#
Probabilistic Estimate of |F_{oa}| from FEL Data

^{*}

## Abstract

**:**

## 1. Introduction

- for the state 1 $\Delta {f}_{j}={[\Delta {f}_{j}^{\prime}]}_{1}$ and ${f}_{j}^{\u2033}={[{f}_{j}^{\u2033}]}_{1}$;
- for the state 2 $\Delta {f}_{j}={[\Delta {f}_{j}^{0}]}_{2}+{[\Delta {f}_{j}^{\prime}]}_{2}$ and ${f}_{j}^{\u2033}={[{f}_{j}^{\u2033}]}_{2}$.

## 2. The Mathematical Model

**h**and −

**h**, respectively.

- (1)
- For light atoms, $\Delta {f}_{j}$ is assumed to be negligible both for the damaged and for the undamaged crystals. In our model, we set $\Delta {f}_{j}=0$ for j = 1, …, L. Furthermore, ${f}_{j}^{0}$ is assumed to be the same for the jth atom independently of whether this refers to the undamaged or damaged crystal.
- (2)
- For heavy atoms, the scattering factor of the jth atom ${f}_{j}^{0}$ is described by two different functions according to whether the atom is considered in the undamaged or in the damaged crystal. As a result, $\Delta {f}_{j}$ will assume different values.
- (3)
- The atomic positions in the damaged and in the undamaged crystals coincide [26,32]. Indeed, there is no evidence so far that the pulse duration (about a few femtoseconds) produces detectable changes in the heavy atom positions in femtosecond X-ray nanocrystallography even if the simulations of radiation dynamics in proteins suggests some correlated movement of the heavy atoms [33].
- (4)
- The observed amplitudes ${F}_{}^{+},{F}_{d}^{+},{F}_{}^{-},{F}_{d}^{-}$ are affected by errors, which are calculated as:$${\mu}^{+}=\left|{\mu}^{+}\right|exp\left(i{\vartheta}^{+}\right)\mathrm{and}{\mu}^{-}=\left|{\mu}^{-}\right|exp\left(i{\vartheta}^{-}\right),$$$${\mu}_{d}^{+}=\left|{\mu}_{d}^{+}\right|exp\left(i{\vartheta}_{d}^{+}\right)\mathrm{and}{\mu}_{d}^{-}=\left|{\mu}_{d}^{-}\right|exp\left(i{\vartheta}_{d}^{-}\right)$$

## 3. The Joint Probability Distribution $\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{0}\mathit{H}}\mathbf{,}{\mathbf{E}}_{}^{\mathbf{+}}\mathbf{,}{\mathbf{E}}_{\mathit{d}}^{\mathbf{+}}\mathbf{,}{\mathbf{E}}_{}^{\mathbf{-}}\mathbf{,}{\mathbf{E}}_{\mathit{d}}^{\mathbf{-}}\mathbf{\right)}$

**K**is the symmetric square matrix. The elements of

**K**are specified as follows:

**K**

^{−1}.

^{2}= Q

_{1}

^{2}+ Q

_{2}

^{2}

_{11}is always expected to be positive, the expected values of $<{R}_{oa}^{}>$ and ${\sigma}_{{R}_{oa}^{}}$ are always positive. The reflections with the largest values of the ratio found in Equation (9) are likely to be the most useful ones.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviation

$\epsilon $ | statistical Wilson coefficient (corrected for expected intensities in reciprocal lattice zones). |

$L\text{},\text{}H$ | number of non-hydrogen light and heavy atoms in the unit cell, respectively. Their values do not vary when considered in the undamaged or damaged crystals. |

$N=L+H$ | number of non-hydrogen atomic positions in the unit cell, for the undamaged and the damaged crystals. |

${f}_{j}^{}={f}_{j}^{0}+\Delta {f}_{j}+i{f}_{j}^{\u2033}={f}_{j}^{\prime}+i{f}_{j}^{\u2033}$ | scattering factor of the jth atom. f′ is its real and f″ is its imaginary part. The thermal factor is included. |

${\sum}_{N}={\displaystyle \sum _{j=1}^{N}\left({f}_{j}^{\prime 2}+{f}_{j}^{\u20332}\right)}$, | the summation is calculated for the undamaged crystal and is extended to all the atoms in the unit cell. |

