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Molecular and laboratory frame photofragment angular distributions from oriented and aligned molecules
Chemical Physics Letters 372 (2003) 187–194 www.elsevier.com/locate/cplett
Molecular and laboratory frame photofragment angular distributions from oriented and aligned molecules T. Peter Rakitzis b
a,*
, Alrik J. van den Brom b, Maurice H.M. Janssen
b
a Department of Physics, University of Crete and IESL-FORTH, 711 10 Heraklion-Crete, Greece Laser Centre and Department of Chemistry, Vrije Universiteit, de Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
Received 14 January 2003; in final form 5 March 2003
Abstract We present explicit expressions for the molecular frame and laboratory frame photofragment angular distributions from oriented parent molecules, in terms of the dynamically significant molecular frame angles between the recoil direction and the transition and permanent dipole moments of the molecule. We discuss how these angles can be measured from distinct experimental geometries. Explicit examples are given on the extracted information of the molecular frame photodissociation, especially in case of non-axial recoil dynamics. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The orientation or alignment of molecules in full and half collisions has been used to study the spatial aspects of these processes [1,2]. Molecular orientation has been achieved in strong electric fields by the Ôbrute forceÕ technique [1], involving the orientation of the molecular permanent dipole, d, and by molecular quantum-state selection, using hexapole focusing [3]. Molecular alignment has been achieved by quantum-state preparation by laser excitation [4], and temporary alignment and orientation has been predicted and achieved during intense laser pulses [5–9].
*
Corresponding author. Fax: +30-2810-391-305. E-mail address: [email protected] (T. Peter Rakitzis).
The theoretical and experimental studies of photofragment angular distributions have been used to understand the photodissociation process for many years [10]. The cornerstone of these studies is the well-known expression for the angular distribution of unpolarized photofragments in the laboratory frame [11] IðhÞ ¼
r ½1 þ bP2 ðcos hÞ; 4p
ð1Þ
where h is the angle between the photofragment velocity and the photolysis linear polarization direction and P2 ðcos hÞ the second order Legendre polynomial (the normalization factor r=4p will be omitted from all subsequent equations for convenience). This formula may seem very simple, however many details of the dissociation dynamics are encompassed into a single observable, the b parameter. The b parameter ranges from +2, for a
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00399-3
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T. Peter Rakitzis et al. / Chemical Physics Letters 372 (2003) 187–194
pure parallel dissociative transition, to )1, for a pure perpendicular dissociative transition, using the dipole approximation and in the axial recoil limit. An intermediate value of the b parameter (for example b ¼ þ1) can be caused by many factors [12]: (a) coherent or incoherent excitation of pure parallel and pure perpendicular dissociative transitions, (b) transition to a dissociative state that is neither parallel nor perpendicular in character, which occurs in polyatomic molecules only, and (c) breakdown of the axial recoil approximation due to an anisotropic potential energy surface of the dissociative state, or due to a dissociative life time larger or comparable to the rotational period (i.e., for non-prompt dissociation), or any combination of (a), (b), or (c). Measurements other than those directly related to Eq. (1) have been used to study the photodissociation dynamics in more detail. For example, the measurements of interference effects in photofragment angular momentum polarization investigate the coherent excitation of parallel and perpendicular transitions [13], as in case (a); the measurement of vector correlations of molecular fragments investigate the anisotropic forces during dissociation [14], and the lifetimes of non-prompt dissociation processes can be probed with ps and fs measurements, as in cases (b) and (c). It has been shown that the details of the dynamics of non-axial recoil dissociation can be studied from the photodissociation of oriented molecules. When the dissociation is rapid and in the absence of dynamical effects, the angular distribution is described by the Choi and Bernstein [15], and Zare expression [16] J 2 IðhÞ ¼ dKM ðhÞ ½1 þ bP2 ðcos hÞ; 2
ð2Þ
J where ½dKM ðhÞ represents the laboratory frame orientation distribution of a symmetric top like rotational state jJKMi. General treatments of the photofragment angular distributions have been presented for nonaxial recoil and arbitrary geometry of orientation field and polarization direction. It has been shown that, in principle, more details of the dynamics of dissociation can be studied from the photodissociation of oriented molecules [17]. Seideman
showed [18] how the exact quantum mechanical expressions for the angular distribution in the limit of a rapid dissociation reduce to the more phenomenologically derived expressions of [15,16]. A direct experimental verification of Eq. (2) was presented in a photodissociation experiment on methyl iodide [19] combining the ion-imaging technique [20] with hexapole state-selection and orientation. Using the technique of electric hexapole state-selection, it is possible to prepare beams of symmetric top molecules in essentially a singlestate, which can be oriented in space. With these hexapole state-selected parent beams, in combination with ion-imaging, one can also study details of the dissociation dynamics as reflected in the rotational state distribution of fragments and the effect of the initially selected quantum state of the parent molecule [11,21–23]. The aim of this paper is to derive and present detailed formulas of the photofragment angular distributions, for various experimental geometries, in terms of dynamically significant parameters of the molecular frame dissociation. We will formulate explicit expressions in terms of the angle v between the asymptotic recoil direction v and the transition dipole moment l and the angle a between v and the permanent dipole d of the molecule. In particular, we show that when the angle between the orientation field and the photolysis linear polarization is 45°, orientation in the angular distribution of the photofragments, perpendicular to the orientation field, occurs only for nonzero values of both v and a. This symmetry breaking in the photofragment angular distribution is an important experimental probe of non-axial recoil dynamics, and may be used to extract directly the molecular frame angles v and a. We will report such symmetry-breaking experiments elsewhere, for the photodissociation of oriented OCS.
