

\newcommand{\chplus}{CH$^{+}$}
\newcommand{\sihplus}{SiH$^{+}$}
\newcommand{\heneplus}{HeNe$^{+}$}
\newcommand{\mone}{m_{1}}
\newcommand{\mtwo}{m_{2}}

\newcommand{\singpi}{$^{1}\Pi$}
\newcommand{\singsig}{$^{1}\Sigma$}
\newcommand{\trippi}{$^{3}\Pi$}
\newcommand{\tripsig}{$^{3}\Sigma$}
\newcommand{\trippiz}{$\rm ^{3}\Pi_{0^{+}}$}
\newcommand{\trippio}{$\rm ^{3}\Pi_{1}$}

\newcommand{\doubpoh}{${\rm ^{2}P_\frac{1}{2} + ~^{2}S}$}
\newcommand{\doubpth}{${\rm ^{2}P_\frac{3}{2} + ~^{2}S}$}

\chapter{Spectroscopy of the \gehplus ~Molecular Ion}

\section{Context}
\subsection{Introduction}
In contrast to the isovalent species \chplus~and \sihplus~relatively little is known about the spectroscopy of GeH$^{+}$. 
Only one experimental technique has been applied to \gehplus and prior to this work only one electronic transition verified extensive ab-initio studies of many states. New experimental work detailed in this Chapter will attempt to redress this balance by outlining the recording, assignment and analysis of a transition between the ground electronic state and the previously unobserved \singpi~ state.
 
\subsection{Spectroscopy of \gehplus} 
Four electronic states of \gehplus correlate to the first (({$ \rm ^{2}P) Ge^{+} +  (^{2}S) H$}) dissociation asymptote (neglecting spin-orbit splitting): \singsig, \trippi, \singpi ~and \tripsig. The potential energy surfaces calculated for these states are given in Figure ~\ref{figure:abinit}. \\
The first spectrum of \gehplus was observed by Tsuji et al. in 1982 \cite{tsu1} using a helium afterglow experiment \cite{heliumag}. Ions are created using charge exchange between ionised helium (99\%) and GeH$_{4}$ (germane) (1\%). A spectrum is recorded by observing the emission of flowing gases using a monochromator and photomultiplier.\\  An initial experiment resolved 3 vibrational bands which were assigned to [0,1] and [0,0] of the (spin-forbidden) a\trippi$_{0}$$\rightarrow$X\singsig~ transition  and [0,0] of the \trippi$_{1}$$\rightarrow$\singsig~ transition. The data  was least squares fitted yielding  spectroscopic constants for v$^{\prime}$=0 of the \trippi$_{0,1}$ states and v$^{\prime\prime}$=0,1 of the \singsig~ state. \\

         Transitions  between singlet and triplet states are forbidden by the $\Delta$S=0  selection rule [Hund's case (a)]. However, transitions are allowed if the dark states can `borrow intensity' from the singlet states through the spin-orbit interaction \cite{hollas}. Spin-orbit coupling in Ge$^{+}$ is relatively large (1761 \waveno) resulting in strong coupling for \gehplus.\\
The explanation that intensity borrowing from one transition to another can account for this behaviour is not strictly correct. It only arises due to the initial exclusion of spin-orbit coupling from the Hamiltonian. A correct understanding for these spectra requires a Hund's case (c) formalism.

Using apparatus of greater sensitivity,  \gehplus~was studied by Tsuji et al. to record extra emission bands \cite{tsu2}. Five new vibrational bands were vibrationally assigned. A full rotational assignment required still greater sensitivity; achieved in a third experiment \cite{tsu3}. This yielded rotational constants for the v=0, 1 and 2  of the \singsig~ state, v=0 and 1 of the \trippi$_{0}$~ state and v=0 of the \trippi$_{1}$~ state. Despite the large spin-orbit coupling in \gehplus, no evidence of an interaction between triplet and singlet~$\Pi$~states was found; the emission spectra were regular and unperturbed.
Confirmation of the vibrational assignment was achieved through the analysis of the GeD$^{+}$ isotope recorded in emission.\\ 




\subsection{Ab-initio Studies of \gehplus}
The first ab-initio study of \gehplus was undertaken by Binning and Curtis who  calculated the ground \singsig~state \cite{binning}. The first 13 electronic states of \gehplus~ were studied shortly afterwards by  Das and Balasubramanian \cite{kal}. Potentials were constructed and the \singsig~ surface was calculated using second order CI calculations to achieve higher accuracy. The lowest lying potentials are reproduced in Figure ~\ref{figure:abinit}.\\
        The predicted electronic states were found to have good agreement to experimental studies of \singsig~ and \trippi~ states, and matched reasonably to extrapolations of the states of GaH and \sihplus.
 A shallow (almost entirely repulsive) \singpi~ state was predicted. A large uncertainty in the r$_{\rm e}$ was given, which was attributed to the shallow nature of the well.\\
 A long range minimum was predicted for the \tripsig~ state, and the interaction between this state and the \singpi~ and \trippi~ states was predicted to increase significant at long range.


\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,6)(0.5,0)
\put(0,0){\special{psfile=figures/abcvs.eps}}
\end{picture}
\caption{Ab-initio results for the  first four potential surfaces of the germanium hydride cation as calculated by Das and Balasubramanian (Reproduced from \cite{kal})}
\label{figure:abinit}
\end{center}
\end{figure}



\section{Experimental considerations}


\subsection{Spectroscopic Experiments}
A precursor gas (5$\times 10^{-6}$ torr) of Germane\footnote{With thanks to Professor Paul Davies} (GeH$_{4}$) was electron impact ionised by a thoriated tungsten filament operating with a trap current of 500 $\mu$A, and accelerated from the ion source at 1425 V. The mass spectrum was recorded (Figure \ref{figure:masspec}) and assigned.\\
 Two isotopes of \gehplus~ were chosen for fast ion beam experiments, \mbox{\gehzero~and \gehfour.} Laser photofragment spectra were recorded by detecting the Ge$^{+}$ fragment arising from predissociated electronic states.\\
 The ions were coaxially irradiated by the focused output of a CR699-29 continuous wave dye laser. Typical magnet currents used to select the parent/daughter species were  2.8 $\rightarrow$ 3.0 A. \\
Metastable fragment ions (Ge$^{+}$) due to unimolecular and collisional dissociation were observed using the second electromagnetic sector and off-axis electron multiplier with no laser radiation. The appearance of fragment ions due to collisions was verified by the observed increase in Ge$^{+}$ ions with the diffusion pump in the flight region capped using the baffle. \\

A 1 m focal length lens was inserted 40 cm before the laser entered the vacuum apparatus to maximize the ion beam/laser overlap. \\
Scan parameters were as follows: a 30 MHz sampling rate and a  scan speed of 0.5 \waveno/min. Lock-in amplifier settings: time constant 300 ms, typical sensitivity 20 mV (depending on local conditions such as ion beam current, laser power, dye age and region scanned) \\

The  mass/charge ratio ({\it m/z}) = 71 line is mono-isotopic (attributable to \gehzero), it was preferable to record the spectroscopy of this species (rather than other lines [e.g. {\it m/z} = 75] which contain many species  possibly congesting the spectrum.)\\
 However, the overwhelming majority of spectroscopic lines were found in both {\it m/z}= 71 and 75. The \gehfour~isotope has a greater abundance (corresponding to greater ion current Figure~\ref{figure:masspec}) and was hence used  for continuous scanning between 16300 \waveno and 18500 \waveno. A selective spectrum of  \gehzero~ isotope was recorded by scanning regions of interest predicted from the \gehfour~ spectrum.
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(0,3)(3,0)
\put(0,0){\special{psfile=figures/masspec.eps}}
\end{picture}
\caption{Mass Spectrum of GeH$_{4}$}
\label{figure:masspec}
\end{center}
\end{figure}



\begin{figure}
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(0,8)(5,0.75)
\put(0,0){\special{psfile=figures/stickspec.epsi}}
\end{picture}
\caption{Stick Spectrum of all recorded lines of \gehfour. Note clusterings of lines around 16700,17600 and 18400 \waveno.}
\label{figure:allline}

\end{center}

\end{figure}


\subsection{Momentum Release Experiments}

The apparatus for momentum release experiments was modified as follows: two slits (width 30 $\mu$m and  50  $\mu$m) were placed at the entrance and exit lens stacks of the second electromagnet respectively. This achieved sufficient sensitivity to resolve momentum releases from the intrinsic experimental profile. The slits were manufactured from razor blades using a spot welder.
The energy releases for \gehfour were found to be congested with the releases due to:\\
 $^{73}{\rm GeH}_{2}^{+}\rightarrow ^{73}{\rm Ge}^{+}$, \\
$^{73}{\rm GeH}_{2}^{+}\rightarrow ^{73}{\rm GeH}^{+}$,\\
$^{72}{\rm GeH}_{3}^{+}\rightarrow ^{72}{\rm GeH_{2}}^{+}$,\\
$^{72}{\rm GeH}_{3}^{+}\rightarrow ^{72}{\rm GeH}^{+}$ and \\
 $^{72}{\rm GeH}_{3}^{+}\rightarrow ^{72}{\rm Ge}^{+}$, \\
The number of fragments arising from the {\it m/z} = 75 peak make recording momentum releases of the ${\rm ^{74}Ge^{+}}$ fragment difficult, hence  the isotopically pure \gehzero~ was used, and the  ${\rm ^{70}Ge^{+}}$ fragment collected.\\
The scan voltage, which controls current through the electromagnet, and the ion current arriving at the electron multiplier were recorded simultaneously on the AUTOSCAN data acquisition system. The scan voltage was calibrated against the magnet current, and a momentum abscissa calculated by comparison with an accurate mass spectrum.


