Trapping of Neutral Molecules by the Electromagnetic Beam Field
G. Franchetti
GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany,
Goethe University Frankfurt, Max-von-Laue-Straße 7, 60438 Frankfurt am Main, Germany ∗F. Zimmermann
European Organization for Nuclear Research (CERN),
Espl. des Particules 1 1217 Meyrin, Switzerland

M.A. Rehman
High Energy Accelerator Research Organization (KEK),
1-1 Oho, Tsukuba, Ibaraki, Japan, 305-0801, Japan(Dated: January 14, 2021)
Neutral uncharged molecules are affected by the electromagnetic field of a charged particle beam
if they carry either an electric or a magnetic dipole moment. The residual gas in an accelerator beam
pipe consists of such molecules. In this paper we study their dynamics. Under a few approximations,whose validity we explore and justify, we derive the equations of motion of neutral molecules andtheir invariants, determine the conditions for these neutral molecules to become trapped in thefield of the beams a function of beam-pipe temperature, and compute the resulting enhancement ofmolecule density in the vicinity of the beam. We demonstrate that large agglomerates of molecules,“flakes,” are much more likely to be pulled into the beam than single molecules, and suggest that
this phenomenon might help explain some beam observations at the Large Hadron Collider.

I.

INTRODUCTION

During the 2017 and 2018 runs of the Large Hadron
Collider (LHC), abnormal beam losses, some leading toemergency beam aborts, were observed at one particular location, in half-cell no. 16 left of LHC Interaction
Point 2 (“16L2”) [1, 2]. These losses, occasionally accompanied by a fast beam instability [3], were later attributed to an accidental in-leak of air into the beamvacuum, which had occurred while the beam screen was
at a temperature of 20 K. This in-leak had resulted ina surface layer of frozen air molecules, including oxygen,nitrogen, and water [4, 5]. Molecules of such a frozen
surface layer could have been detached by beam-induced
heating, synchrotron radiation, or by the beam field. Detached molecules which approach the beam may interact
with the circulating protons.
Molecules of oxygen or nitrogen are either paramagnetic or diamagnetic, respectively, having either a permanent or an induced magnetic dipole moment. Water
molecules, on the other hand, carry a permanent electric
dipole moment, which makes them ‘polar’.
The electromagnetic force exerted on molecules exhibiting an induced magnetic or electric dipole moment
is much weaker than the force experienced by paramagnetic or polar molecules, i.e., molecules with a permanent
dipole moment. For this reason, in the following, we will
not further discuss the motion of molecules with only induced dipole moments. Instead, our objective is to simulate and analyze the motion of neutral molecules carrying∗ g.franchetti@gsi.dea permament magnetic of electric dipole moment, in particular of the paramagnetic oxygen and of the polar water molecules, under the influence of the charged-particle
beam field.
If interactions with the residual gas limit the beam lifetime, the motion of neutral molecules inside the vacuum
chamber becomes important. The motion of paramagnetic or polar molecules in an electromagnetic field isgoverned by their electric and magnetic dipole moments.
We start this article by surveying the electric andmagnetic dipole moments of a few relevant molecules.
Then we construct the equations of motion for such neutral molecules subjected to the electromagnetic field of
the beam, at first crudely approximated by a thin-wiremodel. We solve these equations numerically, using a
Runge-Kutta method, for different initial conditions, inorder to identify the critical parameters for the moleculemotion. Next, considering a more realistic Gaussian
beam distribution, and recognizing important constantsof motion, we estimate the position-dependent criticaltemperature for molecules to be trapped by the electric or
magnetic field of the charged-particle beam. Expanding
this analytical treatment, we derive the general trapping
efficiency as a function of temperature, and the local density enhancement resulting from the gas molecules’ permanent dipole moments, which depends on beam current
and temperature. Finally, we consider the special case of“flakes”, i.e. agglomerates or clusters of many molecules
carrying a dipole moment, and show that such flakes are
more easily trapped in the field of the beam than theirindividual constituent molecules.
To simplify the equations, we are assuming the electric or magnetic dipole moments to always be alignedwith the corresponding local field of the beam. In the

2appendix we investigate the validity of this assumption.

II.

PERMANENT DIPOLE MOMENTS OF
NEUTRAL MOLECULESThe electric dipole moment (EDM) p~ for a pair ofcharges of the same magnitude but opposite sign is defined as~ ,p~ = QL(1)~ is thewhere Q denotes the magnitude of charges and L
displacement vector pointing from the negative charge
to the positive charge. The SI unit of the EDM is the
Coulomb meter (C m). In atomic physics also the unit
Debye (D) is used (1 D ≈ 0.21 eÅ with e the electron
charge and 1 Å = 0.1 nm).
If a molecule with nonzero EDM is exposed to an external electric field the molecule experiences a torque ~τ
given by~ .
~τ = p~ × E(2)Through the effect of τ , the electric dipole of themolecule, hence the molecule itself, tends to align its
axis with the direction of the applied electric field. The
amount of the torque is maximum if the dipole moment
is oriented at an angle of 90 degree with respect to theexternal electric field.
Molecules with an asymmetric internal charge separation like H2 O or CO feature a permanent non-zero
EDM. In fact, we can decompose the dipole moment of
H2 O, into an effective charge Q and distance L, for ex~ = 3.9 × 10−12 m, and hence
ample, as Q = 10e and |L|−30|~
pH2 O | = 6.2×10
Cm, where e ≈ 1.6×10−19 C denotes
the elementary electron charge. For the water molecule,it should be noted that L is a distance chosen to reproduce the measured dipole moment, for a given charge Q.
However, the actual geometry of the H2 O molecule resembles an isosceles triangle with two equal sides of about
1 A each, and one longer side of 1.6 A. This circumstanceshould be carefully considered in future modeling.
~ =Similarly, for CO, we can assign Q = 14e, and |L|−13−31
1.8×10m, to obtain |~pCO | = 4.06×10
Cm. On theother hand, molecules with a mirror-symmetric charge
distribution, like N2 , O2 , or CO2 , exhibit zero permanent
EDM [6, 7][8, Ch. 3].
Like the electric dipole moment, also the magnetic
dipole moment (MDM) µ~ is known as a property of both
fundamental and less fundamental particles. The MDM
can be expressed as a vector µ
~,
µ
~=

e
η~v ,
2m

(3)

where e, m and η are the charge, mass and reduced
Planck constant of a particle, respectively. The unit vector ~v represents the direction of the MDM. In case of theelectron, µ~ e is known as Bohr Magneton (BM) and its
value is 9.27 × 10−24 J/T .
Molecules can also feature a nonzero MDM. Similar tothe case of the EDM in an electric field, a molecule witha certain MDM exposed to an external magnetic field issubject to a torque ~τ , according to~ .
~τ = µ~ ×B(4)For both EDM and MDM, the torque is zero only if thedipole-moment axis is aligned with the field direction.
The magnitude of the molecular MDM depends on theelectron configuration in the valence shell. The magnetic
dipole moment arising due to unpaired electron spins is[9]pµs = n(n + 2)µe ,(5)where n denotes the number of unpaired electrons. In
the case of an oxygen molecule, O2 , two unpaired electrons are present, n = 2, so that the µs for oxygen is 2.8µe . Due to this nonzero value of its MDM, the oxygen
molecule exhibits paramagnetic properties.
Other molecules, like N2 , CO2 , CO and H2 O, do notpossess any unpaired electron in their respective valenceshell. Therefore, the spin magnetic dipole moment ofthese molecules is zero [6, 8, 10]. These molecules arecalled diamagnetic.
In the case of an air in-leak into the vacuum system ofan accelerator beam pipe, air molecules flowing into thevacuum pipe consist of nitrogen, oxygen, carbon dioxide,
and water vapors, which carry specific electric and magnetic dipole moments. Table I shows relevant dipole moments for such air molecules, with their respective magnitude.
TABLE I. Permanent electric (EDM) and magnetic dipole
moments (MDM) of various air molecules in units of Debye(D) and Bohr Magneton (BM), respectively, along with themolecule mass m in atomic mass units (amu).
Molecule EDM [D] MDM [BM] m [amu]
H2 O
0.39
0
18
O2
0
2.8
32
CO
0.025
0
28
N2
0
0
28
CO2
0
0
44

III.

MOLECULE MOTION ARISING FROM
THE DIPOLE MOMENTS~
If a neutral molecule is exposed to an electric field E~
and to a magnetic field B, its center of mass experiences
the force~ + (~~ ,F~t = (~
µ · ∇)Bp · ∇)E(6)

3~ = µ0 I ~r × ẑ ,
B
2πr r

(7)

where ẑ is a unit vector in the direction of the beam motion and I denotes the average beam current. The motion of the neutral molecules typically is extremely slow

compared with the time interval between the passage ofindividual bunches of the beam. For this reason, it is sufficient to consider the average beam current I, without
taking into account any bunch structure. The magnetic
field strength and field pattern are illustrated in Fig. 1.
3

0.05
0.05

0.075

2

B (mT)

0.250
0.225
0.175

1

Y (mm)

where p~ and µ~ denote the molecule’s electric and magnetic dipole moment, respectively. In the following weassume that the magnetic and electric dipole moments
are independent of location and, in particular, independent of the magnitude of the external fields.
As a next step, we now consider the force experienced~ and magby a molecule when subjected to the electric E~
netic B field generated by the beam. As p~ and µ~ oscillate~ and B,~ respectively, the moaround the direction of E,
tion of molecules may become quite complex.
The dynamics of the molecules is characterized by multiple time scales. The shortest time scale corresponds tothe vibrations of the atoms around their equilibrium position. The next, longer, time scale refers to the frequency
at which the molecule’s electric of magnetic dipole moment oscillates around the respective field. Lastly, thelongest time scale refers to the periodicity of a molecule’s
center-of-mass motion under the influence of the field gradient.
If the period of oscillation of p~ and µ~ is shorter than the~ and E
~ — this change
characteristic time of change of B
being due either to the molecule dynamics or due thevariation of the beam fields (individual bunch passages,
injection, etc.) —, we may assume that p~ and µ~ are andremain aligned with their respective beam field. In Appendix A we present a detailed discussion on the validityof this approximation.
Our simplifying assumption would also imply a highly
specific relation between p~ and µ~ , if both were simultaneously present, which, however, is not the case for anyof the air molecules in question (compare Table I).
We note that in the case of a complete alignment of
their magnetic and electric dipole moment with the corresponding beam field, the force on the molecules will be
maximum, hence leading to the largest possible excursionof the molecule motion, and therefore, to a conservative
assessment of the potential impact. In Appendix A, we
examine the validity of the alignment assumption.
Without any other external field, or in close proximity to the beam, the dominant component of the electricor magnetic field is the one produced by the circulating
beam.
In this section, we model the magnetic field by approximating the beam current as concentrated in a deltafunction-like manner into one transverse point at the center of the vacuum chamber, corresponding to the magnetic field of an extremely thin current-carrying wire. In
later sections, we will consider the field of a more realistic
beam distribution.
The thin-wire model properly describes the field at aradial distance r outside of a round (or approximately
round) beam. This field reads

0.200
0.15

0.25

0.175

0.125

0
0.150
0.225

0.2

0.125

-1

0.100
-2

0.075

0.1

0.050
0.05

-3
-3

0.05

-2

0

-1

1

2

3

X (mm)

FIG. 1. Magnetic field line pattern around a beam. Magnetic
field strength decreases as 1/r with radial distance r fromthe origin. Red spot at the center indicates the position of a
proton beam leaving the picture plane towards the reader.

