Classical over-the-barrier model for ionization of poly-cyclic aromatic hydrocarbons in keV-
collisions with atomic ions
Bachelors thesis
Department of Atomic Physics Stockholm University
Author Bj¨orn Forsberg
Advisor
Henning Zettergren
Contents
1 Abstract 3
2 Introduction 3
3 The classical over-the-barrier model concept 4
3.1 Electron transfer in ion-atom collisions . . . 5 3.2 Electron transfer in ion-cluster collisions . . . 6 4 A classical over-the-barrier model for infinitely thin conducting
discs 8
4.1 Ionization potentials for an infinitely thin, conducting disc . . . . 9 4.2 Polarization of a circular conducting disc by a point charge . . . 9 4.3 Potential barrier for the active electron . . . 12 4.3.1 Point charge along Normal Symmetry Axis (NSA) . . . . 12 4.3.2 Point charge in Disc Tangent Plane (DTP) . . . 14 5 Comparisons with earlier over-the-barrier potential barriers and
DFT calculations 14
6 Results 16
6.1 Critical electron transfer distances . . . 17 6.2 Absolute ionization cross-sections . . . 17
7 Outlook 18
7.1 General angular dependence . . . 18 7.2 Extension to elliptic discs and spherical caps . . . 19
A Infinte plane model 22
B Relevant trigonometric identities 22
C Evaluation of VD0(R) 23
1 Abstract
We are developing a novel classical over-the barrier model for electron transfer from an infinitely thin conducting disc to a point charge projectile to model multiple electron capture in e.g. keV collisions of atomic ions with poly-cyclic aromatic hydrocarbons (PAHs). In its final form, the present model will incor- porate the polarization of the PAH molecules due to the active electron and the point charge projectile at a general angle of incidence. This will drastically improve the description of the potential barrier in comparisons with simpler ver- sions of the model where the finite size and polarizability of the target molecule is neglected or treated in an averaged fashion. In this work we arrive at ex- pressions for the electrostatic potential energy barrier experienced by the active electron 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 simpler versions of the classical over-the-barrier models for atomic and spherical cluster targets. These results thus strongly supports the conducting disc approximation of 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 spherical caps.
2 Introduction
Poly-cyclic Aromatic Hydrocarbons (PAHs) are molecules that are built from two or more aromatic rings. The simplest PAH is naphthalene (C10H8), con- sisting of only two such hexagonal rings. Other small PAHs are for instance pyrene (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 the most widespread types of organic pollutants, produced in the combustion of almost any organic material. Many PAHs, for instance naphthalene, has also been identified as carcinogenic [1], PAHs are also of interest in astrophysical research as they are believed to be present in the interstellar medium [2] and responsible for the dominant features of the infrared emission spectra of many interstellar objects [3].
Recent studies of the multiple ionization and fragmentation behaviour of small PAH-molecules in keV-collisions with highly charged molecular ions have
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 PAHs are neglected. Interestingly, the resulting relative ionization cross-sections from such 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 account by treating them as infinitely thin circular conducting discs. Consequently the spatial analysis of the collisions will have an angular dependency which will be taken into account in the modeling of the potential barrier. The thesis is organized as follows. In Chap. 2, we present established over-the-barrier models for ion-atom [5] and ion-cluster [6],[7] collisions to illustrate the concept of the model in cases of spherical symmetry.
The present model is described in Chap. 3, where we first demonstrate that the sequence of ionization potentials for coronene calculated using density functional theory (DFT) follows a linear trend as functions of charge state, in agreement with the expected behaviour for acircular conducting disc. This justifies the use of such a model and we deduce a model radius for coronene from a linear fit to the DFT data.
