Latest Advances In Atomic Cluster Collisions: Fission, Fusion, Electron, Ion And Photon Impact

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Buffat and J. Borel, Phys. A 13, Borel, Surf. Garzon and J. Jellinek, Z. D 20, Valkealahti and M. Manninen, Comp. Ju and A. Bulgac, Phys. B 4 8, Poteau, F. Spiegelmann, and P. Labastie, Z. D 30, 57 Gronbeck, D. Tomanek, S. Kim, and A. Rosen, Z. D 40, Cleveland, W. Luedtke, and U. Landman, Phys. Calvo and F. Spiegelmann, J. Schmidt, R. Kusche, W. Kronmiiller, B. Haberland, Phys. Haberland, Nature , Kusche, Th.

Hippler, M. Schmidt, B. Haberland, Eur. D 9, 1 Martin, U. Naher, H. Schaber, and U. Zimmermann, J. Mukherjee, and K. Bennemann, Phys. B 28, Li, E. Blaisten-Barojas, and D. Papaconstantopoulos, Phys. B 57, Nose, Mol. Hoover, Phys. A 31, Barnett and U. B 48, Rytkonen and M. Manninen, Eur. D 23, Rytkonen, H. Hakkinen, and M. Manninen, Phys. D 9, Amusiaa'6, A. Msezane0, and V. An exact functional equation for the effective interaction, from which one can construct the action functional, density functional, the response functions, and excitation spectra of the considered systems, is outlined.

In the context of the density functional theory we consider the single particle excitation spectra of electron systems and relate the single particle spectrum to the eigenvalues of the corresponding Kohn-Sham equations. We find that the single particle spectrum coincides neither with the eigenvalues of the Kohn-Sham equations nor with those of the Hartree-Fock equations. Introduction The density functional theory DFT , that originated from the pioneering work of Hohenberg and Kohn1, has been extremely effective in describing the ground state of finite many-electron systems.

Such a success gave birth to many papers concerned with the generalization of DFT, which would permit the description of the excitation spectra also. Unfortunately, the one-to-one correspondence establishes only the existence of the functionals in prin- 41 42 M.

Shaginyan ciple, leaving aside a very important question on how one can construct them in reality. This shortcoming was resolved to a large extent in2' where exact equations connecting the action functional, effective interaction and linear response function were derived. But the linear response function, containing information of the particle-hole and collective excitations, does not directly present information about the single particle spectrum.

In this Report, the self consistent version of the density functional theory is outlined, which allows to calculate the ground state and dynamic properties of finite multi-electron systems starting with the Coulomb interaction. An exact functional equation for the effective interaction, from which one can construct the action functional, density functional, the response functions, and excitation spectra of the considered systems, is presented.

The effective interaction relating the linear response function of non-interacting particles to the exact linear response function is of finite radius and density dependent. We derive equations describing single particle excitations of multi-electron systems, using as a basis the exact functional equations, and show that single particle spectra do not coincide either with the eigenvalues of the Kohn-Sham equations or with those of the Hartree-Fock equations. For Eq. According to Eq. It is evident that the linear response function xid tends to the linear response function of the system in question as g goes to 1.

Here R r1,v2,uJ,g is the effective interaction depending on the coupling constant g of the Coulomb interaction. The coupling constant g in Eq. Shaginyan These constitute the linear response function x o r i,r 2 ,a-0 entering Eq. The Effective Interaction The above equations solve the problem of calculating Exc, the ground state energy and the particle-hole and collective excitation spectra of a system without resorting to approximations for Exc, based on additional and foreign inputs to the considered problem, such as found in calculations such as Monte Carlo simulations.

We note, that using these approximations, one faces difficulties in constructing the effective interaction of finite radius and the linear response functions J. On the basis of the suggested approach, one can solve these problems. For instance, in the case of a homogeneous electron liquid it is possible to determine analytically an efficient approximate expression RRPAE for the effective interaction R, which essentially improves the well-known Random Phase Approximation by taking into account the exchange interaction of the electrons properly, thus forming the Random Phase Approximation with Exchange 4 ' 5.

