Classical over-the-barrier modelfor ionization of poly-cyclic

aromatic hydrocarbons in keV-collisions with atomic ions

Bachelors thesis

Department of Atomic PhysicsStockholm University

AuthorBjörn Forsberg

AdvisorHenning Zettergren

Contents

1 Abstract 3

2 Introduction 3

3 The classical over-the-barrier model concept 43.1 Electron transfer in ion-atom collisions . . . . . . . . . . . . . . . 53.2 Electron transfer in ion-cluster collisions . . . . . . . . . . . . . . 6

4 A classical over-the-barrier model for infinitely thin conductingdiscs 84.1 Ionization potentials for an infinitely thin, conducting disc . . . . 94.2 Polarization of a circular conducting disc by a point charge . . . 94.3 Potential barrier for the active electron . . . . . . . . . . . . . . . 12

4.3.1 Point charge along Normal Symmetry Axis (NSA) . . . . 124.3.2 Point charge in Disc Tangent Plane (DTP) . . . . . . . . 14

5 Comparisons with earlier over-the-barrier potential barriers andDFT calculations 14

6 Results 166.1 Critical electron transfer distances . . . . . . . . . . . . . . . . . 176.2 Absolute ionization cross-sections . . . . . . . . . . . . . . . . . . 17

7 Outlook 187.1 General angular dependence . . . . . . . . . . . . . . . . . . . . . 187.2 Extension to elliptic discs and spherical caps . . . . . . . . . . . 19

A Infinte plane model 22

B Relevant trigonometric identities 22

C Evaluation of V 0D(R) 23

2

1 Abstract

We are developing a novel classical over-the barrier model for electron transferfrom an infinitely thin conducting disc to a point charge projectile to modelmultiple electron capture in e.g. keV collisions of atomic ions with poly-cyclicaromatic hydrocarbons (PAHs). In its final form, the present model will incor-porate the polarization of the PAH molecules due to the active electron andthe point charge projectile at a general angle of incidence. This will drasticallyimprove the description of the potential barrier in comparisons with simpler ver-sions of the model where the finite size and polarizability of the target moleculeis neglected or treated in an averaged fashion. In this work we arrive at ex-pressions for the electrostatic potential energy barrier experienced by the activeelectron in the two spatial orientations where the point charge projectile is lo-cated along the normal symmetry axis and in the tangent plane of the disc.Applied to coronene (C24H12) such barriers compare better with high level den-sity functional theory (DFT) calculations than with the results from the simplerversions of the classical over-the-barrier models for atomic and spherical clustertargets. These results thus strongly supports the conducting disc approximationof PAHs. Finally we discuss the final steps in the model development and possi-ble extensions of the model to include less symmetric elliptical discs or sphericalcaps.

2 Introduction

Poly-cyclic Aromatic Hydrocarbons (PAHs) are molecules that are built fromtwo or more aromatic rings. The simplest PAH is naphthalene (C10H8), con-sisting of only two such hexagonal rings. Other small PAHs are for instancepyrene (C16H10) and coronene (C24H12), all shown in fig 1.

Figure 1: Molecular structures of some small PAH-molecules

PAHs have attracted a lot of attention in recent years, as they are one of themost widespread types of organic pollutants, produced in the combustion ofalmost any organic material. Many PAHs, for instance naphthalene, has alsobeen identified as carcinogenic [1], PAHs are also of interest in astrophysicalresearch as they are believed to be present in the interstellar medium [2] andresponsible for the dominant features of the infrared emission spectra of manyinterstellar objects [3].

Recent studies of the multiple ionization and fragmentation behaviour ofsmall PAH-molecules in keV-collisions with highly charged molecular ions have

3

been carried out with the purpose of understanding their inherent properties[4]. The experimental results were guided by means of the classical over-the-barrier model in a form in which the polarizability and finite size of the PAHsare neglected. Interestingly, the resulting relative ionization cross-sections fromsuch a model coincide surprisingly well with the experimental results[4].In this thesis we present the intial steps in the development of a novel over-the-barrier model that takes the polarizability of small, flat PAHs into accountby treating them as infinitely thin circular conducting discs. Consequently thespatial analysis of the collisions will have an angular dependency which willbe taken into account in the modeling of the potential barrier. The thesis isorganized as follows. In Chap. 2, we present established over-the-barrier modelsfor ion-atom [5] and ion-cluster [6],[7] collisions to illustrate the concept of themodel in cases of spherical symmetry.The present model is described in Chap. 3, where we first demonstrate that thesequence of ionization potentials for coronene calculated using density functionaltheory (DFT) follows a linear trend as functions of charge state, in agreementwith the expected behaviour for acircular conducting disc. This justifies the useof such a model and we deduce a model radius for coronene from a linear fit tothe DFT data.For completeness we state the angular dependent potential caused by an isolatedconducting disc in the presence of a point charge [8],[9]. We then proceed todeduce the potential barrier experienced by the active electron in the specialcases where the ion is located along the normal symmetry axis of the disc and inthe disc’s tangent plane. In Chap. 4 we show that the present potential barrierscompare well with computationally demanding DFT calculations, asserting itsvalidity as a description of the essentials of the ion-PAH interaction. In Chap5, the critical distances for electron transfer and the absolute ionization crosssections are compared with the corresponding results from the classical over-the-barrier models for ion-atom and ion-cluster collisions. We find that thereare significant differences, suggesting that absolute (And relative) ionizationcross-sections from simpler over-the-barrier models should be taken with somecaution.Finally, in Chap. 6 we present further developments and possible extensions ofthe present model, including a general angular dependency and how accuratepredictions of absolute ionization cross-sections could be calculated using Monte-Carlo simulations. We also consider the possible extensions to elliptical discsand spherical sections.

