exist for multiply charged $\rm(CO)^{n\geq 3+}$
cations [Kumar, 2010] and sequential fragmentation is thus expected to be
observed only in the (1,[1,1]) channel.

####  III.2.1  The (1,[1,1]) channel: concerted and sequential fragmentation

Since its first observation within a mass spectrometer in 1932 [Freidlander,
1932], the lifetime of metastable states of $\rm CO^{2+}$ has been widely
studied both experimentally and theoretically [Andersen, 1993-Mrugala, 2008].
When such metastable states are populated, the late fragmentation of the
covalently bond molecule washes most of the information on the initial
geometry. In order to identify these events and exclude them from the
analysis, we will first study the Dalitz representation of the 3-body
fragmentation.

Fig. 4 shows the Dalitz diagram of the 3-body dissociation of the dimer. This
two-dimensional graph represents the reduced squared momenta of each fragment
$\rm\varepsilon_{i}=\frac{p_{i}^{2}}{\sum_{i=1}^{3}p_{i}^{2}}$ (where the
subscript i stands respectively for the $\rm C^{+}$, $\rm O^{+}$ and $\rm
CO^{+}$ recoiling ions) along the three axis of the diagram. The corresponding
Cartesian coordinates are calculated as : $\rm
x_{D}=\frac{\varepsilon_{1}-\varepsilon_{2}}{\sqrt{3}}$ and $\rm
y_{D}=\varepsilon_{3}-\frac{1}{3}$ where i is the number of the fragment
sorted by ascending order of mass-to-charge ratio. Two main contributions are
clearly visible and are assigned respectively to a concerted or a sequential
dissociation depending on the lifetime of the transient $\rm CO^{2+}$ dication
[Ding, 2017,Song, 2019]. The quasi-horizontal line in the upper part of the
graph is associated to the sequential fragmentation. In such a two-step
process, the $\rm CO^{2+}$ dication breaks up only after the two recoiling
molecular ions have reached the asymptotic limit for which their relative
Coulomb repulsion vanishes. As shown in the next section (Fig. 6.b), such a
limit is reached after about 10 ps and only metastable states of $\rm CO^{2+}$
with lifetimes longer than 10 ps contribute to the sequential dissociation
process. The recoiling $\rm CO^{2+}$ ion has thus enough time to rotate before
fragmenting and completely loses the information on its initial orientation
within the dimer. Contrarily, in a concerted fragmentation the $\rm CO^{2+}$
dication dissociates within a timescale shorter than 10 fs (see Fig. 6.b). The
corresponding events are located close to the bottom of the vertical axis.
Here, the $\rm CO^{+}$ acquires a smaller kinetic energy because of the fast
repulsion of the two $\rm C^{+}$ and $\rm O^{+}$ atomic ions following the
dissociation of the $\rm CO^{2+}$ dication. The exact distribution of these
events is directly correlated to the initial structure of the dimer but also
to the $\rm CO^{2+}$ fragmentation dynamics. Only these concerted
fragmentation events contain the desired information about the initial three-
dimensional structure of the dimer. This lower part of the Dalitz distribution
will later be compared to classical mechanics calculations with the aim to
infer the relative molecular orientation in the dimer. Note that a ”V shape”
contribution is also visible on the Dalitz plot but with a very small
intensity (expending from $\rm y_{D}\cong-0.2$ up to the upper part of the
plot). These events are also assigned to a concerted fragmentation and will be
discussed in the last section.

Figure 4: Dalitz plot of the (1,[1,1]) dissociation channel and identification
of the concerted and sequential fragmentation processes.

