Analysis of Industrial Granular Flow Applications by Using Advanced Collision Models

Granular flow phenomena are frequently encountered in the design of process and industrial plants in the traditional fields of the chemical, nuclear and oil industries as well as in other activities such as food and materials handling. Multi-phase flow is one important branch of the granular flow. Granular materials have unusual kinds of behavior compared to normal materials, either solids or fluids. Although some of the characteristics are still not well-known yet, one thing is confirmed: the particle-particle interaction plays a key role in the dynamics of granular materials, especially for dense granular materials. At the beginning of this thesis, detailed illustration of developing two models for describing the interaction based on the results of finite-element simulation, dimension analysis and numerical simulation is presented. The first model is used to describing the normal collision of viscoelastic particles. Based on some existent models, more parameters are added to this model, which make the model predict the experimental results more accurately. The second model is used for oblique collision, which include the effects from tangential velocity, angular velocity and surface friction based on Coulomb's law. The theoretical predictions of this model are in agreement with those by finite-element simulation. I n the latter chapters of this thesis, the models are used to predict industrial granular flow and the agreement between the simulations and experiments also shows the validation of the new model. The first case presents the simulation of granular flow passing over a circular obstacle. The simulations successfully predict the existence of a parabolic steady layer and show how the characteristics of the particles, such as coefficients of restitution and surface friction affect the separation results. The second case is a spinning container filled with granular material. Employing the previous models, the simulation could also reproduce experimentally observed phenomena, such as a depression in the center of a high frequency rotation. The third application is about gas-solid mixed flow in a vertically vibrated device. Gas phase motion is added to coherence with the particle motion. The governing equations of the gas phase are solved by using the Large eddy simulation (LES) and particle motion is predicted by using the Lagrangian method. The simulation predicted some pattern formation reported by experiment.

Investigation of interaction between native and impurity defects in ZnSe

Zinc selenide is a prospective material for optoelectronics. The fabrication of ZnSe­based light-emitting diodes is hindered by complexity of p-type doping of the component materials. The interaction between native and impurity defects, the tendency of doping impurity to form associative centres with native defects and the tendency to self-compensation are the main factors impeding effective control of the value and type of conductivity. The thesis is devoted to the study of the processes of interaction between native and impurity defects in zinc selenide. It is established that the Au impurity has the most prominent amphoteric properties in ZnSe among Cu, Ag and Au impurities, as it forms a great number of both Au; donors and Auz„ acceptors. Electrical measurements show that Ag and Au ions introduced into vacant sites of the Zn sublattice form simple single-charged Agz„+ and Auzn+ states with d1° electron configuration, while Cu ions can form both single-charged Cuz„ (d1) and double-charged Cuzr`+ (d`o) centres. Amphoteric properties of Ag and Au transition metals stimulated by time are found for the first time from both electrical and luminescent measurements. A model that explains the changes in electrical and luminescent parameters by displacement of Ag ions into interstitial sites due to lattice deformation forces is proposed. Formation of an Ag;-donor impurity band in ZnSe samples doped with Ag and stored at room temperature is also studied. Thus, the properties of the doped samples are modified due to large lattice relaxation during aging. This fact should be taken into account in optoelectronic applications of doped ZnSe and related compounds.

Compositional influence on the electrical performance of zinc indium tin oxide transparent thin-film transistors

In this work, zinc indium tin oxide layers with different compositions are used as the active layer of thin film transistors. This multicomponent transparent conductive oxide is gaining great interest due to its reduced content of the scarce indium element. Experimental data indicate that the incorporation of zinc promotes the creation of oxygen vacancies. In thin-film transistors this effect leads to a higher threshold voltage values. The field-effect mobility is also strongly degraded, probably due to coulomb scattering by ionized defects. A post deposition annealing in air reduces the density of oxygen vacancies and improves the fieldeffect mobility by orders of magnitude. Finally, the electrical characteristics of the fabricated thin-film transistors have been analyzed to estimate the density of states in the gap of the active layers. These measurements reveal a clear peak located at 0.3 eV from the conduction band edge that could be attributed to oxygen vacancies.

