A ballistic model for ion breakdown

Internotional JOZUM~ of Mass Spectromerry Elsaier Publishing Company, Amsterdam.

’ A BALLISTIC

P. F.

MODEL

and Ion Physics

217

Printed in the Netherlands

FOR ION BREAKDOWN

KNEWSTUBB

Deparrmenr of Ph.vsical Chemisrry,

Umkvsity

of Cambridge,

Lerzfieeld Road, Cambridge

CBZ IEP

(figland)

(Received

December

Znd, 1970)

ABSTRACT

A model of ion fragmentation is developed in which orbital motion of the fra_ements is considered. This leads to the description of a loose transition state and the development of a transmission coefficient. Values of the latter are given for selected examples. It is possible to include the participation of initial rotational ener_q of the ion and to demonstrate an apparent lowering of threshold which should result. Rate coefficientsare calculated from this model and from an earlier viewpoint and are compared_

IN-lRODUCTION

Almost since the foundation of the study of ion-molecule reactions as 5 distinct field of research; the classical theory of such encounters has been a cornerstone. This theory was first discussed by Langevin in 1905’, in connection with the mobility of ions in gases, and was applied to ion-molecule reactions in a famous paper of Gioumousis and Stevenson in 1958’. The concept of trajectories controlled by long-range, ion-induced dipole forces has been recognised as providing a somewhat imperfect description,but neverthelessis still taken as forming a useful basis for comparison with experimental results3. Superficially,one might consider the breakdown of an internallyexcited ion to be related to the process of ion-molecule reaction by simple reversal of the time variable. This paper attempts tc set up a model and to examine the relevance of the theory of classical trajectories to ion fragrnertation and to the calculation of rate coefficientsfor such processes. The deveIopment of the modei makes possible some study of the effects of initial rotation of the molecule on its fragmentation. The probable occurrence of such effects has been recognised by ChupkaJ and by Ottinger’, who invoked some considerations of k,rational effects in discussing the results of their experiments Inr. .F_ Ma-

Spectrom.

Ion P&s.,

6 (1971)

217-223

218

P_ F_ KNEWSTUBB

with methane and ethane- Within the approximations of the model, quantitative predictions about the importance of such eff&cts are now produced, which generally support the earlier sugges!tions.

A MODEL

FOR ION BREAKDOWN

A model of the process is set up by considering that effective scission of two parts of an ion occurs at a separation (r) of ‘*d” of the respective centres of mass. In this modei, the parent ion would rotate as a rigid, near-spherical body of moment of inertia 1, for separations iniinitesimally less than d, and as two independent bodies for separations greater than d. The moments of inertia are then IA and Ia, as indicated in Fig. 1. The motion for all r > d is supposed to be governed by a centra1 attractive force corresponding to the potential f0

Fig_ l_ The simple model

for fI-agznentatios_

Here a is the mean polarisabiiity of the neutral .fragment and e the electronic charge_ We now suppose that the internal energy of the ion becomes so arranged that the separation d is approached with a kinetic energy of E present in the coordinates which correlate with unbound motion of A and B. This energy is regarded as a vector making an angle q3with the separation vector d. As the system passes discontinuously (in the model) into the separated =+ate there is a partition of angular momentum into an orbital component of the resultant motion and a recoil rotation of A and E; as indicated in the figure. Int. J.

Moss

Specrrom.

Ion

Phys., 6 (!971)

217433

BALLISTIC

MODEL

FOR

ION

219

BREAKDOWN

so that

(3)

IO = I,+pd' then it may readily be shown that initial radial ener,oy = E co&$

(4)

initial orbital energy = E sin2# x 1,/l,.

(9

The particles are now assumed to move in the central force field, and a condition may *befound which determines whether or not they will separate. The condition is essentially that which gives the “orbiting” condition in the collision theory of Langevin, and with the present model may be expressed by 2

(Fx‘;I,)

sin4 +--4g

V (

cosZ++

I 2 sin’+ 10

>

-t-4=0

for an ion which initially has no rotational motion_ The energy represented by V is discussed below. An ion which has initial rotation represented by a vector ER perpendicular to d and at an angle CLto the plane of d and E may be treated by defining

when the condition for separation becomes

v1_ (9 r iJ

r,

-4E V\

sd

&-+2

It 0 -1

ICI

fcos2~+Ln~~+~ Ic_

2

sin 4 cos xi-e2 3 - 0 I,

)

fsin~c”scr+sz

+4

=o.