####  III.2.2 The multiply charged (1,[p,q]) channels

If one electron is captured on the first molecule and more than two electrons
are captured from the other molecule, the dimer undergoes a concerted 3-body
explosion as there are no long-lived states of the $\rm CO^{n+}(n\geq 3)$
cations. In the present experiment, events corresponding to n=3 and n=4 have
been recorded. The cross sections for multiple ionization decrease with the
number of captured electrons and result in low statistics for these highly
charged channels. For n=3, two distinct channels compete depending on the
charge sharing between the emitted carbon and oxygen ions: (1,[1,2]) and
(1,[2,1]). We found that the relative yield of the (1,[1,2]) is approximately
five times lower than the dominant (1,[2,1]) channel. This ratio is compatible
with previous measurements for monomer targets and results from the lower
ionization potentials of the carbon ions [Ben-Itzhak, 1993]. For n=4, the
symmetric (1,[2,2]) channel is found to be by far the dominant channel and no
significant contribution of the asymmetric channels (1,[1,3]) and (1,[3,1])
has been observed. The preference for symmetric charge sharing in the
fragmentation of the $\rm CO^{4+}$ ion has also been reported for monomers
targets [Ben-Itzhak, 1993] and agrees with our measurement.

The Dalitz plots for the (1,[1,2]), (1,[2,1]) and (1,[2,2]) channels are shown
in Fig. 5. For these channels, no sequential fragmentation is observed but a
weak ”V shape” contribution can be identified in the (1,[2,1]) channel. The
main density distribution stands on the right or left side of the graph
depending on the mass-over-charge ratio of each ion. A more detailed
description of these graphs will be given when comparing to the simulated data
in the next section.

Figure 5: Dalitz diagrams for the (1,[2,1]), (1,[1,2]) and (1,[2,2]) channels.
The Cartesian coordinates are: $\rm
x_{D}=\frac{\varepsilon_{1}-\varepsilon_{2}}{\sqrt{3}}$ and $\rm
y_{D}=\varepsilon_{3}-\frac{1}{3}$ where $\rm\varepsilon_{i}(i=1,2,3)$ are the
reduced squared momenta of the fragments, sorted by ascending order of mass-
to-charge ratio. Using this convention, the momentum of the $\rm CO^{+}$ ion
is plotted along the vertical axis but the carbon and oxygen ions are plotted
the right or left diagonals depending on their respective charge states.

##  IV COMPUTATIONAL METHOD

The momentum sharing in the 3-body break-up of the dimer has been investigated
using Monte Carlo simulations based on a classical model where each charged
fragment is considered as a point-like particle located at its center-of-mass.
The complex fragmentation dynamics of the molecular $\rm CO^{n+}(n\geq 2)$ has
been simply accounted for using the experimental KER spectra. For each
simulated event, the classical trajectories of the three emitted ions are then
numerically integrated to access the final momentum of each fragment. Despite
its simplicity, the resulting model gives a good quantitative description of
the dependence of the experimental observables on the geometrical structure of
the dimer.

####  IV.0.1 Description of the model

The model assumes a concerted 3-body fragmentation in which both the van der
Waals and the covalent bonds break at the same time. The three ions are
considered as three point-like particles with zero initial velocity and are
initially positioned accounting for the intermolecular distance $\rm
R_{e}=4.5~{}\text{\AA}$, the molecular bond length $\rm
R_{CO}=1.13~{}\text{\AA}$ of the neutral CO and the orientation $\rm\uptheta$
of the dissociating $\rm CO^{n+}$ with respect to the dimer axis (see Fig. 1).
Note that the qualitative description of the simulated Dalitz graphs is only
weakly sensitive to the bond length and we used here the value $\rm
R_{e}=4.5~{}\text{\AA}$ deduced from the pure 2-body Coulomb model (see
section III.A). Then, the equations of motion are integrated using a standard
Runge-Kutta method and assuming a pure Coulomb repulsion between both atomic
$\rm C^{p+}$ and $\rm O^{q+}$ ions and the non-dissociating $\rm CO^{+}$
molecular ion. Contrarily, the mutual interaction between the $\rm C^{p+}$ and
$\rm O^{q+}$ ions is known to be not purely Coulombic as it can be obviously
demonstrated by the large KER distributions for these channels. Accounting for
the KER distributions by the mean of ab-initio calculations of the potential
energy curves would be a tedious task. We thus use a scaled Coulomb energy
curve directly inferred from the experimental data. For each simulated event,
the total KER is first sampled randomly according to the experimental KER
distribution. The 2-body KER for $\rm C^{p+}$ and $\rm O^{q+}$ ions is then
obtained by subtracting the potential energy due to Coulomb interaction with
the remaining $\rm CO^{+}$ molecular ion at a distance $\rm R_{e}$. The 2-body
repulsive potential between the $\rm C^{p+}$ and $\rm O^{q+}$ ions is then
approximated by : $\rm V_{CO}(r)=\frac{p\times
q}{4\pi\epsilon_{0}}\frac{A_{CO}}{r}$ where $\rm r$ is the distance between
the $\rm C^{p+}$ and the $\rm O^{q+}$ ions and where the unit-less parameter
$\rm A_{CO}$ is set to obtain the desired KER value. Such a potential curve
gives rise to a 2-body KER of $\rm\frac{p\times
q}{4\pi\epsilon_{0}}\frac{A_{CO}}{R_{CO}}=p\times q\times A_{CO}\times
12.74eV$ for the monomer cation as the bond length is fixed at $\rm
R_{CO}=1.13~{}\text{\AA}$. The parameter is chosen either $\rm A_{CO}>1$ or
$\rm A_{CO}<1$ in order to reproduce the experimental KER spectrum and typical
values range from $\rm A_{CO}=0.4$ to $\rm A_{CO}=4$. The 3-dimensional
coupled differential equations of motion are then deduced from the forces
acting between each partner:

$\rm\left\lVert\overrightarrow{F_{12}}\right\rVert=\frac{p}{4\pi\epsilon_{0}}\frac{1}{r_{12}}$

$\rm\left\lVert\overrightarrow{F_{13}}\right\rVert=\frac{q}{4\pi\epsilon_{0}}\frac{1}{r_{13}}$

$\rm\left\lVert\overrightarrow{F_{23}}\right\rVert=\frac{p\cdot
q}{4\pi\epsilon_{0}}\frac{A_{CO}}{r_{23}}$

where the subscript 1, 2 and 3 stand for the $\rm CO^{+}$, $\rm C^{p+}$ and
$\rm O^{q+}$ ions respectively.

Finally, the numerical integration interval is set to 1 ns which is a
sufficiently long time to ensure that all ions have reached their asymptotic
momentum vector limit. Fig. 6 shows three examples of such a 3-body
dissociation for the (1,[1,1]) channel, with $\rm A_{CO}=1$ but using
different initial geometries $\rm\uptheta=90^{\circ},45^{\circ},135^{\circ}$.
For all cases, all ions reach their kinetic energy limit after few picoseconds
(see Fig. 6.b).

Figure 6: a), c) and d) : calculated ion trajectories of the 3-body
dissociation of the (1,[1,1]) channel for
$\rm\uptheta=90^{\circ},45^{\circ},135^{\circ}$. b) : Evolution of the kinetic
energy and total KER as a function of time for $\rm\uptheta=90^{\circ}$. The
asymptotic limit is reached after about 10 ps.

For a qualitative understanding of the 3-body fragmentation, Fig. 7 shows the
Dalitz plots associated to the three intended geometrical configurations
$\rm\uptheta=90^{\circ},45^{\circ},135^{\circ}$ and for KER ranging from 10 to
50 eV ($\rm 0.39\leq A_{CO}\leq 3.9$). First, the figure indicates that,
independently of the initial geometry, large kinetic energy released in the
dissociation of the molecular $\rm CO^{2+}$ dication results in smaller energy
of the $\rm CO^{+}$ ions. Indeed, the $\rm CO^{+}$ spends a shorter time in
the field of the emitted $\rm C^{+}$ and $\rm O^{+}$ atomic ions and thus
acquires less kinetic energy. Secondly, for a given KER of the $\rm CO^{2+}$
dication, the $\rm CO^{+}$ acquires a significantly larger kinetic energy in
the case of the C-bonded or O-bonded ($\rm\uptheta=45^{\circ}$ or $\rm
135^{\circ}$) geometries because the $\rm CO^{+}$ ion is pushed away by the
$\rm C^{+}$ or $\rm O^{+}$ ion emitted towards him. Finally, the dots
associated to the $\rm\uptheta=45^{\circ}$ initial orientation are located on
the right side of the points for $\rm\uptheta=90^{\circ}$ because the $\rm
O^{+}$ atomic ion experienced a smaller Coulomb repulsion due to its central
position within the fragmenting dimer. A similar argumentation justifies that
the dots for $\rm\uptheta=135^{\circ}$ stand on the left side of the Dalitz
plot. Comparing with the experimental data, where a large majority of events
are located on the left part of the Dalitz plot (Fig. 4), the calculated $\rm
45^{\circ}$ (O-bonded) structure seems to be excluded since it does not match
the experimental momentum repartition. On the other hand, both the $\rm
135^{\circ}$ (C-bonded) and $\rm 90^{\circ}$ structures may reproduce the
experiment.