Also the electric field generated by the round or approximately round beam, outside the beam core can approximately be described in a thin-wire approximation,
as the field of a line charge distribution, namely as~ =E

λ 1 ~r,2π0 r r(8)

where λ = Q/C is the line charge density, with Q thetotal charge in the accelerator and C the accelerator
circumference, 0 = 8.85 × 10−12 F/m the permittivityof free space, and r the radial distance. For an ultrarelativistic beam we have I = cλ, and hence~ =E

I 1 ~r.
2π0 c r r(9)Figure 2 displays the magnitude of the magnetic and electric fields for a beam with I = 1 A current as a functionof radial position and Figure 3 illustrates the electric field
pattern around the beam.
The equations of motion for molecules with electricand magnetic dipole moment are constructed under theapproximation previously discussed. Namely we use ouransatz that the EDM and MDM are aligned with theelectric and magnetic field generated by the beam, respectively. Namely we write
p~ = p

~r
r

and

~r
µ
~ = µ × ẑ .
r

(10)

Inserting these equations, together with Eqs. (9) and (7),

4
200

0.70

Electric Field

Magnetic Field 0.60
0.50
0.40
100

0.30

B (mT)E (V/mm)

150

0.20

50

0.10
0

0.00
0.5

1.0

1.5

2.0

2.5

3.0

r (mm)

FIG. 2. Electric and magnetic field of the beam as functionof radial distance as generated by a point-like thin-wire beamof current 1 A.

For our study we consider molecules having an initial
velocity consistent with a certain temperature T , andhaving random component in the x, y directions. The
rms thermal transverse velocity of a molecule, in the x-yplane, can be estimated by the classical formula
r
kb T
vth =
,
(14)
M
where kb designates the Boltzmann constant, T the temperature and M the mass of the molecule.
The top picture of Fig. 4 represents the variation of vth
as a function of temperature. The bottom picture shows
vth in units of energy (eV).
100

O2
H2 O

80

3

Vth (m/s)

2
70
50

1

Y (mm)CO

E (V/mm)

60

40

60

70

20
50

0

0

40

0

60

-1

10

15

20

Temperature (K)

30

O2

-2

-1

0

1

CO
2

3

X (mm)

FIG. 3. Electric field lines around the round beam for a beam
current of 1 A.

into Eq. (6), we obtain


pI
µµ0 I
x
+
,
Fx = −22π
2π0 c (x + y 2 )3/2


µµ0 I
pI
y
Fy = −+.
22π
2π0 c (x + y 2 )3/2

µ
.
c

0.5

0

20

40

60

80

100

Vth (m/s)(11)

FIG. 4. The rms transverse thermal velocity of a molecule,vth , as a function of temperature (top) and the average kinetic
energy versus the rms thermal velocity vth (bottom).

(12)

where I is the beam current, M is the molecule mass,and p̃ is an “effective” dipole moment, defined asp̃ = p +1.0

0.0

Hence, the equations of motion become
dvx
p̃I
x
=−
,
dt
2π0 M c (x2 + y 2 )3/2
p̃I
y
dvy
=−
,
dt
2π0 M c (x2 + y 2 )3/2

E (meV)

-2

H2 O

1.5

20

20

-3
-3

5

30

40

(13)

In order to integrate the equations of motion, the initialconditions of a molecule are required, namely its initial
velocity and initial position, ~r0 and ~v0 .

IV.

MOTION PHENOMENOLOGY

A numerical solution of the equations of motion has
been implemented using the software package MATHEMATICA [11]. SI units are used for all physical quantities. Different initial thermal velocities and positions are
considered for the simulation of molecules motion in theelectromagnetic field of the beam in order to identify thecritical parameters. In the simulation we can vary, forexample, the initial velocity and the initial position of amolecule.

5
6

4

2
Electric Field
Y (mm)The beam current is taken to be 1 Ampere. In all cases,
we assume that a molecule’s electric or magnetic dipole
moment is always completely aligned with the electric or
magnetic field of the beam, respectively (see appendix
A).
We first consider molecules at 1 mm distance from thebeam with zero initial velocity. Such molecules will get
trapped in the electromagnetic field of the beam and undergo an oscillatory motion. Figure 5 shows the horizontal motion of such H2 O, CO, and O2 molecules, initiallyat rest, as a function of time. The oscillation perioddepends on the dipole moment and on the mass of themolecules. In particular, the oxygen molecules perform
oscillatory motion, if subjected to the magnetic field of
the beam, due to their magnetic dipole moment, while
they are not affected by the electric beam field, since
they possess zero EDM. By contrast, water and carbon
monoxide molecules undergo an oscillatory motion due tothe electric field of the beam, since they carry a non-zero
electric dipole moment. Figure 6 displays the moleculemotion in the x-y plane.

Magnetic Field

0

O2
H2 O
-2

CO

-4

-6
-6

-4

-2

0

2

4

6

X (mm)

FIG. 6. The motion of neutral oxygen, water and carbon
monoxide molecules initially at rest, 1 mm from the beam,and subjected to the electromagnetic field of a beam, in thetransverse plane.With zero initial velocity, all molecules possessing either a magnetic or electric dipole moment undergo
an oscillatory motion.

FIG. 5. Motion of neutral molecules initially 1 mm from thebeam, at rest, and subjected to the electromagnetic field of
the beam, as seen in the horizontal direction. Different colors
correspond to different molecules as indicated.

We now add an extremely small random initial velocity
corresponding to a temperature of order 1 mK. Since
particles are launched with different initial velocities inthe x and y direction, the vertical and horizontal motion
are not identical. Figure 7 presents some examples of x
and y motion with respect to time, for this case of a small
initial velocity, and Fig. 8 the corresponding picture of
motion in the transverse plane.
We next increase the temperature to 15 K, a typicaltemperature of the LHC beamscreen (which is held between about 5 and 20 K). Molecules are again launchedat a distance of 1 mm from the beam. Figures 9 and 10
present the molecule motion for this case. Under these
conditions, single molecules of oxygen, water, and carbon
monoxide do not get trapped in the electromagnetic fieldof the beam, as their large thermal velocity overcomesthe effective potential created by the beam field. Insteadof getting trapped in the beam field, these molecules hit

FIG. 7. Motion of neutral molecules initially at 1 mm fromthe beam with a thermal velocity corresponding to a temperature of 0.9 mK, and subjected to the electromagnetic fieldof the beam, in horizontal (top) and vertical direction (bottom). Different colors correspond to different molecules as
indicated.

the vacuum chamber wall. In our simple model here, we
assume that the molecules then bounce back from thewall elastically. The wall is set at 6 mm, about threetimes closer than the actual LHC beam pipe, which has

6
6

6

4

4

2

2

Electric Field

Magnetic Field

0

❖2
❍2 O

-2

Y (mm)Y (mm)

Electric Field

Magnetic Field

0

O2
H2 O
-2

CO

CO

-4

-4

-6

-6
-6

-4

-2

0

2

4

-6

6

-4

-2

0

2

4

6

X (mm)X (mm)

4

X (mm)

FIG. 8. The motion of neutral oxygen, water and carbon
monoxidemolecules initially at 1 mm from the beam with a2thermal velocity corresponding to a temperature of order 1
mK, and subjected to the electromagnetic field of a beam,inO2
0
the transverse
plane. The initial thermal velocities were for
H2 O
O2 vth = 0.2 m/s (0.1 mK), for water vth = 3 m/s (20 mK),CO
-2
and for CO vth = 0.3 m/s (0.3 mK).

FIG. 10. The motion of neutral oxygen, water and carbon
monoxide molecules initially at 1 mm from the beam witha thermal velocity corresponding to a temperature of 15 K,
and subjected to the electromagnetic field of a beam, in thetransverse plane. The initial thermal velocities were vth =62 m/s for O2 ,vth = 82 m/s for water, and vth = 67 m/s for
CO. Note the assumption of perfect elastic collisions with thechamber wall.

-4

a radius
20 mm.
0.0 of about
0.2
0.4

0.6

0.8

1.0

Time (ms)4

Y (mm)

2
O2

0

H2 O
-2

CO

-4
-6
0.0

0.2

0.4

0.6

0.8

1.0

Time (ms)FIG. 9. Motion of neutral molecules initially at 1 mm fromthe beam with a thermal velocity corresponding to a temperature of 15 K, and subjected to the electromagnetic field of
the beam, in the vertical direction, with a hypothetical chamber wall at a radius of 6 mm. Different colors correspond todifferent molecules as indicated.

In order to obtain trapping at this temperature, we
can launch molecules closer to the beam. Figures 11 and12 illustrate the case when molecules are considered in a
15 K beam pipe and their initial distance from the (thinwire, delta-function-like) beam is only 1 nm. In this case,the molecules again get trapped in the electromagnetic
field of the beam because here the beam field is strong
enough.
Another possibility for molecules to be trapped is toincrease their mass M , and, thereby, reduce their initial
thermal velocity, according to Eq. (14). Indeed, it is extremely likely that molecules with a permanent dipole

FIG. 11. Motion of neutral molecules initially at 1 nm fromthe thin-wire beam with a thermal velocity corresponding toa temperature of 15 K, and subjected to the electromagnetic
field of the beam, in the horizontal direction. Different colors
correspond to different molecules as indicated.

moment form flakes or clusters, consisting of 10,000 or
many more oxygen or water molecules. We should thenexplore the motion of such flakes. Again we may assume
that the magnetic or electric dipole moment of a flakeof molecules is completely aligned with the direction of
the beam’s electric or magnetic field. Our simple simulation shows that, in equilibrium at 15 K, a flake of
104 molecules will be captured by the beam field, evenwhen its starting position is at a radial distance of 1 mm.
Figures 13 and 14 display the time evolution of x and ycentroid coordinates and the corresponding phase space
for a flake of molecules.

7
2

2

1

1

Magnetic Field❖2

Electric Field
Y (mm)Y (nm)

Electric Field
0

Magnetic Field

0

❖2

❍2 O

❍2 O

CO
-1

CO
-1

-2

-2
-2

-1

0

1

2

-2

-1

0

X (nm)

FIG. 12. The motion of neutral oxygen, water and carbon
monoxide molecules initially at 1 nm from the beam witha thermal velocity corresponding to a temperature of 15 K,
and subjected to the electromagnetic field of a beam, in thetransverse plane. The initial thermal velocities were vth =62 m/s for O2 , vth = 82 m/s for water, and vth = 67 m/s for
CO.