2. Molecular frame photofragment angular distributions The distribution of the permanent dipole moment d about the orientation or alignment axis O (which is defined as the direction of the DC electric
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189
orientation field, or as the direction of the polarization of the laser that induces alignment) is cylindrically symmetric, and can be described by an expansion of Legendre polynomials, Pk ðcos hdO Þ [15] Dðcos hdO Þ ¼
N 1 X þ ck Pk ðcos hdO Þ; 2 k¼1
ð3Þ
where hdO is the angle between d and O. Notice that the distribution Dðcos hdO Þ is normalized so that integration over cos hdO from )1 to 1 yields unity. The coefficients ck are dependent on the orientation technique and the particular rotational states selected. In the case of a symmetric top like molecule in state jJKMi ¼ j111i; c1 ¼ 3=4 and c2 ¼ 1=4, and in general k 6 2J. We will first derive the molecular frame angular distribution, using a simple model that calculates the probability of photodissociation of a particular initial molecular geometry by multiplying the initial parent molecule orientation distribution by the probability of photon absorption. Let us consider a photofragment with velocity v, which defines the molecular z-axis. We will now define the directions of the polarization of the dissociating electric field, e, of the DC electric orientation field, O, of the permanent dipole moment, d, and of the transition dipole moment, l, with respect to the molecular coordinate system, where the (x; z)-plane is defined by the (v, e)-plane. The spherical polar angles about v are given by ðc; 0Þ for e, ðd; uOe Þ for O, (a; ude ) for d, and (v; uld þ ude ) for l (see Fig. 1). The probability of producing molecules with velocity v, i.e., the molecular frame angular distribution Iðe; OÞ, is determined by the direction of the transition dipole l and the permanent dipole d at the time of excitation by the photolysis light. This angular distribution, Iðc; d; uOe Þ (which depends explicitly on the angles of e and O about v), is given by the product of the parent molecule bond distribution, Dðcos hdO Þ, with the probability 2 of photon absorption, jl ej , integrated over all possible molecular geometries that yield photofragments with velocity v 1 Iðc; d; uOe Þ ¼ 2p
Z 0
Fig. 1. Molecular frame polar coordinates of the various vectorial quantities. The z-axis is chosen along the direction of the velocity v. The (x; z)-plane is defined by the plane of the polarization of the photolysis laser and the velocity v.
where ude is the azimuthal angle between d and e, which, when increased between 0 and 2p, sweeps d and l about v, through all possible molecular geometries. The factor of 6 is added for convenience and normalization of subsequent equations. Note that the derivation that proceeds from Eq. (4) assumes that the initial directions of d and l about v are fixed (thus defining the molecular frame), and that they do not follow a distribution in the molecular frame. This approach was used before by Taatjes et al. [17]. Substituting Eq. (3) into Eq. (4), and expressing jl ej2 as cos2 hle , and then in terms of P2 ðcos hle Þ, gives # Z 2p " N X 1 Iðc; d; uOe Þ ¼ 1þ2 ck Pk ðcos hdO Þ 2p 0 1
1 þ 2P2 ðcos hle Þ dude : ð5Þ The Legendre polynomials of cos hdO and cos hle can be expressed in terms of molecular frame angles, using the spherical harmonic addition theorem, by k X k k Pk ðcos hdO Þ ¼ CM ðdÞCM ðaÞ cos Mðude uOe Þ; M¼ k
ð6aÞ P2 ðcos hle Þ ¼
2p 2
6Dðcos hdO Þjl ej dude ;
ð4Þ
2 X
2 2 CM ðcÞCM ðvÞ cos Mðude þ uld Þ;
M¼ 2
ð6bÞ
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k where the CM ðhÞ are equivalent to the modified spherical harmonics with u ¼ 0, given by ½4p= 1=2 ð2k þ 1Þ YMk ðh; 0Þ. Substitution of Eqs. (6a) and (6b) into Eq. (5), and integration over ude yields the molecular frame angular distribution:
Iðc; d; uOe Þ ¼ ½1 þ 2P2 ðcos vÞP2 ðcos cÞ # " N X k k ck C0 ðaÞC0 ðdÞ
1þ2
3. Laboratory frame photofragment angular distributions
k¼1
þ 12 sin c cos c sin v
cos v cosðuOe þ uld Þ
N X
ck C1k ðaÞC1k ðdÞ
k¼1
þ 3 sin2 c sin2 v cos 2ðuOe þ uld Þ N X ck C2k ðaÞC2k ðdÞ:
Eq. (8), however generalization to larger N using Eq. (7), which is necessary for the description of intense laser-pulse bond alignment, is straightforward.