3 The classical over-the-barrier model concept

The Classical Over-the Barrier (COB) model is based on a criterion for elec-tron transfer between two interacting objects and has been successfully used tomodel keV collisions between for instance atomic ions and atoms or sphericalclusters. The criterion states that when the maximum height of the potentialenergy barrier between two objects equals the stark-shifted ionization energy,an electron will be transferred. Note that this criterion relies on the assump-tion that there is always a resonant state on the projectile (quasi-continuumapproximation), which is well-justified at least for highly charged projectiles.

4

3.1 Electron transfer in ion-atom collisions

As a first example of the COB model, consider a neutral point target in spacelocated at the origin. Consider in addition a point projectile with a positive netcharge of magnitude q at a distance R along the x-axis from the origin. Followingthe idea of the COB model we would like to know the potential experienced byan electron that is moving from the target and find the height of the resultingbarrier

+1 −1x

qR

Figure 1.a: Schematic of electron transfer between atomic target and projectile

We then let an electron move from the target at the origin to the point x, asdepicted in figure 1.a. The electrostatic potential felt by the electron due to thetwo points is then

V (x) =1

x+

q

R− x(1)

This is easily generalized to the potential experienced by the nth sequentialelectron being transferred, as shown in figure 1.b:

+n −1x

q − n+ 1R

Figure 1.b: Multiple electron transfer between atomic target and projectile

Vn(x) =n

x+q + n− 1R− x

(2)

One now simply finds the maximum of the potential energy Un = qeVn = −Vn,at distances between the points and compares that to the stark-shifted ionizationenergy qeIn+1 = −In+1. The Stark-shift of the ionization potential is in thistrivial case the shift in potential at the origin due to the presence of the pointcharge. This is simply magnitude over distance, thus I∗n+1 = In+1 + q/R.Effectively this will increase the ionization potential, making it harder to movethe electron from the origin to infinity.When one has built the necessary tools to examine whether the COB conditionis fulfilled, it is straight forward to gradually reduce the distance R until it ismet. In figure 2 this is exemplified by comparing a Stark-shifted first ionizationpotential I1 = 7.1eV with the potential barrier expressed in eq.(2) for q =+15. At distances larger than about 35a0 the criterion is not met, but whenthe distance is reduced, eventually the COB criterion allows an electron to betransferred. After the transfer the barrier and ionization potential will thenboth be immediately changed due to the reduction and increase in magnitudeof the point charge and target charge respectively, whereupon R again needsto be reduced until a second electron may be transferred. From subsequenttransfer distances one may find a cross-sectional area for each ionization in theform of an annulus, due to the spherical symmetry1 of the target

1Though a point is technically a one-dimensional object, it is identical viewed from anyangle

5

Figure 2: Examples of the potential energy barriers and the stark shifted ion-ization energies for different distances between a point projectile q = 15 and aninitially neutral target atom (cf. text).

This model was recently used by Lawicki et al.[4] to make first predictions ofrelative ionization cross-sections of coronene, and similarly by Seitz et al.[10] forpyrene and flouranthene.

3.2 Electron transfer in ion-cluster collisions

We now illustrate target polarization with another example, this time consid-ering a point charge of magnitude q located a distance R from the center of agrounded conducting sphere of radius a. By the method of electrostatic imagesit can be shown [11] that a single point-charge q1 located a distance R1 fromthe center of the sphere (cf fig2.1) according to eq. (3) is required to fulfill thespherical equipotential condition in the presence of the point charge.

a

r

−q1 q1R1

qR

Figure 2.1:Images point charges used to describe the polarization o aconducting sphere due to an external point charge q

q1 =−aqR

R1 =a2

R(3)

However, we wish to model an isolated sphere as opposed to a grounded. Dueto the spherical symmetry one may change the magnitude of the potential on

6

the sphere surface by inserting a point charge of arbitrary magnitude at thecenter of the sphere. An image charge of equal and opposite magnitude toq1 can therefore be placed in the center of the sphere to preserve the neutralcharge without violating the boundary conditions. These two image chargesthen describe the polarization of the conducting sphere. At a distance r fromthe center of the sphere along the radial line to the point charge, the electrostaticpotential due to the two image charges is then

