ANGULAR DISTRIBUTION FUNCTIONS OF IONIZED PHOTOFRAGMENTS IN THE DETECTION REFERENCE FRAME

THE DETECTION REFERENCE FRAME

V. ANGULAR DISTRIBUTION FUNCTIONS OF IONIZED PHOTOFRAGMENTS IN THE DETECTION REFERENCE FRAME

A. Parallel transition

From Eq. (12) and Table I, the angular distribution function

N

_||(

θ

φ

χ

) of ionized photofragments in the detection reference frame for the case of a parallel transition in the initial photodissociation process is given by

}

The 2D projection of the image pattern is treated in a companion paper. Without executing the Radon transform,⁵² many interesting features still can be learned from the above equation. In Fig. 3a, a schematic diagram (the symmetry pattern) of the function

φ θ

cos 2

sin is depicted on the surface of a sphere. It exhibits a “left-right” asymmetry with respect to the XZ-plane by viewing this pattern along the direction of the X-axis (in the SFF). When the projection onto the XZ-plane is considered, the net contribution from the

φ θ

cos 2

sin -related terms is null. Thus, the projected image pattern is expected to exhibit a

left-right and a “top-bottom” symmetry with respect to the vertical and the horizontal lines of the XZ-plane at any chosen angle

χ

, respectively. In particular,

]

,47

transition from Eq.(13). Similarly,

(v || J, and for

Eq. (21) tells us that the 3D angular distribution is not cylindrically

symmetric with respect to the Z-axis when . On the other hand, the projected 2D image displays a symmetric pattern with respect to its vertical and horizontal lines no matter which probe angle is chosen. The determination of the anisotropy parameter

β

from projected photoion images by a forward-projection simulation scheme will be presented in a companion paper. It is based on Eq. (22c) for the present case of a parallel transition. As soon as the parameter is determined by the forward-projection scheme, can be calculated from the following equation,

β

, (23a)

where the integrated (total) intensities and are given by

(23b) and

improve the accuracy of the measured

β

and parameters from Eqs. (22c) and (23a), Eq. (22a) or (22b) can be utilized in an iterative simulation. It should be noted that Eq. (23a) is a parametric relationship between

β

and that is imposed by the connection between integrated intensities at different probe angles. There is no functional dependence of on

Alternatively, the

β

₀² parameter can be measured by the slice imaging techniques.

≠90

53 In practice, one has to set the probe laser at an angle or and to choose a non-central slice ( ) in order to observe the contribution from the term

cos + The slice image under the

above-mentioned conditions exhibits a top-bottom asymmetry (cos angular variations).

φ

Without

a priori knowledge on σ

′ and it is difficult to extract quantitative information on from the slice image. Thus, the conventional photoion imaging techniques are recommended to determine and the advantages can be appreciated by utilizing Eqs. (22c) and (23a).

β

B. Perpendicular transition

From Eq. (14) and Table I, the angular distribution function

N

_⊥(

θ

φ

χ

)of ionized photofragments in the detection reference frame for the case of a perpendicular transition in the initial photodissociation process is given by

′

β

σ

)

any probe angle. In Fig. 3b, a schematic diagram (the symmetry pattern) of the function

φ

θ

cos2

cos is depicted on the surface of a sphere. In analogy to the left-right asymmetry

argument presented in the last subsection, the projected image onto the XZ-plane has no contribution from the sin2

θ

cos

φ

- and the cos

θ

cos2

φ

-related terms. Thus, Eq. (24) can be reduced to the following forms at a few selected

χ

angles. They are:

)

From the above equations, we can prove that

Eq. (26) infers that there are only two independent measurements of the angular distributions or the integrated intensities for a given probe transition. Because there are four unknown parameters

β

₀² ,

β

₂² and

σ

′ in Eq. (25), two independent image acquisitions at two different probe transitions should be executed in order to establish the relationship between the unknown parameters. For example, Eq. (26c) can be employed to

P

−branch) and −branch) is called , the following expression can be

When the above

β

₂² expression is substituted into Eq. (26c), the modified cross section is found to be

To establish the parametric dependence of

β

₀² on

β

, we define another ratio and

transition. From Eqs. (26), (28) and (30), we can prove that

When experimentally measured total intensities at various probe angles and probe transitions, as well as the numerical values of a are substituted into the and expressions, Eq. (25) depends solely on a single parameter

β β

₂²

β

in a non-linear fashion.