\section{Spectroscopic and Momentum release experiments of \gehplus}

The spectroscopic data was analysed in terms of line position, FWHM, and intensity. Line positions and intensities (unnormalised) are shown in Figure \ref{figure:allline}. A sample 50 \waveno~ spectrum is shown in  Figure \ref{figure:recspec}. 


\begin{itemize}

\item{Over 160 lines were recorded between 16300 and 18500 \waveno~ and are shown in Figure \ref{figure:allline}. The region between 17800 and 18050 \waveno was not recorded.\footnote{This is due to low laser power achievable with a 699-29 dye laser in this region. This portion of the spectrum lies between the Rhodamine 110 and Rhodamine 6G laser dyes (see Table ~\ref{table:dye}), so a mix of dyes was necessary. After mixing various recipes of dyes, it was decided not to attempt this region, as the laser cavity was unstable, with low power output, increasing scan time beyond an acceptable limit.}}
\item{Linewidths vary between 0.001 \waveno ~(at the lower limit of experimental detection) and $>$1.5 \waveno. Two broadened lines are shown in Figures \ref{figure:q8v0} and \ref{figure:q14v1}. Two of the narrowest assigned lines are given in Figures \ref{figure:q17v0} and \ref{figure:p13v1}. }
\item{The majority of the  spectroscopic transitions of ~\gehfour~ could be found in ~\gehzero~ accompanied by an isotopic  shift of between -1 to -3 \waveno~ (i.e. ~\gehzero~ lies to lower energy.)}
\item{Lines can be seen to cluster around three regions 16500, 17650 and 18300 \waveno~. See Figure~\ref{figure:allline}}
\item{Spectra of ~\gehfour~ and ~\gehzero~ are sufficiently resolved to see narrow \mbox {$<$0.01 \waveno~  splittings}  between pairs of lines (see Figure \ref{figure:hysplit}). }
\end{itemize}



\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(3,3.6)(2.5,0.8)
\put(0,0){\special{psfile=figures/q8v0.epsi}}
\end{picture}
\caption{Line exhibiting large linewidth for low J in the v$^{\prime}$=0 Q branch of \gehfour}
\label{figure:q8v0}
\end{center}
\end{figure}


\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(3,3.6)(2.5,0.8)
\put(0,0){\special{psfile=figures/q17v0.epsi}}
\end{picture}
\caption{Narrow line at intermediate J in the v$^{\prime}$=0 Q branch of \gehfour.}
\label{figure:q17v0}
\end{center}
\end{figure}



\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(3,3.6)(2.5,0.8)
\put(0,0){\special{psfile=figures/p13v1.epsi}}
\end{picture}
\caption{Line exhibiting lifetime narrowing at intermediate J in the v$^{\prime}$=1 state of \gehfour}
\label{figure:p13v1}
\end{center}
\end{figure}




\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(3,3.6)(2.5,0.8)
\put(0,0){\special{psfile=figures/q14v1.epsi}}
\end{picture}
\caption{Line exhibiting broadening for high J in the v$^{\prime}$=1 state of \gehfour}
\label{figure:q14v1}
\end{center}
\end{figure}




\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(3,3.6)(2.5,0.8)
\put(0,0){\special{psfile=figures/hysplit.epsi}}
\end{picture}
\caption{Line (unassigned) exhibiting  splitting consistent with proton hyperfine splitting.}
\label{figure:hysplit}
\end{center}
\end{figure}





\begin{figure}
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(5,8)(2.7,0.75)
\put(0,0){\special{psfile=figures/rawdat.eps}}
\end{picture}
\caption{A sample portion of the ~\gehfour~ spectrum recorded with 800 mW of laser power, 20 mV sensitivity, 300 ms time constant, $3\times 10^{-6}$ torr of germane and with a 30 MHz sampling rate (vacuum and lab waveno. abscissa shown). Assignments correspond to those described later in this chapter.}
\label{figure:recspec}
\end{center}
\end{figure}

\newpage

\section{Assignment}

Assignment of the laser photofragment spectrum of \gehplus~ was achieved through several parallel methods:
\begin{itemize}
\item{Experimental line positions}
\item{Line signatures [linewidths, hyperfine splittings and intensities]}
\item{Ground state differences}
\item{Isotope shifts}
\item{Calculated line positions}
\end{itemize}
 In this section, these methods will be outlined, the obtained results displayed and an analysis of the assigned absorption spectra is given. 




\subsection{Experimental line positions}
 The spectrum was believed to consist of transitions from the ground (X\singsig) to various predissociated upper states, in particular the  $\Pi$ states dissociating to the $^{2}$P$_{\frac{3}{2},\frac{1}{2}}$. These states are energetically accessible to the wavelength of the laser radiation used and are estimated to have reasonable Frank-Condon factors. An initial attempt was made to assign the spectrum to these electronic states.\\
 The first method used in the assignment was a search for regularities in the spectrum, i.e. comparing differences between line positions, intensities and linewidth signatures. Initial attempts centred around 17600 \waveno~ due to the high density of lines. 
However, this region proved too congested to find regularities. Many lines in this region vary dramatically in intensity, linewidth  and statistically significant regularities between three lines could always be found. However, none of these could be extrapolated further, making it impossible to correctly assign the spectrum using this method alone.

\subsection{Calculation of transitions between electronic states}
Another technique used  to assign \gehplus was to generate a full spectrum for all states using data from the  ab-initio and emission studies in conjunction with the programs of Leroy \cite{leroyrkr},\cite{leroylevel}.\\
Predicted spectra could be compared  directly to the experimental data, allowing for small shifts due to changes in spectroscopic constants, perturbations etc. This method has an inherent disadvantage of extrapolating data from the lowest lying states (observed in the emission studies) to higher lying states. It is not possible to accurately predict spectra in this way, due to fluctuations in the potential surfaces near dissociation and interactions between states. Despite these difficulties, it  was hoped that an approximate spectrum could be generated.\\
The small number of parameters taken from the ab-initio study \cite{kal} for the \singpi~ state  made the construction of potentials for this state difficult. Large uncertainties were associated with the given parameters, adding to these difficulties.\\

As a first step towards assignment, simulated spectra for the \trippi-\singsig~ transitions were created using parameters obtained from emission studies by Tsuji et al. on transitions between $^{3}\Pi_{0^{+}}$, ~$^{3}\Pi_{1}$ and $^{1}\Sigma$ states.\\
The $^{3}\Pi_{2}$ states were not observed in emission. It is known however, that both parity components correlate to the lower dissociation asymptote (\doubpoh) and hence would not be observed in a photofragmentation experiment.\cite{williams1}\\


 Construction of a realistic \singsig~ surface was possible due to the availability of  molecular parameters for the three lowest vibrational states. A small modification to the molecular parameters from \cite{tsu3} was  made before the behaviour of the \singsig~ state was  considered physical:\\
If  the parameters $\gamma_{\rm e}$, B$_{\rm e}$   and $\alpha_{\rm e}$ are  used in Equation ~\ref{equation:fullbv} to calculate the B values for each state with vibrational quantum number v,  B$_{\rm 5}$ is seen to be larger than B$_{4}$, clearly  a counter intuitive situation, (bond length would be expected to increase with v, therefore B decreases).

\begin{equation}
{\rm B_{v}= {B}_{e}-\alpha_{e}\left({v} +\frac{1}{2}\right)+\gamma_{e}\left({v} +\frac{1}{2}\right)^{2}}
\label{equation:fullbv}
\end{equation}

This yields:\\
B$_{3}$=6.3275, ~B$_{4}$=6.2875 and B$_{5}$=6.3075, \waveno\\ when  B$_{\rm e}$=6.94, $\alpha_{\rm e}$=0.28 ~and $\gamma_{\rm e}$=0.03 \waveno.\\

 Due to this non-physical behaviour of B$_{v}$,  a  least squares fit of B$_{\rm v}$  against v+1/2 was used to determine the value of ${\rm B_{e}~ and ~\alpha_{e}}$ from Equation \ref{equation:halfbv}: 
\begin{equation}
{\rm B_{v}= {B}_{e}-\alpha_{e}\left({v} +\frac{1}{2}\right)}
\label{equation:halfbv}
\end{equation}
Fitting two parameters to three data points is ambiguous, however the resulting B$_{\rm v}$ values decreased with increasing vibrational quantum number as it must for a physical system. Values calculated for B$_{\rm e}$ and $\alpha_{\rm e}$ are 6.90(8) and 0.19(6) \waveno~ respectively (the numbers in brackets indicate 3 $\sigma$ in the last quoted digit). Potential energy surfaces constructed from these parameters (using the {\bf RKR1} program \cite{leroyrkr}) were considered to be the best possible using only the data from the emission studies. Any resulting predictions for higher vibrational states in the \singsig~ state are therefore more likely to correlate to the recorded spectrum than if the unadjusted parameters were used.
However, despite this correction,  no assignment could be made between the laser photofragment spectrum and the calculated spectrum for \trippi~- \singsig~ states. 



