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- The equi-probability surface of s states in the hydrogen atom are usually in the shape of a sphere, which is altered roughly to the form of a spittpoon in the presence of static electric field (Figure 13-02m).
- The graph in Figure 13-02m shows the parabolic coordinates of the system, and the boundary between the classically allowed and
- The stationary Schrodinger equation is separable and solvable in the parabolic coordinates (, , ) with corresponding quantum numbers (n
_{1}, n_{2}, |m|) in the presence of static electric field (Figure 13-02m). This new system is called the Stark states. The principal quantum number n for stand-alone hydrogen atom is related to these quantum numbers by n = n_{1}+ n_{2}+ |m| + 1, where n_{1}and n_{2}associate with the and coordinate respectively (with corresponding number of nodes) while leads to the azimuthal quantum number m. - The stand-alone potential of the hydrogen atom is modified by the static electric field
*F*: V = e*F*z. Figure 13-02n depicts the resulting potential along the z-axis. It shows that the electron motion is always bound in the coordinate whereas the motion along the coordinate depends on the energy E. It could escape either by tunneling or as a free particle if the available E is over the barrier. Since the and coordinates are orthogonal to each other, the electron can be ionized only along the direction with constant reflecting the probability distribution on the coordinate (see insert in Figure 13-02o) when a snapshot is taken on the ionizing electrons from many similar hydrogen atoms.

## Figure 13-02m H Atom in E Field [view large image] |
## Figure 13-02n Energy Landscope |
forbidden region for four excitation levels (to be populated in the experiment and indicated by the energy in unit of cm^{-1}). |

- An atomic hydrogen beam is prepared by photodissociating H
_{2}S (by the 213 nm laser beam), and passing through a 3 mm aperture 65 mm downstream - (a) in Figure 13-02o. - The atoms then enter the velocity map imaging (VMI) spectrometer, an example of which is shown in Figure 13-02p.
- The ground state H atoms are resonantly excited in there to a mixture of 2s and 2p superposition state by two-photon transition with the 243 nm laser. After turning on the static electric field, it is further pumped by a tunable (365-367 nm) UV laser to the quasi-bound states (highly excited Rydberg states) of (n
_{1}, n_{2}, |m|) = (0, 29, 0), (1, 28, 0), (2, 27, 0) or (3, 26, 0) corresponding to n = 29. The energy at these levels is in the order of - 0.002 ev (very close to the ionization level of E = 0). - The electron at such quasi-bound states can become ionized along the coordinate. The huge number of hydrogen atoms in the active region will together show a profile corresponding to the probability distribution on the coordinate in the form of a photocurrent density j
_{n1}(,=_{o}) |_{n1}()|^{2}, where_{n1}() denotes the wave function along the coordinate and n_{1}is the corresponding quantum number. - The function of photoionization microscopy is to reproduce an image reflecting the structure of this electron flux. It is based on a standard VMI spectrometer to map a spectrum according to the velocity of the charged particles from different places. It captures the angular pattern of the probability distribution perpendicular to the z-axis (see insert in Figure 13-02o) by selecting the x and y components of the velocity from various ionized electrons (see "mathematics" for detail).
- At the completion of the ionization process, a voltage difference is applied across the repeller (b) and extractor (c) electrodes, the photoelectrons are accelerated toward the detector (d) consisting of a set of microchannel plates (MCPs).
- Before hitting the detector, the photoelectrons image is magnified by about one order of magnitude with an electric lens (e).

## Figure 13-02o Microscope, Photoionization |
## Figure 13-02p VMI Spectro-meter [view large image] |

- The images of the hydrogen atom are displayed in the middle of Figure 13-02q. The nodal structures for the four cases (as mentioned above) are viewed with the z-axis pointing toward the observer. The size scale is the actual dimension of the MCP detector.
- The transverse nodal structures (on the y-z plane) are shown on the left as the beam exits from the active region of the VMI spectrometer. The size scale is therefore much smaller.
- A comparison of the experimentally measured (solid lines) and calculated radial probability distributions (dashed lines) is shown to the right of the experimental results.

## Figure 13-02q H Atom Images [view large image] |
The theoretical curves are scaled up to the macroscopic dimension of the MCP detector. These curves do not correspond to the images taken by the experiment. |

The inverse transformation from the parabolic coordinates to Cartesian coordinates is :

x = cos()

y = sin()

z = (

At =

dx/dt =

dy/dt =

The (d/dt) is determined by the available energy [E - V()] as shown in Figure 13-02n. The value of (d/dt) increases from zero at the edge of the barrier with the probability distribution varies up and down. It is the (d/dt) together with the angular coordinate that traces a circular pattern on the detecting screen (see insert in Figure 13-02p). The intensity on each circle reflects the probability distribution which is proportional to the number of ionized electron having the particular values of (dx/dt), (dy/dt).