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2.6.1 Fit Functions

Fitting of time resolved photoelectron and H+ _{transients in Chapters 3 and 4 was all} performed by integrating slices of the spectra and fitting these to appropriate functions. After a feature of interest is identified, a suitable spectral window is chosen, and the intensity of the feature, I(t), is found by numerically integrating in this window as a function of pump-probe delay, t. In general, there are two processes that may be observed in I(t). Firstly, in cases where a signal arises at t0 (when pump and probe temporally overlap), which then drops to zero, an exponential decay function is used:

0 τ

( ) ( ) exp t t

I t G t A (2.8)

where τ is the lifetime of the decay and A gives the intensity of this feature. G(t) is the

Figure 2.10 An example of the image (left, convoluted; right, deconvoluted central slice) and resultant spectrum for photodetachment from I– _{with 4.66 eV photons. The spacing between the}

peaks allows the VMI setup to be calibrated.

0 0.5 1 1.5 2 2.5 3 3.5 4 x 104 0 250 500 (radius / pixel)2 In te n s it y 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 400

ε

Experimental|Data Fitting and Analysis

58 instrument response function, discussed below. Exponential decays are usually the appropriate function for photoelectron signals, as well as ionic fragments produced from absorption-ionization-detachment processes (see Chapter 1). The parent ion spectra presented in Chapter 3 are fit with the exponential decay, as these are essentially photoelectron yields. On the other hand, in situations where a signal is produced which does not decay away, an exponential rise is more appropriate:

0

1

τ

( ) ( ) exp t t

I t G t A (2.9)

The exponential rise is typically used for signals arising from ionization of neutral hydrogen atoms. Note that often several processes will overlap within the same spectral window, so a combination of exponential rises and exponential decays may be appropriate to fit the entire transient.

In each case, the function was convoluted with G(t), the instrument response function. G(t) is simply a Gaussian function, centred at t0:47

2 0 2 2

σ

where the width is given by σ,related to the pump-probe cross correlation FHWM by:

2 2 2 σ

ln

FWHM _(2.11)

Fitting was then performed by the Levenberg-Marquardt algorithm (as implemented by the Origin software package) to minimize the squares of the difference between the data and fits from Eqs. (2.8) and (2.9).

Figure 2.11 Simulated intensity plot (a) and component decay associated spectra (b) of a TRPES experiment with two spectral components, a feature ~ 3 eV which transfers to < 2 eV on a 250 fs timescale, and a 1000 fs decay of this secondary feature. The dashed line in b) shows the sum of the two DAS componenents. The negative going signal in the DAS shows population transfer, which can also be seen as a later onset in the signal maximum in the TRPES at low eKE.

0 1 2 3 4 5 -0.5 0 0.5 1 eKE / eV In ten si ty τ= 1000 fs τ= 250 fs t / fs eK E / eV -5000 0 500 1000 1500 2000 2500 1 2 3 4 5 0 0.5 1 a) b)

Experimental|Data Fitting and Analysis

59

2.6.2 Global Fitting

For the TRPES fits in Chapter 5, a global fitting algorithm was employed.48, 49 _{In essence,} global fitting operates by fitting all energies at all time delays simultaneously to the same set of time constants for a sum of i exponential decays:

0 τ ( ) ( , ) ( ) ( ) i t t i i S eKE t G t c eKE e (2.12)

where ci(eKE) is the intensity of the ith exponential decay component as a function of

kinetic energy, which has a decay constant of τi. The global fit is performed by a

squares fit of S(eKE, t) to the experimental data, with ci(eKE) and τi as parameters to be determined by the fit. The number of decay components to include, i, must be selected as a parameter of the fitting procedure; in general the smallest i which produces an acceptable fit is desirable.

The coefficients ci(eKE) give the spectral shape of the feature associated with the

exponential decay time constant τi, and for this reason are also known as decay associated spectra (DAS). Intriguingly, there is no stipulation that DAS must be universally positive; negative going DAS may be produced by the fitting procedure. A negative DAS feature in isolation is unphysical – it corresponds to a gain in photoelectron signal from a negative starting position. On the other hand, if it is superimposed on top of a positive feature with a different τ, then we can interpret it in terms of a flow of population. The negative component means that this spectral region is being populated with one time constant, and then a DAS with a positive component at this region shows this population decaying away at a second time constant. In the DAS with a negative going feature, positive features then likely correspond to decay away from that spectral region, into the negative going region. An example simulated TRPES experiment is shown in Fig. 2.11, along with the constituent DAS. In this example, there is a negative going feature in the DAS of the faster component, representing population transferring from the high eKE region to the lower eKE region. The second time component is responsible for this lower eKE feature decaying away again. The population transfer is visible by eye in the TRPES intensity plot as a shift to later time of the feature of maximum intensity. Whilst powerful, it is important to keep in mind that global fitting can be a delicate process; as the number of parameters employed in the fitting process is so high, a good fit can often be found even if it is not physically meaningful. The fit is always constrained by the initial model (number of time constants and starting parameters), and so care should be taken if there is not an obvious physical explanation for fit results. Additionally, this algorithm can only fit data

Experimental|References

60 for which each state has a static spectrum. If the eKE of a given state shifts over time (e.g. due to wavepacket motion), then the global fitting algorithm cannot exactly fit the data, instead approximating it with a static state model. For datasets with large amplitude motions, the global fitting procedure is not applicable.50

To obtain confidence intervals for parameters produced by global fits, the support plane analysis technique can be used.51-53 _{To perform this procedure, the parameter} whose confidence interval we are interested in is fixed to a set value and the square of the residuals between the data sat and model (χ2_{) is minimized. The fixed parameter is} scanned   across   the   region   of   interest,   and   χ2 _{is computed at each step. This then} provides insight as to the region in parameter space (generally time constants) where an acceptable fit is produced. A numerical confidence interval can also be determined by  the  points  at  which  χ2 _{deviates statistically significantly from its minimum point. The} point, x, of significant deviation for 95% confidence intervals can be calculated by: 53

2 2

1

(0.95, , )

p

F

p

where p is the number of parameters in the fit, i.e. the sum of the number of time constants, components in each DAS, and t0and σ  values. ν gives the degrees of freedom

(total number of data points – p), and F is the inverse F cumulative distribution function (implemented in Matlab as finv).

In practical terms, the global fitting algorithm has been implemented using Matlab. The program takes as parameters the experimental matrix of intensity as a function of time and energy, the two respective axes for this matrix, and the cross correlation for the experiment. The number of time components to fit is determined by the length of a τ  vector, where each element is the initial trial value. The DAS are represented by a

𝑚   × 𝑛 matrix, where m is the size of the eKE scale used, and n is the number of decay components to fit. The DAS are initially flat. From these, a trial S(eKE, t) is generated, and the Levenberg-Marquardt algorithm is used, as implemented by the Matlab optimization  toolbox,  to  minimize  χ2_{. If the output of an imaging experiment is to be} used directly in the global fitting algorithm, then the nonlinear eKE axis (proportional to r2_{) means that a weighting factor of r should be used to prevent overfitting at lower} energies. The code for the global fitting algorithm is presented in Appendix B.