I

L

(7)

Eqns. (6) and (7) yield values of E and # relevant to the critical condition for separation and hence, for a given E, define a range of #J values over which fragmen’ation of the icn will result. A typical example of this is shown in Fig. 2 for the case of H-atom Ioss from the ethane ion when the latter has initially no rotation. This fragmentation is used to iliustrate several points in this account. In the classical terms of the model adopted, one may now view a single excursion into extension of the C-H bond and consider the angle 9 to be determinable. Representing this event by a point on the E, 4 diagram, it may be seen whether or not it lies in the region for dissociation. If it does, it is convenient to associate with it a weighting factor which represents the relative density of orbital state: in tbe products of reaction. Once the particIes are assumed to enter a central ht.

I. Mass

Qectrom.

Ion Phys.,

6 (1971)

217-228

220

P. F. KNEWSTUBB

20

E v j

t5

I i I

t

f--_--2

a

EJECTION

ANGLE

@

so

Fig. 2. Variation of limiting angle for dissuciation versus reduced ener,T. Actual numerical data derived from the calculation described in the text; the cut-off at high angle is due to the “well” shown in Fig. 3.

field of force, &he angular momentum of orbital motion will be quantised and will lie in a piane perpendicular to the d vector. Thus one may consider that only m = 0 states are produced, and using the notation given above, the density of orbital states in the coordinate 4 may be given asi k)orb

[E,

Q)

=

ffR

=O=

4

where N,

=

j2=1 \fr’I 0

vo-q

under the assumption of high orbit& quantum numbers so that (I+$)’ e Z(Z+ i)The use of a weighting factor of cos Q wil! thus aliow for the effect of availability of orbitaf stares on the dissociation. A further assumption must be made about the distribution of bond extension events as a function of 4. The concept is implicitly retained that there are various stable but “active” states belonging to the three oscillators involved. These states can then receive an energy E* from some other oscillator to raise the total to a level t A preliminl~y account of the model to be found in hftxmces in Mzss Spectrometry, press) is in error on this point.

Vol. 5 Cm

BALLISTiC

MODEL

FOR

ION

221

BREAKDOWN

E at which dissociation may occur. Any one of these resulting conditicns will provide events covering a range of 4 values, which could be imagined as the result of a three-dimensional Lissajcus motion. If, in addition, it is supposed that several such states exist between energy values E and Et&Y, the only workable (and not unreasonable) assumption is that events occur uniformly in the Q variable. The mean effect of all the states in the band bE is then obtained simply and is expressed by a transmission coefficient

s 91.

h_cE, =

cos &d+

(10)

0

where 411 is the limiting value set by eqns. (6) or (7). Euch values of transmission coefficient will be obtained from eqn. (7) as a function also of &. If one assumes still the case of a near-spherical molecule, the general case of an initial rotational vector R randomly oriented with respect to d may be treated, the angle between these vectors being expressed as /3. The effect of the randomly oriented rotation is then expressed by

Values of transmission coefficient calculated on this basis are shown in Fig. 5 and are discussed later.

THE POTEETIAL

PARAMETER

Abii

REAL

MOLECULES

The ener_gy scale of the effects is determined by the value selected for the parameter V_ This factor is defined in the model analysis as the polarisation energy at the instant of separation, i.e.

In trying to apply this approach to real molecules, it is supposed that some potential energy curve for the elongation of the relevant bond can be drawn. It is of course true that this cannot be done precisely, since such a curve represents a section through a many-dimensional surface, but an average or most probable curve can be expected to show the general features familiar in diatomic species. The curve should tend towards r-4 behaviour at large distances, while departing severely from this at the equilibrium distance. There is a compromise invoived in selecting a separation beyond which the model may reasonably be applied_ On the one hand, a stretching of the bond decreases the disagreement between the actual and assumed potentiai functions and reduces the operation of angular restoring forces or directed valency effects. On the other hand the feature of the ballistic model is that the Int- J. Mass

Spectrom.