V (r,R) = −q1r

+q1

r −R1= aq

(1

rR− 1rR− a2

)(4)

The electric field that the point charge q experiences due to the polarized sphereis therefore

E(r = R,R) = − ∂∂r

(V (r,R)

)∣∣∣∣r=R

=q1R2− q1

(R−R1)2=aq

R3− aqR

(R2 − a2)2(5)

In modeling electron transfer one is interested in the potential due to the po-larization caused by an electron and the point charge q at the position of theelectron. To find the contribution from the electron self-image interaction Eewe set q = −1 in eq. (5), and thus consider the image charges caused by anegative unit point charge. Since the electric field due to the image chargesdepend explicitly on the distance R to the electron, the potential due to themat the position x of the electron2 can be obtained by integration of the electricfield Ee expressed in eq. (5) over the radial line from x to infinity:

Ve(x ∈ R > a) = −∫ ∞x

Ee(R)dR = · · · =a

2

(1

x2− 1x2 − a2

)(6)

Before proceeding further we will state a mathematical result briefly mentionedby Friedberg [15] that will be very useful later. Consider a function f of twovariables s and t at the point (s, t = s). Then

d

ds

(f(s, t = s)

)=

∂

∂sf(s, t)

∣∣∣∣t=s

+∂

∂tf(s, t)

∣∣∣∣t=s

(7)

this makes it clear that if f(s, t) = f(t, s) then

∂

∂sf(s, t)

∣∣∣∣t=s

=1

2

d

ds

(f(s, s)

)(8)

As we will show in Chap. 3 this will transform complicated derivatives andintegrations to (at the most) limits of functions. In the present case for instance,consider eq. (4) and the integral formulation eq. (6). The general formulationwe utilize to find the potential at r = R is

Vq(x) = −∫ ∞x

Eq(r = R,R)dR = −∫ ∞x

∂

∂r

(V (r,R)

)∣∣∣∣r=R

dR (9)

2Of course x = R, but the distinction is made so as to keep the integral formulation clear

7

Eq (4) is identical with respect to exchange of the variables r and R, which isall we need to greatly simplify the calculations according to eq. (8). This gives

Vq(x) = −1

2

[V (R,R)

]∞R=x

(10)

= −aq2

(1

x2− 1x2 − a2

)(11)

which is in agreement with eq. (6) when we set q = −1. This result generalizesfor the type of interaction we are considering since the potentials are functions ofthe distance between the two positions considered necessarily interchangeable.This constitutes a plausible control for any form of potential resulting from theinteractions considered, and gives a mathematical explanation to the factor 12observed by Bárány et al.[5] while first establishing this point-sphere model.

We now need to add the potential experienced by the electron due to the pro-jectile (qn = q − n + 1), its induced image charges, and the target charge (n).After some algebraic exercise this gives the following expression for the potentialbarrier for the transfer of the nth electron:

Un(x) = −qn

R− x︸ ︷︷ ︸projectile

+aqn

R(x− a2/R)︸ ︷︷ ︸image

−aqn/R+ nx︸ ︷︷ ︸

center charge

+1

2

(a

x2− ax2 − a2

)︸ ︷︷ ︸electron self-potential

(12)

where qn = q − n+ 1

In this case the stark shifted ionization potential is given by

I∗n = In +qnR

(13)

which is identical to the stark-shift for the ion-atom model stated in the previoussection. With the sequential transfer of charge from the sphere to the point-charge taken into account in both eq. (12) and eq. (13) one is able to find thecritical distances rn and the absolute ionization cross-sections σn = π(r

2n−r2n+1).

This has been successfully used for slow interactions between spherically shapedfullerenes and highly charged atomic ions [7].

4 A classical over-the-barrier model for infinitelythin conducting discs

In this section we will, to our knowledge, for the first time deduce the potentialbarriers experienced by the active electron moving from a conducting disc to apoint charge located along the normal symmetry axis and in the disc tangentplane. These results form the foundation for our novel classical over-the-barrier,which in its final form will be generalized to include angular dependent potentialenergy barriers which may e.g. be used to model keV ion-PAH collisions. Forcompleteness we will first discuss the ionization energies for a conducting circulardisc in view of DFT calculations followed by the the angular dependent discpolarization due the presence of a point charge [8],[9],[12],[13].

8

4.1 Ionization potentials for an infinitely thin, conductingdisc

The theoretical sequence of ionization potentials for a conducting object is

In+1 = W +n+ 1/2

C(14)

where W is the work function and C the capacitance of the object. For aconducting circular disc of radius a the capacitance is given by C = 2a/π.The aim of the present work is to investigate the validity of treating PAHsas conducting discs, as we will illustrate by modeling coronene (C24H12)) assuch. We will utilize ionization potentials for the sequential removal of electronsfrom a coronene molecule calculated by Holm et al.[14] using DFT to establishan approximate linear relationship between the ionization potential and theionization (charge state). In figure 3 we show the linear fit to the sequence of theionization potentials, which compared to eq.(14) using the stated capacitancewill yield the radius of the metal disc.