ð7Þ
k¼1
Equation (7) is a key result of this Letter, as it can be used with straightforward procedures, described below, to calculate the laboratory frame photofragment angular distributions for arbitrary laser-polarization and orientation-field geometries. Notice that for the special case of a ¼ 0 (such as for axial recoil of many types of symmetric molecules), Eq. (7) reduces to the same form as Eq. (2), i.e., the angular distribution from the photodissociation of isotropic molecules multiplied by the parent molecule bond distribution. For the description of ÔDC brute forceÕ orientation the expansion in Eq. (3) is usually terminated at N ¼ 2 [1] (which also suffices for orientation of state-selected jJKMi with J ¼ 1), for which Eq. (7) reduces to
The distribution of the photolysis polarization e, the orientation field O, and the photofragment velocity v about the laboratory Z-axis is shown in Fig. 2. The Z-axis is chosen to be parallel to the symmetry axis of the experiment (i.e., parallel to the Doppler, or time-of-flight direction, or perpendicular to the velocity map imaging plane of the detector). The laboratory ðX ; ZÞ-plane is defined by the (e; Z)-plane. The spherical polar angles about the laboratory Z-axis are given by (C; 0) for e, ðD; UÞ for O, and ðX; HÞ for v. Note that H is the azimuthal angle of v about the Z-axis; in later discussions of slice imaging, this angle will correspond to the polar angle of a 2D-cut when X ¼ 90°. The molecular frame angles c; d, and uvO , can be expressed in terms of the laboratory frame angles by:
Iðc; d; uOe Þ ¼ ½1 þ 2P2 ðcos vÞP2 ðcos cÞ
½1 þ 2c1 cos a cos d þ 2c2 P2 ðcos aÞP2 ðcos dÞ þ 6 sin c cos c sin v cos v sin a sin d
cosðuOe þ uld Þ½c1 þ 3c2 cos a cos d 9 þ c2 sin2 c sin2 v sin2 a sin2 d 8
cos 2ðuOe þ uld Þ:
ð8Þ
For the sake of brevity, the description of laboratory-frame distributions will be derived using
Fig. 2. Laboratory frame coordinate system with the Z-axis along the normal of the detector plane. The plane defined by the polarization e and the Z-axis defines the XZ-plane.
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cos c ¼ cos X cos C þ sin X sin C cos H;
ð9aÞ
cos d ¼ cos X cos D þ sin X sin D cosðH UÞ; ð9bÞ cos uOe ¼ fsin2 X cos C cos D þ sin C sin D cos U sin X cos X½sin D cos C cosðU HÞ þ sin C cos D cos H sin2 X sin C sin D
cosðU HÞ cos Hg=ðsin c sin dÞ;
ð9cÞ
3.2. 2D imaging Velocity-map ion images are the 2D projections of the full 3D photofragment distribution. For a monoenergetic recoil distribution of photofragments forming a normalized 3D sphere with unit radius, the intensity of the 2D projection, I2D ðC; D; U; r; HÞ, is described as a function of the 2D polar coordinates r and H. The 2D projection of the 3D distribution superimposes points with coordinates (X; H) and (p-X; H) I2D ðC; D; U; r; HÞ
sin uOe ¼ fcos X sin C sin D sin U
¼ ½I3D ðC; D; U; X; HÞ
sin X½sin D cos C sinðU HÞ þ sin C cos D sin Hg=ðsin c sin dÞ;
191
ð9dÞ
Substitution of Eqs. (9a)–(9d) into Eq. (7) yields the complete 3D laboratory frame photofragment angular distribution for arbitrary laboratory experimental geometries, I3D ðC; D; U; X; HÞ. This 3D expression, I3D ðC; D; U; X; HÞ, is large and cumbersome to express fully, however experiments almost always measure a 1D projection (such as time-of-flight of Doppler profiles), a 2D projection (such as ion imaging), a 1D sample (such as coresampling), or a 2D slice (such as slice imaging) of the 3D distribution. Therefore, we find that it is more convenient to calculate and express the angular distributions for specific experimental geometries and setups. 3.1. Doppler profiles The Doppler profile 1D expression (or time-offlight profile with no Doppler selectivity) can be obtained straightforwardly from the complete I3D ðC; D; U; X; HÞ distribution by integrating over the azimuthal angle H of v about the detection axis: Z 2p 1 I1D ðC; D; U; xD Þ ¼ I3D ðC; D; U; X; HÞ dH; 2p 0 ð10Þ where the Doppler shift xD is equal to cos X. Expressions for the idealized cases of core-sampling can be obtained by setting X ¼ 0 and 180° in I3D ðC; D; U; X; HÞ.
pffiffiffiffiffiffiffiffiffiffiffiffi þ I3D ðC; D; U; p X; HÞ=2 1 r2 ; ð11Þ pffiffiffiffiffiffiffiffiffiffiffiffi where the factor 1= 1 r2 is the Jacobian of the projection. After the application ofpEq. (11), the ffiffiffiffiffiffiffiffiffiffiffiffi substitutions sin X ¼ r and cos X ¼ 1 r2 can be made so that the 2D image is described radially by r, which ranges from 0 to 1. 3.3. Slice imaging The experimental technique of slice imaging [24] measures the 2D center-slice of the 3D distribution. Such a 2D cut, which is perpendicular to the laboratory Z-axis, is described in our formalism by X ¼ 90°, and H now becomes the 2D polar angle (see Fig. 2). For the 2D ion-imaging technique, the angular distribution of the center slice corresponds to the angular distribution of the outer edge of a 2D image. For simplicity, we will consider molecules for which uld ¼ 0 or 90°, i.e., l is either parallel or perpendicular to the (v,d)-plane. Henceforth, we will present explicit equations for which the upper sign of +/) or )/+ will correspond to uld ¼ 0, whereas the lower sign will correspond to uld ¼ 90°. This constraint applies, but is not limited, to planar molecules (including diatomic and triatomic molecules). For the orientation field parallel to the Z-axis (given by D ¼ 0), for the photolysis polarization parallel to the X-axis (given by C ¼ 90°), setting X ¼ 90°, and using Eqs. (8),(9a)–(9d), the photofragment angular distribution about the photolysis polarization, in the XY-plane, is given by
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IðHÞ ¼ 1 þ
2P2 ðcosvÞ 34c2 sin2 vsin2 a=½1 þ 2c2 P2 ðcosaÞ
!