1

2

X (mm)

FIG. 14. The motion of neutral oxygen, water and carbon
monoxide molecule flakes initially at rest, initially 1 mm fromthe thin-wire beam, and subjected to the electromagnetic
beam field, in the transverse plane, for a pipe temperatureof about 15 K.

V.

TRAPPING BY BEAM POTENTIAL

We observe that the dynamics described by the equations of motion (12) admit a constant of motion:
C=

1 2
p̃I
1
(v + vy2 ) −.
2 x
2π0 M c r(15)

If C is positive the motion is unstable and the trajectoryof the molecule diverges. If instead C < 0 the motion isstable and a molecule oscillates between a maximum anda minimum radius. Therefore, for a molecule initiallylocated at ~r0 and having a temperature T0 defined by its
22
initial velocities as T0 ≡ M (vx,0+ vy,0)/kb , the invariant
C assumes the valueC=

1 kb T
p̃I
1
−
.
2 M
2π0 M c r0

In order for a molecule to exhibit stable motion (trapping), its invariant C must be negative, and hence thetemperature T0 (in the following just written as T ) mustbe smaller than a limiting temperature Tlim (r0 ), according to
T < Tlim (r0 ) =FIG. 13. Motion of neutral flakes of molecules initially at
1 mm from the thin-wire beam with a thermal velocity corresponding to a temperature of about 15 K, and subjectedto the electromagnetic field of the beam, in horizontal (top)and vertical direction (bottom). Different colors correspondto different types of constituent molecules as indicated.

p̃I 1.
π0 kb c r0

(16)

Above the limiting temperature the molecule motion becomes unstable. Evidently, this condition depends on theinitial radial position r0 . This formula also suggests that
if r0 → 0 any temperature is admitted, but this is an
artifact of the thin-wire model for the beam, which we
employ here. In fact, at r → 0 the beam field diverges,
which is non-physical and originates from our adopted
model. With this caveat, we can still roughly estimate
a limiting temperature below which a molecule would be8trapped. Such a limiting temperature can be obtainedby considering an initial position at the beam pipe. Denoting by Rp the radius of the beam pipe, we have
Tlim =p̃I1.
π0 kb c Rp

(17)

However, even some molecules with T < Tlim may still
be lost, and hit the beam pipe, even if they could not
escape to infinity. By requiring that the maximum radius reached by a molecule be less than the beam pipe
Rp , we can further constrain the set of initial positionsof molecules with temperature T , in order for them toremain trapped by the beam field. With a little bit ofalgebra we find that a molecule with initial position r0
and temperature T is trapped inside the beam pipe, if
r0
1
≤
=f .
π
kb c T Rp
0
Rp
1+
p̃

(18)

I

For an initially uniform distribution of molecules,
the fraction of molecules trapped with respect to allmolecules present in the beam pipe is f 2 . Therefore,
if ñ signifies the density of the molecules at temperature
T , the density of molecules trapped isthe hotter molecules have to be closer to the beam center,
where the fields are stronger, in order to be trapped.
Conversely, molecules at a distance r0 = Rb = 3σ would
need to be extremely cold to remain inside the beam.
This, however, is an artifact of the adopted thin-wiremodel for the beam field. A more realistic model will be
developed in the next section.

VI.

Next we consider an axi-symmetric beam with chargeline distribution
 2Ir
ρ(r) =,(19)n
πσ 2 cσ2
defined for an ultra-relativistic beam, with n(u) a function of u satisfying the normalizationZ ∞n (u) du = 1 .
(20)
0

For a Gaussian distribution, this function assumes theform

ntrapped = ñf 2 πRp2n(u) =

Note that π0 kb c = 1.1 × 10−25 C/(K m s).
For example, considering H2 O molecules, with p̃ '6.2 × 10−30 Cm, we findf=1
1 + 18.5 × 103

T Rp
I

ARBITRARY AXI-SYMMETRIC BEAM.

Further, with I = 1 A, and Rp = 0.1 m the expression
for f becomes1f=,
1 + 1.85 × 103 T [K]which implies that only extremely cold molecules canbe trapped by the beam. At T = 54 mK the normalized trapping radius would be f = 0.5, and, consequently, 25% of the molecules of this temperature present
in the pipe would be trapped. If instead we require thata molecule trapped remains inside the beam of radius
Rb = 3σ (with σ the true rms beam size), then we needto demand
r0
1
≤
.
π
0
Rb
1 + p̃kb c T IRb
For σ = 0.333 mm, we find Rb = 1 mm and

1 −u
e 2 .
2

(21)

~ r) =
By applying Gauss’ law, we find the electric field E(~
E(r)~r/r, with
 2
1
1
r
E(r) =I N,(22)
2π0 c r
σ2where for convenience we define the function
Z u
N(u) =n(u0 )du0 .

(23)

0

The magnetic field is computed by applying Ampere’slaw, yielding~ r) = B(r) ~r × ẑ ,
B(~
r

(24)

with
B(r) =

µ0 1
I N
2π r

 2
r
.
σ2

(25)

Next we apply Eq. (6) to the magnetic field case for analigned MDM to obtainX  ~r~r~ =µ(~µ · ∇)B× ẑ ∂i B(r) × ẑ .
(26)
r
r
i
i

r0
1
≤
3σ
1 + 18.5 T [K]With some straightforward algebra we obtain

Therefore, if T = 0.108 K, a molecule initially at r0 /σ ≤1 will be trapped within the radius Rb = 3σ. Basically,~ = −µ(~µ · ∇)B

B(r) ~r
.
r r

(27)

9

which, again with straightforward algebra, becomes~ = pE 0 (r) ~r .
(~p · ∇)E(29)
r
Therefore, the general force acting on a molecule
exposed to the electric and magnetic field of an axisymmetric beam is~ + (~~ = pE 0 (r) − µ B(r) ~r . (30)F~t = (~
p · ∇)Eµ · ∇)Brr
Now we substitute the expressions for E(r) and B(r) tofind"  2 0 2 #
1
r
µ
1
r
~r
1
I p N
−
N
,
F~t =
2π0 c
r
σ2
c r2
σ2
r
that isF~t = 2 

 2
1 I
r
r
σ2
~r
N
,
2pn
−
p̃
2π0 c σ 2
σ2
r2
σ2
r

and

N(u) ≈ n0 u ,1 I~rn0 [2p − p̃]
2π0 c σ 2rand the force for the EDM is of opposite sign to the one
for the MDM.
For the EDM, we obtain
F~t ≈

1 I
~r
n0 p ,
2π0 c σ 2rwhile, for the MDM, we haveF~t ≈ −

0.2
0.0

re

re

0.2
0.4
0.6 10

5

0

x/

5

10

FIG. 15. The horizontal normalized force acting on an EDM(black) or MDM molecule (red), respectively, as a function of
the transverse (x, 0) coordinate. The equilibrium position for
the EDM case is indicated by the label re . Note that for bothtypes of dipole moment the force is discontinuous at x = 0.

molecules leave the beam core, they experience a stabilizing force, corresponding to the force of the thin-wiremodel.
Figure 15 illustrates the horizontal forces experienced
by molecules with either an EDM or a MDM, as a function of the horizontal position x along an axis passing through the center of the Gaussian beam, locatedat x = 0. For convenience, the forces (compare, e.g.,
Eq. (57) below) are normalized by multiplying with 2/T
and with the reference temperature(32)

with n0 = n(0), so that Eq. (31) reads
F~t ≈

0.4

(31)

with the previous definition of the effective dipole moment, p̃ = (p + µ/c) (see Eq. (13)).
We note that for particles outside a beam of rms size σ,
namely at r  σ, we have n(r2 /σ 2 ) ≈ 0 and N(r2 /σ 2 ) ≈1, so that outside the beam the formula for the force
equals the one of the thin-wire model.
In Eq. (31), the term in square brackets is a geometricterm, specific to the type of beam distribution. If in thecore of the beam the radial charge density is uniform, we
expectn(u) ≈ n0

EDM: Fx × 2 × Tp* /T
MDM: Fx × 2 × T * /T

0.6
normalized force

Instead, for the electric field, and, as before, considering the EDM to be aligned with the field, the force on
the molecule becomesX  ~r ~r~∂i E(r),(28)(~
p · ∇)E = p
r i
r
i

1 I
µ ~r
n0
.
2
2π0 c σ
crSo, it seems that molecules with EDM cannot be trappedin the core of the beam where the beam density is uniform. This situation is unusual, because, once EDM

Tp∗ =

1 I
p,
π0 kb c σ

(33)

Tµ∗ =

1 µI
.
π0 kb c c σ(34)

or, respectively,

We will refer to Tp∗ , Eq. (33), and Tµ∗ , Eq. (34), as
the “trapping temperature”. The shape of the force in
Fig. 15 suggests that the EDM molecules may oscillate
not around the beam center, but rather around an equilibrium radius re , which is of order of the beam size.
Incidentally, the trapping temperatures Tp∗ and Tµ∗
resemble the limiting temperature Tlim (r0 ) defined in
Eq. (16) or Tlim of Eq. (17). The various characteristic temperatures simply correspond to different areas oftrapping. While Tp∗ and Tµ∗ refer to trapping over a radial
length scale equal to the rms beam size, the temperature
Tlim indicates trapping within the beam pipe Rp , for thethin-wire model. A universal trapping temperature could

10
be introduced as T ∗ = p̃I/(π0 kb cS), with S denoting thecharacteristic length scale of trapping.
In general, from (31), the equilibrium radius re for the
EDM is the solution of the equation
 2
 2
σ2
r
r
=
N
.
(35)
2n
σ2
r2
σ2
For a Gaussian beamn(u) =

1 −u
e 2,
2

Z u/2
N(u) =d~v
1
I µ ~r=−n0 ,dt
2π0 M c σ 2 c r

2w2 e−w /2 = 1 − e−w /2
with w2 = re2 /σ 2 . Its solution is re /σ = 1.585201 ' π/2.
For a beam with a Kapchinskij-Vladimirskij (K-V) distribution [12, 13], instead, such equilibrium solution does
not exist, and the force acting on an EDM molecule will
be subject to a discontinuity at the beam edge. Namely,
for a K-V beam, at r < Rb the force is constant and defocusing, whereas at r > Rb it is attractive and non-linearlydecreasing with amplitude.
OSCILLATION FREQUENCY

According to (31), and introducing the radial velocity~v ≡ d~r/dt, for EDM molecules the equation of motionreads  2
 2 
d~v
1
I
r
r
σ2
~r
=
p
2n
N
. (36)−dt
2π0 M c σ 2
σ2
r2
σ2
r
Expanding around the equilibrium radius re we find 
 2 
d~v
1
I
re 0 re2σ
rer − re ~r
=
p 4 n
+2 n
dt
2π0 M c σ 2
σ
σ2
re
σ2
σ r
(37)
Using the equilibrium radius re and n(u) for a Gaussian distribution, to good approximation, the term inthe square bracket is equal to −e/10 (the exact number
is −0.2716776662 . . .). Then Eq. (37) becomesd~vep
I (r − re ) ~r≈−.
dt
10 2π0 M c σ 2σr
Therefore, the angular oscillation frequency is
r
e
p
ω≈
I.
10 2π0 M cσ 3

(41)

and so the beam seems to create a constant central
force pointing towards the beam center. Here, an MDM
molecule oscillates at a frequency which depends on thesquare root of the maximum oscillation amplitude.

so that Eq. (35) becomesVII.