Termodinâmica estatística de líquidos com o método de Monte Carlo. I. Metodologia

Statistical mechanics Monte Carlo simulation is reviewed as a formalism to study thermodynamic properties of liquids. Considering the importance of free energy changes in chemical processes, the thermodynamic perturbation theory implemented in the Monte Carlo method is discussed. The representation of molecular interaction by the Lennard-Jones and Coulomb potential functions is also discussed. Charges derived from quantum molecular electrostatic potential are also discussed as an useful methodology to generate an adequate set of partial charges to be used in liquid simulation.

Termodinâmica estatística de líquidos com o método de Monte Carlo. II. O líquido clorofórmio

Thermodynamic properties and radial distribution functions for liquid chloroform were calculated using the Monte Carlo method implemented with Metropolis algorithm in the NpT ensemble at 298 K and 1 atm. A five site model was developed to represent the chloroform molecules. A force field composed by Lennard-Jones and Coulomb potential functions was used to calculate the intermolecular energy. The partial charges needed to represent the Coulombic interactions were obtained from quantum chemical ab initio calculations. The Lennard-Jones parameters were adjusted to reproduce experimental values for density and enthalpy of vaporization for pure liquid. All thermodynamic results are in excelent agreement with experimental data. The correlation functions calculated are in good accordance with theoretical results avaliable in the literature. The free energy for solvating one chloroform molecule into its own liquid at 298 K and 1 atm was computed as an additional test of the potential model. The result obtained compares well with the experimental value. The medium effects on cis/trans convertion of a hypotetical solute in water TIP4P and chloroform solvents were also accomplished. The results obtained from this investigation are in agreement with estimates of the continuous theory of solvation.

Universality of the 1/3 shot-noise suppression factor in nondegenerate diffusive conductors

Shot-noise suppression is investigated in nondegenerate diffusive conductors by means of an ensemble Monte Carlo simulator. The universal 1/3 suppression value is obtained when transport occurs under elastic collision regime provided the following conditions are satisfied: (i) The applied voltage is much larger than the thermal value; (ii) the length of the device is much greater than both the elastic mean free path and the Debye length. By fully suppressing carrier-number fluctuations, long-range Coulomb interaction is essential to obtain the 1/3 value in the low-frequency limit.

Self-consistent theory of current and voltage noise in multimode ballistic conductors

Electron transport in a self-consistent potential along a ballistic two-terminal conductor has been investigated. We have derived general formulas which describe the nonlinear current-voltage characteristics, differential conductance, and low-frequency current and voltage noise assuming an arbitrary distribution function and correlation properties of injected electrons. The analytical results have been obtained for a wide range of biases: from equilibrium to high values beyond the linear-response regime. The particular case of a three-dimensional Fermi-Dirac injection has been analyzed. We show that the Coulomb correlations are manifested in the negative excess voltage noise, i.e., the voltage fluctuations under high-field transport conditions can be less than in equilibrium.

Shot-noise spectroscopy of energy resolved ballistic currents

We investigate the shot noise of nonequilibrium carriers injected into a ballistic conductor and interacting via long-range Coulomb forces. Coulomb interactions are shown to act as an energy analyzer of the profile of injected electrons by means of the fluctuations of the potential barrier at the emitter contact. We show that the details in the energy profile can be extracted from shot-noise measurements in the Coulomb interaction regime, but cannot be obtained from time-averaged quantities or shot-noise measurements in the absence of interactions.

Control and femtosecond time-resolved imaging of torsion in a chiral molecule

We study how the combination of long and short laser pulses can be used to induce torsion in an axially chiral biphenyl derivative (3,5-difluoro-3 ,5 -dibromo-4 -cyanobiphenyl). A long, with respect to the molecular rotational periods, elliptically polarized laser pulse produces 3D alignment of the molecules, and a linearly polarized short pulse initiates torsion about the stereogenic axis. The torsional motion is monitored in real-time by measuring the dihedral angle using femtosecond time-resolved Coulomb explosion imaging. Within the first 4 picoseconds (ps), torsion occurs with a period of 1.25 ps and an amplitude of 3◦ in excellent agreement with theoretical calculations. At larger times, the quantum states of the molecules describing the torsional motion dephase and an almost isotropic distribution of the dihedral angle is measured.We demonstrate an original application of covariance analysis of two-dimensional ion images to reveal strong correlations between specific ejected ionic fragments from Coulomb explosion. This technique strengthens our interpretation of the experimental data

