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Observation of fine structure and hyperfine structure depolarization in the photofragment anisotropy in triplet H2
Chemical Physics ELSEVIER
Chemical Physics 218 (1997) 309-323
Observation of fine structure and hyperfine structure depolarization in the photofragment anisotropy in triplet H 2 Eloy R. Wouters a,l, Laurens D.A. Siebbeles a,2, Katharine L. Reid b, Bart Buijsse a, Wim J. van der Zande a a FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJAmsterdam, The Netherlands b Department of Chemistry, University of Nottingham, Nottingham NG7 2RD, United Kingdom
Received 27 December 1996
Abstract Electron spin and nuclear spin reduce the anisotropy of photofragments in relatively long-lived levels of the g 3~- and i3Hg states of the n = 3 triplet gerade system of molecular hydrogen. The intramolecular forces due to the spin result in a
precession of the plane of rotation around the total angular momentum vector. The photofragment anisotropy reveals whether the excited state lifetime allows for this precession to occur, hence whether the fine and hyperfine structure are resolvable. In partially fine structure resolved experiments, using fast beam photofragment spectroscopy, this fine and hyperfine structure depolarization is observed. The theory behind the depolarization is presented as well.
1. Introduction Since the first observations of fine structure, much has been learned about its depolarizing effects on the spatial anisotropy of fluorescence and photoelectrons [1,2]. Both theoretical and experimental studies have demonstrated depolarization in a variety of atomic and molecular systems [3-6]. For dissociating molecules another and much stronger depolarization can be observed: that of the anisotropy of photofragments. This mechanism originates from the same intramolecular forces, namely those associated with electron spin. In relatively long-lived laser-excited molecules (hence with a reduced photofragment anisotropy), electron spin induced photofragment depolarization may be more pronounced than depolarization of the nearly isotropic fluorescence. An advantage of detecting photofragments is that it allows one to study those dissociations in which neither an electron nor a photon is emitted. In this paper we discuss the depolarization induced by both electron and nuclear spins on the photofragment anisotropy in a diatomic molecule. We will present a theoretical
Present address: Dept. of Chemistry, University of California, Berkeley, CA 94720, USA. 2 Present address: IRI, Mekelweg 15, 2629 JB Delft, The Netherlands. 0301-0104/97/$17.00 Copyright © 1997 Published by Elsevier Science B.V. All rights reserved. PH S0301-0104(97)00082-7
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E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
I
"
-
A
Fig. 1. Vector diagram of Hund's case bt3~.
treatment using angular momentum algebra to calculate the photofragment anisotropy. We will also present experimental data obtained in the photodissociation of highly excited triplet (S = 1), para ( I = 0) and ortho ( I = 1) molecular hydrogen. We will first briefly review why the photofragment anisotropy is affected by spin. The intramolecular forces associated with the electron spin cause the rotational angular momentum N to precess around the angular momentum J = N + S (see Fig. 1). If the total nuclear spin I is non-zero as well, a similar precession of J takes place around the total angular momentum F = J + I. In an aligned or oriented ensemble of molecules, as for example following photoexcitation with polarized radiation, this precession induces reversible exchanges of orientation or alignment among (a) orbital and rotational angular momenta and (b) electronic and nuclear spin. Examples known from experiments are fluorescence depolarization by internal fields (fine structure [4], hyperfine structure [7]) and by external fields (Hanle effect [8-10], the Earth's magnetic field). In general these effects decrease the anisotropy created during the excitation step. The depolarization strongly depends on the relative magnitudes of N and S (or I). The amount of depolarization decreases with increasing J. In this paper fine and hyperfine structure depolarization of the photofragment angular distribution following dissociation of long-lived triplet states of molecular hydrogen is investigated using fast beam photofragment spectroscopy. As was recently demonstrated by Buijsse et al, [11], the anisotropy of the photofragment angular distribution is a powerful tool in the investigation of intramolecular predissociation processes. It is shown that for long-lived excited complexes the observed anisotropy parameters demonstrate the presence of fine and hyperfine structure.
2. The triplet states of molecular hydrogen In this experiment we study photoexcitation from the metastable [12] c3I-Iu Rydberg state to the n = 3 triplet gerade complex, which is accessed by 500-800 nm radiation [13-21]. Rovibrational levels of the complex are excited below the n = 2 dissociation limit. The only available decay pathways are fluorescence to the repulsive b3,~+ state and the bound caHu state (see Fig. 2) which results in excited state lifetimes on the order of 10 ns. Hydrogen exists in two forms, ortho and para; as the latter is characterized by a zero nuclear spin, it exhibits no hyperfine structure. The effect of hyperfine structure can thus be studied selectively by looking at ortho-H 2 transitions. Fig. 3 shows that the electron spin S causes a rotational level N to split into a multiplet resulting (for S = 1) in three levels, historically called F1, F 2 and F 3 corresponding to the J values N + S, N and N - S, respectively. The fine structure and, in the case of ortho-hydrogen, hyperfine structure of the n = 2 caII~ state was measured accurately by Lichten and Wik [22] in the late seventies. The upper component F2 lies approximately 5.5 GHz above the two lower components, F 3 and F l which are separated by only 600 MHz. These splittings decrease both with increasing rotational quantum number N and with increasing vibrational quantum number v. The hyperfine structure is found to be smaller than the fine structure. Therefore, the
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Internuclear distance (0.u.) Fig. 2. Potential energy diagram of H 2 (left panel). Only the pertinent triplet states are displayed, including the four states belonging to the n = 3 triplet gerade complex: g3~g+, h 3 ~ , i31-lg and j ~ g . The two dissociation pathways of bound levels of the triplet gerade complex are shown, as well as the resulting features in the kinetic energy release (ker) distribution (right panel). Molecular emission, v I , to the repulsive b3~ + state gives the undulatory structure, a reflection of the (square of the) vibrational wave function, plotted in the left panel for a v' = 3 level. The other pathway which will not be examined in this study, is bound-bound emission, v 2 to the c311+ u state, followed by predissociation by the b3~u+ state, leading to a sharp peak at high ker.
hyperfine structure multiplet of the upper fine structure level (F 2) is well separated from the lower two multiplets. The fine and hyperfine splittings in the n = 3 triplet gerade states were not resolved until 1979 [13], being an order of magnitude smaller than those in the c3Hu state. Rox and coworkers [20,21] used Doppler free saturation spectroscopy with a resolution of 20 MHz to investigate the fine structure of the complex in more detail. The largest observed splitting is 500 MHz [20]. The fine and hyperfine structure intervals can be very irregular [13] due to the strong dependency of the amount of L uncoupling [18] on the individual rovibrational levels. Moreover, sometimes the fine and hyperfine structure splittings are of the same order of magnitude [21]. In the present experiment, the bandwidth of 3 GHz made it possible to resolve the largest fine structure splitting of the initial c3II~ state, allowing selective excitation of H 2 from the F2 fine structure level (see Fig. 3). The experiments are however not fully state resolved. Within the bandwidth of the laser, hydrogen molecules in different hyperfine levels are excited simultaneously to all allowed excited hyperfine levels.
N' ~ .
=: ~ e r
++-
. ~---
N"
,. fs ~ "
J" N"
F~
J::=N::- S F, J =N +S F,
Fig. 3. Fine structure splitting of the c3//~ triplet state in molecular hydrogen, the initial state for the laser excitation to levels in the n = 3 triplet gerade complex. The two different excitation schemes are also sketched: (i) starting from the F 2 fine structure level and (ii) starting from both the F I and F 3 fine structure levels of the c3II~ state. In either case, all fine and hyperfine structure levels of the excited state are being populated.
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3. Theory For an isotropically populated initial state, the photodissociation differential cross section can be written as
I(0) cx 1 + flP2(cosO). The value of the anisotropy parameter fl depends on the quantum numbers A (the projection of the electronic orbital angular momentum on the internuclear axis), F, J, and N of the initial and excited state. As was shown in Ref. [23], if the excited state is predissociated, /3 is not determined by the A value of the given predissociative excited state, but by that of the continuum. The dissociation of the excited levels of the g3~g and i3/-/g states which we study here, takes place via fluorescence. It was shown in Ref. [23] that the photofragment anisotropy of an excited state which decays via spontaneous emission is equal to the anisotropy that would result from predissociation by a continuum of the same A as the excited state, a so-called homogeneous predissociation. We will now discuss the effect of the lifetime (or linewidth) on the anisotropy. The anisotropy parameter is determined by the ratio of the linewidth and level spacings. In direct bound-free photodissociation, no rotational structure is observed in absorption and the anisotropy parameter fl takes the limiting values of - 1 or 2, for perpendicular and parallel transitions, respectively. If the lifetime increases, the molecule has time to rotate and rotational structure becomes visible in absorption. In this case, different anisotropy parameters are found for P, Q, and R branch transitions, in the absence of fine and hyperfine structure. These are given by the well known expressions for slow predissociation (see, e.g., Ref. [24]):
N"(N"-
1) - 3A '2
P(N"),
U"(2 N" + 1)
N " ( N " + 1) - 3A '2 /3 =
-
N " ( N " + 1)
Q( N"),
(1)
( N " + 1)(N" + 2) - 3A '2 ( N " + 1)(2N" + 1)
R(N").
In general, the presence of a well resolved rotational level results in an anisotropy parameter in between the impulsive extremes ( - 1 and 2). If the lifetime of the levels is very long (in hydrogen a few million rotational Table 1 C a l c u l a t e d values for the a n i s o t r o p y p a r a m e t e r in p a r a h y d r o g e n , for optical transitions starting f r o m different fine structure levels o f the c31I~ state. In all calculations we s u m m e d o v e r the excited state fine structure levels. The calculations u s i n g the treatment in the a p p e n d i x are c o m p a r e d to those using the theory for slow predissociation S l o w pred.
T h e o r y with fine structure F1
F2
F3
averaged
- 0.75 - 0.90
- 0.23 - 0.72
- 0.25 - 0.80
- 0.47 - 0.81
- 1 - 1
-0.I0 0.25
0.13 0.13
-0.11 0.11
-0.028 0.18
-0.10 0.25
- 0.37 - 0.76
- 0.11 - 0.61
- 0.13 - 0.68
- 0.24 - 0.69
- 0.50 - 0.85
0.41 0.51
0.33 0.49
0.60 0,60
0.42 0.52
0.60 0.60
g3Vg Q branch Q(2) Q(4)
i3Hg P
branch
P(2) P(4)
i311g O b r a n c h Q(2) Q(4)
i31Ig R R(2) R(4)
branch
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
313
Table 2 Calculated values for the anisotropy parameter in ortho hydrogen, for optical transitions starting from different fine structure levels of the c3II~ state. In all calculations we summed over the excited state hyperfine structure levels. The calculations using the treatment in the appendix are compared to those using the theory for slow predissociation Slow pred.