Figure 3: A linear fit to the ionization potentials of coronene calculated withDFT as a function of charge state.

In+1 = W +n+ 1/2

C= W +

n+ 1/2

2a/π= 6.759 + 4.059n⇒ (15)

a = 10.53 a0 (16)

Compared with the radius of the molecular cage (8.75 a0), the conducting dischas a slightly larger radius (10.53 a0), reflecting that the conducting disc mayto some extent be viewed as including (part of) the electron cloud of coronene.

4.2 Polarization of a circular conducting disc by a pointcharge

One can now proceed to the main tool of the present work, namely the mathe-matical description of the polarization of a conducting disc in the presence of a

9

point charge at an arbitrary position. Using an exotic coordinate substitutionintroduced by C. Neumann [12] in 1864, E.W. Hobson was able to solve thisproblem [8] as early as 1897, following an idea by Sommerfeld [13]. However amore comprehensive presentation of the solution and further development wasput forth by Davis & Reitz [9] in 1971, and it is mostly upon this paper thatthe present work will rely.

The generalized method of images presented by Davis & Reitz is based on lettingthe disc be the branch membrane between the physical space and a Riemann-space in which complex image charges can be placed. The resulting generalformula to describe the potential (due to the point charge and the resultingpolarization of a grounded disc) along the line joining a point charge q with thecenter of a conducting disc is

V(r,φ)P + V

(r,φ)D =

2q

π

[1

Dtan−1

(σ + τ

σ − τ

)1/2− 1D′

tan−1(σ − τ ′

σ + τ ′

)1/2](17)

Expr. (17) results from a change of coordinates to ’peri-polar’ coordinates,which is based on a circle of radius a centered at the origin. This coordinatesystem introduces a discontinuity of 2π in one coordinate when passing throughthe plane of the disc defined by the basis-circle, a fact that is exploited in thesuperposition of physical and image spaces. The new spatial coordinates aredefined as the angle θ and the ratio eρ of the longest (r2) and shortest (r1)distance to the circle, and relate to the normal spherical coordinates3 through

θ = − tan−1(

2ar sinφ

r2 − a2

)θ ∈ [−π, π] (18)

ρ = ln

[r2r1

]= ln

√

(r2 − a2)2 + 4r2a2 sin2 φr2 + a2 − 2ar cosφ

(19)

(0,0)

r

φ

r2

−θ

r1r

φ

Figure 3.1: The relation between spherical coordinates (left) and theperi-polar coordinates (right) used in the present work

Here φ is the angle of elevation of the line joining the point charge and the disccenter, and r is the parameter of that same line, and the variables in eq. (17)

3Note φ=0 in the plane of the disc

10

are

D(r, θ) =

√2a2(cosh(ρ− ρ′)− cos(θ − θ′))(cosh ρ− cos θ)(cosh ρ′ − cos θ′)

, D′ = D(θ′ → −θ′) (20)

σ(r) =

√cosh (ρ− ρ′) + 1

2(21)

τ(θ) = cos

(θ − θ′

2

), τ ′ = τ(θ′ → −θ′) (22)

Note that ρ′ and θ′ are fixed coordinates in the peri-polar system, used to signifythe position of the point charge, while τ ′ and D′ are simply auxiliary parametersnot directly associated with the point charge or even subject to any physicalinterpretation.

In figure 4, we show the potential along the line joining the center of thedisc and a point charge for every 10◦ of elevation above the plane of the discbetween 0◦ and 90◦ according to eq. (17).

Figure 4: Potentials along the line joining the point charge and the origin. Plot-ted potentials are due to the polarization of a grounded disc and the inducingpoint charge q = +10, R = 50a0 for every multiple of 10

◦ between 0◦ and 90◦,all going to 0 on the surface of the disc.

In order to construct a model of an isolated disc as opposed to a groundedone, one needs to preserve the charge of the disc, analogously to the compen-sating center charge of the point-sphere model. Thanks to the early efforts of

11

E.W. Hobson [8] it is known that the potential due to such a distribution on aconducting disc of net charge q′ (in terms of the least and largest distances tothe periphery of the disc, r1 and r2 respectively) is:

V(r,φ)C =

q′

asin−1

(2a

r1 + r2

)(23)

The total induced charge on the disc due to the point source q was by the sameauthor stated as

qind = −2q

πsin−1

(2a

r′1 + r′2

)(24)

Thus we set q′ = −qind in eq. (23) in order to maintain the neutral net chargeof the isolated disc, which means that the potential due to the compensatingcharge distribution then is given by

V(r,φ)C =

2q

aπsin−1

(2a

r′1 + r′2

)sin−1

(2a

r1 + r2

)(25)

Which may be expressed in terms of our peri-polar coordinates:

V(r,φ)C =

2q

aπcos−1

(cos(θ′/2)

cosh(ρ′/2)

)cos−1

(cos(θ/2)

cosh(ρ/2)

)(26)

4.3 Potential barrier for the active electron

The tools to build a complete picture of the potential barrier experienced by theactive electron being transferred from the center of the conducting disc to anion of arbitrary charge are now in place. As a first step towards a more generalmodel of keV ion-PAH collisions we will consider the special cases when the ionis placed along the normal symmetry axis of the disc and in the plane of the disc,in which the analytical expressions are possible to express in a straight-forwardand somewhat transparent way.