Magnetic and transport properties of II-V diluted magnetic semiconductors doped with manganese and nickel

This thesis is devoted to investigations of three typical representatives of the II-V diluted magnetic semiconductors, Zn1-xMnxAs2, (Zn1-xMnx)3As2 and p-CdSb:Ni. When this work started the family of the II-V semiconductors was presented by only the compounds belonging to the subgroup II3-V2, as (Zn1-xMnx)3As2, whereas the rest of the materials mentioned above were not investigated at all. Pronounced low-field magnetic irreversibility, accompanied with a ferromagnetic transition, are observed in Zn1-xMnxAs2 and (Zn1-xMnx)3As2 near 300 K. These features give evidence for presence of MnAs nanosize magnetic clusters, responsible for frustrated ground magnetic state. In addition, (Zn1-xMnx)3As2 demonstrates large paramagnetic response due to considerable amount of single Mn ions and small antiferromagnetic clusters. Similar paramagnetic system existing in Zn1-xMnxAs2 is much weaker. Distinct low-field magnetic irreversibility, accompanied with a rapid saturation of the magnetization with increasing magnetic field, is observed near the room temperature in p- CdSb:Ni, as well. Such behavior is connected to the frustrated magnetic state, determined by Ni-rich magnetic Ni1-xSbx nanoclusters. Their large non-sphericity and preferable orientations are responsible for strong anisotropy of the coercivity and saturation magnetization of p- CdSb:Ni. Parameters of the Ni1-xSbx nanoclusters are estimated. Low-temperature resistivity of p-CdSb:Ni is governed by a hopping mechanism of charge transfer. The variable-range hopping conductivity, observed in zero magnetic field, demonstrates a tendency of transformation into the nearest-neighbor hopping conductivity in non-zero magnetic filed. The Hall effect in p-CdSb:Ni exhibits presence of a positive normal and a negative anomalous contributions to the Hall resistivity. The normal Hall coefficient is governed mainly by holes activated into the valence band, whereas the anomalous Hall effect, attributable to the Ni1-xSbx nanoclusters with ferromagnetically ordered internal spins, exhibits a low-temperature power-law resistivity scaling.

Magnetic and transport properties of LaMnO3+δ, La1-xCaxMnO3, La1-xCaxMn1-yFeyO3 and La1-xSrxMn1-yFeyO3

Interest to hole-doped mixed-valence manganite perovskites is connected to the ‘colossal’ magnetoresistance. This effect or huge drop of the resistivity, ρ, in external magnetic field, B, attains usually the maximum value near the ferromagnetic Curie temperature, TC. In this thesis are investigated conductivity mechanisms and magnetic properties of the manganite perovskite compounds LaMnO3+, La1-xCaxMnO3, La1-xCaxMn1-yFeyO3 and La1- xSrxMn1-yFeyO3. When the present work was started the key role of the phase separation and its influence on the properties of the colossal magnetoresistive materials were not clear. Our main results are based on temperature dependencies of the magnetoresistance and magnetothermopower, investigated in the temperature interval of 4.2 - 300 K in magnetic fields up to 10 T. The magnetization was studied in the same temperature range in weak (up to 0.1 T) magnetic fields. LaMnO3+δ is the parent compound for preparation of the hole-doped CMR materials. The dependences of such parameters as the Curie temperature, TC, the Coulomb gap, Δ, the rigid gap, γ, and the localization radius, a, on pressure, p, are observed in LaMnO3+δ. It has been established that the dependences above can be interpreted by increase of the electron bandwidth and decrease of the polaron potential well when p is increased. Generally, pressure stimulates delocalization of the electrons in LaMnO3+δ. Doping of LaMnO3 with Ca, leading to La1-xCaxMnO3, changes the Mn3+/Mn4+ ratio significantly and brings an additional disorder to the crystal lattice. Phase separation in a form of mixture of the ferromagnetic and the spin glass phases was observed and investigated in La1- xCaxMnO3 at x between 0 and 0.4. Influence of the replacement of Mn by Fe is studied in La0.7Ca0.3Mn1−yFeyO3 and La0.7Sr0.3Mn1−yFeyO3. Asymmetry of the soft Coulomb gap and of the rigid gap in the density of localized states, small shift of the centre of the gaps with respect to the Fermi level and cubic asymmetry of the density of states are obtained in La0.7Ca0.3Mn1−yFeyO3. Damping of TC with y is connected to breaking of the double-exchange interaction by doping with Fe, whereas the irreversibility and the critical behavior of the magnetic susceptibility are determined by the phase separation and the frustrated magnetic state of La0.7Sr0.3Mn1−yFeyO3.

