Born-Mayer Potential Research Articles

The energy of formation of Schottky defects in ionic crystals

Abstract This paper calculates the energy of formation Ws of Schottky defects in ionic crystals by methods derived from the original method of Mott and Littleton. The main aim of these calculations is to examine the variations in the predicted energies resulting from reasonable variations in the assumed models. The types of variation considered are: (1) changes in the description of the long-range displacement and polarization field around the vacancies, corresponding to the inclusion of (a) purely elastic distortions and (b) electronic deformations of the ions resulting from the polarization of the lattice; (2) changes in the magnitude of the second neighbour Born-Mayer interactions corresponding to different available choices for ionic radii and to the inclusion or otherwise of Van der Waals forces; and, more briefly, (3) changes from the Born-Mayer form to the Born—Mayer-Yerwey form of closed-shell repulsion potential. The results are given in tables but are broadly as follows. For a given closed-shell repulsive potential, the inclusion of elastic terms in the displacement field (in addition to polarization displacements resulting from the effective charges on the defects) raises Ws, while inclusion of electronic deformations lowers it again. The net result of including both is to increase Ws by ∼½ev. For given assumptions about the displacement and polarization field changes in the Born-Mayer constants can be important; thus use of the conventional Goldschmidt radii in place of the more recent radii of Tosi and Fumi leads to a lowering of the Schottky energies by amounts which are larger for the compounds of small cations and large anions—for NaCl by ∼1/2ev. These results are consistent with those of Tosi and result from the importance of the first-neighbour interactions in determining Ws. Use of a Born-Mayer—Yerwey potential increases Ws compared to the values for the corresponding Bom-Mayer form. Experimental values of W s are known for some alkali halides. Reasonable agreement with those for Nad, NaBr and KC1 results from using the Tosi-Fumi constants in a Born-Mayer potential, but for the Li halides and KBr the calculated values are too low. For these substances a somewhat stiffer potential seems necessary. Our results, especially those on the effect of the changes in the closod-shell repulsions, are important for the extension of this model to simple oxides.

Re-examination of the Born-Mayer potential for ion-core interactions in copper

Ion-core interactions between nearest neighbors in copper are approximated by a Born-Mayer potential E c − c = α exp ρ[ (r O−r) r O ] . Experimental values of cohesive energy, lattice parameter and compressibility are used to determine the parameters in the Born-Mayer potential. Thus we are using all the volume dependent terms in the interaction energy rather than just the Fermi energy which has been used in earlier calculations. Also we avoid the use of shear constants which were used in earlier calculations since there is a lot of uncertainty in them. Our results for α and ρ are 0.0958 eV and 13.34 respectively. At short distances our potential is in good agreement with the most commonly used potential due to Huntington ( α = 0.053 eV and ρ = 13.9) whereas it is somewhat harder than a potential based on Thomas-Fermi-Dirac approximation. We compare some results on the basis of our potential with the experiment and the agreement is quite good.

Anharmonicity in Noble Metals: Some Thermal Properties

The room-temperature Gr\"uneisen parameters of copper, silver, and gold have been calculated within the quasiharmonic approximation, using the anisotropic-continuum model and values of the third-order elastic constants recently determined. The agreement between the calculated and the experimental values is good for copper and silver, but only fair (12%) for gold. Expressions for the temperature dependences of the three second-order elastic constants have been derived using the continuum model. These depend upon the second-, third-, and fourth-order elastic constants. The fourth-order elastic constants of copper, silver, and gold have been computed using these expressions, and it was found that they are positive in sign and of the order of ${10}^{14}$ dyn/${\mathrm{cm}}^{2}$ in magnitude. Only the closed-shell repulsive interaction between nearest-neighbor ions was taken into account in computing the fourth-order constants. Previous work has indicated that this might be a good approximation for noble metals. The two parameters in the Born-Mayer potential expressing the repulsive interaction were evaluated, and reasonable values were obtained for copper and silver. Anomalous results for gold indicate that contributions to the higher order elastic constants from other than ion-core repulsion cannot be ignored in that metal.