4.3.1 Point charge along Normal Symmetry Axis (NSA)

−1x

qR

r

Figure 4.a: Schematic of electron transfer in normal symmetry orientation

In analogy with the ion-sphere model we begin by finding the electrostatic ’self-potential’ induced by the electron. From the general expression (17) one mayshow [15] that the induced potential due to only the grounded disc is

V(r,φ)D

∣∣∣∣φ=π/2

= Vπ/2D (r,R) =

2q

π(r2 −R2)

[R tan−1

(ar

)− r tan−1

( aR

)](27)

12

and as for the compensating charge distribution potential4 given by eq. (25)

V(r,φ)C

∣∣∣∣φ=π/2

= Vπ/2C (r,R) =

2q

aπtan−1

( aR

)tan−1

(ar

)(28)

With these rather complicated expressions it becomes clear why the mathe-matical result given by eq. (8) presented through the point-sphere model isappealing to us. Since they are both identicalwith respect to exchange of r andR, the contribution from the electron’s self-induced potential simply becomes

Uπ/2e (x) = −1 · Vπ/2q=−1(x) =

1

2

[Vπ/2D (R,R)

∣∣∣∣q=−1

+ Vπ/2C (R,R)

∣∣∣∣q=−1

]∞R=x

(29)

=−12πx

[tan−1

(ax

)+

ax

a2 + x2

]+

1

aπ

[tan−1

(ax

)]2(30)

The evaluation of Vπ/2D (R,R) is still not trivial, but through consideration in

terms of limits5 it becomes a manageable task compared to the general formu-lation stated by eq. (9).

To consider the potential of an electron being transferred between the discand the point charge projectile q, we simply add the potential from a secondpoint charge and the contribution from its induced charge distribution on thedisc. The electrostatic potential experienced by the nth sequential electron be-ing transferred, at a distance x between an initially neutral conducting disc anda point charge of magnitude +q at a distance R along the symmetry axis of thedisc is thus given by

Uπ/2n (x) = Uπ/2e −

2qnπ(x2 −R2)

[R tan−1

(ax

)− x tan−1

( aR

)]︸ ︷︷ ︸

Uπ/2D

−

−1a

[2qnπ

tan−1( aR

)− n

]tan−1

(ax

)︸ ︷︷ ︸

Uπ/2C

− qnR− x︸ ︷︷ ︸Uπ/2P

(31)

where qn = q − n+ 1

Since the total induced charge is obviously dependent on the angle φ accordingto eq. (24) while the capacitance of a disc of fixed radius is constant, the stark-shift also has an angular dependency, and along the normal symmetry axis it isgiven by

δIπ/2n =qna

tan−1( aR

)(32)

In the limit a→ 0 the disc becomes a point, and the stark-shift δIπ/2n → q/R isconsistent with that of an atomic target.

4sin−1 [cos(θ/2)sech(ρ/2)] = π/2 − sin−1(

2ar1+r2

)5As an example a similar evaluation can be found in section C of the appendix

13

4.3.2 Point charge in Disc Tangent Plane (DTP)

−1x

qR

r

Figure 4.b: Schematic of electron transfer in disc tangent orientation

We now consider the other extreme case, where the point charge q is located inthe plane of the disc. We proceed in exact analogy to the already consideredorientation, by first considering the contribution from the induced potential dueto the electron. Eq. (17) is in this case given by

V 0D(r,R) = −4q

πDtan−1

(σ − 1σ + 1

)1/2(33)

and the compensating charge potential

V 0C(r,R) = −2q

aπsin−1

( aR

)sin−1

(ar

)(34)

Since both of these expressions are identical under exchange of r and R, eq. (8)applies6:

U0e (x) = −1 · V 0q=−1(x) =1

2

[V 0D(R)

∣∣∣∣q=−1

+ V 0C(R)

∣∣∣∣q=−1

]∞R=x

(35)

=1

aπ

[a2

(x2 − a2)−(

sin−1(ax

))2](36)

As before adding the extra terms to describe the extra point charge and itspolarization of the disc completes the mathematical description of the potentialbarrier. The nth electron potential in the plane of the disc becomes

U0n(x) = U0e −

2qnDπ

[π

2− 2 tan−1

(σ − 1σ + 1

)1/2]︸ ︷︷ ︸

U0P+U0D

+

− 1a

[2qnπ

sin−1( aR

)− n

]sin−1

(ax

)︸ ︷︷ ︸

U0C

(37)

where qn = q − n+ 1, and D = D(x) and σ = σ(x) according to eqs. (20) and(21). The stark-shift in this case is

δI0n =qna

sin−1( aR

)(38)

5 Comparisons with earlier over-the-barrier po-tential barriers and DFT calculations

In recent experiments [4] first estimates of the relative ionization cross-sectionshave been made for coronene (as well as pyrene and flouranthene[10]) using the

6For more complete derivation consult appendix

14

classical over-the-barrier model, where the finite size and as a consequence alsothe polarization of the PAHs were neglected. In this section we will compare thepresent model results with those from earlier classical over-the-barrier modelsand DFT calculations to reveal the the importance of including polarization ef-fects for different target geometries, and in extension the validity of the modelsfor PAH-molecules.