Now the equilibrium radius is the beam center r = 0.
Close to this point, with a central beam density n0 , theoscillatory motion follows the equatione−w dw = 1 − e−u/2 ,02

For a molecule featuring an MDM, instead of Eq. (38)
we have
 2
r
~r
d~v
1
I µ σ2
N
=−
.
(40)
dt
2π0 M c σ 2 c r2
σ2 r

(38)

(39)

For a water molecule, H2 O, characterized by M = 3 ×
10−26 kg and p = 6.2 × 1030 Cm, and a beam withσ = 3 × 10−4 m and I = 1 A, we find ω = 11189
rad/s. Hence, in this case, the frequency of oscillation
around the equilibrium radius re is f = ω/(2π) = 1780
Hz, which, for the LHC, is of the order of the fractional
betatron frequency.

VIII.

INVARIANTS AND TRAPPING

Starting again from (31) and multiplying with the radial velocity ~v ≡ v~r/r, we can rewrite the equation of
motion as
d 1 2
v =
dt 2

 2
 2 
1
I
σ d
r
r
σ2
d r
p
N
N
( ).
−
p̃
2
2
2
2π0 M c σr d(r/σ)σr
σdt σ
For EDM molecules we have
 2 

1
I d σ
r
d 1 2
v =
p
N
,
dt 2
2π0 M c σ dt r
σ2
and the integral of motion becomes
 2
r
1 2 kb Tp∗ σ
v −
N
=D,
2
2M r
σ2

(42)

(43)

where we have used the reference temperature Tp∗ , defined in Eq. (33). The constant D is determined by the
initial conditions, namely
 2
r0
1 2 kb Tp∗ σ
v −
N
=D.
(44)
2 0
2M r0
σ2
Now, the condition for a particle with initial conditions(r0 , v0 ) to be trapped within the radius Rb is 2 2 
kb Tp∗ σ
r0
σ
Rb
N
−
N
.
(45)
v02 <
M
r0
σ2
Rb
σ2
Note that not all values of r0 are admitted. In fact, forr0 → 0 we find
 2
σ
r0
→0.
N
r0
σ2
Therefore, as could be guessed by looking at Fig. 15,
there is a minimum r0,min associated with Rb so that!
 22
r0,min
σ
σ
Rb
N
=
N
.
(46)
r0,min
σ2
Rb
σ2

11
Molecules with a smaller initial value of r0 cannot be
Gaussian beam distribution. Indeed, the right picture oftrapped.
Fig. 16 suggests that for a gas of H2 O molecules withtemperature T = 2 K, corresponding to T /Tp∗ = 11.1,
Equation (45) sets a constraint on the trapping for a
the fractional trapping is ' 0.037, or only about 4%. If
gas of molecules of temperature T . At a given value of
instead T = Tp∗ , Fig. 16 (right) shows that the fraction
r0 , the fraction f of the particles that can be trapped isof trapped particles would be ' 0.067 or close to 7%.
obtained by integrating the Maxwell-Boltzmann distribution
For MDM molecules, the equation of motion becomes


Z
Z
M v2
1 M v02
1 M − 2k

1
I µ d r/σ 1
d 1 2
2
e b T dvx dvy = 1−exp −
.
f=
N
u
v
=
−
du ,
2
2
2π
k
T
2
k
T
2 +v 2 <v
b
b
vx
dt 2
2π0 M c σ c dt
u
y
0
0

as

yielding the constant of motion E,
1 Tp∗ σ 2 2 
σr0
Rb−.
f = 1 − exp −NN
2 T r0
σ2
Rb
σ2
(47)
The total amount of gas trapped in the beam at Rb = 3σis determined by integrating over the initial radii: 2Z Rb 
1 Tp∗ σr0
Ntrap = ñ1 − exp −N
2 T r0σ2
r0,min 2 
σRb− N
2πr0 dr0
(48)
Rb
σ2


where ñ denotes the gas density (molecules per volume).
This can be further transformed to 2 
Ntrap
1 Tp∗ σ
Rb
=
1
−
exp
×
N
ñπRb2
2 T Rbσ2!
Z
σ 2 Rb /σ
1 Tp∗ N u2
u du . (49)
× 2
exp −
Rb r0,min /σ
2 TuThis formula can be integrated numerically with results
shown in Fig. 16, for the case of a Gaussian distribution.

0.75

Ntrap /(n 9 2 )Ntrap /(n 9 2 )

1.00
0.50
0.25
0

1

T / Tp*

2

0.06
0.05
0.04
10

T / Tp*20

FIG. 16. Fraction of EDM molecules trapped in the beam asfunction of T /Tp∗ , for T from 0 to 2Tp∗ (left) and through 20Tp∗(right). For this calculation we considered trapping inside a
radius of Rb = 3σ, with σ the rms beam size.

For H2 O with a beam of 1 A beam current and σ =333 µm rms beam size we find Tp∗ = 0.18 K. This temperature is of similar magnitude as the trapping temperature predicted earlier by the thin-wire model, but thetrapping efficiency is much reduced for this more realistic

1 2 kb Tµ∗  r 
=E,
v +
N2
2
2M
σ
where we have used the trapping temperature Tµ∗ , definedin Eq. (34), and the new function
Z u
N2 (u) =0


1
N v 2 dv .
2
v

(50)

In this case, the motion may be confined for E > 0. In
particular, for a molecule to be trapped inside the beam
radius Rb , its phase-space coordinates r, v should satisfy 
kb Tµ∗  r  kb Tµ∗
Rb
v2 +
N2
N2
,
≤
M
σ
M
σ
or
  
 r 
kb Tµ∗
Rb
v ≤
.
N2
− N2
M
σ
σ
2

In this case the fraction of trapped particles in a
Maxwellian gas is   r 
1 Tµ∗Rb
f = 1 − exp −
N2
− N2
(51)
2 T
σ
σ
As before, this result applies to particles located at r/σ.
Again, the total fraction of trapped particles is obtainedby integrating over all locations in the beam pipe, that
is

 
Ntrap
σ2
1 Tµ∗
Rb
=
1
−
2
exp
−
N
×
2
ñπRb2
Rb2
2 T
σ


Z Rb /σ
1 Tµ∗×exp
N2 (u) u du .
2 T
0Figure 17 displays the total fraction of trapped molecules
as a function of T /Tµ∗ for the case of a beam with an axisymmetric Gaussian distribution.
As an example, considering oxygen molecules, O2 , anrms beam size σ = 3 × 10−4 m and an average beam
current I = 1 A, we find Tµ∗ = 2.5 mK. So, for molecules
at a temperature of T = 2 K, we have T /Tµ∗ = 800 andthe fraction of trapped molecules is only 1.25 ' 10−4 or
0.0125%, while for T /Tµ∗ = 1 the fractional trapping is' 0.1 or about 10%.

1.00

0.00100.75

0.0008Ntrap /(n 9 2 )Ntrap /(n 9 2 )

12

0.50
0.25
0

1

T/T *

0.00060.00040.0002

2

500
T/T *

1000

FIG. 17. Fraction of MDM molecules trapped in the beam
as function of T /Tµ∗ , for T from 0 to 2Tµ∗ (left) and through1000Tµ∗ (right). For this calculation we assumed Rb = 3σ.

IX.

PINCH OF NEUTRAL MOLECULESThe fraction of trapped molecules, extracted from theconstant of motion and the temperature, is an important indicator, but it does not provide any prediction for
the average particle density as a function of radial position. To obtain such a prediction, we need to integrate
the equation of motion. For molecules equipped with an
EDM we start from Eq. (36). Using the definition of Tp∗(33), this becomes 2 
  2
d~v /σ1 kb ∗
r
r
σ2
~r/σ
=
T
N
.
2n
−
dt
2 M σ2 p
σ2
r2
σ2
r/σ
(52)
Defining r̂ ≡ ~r/σ, this transforms to


 r̂
 σ2
d2 r̂
1 kb Tp∗
2
2
=
.
(53)
2n r̂ − 2 N r̂dt2
2 M σ2r|r̂|which means that the initial distribution of the normalized velocities dr̂/dt̂ has a standard deviation 1.
Recalling (39), the velocity vth and temperature are
related via v 2
kb Tp∗ T
kb T
th
=
=
=
σ
M σ2
M σ 2 Tp∗
20 2 T
p
I T
=
=
ω ∗ .
π0 M c σ 3 Tp∗
e
Tp
Combining this with the definition of vth , Eq. (55), andintegrating the dynamics up to a certain time tmax , thecorresponding t̂max becomesr s
20 Tωtmax .
(59)t̂max =eTp∗
Note that the product ωtmax /(2π) is the number of oscillations ne performed by molecules near the equilibrium
radius re during the time interval tmax . We now consider time intervals for which the number of oscillations
around re is constant, equal to ne , so that ωtmax = 2πne .
The corresponding t̂max iss
20 Tt̂max =
2πne .
(60)
e Tp∗
This formula allows us to set t̂max as a function of T /Tp∗ ,so as to keep the same number of oscillations around theequilibrium radius re . In this way we can compare themolecular gas response to the beam for different temperatures T .

Employing the rms transverse thermal velocity of the gas
molecules,
r
kb T
vth =
,
(54)
M
we normalize the time coordinate ast̂ = tvth /σ ,(55)

and find


2
 σ2
 r̂
d2 r̂ vth
1 kb Tp∗22=
2n r̂ − 2 N r̂.
2 M σ2r|r̂|dt̂2 σ 2(56)

Note that vth /σ is the rms inverse time a molecule would
need to travel across the beam size σ.
Using the definition of the gas temperature, we obtain


 σ2
 r̂
d2 r̂
1 Tp∗
2
2
=
2n r̂ − 2 N r̂
.
(57)
2 T
r
|r̂|
dt̂2
We observe that, in these dimensionless scaled coordinates, the velocity of the molecule is
dr̂
d~r 1
dt 1
~v
=
= ~v
=
,
vth
dt̂
dt̂ σ
dt̂ σ

(58)

FIG. 18. Simulated pinch of the neutral EDM molecules in acold gas, i.e. T /Tp∗ = 0. The particles are distributed either
uniformly only in the x-plane (left) or randomly distributed(right) in a circular beam pipe of radius Rp = 10σ.