Icn Phys_, 6 (1971) 217-228

P. F. KNEWSTUBB TABLE

1

P-s

UsED IX =

CALCULATIOXS

Razcrion

OF

E?L#fPIEs

SN

FIG.

PO fential parameter

ficape lerei

Ce VI

(eW

5.

Moments of inertii7 X 1O-co g cm2 10

II

2.99

1.54

0.4

5.89

CzHa+ -P C2H3+ +H

2.92

0.4

33.78

30.8

C2HsC +-Ii

1.28

0.3

43.09

40.1

2.08

0.4

53.88

C%+

+ CH3+fH

C2H5+

+

GHs’

+ CH3+ +CH3

5.98

(CH&O+

+ CHsO*+CHS

I.00

0.3

91.9

34.2

(CHdsNf

+

(CH3)2Ni

4.48

- 0.4

127.3

63.6

+-CHI

Mean polarisabilitiess; H atom 0.665 x IO-** cm’; CH3 radical 1.88 x 10m2’ cm*. Well angk for each zase 0.7 radian.

initial rotation /

.-

L

-/

Fig. 3. Separation under polarisation forces. Elaborated model indicating an escape level at 0.25 V where Yis the potential parameter. The labelled &CL&U arcs are potential contours of the assumed central field. Tt. J. M&s Spectrom.

Ion P&s.,

B (1917)

217-228

BALLISTIC

MODEL

FOR

ION

BREAKDOWN

energy in the relevant degrees of freedom is assumed initially to be present as kinetic energy, implying commencement of the trajectory from the equilibrium position of the fragments with respect to each other. The

assumption is made that a 25 T/,stretch of the bond is a reasonable colm-

promise, and the actual binding energy at this point is taken as the experimentallydetermined difference of ionisation potential of the parent molecule and appearance potential for the fragmentation reaction, and so Y is given this value. In the trajectory analysis, V is then associated with a fictitious separation “a” in the purely tm4 potential such that

&C_

03)

The value of “a” so introduced is not, however, required in the final equations. A further elaboration of the model has been included, as shown in Fig. 3, by requiring that the separating portion of the ion should escape from the base of a well. This is defined by the “well angle” and “escape level” as indicated on the figure, and va!ues used in the calculation are indicated in Table 1 with other parameters of the calculation. The relative number of trajectories excluded by this modification is not large with the values as shown, and the correction may be applied simply by imposing an upper limit on #, as is seen in Fig. 2.

CALCULATIONS

OF R_4TE COEFFICIENTS

The calculation of rate coefficients for decomposition of excited ions has for some years been carried out by the procedure f%rstdescribed for isolated systems by Rosenstock, Wallenstein, Wahrhaftig be given as

k,,6E

=

1 -

h

and Eyring6. The formulation

1vN-l(E-En)

adopted may

W)

f-+W3

where kCE)is the &t-order rate coefllcient per unit energy band at a total internal is the number of vibrational states energy E, E, is the activation energy, NN-_l(E_E,_,)

of the N-l oscillators of the transition state in the energy range zero to E-E,, and PN(E) is the density of states of the N oscillators of the ion at the total energy E. By making some assumptions about the likely frequencies of vibration in the activated complex, values of k as a function of energy E can be produced. An example of the results of such a calculation is shown in Fig. 4 for the elimination

of a hydrogen atom from the C2H6+ ion. In&J.

Mass Specmom. Ion P&s.,

6 (1971) 217-228

224

P. F. KNEWSTUBB

I

I -0.05

I l-0

0 Infernal

I

20

energy - Appearance

i 3.0

P~tentkl (eV)

Fig. 4- Rate coefficient calculations for the fragmentation CzHe+ -+ C2H5’ fH. Curve --i cakuiated by one-dimensional eqn_ (14); cakuiated by three-dimensional eqn. (15). Cilwe - _. -. _ m expanded energy scale cakulated for initial rotation energy of 0.103 eV =_ 2.4 kT at 500 “K_

