A new study of the  sigmaA parameter has been undertaken to understand itsbehaviour when the diffraction amplitude distributions are far from the standardWilson distributions. The study has led to the formulation of a new statisticalinterpretation of  sigmaA, expressed in terms of a correlation factor. The newformulas allow a more accurate use of  sigmaA in electron-density modification procedures.

The resolution parameter Sigma (A) is currently used for evaluating the degree of similarity between a model and the target structure. Here, quasi-Wilson distributions are used to represent the local statistics of the normalized amplitudes both for the target and for the model structure. The study uses the joint probability distribution approach to provide (i) a description of the statistical properties of the Sigma (A) parameter; (ii) a deeper insight into the role, for the Sigma (A) estimate, of the high-order moments of the target and of the model structure-factor distributions; and (iii) new statistical formulas for estimating Sigma (A) . The theoretical results have been checked using test proteins.

New methods have been recently developed to improve the structure solution of macromolecules by ab initio (Patterson or Direct Methods) and non ab initio (Molecular Replacement) approaches. Phasing proteins at non-atomic resolution is still a challenge for any ab initio method. The combined use of different algorithms [Patterson deconvolution and superposition techniques, cross-correlation function (C-Map), the VLD (Vive la Difference) approach included in the Direct Space Refinement (DSR) procedure, a new probabilistic formula estimating triplet invariants and capable of exploiting a model electron density maps, the FREE LUCH extrapolation method, a new FOM to identify the correct solution] allow to overcome the lack of experimental information. The new methods have been applied to a large number of protein diffraction data with resolution up to 2.1Å, under the condition that Ca or heavier atoms are in the structure. Results show that solving proteins at limited resolution is a feasible task, achievable even by new Direct Methods algorithms, against the traditional common believe that atomic resolution is a necessary condition for the success of a direct ab initio phasing process.A new procedure (REVAN) , aiming at solving protein structures via Molecular Replacement and density guided optimization algorithms, has been assembled. It combines a variety of programs (REMO09, REFMAC, COOT) and algorithms (Cowtan-EDM, DSR, VLD, FREE LUNCH), and can successfully lead to the structure solution also when the sequence identity between target and model structures is smaller than 0.30 and data resolution up to ~ 3Å. The application to a wide set of test structures (including difficult cases proposed by DiMaio et al. (2011), solved by using MR procedures together with energy guided programs) suggests that REVAN is quite effective even far from atomic resolution and, in combination with EDM techniques and sequence mutation algorithms, it is able to efficiently extend and refine the set of phases, reducing its average error.The final step of the automatic solving process (ab initio or MR approaches) is the application of an Automated Model Building program (i.e. Buccaneer, Nautilus, ARP-wARP or Phenix-Autobuild) in order to recover the correct structure. Results suggest that the quality of the phases at the end of the phasing process is good enough to lead the AMB program to success.These new efficient procedures are implemented in the current version of the software package SIR2014.

The REVAN pipeline aiming at the solution of protein structures via molecular replacement (MR) has been assembled. It is the successor to REVA, a pipeline that is particularly efficient when the sequence identity (SI) between the target and the model is greater than 0.30. The REVAN and REVA procedures coincide when the SI is >0.30, but differ substantially in worse conditions. To treat these cases, REVAN combines a variety of programs and algorithms (REMO09, REFMAC, DM, DSR, VLD, free lunch, Coot, Buccaneer and phenix. autobuild). The MR model, suitably rotated and positioned, is first refined by a standard REFMAC refinement procedure, and the corresponding electron density is then submitted to cycles of DM-VLD-REFMAC. The next REFMAC applications exploit the better electron densities obtained at the end of the VLD-EDM sections (a procedure called vector refinement). In order to make the model more similar to the target, the model is submitted to mutations, in which Coot plays a basic role, and it is then cyclically resubmitted to REFMAC-EDM-VLD cycles. The phases thus obtained are submitted to free lunch and allow most of the test structures studied by DiMaio et al. [(2011), Nature (London), 473, 540-543] to be solved without using energy-guided programs.