Figure 18 shows the simulated dynamics of few particles at zero temperature, T = 0. In the left picture,
the particles are launched only in the x−plane. The dynamics of the molecule pinch resembles the one of theelectron-cloud pinch, e.g. the density evolution of cloud
electrons during a bunch passage [14–16]. However, note
that, in this case for the EDM, the particles oscillate linearly around the equilibrium radius re = (π/2)σ, and notaround 0.

13

15
10

/0

The right picture of Fig. 18 shows the same case of acold molecule gas, but now distributed in the full circularbeam pipe. The picture reveals a complex pattern, noteasily interpreted.
A better way to visualize the process is to carry out
simulations with many more particles and to compute themolecule density as a function of the radius. The resultof such a simulation is shown in Fig. 19.
This simulation, along with all those that follow, considers an ensemble of Nmac = 4 × 105 macro-particles.
The dynamics is computed with a leapfrog scheme applied over 201 steps per oscillation length. At a given
integration step the local density ρ(r) is obtained as
ρ(r) = ∆Nmac /(2r∆r), with ∆Nmac denoting the number of macro-particles found in the radial shell [r −∆r/2, r + ∆r/2]. The number of such shells in the interval [0, rmax ] is 400. Initially the molecules are distributed randomly and uniformly in the (x/σ, y/σ) plane,
with a random initial normalized velocity corresponding to a Gaussian distribution with standard deviation
1. For convenience, we normalize the particle density bydivision with the initial particle density ρ0 , computed as2).
ρ0 = Nmac /(πrmax
In the top picture of Fig. 19, we see the evolution of thelocal density normalized to the initial one. We observe avery pronounced pinch. The bottom picture instead displays the time-average density, normalized to the initial
density ρ0 (which always is equal to the average density), as a function of radial position r, revealing a peakaverage density close to the beam almost 20 times higher
than the initial one. This simulation was performed at atemperature such that T /Tp∗ = 10−5 .
The density evolution strongly depends on the temperature, T /Tp∗ . To illustrate this point, Fig. 20 shows thesame set of pictures for a gas temperature T /Tp∗ = 0.1.
We see that the nonzero temperature limits the effect ofthe pinch, and, in this case, we find a peak of only ' 3ρ0 ,i.e. three times the initial density.
In view of the strong dependence of pinch density enhancement on T /Tp∗ , we summarize the situation in a
global picture presented in Fig. 21, where we have averaged over a time interval corresponding to 10 oscillations near position re . The black markers indicate
the normalized peak molecule density, with black values on the left vertical axis. For example, at T /Tp∗ = 1,
we have ρmax /ρ0 ' 1.4, while for T /Tp∗ > 1 this ratio approaches ρmax /ρ0 ' 1: To good approximation,
for T /Tp∗ > 5 we find ρmax /ρ0 ' 1. The blue markers,
with values on the right vertical axis, indicate the radial
position at which the molecule density assumes its maximum value. We see that for T /Tp∗ > 5 the maximum
molecule density is located around re = (π/2)σ. The
dashed-dot blueR curve is the
R “average radius” r, computed as hri = ρ(r)rdr/( ρ(r)dr), i.e. without weighing the integrand with the radius r (or area of the corresponding circular stripe), over the radial range from0 to 10σ, For T /Tp∗ > 5 this curve is locked at a valueof 5, reflecting the fact that the distribution ρ(r) is uni-

5
0

0.0

2.5

5.0
r/

7.5

10.0

FIG. 19. EDM molecule density as a function of the radiusand “time” (top) and the end time-averaged local molecule
density versus radius (bottom). The molecule density isnormalized with respect to the initial (or space-averaged)
molecule density. The molecules are cold T /Tp∗ = 10−5 . In
this simulation, the particles were distributed throughout thetransverse x-y space. At the bottom, the large peak of density is ∼ 20 times the initial uniform distribution. Note thesmaller peak of density, which reflects the local oscillationsof molecules around the equilibrium radius r/σ ' π/2; thisphenomenon is clearly visible in the top picture as well.

form in r. For T /Tp∗ < 3, this curve tends towards thevalue 2.65, reflecting the presence of a peak in the density. The dashed blue curve shows the standard deviationof r; hence it indicates the width of the local √peak. For a
uniform distribution this value would be 10/ 12 = 2.88,
which indeed is reached at high temperatures in Fig. 21.
For T /Tp∗ < 3 the standard deviation becomes smaller,
assuming a value of ∼ 2.3 for T /Tp∗ = 0, again signaling a
non-uniformity in the distribution, namely the presenceof a density peak,as is seen in Fig. 19.

14
20.0

cold gas limit

17.5

5

r/

15.0

4

12.5
max / 0

r2

r 2/

3
r/

10.0
7.5

2

5.0

1

location of max

2.5
0.0

10 5

hot gas limit
10 3
10 1
Tgas / Tp*

101

0

FIG. 21. Variation of the radial density distribution of EDM
molecules with temperature T /Tp∗ . Shown are the density enhancement due to the pinch in the beam field (black markers,
left axis), the radial location of the maximum (blue markers,
right axis), the average radial position of molecules (dashdotted blue line), and the rms value of the radial position(dashed blue line).

Similar analyses can be carried out for molecules possessing a MDM and results analogous to those for the
EDM are shown in Figs. 22, 23, 24 and 25. In Fig. 25 weobserve that the cold gas limit for the MDM is an orderof magnitude higher than for the EDM.

2.00
1.75
1.50
/0

1.25
1.00
0.75
0.50

0.0

2.5

5.0
r/

7.5

10.0

FIG. 20. Molecule density as a function of the radius and“time” (top), and the end time-averaged local molecule density versus radius (bottom). The center picture shows the x-y
structure of the molecule density at the point of maximum
density. The molecule density is normalized with respect tothe initial (or space-averaged) molecule density. In this simulation, the molecules are at a temperature T /Tp∗ = 0.1, andthe particles initially distributed throughout the transverse xy space. According to the top picture (color scale), the peakdensity is ∼2.25 times the initial uniform density.

FIG. 22. Simulated pinch of the neutral MDM molecules in acold gas, i.e. T /Tµ∗ = 0. The particles are distributed either
uniformly only in the x-plane (left) or randomly distributed(right) in a circular beam pipe of radius Rp = 10σ.

X.

DYNAMICS OF FLAKES

Flakes of molecules are agglomerates composed of alarge number N of single molecules each having a mass
m and EDM p or EDM µ. A flake may be held together
by the forces between the (aligned) molecular dipole moments. A general flake thus has a mass Mf = N M ,and an electric dipole moment pf = N p or a magnetic
dipole moment µf = N µ. In both cases the initial con-

15
cold gas limit

250

4.5
4.0
3.5

5

r/

4

200

/0

3.0
2.5

r2

150

1.5
1.0
0.0

2.5

5.0
r/

7.5

10.0

FIG. 23. MDM molecule density as a function of the radiusand “time” (left) and the end time-averaged local molecule
density versus radius (right). The molecule density is normalized with respect to the initial (or space-averaged) molecule
density. In this simulation, the molecules are at a temperature
T /Tµ∗ = 0.1, and the particles initially distributed throughout
the transverse x-y space.

200

3
r/

max / 0

2.0

r 2/

100

2

50

1

0

location of max
10 5

hot gas limit
10 3
10 1
T/T *

101

0

FIG. 25. Variation of the radial density distribution of MDM
molecules with temperature T /Tµ∗ . Shown are the density enhancement due to the pinch in the beam field (black markers,
left axis), the radial location of the maximum (blue markers,
right axis), the average radial position of molecules (dashdotted blue line), and the rms value of the radial position(dashed blue line).

150
/0

100
50
0

0.0

2.5

5.0
r/

7.5

10.0

FIG. 24. MDM molecule density as a function of the radiusand “time” (left) and the end time-averaged local molecule
density versus radius (right). The molecule density is normalized with respect to the initial (or space-averaged) molecule
density. In this simulation, the molecules are at a low temperature of T /Tµ∗ = 10−5 , and the particles initially distributed
throughout the transverse x-y space.

dition is again determined by the thermal equilibrium
temperature T , which remains the same as for the single molecules. The characteristic temperatures Tp∗ , Tµ∗
defined in Eqs. (33) and (34), respectively, are related
to the corresponding trapping temperature of the flakes(suffix “f”) via∗
Tp,f= N Tp∗ ,∗Tµ,f= N Tµ∗ .

(61)This reflects the fact that, at the same temperature, theflake has a much lower thermal velocity than a single
molecule, and that it can more easily be trapped in the
beam potential. Namely, if in Fig. 16 and Fig. 17 a single
particle has a certain temperature T /Tp∗ , the corresponding flake would have an N times higher trapping temper∗∗
ature Tp,f
= N Tp∗ , or T /Tp,f= (T /Tp∗ )/N . Hence, thepoint describing the flake lies much closer towards zero,on the left side of the diagrams, and, therefore, a significantly larger fraction of the flakes will be trapped, as

indicated in the aforementioned pictures.
As a concrete example, in the previously discussed casewith molecules of H2 O, we found that Tp∗ = 0.18 K andfor this gas at T = 2 K we had T /Tp∗ = 11.1; in the casethese molecules cluster to form a flake, for instance each∗containing N = 10, 000 molecules, we have Tp,f
= 180 K,∗so that at T = 2 K we find T /Tp,f= 0.0011, and the
majority of the flakes will be trapped.
More generally, in Figs. 21 and 25 it is quite evident∗∗
that for T /Tp,f→ 0 or T /Tµ,f→ 0, respectively, themaximum density of the pinched flakes becomes large,
with potential consequences for the effective vacuum
pressure and interaction with the beam. For the case ofpolar H2 O flakes, the peak density in the pinch reaches
about 20 times the initial value, whereas for paramagnetic O2 flakes the peak density even increases by a factorof 250.

XI.