For completeness we state the angular dependent potential caused by an isolated conducting disc in the presence of a point charge [8],[9]. We then proceed to deduce the potential barrier experienced by the active electron in the special cases where the ion is located along the normal symmetry axis of the disc and in the disc’s tangent plane. In Chap. 4 we show that the present potential barriers compare well with computationally demanding DFT calculations, asserting its validity as a description of the essentials of the ion-PAH interaction. In Chap 5, the critical distances for electron transfer and the absolute ionization cross sections are compared with the corresponding results from the classical over- the-barrier models for ion-atom and ion-cluster collisions. We find that there are significant differences, suggesting that absolute (And relative) ionization cross-sections from simpler over-the-barrier models should be taken with some caution.
Finally, in Chap. 6 we present further developments and possible extensions of the present model, including a general angular dependency and how accurate predictions of absolute ionization cross-sections could be calculated using Monte- Carlo simulations. We also consider the possible extensions to elliptical discs and 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 to model keV collisions between for instance atomic ions and atoms or spherical clusters. The criterion states that when the maximum height of the potential energy 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-continuum approximation), which is well-justified at least for highly charged projectiles.
3.1 Electron transfer in ion-atom collisions
As a first example of the COB model, consider a neutral point target in space located at the origin. Consider in addition a point projectile with a positive net charge of magnitude q at a distance R along the x-axis from the origin. Following the idea of the COB model we would like to know the potential experienced by an electron that is moving from the target and find the height of the resulting barrier
+1 −1
x
q R
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, as depicted in figure 1.a. The electrostatic potential felt by the electron due to the two points is then
V (x) = 1
x+ q
R − x (1)
This is easily generalized to the potential experienced by the nth sequential electron being transferred, as shown in figure 1.b:
+n −1
x
q − n + 1 R
Figure 1.b: Multiple electron transfer between atomic target and projectile
Vn(x) = n
x+q + n − 1
R − 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 ionization energy qeIn+1 = −In+1. The Stark-shift of the ionization potential is in this trivial case the shift in potential at the origin due to the presence of the point charge. This is simply magnitude over distance, thus In+1∗ = In+1+ q/R.
Effectively this will increase the ionization potential, making it harder to move the electron from the origin to infinity.
When one has built the necessary tools to examine whether the COB condition is fulfilled, it is straight forward to gradually reduce the distance R until it is met. In figure 2 this is exemplified by comparing a Stark-shifted first ionization potential 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 when the distance is reduced, eventually the COB criterion allows an electron to be transferred. After the transfer the barrier and ionization potential will then both be immediately changed due to the reduction and increase in magnitude of the point charge and target charge respectively, whereupon R again needs to be reduced until a second electron may be transferred. From subsequent transfer distances one may find a cross-sectional area for each ionization in the form 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 any angle
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 an initially neutral target atom (cf. text).
This model was recently used by Lawicki et al.[4] to make first predictions of relative ionization cross-sections of coronene, and similarly by Seitz et al.[10] for pyrene 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 a grounded conducting sphere of radius a. By the method of electrostatic images it can be shown [11] that a single point-charge q1 located a distance R1 from the center of the sphere (cf fig2.1) according to eq. (3) is required to fulfill the spherical equipotential condition in the presence of the point charge.