alpha-Helices are peculiar atomic arrangements characterizing protein structures. Their occurrence can be used within crystallographic methods as minimal a priori information to drive the phasing process towards solution. Recently, brute-force methods have been developed which search for all possible positions of alpha-helices in the crystal cell by molecular replacement and explore all of them systematically. Knowing the alpha-helix orientations in advance would be a great advantage for this kind of approach. For this purpose, a fully automatic procedure to find alpha-helix orientations within the Patterson map has been developed. The method is based on Fourier techniques specifically addressed to the identification of helical shapes and operating on Patterson maps described in spherical coordinates. It supplies a list of candidate orientations, which are then refined by using a figure of merit based on a rotation function calculated for a template polyalanine helix oriented along the current direction. The orientation search algorithm has been optimized to work at 3 A resolution, while the candidates are refined against all measured reflections. The procedure has been applied to a large number of protein test structures, showing an overall efficiency of 77% in finding alpha-helix orientations, which decreases to 48% on limiting the number of candidate solutions (to 13 on average). The information obtained may be used in many aspects in the framework of molecular-replacement phasing, as well as to constrain the generation of models in computational modelling programs. The procedure will be accessible through the next release of IL MILIONE and could be decisive in the solution of new unknown structures.

A method is presented to determine the helix orientation starting from X ray diffraction data. Our method is based on the periodicity properties of alpha helix and on the analysis of the properties of the corresponding Patterson function, obtained directly from raw crystallographic intensities. Knowledge of helix orientation is a useful information within the structure solution process of proteins. © 2011 IEEE.

The energy-dependent cross section of the Be7(n,?)He4 reaction, of interest for the so-called cosmological lithium problem in big bang nucleosynthesis, has been measured for the first time from 10 meV to 10 keV neutron energy. The challenges posed by the short half-life of Be7 and by the low reaction cross section have been overcome at n_TOF thanks to an unprecedented combination of the extremely high luminosity and good resolution of the neutron beam in the new experimental area (EAR2) of the n_TOF facility at CERN, the availability of a sufficient amount of chemically pure Be7, and a specifically designed experimental setup. Coincidences between the two alpha particles have been recorded in two Si-Be7-Si arrays placed directly in the neutron beam. The present results are consistent, at thermal neutron energy, with the only previous measurement performed in the 1960s at a nuclear reactor. The energy dependence reported here clearly indicates the inadequacy of the cross section estimates currently used in BBN calculations. Although new measurements at higher neutron energy may still be needed, the n_TOF results hint at a minor role of this reaction in BBN, leaving the long-standing cosmological lithium problem unsolved.

We report on the measurement of the Be-7(n,p)Li-7 cross section from thermal to approximately 325 keV neutron energy, performed in the high-flux experimental area (EAR2) of the n_TOF facility at CERN. This reaction plays a key role in the lithium yield of the big bang nucleosynthesis (BBN) for standard cosmology. The only two previous time-of-flight measurements performed on this reaction did not cover the energy window of interest for BBN, and they showed a large discrepancy between each other. The measurement was performed with a Si telescope and a high-purity sample produced by implantation of a Be-7 ion beam at the ISOLDE facility at CERN. While a significantly higher cross section is found at low energy, relative to current evaluations, in the region of BBN interest, the present results are consistent with the values inferred from the time-reversal Li-7(p,n)Be-7 reaction, thus yielding only a relatively minor improvement on the so-called cosmological lithium problem. The relevance of these results on the near-threshold neutron production in the p + Li-7 reaction is also discussed.

SIR2014 is the latest program of the SIR suite for crystal structure solution of small, medium and large structures. A variety of phasing algorithms have been implemented, both ab initio (standard or modern direct methods, Patterson techniques, Vive la Différence) and non-ab initio (simulated annealing, molecular replacement). The program contains tools for crystal structure refinement and for the study of three-dimensional electron-density maps via suitable viewers.

The n_TOF neutron time-of-flight facility at CERN is used for high quality nuclear data measurements from thermal energy up to hundreds of MeV. In line with the CERN open data policy, the n_TOF Collaboration takes actions to preserve its unique data, facilitate access to them in standardised format, and allow their re-use by a wide community in the fields of nuclear physics, nuclear astrophysics and various nuclear technologies. The present contribution briefly describes the n_TOF outcomes, as well as the status of dissemination and preservation of n_TOF final data in the international EXFOR library.

The triplet structure invariant is estimated via the method of joint probability distribution functions when a model structure is available. The six-variate probability distribution function P(Eh, Ek, E-h-k, Eph, Epk, Ep,-h-k) is studied under the condition that imperfect isomorphism between the target and model structures exist. The results are compared with those available in the literature, which were obtained under the condition of perfect isomorphism. It is shown that the new formalism is more suitable for real cases, where perfect isomorphism is very rare.