First we consider barriers when the point charge is located above differentplane geometries. This includes the point-point interaction and the interac-tion with an infinite plane conductor7. In figure 5 we present the potentialenergy barriers in the normal symmetry-axis disc orientation according to eq.(31) (a = 10.53a0, R = 20Å) compared to earlier over-the-barrier models andhigh-level DFT-calculations of coronene for q = 15 and n = 1, 2. The modeland DFT-calculations are in good agreement, and show significant improvementover the earlier models8.Mathematical consideration of eq. (31) in the limiting cases where a goes to 0

Figure 5: Present model potential energy-barriers as experienced by the activeelectron between a charged conducting disc of radius 10.53a0 and a point chargeat a distance R = 20Å from the center of the disc along its normal symmetryaxis for charge states (n = 1, qn = 15) (left) and (n = 2, qn = 14) (right) Thecorresponding results from DFT-calculations (asterisks) and the classical over-the-barrier models for a point (dashed) and infinte plane (dashed-dotted) targetare shown for comparison.

and ∞ respectively reduces to the point-point and point-infinite plane modelsillustrated in figure 5. It should be noted that we are considering a potentialbarrier but are mainly interested in its maximum value, which in relation tothe stark-shifted ionization potential determines whether the over-the-barriercriterion is fulfilled. The present model shows a significant improvement in thisrespect compared tosimpler over-the-barrier model, as is evident when consid-ering the DFT-data as a reference.

7For mathematical description consult appendix8The stark-shift of the ionization potential differs between models, so conclusions based

solely on barrier height are not possible

15

In the case of the disc tangent planar model we again consider the differencecompared to a point-point model, but also consider the spherical model of equalradius to that of the disc, which will have the same asymptotic behaviour in theclose limit where x→ a. In figure 6 we show a comparison of potential barriersin disc tangent plane orientation according to eq. (37) (a = 10.53a0, R = 20Å)compared to earlier over-the-barrier models and high-level DFT-calculations ofcoronene for q = 15 and n = 1, 2. In this orientation we also see significantimprovements in barrier height predictions using the present model comparedto earlier models. Note that one should not compare the present model resultswith the DFT results for small distances since the model is only valid for dis-tances larger than the disc radius (10.53a0), while it is possible to set a pointcharge much closer to the coronene cage radius (8.75a0) in the DFT calculations.However, since we are only interested in the barrier heights which are locatedat much larger distances, this effect is not important in our considerations.

Figure 6: Present model energy-barriers as experienced by the active electronbetween a charged conducting disc of radius 10.53a0 and a point charge at adistance R = 20Å from the center of the disc in its tangent plane for chargestates (n = 1, qn = 15) (left) and (n = 2, qn = 14) (right). The correspond-ing results from DFT calculations (asterisks) and the classical over-the-barriermodels for a point (dashed line) and spherical (dashed-dotted) target are shownfor comparisons.

6 Results

In this section we calculate the critical electron transfer distances and the abso-lute ionization cross-sections based on the ionization energies and the potentialenergy barriers shown in Chap. 4 for the present model and the classical over-the-barrier models for a point target and a conducting sphere target. Althoughthe sphere-point model has not been used to model PAHs one might imagine itis a reasonable approach to include polarization effects.

16

6.1 Critical electron transfer distances

The critical electron transfer distances rn at which n electrons are captured bythe projectile ion, corresponding to the formation of a charge state of C24H

n+12

are shown in fig 7 for n ≤ 7. These electron transfer distances are independentof the angle of incidence in the point- and sphere models due to symmetry, whilethe present model (considering here only the two extremes) clearly exhibits anangular dependence.