a
r
−q1 q1
R1
q R
Figure 2.1:Images point charges used to describe the polarization o a conducting sphere due to an external point charge q
q1=−aq
R R1=a2
R (3)
However, we wish to model an isolated sphere as opposed to a grounded. Due to the spherical symmetry one may change the magnitude of the potential on
the sphere surface by inserting a point charge of arbitrary magnitude at the center of the sphere. An image charge of equal and opposite magnitude to q1 can therefore be placed in the center of the sphere to preserve the neutral charge without violating the boundary conditions. These two image charges then describe the polarization of the conducting sphere. At a distance r from the center of the sphere along the radial line to the point charge, the electrostatic potential due to the two image charges is then
V (r, R) = −q1
r + q1
r − R1 = aq
1
rR− 1
rR − a2
(4) The electric field that the point charge q experiences due to the polarized sphere is therefore
E(r = R, R) = −∂
∂r
V (r, R)
r=R
= q1
R2− 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 the electron. To find the contribution from the electron self-image interaction Ee
we set q = −1 in eq. (5), and thus consider the image charges caused by a negative unit point charge. Since the electric field due to the image charges depend explicitly on the distance R to the electron, the potential due to them at the position x of the electron2 can be obtained by integration of the electric field Eeexpressed in eq. (5) over the radial line from x to infinity:
Ve(x ∈ R > a) = − Z ∞
x
Ee(R)dR = · · · = a 2
1
x2 − 1 x2− a2
(6) Before proceeding further we will state a mathematical result briefly mentioned by Friedberg [15] that will be very useful later. Consider a function f of two variables 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 and integrations to (at the most) limits of functions. In the present case for instance, consider eq. (4) and the integral formulation eq. (6). The general formulation we utilize to find the potential at r = R is
Vq(x) = − Z ∞
x
Eq(r = R, R)dR = − Z ∞
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
Eq (4) is identical with respect to exchange of the variables r and R, which is all we need to greatly simplify the calculations according to eq. (8). This gives
Vq(x) = −1 2
V (R, R)
∞
R=x
(10)
= −aq 2
1
x2 − 1 x2− a2
(11) which is in agreement with eq. (6) when we set q = −1. This result generalizes for the type of interaction we are considering since the potentials are functions of the distance between the two positions considered necessarily interchangeable.
This constitutes a plausible control for any form of potential resulting from the interactions considered, and gives a mathematical explanation to the factor 12 observed by B´ar´any 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 potential barrier for the transfer of the nth electron:
Un(x) = − qn
R − x
| {z }
projectile
+ aqn
R(x − a2/R)
| {z }
image
−aqn/R + n x
| {z }
center charge
+1 2
a
x2 − a x2− a2
| {z }
electron self-potential
(12)
where qn = q − n + 1
In this case the stark shifted ionization potential is given by In∗= In+qn
R (13)
which is identical to the stark-shift for the ion-atom model stated in the previous section. 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 the critical distances rnand the absolute ionization cross-sections σn= π(rn2−r2n+1).
This has been successfully used for slow interactions between spherically shaped fullerenes and highly charged atomic ions [7].
4 A classical over-the-barrier model for infinitely thin conducting discs
In this section we will, to our knowledge, for the first time deduce the potential barriers experienced by the active electron moving from a conducting disc to a point charge located along the normal symmetry axis and in the disc tangent plane. These results form the foundation for our novel classical over-the-barrier, which in its final form will be generalized to include angular dependent potential energy barriers which may e.g. be used to model keV ion-PAH collisions. For completeness we will first discuss the ionization energies for a conducting circular disc in view of DFT calculations followed by the the angular dependent disc polarization due the presence of a point charge [8],[9],[12],[13].
4.1 Ionization potentials for an infinitely thin, conducting disc
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 a conducting 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 PAHs as conducting discs, as we will illustrate by modeling coronene (C24H12)) as such. We will utilize ionization potentials for the sequential removal of electrons from a coronene molecule calculated by Holm et al.[14] using DFT to establish an approximate linear relationship between the ionization potential and the ionization (charge state). In figure 3 we show the linear fit to the sequence of the ionization potentials, which compared to eq.(14) using the stated capacitance will yield the radius of the metal disc.
Figure 3: A linear fit to the ionization potentials of coronene calculated with DFT 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 disc has a slightly larger radius (10.53 a0), reflecting that the conducting disc may to some extent be viewed as including (part of) the electron cloud of coronene.
4.2 Polarization of a circular conducting disc by a point charge
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
point charge at an arbitrary position. Using an exotic coordinate substitution introduced by C. Neumann [12] in 1864, E.W. Hobson was able to solve this problem [8] as early as 1897, following an idea by Sommerfeld [13]. However a more comprehensive presentation of the solution and further development was put forth by Davis & Reitz [9] in 1971, and it is mostly upon this paper that the present work will rely.