The newly built second experimental area EAR2 of the n_TOF spallation neutron source at CERN allows to perform (n, charged particles) experiments on short-lived highly radioactive targets. This paper describes a detection apparatus and the experimental procedure for the determination of the cross-section of the 7Be(n,?)? reaction, which represents one of the focal points toward the solution of the cosmological Lithium abundance problem, and whose only measurement, at thermal energy, dates back to 1963.The apparently unsurmountable experimental difficulties stemming from the huge 7Be ?-activity, along with the lack of a suitable neutron beam facility, had so far prevented further measurements. The detection system is subject to considerable radiation damage, but is capable of disentangling the rare reaction signals from the very high background. This newly developed setup could likely be useful also to study other challenging reactions requiring the detectors to be installed directly in the neutron beam.

Following the completion of the second neutron beam line and the related experimental area (EAR2) at the n_TOF spallation neutron source at CERN, several experiments were planned and performed. The high instantaneous neutron flux available in EAR2 allows to investigate neutron induced reactions with charged particles in the exit channel even employing targets made out of small amounts of short-lived radioactive isotopes. After the successful measurement of the 7Be(n,?)? cross section, the 7Be(n,p)7Li reaction was studied in order to provide still missing cross section data of relevance for Big Bang Nucleosynthesis (BBN), in an attempt to find a solution to the cosmological Lithium abundance problem. This paper describes the experimental setup employed in such a measurement and its characterization.

The slow neutron capture process (s-process) is responsible for producing about half of the elemental abundances heavier than iron in the universe. Neutron capture cross sections on stable isotopes are a key nuclear physics input for s-process studies. The72Ge(n, ?) cross section has an important influence on production of isotopes between Ge and Zr during s-process in massive stars and therefore experimental data are urgently required.72Ge(n, ?) was measured at the neutron time-of-flight facility n-TOF (CERN) for the first time at stellar energies. The measurement was performed using an enriched72GeO2sample at a flight path of 185m with a set of liquid scintillation detectors (C6D6). The motivation, experiment and current status of the data analysis are reported.

A novel concept for3He-free thermalization detector is here investigated by means of GEANT4 simulations. The detector is based on strips of solid-state detectors with6Li deposit for neutron conversion. Various geometrical configurations have been investigated in order to find the optimal solution, in terms of value and energy dependence of the efficiency for neutron energies up to 10 MeV. The expected performance of the new detector is compared with those of an optimized thermalization detector based on standard3He tubes. Although an3He-based detector is superior in terms of performance and simplicity, the proposed solution may become more appealing in terms of costs in case of shortage of3He supply.

Organic capped Au nanoparticles (NPs) and PbS quantum dots (QDs), synthesized with high control on size and size distribution, were used as building blocks for fabricating solid crystals by solvent evaporation. The superlattice formation process for the two types of nano-objects was investigated as a function of concentration by means of electron microscopy and X-ray techniques. The effect of building block composition, size, geometry, and concentration and the role of the organic coordinating molecules was related to the degree of order in the superlattices. A convenient combination of different complementary X-ray techniques, namely in situ and ex situ GISAXS and GIWAXS, allowed elucidating the most reliable signatures of the superlattices at various stages of the self-assembly process, since their early stage of formation and up to few months of aging. Significantly different assembly behaviour was assessed for the two types of NPs, clearly explained on the basis of their chemical composition, ultimately reflecting on the assembling process and on the final structure characteristics.

Monte Carlo (MC) simulations are an essential tool to determine fundamental features of a neutron beam, such as the neutron flux or the ?-ray background, that sometimes can not be measured or at least not in every position or energy range. Until recently, the most widely used MC codes in this field had been MCNPX and FLUKA. However, the Geant4 toolkit has also become a competitive code for the transport of neutrons after the development of the native Geant4 format for neutron data libraries, G4NDL. In this context, we present the Geant4 simulations of the neutron spallation target of the n_TOF facility at CERN, done with version 10.1.1 of the toolkit. The first goal was the validation of the intra-nuclear cascade models implemented in the code using, as benchmark, the characteristics of the neutron beam measured at the first experimental area (EAR1), especially the neutron flux and energy distribution, and the time distribution of neutrons of equal kinetic energy, the so-called Resolution Function. The second goal was the development of a Monte Carlo tool aimed to provide useful calculations for both the analysis and planning of the upcoming measurements at the new experimental area (EAR2) of the facility.