Non-Markovian dynamics and quantum information probes

This Thesis discusses the phenomenology of the dynamics of open quantum systems marked by non-Markovian memory effects. Non-Markovian open quantum systems are the focal point of a flurry of recent research aiming to answer, e.g., the following questions: What is the characteristic trait of non-Markovian dynamical processes that discriminates it from forgetful Markovian dynamics? What is the microscopic origin of memory in quantum dynamics, and how can it be controlled? Does the existence of memory effects open new avenues and enable accomplishments that cannot be achieved with Markovian processes? These questions are addressed in the publications forming the core of this Thesis with case studies of both prototypical and more exotic models of open quantum systems. In the first part of the Thesis several ways of characterizing and quantifying non-Markovian phenomena are introduced. Their differences are then explored using a driven, dissipative qubit model. The second part of the Thesis focuses on the dynamics of a purely dephasing qubit model, which is used to unveil the origin of non-Markovianity for a wide class of dynamical models. The emergence of memory is shown to be strongly intertwined with the structure of the spectral density function, as further demonstrated in a physical realization of the dephasing model using ultracold quantum gases. Finally, as an application of memory effects, it is shown that non- Markovian dynamical processes facilitate a novel phenomenon of timeinvariant discord, where the total quantum correlations of a system are frozen to their initial value. Non-Markovianity can also be exploited in the detection of phase transitions using quantum information probes, as shown using the physically interesting models of the Ising chain in a transverse field and a Coulomb chain undergoing a structural phase transition.

Contact with friction using the augmented Lagrangian Method: a conditional constrained minimization problem

This work presents a formulation of the contact with friction between elastic bodies. This is a non linear problem due to unilateral constraints (inter-penetration of bodies) and friction. The solution of this problem can be found using optimization concepts, modelling the problem as a constrained minimization problem. The Finite Element Method is used to construct approximation spaces. The minimization problem has the total potential energy of the elastic bodies as the objective function, the non-inter-penetration conditions are represented by inequality constraints, and equality constraints are used to deal with the friction. Due to the presence of two friction conditions (stick and slip), specific equality constraints are present or not according to the current condition. Since the Coulomb friction condition depends on the normal and tangential contact stresses related to the constraints of the problem, it is devised a conditional dependent constrained minimization problem. An Augmented Lagrangian Method for constrained minimization is employed to solve this problem. This method, when applied to a contact problem, presents Lagrange Multipliers which have the physical meaning of contact forces. This fact allows to check the friction condition at each iteration. These concepts make possible to devise a computational scheme which lead to good numerical results.

« Chantz royaulx, ballades, rondeaulx et epigrammes » présentés au « puy, à l'honneur et reverence de l'Immaculée Conception de la tres sacrée mere de Dieu, nostre redempteur... le dymenche.XIIIIE. jour de decembre, l'an mil cinq centz trente troys, à Rouen, au couvent des Carmes ».