Figure 7: Dalitz representation of the reduced squared momenta in the
concerted 3-body dissociation of the (1,[1,1]) channel for
$\rm\uptheta=90^{\circ},45^{\circ},135^{\circ}$ and for kinetic energy release
ranging from 5 to 50 eV ($\rm 0.39\leq A_{CO}\leq 3.9$).

####  IV.0.2 Convolution with the experimental conditions

To perform a more realistic comparison, the simulation algorithm has been
extended in order to include most of the ingredients of the experimental
conditions that may impact the data. The main causes responsible for the
larger experimental distributions are the vibrational and rotational motion
inside the dimer target and, to a smaller extent, the instrumental resolution
of the spectrometer. To reproduce as closely as possible the experimental
conditions, the 3-body dissociation is now simulated in the electric field of
the recoil ion spectrometer and the ions are collected up to the detector.
Events are generated using a Monte-Carlo approach. For each dissociation
channel, the number of events is taken close to the statistics measured in the
experimental data: respectively 2000, 1700, 500 and 500 events for the
(1,[1,1]), (1,[1,2]), (1,[2,1]) and (1,[2,2]) channels. For each event, the
initial conditions are set randomly and the ions are then propagated up to the
detector. In such a way, the simulated data are created as a set of coincident
detection of the time-of-flight, Y and Z positions on the detector and are
further treated using the same analyzing procedure as for the experimental
data, ensuring a fair comparison between experiment and calculation.

Here is a detailed description of the parameters used in the model. First, the
initial positions of the dimer were generated using random three dimensional
Gaussian distributions with widths (FWHM) $\rm\Delta x=\Delta y=0.6mm$
corresponding to the diameter of the ion beam collimator and $\rm\Delta
z=1.5mm$ according to the size of the gas jet. The initial velocities of the
center of mass of the dimer is dominated by the momentum transferred to the
target by the highly charged projectile ion during the collision [Iskandar,
2018]. These velocity distributions have been inferred from the experimental
data and are found to be $\rm\Delta v_{x}=\Delta v_{y}=450m/s$ (FWHM) in the
transverse plane and about $\rm\Delta v_{z}=45m/s$ (FWHM) in the longitudinal
direction. Then, the initial geometry of the dimer is fixed and the initial
position of the point-like $\rm C^{+}$, $\rm O^{+}$ and $\rm CO^{+}$ ions are
defined as described in the previous section. The orientation of the dimer
with respect to the laboratory frame is then chosen according to an isotropic
distribution. With this last assumption, we neglect any angular dependence of
the electron capture process leading to these fragmentation channels. Such an
angular dependence will however have no influence on the present determination
of the geometrical structure of the dimer. To crudely account for the
vibration of the molecules inside the dimer, a jitter of $\rm\Delta
R_{e}=0.9~{}\text{\AA}$ (FWHM) on the intermolecular distance is used as
deduced from the width of the KER spectra of the 2-body
$\rm(CO)_{2}^{2+}\rightarrow CO^{+}+CO^{+}$ channel. Additionally, an
arbitrary jitter of $\rm\Delta\uptheta=20^{\circ}$ (FWHM) is used around the
mean angles $\rm\uptheta=90^{\circ}$, $\rm 45^{\circ}$, $\rm 135^{\circ}$,
$\rm 0^{\circ}$, $\rm 180^{\circ}$. Finally, to reproduce faithfully the
experimental KER distribution, we randomly sample the experimental KER spectra
associated to each channel and scale the effective repulsion coefficient $\rm
A_{CO}$ between the two $\rm C^{p+}$ and $\rm O^{q+}$ accordingly.