Two new computational methods dedicated to neutron crystallography, called n-FreeLunch and DNDM-NDM, have been developed and successfully tested. The aim in developing these methods is to determine hydrogen and deuterium positions in macromolecular structures by using information from neutron density maps. Of particular interest is resolving cases in which the geometrically predicted hydrogen or deuterium positions are ambiguous. The methods are an evolution of approaches that are already applied in X-ray crystallography: extrapolation beyond the observed resolution (known as the FreeLunch procedure) and a difference electron-density modification (DEDM) technique combined with the electron-density modification (EDM) tool (known as DEDM-EDM). It is shown that the two methods are complementary to each other and are effective in finding the positions of H and D atoms in neutron density maps.

The spent fuel of current nuclear reactors contains fissile plutonium isotopes that can be combined with 238U to make mixed oxide (MOX) fuel. In this way the Pu from spent fuel is used in a new reactor cycle, contributing to the long-term sustainability of nuclear energy. The use of MOX fuels in thermal and fast reactors requires accurate capture and fission cross sections. For the particular case of 242Pu, the previous neutron capture cross section measurements were made in the 70's, providing an uncertainty of about 35% in the keV region. In this context, the Nuclear Energy Agency recommends in its "High Priority Request List" and its report WPEC-26 that the capture cross section of 242Pu should be measured with an accuracy of at least 7-12% in the neutron energy range between 500 eV and 500 keV. This work presents a brief description of the measurement performed at n_TOF-EAR1, the data reduction process and the first ToF capture measurement on this isotope in the last 40 years, providing preliminary individual resonance parameters beyond the current energy limits in the evaluations, as well as a preliminary set of average resonance parameters.

Nuclear data in general, and neutron-induced reaction cross sections in particular, are important for a wide variety of research fields. They play a key role in the safety and criticality assessment of nuclear technology, not only for existing power reactors but also for radiation dosimetry, medical applications, the transmutation of nuclear waste, accelerator-driven systems, fuel cycle investigations and future reactor systems as in Generation IV. Applications of nuclear data are also related to research fields as the study of nuclear level densities and stellar nucleosynthesis. Simulations and calculations of nuclear technology applications largely rely on evaluated nuclear data libraries. The evaluations in these libraries are based both on experimental data and theoretical models. Experimental nuclear reaction data are compiled on a worldwide basis by the international network of Nuclear Reaction Data Centres (NRDC) in the EXFOR database. The EXFOR database forms an important link between nuclear data measurements and the evaluated data libraries. CERN's neutron time-of-flight facility n_TOF has produced a considerable amount of experimental data since it has become fully operational with the start of the scientific measurement programme in 2001. While for a long period a single measurement station (EAR1) located at 185 m from the neutron production target was available, the construction of a second beam line at 20 m (EAR2) in 2014 has substantially increased the measurement capabilities of the facility. An outline of the experimental nuclear data activities at CERN's neutron time-of-flight facility n_TOF will be presented.

Phasing proteins at non-atomic resolution is still a challenge for any ab initio method. A variety of algorithms [Patterson deconvolution, superposition techniques, a cross-correlation function (C map), the VLD (vive la difference) approach, the FF function, a nonlinear iterative peak-clipping algorithm (SNIP) for defining the background of a map and the free lunch extrapolation method] have been combined to overcome the lack of experimental information at non-atomic resolution. The method has been applied to a large number of protein diffraction data sets with resolutions varying from atomic to 2.1 Å, with the condition that S or heavier atoms are present in the protein structure. The applications include the use of ARP/wARP to check the quality of the final electron-density maps in an objective way. The results show that resolution is still the maximum obstacle to protein phasing, but also suggest that the solution of protein structures at 2.1 Å resolution is a feasible, even if still an exceptional, task for the combined set of algorithms implemented in the phasing program. The approach described here is more efficient than the previously described procedures: e.g. the combined use of the algorithms mentioned above is frequently able to provide phases of sufficiently high quality to allow automatic model building. The method is implemented in the current version of SIR2014. © 2014 International Union of Crystallography.

The neutron time of flight (n-TOF) facility at CERN is a spallation source characterized by a white neutron spectrum. The innovative features of the facility, in the two experimental areas, (20 m and 185 m), allow for an accurate determination of the neutron cross section for radioactive samples or for isotopes with small neutron capture cross section, of interest for Nuclear Astrophysics. The recent results obtained at n-TOF facility are presented.