Figure 7: Present model critical electron transfer distances from a charged con-ducting disc of radius 10.53a0 to a point charge projectile q=20 located at thecenter of the disc along its normal symmetry Axis (NSA) and in the disc tan-gent plane (DTP). The corresponding results from the classical over-the-barriermodels for a point and spherical target are shown for comparisons

6.2 Absolute ionization cross-sections

In collisions between objects with spherical symmetry, the absolute (and rela-tive) ionization cross sections is simply the difference in area of circles definedby the critical distances, i.e. π(r2n − r2n+1). However with less symmetric colli-sion geometries, like that of the present model, the ionization cross-section hasa less obvious geometrical interpretation, due to the orientation dependence ofthe disc in relation to the trajectory of the projectile. As a first approach we willuse the above stated definition in spite of its lack of validity, recognizing that amore accurate definition is possible only through a general angular dependenceof disc orientation.These results may be used for comparisons with the experimental relative ion-ization cross-sections[4] obtained through mass spectrometry of the productsproduced in slow collisions of coronene with highly charged ions such as Xe20+.The ion-sphere model seems to reproduce the results for the disc tangent plane,reflecting the similarity in polarizability and spatial extension of the disc in thisorientation. The results from the point target model is intermediate between

17

Figure 8: Absolute ionization cross-sections σφn = π ·(

(rφn)2 − (rφn+1)2

)for

different models with In+1 = 6.759 + 4.059n

the treated orientations of the disc, which may explain why the experimentalrelative ionization cross-sections of coronene coincided well with such a simplemodel. It could be that the averaging of all disc orientations would result intheoretical ionization cross-sections somewhat similar to a simpler point targetmodel, but this needs confirmation by further model developments. However,in any case the description of the potential energy barrier is significantly im-proved by the present model. This clearly illustrates the importance of includingpolarization effects when modeling ion-PAH collisions.

7 Outlook

The present results clearly show that flat, approximately circular PAHs wouldbenefit from being described as discs, rather that points or spheres. In thischapter we present some interesting extensions to the presented material andpossible model developments, in order to be used in direct comparison with theresults from already conducted and future experiments with PAHs.

7.1 General angular dependence

Provided one considers the potential energy along the line joining the center of adisc with a point charge, the general solution to the problem of the polarizationof a disc in the presence of a point charge is presented in previous sections. Inorder to extend this to a potential energy barrier in the general case, one mustconsider the electron ’self-potential’ through the evaluation of

Uφe = −1

2limr→R

{UφD(r) + U

φC(r)

}(39)

18

This constitutes a potentially very complicated problem analytically, but isstraight forward in a physical sense.

The general solution for the isolated conducting disc in the presence of a pointcharge is strictly speaking more complicated, as we this far only have consideredelectrons transferred on a line joinin the center of the disc and the point charge.This is clearly not the preferred path of electron transfer for all angles φ. Toconsider transfer of electrons along other paths one would need to consider sit-uations other than φ = φ′ (see fig. 8.a), which would transform the analysis ofa one-dimensional potential barrier to that of a potential surface.

γ

(r, φ)

(r′, φ′)

Figure 8.a: Shematic of possible generalization of disc potentials to anglesφ 6= φ′ (cf. text)

Out of physical reasoning the symmetry of the disc makes an analysis of theangular parameter γ in the plane of the disc redundant. That is, an activeelectron would only (theoretically) be considered to move in r and φ, since anymovement of that plane would obviously not be preferred in terms of energy.

In typical experiments, the method of measuring the ions produced in theinteractions between highly charged ions and PAHs is by means of mass spec-trometry, from which the relative ionization cross sections can be extractedfrom the yields of intact multiply charged PAHs. Therein lies a problem withthe present results, since the critical distances for the two special cases consid-ered are based on a point charge that is approaching the disc directly towardsits center. Consequently the two will ultimately collide, and the design of theexperiments that might be conducted to validate the predictions of a point-discmodel do not permit measurements in this specific case, since this type of frontalcollisions would completely destroy the PAH-molecule.In order to extract absolute ionization cross-section by means of the presentmodel, Monte-Carlo simulations are required in which the impact parameterand the orientation of the disc are randomly generated and the over-the-barriercriterion is controlled along the projectile ion’s trajectory. This is in princi-ple straight forward once the potential barrier for a general disc orientation isknown.

7.2 Extension to elliptic discs and spherical caps

Pyrene and fluoranthene are two fairly simple PAHs which have the same chem-ical composition but differs in structure, both of which are currently the subjectof experimental studies. These are like most other PAHs flat molecules, butwould most likely benefit from a elliptical description rather than a circular, aswould also for instance antracene, shown in figure 9.

19

Figure 9: Pyrene (C16H10) and antracene (C14H10)

The generalization to elliptical discs would further reduce the collision sym-metry, introducing an angular parameter in the plane of the disc. However itwould be fairly straight forward to utilize measured or calculated ionizationspotentials for these (and indeed any other flat) molecules and under the as-sumption that they follow the same approximate linear relationship calculatean equivalent radius of a circular disc. This is analogous to the method used forcoronene in the present work, and presumably results in a disc radius slightlylarger than the average molecular dimension.

One might also consider PAHs similar to corannulene (C20H10), which in con-trast to most PAHs is not flat but has a slight curvature due to its centralpentagon. In such cases the molecule might benefit from being modeled as asection of a sphere known as a spherical cap. The theoretical results for the po-larization of such a conducting surface due to a point charge has been presentedbriefly by Hobson [8]. The expression equivalent to eq. (17) in this case is thendependent on the angle β, and reduces to the disc model when β vanishes.