The coupled motion of each ion is then integrated up to the detector assuming
a pure concerted fragmentation and accounting for the homogeneous extraction
field E=40 V/m of the spectrometer. Finally, the detector response function
has also been included using Gaussian distributions of $\rm\Delta TOF=1~{}ns$
for the timing resolution and of $\rm\Delta Y=\Delta Z=0.3~{}mm$ for the
position resolution.

##  V COMPARISON TO EXPERIMENT

###  V.1 3-body channels

Figure 8: Experimental and simulated Dalitz plots for the (1,[1,1]),
(1,[2,1]), (1,[1,2]) and (1,[2,2]) channels. For the experimental (1,[1,1])
channel, only the concerted events are shown. For each channel, simulations
have been run with initial orientations of
$\rm\uptheta=90^{\circ},45^{\circ},135^{\circ},0^{\circ},180^{\circ}$ assuming
a concerted fragmentation. Bottom : projection of the horizontal Dalitz
coordinate $\rm x_{D}=\frac{\varepsilon_{1}-\varepsilon_{2}}{\sqrt{3}}$. The
vertical dashed line indicates the centroid of the experimental distribution
for each fragmentation channel.

Simulations have been performed for the four fragmentation channels (1,[1,1]),
(1,[2,1]), (1,[1,2]), (1,[2,2]). For each channel, simulations were performed
for five initial angular structures of the dimer $\rm\uptheta=90^{\circ}$,
$\rm 45^{\circ}$, $\rm 135^{\circ}$, $\rm 0^{\circ}$, $\rm 180^{\circ}$
corresponding to different conformations predicted by theoretical
calculations. The resulting Dalitz plots are summarized on Fig. 8 where they
are compared to the experimental data. For several channels and initial
geometries of the simulation, few events appear as a diffuse background
superimposed to the main contribution. These events are due to an incorrect
reconstruction during the data analysis procedure and originates from the
overlap of the time-of-flight distributions of the $\rm C^{+}$ and $\rm O^{+}$
or the $\rm C^{+}$ and $\rm O^{2+}$ partners due to their close mass-over-
charge ratios. A small portion of these events are not efficiently filtered
out by the momentum conservation law restriction imposed in the data analysis.
They will be ignored in the forthcoming discussion.

The bottom graphs on Fig. 8 represent the projection of the horizontal Dalitz
coordinate $\rm x_{D}=\frac{\varepsilon_{1}-\varepsilon_{2}}{\sqrt{3}}$ and
gives the momentum repartition between the two atomic ions $\rm C^{p+}$ and
$\rm O^{q+}$. Such one dimensional histograms are more appropriate to perform
a quantitative comparison between experiment and calculations, especially when
the statistic is low. Fig. 8 will now be used to deduce the three-dimensional
structure of the dimer through a detailed comparison to simulated events. Even
if the sensitivity to the initial angle $\rm\uptheta$ is limited, the combined
information extracted from the four different fragmentation channels enables
identifying the initial geometrical structure of the dimer.

  * $-$

the simulated Dalitz spectra for $\rm\uptheta=90^{\circ}$ and
$\rm\uptheta=135^{\circ}$ are quite similar except for the (1,[2,1]) channel
where significantly different distributions are observed. A close comparison
of the $\rm x_{D}$ distributions clearly shows a better agreement of the
experimental data with the $\rm\uptheta=90^{\circ}$ structure. For all
fragmentation channels, the centroid of the simulated $\rm\uptheta=90^{\circ}$
data matches almost perfectly with the experiment while the
$\rm\uptheta=135^{\circ}$ curves are significantly shifted to the right or
left side of the measured value.

  * $-$

the simulated O-bonded structure ($\rm\uptheta=45^{\circ}$) does not reproduce
the experimental data. It confirms our previous conclusion obtained from the
2-body dissociation (1,1) and (1,2) channels for which the measured dimer bond
length does not match with theoretical predictions of the O-bonded structure.

  * $-$

simulated data corresponding to $\rm\uptheta=0^{\circ}$ and $\rm 180^{\circ}$
show very specific Dalitz distributions for all fragmentation channels. The
$\rm CO^{+}$ ion acquires a large amount of kinetic energy (events are located
in the upper part of the Dalitz diagram) because of the Coulomb repulsion
between the $\rm CO^{+}$ ion and the $\rm C^{p+}$ ($\rm\uptheta=180^{\circ}$)
or $\rm O^{q+}$ ($\rm\uptheta=0^{\circ}$) ion emitted along the dimer axis.
Such events have a very weak contribution in the experimental data, thus
excluding a linear geometry of the dimer.