Theory with hyperfine structure
g3If~- Q branch Q(1) Q(3) i3IIg P branch P(3) P(5) iaH~ Q branch Q(I ) Q(3) i3IIg R branch R(1) R(3) R(5)
FI
F2
F3
averaged
-0.26 - 0.69
0.17 - 0.39
-0.28 - 0.29
-0.12 - 0.49
- 1 - 1
0.10 0.27
- 0.0034 0.18
- 0.0057 0.14
0.040 0.20
0.14 0.31
0.13 - 0.52
- 0.086 - 0.29
0.14 - 0.22
0.06 - 0.37
0.50 - 0.75
0.18 0.42 0.49
0.071 0.36 0.46
0.14 0.43 0.52
0.14 0.40 0.49
0.50 0.61 0.60
periods) fine and hyperfine structure may be observable if the linewidths are smaller than the fine and hyperfine structure splittings. This is the lifetime range that will be discussed in this paper. The fine and hyperfine structure cause a further depolarization of the anisotropy; the excited state lifetime allows for the precession of the rotational angular momentum N around J or F. A general expression for the anisotropy parameter which includes the electron and nuclear spin angular momentum can be found in the appendix (Eq. (A.11)). It is applicable to any Hand's case (b) diatomic molecule. To the best knowledge of the authors a theoretical framework describing the angular distribution in the presence of fine or hyperfine structure has not been published previously. Several theoretical papers exist in which the alignment of an ensemble of photoexcited molecules is calculated using density matrices (see, e.g., Refs. [25,26] and references therein). This related property is however not readily transformed in an anisotropy of resulting photofragments. The bandwidth of our laser is such that the hyperfine structure of the initial state and the fine and hyperfine structure of the excited state can not be resolved (see Fig. 3). To compare the calculated values to the experimental ones, we have to average the fully resolved expression for fl of Eq. (A.11), weighted by the relative cross section as given in Eq. (A.12). For a number of rotational transitions to the g3Vg and i3IIg states Eq. (A.11) is evaluated substituting S " = S' = 1, and I = 0 or 1 for para (Table 1) and ortho-H 2 (Table 2), respectively. The initial state is the negative electronic parity c3H~, for which the rotational levels with odd N belong to ortho-H 2, the levels with even N to para. To show the depolarization effects caused by the electron spin the weighted average is given of the anisotropy parameter assuming broadband excitation of all fine structure levels of the c3H~ state, see Table 1. Table 2 shows the anisotropy for ortho-H2; the effect of the nuclear spin is a further reduction of the anisotropy. The anisotropy parameters for the transitions populating the g 3~g+ state show the clearest effects of the fine and hyperfine structure, as the slow predissociation theory yields 1, irrespective of the rotational quantum number (see Eq. (1)). The magnitude of the depolarization decreases with increasing rotational quantum number, since the effect of adding or subtracting of one unit of angular momentum to the total angular momentum J or F diminishes with increasing J or F. We note that the averaged anisotropy parameters describe a simple diminution of the anisotropy upon adding the electron spin and nuclear spin. For individual fine structure levels in the c3Hu state however, the anisotropy displays large variations, including changes of sign. In conclusion, if the excited state -
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E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
fine and hyperfine levels are long-lived and observable in absorption experiments, then the photofragment anisotropy carries the signature of the fine and hyperfine structure even in the case of broad band unresolved excitation.
4. Experiment The technique of fast beam photofragment spectroscopy [27] was used to measure the angular dependence of the photodissociation products. A sketch of the experimental setup is presented in Fig. 4. A beam of 6.8 keV H~- is sent through a cell containing Cs vapor, in which a fraction of the ions are neutralized. This near-resonant charge exchange efficiently populates the metastable c3H~ state in H 2. The remaining ions are deflected out of the beam whereas the molecules formed in different, short-lived states dissociate out of the beam so that their atomic fragments are captured behind the 300 ~ m slit. The metastable molecules then cross the cavity of a 30 GHz bandwidth tunable cw dye laser operating with Rhodamine 6G dye. To resolve the fine structure in the c3H~ state, the bandwidth of the dye laser has been reduced to 3 GHz by positioning an etalon (FSR = 75 GHz) in the cavity. The laser beam is perpendicular to the molecular beam axis and its polarization vector is parallel to the molecular beam. A chopper is positioned in the cavity of the laser to allow subtraction of background signal. The molecules which are excited by the laser dissociate within a few nanoseconds, having traveled less than 1 cm and their atomic fragments fly out of the beam to a detector, positioned 220 cm downstream from the point where the laser crosses the molecular beam. To reduce background signal, the non-excited remainder of the beam is blocked by a beam flag, positioned 20 cm downstream. The photofragments are detected in coincidence on a time and two-dimensional position sensitive detector. This novel detector is described in detail in Ref. [28]. The fast detection electronics provide the position in two dimensions of both the atoms, as well as their arrival time difference. From this data the center-of-mass kinetic energy, and the two Euler angles of the molecular axis at the moment of dissociation can be determined to a high degree of accuracy. From the kinetic energy distribution the dissociation pathway can be determined (see Fig. 2). The photofragrnent angular distribution is measured as well. The experimental conditions were chosen such that the range of the ker distribution which could be observed only contains photofragments via the pathway of molecular fluorescence to the b3Z~+ state [process (1)]. The small bandwidth
2D Position
Fig. 4. This schematic shows various parts of the experimental setup along the beamline. Note that this diagram is not to scale: the flight length L from the interaction with the laser to the detector is 220 cm, whereas the diameter of the detector is only 6 cm. The released kinetic energy can be calculated from e = ( E o/4L2)[R 2 +(vor)2], where E o and vo are the energy (6.8 keV) and velocity of the molecules, R is the distance with which the fragments hit the detector, with time difference r. One of the Euler angles, 0 can be determined from tan O = R / vor.
E.R. Wouters et al./ Chemical Physics 218 (1997) 309-323
315
of our laser (3 GHz), in combination with the high velocity of the molecules (v 0 = 8 • 105 m/s), make our experiment sensitive to very small misalignments. A quarter of a degree misaiignment leads to a Doppler shift induced splitting of 6 GHz, which is of the same order as the largest fine structure splitting of the c3II~ state. To avoid loss of spectral resolution the beam was deliberately misaligned giving a Doppler shift of 10-20 GHz.
5. R e s u l t s a n d d i s c u s s i o n
We measured the anisotropy parameter after photoexcitation of a number of levels (v' ~< 3) of the g 3~g and states. The excitation frequencies for the diagonal (Av = 0) bands were obtained from Rox and coworkers [19~. Narrow (3 GHz) bandwidth laser light was applied to excite hydrogen molecules from the F 2 fine structure level, or from the F t and F 3 levels in the c3II~ state (see Fig. 3). Broad band (30 GHz) light was applied to excite molecules from all fine structure levels. As an example, the angular dependence of the photofragment yield is presented in Fig. 5 for a R(2) transition from the c3H~ (u" = 2) to the fine structure multiplet of the i3H~ (v' = 2, N' = 3). The full curve is the result of a least squares fit to the expression 1 + flP2(cosO), scaled by an arbitrary factor. The photofragment distribution of the left panel (from the c3II~ F 1 and F 3 fine structure levels) is clearly different from the one in the right panel (starting on the F 2 level); both transitions involve para hydrogen. This is directly showing the large depolarizing effect of the electron spin on the anisotropy of the photofragments. The measured anisotropy parameters are listed in Tables 3 and 4 for para and ortho hydrogen, respectively. The Q(2) transition to t h e g3~; state starting from the F 2 level is very weak, and was not detectable in the experiment. Tables 3 and 4 also list the anisotropy parameters calculated using Eqs. (A.11) and (A.12), weighted by 2 J" + 1 over the initial fine structure components for the combined F t + F 3 and F] + F 2 + F 3 values. For transitions involving ortho hydrogen, the theoretical results without hyperfine structure (setting I = 0 in the calculations) are included in Table 4. As explained before, the depolarization is only observable if the excited state is sufficiently long lived. For some excited state levels the hyperfine levels partially overlap, indicating that the anisotropy parameter may be better described when the hyperfine structure is neglected, and only fine structure is included. On examining the experimental anisotropy parameters for transitions starting on different fine structure levels within the same multiplet one clearly observes a difference between the F: and the F 1 + F 3 values for the
i3II
2.0x104
I
I
I
I
t
I
i
I
1.5x104
i
---------4+1.5
+ +4- + ++
+
1.0 Q (J 1.0
0.5
0.0 0
I 15
I 30
--$~0.05 t
45
I 60
I 75
9O 0
I 15
I 30
0.5
0
+ R(2) %
+ R(2) F~+F 3 - - ~'=0.23
I 45
I 60
I 75
0.0 90
O(degrees) Fig. 5, Cross section as a function of dissociation angle 0, for the R(2) transition from the F 1 and F3 fine structure levels of the c3H~ state (left panel) and from the F2 fine structure level (right panel), populating the fine structure multiplet of the i311g (v' = 2, N' = 3) level. The plus signs represent the experimental data, the full curves are the result of a least squares fit to 1 +/3P2(cos0). The vertical scale is in number of counts. The right half of the figure represents an essentially isotropic distribution (/3 = 0.05 + 1). The insets show a polar representation of the angular dependence of the cross section for the best-fit value, in which the laser polarization is along the vertical axis
(o = o).
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
316
Table 3 Experimental and calculated values for the anisotropy parameter in para hydrogen, for transitions starting from different fine structure levels of the c3IIu state. The error in 13 is approximately 0.1, except where indicated. For these / i v = 0 transitions, the vibrational quantum number is also indicated. They are compared to calculations using Eq. (A.11). The F I + F 3 value is determined by averaging the F I and F 3 values, weighted by 2 J " + 1, whereas the F 1 + F 2 + F 3 value is the weighted average over all three fine structure components v
g3~ Q(2)
i 3//s
Initial fs
Exp
Calc w / o fs
with fs
transition Fl + F3 F2 F~ + F 2 + F 3 a
- 0.55 - 0.30 -0.46
- 1 - 1 - 1
- 0.60 - 0.23 -0.47
F~ + F 3 F2 Fl + F 2 + F 3 a Fl +F 3 F~ + F 3 F2
-0.1(2) -0.2(2) 0.10 -0.52 0.23 0.05
-0.10 -0.10 -0.10 -0.50 0.60 0.60
-0.10 0.12 -0.03 -0.30 0.47 0.33
transition
P(2)
3
Q(2) R(2)
2 2
a Data taken in a separate experiment with a 30 GHz bandwidth laser which could not resolve the fine structure of the
c3H~ state.
majority of the measured lines. This demonstrates the influence of fine and hyperfine structure on the photofragment anisotropy. The observations in Table 3 agree largely with the theoretical predictions. This indicates that the different fine structure levels are well-separated, an assumption in the derivation of the theory.