Note that the effective interaction RRPAE Q, p permits the description of the electron gas correlation energy e c in an extremely broad range of the variation of the density. Note that these eigenvalues Si do not have a physical meaning and cannot be regarded as the single-particle energies see e. Shaginyan method nor with et of Eq. To proceed, we use a method developed in 5.

The linear response function Xo and density p r , given by Eqs. As it follows from Eq. E , , 16 Upon using Eq. Substituting Eq. To calculate the derivative we consider an auxiliary system of non-interacting particles in a field U r. The third term on the right hand side of Eq. In the considered simplest case when we approximate the functional Exc by Ex, the coupling constant g enters Ex as a linear factor.

We employ Eq. Approximating the correlation functional Ec[p, m] by Eq. This condition is of crucial importance when calculating the wave functions and eigenvalues of vacant states within the framework of the DFT approach 5. Note, that these functions and eigenvalues that enter Eq.

This spectrum has to be compared with the experimental results. The single particle levels ej, given by Eq. But this is not the case, since , are solutions of Eq. We also anticipate that Eq. In the case of solids, we expect that the energy gap at various highsymmetry points in the Brillouin zone of semiconductors and dielectrics can also be reproduced. Shaginyan 5. Conclusions We have presented the self consistent version of the density functional theory, which allows calculation of the ground state and dynamic properties of finite multi-electron systems.

An exact functional equation for the effective interaction, from which one can construct the action functional, density functional, the response functions and excitation spectra of the considered systems, has been outlined. We have shown that it is possible to calculate the single particle excitations within the framework of DFT.

The developed equations permit the calculations of the single particle excitation spectra of any multielectron system such as atoms, molecules and clusters. We also anticipate also that these equations when applied to solids will produce quite reasonable results for the single particle spectra and energy gap at various high-symmetry points in the Brillouin zone of semiconductors and dielectrics. We have related the eigenvalues of the single particle KohnSham equations to the real single particle spectrum.

In the most straightforward case, when the exchange functional is treated rigorously while the correlation functional is taken in the local density approximation, the coupling equations are very simple. The single particle spectra do not coincide either with the eigenvalues of the Kohn-Sham equations or with those of the Hartree-Fock equations, even when the contribution coming from the correlation functional is omitted.

MYaA is grateful to the S. Shonbrunn Research Fund for support of his research. Hohenberg and W. Kohn, Phys. B , ; W. Kohn and L. Sham, Phys. A , ; W. Kohn, P. Washishta, in: Theory of the Inhomogeneous Electron Gas, eds. Lundqvist, N. March Plenum, New York and London, p. Garbo, T. Kreibich, S. Kurht, E. Anisimov Gordon and Breach, Tokyo, Khodel, V. Shaginyan, and V. Khodel, Phys. Runge and E.

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Gross, Phys. New Approach to Density Functional Theory 49 4. Shaginyan, Solid State Comm. Amusia and V. Shaginyan, J. B 25, L ; M. II Prance 3, Shaginyan, Phys. A , ; M. Amusia, V. Shaginyan, and A. A 4 7, Ceperly and B. Alder, Phys.

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  • Landau, Sov. JETP 3, Ioffe Physical-Technical Institute, St. A new theoretical framework for modelling the fusion process of noble gas clusters is presented. We report the striking correspondence of 51 52 LA. Solov'yov et al. Introduction There are many different types of clusters, such as metallic clusters, fullerenes, molecular clusters, semiconductor clusters, organic clusters, quantum dots, positively and negatively charged clusters. All have their own features and properties. Comprehensive survey of thefieldcan be found in review papers and books.

    In our paper we want to demonstrate this feature on a few examples. Namely, we will discuss sodium, magnesium and noble gas clusters and will show the principal differences in their structure and properties. In this work we consider the optimized ionic structure and the electronic properties of small sodium8 and magnesium9 clusters within the size range N Ab Initio Calculations and Modelling of Atomic Cluster Structure 53 clusters were used in such important experimental work as the discovery of metal cluster electron shell structure and the observation of plasmon resonances.