The formulation appropriate to the case of a three-dimensional reaction coordinate has been given consideration’ and may, by contrast, be set out as

where K(~~, is the transm>ssion coefEcient discussed above, NxcE3j is the total possible number of orbital states as given by eqn. (9), pN-3 signifies the density of states, or an equivalent function, for the remaining N-3 oscillators of the activated compIex. The asterisk notation signifies a convolution integral (see ref. 7). Calcuiations of rate coefficient on this basis are also shown in Fig. 4 for the same fragmentation process, both for the condition that the C&,’ ion has initially no rotational energy and for a randomly oriented initial rotation vector_ Further illustrations of the application of the model to the production of transmission coefficients for various cases are shown in Fig. 5. These values include the appropriate weighting factor for avaiiability of rotational states, and are given for the escape of H atom and of CH3 radical each from three different ions. The parent ions were chosen as having well-defined and near-spherical shapes. The effect of initiat rotation is indicated by one extra curve above the nominal appearance potential, and by a set of three curves, on an expanded energy scale, below the nominal appearance potential for each reaction. f.. J- Mass S~ctrom.

Ion Pi&s.,

6 (1971)

217-228

BALLISTiC

MODEL

FOR

ION

225

BREAKDOWN

Fig_ 5. Transmission coefficient calculations for emission of H atom from (a) CH,*,

(b) &Ha+,

and for e-mission of CH3 radical from (d) C,H,+, (e) (CH&O’, (f) (CH,),N+. Curves above threshold (positive energy scale) are for zero initial rotation (lower) and initial rotation energy 2.4 kT. Curves below threshold (expanded negative energy scale) are for initial rotation energies 1.2 kT, 2.4 kT, 3.6 kT (upperrr_ost). Temperature T set to 500c K. (4

CzHs+

DISCUSSION

1. Trarrsmission coefficients The transmission coefficient values shown in Fig. 5 are to be taken as illustrating the relative ease of escape of the neutral fragment under the conditions indicated. In view of the many approximations involved, they are obviously not to be regarded as of great accuracy, but they serve to illustrate two main features. The first is that the energy range over which the restrictions appear is related to, and in fact approximately equal to the value assigned to V(Table I, second column). Secondly, at the nominal threshold ener_gg,and below it, the effects of thermailyexcited rotation of rhe parent ion can be considerable. The scale of the effects is quite large in the case of the CH4 + ion with its low moment of inertia, as anticipated by Chupka* and by 0ttinger5. However, large effects also appear in Fig. 5 (d, e, f) due to the considerable difference between moment of inertia of the parent ht.

J. Mass

Spectrom.

lox Phys-, .6 (1971) 217-225

P. F. KNEWSTUBB

ion and the sum of these quantities for the two separated portions and hence the effective coupling of rotational energy into the reaction coordinate. Variation of the parameter f/ while keeping other parameters constant confirmed the behaviour above the appearance potential, as already stated- However, it also demonstrated that the energy range of the rotational lowering of threshold was not markedly affected b!r this. Thus the participation of initial rotational ener,gy becomes increasingly important as lower values of Y are introduced_ The rotational effect is indicated further for one example (c) in the calculation of rate coefficients. 2. Rate coeficients The calculation of rate coefficients for the se&ted example shows an interesting comparison between the two formulations which are tested. The results provide quantitative con&mation of trends suggested in ref. 7. At higher internal energies the rate coefficient caIcuIated acccrding to eqn. 14 rises steadily;while that given by eqn. 15 is Iower and rising much less rapidly. The trend of the latter curve is such that at higher energies a maximum might be indicated. By contrast, the reverse comparison is found at threshold, with the calculations according to eqn. 15 giving, at least for this example, the higher result. It is also seen that whereas the two curves which include no rotation effects fall sharply at the appearance potential, the effect of including a modest degree of rotation gives a marked extension of the curve below this ener,o and actually raises it slightly at all energ& The trend to lower thresholds continues if higher rotational energies are inserted into the calculation as is indicated by the curves of Fig_ 5. It will also be seen that of the examples shown in Fig. 5, that leading to Fig. 4 shows the least effect of rotation. It is notabIe that while experimental measurements indicate the presence of “metastable ions” in the mass spectrum of even such a small molecule as methaneg, calculations do not predict rate coefficients for fragmentation which are suitable for their appearance (see also refs. 5 and 8). 1 he !Ggh values of rate coefficient, as shown in Fig. 4, result from the iow values OFthe state density function ix small ions. The inclusion of rotational effects, as indicated here, goes some way towards solution of this apparent inconsistency. A furGrerstep towards agreement is likely to be made if and when a treatment of processes in the region of the threshold For fragmentation can be given in quantum-mechanical terms. A better understanding of these effects is obviously very desirable in any detaiIed study of breakdown diagrams of excited ions and of possible shifts of ap.zarance potential by rotational and kinetic effects, for the further rehnement of bond energy estimations.

ht.