Table 4 Experimental and calculated values for the anisotropy parameter in ortho hydrogen, for transitions starting from different fine structure levels of the c3H~ state. The error in fl is approximately 0.1, except where indicated. For these /iv = 0 transitions, the vibrational quantum number is also indicated. They are compared to calculations using the theory for slow predissociation ( w / o fs) and Eq. (A. 11), where we used S = 1, I = 0 (with fs) and S = 1 = 1 (with hfs). The F 1 + F 3 calculation is determined by averaging the e I and e 3 values, weighted by 2 J " + 1, whereas the F 1 + F 2 + F 3 is the weighted average over all three initial fine structure components v
g 3 ~ g transition Q(1)
Q(3)
1
Initial fs
Exp
Calc w / o fs
with fs
with hfs
F~ + F 3 F2 F~ + F z + F 3 a Fl +F 3 F2
-0.71 -0.42 -0.51 -0.32 -0.38
-
1 1 1 1 1
-0.63 0.42 -0.28 -0.77 -0.56
-0.26 0.17 -0.12 -0.55 -0.39
FI +F 3 FI + F 3 F2 Fj + F 3 Fz F1 + F 2 + F 3 a F I + F3 F2
-0.14 -0.12 - 0.09 0.04 O. 13 0.40 0.30 0.36
0.14 0.50 0.50 0.50 0.50 0.50 0.61 0.61
0.09 0.31 - 0.21 0.29 O. 13 0.24 0.52 0.44
0.06 0.13 - 0.09 0.17 0.07 0.14 0.42 0.36
i311s transition P(3) Q(1)
3 2, 3 b
R(1)
2
R(3)
3
a Data taken in a separate experiment with a 30 Gl-lz bandwidth laser which could not resolve the fine structure of the b The anisotropy parameters for both vibrational levels were equal to within the error limit.
c3II~ state.
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
317
Indeed, Ottinger and coworkers [20,21] could resolve all the fine structure levels using high-resolution spectroscopy. The anisotropy parameters for the Q(2) transition to the g3Vg state, a para line, are in good agreement with the calculations. The weighted average of the experimental values of the F 1 + F 3 and the F2 yields a value identical to the experimental anisotropy using the broadband laser, listed as F~ + F 2 + F 3. Transitions to para state levels are not without problem. The P(2) transition is already nearly isotropic without taking fine structure into account. The R(2) transitions suffer from spectral contamination; the R(5) and Q(4) lines of the (v = 2) state lie within one wavenumber [19,29]. The negative anisotropy parameters for the Q(4) line (Table 1) could cause the observed reduced values. A quantitative interpretation of the experimental anisotropies for the transitions involving ortho hydrogen (Table 4) is more complicated. The Q(1) transition to the g 3Vg state yields a very interesting result. The broadband experimental value is in agreement with the weighted average over the anisotropy parameters measured with the narrowband laser. But a clear discrepancy is seen between the experiment and the calculations, almost suggesting a misassignment. The depolarization due to the combination of electron spin and nuclear spin is rather extreme, since only one unit of rotational angular momentum is affected by two units of spin angular momentum. We do not exclude the possibility that the used hierarchical formalism may break down in very low angular momentum states. The Q(1) transition to the levels has been measured for both the v = 2 and 3 levels. The anisotropy parameters were found to be the same, which confirms the expectation that the vibrational quantum number does not affect the anisotropy. As with the Q(1) transition to the g3~g state, the comparison of experiment and (hyperfine structure) calculation is not very convincing, as a difference in sign is again observed. Although the depolarizing effect of the intramolecular structure is clearly observed, the present level of theory may be insufficient. In the R(1) transition the broadband excitation result is not in agreement with the narrow band excitation results; using broadband excitation another level has been photodissociated as well with a large positive anisotropy parameter. As yet we have not been able to identify such a level. Our analysis and calculations involve a number of assumptions. Minor problems arise from: (1) A statistical population of fine and hyperfine structure states in our fast neutral H 2 beam. Indeed the lifetimes of the different fine structure levels (against predissociation) were found to be different [30]. However, the time between the population of the state in the charge transfer collision and the photoexcitation amounts to 400 ns which is much shorter than the observed lifetimes (10-1000 /xs). Therefore we do not believe that the initial beam population has become spin-aligned. (2) Charge exchange of H~- with Cs is assumed to produce isotropic neutral beam (all M~ levels equally populated). In one case this was experimentally verified [31]. More serious and less understood effects are: (1) The theory is based on the hierarchical coupling scheme [2]; hyperfine interactions are assumed to be much smaller than fine structure interactions. This is not always a good approximation for the excited states under study. The hyperfine and fine structure intervals are in fact comparable which invalidates a strict hierarchical treatment. A full treatment (see, e.g., Refs. [2,21]) is beyond our capabilities since the coupling scheme of the electron and nuclear spin will be different for each level as the hyperfine structure splittings have been observed to be highly irregular [21]. (2) When the hyperfine structure splittings are on the order of the radiative width of the level, the depolarization by the nuclear spin will be incomplete and their excitation should be treated as partially coherent [2]. In the case of fully overlapping hypeffine levels the theoretical values including fine structure only may apply. Rox and coworkers observed that for a few rovibrational levels of the g3~g and states the smallest hyperfine structure interval is on the order of 10 MHz [21]. The smallest fine structure splittings measured by Rox and coworkers [20] for para hydrogen are approximately 30 MHz. In para hydrogen Eyler and Pipkin have determined radiative lifetimes between 10 ns (16 MHz) and 15 ns (11 MHz) for a number of fine structure levels [32]. Hence for the transitions involving para hydrogen, the calculated anisotropy parameters are expected to agree with the experimental ones. (3) The calculations were performed for excited states that are pure H or pure 2 states. The Born-Op-
i3IIg
i3IIg
i3IIg
c3Hu
i3Hg
318
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
penheimer approximation is known to break down in the n = 3 triplet gerade complex [33,34]. Mixing parameters have been obtained from level positions [19,35,36] and from decay behavior [37,38]. The amount of electronic state mixing depends on the level separation and increases with the rotational quantum number. The levels with positive electronic parity of the g3~- and t.3T/g+ states (populated in Q branch transitions) are found to mix their electronic character [33]. The eigenstates of both N = 2 levels consist of a 75%-25% mixture [19]. The anisotropy parameters in pure states are /3 = - 0 . 6 (,~) and /3 = - 0 . 3 ( / 7 ) both for Q transitions from F~ + F 3 fine structure levels. Indeed, the experimentally observed values for these levels and transitions are in between these values. The t.3T/g+ ( N = 1) level shows good agreement between experiment and theory for the F 2 level, and indeed has a very small ,~g admixture. For the P and R branch transitions populating the state, mixing in of ,~g+ character is strictly forbidden. The above aspects show that the anisotropy parameter contains a wealth of information. The/3 parameter is determined by the molecular coupling scheme and the level structure. In the case of triplet n = 3 hydrogen, non-adiabatic effects contribute as well. The effects mentioned above explain why we have not yet reached quantitative agreement with calculations. Nevertheless, the results clearly indicate the depolarizing effects of the nuclear and electron spin on photofragment angular distributions and point at the power of observing anisotropy parameters to reveal intramolecular structure.
i3/7~
6. Conclusion We have excited several bound levels in the n = 3 triplet gerade complex, and measured the anisotropy of the dissociation products. It has been found that the anisotropy depends on the initial fine structure levels for the transitions; the electron and nuclear spin depolarize the anisotropy of photofragments. This observation clearly indicates that the excited state fine and hyperfine structure can (at least partially) be resolved. We have presented a theory to calculate the photofragment anisotropy which includes electron and nuclear spin. It has become possible to calculate the/3 parameter as a function of the initial and final fine and hyperfine structure level. The fine and hyperfine structure are shown to decrease the anisotropy given by the theory calculated in the limit of slow predissociation, especially for the low rotational levels. The comparison of theory and experiment indicates that there are still aspects for both the theoretical and experimental molecular spectroscopists to be found in molecular hydrogen. For instance, the strongest lines belong to ortho hydrogen, where the presence of nuclear spin complicates the picture. The present calculation encompasses two extremes, fully separated upper state fine and hyperfine structure levels, and fully coherently excited upper state fine and hyperfine structure levels. In reality the hyperfine structure level separation and the linewidths can be of the same order of magnitude, requiring a more elaborate calculation as in Ref. [10]. The actual hyperfine and fine structure splitting suggest models beyond the hierarchical coupling scheme. The breakdown of the Born-Oppenheimer approximation in the n = 3 triplet manifold is another complication to be considered. We believe we have made clear why depolarization occurs, what the effect is on the photofragment anisotropy and under which circumstances it does or does not play a role.
Acknowledgements We are highly indebted to Prof. R.N. Zare for his suggestions which lead to this paper. KLR would like to thank EPSRC for an Advanced Fellowship and for supporting a two month stay at the FOM Institute. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for
E.R. Wouterset al./ ChemicalPhysics218 (1997)309-323
319
Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Research).
Appendix A In the following we will derive an expression for the anisotropy parameter describing the photofragment angular distribution resulting from the laser excitation of a hyperfine structure manifold. The initial level will be a single fine structure level. We will confine ourselves to triplet states in a diatomic molecule. Our choice of having a single initial fine structure level, whereas the excitation populates a whole manifold, which corresponds to the experimental limitations we have on the bandwidth of the laser (see Section 4), does not limit the applications of this expression for the anisotropy because the removal of the summation over the excited fine and hyperfine structure levels suffices to produce a general expression. The usual expression describing the angular dependence of photofragments is [39] I ( 0 ) = ° ' ° [1 + ~3P2(cos0)] 47/-L
(A.I)
with ~r0 the total cross section, /3 the anisotropy parameter ranging between - 1 and 2, P2(cos0) the second Legendre polynomial in cos O, with 0 the dissociation angle with respect to the polarization vector of the incident linearly polarized photon. We will derive an expression for l(O), from which the anisotropy parameter can be determined. We begin by writing down the wave function of a single hyperfine structure level in a diatomic molecule as IFMF). The states we consider are best approximated in a Hund's case (b) basis [40]. We will use the hierarchical coupling scheme ((bjsj), see Fig. 1). We can write the state [FMF) in terms of the uncoupled state quantum numbers N,M m S,M s and I,M 1 as follows:
[FM~) =
E ( -)~+s-MJ+J+'-M~2~CFV-4-F 2~CFYTY 2vTF-+1 MNMsMjMI ( N × M
S Ms
J )( J -Mj..Mj
I M,
F Ix(R) N * -MI: J R [DMNA(¢'O'O)] ¢e(r)ChsCh"
(A.2)
where (:::) represents a Wigner 3j symbol and D~NA($,O,O) a Wigner rotation matrix. The various M quantum numbers are the projection of the subscripted angular momenta on the laboratory Z axis, and X, ~be, ~bs and ~bt are the vibrational, electronic, electron spin and nuclear spin wave functions, respectively. We will now express the wave function of H 2 in the c3IIu state as IF"M~,c) and that in the excited state in the n = 3 triplet gerade complex as IF'M'r, x). The functions ¢ are labelled correspondingly. Note that in this description of the system parity is ignored (see Ref. [41] for the wave functions including parity), but this has no implication for the validity of our results. The molecule is photoexcited from the metastable c3IIu state to a state x in the n = 3 triplet gerade complex:
IF"M;,c) + hv ---*]F'M'~,x).