1 34c2 sin2 vsin2 a=½1 þ 2c2 P2 ðcosaÞ
P2 ðcosHÞ:
ð12Þ
Notice that for a ¼ 0, Eq. (12) reduces to the form of Eq. (1), so that the coefficient of P2 ðcos HÞ is +2 for a parallel transition (v ¼ 0), and )1 for a perpendicular transition (v ¼ 90°). As discussed by van den Brom et al. [22], this geometry can be used to measure the spatial anisotropy parameter b directly for strongly oriented molecules, while effectively ignoring the degree of parent orientation. This geometry decouples the spatial anisotropy measurement from the initial parent orientation, as long as all excited states involved in the photodissociation are accessed via either approximately parallel (v 0) or perpendicular (v 90°) transitions. For the geometry for which the electric orientation field and the photolysis linear polarization are both perpendicular to the Z-axis (and parallel to the X-axis, described by C ¼ 90°; D ¼ 90°, and U ¼ 0), the photofragment angular distribution can be calculated as before, using Eqs. (8), (9a)– (9d), and is described by a Legendre polynomial expansion terminated at the fourth order: IðHÞ ¼ 1 þ b1 P1 ðcos HÞ þ b2 P2 ðcos HÞ þ b3 P3 ðcos HÞ þ b4 P4 ðcos HÞ:
ð13Þ
The coefficients bi are given by bi =b0 , where the bi are expressed by:
b1 ¼
c1 ½ cos að5 þ 4P2 ðcos vÞÞ þ 6 sin v cos v sin a; 5 ð14bÞ
b2 ¼ 2P2 ðcos vÞ þ 2c2 P2 ðcos aÞ 8c2 P2 ðcos vÞP2 ðcos aÞ þ 7
3 3 þ sin v cos v sin a cos a sin2 v sin2 a ; 2 4 ð14cÞ b3 ¼
6c1 ½2P2 ðcos vÞ cos a 3 sin v cos v sin a; 5 ð14dÞ
b4 ¼
72c2 P2 ðcos vÞP2 ðcos aÞ 35 2 sin v cos v sin a cos a
ð14eÞ The four experimental observables, bi , yield four independent equations, which allow the overdetermination of the angles v and a. A particularly interesting geometry is one for which the orientation field is parallel to the Zaxis (D ¼ 0), and the photolysis polarization is at 45° to the Z-axis (C ¼ 45°). This geometry is convenient for ion-imaging experiments, as the extraction field (which also acts as the orientation field) is parallel to the Z axis. Calculation of the laboratory frame angular distribution then yields:
! 3c1 sin v cos v sin a IðHÞ ¼ 1 þ P1 ðcos HÞ ½1 þ 2c2 P2 ðcos aÞ 1 12 P2 ðcos vÞ 163 c2 sin2 v sin2 a ! ½1 þ 2c2 P2 ðcos aÞP2 ðcos vÞ 38 c2 sin2 v sin2 a þ P2 ðcos HÞ: ½1 þ 2c2 P2 ðcos aÞ 1 12 P2 ðcos vÞ 163 c2 sin2 v sin2 a
4c2 P2 ðcos vÞP2 ðcos aÞ 5 3 þ 3 sin v cos v sin a cos a sin2 v sin2 a ; 4
b0 ¼ 1 þ
ð14aÞ
1 sin2 v sin2 a : 8
ð15Þ
Notice that the initial parent molecule orientation is produced parallel to the Z-axis, whereas photofragment orientation (described by the P1 ðcos HÞ term) occurs parallel to the X-axis, perpendicular to the original orientation direction. This orientation term vanishes for either v ¼ 0 or
T. Peter Rakitzis et al. / Chemical Physics Letters 372 (2003) 187–194
90°, or a ¼ 0 (which is the case for axial recoil dissociation of many symmetric molecules). Fig. 3a shows the variation of the orientation term on the angles v and a. In contrast, Fig. 3b shows that the b2 term is largely insensitive to the value of a, and can thus be used to measure the value of v directly. The b1 term can then be used to determine the value of a. We conclude that the observation of orientation in this geometry is a sensitive probe of non-axial recoil photodissociation, allowing the measurement of the two dynamical angles v and a. We will report such orientation measurements
193
elsewhere, for the photodissociation of OCS at 230 nm.
Acknowledgements This research has been financially supported by the councils for Chemical Sciences and Physical Sciences of the Dutch Organization for Scientific Research (NWO-CW, NWO-FOM). The authors would like to thank Prof. S. Stolte for support. TPR thanks the EU for support to access the experimental facilities of the Laser Centre Vrije Universiteit through the EU programme HPRICT-1999-00064. MHMJ thanks the CommunityÕs Human Potential Programme under Contract HPRN-CT-1999-00129 (COCOMO) for support at FORTH. AJvdB thanks the EU for support as a Marie Curie Training Institute fellow at FORTH under Contract HPMT-CT-2000-00201.
References
Fig. 3. The dependence on the angle a of the degree of (a) photofragment orientation, b1 , and (b) photofragment alignment, b2 , in the plane perpendicular to the initial parent molecule orientation, for the geometry described by Eq. (12) (for C ¼ 45° and D ¼ 0), for various values of the angle v.
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[18] T. Seideman, Chem. Phys. Lett. 253 (1996) 279. [19] M.H.M. Janssen, J.W.G. Mastenbroek, S. Stolte, J. Phys. Chem. A 101 (1997) 7605. [20] D.W. Chandler, P.L. Houston, J. Chem. Phys. 87 (1987) 1445. [21] J.M. Teule, G.C. Groenenboom, D.W. Neyer, D.W. Chandler, M.H.M. Janssen, Chem. Phys. Lett. 320 (2000) 177.