CONCLUSIONS AND OUTLOOKMany neutral molecules possess a permanent electric or
magnetic dipole moment. Their motion in an accelerator
vacuum system will be perturbed by the electromagnetic
field of the beam, leading to a possible trapping and density enhancement of such particles in the vicinity of thebeam, especially in cold environments.
In this paper, we have analyzed the equations of motion of electrically or magnetically polar molecules, andidentified the respective constants of motion. We derived the fraction of molecules, with either electric or
magnetic dipole moment, trapped by the beam field, asa function of temperature, expressed in terms of a char-

16
acteristic trapping temperature. The resulting local density enhancement was calculated as a function of radial
position and time.
In particular, we have shown that molecules with amagnetic dipole moment oscillate around the transverse
center of the particle beam, whereas molecules with anelectric dipole moment oscillate around an equilibrium
point located at the edge of the beam.
Observations of beam loss and beam instabilities in
the 2017 and 2018 LHC runs cannot be explained by the
motion of single neutral molecules, which, at a temperature of 5 K, would mostly not be trapped by the fieldof the beam. However, the formation and trapping oflarger neutral flakes, or agglomerates of a large numberof polar water or paramagnetic oxygen molecules is indeed possible, and this could well have contributed to themagnitude of the observed phenomena.
Throughout this article, we have analyzed the motionof uncharged objects carrying a dipole moment. Evidently, once a molecule or a flake comes close to thebeam it may be ionized [17], from which point onward
its dynamics is radically altered. For ionized molecules
or ionized single atoms interacting with a charged particle beam the equations of motion are well known [18].
On the other hand, a larger flake staying near the coreof the beam would heat up, be charged, and then either evaporate or melt and explode [19], leaving behinda localized high-density mixture of ions, electrons, andmolecules or atoms. A software package is under development at CERN, to model the interaction of such acomplex mixture of species with the LHC proton beam
[20].
In general, the trapping and accumulation of individual neutral molecules or flakes of molecules in the vicinityof the beam enhances the effective gas density and canaggravate ion-induced beam instabilities, such as those
discussed in Refs. [21–24]. The effect considered is particularly important in cryogenic vacuum systems, for highbeam currents or for small beam sizes. Consequently,
it will become more important for future generations ofaccelerators.

ACKNOWLEDGMENTS

This work was supported, in part, by the European Commission under the HORIZON2020 Integrating Activity project ARIES, grant agreement 730871.
M. A. Rehman’s research stay at CERN was supportedby the Graduate University for Advanced Studies (SOKENDAI), Japan, by K. Furukawa of KEK, and by G. Arduini, M. Giovannozzi, and M. Vretenar of CERN.

Appendix A: On the Alignment Condition

In this appendix, we discuss the simplifying ansatz
made throughout the main text regarding the alignment~ or of the MDM µ~of the EDM p~ with E,~ with B.
We consider a simple model of a molecule composed of
two atoms: atom 1 with charge +Q located at ~r1 , andatom 2 with charge −Q located at ~r2 , each of the same
mass m, and kept apart at an equilibrium distance of
L meters, by a linear force. The distance L is typicallya small number. The center of mass of the molecule is~rcm = (~r1 +~r2 )/2, and the location of atom 1 with respect
the center of mass is ~l1 = ~r1 − ~rcm , while for the atom 2
the equivalent distance vector from the center is ~l2 = −~l1 .
Each atom is subject to a binding force that disappears
when the atoms are separated by the equilibrium distance
L. For the atom 1 the binding force is1~ 1 = −k 1 −N(~r1 − ~r2 ) ,(A1)
L |~r1 − ~r2 |~ 2 = −N
~ 1 . The constant k
and for the atom 2 it is N
characterizes the strength of the binding force keeping
the two atoms together.
In addition, for each atom we also apply the external
force F~1 = F~2 = 1/2F~cm , with F~cm the force acting on
the center of mass.
Finally, forces related to a torque ~τ are assigned to thetwo atoms as
1
1~τ × ~l1 = 2 ~τ × ~l ,2(l1 )2l

F~1,τ =

(A2)

and F~2,τ = −F~1,τ , with ~l = (~r1 − ~r2 ) = 2~l1 and l ≡ |~l|.
The EDM is defined as p~ = Q~l. The full set of equations necessary to characterize the motion of the moleculeis d2 ~rcm
1 ~ 2dt2 = 2m Fcm
2
 d p~
Q
1 Q
1= 2m
τ × p~ − m
2k QL− |~p1| p~dt2
p2 ~ F~cm = (~~
p · ∇)E~~τ= p~ × E
Next, we define |p| = QL, p~n = p~/|p|, pn = |~pn |, andconsider the electric field generated by a round beam as  r  ~r
 r  ~r/σ~
~ n ~rE(r)= E0 A= E0 A= E0 Eσ r
σ r/σσwith E0 the maximum electric field. For convenience we~ n = E/E
~ 0 , which reads
have defined E~ n (~v ) = A(v) ~v ,Ev
with, for a Gaussian beam,A(v) =1 2
1
1(1 − e− 2 v ) .
ln(π/2) v

The coefficient ln(π/2) is a good approximation to thereal normalization coefficient (Amax = 0.99927 instead of

17
Amax = 1 and is reached at v = 1.585201.. approximately
at v = π/2). We also define
2
ωE
=

|p|E0
Ii

(A3)

with Ii = mL2 /2 the nominal momentum of inertia ofthe molecule. For a coasting beam the maximum electricfield isπI
E0 = ln,(A4)2 2π0 cσ
with I the beam current and σ the transverse rms beamsize.
By transforming the time t to τ = tωE /(2π) and normalizing the position with the rms beam size σ, i.e. ~rn =~rcm /σ, the equations of motion become d2 ~r2n~ n (~rn )= π2 Lpn · ∇ r n )E2
σ 2 (~
 dτ
2
d p
~n
2
~
rn )] × p̂n −dτ 2 = (2π) [p̂n ×En (~
2 k− (2π) QE0 1 − p1n p~n .
A similar approach can be adopted for the equationsof motion of binary molecules with a MDM. However, in
this case we choose the molecule with ~l orthogonal to ẑ,and also µ~ orthogonal to ẑ. We further define ~ln = ~l/L.
We can decompose µ~ as µ~ = |µ|(µp~l + |µo~l × ẑ) with22
µp + µo = 1.
In analogy to the EDM case, we introduce a normalized
magnetic field~ n = A(v)r̂n × ẑ ,B

(A5)

~ = B0 B~ n , with B0 designating the maximum
so that B
magnetic field generated by the beam. Setting
2
ωB
=

|µ|B0
,
Ii

(A6)

we can write the equations of motion of the molecule as
 2
2
d ~
rn
~

= π2 L2 dτ
σ 2 [µ̂ · ∇n ]Bn2~ d l2n = (2π)2 12 [µ̂ × B~ ] × ~ln −
dτ
ln
 n

1
2 kL
~ln

−
(2π)
1
−

|µ|B0

|~ln |µ̂= µp ˆln + µo ˆln × ẑ ,where now τ = tωB /(2π). We note that τ is a dimensionless variable, which counts the phase advance in units of
2π.
Following the same argumentation, as for the electric field, we estimate the maximum magnetic field fora coasting beam to be π  Iµ0
B0 = ln
.
(A7)
2 2πσ
From the above systems of equations, it is straightforward to show that if p~n is slightly tilted with respect to~ n , it oscillates around E~ n at a frequency ωE . The idenEtical behavior occurs for µ~ , which oscillates at a frequency~ n.
ωB around B
The two sets of equations, (A5) and (A7), also revealthat the acceleration of the center of mass ~rn is proportional to π 2 L2 /σ 2 . This means that the motion of thecenter of mass is slow. During one oscillation period of
the EDM or MDM, the center of mass practically does
not move. We also see that the binding force enters into
the dynamics. However, we assume that this internal
restoring force is much bigger than the additional forces
exerted by the external beam magnetic or electric fields.
This assumption implies that, to good approximation,|~ln | is constant, equal to 1. Expanding the force on thecenter of mass acting on a molecule with MDM we findB(r)~ = [~
F~ = (~µ · ∇)B
µ · r̂]B 0 (r)B̂ − [~µ · B̂]r̂ .
r
If instead the molecule has an EDM we find the analogous
equationE(r)~ = [~
F~ = (~
p · ∇)E
p · r̂]E 0 (r)r̂ + [~p · B̂]B̂
rwhere r̂ = ~r/r, B̂ = ~r/r × ẑ and p~ = px x̂ + py ŷ and~ r) =µ~ = µx x̂+µy ŷ and the electric and magnetic fields E(~~ r) = B(r)B̂. During the time of one oscillation
E(r)r̂, B(~of µ~ or p~ around their equilibrium position, the maximum
displacement of the center of mass is proportional to theratio π 2 L2 /σ 2 which is immensely small, typically L =10−12 m, and σ = 10−3 m, hence π 2 L2 /σ 2 ∼ 10−17 .
Therefore, we can look at the system on a time scale T
during which many oscillations of µ
~ n or p~n around theirequilibrium position occur, while the center of mass stillmoves little. This, in turn, implies that the quantities
B(r), E 0 (r), and r are practically constant over the interval of time T . Hence, defining h·i as an average over
the time interval T , we findhF~ i = h~µ · r̂iB 0 (r)B̂ − h~µ · B̂iB(r)r̂r

for the MDM andhF~ i = h~p · r̂iE 0 (r)r̂ + h~
p · B̂iE(r)B̂rfor the EDM. However, because the number of oscillations of µ
~ , p~ is huge on the time scale T , we can simplify
some terms as h~µ · r̂i = 0, h~µ · B̂i = µeff and h~
p · r̂i = peff ,h~p · B̂i = 0, and we only remain with
hF~ i = −µeff

B(r)
r̂
r

and
hF~ i = peff E 0 (r)r̂ .
In other words, we can replace the force acting on thecenter of mass by an effective force resulting from an18“effective” EDM, and an “effective” MDM aligned withthe respective fields. By making this approximation wegive up any attempt at resolving the tiny fluctuationsof the center of mass deriving from the coupling withthe fast oscillations of the MDM, or EDM,in the planeorthogonal to the field lines.
In addition, we note that the electric or magnetic field
generated by the stored particle beam never appears suddenly, but rather is turned on slowly as the beam particles
are injected or accumulated. This means that the fields~ and B~ appear mostly adiabatically. This additional efEfect further fosters the alignment of the EDM and MDM
to their respective field. We show this by a simulation,
which illustrates that µ~ , and p~ spontaneously align withthe local magnetic or electric field, respectively. Unfortunately, simulations for the real parameters are beyond
computational reach, as they would require resolving thevibrational motion and the center of mass dynamics for
π 2 L2 /σ 2 ∼ 10−17 . Instead, we test the model dynamics for the previously described bi-atomic model with the“relaxed simulation parameters” of L2 /σ 2 = 10−4 andk/(QE0 ) = 103 .
The two top pictures in Fig. 26 compare the angle ofthe electric field at the location of the molecule (red) andthe angle of the electric dipole moment p~ (blue), bothwith respect to the horizontal direction, on two different
time scales. Figure. 26 top-left shows, in blue colour theangle of the electric dipole moment. For this simulation,
the initial angle of p~ was chosen at 180◦ with respect to~ n , and the molecule was launched with zero rotationalEenergy. This is the worst possible scenario. The picture
reveals an oscillation of p~, whose angle more and moreclosely approaches the angle of the electric field. The
picture on the top-right presents the situation at theend of the adiabatic turn-on, clearly demonstrating an~ n . Note that, although the rangealignment of p~ with Eof the time window is the same as on the left, the red
curve (the electrical field at the location of the molecule)
now exhibits a faster change of angle. This feature stems
from the dynamics experienced by the molecule’s centerof mass. At the end of the adiabatic ramping, the gradient of E(r) is much larger than at the start of the fieldramp, leading to a faster motion of the molecule aroundand across the beam. Consequently the red curve appears with a more pronounced slope. Exactly the same
reasoning applies to the example of a molecule equippedwith a magnetic dipole moment and the simulation results for such a case are quire similar, as is shown in thetwo bottom pictures of Fig. 26. Also here a nearly perfectalignment is reached at the end of the adiabatic ramp-up
of the magnetic field.
The simulations of Fig. 26 consider a slow turn-on of
the electric field over a time interval T corresponding
to τmax = 105 , or to the equivalent of 105 oscillation
periods for the maximum field at the end of the adiabatic~ n | = 1. Note that although this number
ramp, where |Eof oscillations seems large, it does correspond to a rather
short actual time interval, typically below a microsecond.