(A.3)
Note that the excited state vibrational wave function xx(R) should be replaced with a energy normalized continuum wave function. We will confine ourselves to light which is linearly polarized along the laboratory Z axis, hence M)' = M). We will follow the time-dependent approach of Siebbeles et al. [41], in which the cross section is expressed in the amplitude of the excited state wave function at large internuclear distance R, where
320
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
the nuclei have become separated. The time-dependent excited state wave function in the limit of large internuclear distance is written as R.-.* ~
~b(t) ~
e ikR
~., (F'M'F,Xl,,~i,tlF"M~-,c)lF'M'F,X)(--i)F'eiSr--e -ie, F'M~
(m.4)
R
Here E is the total energy of the system in the initial state and the photon; k = ~ / h , with /x the reduced mass of the molecule, and • the difference between the energy of the excited state and the dissociation limit. In writing out this expression we have used the fact that a single, non-overlapping F' level is excited. The phase shift 6r, of the outgoing wave equals that of the vibrational part of the dissociative wave function, which for R ~ ~ has the asymptotic form sin(kR + 6r, - ~TrF 1 , ), as can be found in quantum collision theory [42]. Note that only the outgoing part e ilcR of the wave function is present in Eq. (A.4). The matrix element (F'n'~,xl,g'intlF"M~,c) is evaluated using the expression for the wave functions from Eq. (A.2) and the dipole interaction Hamiltonian from Ref. [43]. Using the fact that the integration over the spin coordinates merely results in the well known selection rules for a one-photon transition with linear polarized light (AS = AI = 0), one obtains
( F'M'F,XI,'~intlF"M~,c)
=Vfda
E
E
M ~.,.M . . ~. .M ...tMI
M ' I' v M ~ ,M s M, !
tN" S' Nt~
s
J"llu'
x l m ;, M'; X,
( - ) P ~ / ( 2 J " + 1)(2J' + 1)(2F" + 1)(2F' + 1)(2N" + 1)(2N' + 1) ,
-M'; ) I
,
M's
,
- M ; ] [ M'j
M'l'
, - M ; ] [ M'j
M'1
-M' FJ
*
(,'DM;AnlV' "I r~l x-" M ' v A ' L.'lXolx
(A.5)
•
In this equation, ~0 is the polarization of the light in the lab frame (/x 0 = 0 for linear polarized light), and /z represents the polarization in the molecular frame which equals 0 or 1 for a parallel (AA = 0) or perpendicular (AA = 1) transition, respectively. The phase P equals
P = N " + N' + S' + S " - M ' j - M ' j
+ J" + J' + I" + I ' - M ; - M '
F.
(A.6)
V in Eq. (A.5) is the result of the integration over the internuclear separation R and the electronic coordinates r, hence it is the product of a Franck-Condon factor (Xcl Xx) and the electronic dipole matrix element (¢~lle" rll¢~), from which the angular part is written explicitly as Dl0z in Eq. (A.5). V yields no angular dependence of the photofragment distribution, and for clarity will be left out from here on. The integration over the two Euler angles /-2 is performed using Eq. (3.113) from Ref. [39] (note that, for a diatomic molecule, in this equation the third Euler angle X has to be set to zero). This results in
( F'M'F,xI,C~intIF"M~,c ) ot
Y'. pc n¢ tr M~M~MjMI
(N
• pf
s
X M~
M s'
×
0
M~
( - ) P ~ / ( 2 J " + 1)(2J' + 1)(2F" + 1)(2F' + 1)(2N" + 1)(2N' + 1)
. M I v. M.' s M. j M
l
,
s
,)(,
- M ' j ]I M'lv M's
N)(N,
-M~,
A
/.x
-M)
N)
-A'
"
, M'j
M't'
, -M'~ ) M)
lift
-M't: (A.7)
E.R. Wouters et al. / Chemical Physics 218 (1997) 3 0 9 - 3 2 3
321
Substituting Eqs. (A.2) and (A.7) in Eq. (A.4) yields R--*~
&(t)
, .
e ikR
( - - i ) F e '~'
~
e-ie' E
E
E
(_)v
M'~ M~M~'M~'M' t' M~MkM'~M ~
R
×~/(2J" + 1)(2J' + 1)(2F" + 1)(2F' + 1)(2N" + 1)(2N' + 1) ×tM~
M~'
-M;',,M~
M~
-M)]tM~'
[N" 1
N'IIN"
1
N' /
XIM ~ 0
-M'uJ[A"
t~
-A'
-M/,,M)
M'Iv, M ~ , M 'y, M t'
J'
I'
× M'j M;
F'
I' M~
F' ) -M~
I
. . . . . . ( - ) N+s-M'+J+'-M" 2F1/-~ 2 J ' v ~ v l - ~
E
X
M;'
)
-M'v ~e(r)~s'~I'(DNM~ A')
*]
( N' S' M'N M~
J' ) -M)
"
(A.8)
Following Siebbeles et al. [41], the probability of a photofragment being ejected with an angle 0 with respect to the laser polarization is 1(0) =
do-(0,~b)dj2 = Rlim . . . .f
drf d(spin)14~(t)l2,
(A.9)
where the integration has to be performed over the coordinates of all electrons, denoted by r and over the total spin space. After substituting Eq. (A.8), the limit and integrals are readily performed and the following expression is obtained
l( O) ot ~.,
~.,
Y'~
(2F" + l)(2J" + l)(2N" + l)(2F' + l)(2J' + l)(2N' + l)
M'r M'~M's'M'JM'/ M'~M'sM'jM ~
(N" S' × M~ Ms' x x
0 E
j,, ]2[ N' S' -M~'] t M'N M's -M;,
J' 12[ J" -M'j] I M7
I' F" 12[ J', M'I' - M ; ] ~Mj
I' F' 12 M~ -M'F]
A
(2r+0(2J'+l)(2N'+0
M;,
-M)
M)
-M)]
M'NM'sM'jM i N' ×(DM,uA, ) * DN',uA,] ,
(A.IO)
where we from now on use the fact that the terms contributing to the phase P are all integer, which is appropriate for molecular hydrogen because I - - 0 or 1 (para and ortho) and S = 0 or 1 (singlet and triplet states). We also use the fact that the phase factor in the excited state wave function ( - i)r'e ~ ' only depends on the excited state quantum number F'. For the extreme cases of one, well separated hyperfine structure level or of a completely overlapping hyperfine structure multiplet, interference effects between different F' levels are negligible and the phase factor equals 1 when the norm of the wave function 4~(t) is squared. However, for the case of partially overlapping hyperfine structure levels, which we have omitted in this appendix, the factor is
322
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
responsible for the interference effects, as observed by Siebbeles and coworkers between overlapping rotational levels [44]. Collapsing all the summations in Eq. (A.10) using Eqs. (3.95), (3.116), and (4.15) from Zare [39], and rewriting in the form of Eq. (A.1) leads to
/3(F',F",J',N',S',I',A')=(-)A'-F'-r-s'I/3-O(2F'
×
'
-A'
J'
+ I)(2J' + I)(2N' + I) F2'' 2
F'
N'
2
J'
'
1
1 (A.11)
with {:::} a Wigner 6j symbol. We used the fact that both F' and F" are integer. Note that the necessary algebra to evaluate this expression can nowadays also be found in computer programs using symbolic notation as, e.g., Mathematica [45]. We now go from this general expression for the anisotropy parameter for a one-photon transition from one hyperfine structure level to another to our experiment with a broad band laser. In that case we are unable to resolve the hyperfine structure of the initial state and both the hyperfine and fine structure of the excited state. To obtain values comparable to the experimental data, weighted summations of/3 in Eq. (A.11) are needed over F', F" and J'. The relative cross sections O'0rF"J' ~ ( 2 F " + 1 ) ( 2 J ' + 1 ) ( 2 F ' + 1) N'
J'
J'
F'
1] '
(A.12)
are the weighting factors. Note that we have only listed the factors depending on J', J", F' and F", as only they are required for the weighted average. Then the phase factor drops out as well. As a final remark we note that if both S and I are taken to be zero in Eqs. (A.11) and (A.12), the expressions for slow predissociafion result (see Eq. (1)).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
I.C. Percival, M.J. Seaton, Philos. Trans. R. Soc. London Ser. A 251 (1958) 113. U. Fano, J.H. Macek, Rev. Mod. Phys. 45 (1973) 553. I.M. Bell, C.J.K. Quayle, K. Burnett, J. Chem. Phys. 102 (1995) 1889. P.J. Brueat, R.N. Zare, J. Chem. Phys. 78 (1983) 100. A.J. Orr-Ewing, W.R. Simpson, T.P. Rakitzis, R.N. Zare, Isr. J. Chem. 34 (1994) 95. H. Joswig, M.A. O'Halloran, R.N. Zare and M.S. Child, Faraday Discuss. Chem. Soc. 82 (1986), paper 3. K.L. Reid, S.P. Duxon, M. Towrie, Chem. Phys. Lett. 228 (1994) 351. W. Hanle, Z. Phys. 30 (1924) 93. G. Gou6dard, Ann. Phys. (Pads) 7 (1972) 159. M. Broyer, M. Larzilli~re, M. Cat're, M.L. Galliard, M. Velghe, J.B. Ozenne, Chem. Phys. 63 (1981) 445. B. Buijsse, E.R. Wouters, W.J. van der Zande, Phys. Rev. Lett. 77 (1996) 243. , W. Lichten, Phys. Rev. 120 (1960) 848; R.P. Freis and J.R. Hiskes, Phys. Rev. A 2 (1970) 573; C.E. Johnson, Phys. Rev. A 5 (1972) 1026. W. Lichten, T. Wik, T.A. Miller, J. Chem. Phys. 71 (1979) 2441. D.P. de Bruijn, H. Helm, Phys. Rev. A 34 (1986) 3855. N. Bjerre, H. Helm, Chem. Phys. Lett. 134 (1987) 361. L.J. Lembo, D.L. Huesfis, S.R. Keiding, N. Bjerre, H. Helm, Phys. Rev. A 38 (1988) 3447. W. Koot, W.J. van der Zande and J. Los, Phys. Rev. Lett. 58 (1987) 2746; 59 (1987) 3015(E). W. Koot, W.J. van der Zande, J. Los, S.R. Keiding, N. Bjerre, Phys. Rev. A 39 (1989) 590. L. J6zefowski, Ch. Ottinger, T. Rox, J. Mol. Spectrosc. 163 (1994) 381. L. J6zefowski, Ch. Ottinger, T. Rox, J. Mol. Spectrosc. 163 (1994) 398. Ch. Ottinger, T. Rox, A. Sharma, J. Mol. Spectrosc. 163 (1994) 414. W. Lichten, T. Wik, J. Chem. Phys. 69 (1978) 5428.