SIR2011, the successor of SIR2004, is the latest program of the SIR suite. It can solve ab initio crystal structures of small- and medium-size molecules, as well as protein structures, using X-ray or electron diffraction data. With respect to the predecessor the program has several new abilities: e. g. a new phasing method (VLD) has been implemented, it is able to exploit prior knowledge of the molecular geometry via simulated annealing techniques, it can use molecular replacement methods for solving proteins, it includes new tools like free lunch and new approaches for electron diffraction data, and it visualizes three-dimensional electron density maps. The graphical interface has been further improved and allows the straightforward use of the program even in difficult cases.

Research and development on alternative thermal neutron detection technologies and methods arenowadays needed as a possible replacement of 3He-based ones. Commercial solid state silicon detectors,coupled with neutron converter layers containing 6Li, have been proved to represent a viable solution forseveral applications as present in the literature. In order to better understand the detailed operation andthe response and efciency of such detectors, a series of dedicated GEANT4 simulations were performedand compared with real data collected in a few different congurations. The results show an excellentagreement between data and simulations, indicating that the behavior of the detector is fully understood.

SUNBIM (Supramolecular & SUbmolecular Nano & Bio Materials X-ray IMaging Project) is a suite of integrated programs developed, in collaboration with Rigaku Innovative Technologies, to treat Small and Wide Angle X-ray Scattering data, collected either in transmission geometry (SAXS/WAXS) or in reflection geometry (GISAXS/GIWAXS). In addition, a specific routine to collect and analyze data in SAXS scanning transmission microscopy has been developed as additional tool to investigate tissues or material science samples through a focused X-ray beam which is used to raster scan a specimen while acquiring SAXS scattering patterns with a 2D detector. Indeed, a first-generation-synchrotron-class FrE+ SuperBright Rigaku microsource, coupled to a three pinhole S-MAX3000 camera, was recently installed at the X-ray MicroImaging Laboratory (XMI-L@b) and used with success in SAXS/WAXS/GISAXS/GIWAXS experiments (De Caro et al, 2012; 2013)and for SAXS scanning microscopy (Altamura et al, 2012; Giannini et al, 2013).

SUNBIM (supramolecular and submolecular nano- and biomaterials X-rayimaging) is a suite of integrated programs which, through a user-friendlygraphical user interface, are optimized to perform the following: (i) q-scalecalibration and two-dimensional ! one-dimensional folding on small- andwide-angle X-ray scattering (SAXS/WAXS) and grazing-incidence SAXS/WAXS (GISAXS/GIWAXS) data, also including possible eccentricity correctionsfor WAXS/GIWAXS data; (ii) background evaluation and subtraction,denoising, and deconvolution of the primary beam angular divergence onSAXS/GISAXS profiles; (iii) indexing of two-dimensional GISAXS frames andextraction of one-dimensional GISAXS profiles along specific cuts; (iv) scanningmicroscopy in absorption and SAXS contrast. The latter includes collection oftransmission and SAXS data, respectively, in a mesh across a mm2 area,organization of the as-collected data into a single composite image oftransmission values or two-dimensional SAXS frames, analysis of the composeddata to derive the absorption map and/or the spatial distribution, andorientation of nanoscale structures over the scanned area.

Neutron-induced reaction cross sections are important for a wide variety of research fields ranging from the study of nuclear level densities, nucleosynthesis to applications of nuclear technology like design, and criticality and safety assessment of existing and future nuclear reactors, radiation dosimetry, medical applications, nuclear waste transmutation, accelerator-driven systems and fuel cycle investigations. Simulations and calculations of nuclear technology applications largely rely on evaluated nuclear data libraries. The evaluations in these libraries are based both on experimental data and theoretical models. CERN's neutron time-of-flight facility n_TOF has produced a considerable amount of experimental data since it has become fully operational with the start of its scientific measurement programme in 2001. While for a long period a single measurement station (EAR1) located at 185 m from the neutron production target was available, the construction of a second beam line at 20 m (EAR2) in 2014 has substantially increased the measurement capabilities of the facility. An outline of the experimental nuclear data activities at n_TOF will be presented.