\subsection{Ground State Differences}
Using experimental and ab-initio molecular parameters with the programs of Leroy, \cite{leroy}, it was possible to predict ground state splittings between rotational levels.

This method for assigning a spectrum is useful where the upper electronic state is strongly perturbed \cite{sarresih}, as it relies solely on the lower state splittings.
 Two transitions labelled  P(J+1)  and  R(J-1) will both arrive at a given J in the excited electronic state (P and R transitions retain the same e-e parity). See Figure \ref{figure:gehpqr}. The lower state splitting is given (to first order) \cite{bernathbook} by:
\begin{equation}
 {\rm \Delta E=B_{v}''(5J+6)}
\end{equation}
If the upper electronic state is strongly perturbed, the position of the line will move. However, the ground state splittings remain unchanged, as both the P(J+1) and R(J-1) lines will be perturbed in the same direction and by the same amount. The ground state splittings can be used as an assignment tool provided that the states of the \singsig~ from which the transitions arise are sufficiently removed from the perturbing effects of the upper states. \\
 Two transitions sharing the same upper state have identical lifetimes and hence have equivalent linewidths. A search was undertaken for all pairs of lines having the same width, to match with the predicted ground state splittings.\\
 The splittings for each vibrational state were calculated from recorded transitions of \gehplus~\cite{nishpersonal}, or extrapolated from Equation~\ref{equation:halfbv}



\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(0,3)(2.5,0)
\put(0,0){\special{psfile=figures/gehpqr.eps}}

\end{picture}

\caption{Ground state (J$\rightarrow$J+2) differences (lambda doubling of the \singpi~ state exaggerated for clarity)}
\label{figure:gehpqr}
\end{center}

\end{figure}




\subsection{Isotope Shifts}
The technique outlined here proved decisive in finding regularities in the spectroscopic data which lead to assignment. It is known that (to a first approximation), the isotope shift in a given vibrational band is approximately constant \cite{herzberg}. When lines from  \gehzero were matched to \gehfour (through matching linewidth, intensity and position), isotope shifts could be calculated. Lines having similar shifts could then be grouped, allowing regularities in the line positions to be found.\\

It was found that the majority of the groups had splittings which corresponded closely to predictions for v$^{\prime\prime}$=3 of the  X\singsig~ state.\\
No degradation of the `R' branch (the highest lying groups were initially postulated as `R' branches) occurred. This indicated that the R branch formed a band head rapidly and hence the transitions are associated with a large $\Delta$ B. This is not observed in the emission studies (where B$^{\prime}$=88\% of B$^{\prime\prime}$), suggesting that an electronic state other than the \trippi~ was involved.\\
 The initial B$^{\prime}$ value was estimated to be  3 \waveno ~(corresponding to an internuclear separation of 2.4 \AA). This  matches the predicted minimum of the \singpi~ state from the ab-initio study by Das and Balasubramanian \cite{kal}. It should be noted that this value is too large for the upper state to be the \trippi~. It order to quantify this B$^{\prime}$ value and other molecular parameters accurately, a least square fitting routine was used.



\subsection{Non linear least squares fitting}
Lines were initially assigned to vibrational and rotational states, using results from ground state splittings. Line positions were then used as the input into  a non-linear least squares fitting routine (using a REFINE algorithm), fitting to the following expressions:\\ 
\begin{eqnarray}
{\hspace{-0.8cm}\rm A^{1}\Pi:\hspace{0.5cm}} {~\rm F_{v}(J)}&=& {\rm B_{v}\prime(J(J+1)-1)-D_{v}\prime[J(J+1)-1]^{2}} \nonumber \\
\hspace{0.4cm} &+&{H_{v}\prime[J(J+1)-1]^{3}\pm q_{v}[J(J+1)}] +{\rm T_{v\prime,v\prime\prime}} 
\label{equation:singpi}
\end{eqnarray}

\begin{equation}
{\rm X^{1}\Sigma:\hspace{0.5cm} F_{v}(J)= B_{v}\prime\prime(J(J+1))-D_{v}\prime\prime[J(J+1)]^{2}+H_{v}\prime\prime[J(J+1)]^{3}}
\end{equation}

Where B, D and H are the rotational terms defined in Chapter 2, q$_{\rm v}$ is the lambda doubling parameter for a particular vibrational state, (+ or -) refers to levels of  e and f parity respectively, ${\rm T_{v^{\prime},v^{\prime\prime}}}$ is the separation between the vibrational states v$\prime$ and v$\prime$' in  the \singsig~ and \singpi~ electronic states.\\

One disadvantage of fitting lines using a least squares method is that a systematic error can occur in the numbering of the rotational quantum number J. This was however eliminated,  using the method of ground state splittings outlined above.\\
 The Q transitions were not initially included in the fit, due to the uncertainty in the magnitude of the lambda doubling for the \singpi~ state. If a large lambda doubling parameter exists for \gehplus, this would produce a significant movement of the Q branch with respect to P and R branches. Through trying different fits of the Q branch lines to different J quantum numbers, a definite assignment could be determined and a physical q parameter deduced.\\

 The assigned lines are plotted in Figure~\ref{figure:assdat} and the line positions given in Tables \ref{table:70assign} and \ref{table:74assign}. The overall standard deviation of the fits for the \gehzero and \gehfour data  are $3.89\times10^{-3}$ and $3.88\times10^{-3}$ respectively. Molecular parameters from the least squares fitting procedure are given in Table~\ref{table:lsqfit}. The magnitude of the  q parameter is seen to be consistent with that found in \chplus \cite{carrramsay} and \sihplus \cite{lutz}. The lambda doubling is shown to be linear with ~J(J+1), for the low J values assigned here, suggesting that higher order terms are unnecessary. See Figure \ref{figure:lambdoub}. 

\newpage
\begin{table}[!h]
\centering
 \begin{tabular}{|ccc|}

\hline
\hline
{ \normalsize \bf Spectroscopic}&{\bf \gehfour} & {\bf  \gehzero}\\
{\normalsize \bf Constant}& {\bf \waveno}& {\bf \waveno}\\
\hline
\hline
&&\\
\underline{\bf  \singsig~ state} &  &  \\
&&\\

B$_{3}$ & 6.2104(9) &6.2148(8) \\
D$_{3}\times10^{4}$ & 3.02(2)& 3.02(2)\\
&&\\
B$_{4}$ & 6.042(8) &- \\
D$_{4}\times10^{4}$ &5(1)&-\\
&&\\



&&\\
\hline
\hline
&&\\

\underline {\bf \singpi ~state} &  &  \\
&&\\
B$_{0}$ &$3.319(1)$  &3.321(1)\\
D$_{0}\times 10^{4}$ &$ 9.30(6)$ & 9.26(6)\\     
H$_{0}\times10^{7} $& $2.5(1)$ &  2.4(1)  \\     
q$_{0}\times10^{3}$ & $5.9(1)$&5.80(8)\\
&&\\
B$_{1}$ & 3.131(8) & 3.110(6)    \\
D$_{1}\times 10^{4}$ & 11.4(6)    &  9.7(1)   \\
H$_{1}\times10^{7}$ & 9(1)    & 5.1(9)   \\
q$_{1}\times 10^{3}$& 7.2(2) &  7.6(1)      \\
&&\\
B$_{2}$ &3.000(3) & 2.998(4)  \\
D$_{2}\times 10^{4}$ & $19.1(6)$ &18.1(6) \\
q$_{2}\times 10^{3}$ &4.9(9)  &    6.0(8)\\
&&\\
T$_{0,3}$&  17673.64(2)  & 17671.32(3)\\
T$_{1,3}$&  18055.7(3) &   18054.5(3)\\
T$_{2,3}$& 18397.64(5)&    18395.86(6)\\

&&\\

\hline
\hline
\end{tabular}
\caption{Molecular parameters from the least squares fitting to \singpi-\singsig~ transitions. (Bracketed values give 3$\sigma$ in the last quoted value).}
\label{table:lsqfit}
\end{table}

\newpage


\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(5,3.8)(1.8,0.5)
\put(0,0){\special{psfile=figures/lambdoub.eps}}
\end{picture}
\caption{Lambda doubling in \gehfour. v$^{\prime}$=0,1 and 2  states are represented by +, $\times$  and $\ast$ respectively.}
\label{figure:lambdoub}
\end{center}
\end{figure}





\subsection{Vibrational assignment}
\subsubsection{Lower state}

The rotational constant B for the lower state of the majority of transitions (See Table ~\ref{table:lsqfit}) is calculated to be  6.21 \waveno. This is consistent with the calculated B$_{3}$ value for this state (6.23 \waveno~ from Equation ~\ref{equation:halfbv}). It is therefore proposed that the majority of transitions arise from rovibrational levels the v$^{\prime\prime}$=3 of the \singsig~ electronic state.
 This vibrational state is responsible for three of the observed bands, it's J = 0 level lying ~$\approx$~6650 \waveno~ above the minimum of the \singsig~ electronic state. \\ An estimate of the minimum energy required to dissociate the \gehplus~can therefore be made. This value is calculated to be (6650 + 17673 = ) 24324 \waveno. The \doubpoh~ asymptote must lie below this value, in agreement with the ab-initio study, which gave this limit at 23793 \waveno.