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[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
323
L.D.A. Siebbeles, E.R. Wouters, W.J. van der Zande, Phys. Rev. A 54 (1996) 522. J.A. Beswick, in: Dynamical processes in molecular physics, ed. G. Deigado-Barrio (Institute of Physics, Bristol, 1993). K. Blum, Density matrix theory and applications (Plenum, New York, 1981). A.J. Orr-Ewing, R.N. Zare, Annu. Rev. Phys. Chem. 45 (1994) 315. D.P. de Bruijn, J. Los, Rev. Sci. Instrum. 53 (1982) 1020. E.R. Wouters, L.D.A. Siebbeles, P.C. Schuddeboom, B.R. Chalamala, W.J. van der Zande, Phys. Rev. A 54 (1996) 531. H.M. Crosswhite, ed., The hydrogen molecule wavelength tables of Gerhard Heinrich Dieke (Wiley-Interscience, New York, 1972). Ch. Berg, Ch. Ottinger, J. Chem. Phys. 100 (1994) 8746. L.D.A. Siebbeles, J.M. Schins, J. Los, Phys. Rev. A 44 (1991) 1584. E.E. Eyler, F.M. Pipkin, Phys. Rev. Lett. 47 (1981) 1270. J.M. Schins, L.D.A. Siebbeles, J. Los, W.J. van der Zande, Phys. Rev. A 44 (1991) 4162. J.M. Schins, L.D.A. Siebbeles, W.J. van der Zande, J. Los, Chem. Phys. Lett. 182 (1991) 69. S.A. Astashkevich, B.P. Lavrov, Opt. Spektrosk. 76 (1994) 42. Opt. Spectrosc. (USSR) 76 (1994) 38. S.A. Astashkevich, B.P. Lavrov, Opt. Spektrosk. 77 (1994) 369. Opt. Spectrosc. (USSR) 77 (1994) 327. S.A. Astashkevich, B.P. Lavrov, Opt. Spektrosk. 75 (1993) 1196. Opt. Spectrosc. (USSR) 75 (1993) 703. J.M. Schins, L.D.A. Siebbeles, J. Los, W.J. van der Zande, J. Rychlewski, H. Koch, Phys. Rev. A 44 (1991) 4171. R.N. Zare, Angular momentum (Wiley, New York, 1988). G. Herzberg, Molecular spectra and molecular structure, Vol. 1 (Van Nostrand Reinhold, New York, 1950). L.D.A. Siebbeles, J.M. Schins, W.J. van der Zande, J.A. Beswick, Chem. Phys. Lett. 187 (1991) 633. C.J. Joachain, Quantum collision theory (North-Holland, Amsterdam, 1975). R. Loudon, The quantum theory of light, 2nd ed. (Clarendon, Oxford, 1983). L.D.A. Siebbeles, J.M. Schins, W.J. van der Zande, J. Los, M. Glass-Maujean, Phys. Rev. A 44 (1991) 343. Mathematica by Wolfram Research, Inc., Champaign, Ill.
Chemical Physics 218 (1997) 309-323
Observation of fine structure and hyperfine structure depolarization in the photofragment anisotropy in triplet H 2 Eloy R. Wouters a,l, Laurens D.A. Siebbeles a,2, Katharine L. Reid b, Bart Buijsse a, Wim J. van der Zande a a FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJAmsterdam, The Netherlands b Department of Chemistry, University of Nottingham, Nottingham NG7 2RD, United Kingdom
Received 27 December 1996
Abstract Electron spin and nuclear spin reduce the anisotropy of photofragments in relatively long-lived levels of the g 3~- and i3Hg states of the n = 3 triplet gerade system of molecular hydrogen. The intramolecular forces due to the spin result in a
precession of the plane of rotation around the total angular momentum vector. The photofragment anisotropy reveals whether the excited state lifetime allows for this precession to occur, hence whether the fine and hyperfine structure are resolvable. In partially fine structure resolved experiments, using fast beam photofragment spectroscopy, this fine and hyperfine structure depolarization is observed. The theory behind the depolarization is presented as well.
1. Introduction Since the first observations of fine structure, much has been learned about its depolarizing effects on the spatial anisotropy of fluorescence and photoelectrons [1,2]. Both theoretical and experimental studies have demonstrated depolarization in a variety of atomic and molecular systems [3-6]. For dissociating molecules another and much stronger depolarization can be observed: that of the anisotropy of photofragments. This mechanism originates from the same intramolecular forces, namely those associated with electron spin. In relatively long-lived laser-excited molecules (hence with a reduced photofragment anisotropy), electron spin induced photofragment depolarization may be more pronounced than depolarization of the nearly isotropic fluorescence. An advantage of detecting photofragments is that it allows one to study those dissociations in which neither an electron nor a photon is emitted. In this paper we discuss the depolarization induced by both electron and nuclear spins on the photofragment anisotropy in a diatomic molecule. We will present a theoretical
Present address: Dept. of Chemistry, University of California, Berkeley, CA 94720, USA. 2 Present address: IRI, Mekelweg 15, 2629 JB Delft, The Netherlands. 0301-0104/97/$17.00 Copyright © 1997 Published by Elsevier Science B.V. All rights reserved. PH S0301-0104(97)00082-7
310
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
I
"
-
A
Fig. 1. Vector diagram of Hund's case bt3~.
treatment using angular momentum algebra to calculate the photofragment anisotropy. We will also present experimental data obtained in the photodissociation of highly excited triplet (S = 1), para ( I = 0) and ortho ( I = 1) molecular hydrogen. We will first briefly review why the photofragment anisotropy is affected by spin. The intramolecular forces associated with the electron spin cause the rotational angular momentum N to precess around the angular momentum J = N + S (see Fig. 1). If the total nuclear spin I is non-zero as well, a similar precession of J takes place around the total angular momentum F = J + I. In an aligned or oriented ensemble of molecules, as for example following photoexcitation with polarized radiation, this precession induces reversible exchanges of orientation or alignment among (a) orbital and rotational angular momenta and (b) electronic and nuclear spin. Examples known from experiments are fluorescence depolarization by internal fields (fine structure [4], hyperfine structure [7]) and by external fields (Hanle effect [8-10], the Earth's magnetic field). In general these effects decrease the anisotropy created during the excitation step. The depolarization strongly depends on the relative magnitudes of N and S (or I). The amount of depolarization decreases with increasing J. In this paper fine and hyperfine structure depolarization of the photofragment angular distribution following dissociation of long-lived triplet states of molecular hydrogen is investigated using fast beam photofragment spectroscopy. As was recently demonstrated by Buijsse et al, [11], the anisotropy of the photofragment angular distribution is a powerful tool in the investigation of intramolecular predissociation processes. It is shown that for long-lived excited complexes the observed anisotropy parameters demonstrate the presence of fine and hyperfine structure.
2. The triplet states of molecular hydrogen In this experiment we study photoexcitation from the metastable [12] c3I-Iu Rydberg state to the n = 3 triplet gerade complex, which is accessed by 500-800 nm radiation [13-21]. Rovibrational levels of the complex are excited below the n = 2 dissociation limit. The only available decay pathways are fluorescence to the repulsive b3,~+ state and the bound caHu state (see Fig. 2) which results in excited state lifetimes on the order of 10 ns. Hydrogen exists in two forms, ortho and para; as the latter is characterized by a zero nuclear spin, it exhibits no hyperfine structure. The effect of hyperfine structure can thus be studied selectively by looking at ortho-H 2 transitions. Fig. 3 shows that the electron spin S causes a rotational level N to split into a multiplet resulting (for S = 1) in three levels, historically called F1, F 2 and F 3 corresponding to the J values N + S, N and N - S, respectively. The fine structure and, in the case of ortho-hydrogen, hyperfine structure of the n = 2 caII~ state was measured accurately by Lichten and Wik [22] in the late seventies. The upper component F2 lies approximately 5.5 GHz above the two lower components, F 3 and F l which are separated by only 600 MHz. These splittings decrease both with increasing rotational quantum number N and with increasing vibrational quantum number v. The hyperfine structure is found to be smaller than the fine structure. Therefore, the
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323 --T
1
311
T ("4
i
P~15 > (D
g
>
? c © _10 121 c-
g .o_ % ._c
.4-o
©
2x2 5 0
2
4
6
8
Internuclear distance (0.u.) Fig. 2. Potential energy diagram of H 2 (left panel). Only the pertinent triplet states are displayed, including the four states belonging to the n = 3 triplet gerade complex: g3~g+, h 3 ~ , i31-lg and j ~ g . The two dissociation pathways of bound levels of the triplet gerade complex are shown, as well as the resulting features in the kinetic energy release (ker) distribution (right panel). Molecular emission, v I , to the repulsive b3~ + state gives the undulatory structure, a reflection of the (square of the) vibrational wave function, plotted in the left panel for a v' = 3 level. The other pathway which will not be examined in this study, is bound-bound emission, v 2 to the c311+ u state, followed by predissociation by the b3~u+ state, leading to a sharp peak at high ker.
hyperfine structure multiplet of the upper fine structure level (F 2) is well separated from the lower two multiplets. The fine and hyperfine splittings in the n = 3 triplet gerade states were not resolved until 1979 [13], being an order of magnitude smaller than those in the c3Hu state. Rox and coworkers [20,21] used Doppler free saturation spectroscopy with a resolution of 20 MHz to investigate the fine structure of the complex in more detail. The largest observed splitting is 500 MHz [20]. The fine and hyperfine structure intervals can be very irregular [13] due to the strong dependency of the amount of L uncoupling [18] on the individual rovibrational levels. Moreover, sometimes the fine and hyperfine structure splittings are of the same order of magnitude [21]. In the present experiment, the bandwidth of 3 GHz made it possible to resolve the largest fine structure splitting of the initial c3II~ state, allowing selective excitation of H 2 from the F2 fine structure level (see Fig. 3). The experiments are however not fully state resolved. Within the bandwidth of the laser, hydrogen molecules in different hyperfine levels are excited simultaneously to all allowed excited hyperfine levels.
N' ~ .
=: ~ e r
++-
. ~---
N"
,. fs ~ "
J" N"
F~
J::=N::- S F, J =N +S F,
Fig. 3. Fine structure splitting of the c3//~ triplet state in molecular hydrogen, the initial state for the laser excitation to levels in the n = 3 triplet gerade complex. The two different excitation schemes are also sketched: (i) starting from the F 2 fine structure level and (ii) starting from both the F I and F 3 fine structure levels of the c3II~ state. In either case, all fine and hyperfine structure levels of the excited state are being populated.