The generalized method of images presented by Davis & Reitz is based on letting the disc be the branch membrane between the physical space and a Riemann- space in which complex image charges can be placed. The resulting general formula to describe the potential (due to the point charge and the resulting polarization of a grounded disc) along the line joining a point charge q with the center of a conducting disc is
VP(r,φ)+ VD(r,φ)=2q π
"
1
Dtan−1 σ + τ σ − τ
1/2
− 1
D0tan−1 σ − τ0 σ + τ0
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 coordinate system introduces a discontinuity of 2π in one coordinate when passing through the plane of the disc defined by the basis-circle, a fact that is exploited in the superposition of physical and image spaces. The new spatial coordinates are defined 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 coordinates3through
θ = − tan−1 2ar sin φ r2− a2
θ ∈ [−π, π] (18)
ρ = ln r2 r1
= ln
q
(r2− a2)2+ 4r2a2sin2φ r2+ a2− 2ar cos φ
(19)
(0,0) r
φ
r2
−θ
r1
r φ
Figure 3.1: The relation between spherical coordinates (left) and the peri-polar coordinates (right) used in the present work
Here φ is the angle of elevation of the line joining the point charge and the disc center, and r is the parameter of that same line, and the variables in eq. (17)
3Note φ=0 in the plane of the disc
are
D(r, θ) = s
2a2(cosh(ρ − ρ0) − cos(θ − θ0))
(cosh ρ − cos θ)(cosh ρ0− cos θ0), D0 = D(θ0 → −θ0) (20) σ(r) =
rcosh (ρ − ρ0) + 1
2 (21)
τ (θ) = cos θ − θ0 2
, τ0= τ (θ0→ −θ0) (22)
Note that ρ0and θ0are fixed coordinates in the peri-polar system, used to signify the position of the point charge, while τ0and D0are simply auxiliary parameters not directly associated with the point charge or even subject to any physical interpretation.
In figure 4, we show the potential along the line joining the center of the disc and a point charge for every 10◦ of elevation above the plane of the disc between 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 inducing point charge q = +10, R = 50a0for 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 grounded one, 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
E.W. Hobson [8] it is known that the potential due to such a distribution on a conducting disc of net charge q0 (in terms of the least and largest distances to the periphery of the disc, r1 and r2 respectively) is:
VC(r,φ) =q0 a sin−1
2a r1+ r2
(23) The total induced charge on the disc due to the point source q was by the same author stated as
qind= −2q π sin−1
2a r10 + r02
(24) Thus we set q0 = −qind in eq. (23) in order to maintain the neutral net charge of the isolated disc, which means that the potential due to the compensating charge distribution then is given by
VC(r,φ)= 2q aπ sin−1
2a r01+ r02
sin−1
2a r1+ r2
(25) Which may be expressed in terms of our peri-polar coordinates:
VC(r,φ) = 2q
aπcos−1 cos(θ0/2) cosh(ρ0/2)
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 the active electron being transferred from the center of the conducting disc to an ion of arbitrary charge are now in place. As a first step towards a more general model of keV ion-PAH collisions we will consider the special cases when the ion is 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-forward and somewhat transparent way.
4.3.1 Point charge along Normal Symmetry Axis (NSA)
−1 x
q
R 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 may show [15] that the induced potential due to only the grounded disc is
VD(r,φ) φ=π/2
= VDπ/2(r, R) = 2q π(r2− R2)
h
R tan−1a r
− r tan−1a R
i (27)
and as for the compensating charge distribution potential4 given by eq. (25) VC(r,φ)
φ=π/2
= VCπ/2(r, R) = 2q
aπtan−1a R
tan−1a r
(28)
With these rather complicated expressions it becomes clear why the mathe- matical result given by eq. (8) presented through the point-sphere model is appealing to us. Since they are both identicalwith respect to exchange of r and R, the contribution from the electron’s self-induced potential simply becomes
Ueπ/2(x) = −1 · Vq=−1π/2 (x) = 1 2
"
VDπ/2(R, R) q=−1
+ VCπ/2(R, R) q=−1
#∞ R=x
(29)
= −1 2πx
tan−1a x
+ ax
a2+ x2
+ 1
aπ h
tan−1a x
i2
(30)
The evaluation of VDπ/2(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 disc and the point charge projectile q, we simply add the potential from a second point charge and the contribution from its induced charge distribution on the disc. The electrostatic potential experienced by the nth sequential electron be- ing transferred, at a distance x between an initially neutral conducting disc and a point charge of magnitude +q at a distance R along the symmetry axis of the disc is thus given by
Unπ/2(x) = Ueπ/2− 2qn π(x2− R2)
hR tan−1a x
− x tan−1a R
i
| {z }
UDπ/2
−
−1 a
2qn
π tan−1a R
− n
tan−1a x
| {z }
UCπ/2
− qn R − x
| {z }
UPπ/2
(31)
where qn= q − n + 1
Since the total induced charge is obviously dependent on the angle φ according to 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 is given by
δInπ/2 =qn
a tan−1a R
(32) In the limit a → 0 the disc becomes a point, and the stark-shift δInπ/2→ q/R is consistent with that of an atomic target.