\subsubsection{Upper state}
The upper state vibrational quantum number can be deduced from a consideration of the shift of the observed bands between \gehfour and \gehzero. In general, the isotope shift ($\delta\nu$) is given by:\\
\begin{equation}
{\rm \delta\nu =(1-\rho)\omega_{e}\left(v+\frac{1}{2}\right)-\left(1-\rho^{2}\right)\omega_{e}x_{e} \left(v+\frac{1}{2}\right)^{2}}\\
\label{equation:geniso}
\end{equation}


Where \begin{equation} \rho=\sqrt{\frac{\mu_{1}}{\mu_{2}}}
\label{equation:redmass}
\end{equation} \\

For this case:\\
\begin{equation}
{\rm T_{0}~(^{70}GeH^{+}) - T_{0}~(^{74}GeH^{+})  = \delta\nu(lower) -  \delta\nu(upper)} \\
\label{equation:boriginshift}
\end{equation}






\begin{equation}
{\rm  \delta\nu\left({lower}\right) =\left(1-\rho\right) \omega_{e}^{''} \left(v^{''}+\frac{1}{2}\right) - \left(1-\rho^{2}\right) \omega_{e}x_{e}^{''} \left(v^{''}+\frac{1}{2}\right)^{2}}\\
\label{equation:isolower}
\end{equation}

\begin{equation}
{\rm \delta\nu\left({\rm upper}\right) =\left(1-\rho\right)\omega_{e}^{'}\left(v^{'}+\frac{1}{2}\right)-\left(1-\rho^{2}\right)\omega_{e}x_{e}^{'} \left(v^{'}+\frac{1}{2}\right)^{2}}
\label{equation:isoupper}
\end{equation}


 Where $\omega_{e}^{''} ~{\rm and}~ \omega_{e}x_{e}^{''}$ are taken from the data of Tsuji et al. \cite{tsu3} and  $\mu_{1}$ and $\mu_{2}$ are the reduced masses of \gehzero~ and \gehfour~ respectively.\\
 
An isotope shift for a transition between the lowest state of the \singpi~  and the v$^{\prime\prime}$=3 is calculated from Table ~\ref{table:lsqfit}.
The parameter $\omega_{e}x_{e}^{'}$ is given explicitly through solving three simultaneous equations for G(v). A vibrational assignment was then found by assuming the first observed vibrational state (band origin 17674 \waveno) can be assigned to v$^{\prime}$=i=0. 
An isotope shift and $\omega_{e}'$  were calculated. Comparison was then made  to the experimental result.
The first observed state was then assumed to be the v$\prime$=i=1. The isotope shift and $\omega_{e}'$ were recalculated.\\
In this manner, the isotope shift was calculated assuming an isotope shift for v$^{\prime}$=0$\rightarrow$5. The result of these calculations are found in  Table \ref{table:isoshift}.\\
The best fit  occurs when i=0, i.e. the lowest observed band (17674 \waveno) of the \singpi~ state is the v$^{\prime}$=0 vibrational band.\\

This method of vibrational assignment by isotope shift can be applied to any of the observed states (i.e band origins at 17674, 18056, 18398 \waveno). However the state with a band origin at 17674 \waveno~ was chosen due to the large number of assigned lines in both isotopes and the lowest uncertainty in the band origins (See Figure~\ref{figure:assdat} and Table~\ref{table:lsqfit}).


\begin{table}[!h]
\centering
 \begin{tabular}{|c|c|c|}

\hline
\hline
{ \normalsize \bf i}&{ $\bf \omega_{e}$}&{\bf Shift assuming v$\prime$=i}\\

\hline
\hline
&&\\
{\bf Experimental} &&{\bf 2.32(4)}\\
&&\\
\hline
{\bf Calculated}&&\\
&&\\
0     &   422.227300  &   2.322745  \\
&&\\
1     &   462.370400  &   2.168231  \\
&&\\
2     &   502.513500  &   2.013711  \\
&&\\
3     &   542.656600  &   1.859185  \\
&&\\
4     &   582.799700  &   1.704653  \\
&&\\
\hline
\end{tabular}
\caption{Comparison between the recorded and calculated isotope shifts for the v'=0 state of the \singpi~ electronic state (all values given in \waveno)}
\label{table:isoshift}
\end{table}
 An extensive search for transitions attributable to lower lying vibrational states of the \singpi~  was undertaken using the methods outlined above. However, no lines were found which could be assigned to extra states, verifying the assignment by isotope shift.\\

\begin{figure}[!ht]

\setlength{\unitlength}{1in}
\begin{picture}(4,8.4)(2.5,0.5)
\put(0,0){\special{psfile=figures/lineplot.eps}}
\end{picture}
\caption{Assigned Transitions (\singpi$\leftarrow$\singsig) of \gehplus. Line intensity normalized for clarity.}
\label{figure:assdat}

\end{figure}


\subsection{Evidence for assignments}

Having assigned the lines rotationally, vibrationally and electronically, it was possible to use the fitted experimental data to modify  existing parameters for the \singsig ~state and make comparisons between experimental and ab-initio results for both the \singpi~ and \singsig~ states (Table \ref{table:sin}). 
The B$_{\rm e}$ value can be seen to  agree with the estimation made by Das and Balasubramanian {\cite{kal}, if the estimation of R$_{\rm e}$  is taken from their calculation using a better basis set and inclusion of electron correlation terms.
$\omega_{e}$ is a factor of two greater than that predicted by the ab-initio work. This is probably due to the difference between the shallow \singpi~ surface predicted and the observation that the well is at least 1000 \waveno~ deep (and supports at least three bound vibrational states).\\
 The value of T$_{\rm e}$  needs revising upwards as a consequence of the observed equilibrium of the  \singpi~ state lying 800 \waveno~ higher than that predicted in the ab-initio studies \cite{kal}.\\
Results taken from the emission studies were combined with the fast ion beam results to provide the most accurate set of rotational parameters for the \singsig~ state. The second part of Table \ref{table:sin} gives these values and makes comparison to the original emission study values.
The B$_{\rm e}$ value from the predissociation data is found to lie within the experimental uncertainty of the emission studies (and vice-versa), but $\alpha_{\rm e}$ is lower than that found in emission. This is almost certainly due to the inclusion of the (incorrect) $\gamma_{e}$  term in Equation~\ref{equation:fullbv}.





\begin{table}[!h]
\centering
 \begin{tabular}{|c|c|c|c|}
\hline
{\bf Parameter}&{\bf \gehfour}& {\bf \gehzero}&{\bf Literature}\\
&(\waveno)&(\waveno)&(\waveno)\\
\hline
{\underline {\bf \singpi ~state}} &&&\\
&&&\\
B$_{\rm e}$&  3.40(2)& 3.40(2)& 2.1 \\
&&&\\
&&& 3.1(8)\footnote{Estimated from calculations using an improved basis set}\\
&&&\\
$\alpha_{\rm e}$&0.16(1)&0.16(2)&N/A\\
&&&\\
$\omega_{e}$&422(2)&425.0(8) &230\\
&&&\\
$\omega_{e}x_{e}$&20.1(9)  done& 20.9(6)&N/A\\
&&&\\
T$_{e}$&24115(2) &24113.8(8)&23270\\
&&&\\
\hline
{\underline {\bf \singsig ~state}}&&&\\
&&&\\
B$_{\rm e}$&   6.90(6) & 6.90(5)& 6.94(5) \\
&&&\\
$\alpha_{\rm e}$&0.20(2)& 0.20(1)& 0.28(5)\\
&&&\\


\hline   
\hline



\end{tabular}
\caption{Comparison of the molecular parameters for the electronic states of \gehplus in the fast ion work  to ab-initio and experimental work results \cite{kal} and \cite{tsu3}.}
\label{table:sin}
\end{table}






\begin{table}[!h]
\centering
 \begin{tabular}{|cc|c|c|c|c|c|}
\hline

\hline
  &{\bf v$\prime$'}&0&1&2             &3         &4\\
{\bf v$\prime$}& & & & & & \\ 
\hline
&&&&&&\\
0&          &23320 &21371 & 19488     &{\bf 17673.62} & 15921         \\
&          &{\it 23320} &{\it 21371} & {\it 19488}     &{ \it \bf 17671.32} &{\it 15921}         \\
&&&&&&\\
\hline
&&&&&&\\
1&          &23702 &21753 & 19870     &{\bf 18055.73} & 16303         \\
&          &{\it 23703} &{\it 21754} & {\it 19872}     &{ \it \bf 18054.50} &{\it 16305}         \\
&&&&&&\\
\hline
&&&&&&\\
2&          &24043 &22094 & 20212     &{\bf 18397.65} &{\bf 16644.78} \\
&          &{\it 24045} &{\it 22096} & {\it 20213}     &{ \it \bf 18395.86} & {\it 16646} \\
&&&&&&\\
\hline