312
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
3. Theory For an isotropically populated initial state, the photodissociation differential cross section can be written as
I(0) cx 1 + flP2(cosO). The value of the anisotropy parameter fl depends on the quantum numbers A (the projection of the electronic orbital angular momentum on the internuclear axis), F, J, and N of the initial and excited state. As was shown in Ref. [23], if the excited state is predissociated, /3 is not determined by the A value of the given predissociative excited state, but by that of the continuum. The dissociation of the excited levels of the g3~g and i3/-/g states which we study here, takes place via fluorescence. It was shown in Ref. [23] that the photofragment anisotropy of an excited state which decays via spontaneous emission is equal to the anisotropy that would result from predissociation by a continuum of the same A as the excited state, a so-called homogeneous predissociation. We will now discuss the effect of the lifetime (or linewidth) on the anisotropy. The anisotropy parameter is determined by the ratio of the linewidth and level spacings. In direct bound-free photodissociation, no rotational structure is observed in absorption and the anisotropy parameter fl takes the limiting values of - 1 or 2, for perpendicular and parallel transitions, respectively. If the lifetime increases, the molecule has time to rotate and rotational structure becomes visible in absorption. In this case, different anisotropy parameters are found for P, Q, and R branch transitions, in the absence of fine and hyperfine structure. These are given by the well known expressions for slow predissociation (see, e.g., Ref. [24]):
N"(N"-
1) - 3A '2
P(N"),
U"(2 N" + 1)
N " ( N " + 1) - 3A '2 /3 =
-
N " ( N " + 1)
Q( N"),
(1)
( N " + 1)(N" + 2) - 3A '2 ( N " + 1)(2N" + 1)
R(N").
In general, the presence of a well resolved rotational level results in an anisotropy parameter in between the impulsive extremes ( - 1 and 2). If the lifetime of the levels is very long (in hydrogen a few million rotational Table 1 C a l c u l a t e d values for the a n i s o t r o p y p a r a m e t e r in p a r a h y d r o g e n , for optical transitions starting f r o m different fine structure levels o f the c31I~ state. In all calculations we s u m m e d o v e r the excited state fine structure levels. The calculations u s i n g the treatment in the a p p e n d i x are c o m p a r e d to those using the theory for slow predissociation S l o w pred.
T h e o r y with fine structure F1
F2
F3
averaged
- 0.75 - 0.90
- 0.23 - 0.72
- 0.25 - 0.80
- 0.47 - 0.81
- 1 - 1
-0.I0 0.25
0.13 0.13
-0.11 0.11
-0.028 0.18
-0.10 0.25
- 0.37 - 0.76
- 0.11 - 0.61
- 0.13 - 0.68
- 0.24 - 0.69
- 0.50 - 0.85
0.41 0.51
0.33 0.49
0.60 0,60
0.42 0.52
0.60 0.60
g3Vg Q branch Q(2) Q(4)
i3Hg P
branch
P(2) P(4)
i311g O b r a n c h Q(2) Q(4)
i31Ig R R(2) R(4)
branch
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
313
Table 2 Calculated values for the anisotropy parameter in ortho hydrogen, for optical transitions starting from different fine structure levels of the c3II~ state. In all calculations we summed over the excited state hyperfine structure levels. The calculations using the treatment in the appendix are compared to those using the theory for slow predissociation Slow pred.
Theory with hyperfine structure
g3If~- Q branch Q(1) Q(3) i3IIg P branch P(3) P(5) iaH~ Q branch Q(I ) Q(3) i3IIg R branch R(1) R(3) R(5)
FI
F2
F3
averaged
-0.26 - 0.69
0.17 - 0.39
-0.28 - 0.29
-0.12 - 0.49
- 1 - 1
0.10 0.27
- 0.0034 0.18
- 0.0057 0.14
0.040 0.20
0.14 0.31
0.13 - 0.52
- 0.086 - 0.29
0.14 - 0.22
0.06 - 0.37
0.50 - 0.75
0.18 0.42 0.49
0.071 0.36 0.46
0.14 0.43 0.52
0.14 0.40 0.49
0.50 0.61 0.60
periods) fine and hyperfine structure may be observable if the linewidths are smaller than the fine and hyperfine structure splittings. This is the lifetime range that will be discussed in this paper. The fine and hyperfine structure cause a further depolarization of the anisotropy; the excited state lifetime allows for the precession of the rotational angular momentum N around J or F. A general expression for the anisotropy parameter which includes the electron and nuclear spin angular momentum can be found in the appendix (Eq. (A.11)). It is applicable to any Hand's case (b) diatomic molecule. To the best knowledge of the authors a theoretical framework describing the angular distribution in the presence of fine or hyperfine structure has not been published previously. Several theoretical papers exist in which the alignment of an ensemble of photoexcited molecules is calculated using density matrices (see, e.g., Refs. [25,26] and references therein). This related property is however not readily transformed in an anisotropy of resulting photofragments. The bandwidth of our laser is such that the hyperfine structure of the initial state and the fine and hyperfine structure of the excited state can not be resolved (see Fig. 3). To compare the calculated values to the experimental ones, we have to average the fully resolved expression for fl of Eq. (A.11), weighted by the relative cross section as given in Eq. (A.12). For a number of rotational transitions to the g3Vg and i3IIg states Eq. (A.11) is evaluated substituting S " = S' = 1, and I = 0 or 1 for para (Table 1) and ortho-H 2 (Table 2), respectively. The initial state is the negative electronic parity c3H~, for which the rotational levels with odd N belong to ortho-H 2, the levels with even N to para. To show the depolarization effects caused by the electron spin the weighted average is given of the anisotropy parameter assuming broadband excitation of all fine structure levels of the c3H~ state, see Table 1. Table 2 shows the anisotropy for ortho-H2; the effect of the nuclear spin is a further reduction of the anisotropy. The anisotropy parameters for the transitions populating the g 3~g+ state show the clearest effects of the fine and hyperfine structure, as the slow predissociation theory yields 1, irrespective of the rotational quantum number (see Eq. (1)). The magnitude of the depolarization decreases with increasing rotational quantum number, since the effect of adding or subtracting of one unit of angular momentum to the total angular momentum J or F diminishes with increasing J or F. We note that the averaged anisotropy parameters describe a simple diminution of the anisotropy upon adding the electron spin and nuclear spin. For individual fine structure levels in the c3Hu state however, the anisotropy displays large variations, including changes of sign. In conclusion, if the excited state -
314
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
fine and hyperfine levels are long-lived and observable in absorption experiments, then the photofragment anisotropy carries the signature of the fine and hyperfine structure even in the case of broad band unresolved excitation.
4. Experiment The technique of fast beam photofragment spectroscopy [27] was used to measure the angular dependence of the photodissociation products. A sketch of the experimental setup is presented in Fig. 4. A beam of 6.8 keV H~- is sent through a cell containing Cs vapor, in which a fraction of the ions are neutralized. This near-resonant charge exchange efficiently populates the metastable c3H~ state in H 2. The remaining ions are deflected out of the beam whereas the molecules formed in different, short-lived states dissociate out of the beam so that their atomic fragments are captured behind the 300 ~ m slit. The metastable molecules then cross the cavity of a 30 GHz bandwidth tunable cw dye laser operating with Rhodamine 6G dye. To resolve the fine structure in the c3H~ state, the bandwidth of the dye laser has been reduced to 3 GHz by positioning an etalon (FSR = 75 GHz) in the cavity. The laser beam is perpendicular to the molecular beam axis and its polarization vector is parallel to the molecular beam. A chopper is positioned in the cavity of the laser to allow subtraction of background signal. The molecules which are excited by the laser dissociate within a few nanoseconds, having traveled less than 1 cm and their atomic fragments fly out of the beam to a detector, positioned 220 cm downstream from the point where the laser crosses the molecular beam. To reduce background signal, the non-excited remainder of the beam is blocked by a beam flag, positioned 20 cm downstream. The photofragments are detected in coincidence on a time and two-dimensional position sensitive detector. This novel detector is described in detail in Ref. [28]. The fast detection electronics provide the position in two dimensions of both the atoms, as well as their arrival time difference. From this data the center-of-mass kinetic energy, and the two Euler angles of the molecular axis at the moment of dissociation can be determined to a high degree of accuracy. From the kinetic energy distribution the dissociation pathway can be determined (see Fig. 2). The photofragrnent angular distribution is measured as well. The experimental conditions were chosen such that the range of the ker distribution which could be observed only contains photofragments via the pathway of molecular fluorescence to the b3Z~+ state [process (1)]. The small bandwidth
2D Position
Fig. 4. This schematic shows various parts of the experimental setup along the beamline. Note that this diagram is not to scale: the flight length L from the interaction with the laser to the detector is 220 cm, whereas the diameter of the detector is only 6 cm. The released kinetic energy can be calculated from e = ( E o/4L2)[R 2 +(vor)2], where E o and vo are the energy (6.8 keV) and velocity of the molecules, R is the distance with which the fragments hit the detector, with time difference r. One of the Euler angles, 0 can be determined from tan O = R / vor.
E.R. Wouters et al./ Chemical Physics 218 (1997) 309-323
315
of our laser (3 GHz), in combination with the high velocity of the molecules (v 0 = 8 • 105 m/s), make our experiment sensitive to very small misalignments. A quarter of a degree misaiignment leads to a Doppler shift induced splitting of 6 GHz, which is of the same order as the largest fine structure splitting of the c3II~ state. To avoid loss of spectral resolution the beam was deliberately misaligned giving a Doppler shift of 10-20 GHz.
5. R e s u l t s a n d d i s c u s s i o n
We measured the anisotropy parameter after photoexcitation of a number of levels (v' ~< 3) of the g 3~g and states. The excitation frequencies for the diagonal (Av = 0) bands were obtained from Rox and coworkers [19~. Narrow (3 GHz) bandwidth laser light was applied to excite hydrogen molecules from the F 2 fine structure level, or from the F t and F 3 levels in the c3II~ state (see Fig. 3). Broad band (30 GHz) light was applied to excite molecules from all fine structure levels. As an example, the angular dependence of the photofragment yield is presented in Fig. 5 for a R(2) transition from the c3H~ (u" = 2) to the fine structure multiplet of the i3H~ (v' = 2, N' = 3). The full curve is the result of a least squares fit to the expression 1 + flP2(cosO), scaled by an arbitrary factor. The photofragment distribution of the left panel (from the c3II~ F 1 and F 3 fine structure levels) is clearly different from the one in the right panel (starting on the F 2 level); both transitions involve para hydrogen. This is directly showing the large depolarizing effect of the electron spin on the anisotropy of the photofragments. The measured anisotropy parameters are listed in Tables 3 and 4 for para and ortho hydrogen, respectively. The Q(2) transition to t h e g3~; state starting from the F 2 level is very weak, and was not detectable in the experiment. Tables 3 and 4 also list the anisotropy parameters calculated using Eqs. (A.11) and (A.12), weighted by 2 J" + 1 over the initial fine structure components for the combined F t + F 3 and F] + F 2 + F 3 values. For transitions involving ortho hydrogen, the theoretical results without hyperfine structure (setting I = 0 in the calculations) are included in Table 4. As explained before, the depolarization is only observable if the excited state is sufficiently long lived. For some excited state levels the hyperfine levels partially overlap, indicating that the anisotropy parameter may be better described when the hyperfine structure is neglected, and only fine structure is included. On examining the experimental anisotropy parameters for transitions starting on different fine structure levels within the same multiplet one clearly observes a difference between the F: and the F 1 + F 3 values for the
i3II
2.0x104
I
I
I
I
t
I
i
I
1.5x104
i
---------4+1.5
+ +4- + ++
+
1.0 Q (J 1.0
0.5
0.0 0
I 15
I 30
--$~0.05 t
45
I 60
I 75
9O 0
I 15
I 30
0.5
0
+ R(2) %
+ R(2) F~+F 3 - - ~'=0.23
I 45
I 60
I 75
0.0 90
O(degrees) Fig. 5, Cross section as a function of dissociation angle 0, for the R(2) transition from the F 1 and F3 fine structure levels of the c3H~ state (left panel) and from the F2 fine structure level (right panel), populating the fine structure multiplet of the i311g (v' = 2, N' = 3) level. The plus signs represent the experimental data, the full curves are the result of a least squares fit to 1 +/3P2(cos0). The vertical scale is in number of counts. The right half of the figure represents an essentially isotropic distribution (/3 = 0.05 + 1). The insets show a polar representation of the angular dependence of the cross section for the best-fit value, in which the laser polarization is along the vertical axis
(o = o).