4sin−1[cos(θ/2)sech(ρ/2)] = π/2 − sin−1
2a r1+r2
5As an example a similar evaluation can be found in section C of the appendix
4.3.2 Point charge in Disc Tangent Plane (DTP)
−1 x
q
R 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 in the plane of the disc. We proceed in exact analogy to the already considered orientation, by first considering the contribution from the induced potential due to the electron. Eq. (17) is in this case given by
VD0(r, R) = −4q
πDtan−1 σ − 1 σ + 1
1/2
(33) and the compensating charge potential
VC0(r, R) = −2q
aπsin−1a R
sin−1a r
(34) Since both of these expressions are identical under exchange of r and R, eq. (8) applies6:
Ue0(x) = −1 · Vq=−10 (x) =1 2
"
VD0(R) q=−1
+ VC0(R) q=−1
#∞ R=x
(35)
= 1 aπ
a2
(x2− a2)−
sin−1a x
2
(36) As before adding the extra terms to describe the extra point charge and its polarization of the disc completes the mathematical description of the potential barrier. The nth electron potential in the plane of the disc becomes
Un0(x) = Ue0−2qn
Dπ
"
π
2 − 2 tan−1 σ − 1 σ + 1
1/2#
| {z }
UP0+UD0
+
−1 a
2qn
π sin−1a R
− n
sin−1a x
| {z }
UC0
(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
δIn0=qn
a sin−1a R
(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-sections have been made for coronene (as well as pyrene and flouranthene[10]) using the
6For more complete derivation consult appendix
classical over-the-barrier model, where the finite size and as a consequence also the polarization of the PAHs were neglected. In this section we will compare the present model results with those from earlier classical over-the-barrier models and DFT calculations to reveal the the importance of including polarization ef- fects for different target geometries, and in extension the validity of the models for PAH-molecules.
First we consider barriers when the point charge is located above different plane geometries. This includes the point-point interaction and the interac- tion with an infinite plane conductor7. In figure 5 we present the potential energy barriers in the normal symmetry-axis disc orientation according to eq.
(31) (a = 10.53a0, R = 20˚A) compared to earlier over-the-barrier models and high-level DFT-calculations of coronene for q = 15 and n = 1, 2. The model and DFT-calculations are in good agreement, and show significant improvement over 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 active electron between a charged conducting disc of radius 10.53a0and a point charge at a distance R = 20˚A from the center of the disc along its normal symmetry axis for charge states (n = 1, qn = 15) (left) and (n = 2, qn = 14) (right) The corresponding results from DFT-calculations (asterisks) and the classical over- the-barrier models for a point (dashed) and infinte plane (dashed-dotted) target are shown for comparison.
and ∞ respectively reduces to the point-point and point-infinite plane models illustrated in figure 5. It should be noted that we are considering a potential barrier but are mainly interested in its maximum value, which in relation to the stark-shifted ionization potential determines whether the over-the-barrier criterion is fulfilled. The present model shows a significant improvement in this respect compared tosimpler over-the-barrier model, as is evident when consid- ering the DFT-data as a reference.