\hline
\end{tabular}
\caption{Deslandres table of the  band origins (\waveno) of the \singpi-\singsig~ system of \gehzero (Italicised) and \gehfour. Assigned observed bands are shown in bold. Other bands shown are predicted.}
\label{table:destab}
\end{table}





\subsection{Internal consistency}

The high precision with which line positions were measured, allows accurate determination of the exact ground state splitting between P and R branches. If two pairs of transitions differ only in their upper vibrational state, the line spacings between them should agree (within experimental uncertainty). This is found to be the case for all lines which can be checked. See Table \ref{table:74intcons}.

\begin{table}[!h]
\centering
 \begin{tabular}{|c|c|c|c|}
\hline
{\bf Gnd. State Difference}&{\bf v$^{\prime}$=0}&{\bf v$^{\prime}$=1}&{\bf v$^{\prime}$=2}\\
\hline
R(1)-P(3)&62.0315  & - &  62.0936 \\
&&&\\
R(2)-P(4)&86.8195  & - &  86.8191 \\
&&&\\
R(3)-P(5)&111.509  & - &  111.5631\\
&&&\\
R(4)-P(6)&136.2025 & - &  136.2067\\
&&&\\
R(5)-P(7)&160.7901 & - &  160.7939\\

&&&\\
\vdots&&&\\
&&&\\
R(9)-P(11)& 258.0169 &258.0048 &-\\
&&&\\
R(10)-P(12)&281.9671 &281.9786 &-\\
&&&\\
R(11)-P(13)&305.7762 &305.7687 &-\\
&&&\\
R(12)-P(14)&329.3729 &329.3865 &-\\
&&&\\
R(13)-P(15)&352.8305 &352.80   &-\\
&&&\\

\hline
\end{tabular}
\caption{Ground state splittings for the v$\prime$'=3, \singsig~ state of \gehfour}
\label{table:74intcons}
\end{table}












\subsection{Linewidth information}

Two transitions sharing a common upper state have identical linewidths and signatures (e.g. hyperfine splitting, lifetime broadening). One method to check an assignment is to ensure that the transitions to the same upper state have similar linewidths and profiles. This was found to be true for all assigned lines (vacuum wavenumber listing given in \ref{table:70assign} and \ref{table:74assign}).\\
All assigned lines  were found to be singlet lines with no hyperfine splittings observed. The exception to this is the v$^{\prime}$=2, J$^{\prime}$=4 level. In this state,  the P(5) and the R(3) lines are split by $9.98\times10^{-3}$ and $9.99\times10^{-3}$ (\waveno) respectively [the Q(4) line is not split due to the different parity (`f') of the upper state]. The three observed lines to this state are given in  Figure~\ref{figure:j4}. The R(3) line for the v$\prime$=4 state was observed, with a poorer signal-to-noise ratio than that found for the v$^{\prime\prime}$=3. The P(5) line is almost certainly embedded in the background signal.\\
When predicted singlet lines take on triplet characteristics (such as hyperfine splittings) this can be  attributed to the mixing of the wavefunction for the  singlet state with that of a  wavefunction of a triplet state(s) \cite{whitham}. This is clearly the case for these lines (which have  e' parity), J$^{\prime}$=4, v$^{\prime}$=2, although the same is not true for the Q(4) lines. All three lines are almost certainly arriving in the same upper state, due to the intensity alternation, the splitting between the lines and the linewidth of each line.


The linewidth (and hence lifetime) of rotational levels in a vibrational band of the \singpi~ state can be seen to vary over two orders of magnitude with J (and increasing energy). Lines follow the same general pattern for all three of the  upper vibrational states: The lowest J in the band have widths~$>$0.1 \waveno~; upon increasing  J, a decrease in  width is observed. This continues until a  minimum is reached (at the limit of experimental detection),  followed by rapid increase to linewidth values~$>$0.3 \waveno~. This occurs for both the v$^{\prime}$=1 and v$^{\prime}$=2 states. For the v$^{\prime}$=0, the intensities for the high J lines are too low to detect, but higher lying states almost certainly exhibit this increase.\\
 The linewidths for all vibrational states, for both isotopes of \gehplus~  are plotted in Figures \ref{figure:v0width}-\ref{figure:v2isowidth} . It can be seen that the widths of low J lines (which generally have a large FWHM and small intensity) have deviations between P and R transitions arriving at the same upper state. This is due to the uncertainty associated in measuring the linewidth. Low signal-to-noise ratios (down to 1:1 in \gehplus) make measurement of broad lines difficult as discrimination between background fluctuations and line intensity has to be made. Many of these lines were scanned with long time constants (and correspondingly longer scan times) to overcome this problem, although this proved difficult due to fluctuations in beam strength, magnet current and laser fluctuations.\\ 
 At intermediate J (where linewidth decreases and line intensity increases), all three branches agree. A convergence between P and R lines is expected as both transitions access levels with parity type `e', and hence a P(J-1 `e') and P(J+1 `e') must access the same state J (`e'). The behaviour of the Q branch was unexpected:  Q-type resonances access levels with parity `f'. The different parity blocks couple to separate electronic states and have continuum wavefunctions with inherently different structures \cite{williams1}. This can produce  different linewidths and profiles for the different parity blocks (as found to a certain extent in  \chplus~ \cite{sarrech3} and  \sihplus \cite{sarresih}. For the `e' and `f' parities of the \singpi~ state, the lifetime difference is negligible and linewidths are consistent within experimental uncertainties. It can be postulated that both parity components are predissociated by the same mechanism.\\

The variation of linewidth with rotational quantum number has been studied for many molecules, as the lifetime information about the states can be calculated and an unambiguous assignment of the dissociation mechanism can be found. In some cases the point of crossing between two curves can be calculated or other predissociating states and mechanisms inferred. For examples see: \cite{lefe},\cite{childibr},\cite{childibr2}.\\

 In the case of \gehplus there are three possibilities for predissociation of the \singpi~ state: A curve crossing, an avoided crossing or barrier, or Feshbach resonances. \\
 The most likely candidate for a curve crossing of the \singpi~ state is the $^{3}\Sigma^{+}$ state. However,  all components of this state correlate with the  ${\rm ^{2}P_{\frac{3}{2}}}$ (upper) asymptote. Predissociation by this method alone is unlikely due to the condition that the molecule must dissociate between the atomic spin-orbit limits. If the \singpi~ is embedded in the \tripsig, no predissociation will occur as both states correlate to the same asymptote.
 A predissociating state which crosses a bound state should correlate to a lower (\doubpoh~ for \gehplus) asymptote.
 However, the ab-initio work predicted the  \tripsig~ state to have a long range minimum \cite{kal}. In \chplus, this state has a minimum at 6 \AA~ and a depth of \mbox{$\approx$ 300 \waveno.} If the $^{3}\Sigma^{+}$ surface crosses the \singpi~ state in approximately this region, rapid linewidth variation with rotational quantum number  could  occur. However the \tripsig~ state must couple with the states or a continuum of states correlating to the lower dissociation asymptote.\\

 The second possibility is that a barrier exists on the \singpi~ surface of \gehplus~, and that all states are quasibound. Quantum mechanical tunnelling through the barrier would then give all observed transitions. This is directly analogous to the \singpi-\singsig~ system of GaH  as studied in absorption by Kronek. et al. \cite{kronek}. All transitions to the \singpi~ state were seen as diffuse, due to the short lifetimes (and hence large linewidths) associated with these states.\\
 Feshbach resonances (coupling between bound levels of the \singpi~ state and the continuum of states for the \singsig~ and triplet states) would allow the molecule to dissociate along the \doubpoh~ asymptote. Feshbach resonances are known to be a dissociation mechanism for the isovalent \sihplus \cite{sarresih} and are the dominant dissociation mechanism for the A$^{2}\Pi_{\frac{1}{2}}$ state of HeNe$^{+}$ \cite{carrhene}. \\

To differentiate between these possibilities for the dissociation mechanism of \gehplus, momentum releases were recorded for all possible assigned transitions. A large excess energy for the assigned transitions would be associated with the lower dissociation asymptote, a small excess energy would indicate the upper state. 


\begin{figure}[!h]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,8)(1.5,1.)
\put(0,0){\special{psfile=figures/j4.epsi}}
\end{picture}
\caption{Observed lines corresponding to transitions for v$^{\prime}$=2, J$^{\prime}$=4 from indicated lower states. Abscissa have equal ranges for direct comparison.}
\label{figure:j4}
\end{center}
\end{figure}