The CERN n_TOF neutron beam facility is characterized by a very high instantaneous neutron flux, excellent TOF resolution at the 185 m long flight path (EAR-1), low intrinsic background and coverage of a wide range of neutron energies, from thermal to a few GeV. These characteristics provide a unique possibility to perform high-accuracy measurements of neutron-induced reaction cross-sections and angular distributions of interest for fundamental and applied Nuclear Physics. Since 2001, the n_TOF Collaboration has collected a wealth of high quality nuclear data relevant for nuclear astrophysics, nuclear reactor technology, nuclear medicine, etc. The overall efficiency of the experimental program and the range of possible measurements has been expanded with the construction of a second experimental area (EAR-2), located 20 m on the vertical of the n_TOF spallation target. This upgrade, which benefits from a neutron flux 30 times higher than in EAR-1, provides a substantial extension in measurement capabilities, opening the possibility to collect data on neutron cross-section of isotopes with short half-lives or available in very small amounts. This contribution will outline the main characteristics of the n_TOF facility, with special emphasis on the new experimental area. In particular, we will discuss the innovative features of the EAR-2 neutron beam that make possible to perform very challenging measurements on short-lived radioisotopes or sub-mg samples, out of reach up to now at other neutron facilities around the world. Finally, the future perspectives of the facility will be presented.

An important experimental program on Nuclear Astrophysics is being carried out at the n-TOF since several years, in order to address the still open issues in stellar and primordial nucleosynthesis. Several neutron capture reactions relevant to s-process nucleosynthesis have been measured so far, some of which on important branching point radioisotopes. Furthermore, the construction of a second experimental area has recently opened the way to challenging measurements of (n, charged particle) reactions on isotopes of short half-life. The Nuclear Astrophysics program of the n-TOF Collaboration is here described, with emphasis on recent results relevant for stellar nucleosynthesis, stellar neutron sources and primordial nucleosynthesis.

VLD (vive la difference) is a novel ab initio phasing approachthat is able to drive random phases to the correct values. Ithas been applied to small, medium and protein structuresprovided that the data resolution was atomic. It has neverbeen used for non-ab initio cases in which some phaseinformation is available but the data resolution is usually very ?far from 1 A. In this paper, the potential of VLD is tested forthe first time for a classical non-ab initio problem: molecularreplacement. Good preliminary experimental results encour-aged the construction of a pipeline for leading partialmolecular-replacement models with errors to refined solutionsin a fully automated way. The pipeline moduli and theirinteraction are described, together with applications to a wideset of test cases.

The neutron capture cross section of several key unstable isotopes acting as branching points in the s-process are crucial for stellar nucleosynthesis studies, but they are very challenging to measure due to the difficult production of sufficient sample material, the high activity of the resulting samples, and the actual (n,?) measurement, for which high neutron fluxes and effective background rejection capabilities are required. As part of a new program to measure some of these important branching points, radioactive targets of 147Pm and 171Tm have been produced by irradiation of stable isotopes at the ILL high flux reactor. Neutron capture on 146Nd and 170Er at the reactor was followed by beta decay and the resulting matrix was purified via radiochemical separation at PSI. The radioactive targets have been used for time-of-flight measurements at the CERN n_TOF facility using the 19 and 185 m beam lines during 2014 and 2015. The capture cascades were detected using a set of four C6D6 scintillators, allowing to observe the associated neutron capture resonances. The results presented in this work are the first ever determination of the resonance capture cross section of 147Pm and 171Tm. Activation experiments on the same 147Pm and 171Tm targets with a high-intensity 30 keV quasi-Maxwellian flux of neutrons will be performed using the SARAF accelerator and the Liquid-Lithium Target (LiLiT) in order to extract the corresponding Maxwellian Average Cross Section (MACS). The status of these experiments and preliminary results will be presented and discussed as well

The expected mean-square error of electron-density maps (observed and difference) is traditionally estimated as a function of the variance of the observed amplitudes. The usual purpose is to evaluate the reliability of the structural parameters suggested by the final electron-density maps. Accordingly, such calculations are performed after the refinement stage, when the phases are considered perfectly determined. In this paper a mathematical expression for the variance (observed, difference and hybrid) is obtained for each point of an electron-density map for the space group P1 under a different hypothesis: the current phases are distributed on the trigonometric circle about the correct values, according to von Mises distributions. The variance calculation may then be performed at any stage of the phasing process, starting from a random up to a highly correlated model. It has been shown that the variance does not change dramatically from point to point of the map; therefore emphasis has been given to the concept of map variance, which allows an easier study of its properties. When the model is highly correlated with the target structure the conclusive formulas reduce to those previously described in the literature. The properties of the variance are discussed: it is shown that they are the basis for the most successful phasing procedures.