Figure 10: Corannulene (C20H10) and a spherical cap (cf. text)

20

References

[1] G. Mastrangelo et al., ’Polycyclic aromatic hydrocarbons and cancer inman’, Env. Health. Perspc., 104, 1166-1170, (1996)

[2] A.G.G.M. Tielens, ’The Physics and Chemistry of the Interstellar Medium’,Cambridge U. Press, 173-224, (2005)

[3] F. Y. Xiang et al., ’A Tale of Two Mysteries in Interstellar Astrophysics:The 2175 ÅExtinction Bump and Diffuse Interstellar Bands’, Ast. Phys. J.733 91, (2011)

[4] A. Lawicki et al., ’Multiple ionization and fragmentation of isolated pyreneand coronene molecules in collision with ions’, Phys. Rev. A, 83, 022704-1(2011)

[5] A. Bárány et al., Nuclear Instrument and Methods in Physics Research B9, 397-399 (1985).

[6] S. Diaz-Tendero et al., ’Structure and electronic properties of highlycharged C60 and C58 fullerenes’, J. Chem. Phys. 123, 184306 (2005)

[7] H. Cederquist et al., ’Electronic response of C60 in slow collisions withhighly charged ions’, Phys. Rev. A, 61, 022712-1 (2000)

[8] E.W. Hobson, ’On Green’s function for a circular disc, with application toelectrostatic problems’, Trans. Cambridge Phil. Soc.18, 277-291 (1900)

[9] L.C. Davis & J.R. Reitz, ’Solution to potential problems near a conductingsemi-infinite sheet or conducting disc’, Am. J. Phys. 39, 1255-1265 (1971)

[10] F: Seitz et al., ’Weak isomer effects in PAH-monomer and -cluster ioniza-tion: pyrene versus flouranthene’, yet unpublished (2011)

[11] J.D. Jackson, ’Classical Electrodynamics’ 2:nd edition, 54-56 (1975).

[12] C. Neumann,’Theorie der Elektricitäts- und Wärme-Vertheilung in eniemRinge’, Halle (1864)

[13] A. Sommerfeld, ’Über verzweigte Potential im Raum’, Proc. London Math.Soc. 28, 395-429 (1897)

[14] A.I.S. Holm et al., ’Dissociation and multiple ionization energies for fivepolycyclic aromatic hydrocarbon molecules’, J. Chem. Phys. 134, 044301(2011)

[15] R. Friedberg, ’The electrostatics and magnetostatics of a conducting disk’,Am. J. Phys. 61, 1084-1096 (1993)

21

A Infinte plane model

A useful control for other models, the point-infinite plane interaction relies on avery basic mathematical description due to the simplicity of the image methodfor this symmetry:

r

x

−x

R

−R

Un(x) = −1

4x︸ ︷︷ ︸Ue(x)

+qn

R+ x︸ ︷︷ ︸image

− qnR− x︸ ︷︷ ︸source

(40)

The stark-shift is out of necessity 0, which is verified through the stark shift fora disk of radius a (eq. 32) when the radius grows large, making the stark-shiftgo to 0. A similar consequence is that the ionization potential is replaced bythe work function, which is independent on order of electron transfer. The onlydeciding factor in terms of critical transfer distances in such a model wouldtherefore be the charge of the point.

B Relevant trigonometric identities

2a

R r1r2

tan−1(R

a

)= sin−1

(R

r1

)= cos−1

(a

r1

)(41)

22

C Evaluation of V 0D(R)

While V 0C(R,R) is easily evaluated V0D(R,R) is less obvious as it takes on a

00 -form. By definition

σ =

√cosh(ρ− ρ′) + 1

2(42)

and

ρ− ρ′ = ln[r + a

r − a· R− aR+ a

](43)

so with the substitution

t =

√r + a

r − a· R− aR+ a

→ 1 as r → R (44)

it follows that

cosh(ρ− ρ′) = t2 + t−2

2(45)

and so

cosh(ρ− ρ′) + 1 = t2 + 2 + t−2

2=

(t+ t−1)2

2(46)

so σ becomes

σ =1

2(t+ t−1) (47)

And so further

σ ∓ 1 = 12· (t∓ 2 + t−1) = 1

2· (t1/2 ∓ t−1/2)2 (48)

So the quotient of these is

σ − 1σ + 1

=

(t1/2 − t−1/2

t1/2 + t−1/2

)2=

(t−1/2

t−1/2

)2·(t− 1t+ 1

)2=

(t− 1t+ 1

)2(49)

So one may conclude that √σ − 1σ + 1

=t− 1t+ 1

(50)

in addition the parameter t can be shown to have the useful property

1

D=

2a

(r2 − a2)· (r + a)

(R+ a)(1− t2)(51)

Then the known limit

limt→1

tan−1(t−1t+1

)1− t2

= −14

(52)

yields the result

limr→R

V 0D(r,R) =−2qa

π(R2 − a2)(53)

23