Calculated Heats of Dilute Solid Solution among the Alkali Halides Other Than Cesium Salts

Limiting high-temperature heats (energies) of dilute solid solution are calculated for all 96 cases of one NaCl-type alkali halide (except Cs salts) dissolving in another having a common ion. The direct-summation shell treatment used is based on a slightly modified Born—Mayer potential function and on the polarization energy up to terms square in the ion polarizabilities. Preliminary general expansion of the energy in positive powers (up to cubic) of the lattice-relaxation (ion-displacement) parameters of the first eight shells (98 neighbors) facilitated application to specific cases. In the linear solution for the minimum-potential lattice relaxation, one or two iterations for the first two shells sufficed. The convergence of the energy expansions and the sensitivity of the calculated energies to arbitrary variations in the semiempirical parameters were extensively investigated. Comparison is made with calorimetric heats of solution in 16 cases, the agreement (averaging ±10%) lying within the experimental and computative errors.

A re-examination of the Born-Mayer potential for ion-core interactions in copper

An efficient procedure to access to ortho-SCF3 substituted benzophenones from arynes and trifluoromethylthioesters has been developed under transition-metal free conditions. This difunctionalization process involves the first insertion of arynes into the C–S σ-bonds to achieve vicinal trifluoromethylthiolation-acylation of arenes in one step.

Equation of State, Grüneisen Constant, and Interatomic Forces

The equation of state of solids and the Grüneisen constant are interrelated and these can further be explicitly expressed in terms of the parameters of the interatomic potential and other simple constants. Calculations have been performed in this paper for these quantities on the familiar Born and Born—Mayer potentials. In general, the results reported here on alkali halides exhibit a good qualitative agreement between theory and experiment. This, however, is encouraging in view of the simple and approximate nature of the theory.

A matching procedure for scattering with the Born-Mayer potential at low energy transfers

A matching procedure for scattering with the Born-Mayer potential at low energy transfers

Defect distributions in channeling experiments

A simple collision model allows to calculate energy losses of perfectly channeled particles. The maximum energy loss is related in a simple way to the displacement energy of lattice atoms perpendicular to the channel. From that, one obtains rather definite predictions on the possibility of radiation damage by channeled particles. As an application, one gets a necessary criterion for the occurence of super tails in channeling experiments. The theory involves some assumptions on the behaviour of Born-Mayer potentials which are verified by comparison to experimental displacement energies.

Machine Simulation of Collisions Between a Copper Atom and a Copper Lattice

We have examined a number of collision events in which the target atom is embedded in a crystal lattice. The fcc $〈100〉$, $〈110〉$, and $〈111〉$ orientations were examined in detail for the energy range from 25 eV to 10 keV. Preliminary results for the bcc $〈100〉$ lattice are included. Lattice effects are somewhat smaller than one might anticipate. Orientation effects are significant because the relative atomic separations vary considerably between different orientations. The general features of our results are insensitive to the form of interaction potential, but magnitudes vary considerably. For the Gibson No. 2 parameters in a Born-Mayer potential, lattice effects lose significance around 5-keV bombardment energy; for the Gibson No. 1 they persist to at least 10 keV. No effective mass behavior was detectable. Binary-collision theories are not valid for energies below a few hundred eV.