[22] A.J. van den Brom, T.P. Rakitzis, J. van Heyst, T.N. Kitsopoulos, S.R. Jezowski, M.H.M. Janssen, J. Chem. Phys. 117 (2002) 4255. [23] A.J. van den Brom, M.L. Lipciuc, M.H.M. Janssen, Chem. Phys. Lett. 368 (2003) 324. [24] C.R. Gebhardt, T.P. Rakitzis, P.C. Samartzis, V. Ladopoulos, T.N. Kitsopoulos, Rev. Sci. Instrum. 72 (2001) 3848.
Molecular and laboratory frame photofragment angular distributions from oriented and aligned molecules T. Peter Rakitzis b
a,*
, Alrik J. van den Brom b, Maurice H.M. Janssen
b
a Department of Physics, University of Crete and IESL-FORTH, 711 10 Heraklion-Crete, Greece Laser Centre and Department of Chemistry, Vrije Universiteit, de Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
Received 14 January 2003; in final form 5 March 2003
Abstract We present explicit expressions for the molecular frame and laboratory frame photofragment angular distributions from oriented parent molecules, in terms of the dynamically significant molecular frame angles between the recoil direction and the transition and permanent dipole moments of the molecule. We discuss how these angles can be measured from distinct experimental geometries. Explicit examples are given on the extracted information of the molecular frame photodissociation, especially in case of non-axial recoil dynamics. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The orientation or alignment of molecules in full and half collisions has been used to study the spatial aspects of these processes [1,2]. Molecular orientation has been achieved in strong electric fields by the Ôbrute forceÕ technique [1], involving the orientation of the molecular permanent dipole, d, and by molecular quantum-state selection, using hexapole focusing [3]. Molecular alignment has been achieved by quantum-state preparation by laser excitation [4], and temporary alignment and orientation has been predicted and achieved during intense laser pulses [5–9].
*
Corresponding author. Fax: +30-2810-391-305. E-mail address: [email protected] (T. Peter Rakitzis).
The theoretical and experimental studies of photofragment angular distributions have been used to understand the photodissociation process for many years [10]. The cornerstone of these studies is the well-known expression for the angular distribution of unpolarized photofragments in the laboratory frame [11] IðhÞ ¼
r ½1 þ bP2 ðcos hÞ; 4p
ð1Þ
where h is the angle between the photofragment velocity and the photolysis linear polarization direction and P2 ðcos hÞ the second order Legendre polynomial (the normalization factor r=4p will be omitted from all subsequent equations for convenience). This formula may seem very simple, however many details of the dissociation dynamics are encompassed into a single observable, the b parameter. The b parameter ranges from +2, for a
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00399-3
188
T. Peter Rakitzis et al. / Chemical Physics Letters 372 (2003) 187–194
pure parallel dissociative transition, to )1, for a pure perpendicular dissociative transition, using the dipole approximation and in the axial recoil limit. An intermediate value of the b parameter (for example b ¼ þ1) can be caused by many factors [12]: (a) coherent or incoherent excitation of pure parallel and pure perpendicular dissociative transitions, (b) transition to a dissociative state that is neither parallel nor perpendicular in character, which occurs in polyatomic molecules only, and (c) breakdown of the axial recoil approximation due to an anisotropic potential energy surface of the dissociative state, or due to a dissociative life time larger or comparable to the rotational period (i.e., for non-prompt dissociation), or any combination of (a), (b), or (c). Measurements other than those directly related to Eq. (1) have been used to study the photodissociation dynamics in more detail. For example, the measurements of interference effects in photofragment angular momentum polarization investigate the coherent excitation of parallel and perpendicular transitions [13], as in case (a); the measurement of vector correlations of molecular fragments investigate the anisotropic forces during dissociation [14], and the lifetimes of non-prompt dissociation processes can be probed with ps and fs measurements, as in cases (b) and (c). It has been shown that the details of the dynamics of non-axial recoil dissociation can be studied from the photodissociation of oriented molecules. When the dissociation is rapid and in the absence of dynamical effects, the angular distribution is described by the Choi and Bernstein [15], and Zare expression [16] J 2 IðhÞ ¼ dKM ðhÞ ½1 þ bP2 ðcos hÞ; 2
ð2Þ
J where ½dKM ðhÞ represents the laboratory frame orientation distribution of a symmetric top like rotational state jJKMi. General treatments of the photofragment angular distributions have been presented for nonaxial recoil and arbitrary geometry of orientation field and polarization direction. It has been shown that, in principle, more details of the dynamics of dissociation can be studied from the photodissociation of oriented molecules [17]. Seideman
showed [18] how the exact quantum mechanical expressions for the angular distribution in the limit of a rapid dissociation reduce to the more phenomenologically derived expressions of [15,16]. A direct experimental verification of Eq. (2) was presented in a photodissociation experiment on methyl iodide [19] combining the ion-imaging technique [20] with hexapole state-selection and orientation. Using the technique of electric hexapole state-selection, it is possible to prepare beams of symmetric top molecules in essentially a singlestate, which can be oriented in space. With these hexapole state-selected parent beams, in combination with ion-imaging, one can also study details of the dissociation dynamics as reflected in the rotational state distribution of fragments and the effect of the initially selected quantum state of the parent molecule [11,21–23]. The aim of this paper is to derive and present detailed formulas of the photofragment angular distributions, for various experimental geometries, in terms of dynamically significant parameters of the molecular frame dissociation. We will formulate explicit expressions in terms of the angle v between the asymptotic recoil direction v and the transition dipole moment l and the angle a between v and the permanent dipole d of the molecule. In particular, we show that when the angle between the orientation field and the photolysis linear polarization is 45°, orientation in the angular distribution of the photofragments, perpendicular to the orientation field, occurs only for nonzero values of both v and a. This symmetry breaking in the photofragment angular distribution is an important experimental probe of non-axial recoil dynamics, and may be used to extract directly the molecular frame angles v and a. We will report such symmetry-breaking experiments elsewhere, for the photodissociation of oriented OCS.
