### Improved modelling of ion dynamics in the Thermo Scientific OrbitrapTM mass analyser using Hamiltonian perturbation theory

**Submitting Institution**

Loughborough University**Unit of Assessment**

Mathematical Sciences**Summary Impact Type**

Technological**Research Subject Area(s)**

Physical Sciences: Atomic, Molecular, Nuclear, Particle and Plasma Physics

Chemical Sciences: Physical Chemistry (incl. Structural), Theoretical and Computational Chemistry

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PDF**Summary of the impact**

This case study describes the impact of research at Loughborough
University from 2009-2012 into
the mathematical modelling of the dynamics of ions using perturbation
theory of Hamiltonian
systems of equations. Outcomes from this research have been incorporated
into software used for
the performance modelling of a series of high-precision Fourier Transform
Mass Spectrometers
manufactured by Thermo Fisher Scientific GbmH and branded as Orbitrap^{TM
}with an average price
$0.5 million. The derived methodology reduces the time of numerical
modelling of the behaviour of
charged particles in an Orbitrap^{TM }instrument by a
factor of 100 to 1000. This reduction is of
significant benefit to the Life Science Mass Spectrometry, Scientific
Instrumentation Division of
Thermo Fisher Scientific and indirectly the users of the instrument.

**Underpinning research**

Perturbation theory of Hamiltonian systems is a well-developed
mathematical subject with
immense applications. One classical domain of application of this theory
is the description of the
dynamics of charged particles in electromagnetic fields. Professor Anatoly
Neishtadt, an employee
of Loughborough University from March 2007, has expertise both in the
mathematical aspects of
Hamiltonian perturbation theory and in its applications. References **3.1-3.5**
describe research into
the motion of charged particles in different electromagnetic field
configurations using perturbation
theory of Hamiltonian systems that was conducted at Loughborough
University in 2009-2012.

In references **[3.1-3.4]** the dynamics of a charged particle under
the action of an electromagnetic
wave in plasma with a background uniform magnetic field is studied. The
possibility (in the
framework of the considered model) of unlimited acceleration of charged
particles due to capture
into resonance with such a wave is established for the first time. The
methodology of **[3.1-3.4]** was
used in (the most significant for this case study) paper **[3.5]** for
the mathematical modelling of ion
dynamics in the electrostatic field of the Orbitrap^{TM} mass
spectrometer. This paper is the result of a
collaboration with a group of physicists and applied mathematicians from
Thermo Fisher Scientific
GbmH (D. Grinfeld, A. Makarov, E. Denisov) and A.M. Prokhorov General
Physics Institute,
Russian Academy of Sciences (M. Monastyrskiy, M. Skoblin). The authors
have suggested the use
of Hamiltonian perturbation theory for the description of the motion of
charged particles in Fourier
Transform mass spectrometers manufactured by Thermo Scientific GbmH and
branded as
Orbitrap^{TM}.

Orbitrap^{TM} is a trap for ions. In this trap ions cycle around
the axis of the trap and move back and
forth along this axis. The use of Orbitrap^{TM} in mass
spectrometry is based on measuring the ions'
axial oscillation frequency. Peaks of the Fourier spectrum of the
registered electric signal
correspond to assortments of ions with different mass/charge ratios. In an
ideal Orbitrap^{TM}, axial
motion of an ion is harmonic with the frequency independent of the ion's
energy and inversely
proportional to the square root of the mass-to-charge ratio. This allows
the identification of different
sorts of ions. This fundamental property of an ideal ion trap is corrupted
by perturbations arising
from the presence of the ion injection aperture, inaccuracies of electrode
manufacture and
Coulomb interaction between the ions (space-charge effects). Deviations of
the real field from the
ideal one can be treated as small perturbations. The dynamics of ions in
an Orbitrap^{TM} in the
presence of perturbations may be described by a Hamiltonian system of
ordinary differential
equations with slow and fast variables. In reference **[3.5]** the
averaged Hamiltonian motion
equations are derived through averaging over fast variables. The obtained
equations contain only
slowly changing variables. The use of these derived equations
significantly reduces the time of
numerical modelling of the motion of the ions in the Orbitrap^{TM}
by a factor of between 100 to 1000,
with the simulation accuracy exceeding that of direct trajectory tracing.

**References to the research**

**3.1.** Neishtadt A, Artemyev A, Zelenyi L, Vainchtein D, (2009),
Surfatron acceleration in
electromagnetic waves with low phase velocity, *JETP Letters*,
89(9), 441-447, DOI:
10.1134/S0021364009090045

**3.2. Artemyev A, Neishtadt A, Zelenyi L, Vainchtein D, (2010),
Adiabatic description of
capture into resonance and surfatron acceleration of charged particles
by
electromagnetic waves,** ** Chaos**,

**20(4), 043128, DOI: 10.1063/1.3518360**

**3.3.** Artemyev A, Vainchtein D, Neishtadt A, Zelenyi L (2011),
Resonant acceleration of charged
particles in the presence of random fluctuations, *Physical Review E*,
84(4), 046213, DOI:
10.1103/PhysRevE.84.046213

**3.4. Vasiliev A, Neishtadt AI, Artemyev A, (2011), Nonlinear dynamics
of charged particles
in an oblique electromagnetic wave,** ** Physics Letters A**,