    Our calculations have been performed with the use of the Gaussian 98 software package. Results of the cluster geometry optimization for neutral sodium clusters consisting of up to 20 atoms are shown in Fig. The cluster geometries have been determined using the methodology described in Ref. This result is in 54 I. According to the jellium model see Refs.

    Let us now consider how the ionization potentials of sodium clusters evolve with increasing cluster size. Experimentally, such dependence has been measured for sodium clusters in Refs. The ionization potential of a cluster is defined as a difference between the energy of the singlycharged and neutral clusters. Figure 2 shows the dependence of the clusters ionization potential on N. It demonstrates that the results of the B3LYP calculation8 are in a reasonable agreement with the experimental data.

    Ionization potentials of neutral sodium clusters calculated in the deformed jellium model 12 ' 13 rhomboids and compared with ab initio results8 triangles and with experiment1 circles. The dependencies derived by the B3LYP method as well as the experimental one have a prominent odd-even oscillatory tendency. The maxima in these dependences correspond to the even-N-clusters, which means their higher stability as compared to the neighbouring odd-N-clusters. This happens because the multiplicities of the even and odd-N-clusters are different, being equal to one and two correspondingly.

    A significant step-like decrease in the ionization potential value happens at the transition from the dimer Ab Initio Calculations and Modelling of Atomic Cluster Structure 55 to the trimer cluster and also in the transition from Nag to Nag. Such an irregular behaviour can be explained by the closure of the electronic Is- and lp-shells of the delocalized electrons in the clusters Na,2 and Nag respectively. The comparison of the jellium model result with the ab initio calculation8 demonstrates that the jellium model reproduces correctly most of the essential features of the ionization potential dependence on N.

    Some discrepancy, like in the region 11 56 LA. The label above each cluster image indicates the point symmetry group of the cluster. Starting from Mgg a new element appears in the magnesium cluster structures. This is the six atom trigonal prism core, which is marked out in Fig. The formation of the trigonal prism plays the important role in the magnesium cluster growth process. Adding an atom to one of the triangular faces of the trigonal prism of the Mgg cluster results in the Mgio structure, while adding an atom to the remaining triangular face of the prism within the Mgw cluster leads to the structure of Mgn, as shown in Fig.

    Overall, obtained structures agree with those from Refs. It is worth noting that the formation of hexagonal ring for iV — 15 plays the important role in the evolution of the magnesium cluster structure towards to the bulk lattice, because the hexagonal ring is one of the basic elements of the hexagonal closest-packing hep lattice which is the lattice of bulk magnesium. A single deformed hexagonal ring is the common element in the structures of the Mg16 and Mgn clusters. For the Mgis clusters, two deformed hexagonal rings appear.

    Noble Gas Clusters Both sodium and magnesium clusters are metal clusters. For their description it is important to take into account the quantum effects. The situation is different for noble gas clusters, for example Ar, Kr, Xe. Noble gas clusters are formed by the long range van der Waals forces. This fact allows one to describe geometries of such systems using classical molecular dynamics approach. Relatively simple interaction between atoms in the system allows one to investigate clusters within the much larger size range, up to several hundreds atoms in a cluster, and to tackle the more sophisticated problems.

    Within the classical approximation, the motion of the atoms in a cluster is described by the Newton motion equations with a pairing potential. In our work we use the Lennard-Jones LJ potential. With the growth of the atom number of atoms in the system the problem of searching for the global energy minimum on the cluster multidimensional potential energy surface becomes more complicated.

    The number of local minima on the potential energy surface increases exponentially with the growth cluster size and is estimated5 to be of the order of for N — There are different algorithms and methods of the global minimization, which have been employed for the global minimization of atomic cluster systems.