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
316
Table 3 Experimental and calculated values for the anisotropy parameter in para hydrogen, for transitions starting from different fine structure levels of the c3IIu state. The error in 13 is approximately 0.1, except where indicated. For these / i v = 0 transitions, the vibrational quantum number is also indicated. They are compared to calculations using Eq. (A.11). The F I + F 3 value is determined by averaging the F I and F 3 values, weighted by 2 J " + 1, whereas the F 1 + F 2 + F 3 value is the weighted average over all three fine structure components v
g3~ Q(2)
i 3//s
Initial fs
Exp
Calc w / o fs
with fs
transition Fl + F3 F2 F~ + F 2 + F 3 a
- 0.55 - 0.30 -0.46
- 1 - 1 - 1
- 0.60 - 0.23 -0.47
F~ + F 3 F2 Fl + F 2 + F 3 a Fl +F 3 F~ + F 3 F2
-0.1(2) -0.2(2) 0.10 -0.52 0.23 0.05
-0.10 -0.10 -0.10 -0.50 0.60 0.60
-0.10 0.12 -0.03 -0.30 0.47 0.33
transition
P(2)
3
Q(2) R(2)
2 2
a Data taken in a separate experiment with a 30 GHz bandwidth laser which could not resolve the fine structure of the
c3H~ state.
majority of the measured lines. This demonstrates the influence of fine and hyperfine structure on the photofragment anisotropy. The observations in Table 3 agree largely with the theoretical predictions. This indicates that the different fine structure levels are well-separated, an assumption in the derivation of the theory.
Table 4 Experimental and calculated values for the anisotropy parameter in ortho hydrogen, for transitions starting from different fine structure levels of the c3H~ state. The error in fl is approximately 0.1, except where indicated. For these /iv = 0 transitions, the vibrational quantum number is also indicated. They are compared to calculations using the theory for slow predissociation ( w / o fs) and Eq. (A. 11), where we used S = 1, I = 0 (with fs) and S = 1 = 1 (with hfs). The F 1 + F 3 calculation is determined by averaging the e I and e 3 values, weighted by 2 J " + 1, whereas the F 1 + F 2 + F 3 is the weighted average over all three initial fine structure components v
g 3 ~ g transition Q(1)
Q(3)
1
Initial fs
Exp
Calc w / o fs
with fs
with hfs
F~ + F 3 F2 F~ + F z + F 3 a Fl +F 3 F2
-0.71 -0.42 -0.51 -0.32 -0.38
-
1 1 1 1 1
-0.63 0.42 -0.28 -0.77 -0.56
-0.26 0.17 -0.12 -0.55 -0.39
FI +F 3 FI + F 3 F2 Fj + F 3 Fz F1 + F 2 + F 3 a F I + F3 F2
-0.14 -0.12 - 0.09 0.04 O. 13 0.40 0.30 0.36
0.14 0.50 0.50 0.50 0.50 0.50 0.61 0.61
0.09 0.31 - 0.21 0.29 O. 13 0.24 0.52 0.44
0.06 0.13 - 0.09 0.17 0.07 0.14 0.42 0.36
i311s transition P(3) Q(1)
3 2, 3 b
R(1)
2
R(3)
3
a Data taken in a separate experiment with a 30 Gl-lz bandwidth laser which could not resolve the fine structure of the b The anisotropy parameters for both vibrational levels were equal to within the error limit.
c3II~ state.
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
317
Indeed, Ottinger and coworkers [20,21] could resolve all the fine structure levels using high-resolution spectroscopy. The anisotropy parameters for the Q(2) transition to the g3Vg state, a para line, are in good agreement with the calculations. The weighted average of the experimental values of the F 1 + F 3 and the F2 yields a value identical to the experimental anisotropy using the broadband laser, listed as F~ + F 2 + F 3. Transitions to para state levels are not without problem. The P(2) transition is already nearly isotropic without taking fine structure into account. The R(2) transitions suffer from spectral contamination; the R(5) and Q(4) lines of the (v = 2) state lie within one wavenumber [19,29]. The negative anisotropy parameters for the Q(4) line (Table 1) could cause the observed reduced values. A quantitative interpretation of the experimental anisotropies for the transitions involving ortho hydrogen (Table 4) is more complicated. The Q(1) transition to the g 3Vg state yields a very interesting result. The broadband experimental value is in agreement with the weighted average over the anisotropy parameters measured with the narrowband laser. But a clear discrepancy is seen between the experiment and the calculations, almost suggesting a misassignment. The depolarization due to the combination of electron spin and nuclear spin is rather extreme, since only one unit of rotational angular momentum is affected by two units of spin angular momentum. We do not exclude the possibility that the used hierarchical formalism may break down in very low angular momentum states. The Q(1) transition to the levels has been measured for both the v = 2 and 3 levels. The anisotropy parameters were found to be the same, which confirms the expectation that the vibrational quantum number does not affect the anisotropy. As with the Q(1) transition to the g3~g state, the comparison of experiment and (hyperfine structure) calculation is not very convincing, as a difference in sign is again observed. Although the depolarizing effect of the intramolecular structure is clearly observed, the present level of theory may be insufficient. In the R(1) transition the broadband excitation result is not in agreement with the narrow band excitation results; using broadband excitation another level has been photodissociated as well with a large positive anisotropy parameter. As yet we have not been able to identify such a level. Our analysis and calculations involve a number of assumptions. Minor problems arise from: (1) A statistical population of fine and hyperfine structure states in our fast neutral H 2 beam. Indeed the lifetimes of the different fine structure levels (against predissociation) were found to be different [30]. However, the time between the population of the state in the charge transfer collision and the photoexcitation amounts to 400 ns which is much shorter than the observed lifetimes (10-1000 /xs). Therefore we do not believe that the initial beam population has become spin-aligned. (2) Charge exchange of H~- with Cs is assumed to produce isotropic neutral beam (all M~ levels equally populated). In one case this was experimentally verified [31]. More serious and less understood effects are: (1) The theory is based on the hierarchical coupling scheme [2]; hyperfine interactions are assumed to be much smaller than fine structure interactions. This is not always a good approximation for the excited states under study. The hyperfine and fine structure intervals are in fact comparable which invalidates a strict hierarchical treatment. A full treatment (see, e.g., Refs. [2,21]) is beyond our capabilities since the coupling scheme of the electron and nuclear spin will be different for each level as the hyperfine structure splittings have been observed to be highly irregular [21]. (2) When the hyperfine structure splittings are on the order of the radiative width of the level, the depolarization by the nuclear spin will be incomplete and their excitation should be treated as partially coherent [2]. In the case of fully overlapping hypeffine levels the theoretical values including fine structure only may apply. Rox and coworkers observed that for a few rovibrational levels of the g3~g and states the smallest hyperfine structure interval is on the order of 10 MHz [21]. The smallest fine structure splittings measured by Rox and coworkers [20] for para hydrogen are approximately 30 MHz. In para hydrogen Eyler and Pipkin have determined radiative lifetimes between 10 ns (16 MHz) and 15 ns (11 MHz) for a number of fine structure levels [32]. Hence for the transitions involving para hydrogen, the calculated anisotropy parameters are expected to agree with the experimental ones. (3) The calculations were performed for excited states that are pure H or pure 2 states. The Born-Op-
i3IIg
i3IIg
i3IIg
c3Hu
i3Hg
318
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
penheimer approximation is known to break down in the n = 3 triplet gerade complex [33,34]. Mixing parameters have been obtained from level positions [19,35,36] and from decay behavior [37,38]. The amount of electronic state mixing depends on the level separation and increases with the rotational quantum number. The levels with positive electronic parity of the g3~- and t.3T/g+ states (populated in Q branch transitions) are found to mix their electronic character [33]. The eigenstates of both N = 2 levels consist of a 75%-25% mixture [19]. The anisotropy parameters in pure states are /3 = - 0 . 6 (,~) and /3 = - 0 . 3 ( / 7 ) both for Q transitions from F~ + F 3 fine structure levels. Indeed, the experimentally observed values for these levels and transitions are in between these values. The t.3T/g+ ( N = 1) level shows good agreement between experiment and theory for the F 2 level, and indeed has a very small ,~g admixture. For the P and R branch transitions populating the state, mixing in of ,~g+ character is strictly forbidden. The above aspects show that the anisotropy parameter contains a wealth of information. The/3 parameter is determined by the molecular coupling scheme and the level structure. In the case of triplet n = 3 hydrogen, non-adiabatic effects contribute as well. The effects mentioned above explain why we have not yet reached quantitative agreement with calculations. Nevertheless, the results clearly indicate the depolarizing effects of the nuclear and electron spin on photofragment angular distributions and point at the power of observing anisotropy parameters to reveal intramolecular structure.
i3/7~
6. Conclusion We have excited several bound levels in the n = 3 triplet gerade complex, and measured the anisotropy of the dissociation products. It has been found that the anisotropy depends on the initial fine structure levels for the transitions; the electron and nuclear spin depolarize the anisotropy of photofragments. This observation clearly indicates that the excited state fine and hyperfine structure can (at least partially) be resolved. We have presented a theory to calculate the photofragment anisotropy which includes electron and nuclear spin. It has become possible to calculate the/3 parameter as a function of the initial and final fine and hyperfine structure level. The fine and hyperfine structure are shown to decrease the anisotropy given by the theory calculated in the limit of slow predissociation, especially for the low rotational levels. The comparison of theory and experiment indicates that there are still aspects for both the theoretical and experimental molecular spectroscopists to be found in molecular hydrogen. For instance, the strongest lines belong to ortho hydrogen, where the presence of nuclear spin complicates the picture. The present calculation encompasses two extremes, fully separated upper state fine and hyperfine structure levels, and fully coherently excited upper state fine and hyperfine structure levels. In reality the hyperfine structure level separation and the linewidths can be of the same order of magnitude, requiring a more elaborate calculation as in Ref. [10]. The actual hyperfine and fine structure splitting suggest models beyond the hierarchical coupling scheme. The breakdown of the Born-Oppenheimer approximation in the n = 3 triplet manifold is another complication to be considered. We believe we have made clear why depolarization occurs, what the effect is on the photofragment anisotropy and under which circumstances it does or does not play a role.