~ (red), with reFIG. 26. Simulated angles of p
~ (blue) and E
spect to the horizontal direction, at the start (top left) andend (to mid-left) of an adiabatic increase of the electric fieldof the beam, over a total time T that would correspond toτ = 105 periods of phase advance ωB T . The bottom-left picture show the particle trajectory, and in blue the equilibrium
circle. The pictures show snapshots, each taken over a timeinterval amounting to τ = 200. The pictures on the rightpresent the simulated angular direction of a magnetic dipole~ (red) durmoment µ~ (blue) and of the local magnetic field B
ing an analogous ramp up of the beam’s magnetic field.

Using Eq. (A4), which yields the maximum field for a
Gaussian beam, and the definition of Tp∗ we find for acoasting beam
2
ωE
' ln

π k
b
T ∗.
2 mL2 p(A8)Then, as an example, considering H2 O and Tp∗ = 0.18 K,
and by choosing the L so as to obtain the correct momentof inertia for the equivalent bi-atomic model, namely L =0.65 A, we find ωE = 1.33 × 1011 rad/s; one oscillation
period is δt = 4.7 × 10−11 s. Therefore, in this example
105 oscillations correspond to ∆t = 4.7 × 10−6 s, and wehave shown this to be sufficiently adiabatic for ensuring~ in the case of the relaxed
the alignment of EDM with E,

19
simulation parameters. Note that even with a ten times
faster ramp as in ∆t = 1.78×10−8 s, the alignment would
still be reached to a good approximation. This time scale
is comparable to the bunch spacing in the LHC.
The analysis for the MDM molecules can follow thesame line of thought. Considering the maximum B fieldgiven by Eq. (A7), and using the definition of Tµ∗ , we find
2
ωB
' ln

π k
b
T∗
2 mL2 µAs an example, for O2 we had Tµ∗ = 2.5 mK; henceωB = 6.34 × 109 rad/sec and one oscillation period at thepeak field is δt = 9.91×10−10 s. The duration of an adiabatic ramp over 105 such oscillation periods corresponds
to ∆t = 9.91×10−5 s. This suggests a stiffer behaviour ofthe MDM, compared with the EDM. In consequence, theEDM motion may not resolve individual bunch passages,
but will align to a mean B field.
We can analyze the process of alignment by using theconcept of adiabatic invariants. In fact, the MDM andEDM oscillate around their respective equilibrium direction with angular frequencies ωE or ωB , Eqs. (A3) and(A6). The corresponding equations of motion are
d2 p⊥
2
+ ωE
p⊥ = 0 ,
dt2

d2 µ⊥
2
+ ωB
µ⊥ = 0 .
dt2(A9)

After transforming (A9) to the scaled time coordinate
τ , and, for convenience using the normalized variables
p~n = p~/|~p|, µ~n = µ~ /|~µ|, the equivalent set of equations
reads
2

d pn,⊥+ Ω2E pn,⊥ = 0 ,dτ 22d µn,⊥+ Ω2B µn,⊥ = 0 , (A10)dτ 2where ΩE and ΩB vary with the normalized time τ andat maximum field (in our examples typically reached atτ = τmax ) assume the value 2π.
Therefore, for small oscillation amplitudes, thedimension-less scaled time τ indicates how many oscillations the EDM, or MDM, would perform around theequilibrium direction if the field were maximum, i.e., for
E = E0 or B = B0 .
For example, taking the example of the electric field
E, the adiabatic condition is
~
|E|
2π

,
˙~
ωE
|E|

(A11)

where Ė is the time derivative. Considering an electric field that linear rises from 0 to a value E0 during
the time interval tmax , the adiabatic condition is fulfilled
1/2
if t3/2 ωE0 /(2π)  tmax . This inequality constrains thetime t = t where the field change becomes adiabatic.
~ n or B~ n slowly changes theAn adiabatically rising E
angular frequencies ωE and ωB until they approach their
final values. In the new coordinate frame, the slowly
varying ΩE or ΩB will grow from an initial value of zeroto adiabatically reach the final value ΩB = 2π or ΩE =2π at τ = τmax .
We have introduced the symbols pn,⊥ and µn,⊥ , which
denote the projections of p~n , µ~ n onto the transverse axis
~ or B,~ respectively. In the orthogperpendicular to E
onal dipole-moment “phase space” (pn,⊥ , dpn,⊥ /dt) or(µn,⊥ , dµn,⊥ /dt), we can recognize the key features ofadiabatic invariance. For the case of an EDM molecule,the top-left picture of Fig. 27 presents the dipole-moment
evolution in the perpendicular “normalized” phase spaceobtained by using the time coordinate τ . The model
parameters for this simulation are the same as thosefor Fig. 26. The picture clearly reveals that the traceof the phase-space coordinate (pn,⊥ , dpn,⊥ /dτ ) converges
towards an ellipse. Based on the same set of simulation
data, the top-right picture of Fig. 27 illustrates the evolution of the normalized dipole-moment total energy
" 
#
2
1 2 2
E⊥
1 1 dpn,⊥+ ΩE pn,⊥ .
=
(A12)
ΩE
ΩE 2
dτ
2
This picture confirms that, to good approximation,
Ainv ≡ E⊥ /ΩE ' constant, once the field change has
become adiabatic.
The actual value of Ainv is determined by the initialconditions of the particle’s dipole moment considered,i.e. by the initial value of ~rn , d~rn /dτ, p~n , d~
pn /dτ , and by
the initial speed of turning on the electric of magnetic
field, proportional to 1/τmax . The field turn-on corresponds to a physical ramp time ofTramp =

2π
τmax ,
ωE(A13)

which unveils the equivalence of the dynamics for corresponding pairs of (Tramp , E0 ). The dependence of Ainvon τmax is examined in Fig. 27 (bottom-left) for a linear
ramp considering a particle with the same initial condition (in the τ frame). The bottom-right picture of Fig. 27
illustrates the dependence of Ainv on τmax for a Gaussian
ramp starting at time of 5σz /c before the arrival of thebunch center, where σz denotes the rms bunch length.
Comparing the evolution for several different τmax , thetop-right picture of Fig. 27 shows that the value of Ainv
decreases as τmax increases. These findings reveal that
τmax directly relates to the degree of alignment with respect to the field. In fact, the maximum excursion of pn,⊥at the end of the ramp, which we designate as pn,⊥,max ,is given byr
Ainv (τmax ).
(A14)pn,⊥,max =π
To explain the dependence of Ainv on τmax we again
consider the case of a linear ramp, where the electric field
increases linearly in time, and the frequency ΩE evolvesasrτ
Ω(τ ) = 2π
.
τmax

20
2

0.6
0.5
0.4
/ E

dp /d

1
0
1
2 1.0

0.3

max = 50000

0.2

max = 100000

0.5

0.0

0.5

p

1.0

0.0 0

50000

1000000.5

τ = f 2/3

/ B

0.4
0.3

d

0

0.2

1
2 1.0

Φ(τ ) = 2πf ,that is, by the time

0.6

1

0.1
0.5

0.0

0.5

1.0

0.0 0

50000

100000

100

MDM
EDM

 2/3
3
1/3
τmax
.
2

At this moment the corresponding ΩE isr 1/3τ
3−1/3ΩE (τ ) = 2πτmax
= 2πf 1/3.
τmax
2This corresponds to the value of Ω which sets the valueof Ainv . In fact, from
#
" 
2
1 2 2
1 1 dpn,⊥+ ΩE p⊥ ,(A17)
Ainv =
ΩE 2
dτ
2

MDM
EDM

/

/

100

(A16)

We denote by f the phase advance in unit of 2π required
to enter into the adiabatic domain, so that the adiabatic
process will start when

0.1

2

/d

of this oscillation is estimated asZ τ
2 τ 3/2ΩE (x)dx = 2π √Φ(τ ) =.
3 τmax0

max = 10000

10 1

and neglecting the kinetic term, which is small for our
test particle (defined below), we find

10 1
103

max

105

107

103

max

105

107

FIG. 27. Electric dipole moment phase space orthogonal to
the local field direction, (p⊥ , dp⊥ /dτ ) (top left), and the adiabatic invariant Ainv as a function of the number of oscillations(top right). The right picture presents data for three different values of τmax . The middle two pictures present analogous
simulation results for the case of the magnetic dipole moment.
At the beginning of each simulation, large oscillations of p⊥or µ⊥ are seen, which later on converge onto an ellipse. The
width of this ellipse in the direction of p⊥ , or µ⊥ , becomes
the smaller the larger τmax is. Recalling Eq. (A13), τmax canbe expressed as τmax = Tramp /(2π)(|p|E0 /Ii )1/2 , and, therefore, the width of the ellipse in the direction of p⊥ becomes
the smaller the more adiabatically the field changes (larger
Tramp ) and the higher the final field is (higher E0 ). Thisis demonstrated in the bottom picture by the dependence of
Ainv on τmax , both for a linear (left) and for a Gaussian ramp
(right).

At the beginning of the ramp, the fields are too weak
to meet the adiabatic condition, (A11), which in thesecoordinates assumes the particularly simple form
1/3
τ  τmax.