3. &fnrs Specrrom.

Zon PI’ZJX.,6 (1971) 217-228

BALLISTIC

MODEL

FOR

ION

227

BREAKDOWN

ACKNOWLEDGEhlENT

author

The

wishes to acknowledge

him by the Director

gratefully

and Staff of the University

the time and facilities afforded

Computer

Laboratory.

APPEhJIX

Notes on the calculations Appropriate

frequencies

largely

taken from an account

mental

frequencies

as quoted

other

ate vg from

parameters

for the ethane ion were as given

in ref. 8, with the replacement

tics for the C,Hsi

TABLE

and

of a similar calculation

the calculations

and by taking

by lowering

averages

and Forst’.

by Hansen

of v4 by a free internal

ion (see Table 2), were derived

820 to 700 cm-l

for

by P&ii

were Funda-

and Dennison”, rotation.

Frequen-

the doubly

degener-

of appropriate

in-phase

and

2

F’ARA.METERS ASSUMED

FOR THE

C2H5’

ION AS TRANSlTION

COMPLEX

Freqwncies in cm-’

Doubly degenerate: 2975

2915

1390 993 2994 3188 I392 Constant

For eqn. (14) only for internal rotation

out-of-phase

group

8.06 cm-‘.

frequencies_

bending frequencies of 300 cm-’ simplicity,

1466 700 1180 300

treated as having

For calculations on the basis of eqn. (14), two were included.

the same rotational

The internal constant

rotation

was,

for

parent ion and complex

and complex and was accounted for by a singie convoIution routine. The rate coefficients were derived through enumeration of states of the complex and a summation routine for the range G-O5 eV above the threshold, and by a density function calculated by the Whitten and Kabinovitch’ 1 method for higher internai energies. Program,nes were writien in Titan Autocode.

REFERENCES 1 P. LAXGEVIN, Amt. Chim. Phys-, [S], 5 (1905) 245; see a& E_ W. MCDANIEL, in &l&ion Phenomena in Zonised Gores, Wiley, London, 1964, Appendix II. 2 G. GIOUMOUSISAXD D. F_ S~~nxsox J. Chem. Phys., 29 (1958) 294. hr. 3. Mass Specrrom. Ion PZzys., 6

1971) 2 17-228

22s

P. F_ KNEWSTUBB

3 E. W_ MCDAMEL, V. &RM.&K, A_ DALGARNO. E_ E. FERGUSONAXD L. FRIEDMAN,ionMolecsc~eReackms, Why-Lnterscienct, London, 1970, p_ 321, ff. 4 W- A_ CUUP~ J_ Chcm. Pkys., 45 (:968) 2337. 5 ti. -GER, 2. .i~atur/oncfr., % (1967) 20. 6 H. I& R osExwocK, M. 8. WALLEXSTEIN,-4. L. WM-lRHAFTIG AND H. EYRING. Pmt. Nar_ Acad. Sci. U.S., 38 (1952) 667. 7 P. F. KX~~YIXJBB, IFS. J. Mass Speczrom. Ion Phys., 6 (1971) 223. 8 Z. PR%IL am W_ Fom, J. Pkys. Chem_,?i (1967) 3!66. 9 3. H_ REYXOE;. R_ AM_ CAPRIOU, TV_E_ BA~TTNGER AND J. W_ AMY. Org. Mass Spectrom., 3 (1370) 479. 10 G. E- HAXSENAND D. hi. DENNISOX,J. Ckem. Pkys.. 20 (1952) 3 i3. 11 G. Z ‘#HXTiEX AXD B. S. Rmmovmx. J. Ckem. Pkys., 38 (1963) 2466. Iat_ J_ Mass Sjwcrrom. ion Pkys., 6 (I 97 1) 2 17-228