\newpage


\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,3.5)(2,1.1)
\put(0,0){\special{psfile=figures/just0.eps}}
\end{picture}
\caption{Linewidths of the rotational levels of the (v=0) A$^{1}\Pi$ state of \gehfour. P, Q and R transitions are indicated with +, $\times$ and $\ast$ respectively.}
\label{figure:v0width}
\end{center}
\end{figure}


\begin{figure}[!hb]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,3.5)(2,1.1)
\put(0,0){\special{psfile=figures/just0_iso.eps}}
\end{picture}
\caption{Linewidths of the rotational levels of the (v=0) A$^{1}\Pi$ state of \gehzero. P, Q and R transitions are indicated with +, $\times$ and $\ast$ respectively.}
\label{figure:v0isowidth}
\end{center}
\end{figure}


\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,3.5)(2,1.1)
\put(0,0){\special{psfile=figures/just1.eps}}
\end{picture}
\caption{Linewidths of the rotational levels of the (v=1) A$^{1}\Pi$ state of \gehfour. P, Q and R transitions are indicated with +, $\times$ and $\ast$ respectively.}
\label{figure:v1width}
\end{center}
\end{figure}


\begin{figure}[!hb]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,3.5)(2,1.1)
\put(0,0){\special{psfile=figures/just1_iso.eps}}
\end{picture}
\caption{Linewidths of the rotational levels of the (v=1) A$^{1}\Pi$ state of \gehzero. P, Q and R transitions are indicated with +, $\times$ and $\ast$ respectively.}
\label{figure:v1isowidth}
\end{center}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,3.5)(2,1.1)
\put(0,0){\special{psfile=figures/just2.eps}}
\end{picture}
\caption{Linewidths of the rotational levels of the (v=2) A$^{1}\Pi$ state of \gehfour. P, Q and R transitions are indicated with +, $\times$ and $\ast$ respectively.}
\label{figure:v2width}
\end{center}
\end{figure}


\begin{figure}[!hb]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,3.5)(2,1.1)
\put(0,0){\special{psfile=figures/just2_iso.eps}}
\end{picture}
\caption{Linewidths of the rotational levels of the (v=2) A$^{1}\Pi$ state of \gehzero. P, Q and R transitions are indicated with +, $\times$ and $\ast$ respectively.}
\label{figure:v2isowidth}
\end{center}
\end{figure}








 
\newpage

\subsection{Momentum releases}

Momentum releases were recorded for fourteen \singpi-\singsig~ transitions of the ~\gehzero~ isotope, as well as several intense unassigned (or hyperfine split) lines. Attempts to record a greater number of lines proved fruitless due to the  narrow slits and low ion current associated with the \gehzero~ isotope (parent ion) (See Figure \ref{figure:masspec}). Increasing the width of the slits would have increased the number of attainable momentum releases, but would have  decreased the resolution of the magnetic sector to an unacceptable level.\\

Three examples of the recorded momentum release profiles are given in Figure \ref{figure:momrel}. All the profiles are transitions to the J$^{\prime}$=12, v$^{\prime}$=1 state. The apparent width of the Q(12) lines is greater, and the P(13) and R(11) lines have similar widths. The P(13) and R(11) lines have profiles which appear to be made up of one component, while the Q(12) lines have a local minimum in the center of the profile.
The apparently greater width for  a Q transition momentum release is not thoroughly understood, however, this has been observed in previous studies in the ion beam apparatus, \cite{walmsley}. It could be attributed to a different instrumental response between parallel and perpendicular transitions when using an electron multiplier with a vertical slit arrangement. This could be exaggerated by the large atomic mass of Germanium and the use of a magnetic sector rather than an Electrostatic Analyser.\\

\subsection{Modeling the momentum releases}

Due to the different responses for parallel and perpendicular transitions, it was decided to model the momentum releases using the equations for the laboratory frame energy release, convolute this with the instrumental profile, and convert to momentum releases.
The equation used to calculate the profiles of the momentum releases uses the laboratory energy taken from the study by Huber  et al. \cite{huber}:\\
\begin{equation}
{\rm E_{1}=\frac{\mone \mtwo}{\mone + \mtwo }\left(\frac{E_{0}}{\mtwo} + 2{\left(\frac{E_{0} W}{\mone \mtwo}\right)}^{\frac{1}{2}}cos(\phi) + \frac{W}{\mone}\right)}\\
\label{equation:enmom}
\end{equation}

where m$_{1}$ and m$_{2}$ are the masses of the fragments\\
\hspace{1cm} E$_{0}$ is the initial kinetic energy of the parent molecule (=qV$_{\rm accn}$)\\
\hspace{1cm} W is the Centre of mass kinetic energy release\\
\hspace{1cm} $\phi$ is the angle of fragment ejection with respect to the beam direction.\\

The probability of a fragment being ejected at an angle $\phi'$ is given by:\\
\begin{equation}
{\rm P(\phi') = \frac{1}{2\pi}[1 + \beta P_{2}cos(\phi')]}\\
\label{equation:momprob}
\end{equation}

where \begin{equation}
{\rm P_{2}(cos(\phi')=\frac{1}{2}[3cos^2(\phi')-1]}
\label{equation:momprob2}
\end{equation}\\

$\beta$ depends on the symmetry of the transition. For the case of predissociated it can be approximated by: 0.5 for a P or R transition and -1 for a Q transition.

The laboratory energy releases were calculated using equations \ref{equation:enmom}-\ref{equation:momprob2},  converted to a momentum release and then convolved with the intrinsic experimental profile (taken from the mass spectra recorded using the second magnetic sector). Initial estimates were entered for $\beta$, W and  E0 giving a predicted profile for comparison with the experimental profile.\\
A series of programs based on {\it amoeba} \cite{numrec} were used to fit the experimental and predicted profiles. This program varied the parameters W, E$_{0}$ and (optionally) $\beta$ to achieve a best fit to the experimental data. The data from fixed $\beta$ fit is given in Table ~\ref{table:momrel}.\\

 As the momentum releases were recorded using a magnetic sector (as opposed to an electrostatic analyser (ESA)) with very narrow slits, signal to noise ratios were often low. This made the modelling of the profiles difficult and so $\beta$ was generally fixed, but allowed to float where high signal to noise ratios allowed accurate modeling. These can be found in Table ~\ref{table:floatbeta}. The $\beta$ parameter for P and R transitions of \gehzero with the best signal to noise ratio is between 0.5-0.57. This suggests that at least four of the transitions have a $\beta$ parameter associated with a parallel transition, hence four lines are correctly assigned, and therefore all P and R transitions are correctly assigned. The Q transitions had generally poorer signal to noise than the P and R transitions and exhibited large variations in their $\beta$ values. The fit to the Q branches was poorer, with a chi squared value generally an order of magnitude greater than those for the P and R transitions.
The momentum releases for the transitions in this region (which all correspond to the 25000-25400 \waveno~ region above the minimum of the \singsig~ state) are consistent in their values for releases.





\begin{figure}[!h]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(4,8)(2.0,1.6)
\put(0,0){\special{psfile=figures/momrel.eps}}
\end{picture}
\caption{Momentum releases for transitions to J$^{\prime}$=12, v$^{\prime}$=1 of the \singpi~ state of ~\gehzero .}
\label{figure:momrel}
\end{center}
\end{figure}







\newpage


\begin{table}[!ht]
\centering
 \begin{tabular}{||c|c|r||c|r||c|r||}
\hline
{\bf v$^\prime$}&Line&{\bf W(\waveno)}&Line&{\bf W(\waveno)}&Line&{\bf W(\waveno)}\\
\hline   
\hline

&&&&&&\\
1&P(11)&963&     &      &     &     \\
&&&&&&\\
1&P(12)&1061&Q(11)&1036&     &     \\
&&&&&&\\
1&P(13)&1124&Q(12)&1074&R(11)&1143\\
&&&&&&\\
1&     &    &Q(13)&1097&R(12)&1117\\
&&&&&&\\
1&     &      &Q(14)&680&R(13)&963\\
&&&&&&\\

\hline
\hline
&&&&&&\\
2&     &      &Q(3)&1090&      &     \\
&&&&&&\\
2&     &      &    &    &R(3)  &1048\\
&&&&&&\\
2&     &      &Q(5)&1129        &R(4)  &1040 \\
&&&&&&\\
\hline
\hline

\end{tabular}
\caption{Molecular frame energy releases (W) for all possible lines of \gehzero Q lines were fitted to $\beta$=-1, P and R lines fitted to $\beta$=0.5.}
\label{table:momrel}
\end{table}

\vspace{2cm}

\begin{table}[!hb]
\centering
 \begin{tabular}{||c|c|c|r||c|c|r||}
\hline
{\bf v$^{\prime}$}&Line&{\bf $\beta$}&{\bf W(\waveno)}&Line&{\bf $\beta$}&{\bf W(\waveno)}\\
\hline   
\hline
&&&&&&\\
1&P(11)&0.57&843&&&\\
&&&&&&\\
1&P(13)&0.55&1003&R(11)&0.50&1136\\
&&&&&&\\
1&&&&R(12)&0.57&1152\\
&&&&&&\\
\hline
\end{tabular}
\caption{Molecular frame energy releases (W) for all possible lines of \gehzero $\beta$ is floated as a free parameter. Only lines with a chi-squared fit of less than 40 are included.}
\label{table:floatbeta}
\end{table}