Temperature dependence of the range of focused collision sequences in copper single crystals

The well-known focusing model, which is responsible for the [110] spots, observed in the sputtering of a fcc monocrystal describes a series of collisions along a closed packed atom row in fcc metals. The range of such a focusing sequence has been calculated by Leibfried for copper, using a Born-Mayer potential. For the colliding atoms a hard-sphere model is used. This range is determined from the energy lost at every collision in the sequence by potential interaction with the neighbouring [110] rows. In this article we have generalized Leibfried's treatment by considering the influence of the thermal vibrations of the crystal atoms on the energy loss per collision above the Debye temperature. In doing so, we have made the approximation that the thermal vibrations are harmonic and the coupling between nearest and next-nearest neighbours only has been taken into account. For the coupling constants we have taken values, derived by Jacobsen from experimental results. Using these values in a machine calculation, we compute the mean square of the difference of displacements of two nearest neighbours as a function of temperature. The relative energy loss factor can now be calculated as an average over these displacement differences. The result is ε( T) = ε(0). exp(23.10 −4 T), where ε(0) is the relative energy loss factor as calculated by Leibfried for a non-vibrating lattice. In the case of gold Nelson, Thompson and Montgomery have calculated the relative energy loss per collision due to a different mechanism, namely the scattering of momentum out of the sequence, because of the fact that the thermal vibrations cause the centres of the colliding atoms to be not exactly in one line. We have repeated their calculation for copper and compared the result with the one obtained above for the potential interaction with the neighbouring [110] rows. By treating the two effects as additive we obtain the range of focusing sequences as a function of temperature and initial energy. It turns out that these ranges are much smaller than was previously expected. We find a maximum range of 12 at 315°K and of 4 at 900°K (cf. fig. 5). On the basis of these results we can give a qualitative explanation of some features of the sputtering of single crystals. Because of the shortness of the average focusing range for sputtering which is 5 at 315°K and 2 at 900°K there will be a significant competition between focusing and defocusing mechanisms in ejecting the sputtered particles. From this we conclude that the amount of material, deposited in the [110] focused spots will have a weaker dependence on the range of the focusing sequences than a direct proportionality. We can because of this, expect the total sputtering ratio to vary only slightly with temperature above the Debye temperature.

Computer Studies of the Slowing Down of Energetic Atoms in Crystals

The slowing down of 1- to 10-keV Cu atoms in fcc, bcc, and diamond-structure crystals has been investigated by high-speed digital computer techniques, using exponentially screened Coulomb and Born-Mayer potentials to describe the interatomic repulsion. The ranges of these atoms prove to be strongly dependent on their initial directions of motion, the order of ranges being fcc [011]>[001]>[111]\ensuremath{\approx}isotropic, bcc [111]>[001]>[011]>isotropic, diamond [011]>[001]\ensuremath{\approx}[111]>isotropic. This orientation dependence of the range, which has also been observed experimentally, is a consequence of the tendency of the lattice to focus moving particles into channels bordered by relatively closely packed atomic rows. Inclusion of zero-point vibrations reduces the degree of channeling, but does not modify the order of ranges for fcc. The experimental evidence for channeling and some of its implications for radiation damage theory are discussed.

Long-Range Channeling Effects in Irradiated Crystals

If an energetic atom, a primary, slows down in a lattice, it has the possibility of traveling long distances without essential interaction along almost force-free channels bordered by close-packed atomic chains. This effect has not been considered until recently, because, for the sake of simplicity, the lattice has been replaced mostly by a corresponding random arrangement. This ``channeling'' behavior has been found in machine calculations by Robinson, Holmes, and Oen which take the lattice structure into account. In this paper, the behavior of a primary moving along a channel bordered by the most densely packed directions is investigated analytically. Numerical results are given for a Cu primary moving in a Cu crystal, but the results can easily be extended to other cases. Two potentials are used: an exponentially screened Coulomb potential after Bohr, used also in the machine calculations and thought to give an adequate description for relatively high energies and small interatomic distances; and a purely exponential potential after Born—Mayer, better suited for relatively low energies and large atomic distances. The maximum ranges are very large, for 10 keV in the order of 103 lattice parameters for the Born—Mayer potential and up to 107 for the Bohr potential. Presumably, the Born—Mayer potential is a better description for these events. The investigation is confined to motions near the channel axis. Therefore, we only obtain the range distribution near the maximum range. From an adequate analysis of experimental data on long ranges of primaries, shot onto crystals in low indexed directions, one can obtain information about the potential at distances of about half the lattice spacing.