  * $-$

T-shaped geometries are associated to two distinct $\rm\uptheta$ values
($\rm\uptheta\approx 0/180^{\circ}$ and $\rm\uptheta\approx 90^{\circ}$) [Han,
1997]. They would result in two distinct contributions in the Dalitz plot
depending on either the molecule perpendicular or aligned with the dimer axis
breaks. The relative amplitude of these two contributions is expected to be
about 50 % because the electron capture probability on each molecular site of
the dimer does not depend much on its relative orientation with respect to the
dimer axis and that no rapid charge transfer between the two molecules is
expected before fragmentation. As already discussed, very few events
corresponding to $\rm\uptheta=0$ or $\rm 180^{\circ}$ are observed in the
experiment thus excluding a large contribution of T-shaped geometries.

The above detailed analysis of the 3-body fragmentation channels seems to
reveal that the CO dimer structure consists mainly of two molecules oriented
perpendicularly to the dimer axis ($\rm\uptheta=90^{\circ}$). The weak ”V
shape” observed in the (1,[1,1]) and (1,[2,1]) channels seems to arise from a
small contribution of geometries in which the fragmenting molecule is aligned
with the dimer axis. However, for $\rm\uptheta=90^{\circ}$, the simulation
does not reproduce the experimental widths of the Dalitz distributions which
may indicate that the molecular orientation in the dimer $\rm\uptheta$ has a
wider distribution than accounted for in the simulation.

Surprisingly our data show no evidence for the C-bonded or O-bonded structures
previously predicted by the calculations and associated to the two isomers
observed in infrared spectroscopy measurements [Brookes, 1999,Surin,
2003,Rezaei, 2013]. As discussed in [Brookes, 1999], these two isomers are not
fully distinct but rather overlapping and are associated to a wide angular
extent. Transitions between these two conformations have also been observed in
infrared spectroscopy presuming for a wide range of angular orientation
ranging from C-bonded to O-bonded structure. Similar behaviors could also be
deduced from the calculated PES of the dimer due to the low barrier between
the two main minima. Vibrational excitation allows a dis-rotatory motion from
the global C-bonded minimum towards the local O-bonded minimum while passing
through parallel conformations. However according to the PES calculation, the
rotatory motion is supposed to be accompanied by a stretching of the dimer
because each angular orientation is associated to a different intermolecular
distance.

As already mentioned, the actual vibrational temperature of the gas jet could
be as large as 300 K (about $\rm 200~{}cm^{-1}\approx 25~{}meV$). The
amplitude of the corresponding motion of each CO molecule in the dimer may be
very wide [Brookes, 1999] and obviously much larger than the arbitrary jitter
of $\rm\Delta\uptheta=20^{\circ}$ used in the previous simulations. A precise
description of this motion and of the corresponding dimer conformations is by
far out of the scope of our classical model. For the sake of simplicity, we
ran a new set of simulations in the most extreme approximation of a fully
random initial orientation of the two molecules with respect to the dimer
axis. This approximation includes inherently a mixture of any of the previous
initial angles ($\rm\uptheta=90^{\circ}$, $\rm 45^{\circ}$, $\rm 135^{\circ}$,
$\rm 0^{\circ}$, $\rm 180^{\circ}$) but with a relative weight given by the
sine dependance of the solid angle i.e. with a largely favored contribution of
angles close to $\rm 90^{\circ}$. The resulting Dalitz plots are shown on Fig.
9 for the four fragmentation channels.

Figure 9: Experimental and simulated Dalitz plots for the (1,[1,1]),
(1,[2,1]), (1,[1,2]) and (1,[2,2]) channels. For the experimental (1,[1,1])
channel, only the concerted events are shown. For each channel, simulations
have been run with a random initial orientation and assuming a concerted
fragmentation. Bottom : projection of the horizontal Dalitz coordinate $\rm
x_{D}=\frac{\varepsilon_{1}-\varepsilon_{2}}{\sqrt{3}}$. The vertical dashed
line indicates the centroid of the experimental distribution for each
fragmentation channel.