\newpage





\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(6,5.2)(1.7,0.9)
\put(0,0){\special{psfile=figures/q11fitted.epsi}}
\end{picture}
\caption{Fitted momentum release for J$^{\prime}$=11(f), v$^{\prime}$=1  of the \singpi~ state of ~\gehzero ($\times$ and + correspond to model and recorded data respectively).}
\label{figure:q11mom}
\end{center}
\end{figure}

\begin{figure}[!h]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(6,5.2)(1.7,0.9)
\put(0,0){\special{psfile=figures/r11fitted.epsi}}
\end{picture}
\caption{Fitted momentum releases for  J$^{\prime}$ =12(e), v$^{\prime}$=1 of the \singpi~ state of ~\gehzero~ ($\times$ and + correspond to model and recorded data respectively).}
\label{figure:r11mom}
\end{center}
\end{figure}



\subsection{Analysis of fitted momentum releases}


Comparison between experimental and calculated energy releases can be made using knowledge about the ground state energy, the photon energy and Equation \ref{equation:momrel}. Results obtained are given in Table ~\ref{table:momtheexp}. All transitions of up to J=13,  v$^{\prime}$=1 (of `e' and `f' type parities) are seen to fit to a mean dissociation energy of 24060 $\pm$ 14\waveno.



\begin{equation}
{\rm D_{e}=h\nu+E_{v^{\prime\prime},J^{\prime\prime}} - T}
\label{equation:momrel}
\end{equation}

 where ${\rm D_{e}}$ is the dissociation limit, ~${\rm E_{v^{\prime\prime}, J^{\prime\prime}}}$ is the ground state energy, ${\rm h\nu}$ is the photon wavenumber and T is the energy release. 
 To a first approximation,  the spectroscopy of \gehplus~ is similar to that of \sihplus~ (i.e. the predissociation is dominated by Feshbach resonances), the overwhelming majority of transitions dissociate along the lowest dissociation asymptote. Hence a mean value of 24060 $\pm$14 \waveno~ for the \doubpoh~ dissociation limit is obtained. From Moore's tables \cite{moores}, it is known that the \doubpth~ and the \doubpoh~ are split by the atomic-spin orbit splitting of 1767.3 \waveno. The other  asymptote must therefore lie at {\mbox 25827 \waveno.}\\

However, the possibility exist that all transitions are dissociating along the upper asymptote (which must lie at 24060 $\pm$14 \waveno), as happens in GaH. For this to be  the case, the lower asymptote (\doubpoh) must lie at  {\mbox {22293  \waveno~}}. For the construction of potential energy surfaces for the \singpi~ and \singsig~ states, it is necessary to consider the possibility of the transition fragmenting along the  the different (spin-orbit split) atomic limits.\\

\newpage

\begin{table}[!h]
\centering
 \begin{tabular}{|c|c|c|c|c|}
\hline

{\bf Line (\waveno)}&{\bf v$^{\prime}$}&{\bf J$^{\prime}$(parity)}&{\bf Expt.(\waveno)}&{\bf Calc. \waveno}\\
\hline   
\hline


&&&&\\
17567.9081&1&10(e)&963&973\\
&&&&\\
\hline
&&&&\\
17484.7117&1&11(e)&1061&1036\\
17630.7890&1&11(f)&1036&1036\\
&&&&\\
\hline
&&&&\\
17394.5920&1&12(e)&1124&1105\\
17700.6046&1&12(e)&1143&1105\\
17552.3319&1&12(f)&1074&1104\\
&&&&\\
\hline
&&&&\\
17627.1202&1&13(e)&1117&1179\\
17366.8493&1&13(f)&1097&1177\\
&&&&\\
\hline
&&&&\\
17546.4077&1&14(e)&963&1257\\
17374.0944&1&14(f)&680&1255\\
&&&&\\
\hline
\hline
&&&&\\
18354.0207&2&3(f)&1090&1019\\
&&&&\\
\hline
&&&&\\
18377.7180&2&4(e)&1048&1042\\
&&&&\\
\hline
&&&&\\
18357.1914&2&5(e)&1040&1071\\
18295.0117&2&5(f)&1129&1071\\
&&&&\\
\hline

\end{tabular}
\caption{Comparison between calculated and experimental energy releases, assuming that transitions correspond to \gehplus dissociating along the \doubpoh~ asymptote at 24060 \waveno}
\label{table:momtheexp}
\end{table}


\newpage
\section{Analysis and Discussion}

\subsection{Dissociation due to Feshbach resonances}

Using momentum release data, dissociation energies of 24060 and 25827 $\pm$ 14 \waveno~ have been calculated for the \singsig~ and \singpi~ electronic states. This compliments the ab-initio study, showing close agreement ($<$2\%) with the previously published dissociation limit \cite{kal}.
All observed transitions had relatively large momentum releases ($\approx$ 1000 \waveno), suggesting that the majority of transitions predissociated along the lower (\doubpoh) asymptote. Therefore, predissociation for these levels (in the absence of a curve crossing) must be due to Feshbach resonances.
This occurs when bound levels of the \singpi~ state couple with the continuum of states correlating to the lower asymptote, leading to dissociation of \gehplus~ into \doubpoh. \\

The lifetime behaviour for \gehplus is unusual, in that the spectrum appears to be regular (non-perturbed), but exhibits large variation in linewidth. Lifetime behaviour such as this is usually attributed to curve crossing of a bound state by a repulsive state, such as that occurring in OH, or OD or IBr. However usually this behaviour is observed simultaneously with large perturbations in line positions. However, as discussed below, behaviour of this type has certainly been observed in one (and possibly two) molecules.

Carrington and Softley \cite{carrhene}, outlined the recording and assignment of the predissociation spectrum of HeNe$^{+}$. In total 78 of the lines were assigned to electronic transitions between \mbox{${\rm (v^{\prime}=0,1)~ A_{2}~^{2}\Pi_{\frac{1}{2}}-(v^{\prime\prime}=7,8)~ X^{2}\Sigma}$} states. These electronic states are split by spin-orbit coupling, with the unobserved A$_{1}~^{2}\Pi_{\frac{3}{2}}$~ lying between the states and correlating to the same asymptote as the X$^{2}\Sigma$ state. \\
  Observed linewidths for rotational levels of the v$^{\prime}$=0 decreased with increasing J and then rapidly increased. In the case of the v$^{\prime}$=1 state, the linewidth variation was erratic, but generally decreased with increasing J to a lower limit then increased. \\ 

As no crossing (or avoiding crossing) of the ~A$_{2}$~ state occurs, for transitions to be observed in a photofragment spectrum, this state must dissociate via Feshbach resonances, i.e. the bound levels of this state couple with the continua of the  ~$X^{2}\Sigma~ {\rm and~A}_{1}^{2}\Pi_{\frac{3}{2}}$ states.
 This spectrum provided the first evidence for an electronic predissociation of a  diatomic molecule in the absence of curve or avoided crossings. Energy release measurements were made for each transition, yielding results consistent with dissociation along the lowest asymptote.\\

 
 Walmsley noted \cite{walmsley} that the predissociation of \sihplus was analogous to that in \heneplus. The predissociation of \sihplus in the fast ion beam spectra was attributed due to Feshbach resonances \cite{sarresih} and large variations in linewidth with rotational quantum number were observed. An increase from 0.018 to 4.5 \waveno~ can be observed when J$^{\prime}$~ increases from 17 to 19. Both studies are complementary to the work outlined in this chapter on ~\gehplus. \\

\subsection{Dissociation through a barrier}

An alternative argument to predissociation via Feshbach resonances is to consider that molecules in the \singpi~ state can dissociate via quantum tunnelling through a barrier.\\
For this argument to be feasible,  two possibilities exist for the observed behaviour of the linewidth and momentum releases. First of these is that a barrier arises from the introduction of a  centrifugal term in the effective potential of the non-rotating molecule U$_{0}$(R) to yield an effective potential, U$_{eff}$(R) which may be written as:

\begin{equation}{\rm 
U_eff(R)=U_{0}(R) + \left(\frac{\hbar^{2}}{2\mu R^{2}}[J[J+1]-\Omega^2]\right)}\end{equation}

Solution of the Schr\"odinger equation U$_{eff}$(R) yield levels which lie above and below the dissociation limit of U$_{0}$(R). Those which lie above are termed shape resonances. In the case the \singpi~ state of  \gehplus, the dissociation limit in question is the ${\rm ^{2}P_\frac{3}{2} + ^{2}S}$~asymptote.
A momentum release of 1000 \waveno~ requires a centrifugal barrier of 1000\waveno~, a clearly unfeasible situation for the low J (between 1 and 20) found in this experiment.\\