Contient : 1 « Ballade » ; 2 « Rondeau » ; « Chantz royaulx » ; 3 « Concept sans coulpe en heureuse innocence ». « NICOLE BOYSSEL » ; 4 « La fleur du thym preparée à la mouche ». « FRANÇOYS SAGON » ; 5 « Manne rendant espoir, santé et vie ». « GUILLAUME THIBAULT » ; 6 « Bien souverain de bonté souveraine ». « THOMAS LE PREVOST » ; 7 « Signe luysant à la ligne ecliptique ». « GUILLAUME LE TELLYER » ; 8 « Par droict divin divinement conceue ». « GEFFROY LE PREVOST » ; 9 « L'heronde où gist du printemps le vray signe ». « SANSON DE CAGROFY » ; 10 « Le rond aneau faict pour le cours solaire ». « JEHAN DE BOISSAY » ; 11 « Promesse en grace et purité concene ». « LESCARRE » ; 12 « La verité sur tous est la plus forte ». « SEVYNGUEHEN » ; 13 « La saincte vierge où Dieu print chair humaine ». « PIERRES GAULTIER » ; 14 « La clere estelle, en plain jour lumyneuse ». « ROMMAIN DU VAL » ; 15 « L'estoille clere, en lumyere parfaicte ». « HERNEUS FIERABRAS » ; 16 « Au cueur d'yver le rossignol qui chante ». « GUILLAUME DE MONTIVILLER » ; 17 « Conception où grace est prevenue ». « G. HAUDENT » ; 18 « Des serfz conceue et jamais ne fut serve ». « P. DE VILLY » ; 19 « L'eau vive et pure, aux humains salutaire » ; 20 « Le cler vaisseau de pur verre infrangible ». « JAQUES SYREULDE » ; 21 « Contre peché pure et saincte par grace ». « LE TELLYER » ; 22 « Feu conservé en purité ardante ». « NYCOLE DUPUYS » ; 23 « Ung corps humain, exempt de tache humaine ». « FLORENT COPPIN » ; 24 « Claire phiale, entiere et infrangible ». « JEHAN HEUZE » ; 25 « D'humeur impur ambre pure formée ». « P. BERNARD » ; 26 « La cyre vierge où Dieu print forme humaine ». « LE GALLOYS » ; 27 « Ame divine, en corps humain infuse ». « JEHAN BOUST » ; 28 « Le blanc coulomb portant l'espoir de vie ». « JEHAN LE ROUX » ; 29 « L'isle royalle en vertuz florissante ». « MICHEL HEURTAULT » ; 30 « Ung corps humain franc de macule humaine ». « JEHAN COUPPEL » ; 31 « L'honneur du filz, de la mere et du pere ». « SYMON LE DOULX » ; 32 « La haulte clef de nature parfaicte ». « PIERRE FERE » ; 33 « Seule conceue en parfaicte semence ». « JAQUES SPALLART » ; 34 « Le corps formé à divine semblance ». « LAMBERT MESNARD » ; 35 « Corps oultre loy exempt de tache humaine ». « JEHAN THYBAULD » ; 36 « L'aube du jour, resjouyssant nature ». « HENRY MALLET » ; 37 « Sans offenser humanité divine ». « JEHAN OUYN » ; 38 « Port gracieux et havre salutaire ». « GUIEFFROY BARBEL DE HONNEFLEU » ; 39 « Basme excellent rendant aux mortz la vie ». « P. DU VAL » ; 40 « Le riche mont, le plus complect du monde ». « JEHAN DU MONT » ; 41 « La palme, lys, le signet ou la rose ». « CANNILLET » ; 42 « Le grand blason des bottines Gaultier ». « CACHELEU » ; « Ballades » ; 43 « Femme humaine en divine grace ». « THOMAS LE PREVOST » ; 44 « De deux pierres une estincelle ». « AVRIL » ; 45 « Concept et grace tout ensemble ». « GUILLAUME TYBAULD » ; 46 « La forte espouse du tres fort ». « GUILLAUME LE TELLYER » ; 47 « La plus excellente du monde ». « SANSON DE CAGROFY » ; 48 « Plus necte que pupille d'oeil ». « FR. SAGON » ; 49 « Grace est entierement pour moy ». « JEHAN HEUZE » ; 50 « Fille à mon filz, mere à mon pere ». « JAQUES SIREULDE » ; 51 « Possible à luy et non à moy ». « P. BENARD » ; 52 « Purité, innocence et grace ». « FLORENT COPPIN » ; 53 « Mort ne peché n'ont mors sur moy ». « TOURMENTE » ; 54 « Pour porter le fruict de salut ». « JEHAN LE ROUX » ; 55 « La grace de Dieu me suffict ». « LESCARRE » ; 56 « Volla saine et sauve au desert ». « LE TELLYER » ; 57 « Amour et vertu joinctz ensemble ». « SYMON LE DOULX » ; 58 « Cerve entre chiens de mort exempte ». « JOHANNES BOUST » ; 59 « Pour veoir en chair divinité ». « GEFFROY LE PREVOST » ; 60 « Conceue en grace immensurable ». « SEVYNGUEHEN » ; 61 « L'horloge pour l'heure certaine ». « JEHAN DE BOISSAY » ; 62 « Grace divine te conduict ». « ADRIEN LE PREVOST » ; 63 « Marie fut conceue en grace ». « GUILLAUME HAUDENT » ; 64 « Femme sans macule conceue ». « ROMMAIN DU VAL » ; 65 « Amour, vertu, grace et honneur ». « MYSONART » ; 66 « La fleur du lys entre l'espine ». « JEHAN OUYN » ; 67 « Corps où Dieu print humanité ». « PHILIPPES COUPPEL » ; 68 « Devant le soleil et la lune ». « JEHAN COUPPEL » ; 69 « Conceue en purité parfaicte ». « LE GALLOYS » ; 70 « La plus belle de tout le monde ». « P. GAULTIER » ; 