(A15)

This condition will start to hold true after a time τ large
enough to allow for a significant phase advance of the oscillation of the perpendicular dipole-moment component
around the electric or magnetic field. The phase advance

Ainv =1ΩE (τ )p2⊥ .
2

(A18)

We note that our test particle is launched at a distancefar from the beam center and with a velocity sufficiently
low such that at the time by which the phase advance
becomes 2πf , i.e. by when the adiabatic condition is met,
the particle still remains far away, with p⊥ unchanged.
Substituting for ΩE the value at which the adiabatic
process starts, we obtain
Ainv = πf 1/3 p2⊥

 1/3
3
−1/3
τmax
.
2

(A19)

Our simulation results are shown by the markers in
Fig. 27 (bottom left). Although the previous model isvalid only for small oscillations around the equilibrium
angle of the dipole moment, we find that choosing thevalues f = 20 and p⊥ = 1, Eq. (A19) fits these simulation results extremely well.
Note that this discussion, including the results shownin Fig. 27, is carried out in the reference system withnormalized time τ . This implies that the initial conditionin physical coordinates, d~rn /dt, is related to the initialcondition in normalized space, d~rn /dτ , according tod~rn
d~rn dτd~rn ωE
=
=
.
dt
dτ dt
dτ 2π

(A20)

The initial rms velocity of a molecule follows from thethermal equilibrium with the environment as vth =21pkB T /M , and ωE is given by Eq. (A8). Therefore,
Eq. (A20) becomes
s


m T
drn
2π
L
.
(A21)
=p
dτ rmsσMTp∗ln(π/2)

As L is of order of magnitude ∼ 10−10 m and σ ∼ 10−3 m,and as m/M ≤ 1 is always fulfilled, even for T /Tp∗ = 106
we would find (drn /dτ )rms ∼ 6 × 10−4 . Hence, we may
consider the initial condition of molecules in the systemof coordinates τ as effectively frozen. The same argument
applies to the MDM case, after replacing Tp∗ with Tµ∗ .
It follows that when the electric field rises adiabatically, the oscillation amplitude of p⊥ slowly decreases, so~ independently ofthat p~ more and more aligns with E,
the motion of the center of mass. The same argument
applies to the MDM case as is illustrated in the center
pictures and by the green markers in the bottom picturesof Fig. 27.
The previous analysis is now discussed for bunchedbeams. Considering the passage of a single round bunch,
the maximum electric field is obtained from Eq. A4 with
I now the maximum current at the bunch center andσ the transverse rms beam size. For a Gaussian bunch
profile, the longitudinal charge distribution is
qNb − 21 σz22z ,ρz (z) = √
e
2πσz

(A22)

with Nb the number of particles in the bunch, q
the charge per beam particle, and σz the rms bunch
length. Hence,for this case the peak current reads√
I = cqNb /( 2πσz ), and, finally, we can express the maximum electric field asE0 =ln(π/2) qNb.
(2π)3/2 0 σσz(A23)

For LHC bunches at top energy with σ = 2 × 10−4 m,
σz = 0.076 m, Nb = 1.2 × 1011 protons, we obtain amaximum electric field of E0 ≈ 4 × 106 V/m.
Following the same argumentation, as for the electricfield, we estimate the maximum magnetic field during abunch passage from Eq. (A7), and using the maximum
beam current, we now findB0 =ln(π/2) µ0 cqNb.
(2π)3/2 σσz(A24)

Inserting the same reference LHC beam parameters as
before yields the estimate B0 = 137 × 10−4 T =137 Gauss.
Considering the LHC bunches, for H2 O molecules
Eq. (A3) yields a maximum angular oscillation frequencyωE ≈ 9.0 × 1011 rad/s, corresponding to Tp∗ ≈ 8.1 K,
and, with a ramp time Tramp = 5σz /c, Eq. (A13) yields
τmax ≈ 181. On the other hand, for an O2 molecule, thepeak angular oscillation frequency is ωB = 4.27 × 1010
rad/s, corresponding to Tµ∗ = 0.11 K and τmax ' 8.6.

Inspecting Fig. 27, for example, at τmax ' 181 we findAinv ' 0.5, and, for the perpendicular dipole moment at
the end of the field ramp, i.e., at the passage of the bunch
center, this implies pn,⊥,max ' 0.4, which corresponds toa residual misalignment of ∆θ ' 23.5 degree.

FIG. 28. Simulated density based on a rigid-molecule model,
for a temperature T /Tp∗ = 0.1, at the normalized time nosc. =1, which is approximately the moment of maximum peak density. The picture presents a snapshot of the molecule distribution in the x-y plane. The molecule density is normalizedwith respect to the initial (or space-averaged) molecule density. The particles are initially distributed uniformly throughout the transverse x-y plane. This simulation tracked 4 × 105
molecules with L2 /σ 2 = 10−5 .

These numbers suggest that for the short bunches of
the LHC the alignment condition is not strictly fulfilled
and that, therefore, a residual oscillation of the dipole
moments may affect the dynamics. However, the simulations presented in this appendix where performed primarily in order to confirm that a process of alignment
takes place at all, and these numerical studies were carried out for unrealistically “relaxed” model parameters,
which allowed a full exploration of the complex dynamics,
but did not correspond to the actual parameter regime.
Against this background, as a next step to further examine our alignment ansatz, we now take into accountthat the force responsible for the molecular vibrations ismuch larger than all the other forces, and, consequently,
the time scale of these vibrations far shorter than allother time scales. Hence, we may take the distance between a molecule’s atoms to be frozen and consider a rigid
model of the molecule. Simulation using a rigid model are
numerically much easier to carry out. For the situationin question, where the electric and magnetic fields are alltransverse, the motion of one molecule is described by
3 dynamical variables, namely the normalized transverse
center-of-mass coordinates xn and yn , and the angle ofthe dipole moment with respect to the relevant field θ,and their conjugated momenta.

22
In simulations using the rigid molecule model, the initial centroid velocity spread of the molecules is chosen inaccordance with the thermal equilibrium temperature,and so is the initial rotational velocity spread of thedipole-moment angle, dθ/dτ , while the starting angle θitself is taken to be initially aligned with θE (for thecase of an electric dipole moment). Simulations result for
the rigid-molecule model with the parameters of Fig. 20(center) are shown in Fig. 28, at a time corresponding
to nosc. = 1. Both figures, obtained with the aligned
dipole-moment model and with the rigid-molecule model,
respectively, exhibit a similar annular structure in phasespace, with a density maximum located at the same radial position. This similarity of results lends further support to our simplifying alignment ansatz.

In this appendix, we have shown that the molecularelectric or magnetic dipole moments p~ or µ
~ tend to alignwith the corresponding external field, as the external field
is increased. In particular, we have demonstrated thevalidity of the alignment assumption in various simulations performed under different assumptions (e.g., withand without internal molecule vibrations) and time scales(mean field approximation or resolving individual bunchpassages).
However, an exact simulation of repeated passages ofshort bunches through a region populated by EDM or
MDM molecules with internal vibration modes requires
the development of additional methods and tools to investigate the complete nonlinear dynamics of this complicated multi-scale system.

[1] J. Wenninger for the Operations group. LHC Report:
Operation with Holes. CERN News.
[2] B. Salvant. 16L2. BE-ABP information meeting 14
September 2017.
[3] Benoit Salvant, Sergey Antipov, Gianluigi Arduini, Nicolo Biancacci, Xavier Buffat, Lee Carver, Paul Collier,Arkadiusz Gorzawski, Wolfgang Hfle, Giovanni Iadarola,
Gerd Kotzian, Anton Lechner, Thomas Levens, Lotta
Mether, Daniele Mirarchi, Elias Mtral, Giovanni Rumolo,and Daniel Valuch. Experimental Characterisation ofa Fast Instability Linked to Losses in the 16L2 Cryogenic Half-Cell in the CERN LHC. (CERN-ACC-2018175):THPAF058. 4 p, 2018.
[4] F. Bordry. Summary of LHC Performance Workshop(Chamonix’18).
[5] José Miguel Jimenez et al. Observations, Analysis and
Mitigation of Recurrent LHC Beam Dumps Caused by
Fast Losses in Arc Half-Cell 16L2. (CERN-ACC-2018164):MOPMF053. 4 p, 2018.
[6] Encyclopædia Britannica. Addison-Wesley, Reading,
Massachusetts, 1993.
[7] David Halliday, Robert Resnick, and Jearl Walker. Fundamentals of Physics; 10th ed. Wiley, New York, NY,
2004.
[8] David J Griffiths. Introduction to electrodynamics; 3rded. Prentice-Hall, Upper Saddle River, NJ, 1999.
[9] Mathias Getzlaff. Fundamentals of Magnetism. SpringerVerlag, 2008.
[10] W.S.C Williams. Nuclear and Particles Physics. Oxford
University Press, 1991.
[11] Wolfram Mathematica,
https://www.wolfram.com/mathematica.
[12] I.M. Kapchinskij and V.V. Vladimirskij. Limitations Of
Proton Beam Current In A Strong Focusing Linear Accelerator Associated With The Beam Space Charge. In
2nd International Conference on High-Energy Accelerators, pages 274–287, 1959.
[13] I Hofmann, L J Laslett, L Smith, and I Haber. Stability of the Kapchinskij-Vladimirskij (K-V) distribution inlong periodic transport systems. Part. Accel., 13(LBL14922):145–178. 61 p, Aug 1982.
[14] E. Benedetto and F. Zimmermann. Dynamics of the electron pinch and incoherent tune shift induced by electron
cloud. In 9th European Particle Accelerator Conference(EPAC 2004), 7 2004.
[15] E. Benedetto, G. Franchetti, and F. Zimmermann. Incoherent effects of electron clouds in proton storage rings.
Phys. Rev. Lett., 97:034801, Jul 2006.
[16] F. Zimmermann and G. Franchetti. Electron-cloud Pinch
Dynamics in Presence of Lattice Magnet Fields. Conf.
Proc. C, 110904:586–588, 2011.
[17] Y. Baconnier, A. Poncet, and P. Tavares. Neutralizationof accelerator beams by ionization of the residual gas. In
CERN Accelerator School: Course on General Accelerator Physics, pages 525–564, 1 1994.
[18] Y. Baconnier and G. Brianti. The Stability of Ions in
Bunched Beam Machines. In 1983 CERN 1st Accelerator
School: Antiprotons for Colliding Beam Facilities, page369, 3 1980.
[19] Frank Zimmermann, J. T. Seeman, M. S. Zolotorev, and
W. Stoeffl. Trapped macroparticles in electron storagerings. (SLAC-95-6788):3 p, Jun 1995.
[20] Lotta Mether, Giovanni Iadarola, Kyle Poland, Giovanni
Rumolo, and Galina Skripka. Multi-species electron-ion
simulations and their application to the LHC. (CERNACC-2019-113):WEPTS050. 4 p, 2019.
[21] D.G. Koshkarev and P.R. Zenkevich. Resonance of coupled transverse oscillations in two circular beams. Part.
Accel., 3:1–9, 1972.
[22] T.O. Raubenheimer and F. Zimmermann. A Fast beam
- ion instability in linear accelerators and storage rings.
Phys. Rev. E, 52:5487–5497, 1995.
[23] G.V. Stupakov, T.O. Raubenheimer, and F. Zimmermann. Fast beam ion instability. 2. Effect of ion decoherence. Phys. Rev. E, 52:5499–5504, 1995.
[24] R.A. Bosch. Spread-frequency model of the fast beamion instability in an electron storage ring. Phys. Rev. ST
Accel. Beams, 3:034402, 2000.