**375(34), 3075-3079, DOI: 10.1016/j.physleta.2011.06.055**

**3.5. Grinfeld D, Makarov A, Skoblin M, Monastyrskiy M, Denisov
E, Neishtadt A, (2012),
Perturbation theory and space-charge ion dynamics in Orbitrap mass
analyser,
Proceedings of 13**

^{th}

**,**

*Seminar "Recent Trends in Charged Particle Optics and Surface**Physics Instrumentation"***Brno, 2012, ISBN 978-80-87441-07-7, 21-24. (Available from: http://www.trends.isibrno.cz/)**

The quality of this research is recognised internationally in terms of originality, significance and rigour. Papers 3.1 - 3.4 are published in the respectable academic journals, paper 3.5 was presented at one of the main international conferences in this particular area.

**Details of the impact**

Orbitrap^{TM }is a brand name for a series of high-precision
Fourier Transform Mass Spectrometer
systems produced by Thermo Fisher Scientific. The electrostatic orbital
ion trap with synchronous
oscillation properties was invented in 1996 by Dr. A. Makarov, who
received the American Society
for Mass Spectrometry Distinguished Contribution in Mass Spectrometry
Award in 2008 for this
development. Currently, the Thermo Scientific Orbitrap^{TM}_{ }mass
spectrometer has been produced in
several versions, with an average price about US$0.5 million for one
system. (Further details are
commercially sensitive.) Many Orbitrap^{TM} mass analysers have
been sold to the hospitals and
laboratories worldwide, but the precise number is commercially
confidential.

A recent picture and description of an Orbitrap^{TM} mass
spectrometer is shown below
(www.thermofisher.com).

Thermo Fisher Scientific introduces the Q Exactive high-performance
benchtop
quadrupole-Orbitrap LS-MS/MS. It is the first commercially available
instrument to bring
together quadrupole precursor selection and high-resolution accurate mass
(HR/AM)
Orbitrap^{TM} mass analysis to deliver high confidence
quantitative and qualitative
(quan/qual) workflows. With innovate HR/AM Quanfirmation^{TM}
capability, the Q
Exactive^{TM} mass spectrometer makes it possible to identify,
quantify and confirm more
trace-level metabolites, contaminants, peptides and proteins in complex
mixtures in one
analytical run. Unlike other technologies, high confidence results are
obtained without
sacrificing MS/MS sensitivity, mass resolution or quantitative
reproducibility.

Features

- Resolving power up to 140.000
- Maximum scan speed 12 Hz
- Intra — scan dynamic range > 5000:1
- Quadrupole mass filter
- Spectral multiplexing for enhanced duty cycle
- S-Lens ion source for increased sensitivity

Mathematical modelling has had a significant impact on the design of the
Orbitrap^{TM} device.

The results of reference **[3.5]** on Hamiltonian perturbation theory
of the motion of charged particles
had been used since 2011 by the Life Science Mass Spectrometry,
Chromatography and Mass
Spectrometry Division (CMD) of Thermo Fisher Scientific (Bremen) GmbH, for
Orbitrap^{TM} mass
spectrometer performance modelling. The mathematical techniques were
implemented in the
MASIM-3D software package in 2012. The averaged motion equations describe
the resonant
interactions of many ions of the same or close masses. The calculation
time decrease gained from
averaging ensures the modelling of a reasonably large number of ion
macro-particles (up to ~
2x10^{3}) to reveal the sophisticated space charge effects of
self-bunching and coalescence in an
Orbitrap^{TM} mass spectrometer. The coalescence is a
synchronization phenomenon when ions of
close masses move with the same frequency. Thus they create one peak on
the observed Fourier
spectrum and cannot be identified/distinguished using this spectrum.
Numerical analysis of ionic
motion close to the threshold of synchronisation allows reliable
estimation of the mass resolving
power limitation in an Orbitrap^{TM} mass spectrometer. The
calculated dependence of the
synchronisation threshold as a function of intentionally introduced small
static perturbations allows
the coalescence effect to be controlled.

The modelling of space-charge effects in the Orbitrap^{TM} mass
spectrometer by direct tracing of
multiple interacting ions is an extremely computationally demanding and
time-consuming task.
Obtaining the required accuracy is a major challenge. However, using the
averaged Hamiltonian
equations derived in **[3.5]** allows specialists at Thermo Fisher
Scientific to model the space-charge
effects in reasonable computer time with appropriate accuracy. This allows
them to try many
variations of the parameters in order to determine the necessary values to
optimise the Orbitrap^{TM}
mass spectrometer's performance.

The beneficiaries of the research are Thermo Fisher Scientific and
indirectly their customers in
healthcare research and industry. Orbitrap^{TM} mass spectrometer
is used in many hospitals and
laboratories in the UK and worldwide, in particular for drug monitoring
and food safety analysis.

Thermo Fisher Scientific is a large precision healthcare equipment global company with offices and operations in most countries around the world, and 2013 revenue guidance between $12.83 billion and $12.95 billion.

**Sources to corroborate the impact **

The following sources of corroboration can be made available at request:

**5.1.** Letter from: Director of Global Research for Life Science
Mass Spectrometry,
Thermo Fisher Scientific (Bremen) GmbH

Advanced Mass Spectrometry

Bremen

Germany