7For mathematical description consult appendix
8The stark-shift of the ionization potential differs between models, so conclusions based solely on barrier height are not possible
In the case of the disc tangent planar model we again consider the difference compared to a point-point model, but also consider the spherical model of equal radius to that of the disc, which will have the same asymptotic behaviour in the close limit where x → a. In figure 6 we show a comparison of potential barriers in disc tangent plane orientation according to eq. (37) (a = 10.53a0, R = 20˚A) compared to earlier over-the-barrier models and high-level DFT-calculations of coronene for q = 15 and n = 1, 2. In this orientation we also see significant improvements in barrier height predictions using the present model compared to earlier models. Note that one should not compare the present model results with 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 point charge 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 located at much larger distances, this effect is not important in our considerations.
Figure 6: Present model energy-barriers as experienced by the active electron between a charged conducting disc of radius 10.53a0 and a point charge at a distance R = 20˚A from the center of the disc in its tangent plane for charge states (n = 1, qn = 15) (left) and (n = 2, qn = 14) (right). The correspond- ing results from DFT calculations (asterisks) and the classical over-the-barrier models for a point (dashed line) and spherical (dashed-dotted) target are shown for 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 potential energy 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. Although the sphere-point model has not been used to model PAHs one might imagine it is a reasonable approach to include polarization effects.
6.1 Critical electron transfer distances
The critical electron transfer distances rn at which n electrons are captured by the projectile ion, corresponding to the formation of a charge state of C24Hn+12 are shown in fig 7 for n ≤ 7. These electron transfer distances are independent of the angle of incidence in the point- and sphere models due to symmetry, while the present model (considering here only the two extremes) clearly exhibits an angular 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 the center 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-barrier models 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 defined by the critical distances, i.e. π(rn2− r2n+1). However with less symmetric colli- sion geometries, like that of the present model, the ionization cross-section has a less obvious geometrical interpretation, due to the orientation dependence of the disc in relation to the trajectory of the projectile. As a first approach we will use the above stated definition in spite of its lack of validity, recognizing that a more accurate definition is possible only through a general angular dependence of disc orientation.
These results may be used for comparisons with the experimental relative ion- ization cross-sections[4] obtained through mass spectrometry of the products produced 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 this orientation. The results from the point target model is intermediate between
Figure 8: Absolute ionization cross-sections σnφ = 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 experimental relative ionization cross-sections of coronene coincided well with such a simple model. It could be that the averaging of all disc orientations would result in theoretical ionization cross-sections somewhat similar to a simpler point target model, 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 including polarization effects when modeling ion-PAH collisions.
7 Outlook
The present results clearly show that flat, approximately circular PAHs would benefit from being described as discs, rather that points or spheres. In this chapter we present some interesting extensions to the presented material and possible model developments, in order to be used in direct comparison with the results 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 a disc with a point charge, the general solution to the problem of the polarization of a disc in the presence of a point charge is presented in previous sections. In order to extend this to a potential energy barrier in the general case, one must consider the electron ’self-potential’ through the evaluation of
Ueφ= −1 2 lim
r→R
n
UDφ(r) + UCφ(r)o
(39)
This constitutes a potentially very complicated problem analytically, but is straight forward in a physical sense.
The general solution for the isolated conducting disc in the presence of a point charge is strictly speaking more complicated, as we this far only have considered electrons 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. To consider transfer of electrons along other paths one would need to consider sit- uations other than φ = φ0 (see fig. 8.a), which would transform the analysis of a one-dimensional potential barrier to that of a potential surface.
γ
(r, φ)
(r0, φ0)
Figure 8.a: Shematic of possible generalization of disc potentials to angles φ 6= φ0 (cf. text)
Out of physical reasoning the symmetry of the disc makes an analysis of the angular parameter γ in the plane of the disc redundant. That is, an active electron would only (theoretically) be considered to move in r and φ, since any movement of that plane would obviously not be preferred in terms of energy.