The experimental spectra are very well reproduced within such an approximation
of a fully random initial orientation of the two CO molecules. The centroids
of the Dalitz projections are similar to the ones obtained previously for
$\rm\uptheta=90^{\circ}$ but the widths are significantly increased due to the
contribution of other angles. It also confirms that the ”V shape” is
associated to initial angles close to $\rm\uptheta\cong 0$ or $180^{\circ}$.
The very nice quantitative agreement of the simulated events with our data
indicates that, rather than a well defined orientation, the molecules have a
quasi-random orientation in the dimer. This observation is partly consistent
with the previous results of infrared spectroscopy where the two isomers were
also assumed to have a wide angular distribution. Moreover, our supersonic
expansion method for the production of CO dimers is similar to the one used in
[Brookes, 1999] and similar jet temperatures are thus expected. Even if the
translational and rotational degrees of freedom are efficiently cooled down in
the expansion, the vibrational temperature is large enough to allow highly
excited vibrational states and results in a very large angular distribution.
Within our experimental resolution, the vibrational motion is so large that it
is compatible with a fully random orientation of the molecules inside the
dimer.

###  V.2 2-body channels

As shown in section III.A, the dimer intermolecular distance can be deduced
from KER spectra of the 2-body (1,1) and (2,1) channels. The pure Coulomb
model gives about $R_{e}=\rm 4.5~{}\text{\AA}$ for the two channels but the
value has to be corrected from the rotational energies of the two emitted
molecular ions. During the dimer dissociation, a rotational motion of both
ions is induced by the repulsion between the two dipoles of the $\rm CO^{+}$
and/or $\rm CO^{2+}$ ions. After dissociation, the resulting final rotational
frequency depends on the initial conformation of the dimer (bond length and
angular orientation). According to the previous analysis of the 3-body
channels, the initial molecular orientation in the dimer is almost random and
we used a Monte Carlo approach to average over all geometrical structures.

We thus adapted the previous model and extend it to account for the molecular
rotational motion in the 2-body dissociation. The four atomic centers of the
dimer are assumed to be point like particles. The partial charge on the carbon
and oxygen atoms inside the $\rm CO^{+}$ ions are deduced from the
electrostatic dipole moment of the $\rm CO^{+}$ ion. We used a repartition of
respectively $\rm 90\%$ and $\rm 10\%$ on the carbon and oxygen atoms inside
the $\rm CO^{+}$ molecular ion. For the (2,1) channel, we assumed the charge
to be equally shared between the two atoms as we found no data on the
electrostatic dipole moment of the metastable $\rm CO^{2+}$ dication in the
literature. A pure Coulomb repulsion is used between each atomic ions from two
different molecular sites (no intramolecular repulsion) and the binding
intramolecular force is described by a restoring force proportional to the
displacement along the molecular axis. The spring constant is chosen to
reproduce vibrational wave numbers of about $\rm 2200~{}cm^{-1}$. This
constant has very little influence on the results as only a negligible part of
the initial energy is converted into intramolecular vibrational energy.
Finally, the initial structure of the dimer is sampled using a Monte Carlo
method by using a dimer bond length of $\rm 4.5~{}\text{\AA}$ with a jitter of
$\rm 0.9~{}\text{\AA}$ (FWHM) and assuming a pure random orientation of the
two molecules.