71 « De tous conceups la myeux conceue ». « HENRY MALLET » ; 72 « Le noir ver du tort confondu ». « PIERRES FEREY » ; 73 « L'excellence de ses merveilles ». « MICHEL HEURTAUT » ; 74 « Basme rendant espoir de vie ». « PIERRES DU VAL » ; 75 « D'un corps imparfaict ung parfaict ». « JEHAN TYBAULD » ; 76 « Je suis l'inception de grace ». « JEHAN [C]OQUEREL » ; 77 « Rigueur en grace et mort en vie ». « JAQUES SPALLART » ; 78 « Cappel propre contre tous ventz ». « THOMAS MAURICE DE CAPPEL » ; 79 « Toute belle en conception ». « CANNYLLET » ; « Rondeaulx », Premiers vers : ; 80 « Par don de grace et par prevention ». « FRANÇOYS SAGON » ; 81 « Seul à part moy qui n'estoit ciel ne lune ». « GUILLAUME THYBAULD » ; 82 « Par droict divin, naturel et humain ». « AVRIL » ; 83 « Ainsy que vous de semblable matiere » ; 84 « Sur tous esleuz qui n'ont qu'en moy facture ». « NICOLE BOYSSEL » ; 85 « Mort et peché pour ung mors trop infame ». « GUILLAUME LE TELLYER » ; 86 « Comme la fleur qui eslieve son oeil ». « SANSON DE CAGROFY » ; 87 « Seule entre tous mon filz m'a tant amée » ; 88 « Pour tous humains j'ay mys à mort la mort ». « TOURMENTE » ; 89 « Pour Dieu, mon filz, second en trinité ». « P. DE VILLY » ; 90 « Seule apres Dieu qui me feist à part mectre ». « SEVYNGUEHEN » ; 91 « Par droict d'amour soubz figure inventée ». « JEHAN LE ROUX » ; 92 « Du bien d'amour qui tout faict et parfaict » ; 93 « Entre deux partz longue a esté la guerre ». « JEHAN DE BOISSAY » ; 94 « Je suis qui suis, commencement et fin ». « GEFFROY LE PREVOST » ; 95 « Au gré du filz le pere omnipotent ». « FLORENT COPPIN » ; 96 « Vierge sans per, qui fuz enluminée ». « LE TELYER » ; 97 « Pour tout certain ung enfant tost lamente ». « JAQUES SPALLART » ; 98 « D'homme et de Dien je prins forme et matiere ». « P. GAULTIER » ; 99 « Le pur bouton en plaisante ornature ». « JEHAN COUPPEL » ; 100 « Gentilz Normandz, soyez fortz et hardys ». « PIERRE TYBERGE » ; 101 « Le roy le veult qu'en mon corps ne soit pris ». « JEHAN COQUEREL » ; 102 « Le basme exquis d'une arbustre distille ». « PIERRE DU VAL » ; 103 « Augré d'amour je prospere et impere ». « ROMMAIN DU VAL » ; 104 « Il fut en luy possible et non en moy ». « P. BERNARD » ; 105 « De trop ne peu mon seul filz et ma vie ». « JO. BOUST » ; 106 « Sus mon honneur, je voys mectre mon ame ». « JEHAN DU MONT » ; 107 « Est il pas vray que je suis toute belle ». « ROBERT BECQUET » ; 108 « De par mon filz qui tout peult en effect ». « JEHAN OUYN » ; 109 « Selon mon gré, ma mere ai retenue ». « GUILLAUMES HAUDENT » ; 110 « Mon Dieu, mon filz, mon espoux et mon pere ». « SYMON LE DOULX » ; 111 « Par mon moyen fut l'erreur confondue ». « JAQUES SIREULDE » ; 112 « Pour l'Eternel qui sur la mort impere ». « MYSONART » ; 113 « Une sans plus au gré de Dieu conceue » ; 114 « Il n'en est plus ne sera de ma race ». « HENRY MALLET » ; 115 « C'est mal parlé de tenir le propoz ». « JEHAN TYBAULD » ; 116 « Viande au feu et nourriture à vers ». « CANNILLET » ; « Chants royaulx » ; 117 « La force en fort monstrant la femme forte ». « JAQUES LE LYEUR » ; 118 « La poulle à fleur du coq noir preservée ». « LESCARRE » ; 119 « Palme, laurier, rose, lys et estelle ». « AVRIL » ; 120 « L'arbre de fresne où le serpent n'attouche ». « DU VAL » ; 121 « Remede en peste et confort en famine ». « SEVYNGUEHEN » ; 122 « Le grand arrest donné en plaine boyse ». « JEHAN BAILLEHACHE » ; 123 « Le sacré corps de la vierge Marie ». « MISTY » ; « Ballades ». Refrains ; 124 « D'ung grand mal j'ay faict ung grand bien ». « JAQUES LE LYEUR » ; 125 « Dieu et nature ont tout bien faict ». « LESCARRE » ; 126 « L'ennemy, la chair et le monde ». « JEHAN DU VAL » ; 127 « La toute belle en chair conceue ». « BOYSSEL » ; 128 « Le fer sans rouil, plus cler qu'argent ». « JEHAN HEUZE » ; 129 « La fleur de beaulté souveraine ». « JEHAN HEUZE » ; 130 « La belle des belles sans sy ». « DE QUIEVRECOURT » ; 131 « La mere Diu, en conchept tres parfouette ». « COPPIN » ; 132 « Fille, femme et mere de grace » ; « Rondeaulx ». Premiers vers ; 133 « En cas d'abuz au grand conseil du roy ». « AVRIL » ; 134 « Il n'est que amour quand il tient en ses laqz ». « JEHAN DU VAL » ; 135 « Quoiqu'on ayct dict, j'ay esté en nature » ; 136 « C'est bien pour moy qui suis mere et pucelle ». « JEHAN HEUZE » ; 137 « Du bien de Dieu, vierge incontaminée »