Modification of the Born—Mayer Potential Function as Applied to the Crystalline Alkali Halides

The repulsion constant ρ in the Born—Mayer potential function is assigned values which are different for like- and unlike-ion pairs in such a way that large deviations from the observed compressibilities of the lithium halides, attributable to marked anion—anion repulsion, are removed. Lattice parameters in excellent agreement with the experimental ones are than obtained using alternative sets of ``basic radii'' based on differently evaluated ionic radii, but an indeterminacy characteristic of the Born—Mayer equation is shown to affect markedly the calculated like-ion overlap-repulsive contributions. The possibility of considerable error in the calculated results through lack of correction for ion asymmetry is discussed. The calculated electron affinities, which are unaffected by these uncertainties, agree approximately with the most precise determinations by other methods.

Energy Dependence of Electron Damage in Copper

Electron irradiation below 1-Mev and subsequent recovery in copper have been investigated. The minimum threshold displacement energy is found to be less than 22 ev. The details of the recovery in stage I, as a function of electron energy, are found to be more complex than anticipated. Specifically, ${\mathrm{I}}_{a}$ decreases, ${\mathrm{I}}_{b}$ increases slowly, and ${\mathrm{I}}_{c}$ increases rapidly, respectively, in relative importance as the electron energy is reduced. A calculation of the cross section for displacement of copper atoms by electrons is also presented. A Born-Mayer potential with standard parameters is used. Only displacements in or near the close-packed directions are evaluated in detail. Other directions do not appear to contribute significantly in the near threshold energy region being considered. The agreement between this theory and experiment is found when a minimum threshold displacement energy between 19 and 20 ev and a Frenkel resistivity between 2 and 3 \ensuremath{\mu}ohm cm per atomic percent of interstitial-vacancy pairs is assumed.

Zwischengitteratome in kubisch-flächenzentrierten kristallen, insbesondere in kupfer

We give a comprehensive discussion of the geometry of all the possible equilibrium or non-equilibrium configurations of interstitial atoms in f.c.c. crystals. It is shown that these occur within a polyhedron which occupies only a small fraction of the atomic volume. The interstitial in the stable equilibrium configuration, the so-called 〈100〉 dumb-bell, has two elementary modes of movement (migration and rotation). We characterize them by specifying the side of the polyhedron through which the interstitial atom has to move. The reorientation process of the dumb-bells under external strains gives rise to a relaxation effect, which is discussed in some detail. This relaxation process has been experimentally verified for cold-worked and irradiated nickel. The quantitative calculations of energies and displacements in seven symmetric equilibrium configurations is carried out for copper, employing an improved version of the model due to Seeger and Mann. The improvements allow, in particular, for the anisotropy of the elastic displacement field and for the large relative displacements of neighbouring atoms. The influence of the choice of the Born-Mayer potential and of various cut-off and correction procedures are discussed and allowed for in some detail. The more important numerical results are: energy of migration of an interstitial atom in copper, 0·5 ev; energy of formation, 2·8 ev; increase of the volume of a finite crystal due to one interstitial atom, 0·4 atomic volumes. The expansion of the crystal in the environment of the interstitial atoms was found to be slightly less than one atomic volume. Contrary to previous views, the region of the crystal containing the interstitial atom appears to be more densely packed than the ideal crystal. The relative change in the elastic constants due to the relaxation process of an atomic concentration c of interstitial atoms in copper is calculated to be of the order 20 c. As far as they could be checked, these results are in fair agreement with the experimental data. By introducing additional data, we find for the electrical resistivity in copper 2·5 μΩ cm/per cent Frenkel pairs and 0·9 μΩ cm/per cent interstitials.

New Potential Function for Alkali Halide Molecules

A new potential-energy function has been suggested for the alkali halide molecules, U=(−e2/r)+Pexp(−kr2),where P and k are constants. Values of αe, ωeχe, and the ionic binding energy Di have been derived for the following three functions: (1) simple Born-Mayer potential, (2) Rittner potential, and (3) the above suggested potential. The results obtained with the new function are much better than those obtained with the Born-Mayer potential, but slightly inferior to those obtained with the Rittner potential.