Under this circumstance,

β

can be determined by a forward-projection simulation scheme from the experimental images. The alignment parameters and are calculated according to Eqs. (28) and (31), provided that

02 that the integrated angular distribution is positive definite. Thus, we obtain

22 forward-projection simulation scheme.

β β

₂²

C. Mixed transition

From Eq. (16) and Table I, the angular distribution function

N

_||,_⊥(

θ

φ

χ

) of ionized photofragments in the detection reference frame for the case of a mixed transition in the initial photodissociation process is given by

)]

any probe angle. In Fig. 3c, a schematic diagram (the symmetry pattern) of the function

φ

θ

cos 2

cos is depicted on the surface of a sphere. Contrary to patterns in Figs. 3a and 3b, it displays a left-right symmetry with respect to the XZ-plane. Thus, there is a net contribution to the 2D-projected ion image from the term )

β

sin2

χ

cos2

θ

cos

φ

45o cos2

φ

-related terms, we reach an important step that

)]

asymmetry on the projection plane. If the difference between the top and the bottom

The anisotropy parameter

β

can be determined from the above equation by a

forward-projection simulation scheme. Because the contribution of all the -related terms to the integrated intensities

Thus, the coherent alignment parameter can be calculated from Eq. (36a), provided that

β

is determined from the hemisphere distribution in Eq. (34) and

σ

′ is calculated according to Eq. (35) where experimentally measured total image intensities at various probe angles and probe transitions are utilized. The determination of and in the present case follows naturally from Eqs. (28) and (31). This top-bottom asymmetry of

opens a window to determine the anisotropy parameter

coherent alignment parameter in a straightforward fashion. In addition, the allowed ranges of are

− which can be proven by selecting

in Eq. (33).

θ

VI. DISCUSSION

The relationships between state multipoles and polarization parameters in the molecular frame are given by

)

Rakitzis and Zare¹¹ have derived the relationships between and in addition, Alexander

) (kq

A a

⁽_q^k⁾(

p

);

54 has summarized the conversion factors between the polarization parameters

and Zare.¹¹ The physical meanings of the coherent parameters Re[ (||, )] and Im[ (||, )] have been elucidated by Rakitzis and Zare.

a

⊥

a

⊥ ¹¹ They concluded that an

asymptotic phase difference, ∆ in de Broglie waves associated with the two dissociation

φ

pathways can be noticeable, so long as a coherent excitation to two excited states with different symmetry characters leads to the same asymptotic state of the photofragment. In

addition, Re[a (||,₁² ⊥)] is proportional to cos∆

φ

(in-phase components) and Im[a (||,¹₁ ⊥ )]

is proportional to sin∆

φ

(out-of-phase components).¹¹ In the present work, the extraction of the dynamical parameter

ρ

~₁² from photoion images has been analyzed in Sec. V (see Eq.

(36a)). By taking the difference between the top and bottom hemispheres of the experimental image at a probe angle , one can eliminate the contribution from the incoherent parameters

45o

χ

ρ

~ and

ρ

~₂². An accurate determination of

ρ

~₁² is not only a

complementary measurement on the phase difference ∆ but also a necessary step when

φ φ

∆ approaches null. Thus, Rakitzis and Zare’s scheme¹¹ is equivalent to the present method where both schemes are capable in measuring the phase difference ∆ .

φ

ρ

在文檔中 National Sun Yat-sen University Institutional Repository:Item 987654321/30415 (頁 67-78)