On the formulation and the calculation of the harmonic contributions to the Debye-Waller factor in metals (sodium)

The algebraic expressions for the anharmonic contributions to the Debye-Waller factor up to 0(A ) and 0 L% ) £ where ^ is the scattering wave-vector] have been derived in a form suitable for cubic metals with small ion cores where the interatomic potential extends to many neighbours. This has been achieved in terms of various wave-vector dependent tensors, following the work of Shukla and Taylor (1974) on the cubic anharmonic Helmholtz free energy. The contribution to the various wave-vector dependent tensors from the coulomb and the electron-ion terms in the interatomic metallic potential has been obtained by the Ewald procedure. All the restricted multiple whole B r i l l o u i n zone (B.Z.) sums are reduced to single whole B.Z. sums by using the plane wave representation of the delta function. These single whole B.Z. sums are further reduced to the •%?? portion of the B.Z. following Shukla and Wilk (1974) and Shukla and Taylor (1974). Numerical calculations have been performed for sodium where the Born-Mayer term in the interatomic potential has been neglected because i t is small £ Vosko (1964)3 • *n o^er to compare our calculated results with the experimental results of Dawton (1937), we have also calculated the r a t io of the intensities at different temperatures for the lowest five reflections (110), (200), (220), (310) and (400) . Our calculated quasi-harmonic results agree reasonably well with the experimental results at temperatures (T) of the order of the Debye temperature ( 0 ). For T » © ^ 9 our calculated anharmonic results are found to be in good agreement with the experimental results.The anomalous terms in the Debye-Waller factor are found not to be negligible for certain reflections even for T ^ ©^ . At temperature T yy Op 9 where the temperature is of the order of the melting temperature (Xm) » "the anomalous terms are found to be important almost for all the f i ve reflections.