The second possibility for a barrier arises due to an avoided crossing, which as in the \singpi~ state of the isoelectronic molecule GaH \cite{kronek}, where
absorptions v$^{\prime}$=0 were observed as diffuse due to the large linewidth of the rotational levels. No discernible rotational structure could be found in GaH, but could be found in GaD.  The linewidths observed in \gehplus~ however  are comparable to those found in \chplus and \sihplus, where individual rotational lines can easily be observed, and widths are generally $>$2 \waveno. \\
 Such a barrier arises  from an  avoided crossing between states of the same symmetry (\cite{herzberg}) for GaH it is the two lowest \singpi~ states  (separated by 33000 \waveno~) \cite{bala}. For \gehplus, the separation between the staachievedtes is a much larger 61452 \waveno~ \cite{kal}, which implies that a barrier due to an avoided crossing is unlikely. This is entirely analogous to the case of the isoelectronic AlH and \sihplus. AlH has a large barrier in the \singpi~ state entirely due to an avoided crossing (\singpi~ state separations 32400 \waveno~ \cite{moores}). However, no such barrier has been detected in \sihplus, to date, as the separation of the \singpi~ states is too large (55000 \waveno~). It seems likely, therefore, that no barrier due to an avoided crossing  exists in \gehplus.\\

 To quantify the argument of predissociation via tunnelling, calculations using  {\bf LEVEL6.0} program \cite{leroylevel} were initiated, in an effort to correlate the momentum releases and the widths of the lines. It was hoped this could be  through the adjustment of the dissociation limit(s) to higher and lower energies. However, for the momentum releases to be in the region of 1000\ waveno, all observable levels of v$^{\prime}$=2 state should have linewidths greater than 5 \waveno. This is clearly not the case for either recorded isotope (see Figures~ \ref{figure:v2width}-\ref{figure:v2isowidth}). 


 Due to the similarities in the linwidth behaviour between \gehplus~ and \heneplus, the lack of evidence for a barrier and the low probability of the \singpi~ state being crossed by a repulsive state, the dissociation mechanism is almost certainly due to Feshbach resonances. Molecules existing in the \singpi~ state can therefore be assumed to  dissociate along the lower (\doubpoh) asymptote.



\begin{figure}[!h]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(5,8.2)(2.5,0.6)
\put(0,0){\special{psfile=figures/levs.epsi}}
\end{picture}
\caption{Energies of the assigned rotational states of the \singpi~ state. Those lines recorded in momentum release experiments are denoted by m. The v$^{\prime}$=2, J$^{\prime\prime}$=4 state has been marked T as it is hyperfine split.}
\label{figure:levs}
\end{center}
\end{figure}

\subsection{Potential surfaces constructed from spectroscopic data}

After assigning recorded lines to (v$^{\prime}$=0,1,2)\singpi-\singsig (v$^{\prime\prime}$=3) transitions, potential energy surfaces were constructed from the molecular parameters. Other parameters were taken from the ab-initio and emission studies for the \singsig~ state, some of which were modified to include the predissociation data. (see Table ~\ref{table:sin}). A new \singsig~ state was constructed from the refined parameters, to provide information for further assignments and create the most accurate potential possible from all spectroscopic data.\\
Construction of potential energy surfaces for the four experimentally observed electronic states used the following methods:\\
\begin{itemize}


\item{Molecular parameters from the fitting of absorption and emission spectra data were least squares fitted to yield the best values for the first two Dunham vibrational and rotational parameters for each state. The program {\bf RKR1} \cite{leroyrkr} was used to calculated potentials from this data, together with B$_{\rm v}$ and G$_{\rm v}$ for these states.}
\item{Dissociation energies of the \singsig~  and \trippio~ states was calculated to be 24060 \waveno, i.e. the asymptote along which the momentum releases were observed to fragment. The \doubpth~ asymptote (correlating to  \singpi~ and \trippiz~ states) was calculated by adding the atomic spin-orbit splitting from Moore's tables \cite{moores}.}
\item{Separation between the states was taken from the T$_{\rm e}$ value calculated from the \singpi-\singsig~ transition. Triplet state separations were taken from the work of Tsuji et al. \cite{tsu3}.}
\item{Modified potentials were constructed (See Figure \ref{figure:4pots}) using the {\bf LEVEL6.0} \cite{leroylevel} program. The Schr\"{o}dinger Equation was solved for the Potentials giving levels of all v and J. Transitions between the states were calculated using {\bf LEVEL6.0} in two potential mode. Frank-Condon factors and Einstein A coefficients were calculated for all allowed transitions.}
\item{Potentials  were created which included smoothing to asymptotic limits. The long range part of the potentials determined by the ion induced dipole term ${\rm \frac {-C_{4}}{R^{4}}}$ term, where C$_{4}$ is related to the polarizability of the hydrogen atom.}


\end{itemize}
Potential energy surfaces generated using the above methods are displayed in Figure~ \ref{figure:4pots}. Observed vibrational states from the predissociation spectrum are indicated by horizontal lines.\\
 Agreement between the four observed bands and bands predicted from the output of Leroy's programs are good. Figure \ref{figure:einb} shows the match between the line positions, the Einstein B coefficients and the intensities.\\
 {\bf LEVEL6.0} predicts transitions on the assumption that the lambda doubling is zero (i.e. that `e' and `f' levels are degenerate), resulting in the observed  Q transitions drifting from the predicted positions with increasing rotational quantum number J. This effect can be seen in Figure~\ref{figure:einb}.



\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(5,7.8)(2.5,0.6)
\put(0,0){\special{psfile=figures/4pots.epsi}}
\end{picture}
\caption{Potential energy surfaces of \gehzero. (Triplet states taken from study by Tsuji et. al. \cite{tsu3}), Singlet states taken from this work.}
\label{figure:4pots}
\end{center}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\setlength{\unitlength}{1in}
\begin{picture}(5,7.8)(2.5,0.6)
\put(0,0){\special{psfile=figures/einbrec.epsi}}
\end{picture}
\caption{Comparison between the Einstein B coefficients (calculated from the LEVEL6.0 program by Leroy) [plotted above the zero axis] and the assigned transitions of the predissociation spectrum [plotted below the zero axis]. Lambda doubling neglected.}
\label{figure:einb}
\end{center}
\end{figure}





\section{Unassigned lines}

Despite the progress made in assigning a large number of lines to the \mbox{~\singpi-\singsig~} transition, there exists approximately 100 lines which remain unassigned. This section will discuss the origins, possible methods for assignment and the difficulties encountered. The unassigned lines can be found in Tables ~\ref{table:unass1}-\ref{table:unass3}.

\subsubsection{Triplet lines}

Approximately twenty of the lines exhibit splittings of  ~ $<$ 0.01 \waveno~ (300 MHz) which are associated with the hyperfine splitting term in the Hamiltonian:

\begin{equation}{\rm H_{hf}= aI_{z}L_{z} + b_{F}I.S +\frac{c}{3}(3I_zS_z-I.S) -\frac{d}{2}(S_{+}I_{+}+S_{-}I_{-})}
 \end{equation}

For high J levels, the dominant term is the isotropic coupling interaction or Fermi interaction,  b$_{F}$I.S . The Fermi contact parameter b$_{F}$ provides a measure of the electron spin density at the proton nucleus \cite{sarresih}. This is calculated to be 1420MHz for an electron in a 1S orbital of a hydrogen atom. For a singlet state, such as \singpi, S=0 and hence splitting does not occur. In \gehplus~, the H nucleus has a nuclear spin, I, of $\frac{1}{2}$. For a molecule in a pure \trippi~ state arising from a s1,p1 configuration, only half the total spin angular momentum contributes to the spin density at the proton nucleus. Thus the observed splitting should be less than 710MHz. This is found to be the case for all observed split lines.\\
Progressions in the hyperfine split lines cannot be found, however they cluster around 16500, at 17600 and 18300 \waveno~. They arise due to intensity stealing from transitions of the \singpi-\singsig~ transitions, verified by correlation of the strongest singlet-singlet transition positions with those of the hyperfine split lines. 


\subsubsection{Other lines}
If a line exists in isolation a thorough search can be made for all lines which access the same upper state (using the method of ground state differences outlined above). However this method can only work at low J, as the higher rotational terms contribute at high J. As the assigned spectra generally lie between J=1-20, it is not possible to extrapolate to high J. However, it is believed that many of the unassigned lines are due to high rotational states.\\
The isotope shifts for many of the unassigned lines are irregular, and no further patterns can be found from the lines. However, the \gehzero~ isotope was scanned in regions of interest only.  A thorough investigation of ~\gehzero, with continuous scanning could prove invaluable in assigning the remaining transitions.


\section{Conclusions}

 The first isotopically resolved spectra for \gehplus~ have been recorded, with many absorption features found. Many of the recorded lines have been assigned to the allowed \singpi-\singsig~ transition of \gehplus, the first such observation in this molecule . Kinetic energy releases have verified the assignment, and provided the most accurate information to date on the position of the dissociation asymptotes.\\
 Dissociation of \gehplus~ has been shown to occur through Feshbach resonances, and lifetime behaviour for this case has been examined and found to be consistent with previous studies.








