Gitterdeformationen um Zwischengitteratome, Leerstellen, Zwischengitteratom-Paare und Frenkel-Paare in Kupfer

The distortions of the lattice around a number of different point defects in copper are calculated with the help of the electronic digital computer Z 22 using a general method developed by TEWORDT 3. In the case of an interstitial which is sited at the center of an elementary cube about 500 atoms and in the case of a vacancy about 50 atoms are treated as discrete particles. The elastic solutions which are joined to the displacements of the discrete particles are determined for an anisotropic continuum. The changes in volume of the crystal arising from the interstitial are found to be 0.911, 1.219 and 1.441 atomic volumes respectively for the MORSE potential VM and the two BORN-MAYER potentials V1, V2 we have used [see Eqs. (25) — (27)]. The corresponding values for the vacancy are —0.441, — 0.378 and —0.321 atomic volumes. Further we caluclate the relaxation of the lattice around three configurations of interstitial pairs with axes in the (1,0,0), (1,1,0) and (1,1,1) directions considering about 100 atoms as movable in each case. The contributions to the binding energies arising from the potential V1 turn out to be 0.81, —0.18 and —0.26 eV respectively. This strongly indicates that interstitial pairs can attract one another. Finally the stability of the 10 closest interstitial-vacancy pairs (FRENKEL pairs) is examined. All pairs smaller than 1.5 lattice constants in diameter are found to be instable, the other pairs are stable and give a discrete spectrum of BORN-MAYER energies. The results are discussed in connection with recent experiments in the field of radiation damage.

Infrared Eigen frequency and Debye Θ of ionic crystals

Abstract Using a simplified Born-Mayer Potential, u=− ∝e 2 z 2 r +Mb e −σ ( r r 0 ) an expression for the lattice frequency of ionic crystals in terms of cohesive energy, E has been obtained: v= 1 2π E r 0 2 NMμ 1 2 2σ(σ−2) (σ−1) 1 2 A relation, very similar to one given by Blackman from lattice theory of specific heat, connecting Debye characteristic temperature, Θ and compressibility, χ has also been derived: Θ= h k 1 2π 6r 0 Xμ 1 2 for NaCl type crystals. and Θ= h k 1 2π 3.465r 0 Xμ 1 2 for CsCl type crystals.

Theory of the Dielectric Constants of Alkali Halide Crystals

The simple classical theories of the dielectric constants and compressibility of ionic crystals lead to two relations among the experimental quantities from which arbitrary parameters have been eliminated, the Szigeti relations. Neither is satisfied by the data, indicating the inadequacy of these simple theories. The short-range repulsive interaction between ions with closed shell electron configurations is investigated, and an approximate interpretation of the Born-Mayer potential in terms of overlap integrals is developed. These results are applied to the interaction of model ions consisting of rigid charged shells bound to cores by harmonic restoring forces. Using this model, polarization mechanisms neglected in the simple dielectric constant theory, the "short range interaction polarization," and the "exchange charge polarization" are described. Both arise from charge redistributions occurring when the ions move with resulting changes in electron overlaps. Applied to a crystal, these ion models permit the derivation of generalizations of the Szigeti, Clausius-Mossotti, and Lorenz-Lorentz relations. The $\frac{{e}^{*}}{e}$ of the second Szigeti relation can then be calculated and comparison with the $\frac{{e}^{*}}{e}$ values derived from experimental data imply that the above polarization mechanisms must be at least in part responsible for the deviation of this parameter from unity. The failure of the first Szigeti relation is discussed and attributed to the inadequacy of the treatment of compressibility. The additivity feature of the simple theory and its absence in the refined theory are discussed in relation to the so-called vacuum and crystal ion polarizabilities.