In typical experiments, the method of measuring the ions produced in the interactions between highly charged ions and PAHs is by means of mass spec- trometry, from which the relative ionization cross sections can be extracted from the yields of intact multiply charged PAHs. Therein lies a problem with the 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 towards its center. Consequently the two will ultimately collide, and the design of the experiments that might be conducted to validate the predictions of a point-disc model do not permit measurements in this specific case, since this type of frontal collisions would completely destroy the PAH-molecule.
In order to extract absolute ionization cross-section by means of the present model, Monte-Carlo simulations are required in which the impact parameter and the orientation of the disc are randomly generated and the over-the-barrier criterion is controlled along the projectile ion’s trajectory. This is in princi- ple straight forward once the potential barrier for a general disc orientation is known.
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 subject of experimental studies. These are like most other PAHs flat molecules, but would most likely benefit from a elliptical description rather than a circular, as would also for instance antracene, shown in figure 9.
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 it would be fairly straight forward to utilize measured or calculated ionizations potentials for these (and indeed any other flat) molecules and under the as- sumption that they follow the same approximate linear relationship calculate an equivalent radius of a circular disc. This is analogous to the method used for coronene in the present work, and presumably results in a disc radius slightly larger 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 central pentagon. In such cases the molecule might benefit from being modeled as a section 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 presented briefly by Hobson [8]. The expression equivalent to eq. (17) in this case is then dependent on the angle β, and reduces to the disc model when β vanishes.
Figure 10: Corannulene (C20H10) and a spherical cap (cf. text)
References
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The 2175 ˚AExtinction Bump and Diffuse Interstellar Bands’, Ast. Phys. J.
733 91, (2011)
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highly charged ions’, Phys. Rev. A, 61, 022712-1 (2000)
[8] E.W. Hobson, ’On Green’s function for a circular disc, with application to electrostatic problems’, Trans. Cambridge Phil. Soc.18, 277-291 (1900) [9] L.C. Davis & J.R. Reitz, ’Solution to potential problems near a conducting
semi-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-
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A Infinte plane model
A useful control for other models, the point-infinite plane interaction relies on a very basic mathematical description due to the simplicity of the image method for this symmetry:
r
x
−x R
−R
Un(x) = − 1 4x
| {z }
Ue(x)
+ qn R + x
| {z }
image
− qn R − x
| {z }
source
(40)
The stark-shift is out of necessity 0, which is verified through the stark shift for a disk of radius a (eq. 32) when the radius grows large, making the stark-shift go to 0. A similar consequence is that the ionization potential is replaced by the work function, which is independent on order of electron transfer. The only deciding factor in terms of critical transfer distances in such a model would therefore be the charge of the point.
B Relevant trigonometric identities
2a
R r1
r2
tan−1 R a
= sin−1 R r1
= cos−1 a r1
(41)
C Evaluation of V
D0(R)
While VC0(R, R) is easily evaluated VD0(R, R) is less obvious as it takes on a
0
0-form. By definition
σ =
rcosh(ρ − ρ0) + 1
2 (42)
and
ρ − ρ0= ln r + a
r − a·R − a R + a
(43) so with the substitution
t =r r + a
r − a·R − a
R + a → 1 as r → R (44)
it follows that
cosh(ρ − ρ0) = t2+ t−2
2 (45)
and so
cosh(ρ − ρ0) + 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 = 1
2 · (t ∓ 2 + t−1) =1
2 · (t1/2∓ t−1/2)2 (48) So the quotient of these is
σ − 1
σ + 1= t1/2− t−1/2 t1/2+ t−1/2
2
= t−1/2 t−1/2
2
· t − 1 t + 1
2
= t − 1 t + 1
2
(49) So one may conclude that
r σ − 1
σ + 1= t − 1
t + 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
lim
t→1
tan−1
t−1 t+1
1 − t2 = −1
4 (52)
yields the result
lim
r→RVD0(r, R) = −2qa
π(R2− a2) (53)