2. Molecular frame photofragment angular distributions The distribution of the permanent dipole moment d about the orientation or alignment axis O (which is defined as the direction of the DC electric
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orientation field, or as the direction of the polarization of the laser that induces alignment) is cylindrically symmetric, and can be described by an expansion of Legendre polynomials, Pk ðcos hdO Þ [15] Dðcos hdO Þ ¼
N 1 X þ ck Pk ðcos hdO Þ; 2 k¼1
ð3Þ
where hdO is the angle between d and O. Notice that the distribution Dðcos hdO Þ is normalized so that integration over cos hdO from )1 to 1 yields unity. The coefficients ck are dependent on the orientation technique and the particular rotational states selected. In the case of a symmetric top like molecule in state jJKMi ¼ j111i; c1 ¼ 3=4 and c2 ¼ 1=4, and in general k 6 2J. We will first derive the molecular frame angular distribution, using a simple model that calculates the probability of photodissociation of a particular initial molecular geometry by multiplying the initial parent molecule orientation distribution by the probability of photon absorption. Let us consider a photofragment with velocity v, which defines the molecular z-axis. We will now define the directions of the polarization of the dissociating electric field, e, of the DC electric orientation field, O, of the permanent dipole moment, d, and of the transition dipole moment, l, with respect to the molecular coordinate system, where the (x; z)-plane is defined by the (v, e)-plane. The spherical polar angles about v are given by ðc; 0Þ for e, ðd; uOe Þ for O, (a; ude ) for d, and (v; uld þ ude ) for l (see Fig. 1). The probability of producing molecules with velocity v, i.e., the molecular frame angular distribution Iðe; OÞ, is determined by the direction of the transition dipole l and the permanent dipole d at the time of excitation by the photolysis light. This angular distribution, Iðc; d; uOe Þ (which depends explicitly on the angles of e and O about v), is given by the product of the parent molecule bond distribution, Dðcos hdO Þ, with the probability 2 of photon absorption, jl ej , integrated over all possible molecular geometries that yield photofragments with velocity v 1 Iðc; d; uOe Þ ¼ 2p
Z 0
Fig. 1. Molecular frame polar coordinates of the various vectorial quantities. The z-axis is chosen along the direction of the velocity v. The (x; z)-plane is defined by the plane of the polarization of the photolysis laser and the velocity v.
where ude is the azimuthal angle between d and e, which, when increased between 0 and 2p, sweeps d and l about v, through all possible molecular geometries. The factor of 6 is added for convenience and normalization of subsequent equations. Note that the derivation that proceeds from Eq. (4) assumes that the initial directions of d and l about v are fixed (thus defining the molecular frame), and that they do not follow a distribution in the molecular frame. This approach was used before by Taatjes et al. [17]. Substituting Eq. (3) into Eq. (4), and expressing jl ej2 as cos2 hle , and then in terms of P2 ðcos hle Þ, gives # Z 2p " N X 1 Iðc; d; uOe Þ ¼ 1þ2 ck Pk ðcos hdO Þ 2p 0 1
1 þ 2P2 ðcos hle Þ dude : ð5Þ The Legendre polynomials of cos hdO and cos hle can be expressed in terms of molecular frame angles, using the spherical harmonic addition theorem, by k X k k Pk ðcos hdO Þ ¼ CM ðdÞCM ðaÞ cos Mðude uOe Þ; M¼ k
ð6aÞ P2 ðcos hle Þ ¼
2p 2
6Dðcos hdO Þjl ej dude ;
ð4Þ
2 X
2 2 CM ðcÞCM ðvÞ cos Mðude þ uld Þ;
M¼ 2
ð6bÞ
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k where the CM ðhÞ are equivalent to the modified spherical harmonics with u ¼ 0, given by ½4p= 1=2 ð2k þ 1Þ YMk ðh; 0Þ. Substitution of Eqs. (6a) and (6b) into Eq. (5), and integration over ude yields the molecular frame angular distribution:
Iðc; d; uOe Þ ¼ ½1 þ 2P2 ðcos vÞP2 ðcos cÞ # " N X k k ck C0 ðaÞC0 ðdÞ
1þ2
3. Laboratory frame photofragment angular distributions
k¼1
þ 12 sin c cos c sin v
cos v cosðuOe þ uld Þ
N X
ck C1k ðaÞC1k ðdÞ
k¼1
þ 3 sin2 c sin2 v cos 2ðuOe þ uld Þ N X ck C2k ðaÞC2k ðdÞ:
Eq. (8), however generalization to larger N using Eq. (7), which is necessary for the description of intense laser-pulse bond alignment, is straightforward.