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    We assume that atoms in a cluster are bound by Lennard-Jones potentials and the cluster fusion takes place atom by atom. At each step of the fusion process all atoms in the system are allowed to move, while the energy of the 58 I. Solov'yov el al. The motion of the atoms is stopped when the energy minimum is reached.

    The geometries and energies of all cluster isomers found in this way are analysed. For algorithmic details we refer to our recent papers. In this figure we present the cluster geometries within the size range 4 67 in this paper and refer to our recent work. Here, the cluster surface is considered as a polyhedron, so that the vertices of the polyhedron are the atoms and two vertices are connected by an edge, if the distance between them is less than the given value.

    It is interesting that all the cluster geometries calculated have the structure, in which a number of completed and open polygons round the cluster axis. The maximum possible number of atoms in polygons depends on the cluster size. Growth of noble gas global energy minimum cluster structures with N 60 I. The surface rearrangement of atoms should be an essential component of the cluster growth process. Fusion of a single atom to the global energy minimum cluster structure of 26 atoms does not lead to the global energy minimum of the LJ27 cluster first row.

    Rearrangement of surface atoms in the LJ27 cluster leading to the formation of the global energy minimum cluster structure is needed second row. The result of such rearrangement can be obtained if one starts the cluster growth from the excited state of the LJ25 cluster third row. In the first row of Fig. Thus, the rearrangement of surface atoms in the LJ27 cluster leading to the formation of the global energy minimum cluster structure is needed.

    The necessary rearrangement of atoms is shown in the second row of Fig. The result of such rearrange- Ab Initio Calculations and Modelling of Atomic Cluster Structure 61 ment can be obtained if one starts the cluster growth from the excited state of the L J25 cluster. This is illustrated in the third row of Fig. In this particular example only two smaller cluster sizes are involved in the fusion process of the L J27 cluster. However, the cluster fusion via excited states is not always that simple and evident.

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    In some cases, it involves more then 10 intermediate steps. Such a situation occurs, for example, for the fusion of LJQQ cluster, which can be obtained from the excited state of L J55 cluster. In this case the core structure of the cluster remains the same. However, in some cases it changes radically. Below, we call such radical rearrangements of the cluster structure as lattice rearrangements.

    The first lattice rearrangement takes place in the transition from L J 30 to L J31 cluster. Binding energy per atom for LJ-clusters as a function of cluster size calculated for the cluster chains based on the icosahedral, octahedral, tetrahedral and decahedral symmetry. In the insertion to Fig. The third term is the cluster energy arising due to the curvature of the cluster surface.

    The deviations of the energy curves calculated for various chains of cluster isomers from the liquid drop model 1 are plotted in Fig. The curves for the icosahedral and the global energy minimum cluster chains go very close to each other and the peaks on these dependences indicate the increased stability of the corresponding magic clusters. The dependence of the binding energies per atom for the most stable cluster configurations on N allows one to generate the sequence of the cluster magic numbers. In the inset to Fig. We compare the obtained dependence with the experimentally measured abundance mass spectrum for the Ar and Xe clusters33'34 see inset to Fig.

    Indeed, the magic numbers determined from A. Conclusion The optimized geometries and electronic properties of sodium and magnesium clusters consisting of up to 21 atoms have been investigated using the B3PW91, B3LYP and MPA methods accounting for the all electrons in the system. We compared the results of our calculations with the results obtained within the jellium model and with the available experimental data. Energy curves deviations from the liquid drop model 7 calculated for various cluster isomers chains. From these comparisons, we have elucidated the level of applicability of the jellium model to the description of sodium and magnesium clusters.

    We have also developed a new algorithm for modelling the cluster growth process based on the dynamic searching for the most stable cluster isomers. This algorithm can be considered an efficient alternative to the known cluster global minimization techniques. We have demonstrated that the majority of energetically favourable cluster structures can be obtained from the preceding cluster configurations by fusion of a single atom to the cluster surface. However, in some cases the surface and lattice rearrangements of the cluster occur.