This numerical integration of the ion motions is much more time consuming than
in our previous model of the 3-body dissociation because of the fast
vibrational motion of the two molecular ions. As a consequence, the ions are
not propagated up to the detector but the calculation is stopped after $\rm
50ps$ where we consider that the Coulomb repulsion between the two molecular
ions vanishes. After dissociation, the resulting rotational frequency ranges
from 0 to about 2.5 $\rm ps^{-1}$ depending mainly on the initial orientation
of the molecules with respect to the dimer axis. For each event, the total
rotational energy of the two ions is calculated as the difference between the
initial electrostatic potential energy of the system and the sum of the
kinetic energy of the center-of-mass of the two molecular ions. For the (1,1)
channel, we found that in average about 0.25 eV of the initial energy is
converted into rotational energy. The experimental KER spectra is thus
underestimated by 0.25 eV as our KER measurement includes only the energy of
the center-of-mass of each $\rm CO^{+}$ molecular ion. Adding this rotational
energy to the experimental KER leads to a total energy of 3.46 eV which
corresponds to an equilibrium intermolecular distance of the CO dimer of $\rm
R_{e}^{(1,1)}=4.17~{}\text{\AA}$. The uncertainty on the average rotational
energy (0.25 eV) is hard to quantify but we estimate that its contribution to
the error bars on $\rm R_{e}$ is about $\rm 0.2~{}\text{\AA}$ (FWHM). A
similar method has been applied to the (2,1) channel but assuming an equal
charge sharing between the two atoms in the $\rm CO^{2+}$ dication. We found
that the rotational motion of the two molecular ions carries an energy of
about 0.36 eV for this channel. The corrected total energy is then 6.73 eV and
the corresponding bond length is about $\rm R_{e}^{(2,1)}=4.3~{}\text{\AA}$
but with a large uncertainty due to the lack of information on the $\rm
CO^{2+}$ dipole moment. As already discussed, the bond length derived from the
two channels fits very well with the predicted true ground state of the CO
dimer.

According to theoretical calculations, the minimum energy of each angular
conformation is reached for different dimer bond length ranging from $\rm 3.6$
(O-bonded) to $\rm 4.3~{}\text{\AA}$ (C-bonded). We thus could expect very
large KER distributions in the 2-body channels while the widths of our KER
peaks (Fig. 3) do not show any significant contribution of the shortest bond
length. Our interpretation of the 2-body and 3-body channels seems then not
fully consistent with the shape of the PES of the CO dimer. This apparent
contradiction may arise from the production of high excited vibrational states
in the dimer gas jet. The high vibration excitation may partially destroy the
correlation between angular orientation and intermolecular separation
predicted in the calculations as we observe mainly the longest bond length
corresponding to the ground state C-bonded conformation.

##  VI CONCLUSION

We measured the 2-body and 3-body dissociation of CO dimers following multiple
electron capture by slow highly charged $\rm Ar^{9+}$ ions. We employed a
Coulomb explosion imaging approach based on the axial-recoil approximation to
retrieve the initial three dimensional structure of the dimer from the
coincident measurement of the momentum vector of each emitted ions. Molecular
orientation inside the dimer has been probed using the concerted fragmentation
events of four different 3-body channels (1,[p,q]) and compared to 3-body
simulated events assuming different initial geometries. This experimental
method combined with a dedicated classical trajectory model enabled to get a
direct and unambiguous determination of the dimer geometry. The structure of
the CO dimer is found to be very diffuse and the measured Dalitz plots are
found to be compatible with a purely random orientation of each CO molecule in
the dimer. This result is somewhat consistent with infrared spectroscopy
measurements as it confirms the very wide angular extent of the dimer
conformation that may be attributed to a high vibrational temperature of the
gas jet. According to the calculated PES, a dis-rotatory path covering both
C-bonded and O-bonded geometries as well as intermediate parallel
conformations could be responsible for such a large angular distribution at
this temperature. The mean intermolecular distance has also been deduced from
the KER distribution of the 2-body channels. After correction from the
rotational energy acquired during dissociation, we found a bond length of
about $\rm R_{e}=4.2~{}\text{\AA}$. This distance corresponds to the predicted
true ground state of the CO dimer with a planar C-bonded geometry.
Unexpectedly we found no evidence of the second isomer associated to a much
shorter bond length and previously observed in infrared spectroscopy
measurements. The relatively high temperature of the gas jet may be
responsible for the presence of highly excited vibrational states for which
the wave packet is centered around the global minimum of the PES (about $\rm
4.2~{}\text{\AA}$) but with a quasi random orientation of the two CO
molecules.

##  VII ACKNOWLEDGMENTS

Experiment was performed at Grand Accélérateur National d'Ions Lourds (GANIL)
by means of CIRIL Interdisciplinary Platform, part of CIMAP laboratory, Caen,
France. The authors want to thank the CIMAP and GANIL staff for their
technical support.

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