Acknowledgements We are highly indebted to Prof. R.N. Zare for his suggestions which lead to this paper. KLR would like to thank EPSRC for an Advanced Fellowship and for supporting a two month stay at the FOM Institute. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for
E.R. Wouterset al./ ChemicalPhysics218 (1997)309-323
319
Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Research).
Appendix A In the following we will derive an expression for the anisotropy parameter describing the photofragment angular distribution resulting from the laser excitation of a hyperfine structure manifold. The initial level will be a single fine structure level. We will confine ourselves to triplet states in a diatomic molecule. Our choice of having a single initial fine structure level, whereas the excitation populates a whole manifold, which corresponds to the experimental limitations we have on the bandwidth of the laser (see Section 4), does not limit the applications of this expression for the anisotropy because the removal of the summation over the excited fine and hyperfine structure levels suffices to produce a general expression. The usual expression describing the angular dependence of photofragments is [39] I ( 0 ) = ° ' ° [1 + ~3P2(cos0)] 47/-L
(A.I)
with ~r0 the total cross section, /3 the anisotropy parameter ranging between - 1 and 2, P2(cos0) the second Legendre polynomial in cos O, with 0 the dissociation angle with respect to the polarization vector of the incident linearly polarized photon. We will derive an expression for l(O), from which the anisotropy parameter can be determined. We begin by writing down the wave function of a single hyperfine structure level in a diatomic molecule as IFMF). The states we consider are best approximated in a Hund's case (b) basis [40]. We will use the hierarchical coupling scheme ((bjsj), see Fig. 1). We can write the state [FMF) in terms of the uncoupled state quantum numbers N,M m S,M s and I,M 1 as follows:
[FM~) =
E ( -)~+s-MJ+J+'-M~2~CFV-4-F 2~CFYTY 2vTF-+1 MNMsMjMI ( N × M
S Ms
J )( J -Mj..Mj
I M,
F Ix(R) N * -MI: J R [DMNA(¢'O'O)] ¢e(r)ChsCh"
(A.2)
where (:::) represents a Wigner 3j symbol and D~NA($,O,O) a Wigner rotation matrix. The various M quantum numbers are the projection of the subscripted angular momenta on the laboratory Z axis, and X, ~be, ~bs and ~bt are the vibrational, electronic, electron spin and nuclear spin wave functions, respectively. We will now express the wave function of H 2 in the c3IIu state as IF"M~,c) and that in the excited state in the n = 3 triplet gerade complex as IF'M'r, x). The functions ¢ are labelled correspondingly. Note that in this description of the system parity is ignored (see Ref. [41] for the wave functions including parity), but this has no implication for the validity of our results. The molecule is photoexcited from the metastable c3IIu state to a state x in the n = 3 triplet gerade complex:
IF"M;,c) + hv ---*]F'M'~,x).
(A.3)
Note that the excited state vibrational wave function xx(R) should be replaced with a energy normalized continuum wave function. We will confine ourselves to light which is linearly polarized along the laboratory Z axis, hence M)' = M). We will follow the time-dependent approach of Siebbeles et al. [41], in which the cross section is expressed in the amplitude of the excited state wave function at large internuclear distance R, where
320
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
the nuclei have become separated. The time-dependent excited state wave function in the limit of large internuclear distance is written as R.-.* ~
~b(t) ~
e ikR
~., (F'M'F,Xl,,~i,tlF"M~-,c)lF'M'F,X)(--i)F'eiSr--e -ie, F'M~
(m.4)
R
Here E is the total energy of the system in the initial state and the photon; k = ~ / h , with /x the reduced mass of the molecule, and • the difference between the energy of the excited state and the dissociation limit. In writing out this expression we have used the fact that a single, non-overlapping F' level is excited. The phase shift 6r, of the outgoing wave equals that of the vibrational part of the dissociative wave function, which for R ~ ~ has the asymptotic form sin(kR + 6r, - ~TrF 1 , ), as can be found in quantum collision theory [42]. Note that only the outgoing part e ilcR of the wave function is present in Eq. (A.4). The matrix element (F'n'~,xl,g'intlF"M~,c) is evaluated using the expression for the wave functions from Eq. (A.2) and the dipole interaction Hamiltonian from Ref. [43]. Using the fact that the integration over the spin coordinates merely results in the well known selection rules for a one-photon transition with linear polarized light (AS = AI = 0), one obtains
( F'M'F,XI,'~intlF"M~,c)
=Vfda
E
E
M ~.,.M . . ~. .M ...tMI
M ' I' v M ~ ,M s M, !
tN" S' Nt~
s
J"llu'
x l m ;, M'; X,
( - ) P ~ / ( 2 J " + 1)(2J' + 1)(2F" + 1)(2F' + 1)(2N" + 1)(2N' + 1) ,
-M'; ) I
,
M's
,
- M ; ] [ M'j
M'l'
, - M ; ] [ M'j
M'1
-M' FJ
*
(,'DM;AnlV' "I r~l x-" M ' v A ' L.'lXolx
(A.5)
•
In this equation, ~0 is the polarization of the light in the lab frame (/x 0 = 0 for linear polarized light), and /z represents the polarization in the molecular frame which equals 0 or 1 for a parallel (AA = 0) or perpendicular (AA = 1) transition, respectively. The phase P equals
P = N " + N' + S' + S " - M ' j - M ' j
+ J" + J' + I" + I ' - M ; - M '
F.
(A.6)
V in Eq. (A.5) is the result of the integration over the internuclear separation R and the electronic coordinates r, hence it is the product of a Franck-Condon factor (Xcl Xx) and the electronic dipole matrix element (¢~lle" rll¢~), from which the angular part is written explicitly as Dl0z in Eq. (A.5). V yields no angular dependence of the photofragment distribution, and for clarity will be left out from here on. The integration over the two Euler angles /-2 is performed using Eq. (3.113) from Ref. [39] (note that, for a diatomic molecule, in this equation the third Euler angle X has to be set to zero). This results in
( F'M'F,xI,C~intIF"M~,c ) ot
Y'. pc n¢ tr M~M~MjMI
(N
• pf
s
X M~
M s'
×
0
M~
( - ) P ~ / ( 2 J " + 1)(2J' + 1)(2F" + 1)(2F' + 1)(2N" + 1)(2N' + 1)
. M I v. M.' s M. j M
l
,
s
,)(,
- M ' j ]I M'lv M's
N)(N,
-M~,
A
/.x
-M)
N)
-A'
"
, M'j
M't'
, -M'~ ) M)
lift
-M't: (A.7)
E.R. Wouters et al. / Chemical Physics 218 (1997) 3 0 9 - 3 2 3
321
Substituting Eqs. (A.2) and (A.7) in Eq. (A.4) yields R--*~
&(t)
, .
e ikR
( - - i ) F e '~'
~
e-ie' E
E
E
(_)v
M'~ M~M~'M~'M' t' M~MkM'~M ~
R
×~/(2J" + 1)(2J' + 1)(2F" + 1)(2F' + 1)(2N" + 1)(2N' + 1) ×tM~
M~'
-M;',,M~
M~
-M)]tM~'
[N" 1
N'IIN"
1
N' /
XIM ~ 0
-M'uJ[A"
t~
-A'
-M/,,M)
M'Iv, M ~ , M 'y, M t'
J'
I'
× M'j M;
F'
I' M~
F' ) -M~
I
. . . . . . ( - ) N+s-M'+J+'-M" 2F1/-~ 2 J ' v ~ v l - ~
E
X
M;'
)
-M'v ~e(r)~s'~I'(DNM~ A')
*]
( N' S' M'N M~
J' ) -M)
"
(A.8)
Following Siebbeles et al. [41], the probability of a photofragment being ejected with an angle 0 with respect to the laser polarization is 1(0) =
do-(0,~b)dj2 = Rlim . . . .f
drf d(spin)14~(t)l2,
(A.9)
where the integration has to be performed over the coordinates of all electrons, denoted by r and over the total spin space. After substituting Eq. (A.8), the limit and integrals are readily performed and the following expression is obtained
l( O) ot ~.,
~.,
Y'~
(2F" + l)(2J" + l)(2N" + l)(2F' + l)(2J' + l)(2N' + l)
M'r M'~M's'M'JM'/ M'~M'sM'jM ~
(N" S' × M~ Ms' x x
0 E
j,, ]2[ N' S' -M~'] t M'N M's -M;,
J' 12[ J" -M'j] I M7
I' F" 12[ J', M'I' - M ; ] ~Mj
I' F' 12 M~ -M'F]
A
(2r+0(2J'+l)(2N'+0
M;,
-M)
M)
-M)]
M'NM'sM'jM i N' ×(DM,uA, ) * DN',uA,] ,
(A.IO)
where we from now on use the fact that the terms contributing to the phase P are all integer, which is appropriate for molecular hydrogen because I - - 0 or 1 (para and ortho) and S = 0 or 1 (singlet and triplet states). We also use the fact that the phase factor in the excited state wave function ( - i)r'e ~ ' only depends on the excited state quantum number F'. For the extreme cases of one, well separated hyperfine structure level or of a completely overlapping hyperfine structure multiplet, interference effects between different F' levels are negligible and the phase factor equals 1 when the norm of the wave function 4~(t) is squared. However, for the case of partially overlapping hyperfine structure levels, which we have omitted in this appendix, the factor is
322
E.R. Wouters et al. / Chemical Physics 218 (1997) 309-323
responsible for the interference effects, as observed by Siebbeles and coworkers between overlapping rotational levels [44]. Collapsing all the summations in Eq. (A.10) using Eqs. (3.95), (3.116), and (4.15) from Zare [39], and rewriting in the form of Eq. (A.1) leads to
/3(F',F",J',N',S',I',A')=(-)A'-F'-r-s'I/3-O(2F'
×
'
-A'
J'
+ I)(2J' + I)(2N' + I) F2'' 2
F'
N'
2
J'
'
1
1 (A.11)
with {:::} a Wigner 6j symbol. We used the fact that both F' and F" are integer. Note that the necessary algebra to evaluate this expression can nowadays also be found in computer programs using symbolic notation as, e.g., Mathematica [45]. We now go from this general expression for the anisotropy parameter for a one-photon transition from one hyperfine structure level to another to our experiment with a broad band laser. In that case we are unable to resolve the hyperfine structure of the initial state and both the hyperfine and fine structure of the excited state. To obtain values comparable to the experimental data, weighted summations of/3 in Eq. (A.11) are needed over F', F" and J'. The relative cross sections O'0rF"J' ~ ( 2 F " + 1 ) ( 2 J ' + 1 ) ( 2 F ' + 1) N'
J'
J'
F'
1] '
(A.12)
are the weighting factors. Note that we have only listed the factors depending on J', J", F' and F", as only they are required for the weighted average. Then the phase factor drops out as well. As a final remark we note that if both S and I are taken to be zero in Eqs. (A.11) and (A.12), the expressions for slow predissociafion result (see Eq. (1)).
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