    For the energetically favourable cluster configurations we report the striking correspondence of the peaks in the dependence of the second derivative of the binding energy per atom on cluster size with the peaks in the mass abundance spectra measured for the noble gas clusters. The results of this work can be extended in various directions. One can use the similar methods to study structure and properties of various types of clusters.

    It is interesting to extend calculations towards larger cluster sizes and to perform more advanced comparison of model and ab initio approaches, as well as to study collisions and electron excitations in clusters with the optimized geometries. These and many more other problems of atomic cluster physics can be tackled with the use of the methods considered in our work. Garciasand C. Guet, Physics Reports , Jellinek ed. Meiwes-Broer ed. Lyalin, S. Semenov, A. Solov'yov, N. Cherepkov and W. Greiner, J. B 33, Cherepkov, J.

    Connerade, and W. Taipei 48, Lyalin, I. Matveentsev, LA. Greiner, Eur. D 24, 15 Greiner, Submitted to the J. Me Guire and C. Knight, K. Clemenger, W. Saunders, M. Chou and M. Cohen, Phys. Brechignac, Ph. Cahuzac, F. Carlier, J. Leygnier, Chem. Selby, M. Vollmer, J. Masui, V. Kresin, W. Knight, Phys. B 40, Becke, J. Becke, Phys. A 38, ; C. Lee, W. Yang and R. Parr, Phys. B 37, Perdew, in Electronic Structure of Solids '91, edited by P. Ziesche and H.

    Eschrig p. Burke, J. Perdew and Y. Dobson, G. Vignale and M. Das, Plenum James B. Akeby, I. Panas, L. Petterson, P. Siegbahn, U. Wahlgreen, J. Acioli and J. Jellinek, Phys. Jellinek and P. Acioli, J. A , Martins, J. Buttet and R. Car, Phys. B 31, Kohn, F. Weigend and R. Ahlrichs, Phys. Wales, J. Doye, M. Miller, P. Mortenson and T. Walsh, Adv. Wales and J. Doye, J. Leary and J. Doye, Phys.

    Solov'yov, Andrey V.

    E 60, Harris, K. Norman, R. Mulkern, J. Northby, Phys. Miehle, O. Kandler, T. Leisner, O. Edit, J. Phys 91, In this book we introduce and discuss one such novel approach: the focus is on the radiation formed in a Crystalline Undulator, where electromagnetic radiation is generated by a bunch of ultra-relativistic particles channeling through a periodically bent crystalline structure. It is shown that under certain conditions, such a device emits intensive spontaneous monochromatic radiation and may even reach the coherence of laser light sources.

    Readers will be presented with the underlying fundamental physics and be familiarized with the theoretical, experimental and technological advances made during the last one and a half decades in exploring the various features of investigations into crystalline undulators. This research draws upon knowledge from many research fields - such as materials science, beam physics, the physics of radiation, solid state physics and acoustics, to name but a few.

    Accordingly, much care has been taken by the authors to make the book as self-contained as possible in this respect, so as to also provide a useful introduction to this emerging field to a broad readership of researchers and scientist with various backgrounds. Polarization Bremsstrahlung by Andrey V Korol 12 editions published between and in English and German and held by WorldCat member libraries worldwide This book introduces and reviews both theory and applications of polarizational bremsstrahlung, i.

    The subject, following the first experimental evidence a few decades ago, has gained importance through a number of modern applications. Thus, the study of several radiative mechanisms is expected to lead to the design of novel light sources, operating in various parts of the electromagnetic spectrum. Conversely, the analysis of the spectral and angular distribution of the photon emission constitutes a new tool for extracting information on the interaction of the colliding particles, and on their internal structure and dynamical properties.

    Last but not least, accurate quantitative descriptions of the photon emission processes determine the radiative energy losses of particles in various media, thereby providing essential information required for e.