ð7Þ
k¼1
Equation (7) is a key result of this Letter, as it can be used with straightforward procedures, described below, to calculate the laboratory frame photofragment angular distributions for arbitrary laser-polarization and orientation-field geometries. Notice that for the special case of a ¼ 0 (such as for axial recoil of many types of symmetric molecules), Eq. (7) reduces to the same form as Eq. (2), i.e., the angular distribution from the photodissociation of isotropic molecules multiplied by the parent molecule bond distribution. For the description of ÔDC brute forceÕ orientation the expansion in Eq. (3) is usually terminated at N ¼ 2 [1] (which also suffices for orientation of state-selected jJKMi with J ¼ 1), for which Eq. (7) reduces to
The distribution of the photolysis polarization e, the orientation field O, and the photofragment velocity v about the laboratory Z-axis is shown in Fig. 2. The Z-axis is chosen to be parallel to the symmetry axis of the experiment (i.e., parallel to the Doppler, or time-of-flight direction, or perpendicular to the velocity map imaging plane of the detector). The laboratory ðX ; ZÞ-plane is defined by the (e; Z)-plane. The spherical polar angles about the laboratory Z-axis are given by (C; 0) for e, ðD; UÞ for O, and ðX; HÞ for v. Note that H is the azimuthal angle of v about the Z-axis; in later discussions of slice imaging, this angle will correspond to the polar angle of a 2D-cut when X ¼ 90°. The molecular frame angles c; d, and uvO , can be expressed in terms of the laboratory frame angles by:
Iðc; d; uOe Þ ¼ ½1 þ 2P2 ðcos vÞP2 ðcos cÞ
½1 þ 2c1 cos a cos d þ 2c2 P2 ðcos aÞP2 ðcos dÞ þ 6 sin c cos c sin v cos v sin a sin d
cosðuOe þ uld Þ½c1 þ 3c2 cos a cos d 9 þ c2 sin2 c sin2 v sin2 a sin2 d 8
cos 2ðuOe þ uld Þ:
ð8Þ
For the sake of brevity, the description of laboratory-frame distributions will be derived using
Fig. 2. Laboratory frame coordinate system with the Z-axis along the normal of the detector plane. The plane defined by the polarization e and the Z-axis defines the XZ-plane.
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cos c ¼ cos X cos C þ sin X sin C cos H;
ð9aÞ
cos d ¼ cos X cos D þ sin X sin D cosðH UÞ; ð9bÞ cos uOe ¼ fsin2 X cos C cos D þ sin C sin D cos U sin X cos X½sin D cos C cosðU HÞ þ sin C cos D cos H sin2 X sin C sin D
cosðU HÞ cos Hg=ðsin c sin dÞ;
ð9cÞ
3.2. 2D imaging Velocity-map ion images are the 2D projections of the full 3D photofragment distribution. For a monoenergetic recoil distribution of photofragments forming a normalized 3D sphere with unit radius, the intensity of the 2D projection, I2D ðC; D; U; r; HÞ, is described as a function of the 2D polar coordinates r and H. The 2D projection of the 3D distribution superimposes points with coordinates (X; H) and (p-X; H) I2D ðC; D; U; r; HÞ
sin uOe ¼ fcos X sin C sin D sin U
¼ ½I3D ðC; D; U; X; HÞ
sin X½sin D cos C sinðU HÞ þ sin C cos D sin Hg=ðsin c sin dÞ;
191
ð9dÞ
Substitution of Eqs. (9a)–(9d) into Eq. (7) yields the complete 3D laboratory frame photofragment angular distribution for arbitrary laboratory experimental geometries, I3D ðC; D; U; X; HÞ. This 3D expression, I3D ðC; D; U; X; HÞ, is large and cumbersome to express fully, however experiments almost always measure a 1D projection (such as time-of-flight of Doppler profiles), a 2D projection (such as ion imaging), a 1D sample (such as coresampling), or a 2D slice (such as slice imaging) of the 3D distribution. Therefore, we find that it is more convenient to calculate and express the angular distributions for specific experimental geometries and setups. 3.1. Doppler profiles The Doppler profile 1D expression (or time-offlight profile with no Doppler selectivity) can be obtained straightforwardly from the complete I3D ðC; D; U; X; HÞ distribution by integrating over the azimuthal angle H of v about the detection axis: Z 2p 1 I1D ðC; D; U; xD Þ ¼ I3D ðC; D; U; X; HÞ dH; 2p 0 ð10Þ where the Doppler shift xD is equal to cos X. Expressions for the idealized cases of core-sampling can be obtained by setting X ¼ 0 and 180° in I3D ðC; D; U; X; HÞ.
pffiffiffiffiffiffiffiffiffiffiffiffi þ I3D ðC; D; U; p X; HÞ=2 1 r2 ; ð11Þ pffiffiffiffiffiffiffiffiffiffiffiffi where the factor 1= 1 r2 is the Jacobian of the projection. After the application ofpEq. (11), the ffiffiffiffiffiffiffiffiffiffiffiffi substitutions sin X ¼ r and cos X ¼ 1 r2 can be made so that the 2D image is described radially by r, which ranges from 0 to 1. 3.3. Slice imaging The experimental technique of slice imaging [24] measures the 2D center-slice of the 3D distribution. Such a 2D cut, which is perpendicular to the laboratory Z-axis, is described in our formalism by X ¼ 90°, and H now becomes the 2D polar angle (see Fig. 2). For the 2D ion-imaging technique, the angular distribution of the center slice corresponds to the angular distribution of the outer edge of a 2D image. For simplicity, we will consider molecules for which uld ¼ 0 or 90°, i.e., l is either parallel or perpendicular to the (v,d)-plane. Henceforth, we will present explicit equations for which the upper sign of +/) or )/+ will correspond to uld ¼ 0, whereas the lower sign will correspond to uld ¼ 90°. This constraint applies, but is not limited, to planar molecules (including diatomic and triatomic molecules). For the orientation field parallel to the Z-axis (given by D ¼ 0), for the photolysis polarization parallel to the X-axis (given by C ¼ 90°), setting X ¼ 90°, and using Eqs. (8),(9a)–(9d), the photofragment angular distribution about the photolysis polarization, in the XY-plane, is given by
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IðHÞ ¼ 1 þ
2P2 ðcosvÞ 34c2 sin2 vsin2 a=½1 þ 2c2 P2 ðcosaÞ
!
