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# Search & Match

## How it works

QualX performs a multi-step search:

1. Strongest peaks filter: for each database card, only the N strongest reference peaks are compared against the experimental peak list (fast pre-filter).
2. FOM calculation: a weighted Figure of Merit combines peak-position matching and intensity matching.
3. Ranking: cards are sorted by descending FOM; only cards above Min. FOM are shown.

## Figure of Merit

At the end of the search–match step a list of plausible database crystalline phases are ranked according to the decreasing values of a FoM. The FoM includes four contributions, three of them suitably weighted, according to the following formula:

\[ \mathrm{FoM}=\sqrt{\frac{\mathrm{FoM}_{db}\cdot\left(w_{\theta}\,\mathrm{FoM}_{\theta} +w_{I}\,\mathrm{FoM}_{I}+w_{ph}\,\mathrm{FoM}_{ph}\right)}{w_{\theta}+w_{I}+w_{ph}}} \]

where FoMθ takes into account the average difference between the 2θ values of the observed and matched database peaks; FoMI is related to the average difference between the intensity values of the observed and matched database peaks; FoMph depends on the percentage of the matched experimental peaks and on their intensity; and FoMdb depends on the percentage of the matched database peaks and on their intensity.

FoMθ is the contribution coming from the 2θ differences between the experimental and the associated database peaks:

\[ \mathrm{FoM}_{\theta} = 1 - \dfrac{\displaystyle\sum_{i}^{N_{db}^{ass}} \left|2\theta_i^{\mathrm{exp}} - 2\theta_i^{db}\right|}{N_{db}^{ass}\cdot\Delta} \]

where the summation is over the associated database peaks (a database peak is considered associated if its 2θ distance from the experimental peak is less than Δ), 2θexp and 2θdb are the positions of the experimental and database peaks, respectively.

FoMI is the contribution due to the differences between the intensities of the experimental and the associated database peaks:

\[ \mathrm{FoM}_{I} = 1 - \dfrac{\displaystyle\sum_{i}^{N_{\mathrm{exp}}^{ass}} \left|I_i^{\mathrm{exp}} - I_i^{db}\right|}{N_{\mathrm{exp}}^{ass}} \]

where Iexp and Idb are the experimental and database intensity respectively, the summation is over the associated experimental peaks.

FoMph is the contribution due to the intensities of the associated experimental peaks and their percentage:

\[ \mathrm{FoM}_{ph} = \sqrt{\dfrac{\displaystyle\sum_{i=1}^{N_{\mathrm{exp}}^{ass}} I_i^{\mathrm{exp}}}{\displaystyle\sum_{i=1}^{N_{\mathrm{exp}}} I_i^{\mathrm{exp}}} \cdot\dfrac{N_{\mathrm{exp}}^{ass}}{N_{\mathrm{exp}}}} \]

where Nexp is the total number of experimental peaks.

FoMdb is the contribution due to the intensities of the associated database peaks and their percentage:

\[ \mathrm{FoM}_{db} = \sqrt{\dfrac{\displaystyle\sum_{i=1}^{N_{db}^{ass}} I_i^{db}}{\displaystyle\sum_{i=1}^{N_{db}} I_i^{db}} \cdot\dfrac{N_{db}^{ass}}{N_{db}}} \]

where Idb is the database intensity and the summation at the numerator is over the associated database peaks; Ndb is the total number of the database peaks.

The weighting factors wθ, wI, wph and Δ (default heuristic values are set for them) can be adjusted directly by the user via the graphical interface. wph is related to the number of expected phases.

The default value of the weights wθ, wI, and wph is 0.5, but it can be changed by the dialogue window in Figure 1 (via the "2θ", "Intensity" and "Phases" trackbars).

## Search Options

Open Search → Search & Match Options:

Option | Description
--- | ---
Min. FOM | Discard cards below this threshold
2θ / d weight | Contribution of peak-position agreement to FOM
Intensity weight | Contribution of intensity agreement to FOM
Phases weight | Bias towards single-phase vs. multi-phase solutions
Δ2θ | Peak matching tolerance in degrees (Auto = calculated from peak widths)
Max. entries | Maximum number of results to display
Check strongest peaks | Use only the N strongest peaks for the pre-filter
Check deleted cards | Include cards marked as deleted in the database
Residual searching | After accepting a phase, automatically re-search on residual peaks

## Accepting a Phase

1. Select a card in the Results List.
2. Click Accept (✓). The card is moved to the Quantitative dock.
3. If Residual searching is enabled, QualX subtracts the accepted phase contribution and repeats the search on the remaining peaks.