    When Protons Collide

    This book primarily addresses graduate students and researchers with a background in atomic, molecular, optical or plasma physics, but will also be of benefit to anyone wishing to enter the field. Multiscale modeling of complex molecular structure and dynamics with MBN explorer by Ilia A Solov'yov 8 editions published between and in English and held by WorldCat member libraries worldwide This book introduces readers to MesoBioNano MBN Explorer - a multi-purpose software package designed to model molecular systems at various levels of size and complexity.

    In addition, it presents a specially designed multi-task toolkit and interface - the MBN Studio - which enables the set-up of input files, controls the simulations, and supports the subsequent visualization and analysis of the results obtained. The book subsequently provides a systematic description of the capabilities of this universal and powerful software package within the framework of computational molecular science, and guides readers through its applications in numerous areas of research in bio- and chemical physics and material science - ranging from the nano- to the meso-scale.

    MBN Explorer is particularly suited to computing the system's energy, to optimizing molecular structure, and to exploring the various facets of molecular and random walk dynamics. The package allows the use of a broad variety of interatomic potentials and can, e. MBN Studio enables users to easily construct initial geometries for the molecular, liquid, crystalline, gaseous and hybrid systems that serve as input for the subsequent simulations of their physical and chemical properties using MBN Explorer.

    Despite its universality, the computational efficiency of MBN Explorer is comparable to that of other, more specialized software packages, making it a viable multi-purpose alternative for the computational modeling of complex molecular systems. A number of detailed case studies presented in the second part of this book demonstrate MBN Explorer's usefulness and efficiency in the fields of atomic clusters and nanoparticles, biomolecular systems, nanostructured materials, composite materials and hybrid systems, crystals, liquids and gases, as well as in providing modeling support for novel and emerging technologies.

    Last but not least, with the release of the 3rd edition of MBN Explorer in spring , a free trial version will be available from the MBN Research Center website mbnresearch. Nanoscale insights into ion-beam cancer therapy 7 editions published between and in English and German and held by WorldCat member libraries worldwide This book provides a unique and comprehensive overview of state-of-the-art understanding of the molecular and nano-scale processes that play significant roles in ion-beam cancer therapy.

    It covers experimental design and methodology, and reviews the theoretical understanding of the processes involved. It offers the reader an opportunity to learn from a coherent approach about the physics, chemistry and biology relevant to ion-beam cancer therapy, a growing field of important medical application worldwide. The book describes phenomena occurring on different time and energy scales relevant to the radiation damage of biological targets and ion-beam cancer therapy from the molecular nano scale up to the macroscopic level.

    It illustrates how ion-beam therapy offers the possibility of excellent dose localization for treatment of malignant tumours, minimizing radiation damage in normal tissue whilst maximizing cell-killing within the tumour, offering a significant development in cancer therapy. Kresin , J. Huang , V. Kresin , A. Pysanenko , and M. Halder and V. Matter 28, Huang and V. Kresin , Rev. Kresin , Proc. Jena and A. Kandalam ]. B 92, Bellina , D. Halder , A.

    Liang , and V. Kresin , Nano Letters 15, Guggemos , P. Halder , C. Huang , and V. Ovchinnikov , A. Halder , and V. Gomez , E. Loginov , A. Halder , V. Kresin , and A. Vilesov , Int. Prem and V. Kresin , Phys Rev. A 85, Bulthuis and V. Phys , Liang , C. Yin , and V. Matter 24, [ Invited contribution to the G. Benedek Festschrift ]. Rabinovitch , K. Hansen , and V. A , [Invited contribution to the J. Toennies Festschrift ]. Loginov , L. Gomez , N. Chiang , A. Halder , N. Guggemos , V. Vilesov , Phys. Kresin , in Handbook of Nanophysics , ed. Stark and V.

    B 81, Moro , J. Heinrich , and V. Xia , C. Kresin , Eur. D 52, Bulthuis , J. Becker , R. Moro , and V. Rabinovitch , C. Xia , and V. A 77, Ren and V. Hansen , M. Johnson , and V. B 76, A 76, Poterya , M. Buck , and V. A 75, Ren , R. D 43, Moro , R.