1 34c2 sin2 vsin2 a=½1 þ 2c2 P2 ðcosaÞ
P2 ðcosHÞ:
ð12Þ
Notice that for a ¼ 0, Eq. (12) reduces to the form of Eq. (1), so that the coefficient of P2 ðcos HÞ is +2 for a parallel transition (v ¼ 0), and )1 for a perpendicular transition (v ¼ 90°). As discussed by van den Brom et al. [22], this geometry can be used to measure the spatial anisotropy parameter b directly for strongly oriented molecules, while effectively ignoring the degree of parent orientation. This geometry decouples the spatial anisotropy measurement from the initial parent orientation, as long as all excited states involved in the photodissociation are accessed via either approximately parallel (v 0) or perpendicular (v 90°) transitions. For the geometry for which the electric orientation field and the photolysis linear polarization are both perpendicular to the Z-axis (and parallel to the X-axis, described by C ¼ 90°; D ¼ 90°, and U ¼ 0), the photofragment angular distribution can be calculated as before, using Eqs. (8), (9a)– (9d), and is described by a Legendre polynomial expansion terminated at the fourth order: IðHÞ ¼ 1 þ b1 P1 ðcos HÞ þ b2 P2 ðcos HÞ þ b3 P3 ðcos HÞ þ b4 P4 ðcos HÞ:
ð13Þ
The coefficients bi are given by bi =b0 , where the bi are expressed by:
b1 ¼
c1 ½ cos að5 þ 4P2 ðcos vÞÞ þ 6 sin v cos v sin a; 5 ð14bÞ
b2 ¼ 2P2 ðcos vÞ þ 2c2 P2 ðcos aÞ 8c2 P2 ðcos vÞP2 ðcos aÞ þ 7
3 3 þ sin v cos v sin a cos a sin2 v sin2 a ; 2 4 ð14cÞ b3 ¼
6c1 ½2P2 ðcos vÞ cos a 3 sin v cos v sin a; 5 ð14dÞ
b4 ¼
72c2 P2 ðcos vÞP2 ðcos aÞ 35 2 sin v cos v sin a cos a
ð14eÞ The four experimental observables, bi , yield four independent equations, which allow the overdetermination of the angles v and a. A particularly interesting geometry is one for which the orientation field is parallel to the Zaxis (D ¼ 0), and the photolysis polarization is at 45° to the Z-axis (C ¼ 45°). This geometry is convenient for ion-imaging experiments, as the extraction field (which also acts as the orientation field) is parallel to the Z axis. Calculation of the laboratory frame angular distribution then yields:
! 3c1 sin v cos v sin a IðHÞ ¼ 1 þ P1 ðcos HÞ ½1 þ 2c2 P2 ðcos aÞ 1 12 P2 ðcos vÞ 163 c2 sin2 v sin2 a ! ½1 þ 2c2 P2 ðcos aÞP2 ðcos vÞ 38 c2 sin2 v sin2 a þ P2 ðcos HÞ: ½1 þ 2c2 P2 ðcos aÞ 1 12 P2 ðcos vÞ 163 c2 sin2 v sin2 a
4c2 P2 ðcos vÞP2 ðcos aÞ 5 3 þ 3 sin v cos v sin a cos a sin2 v sin2 a ; 4
b0 ¼ 1 þ
ð14aÞ
1 sin2 v sin2 a : 8
ð15Þ
Notice that the initial parent molecule orientation is produced parallel to the Z-axis, whereas photofragment orientation (described by the P1 ðcos HÞ term) occurs parallel to the X-axis, perpendicular to the original orientation direction. This orientation term vanishes for either v ¼ 0 or
T. Peter Rakitzis et al. / Chemical Physics Letters 372 (2003) 187–194
90°, or a ¼ 0 (which is the case for axial recoil dissociation of many symmetric molecules). Fig. 3a shows the variation of the orientation term on the angles v and a. In contrast, Fig. 3b shows that the b2 term is largely insensitive to the value of a, and can thus be used to measure the value of v directly. The b1 term can then be used to determine the value of a. We conclude that the observation of orientation in this geometry is a sensitive probe of non-axial recoil photodissociation, allowing the measurement of the two dynamical angles v and a. We will report such orientation measurements
193
elsewhere, for the photodissociation of OCS at 230 nm.
Acknowledgements This research has been financially supported by the councils for Chemical Sciences and Physical Sciences of the Dutch Organization for Scientific Research (NWO-CW, NWO-FOM). The authors would like to thank Prof. S. Stolte for support. TPR thanks the EU for support to access the experimental facilities of the Laser Centre Vrije Universiteit through the EU programme HPRICT-1999-00064. MHMJ thanks the CommunityÕs Human Potential Programme under Contract HPRN-CT-1999-00129 (COCOMO) for support at FORTH. AJvdB thanks the EU for support as a Marie Curie Training Institute fellow at FORTH under Contract HPMT-CT-2000-00201.
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