${\sum}_{Nd}={\displaystyle \sum _{j=1}^{N}\left({f}_{dj}^{\prime 2}+{f}_{dj}^{\u20332}\right)}$, | the summation is calculated for the damaged crystal and is extended to all the atoms in the unit cell. |

${\sum}_{L}^{0}={\displaystyle \sum _{j=1}^{L}{\left({f}_{j}^{o}\right)}^{2}}$ | the summation is extended to all the light atoms in the unit cell. |

${\sum}_{H}^{0}={\displaystyle \sum _{j=1}^{H}{\left({f}_{j}^{o}\right)}^{2}}$ | the summation is extended to all the heavy atoms in the unit cell. |

${F}_{}=\left|F\right|exp\left(i\phi \right)={\displaystyle \sum _{j=1}^{N}{f}_{j}exp}\left(2\pi i\mathbf{h}{\mathbf{r}}_{j}\right)$ | structure factor of the undamaged crystal |

$E=\left|F\right|exp\left(i\phi \right)/{\left(\epsilon {\sum}_{N}\right)}^{1/2}=Rexp\left(i\phi \right)=A+iB$ | normalized structure factor of the undamaged crystal. |

${F}_{d}=\left|{F}_{d}^{}\right|exp\left(i{\phi}_{d}^{}\right)={\displaystyle \sum _{j=1}^{Nd}{f}_{dj}exp}\left(2\pi i\mathbf{h}{\mathbf{r}}_{j}\right)$ | structure factor of the damaged crystal. |

${E}_{d}=\left|{F}_{d}^{}\right|exp\left(i\phi \right)/{\left(\epsilon {\sum}_{dN}\right)}^{1/2}={R}_{d}^{}exp\left(i{\phi}_{d}^{}\right)={A}_{d}+i{B}_{d}$ | normalized structure factor of the damaged crystal |

${F}_{0H}^{}=|{F}_{0H}^{}|\mathrm{exp}(i{\phi}_{0H}^{})={\displaystyle \sum _{j=1}^{H}{f}_{j}^{o}\mathrm{exp}\left(2\pi \mathbf{h}{\mathbf{r}}_{j}\right)}$ | normal structure factor for the heavy atom substructure (anomalous scattering excluded). |

${E}_{0H}=\left|{F}_{0H}^{}\right|exp\left(i{\phi}_{0H}\right)/{\left(\epsilon {\sum}_{0H}\right)}^{1/2}={R}_{0H}^{}exp\left(i{\phi}_{0H}^{}\right)={A}_{0H}+i{B}_{0H}$ | normalized structure factor of the normal heavy atom substructure (anomalous scattering excluded). |

${F}_{H}^{}=|{F}_{H}^{}|\mathrm{exp}(i{\phi}_{H}^{})={\displaystyle \sum _{j=1}^{H}{f}_{j}^{}\mathrm{exp}\left(2\pi \mathbf{h}{\mathbf{r}}_{j}\right)}$ | structure factor of the heavy atom substructure. |

MIR, MAD, MIRAS | multiple isomorphous replacement, multiple anomalous dispersion, multiple isomorphous replacement combined with anomalous scattering techniques, respectively. For brevity, we include into the above definitions the particular cases of SIR (single isomorphous replacement), SAD (single anomalous dispersion) and SIRAS (single isomorphous replacement combined with anomalous scattering techniques). |

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**MDPI and ACS Style**

Giacovazzo, C.; Carrozzini, B.; Cascarano, G.L.
Probabilistic Estimate of |F_{oa}| from FEL Data. *Crystals* **2018**, *8*, 175.
https://doi.org/10.3390/cryst8040175

**AMA Style**

Giacovazzo C, Carrozzini B, Cascarano GL.
Probabilistic Estimate of |F_{oa}| from FEL Data. *Crystals*. 2018; 8(4):175.
https://doi.org/10.3390/cryst8040175

**Chicago/Turabian Style**

Giacovazzo, Carmelo, Benedetta Carrozzini, and Giovanni Luca Cascarano.
2018. "Probabilistic Estimate of |F_{oa}| from FEL Data" *Crystals* 8, no. 4: 175.
https://doi.org/10.3390/cryst8040175