Spherical charge

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

The intensity of shading at any point represents the magnitude of 1, i.e. the probability of finding the electron at that point. This may also be called a spherical charge-cloud . In helium, with two electrons, the picture is the same, but the two electrons must have opposite spins. These two electrons in helium are in a definite energy level and occupy an orbital in this case an atomic orbital. 

A very simple version of this approach was used in early applications. An alchemical charging calculation was done using a distance-based cutoff for electrostatic interactions, either with a finite or a periodic model. Then a cut-off correction equal to the Born free energy, Eq. (38), was added, with the spherical radius taken to be = R. This is a convenient but ill-defined approximation, because the system with a cutoff is not equivalent to a spherical charge of radius R. A more rigorous cutoff correction was derived recently that is applicable to sufficiently homogeneous systems [54] but appears to be impractical for macromolecules in solution. 

Brode, H. L. 1959. Blast wave from a spherical charge. Physics of Fluids. 2(2) 217-229. 

We therefore require a new way to evaluate Ees, and we will need to retain the full, non spherical, charge density p(r). 

From the viewpoint of quantum mechanics, the polarization process cannot be continuous, but must involve a quantized transition from one state to another. Also, the transition must involve a change in the shape of the initial spherical charge distribution to an elongated shape (ellipsoidal). Thus an s-type wave function must become a p-type (or higher order) function. This requires an excitation energy call it A. Straightforward perturbation theory, applied to the Schroedinger aquation, then yields a simple expression for the polarizability (Atkins and Friedman, 1997) ... 

Fig. 1 (a) Schematic representation of the spherical and non-spherical charge distribution in a nucleus. The value of electric quadrupole moment Q for the quadrupolar nucleus depends on the isotope under consideration, (b) The quadrupolar interaction arises from the interaction of Q with surrounding electric field gradient (EFG)... 

H. L. Brode, Blast Waves from a Spherical Charge, Physics of Fluids (1959), 2 17. 

V - Secondary fragment initial velocity, in/s Re Radius of spherical charge, inches... 

In addition to its rather narrow limits of validity, this expression is strictly applicable only to spherical charges of Composition B. However, until further work is completed, this equation represents the best method available for predicting secondary fragment velocities. 

As Fig. 12 shows, the inner shell electrons of the alkaline ions behave classically like a polarizable spherical charge-density distribution. Therefore it seemed promising to apply a "frozen-core approximation in this case 194>. In this formalism all those orbitals which are not assumed to undergo larger changes in shape are not involved in the variational procedure. The orthogonality requirement is... 

The non-spherical charge density around Cu can be interpreted as due the hybridization of d electrons with higher-energy unoccupied s and p states. Among these states, hybridization is only allowed for dz and 4s by symmetry, and when this happens part of the dz state becomes unoccupied ( d hole )- These states are responsible for the spatial distribution of the deficiency in the map shown in fig. 6. The complementary empty states are important for EELS, which probe empty states. 

The structure-activity relationship for cobalt catalysts in the pyridine synthesis can be summarized in the following manner If the substituent R is a donor, the Co-NMR signals are shifted to higher field and the catalytic activity decreases. If R is an acceptor, the Co-NMR signal is shifted to lower field and the activity increases. Donor substituents are oriented orthogonal to complexed cod in the catalyst precursors acceptors are oriented parallel. The deformation of the spherical charge distribution about cobalt is also dependent on the nature of R. 

When the electrostatic properties are evaluated by AF summation, the effect of the spherical-atom molecule must be evaluated separately. According to electrostatic theory, on the surface of any spherical charge distribution, the distribution acts as if concentrated at its center. Thus, outside the spherical-atom molecule s density, the potential due to this density is zero. At a point inside the distribution the nuclei are incompletely screened, and the potential will be repulsive, that is, positive. Since the spherical atom potential converges rapidly, it can be evaluated in real space, while the deformation potential A(r) is evaluated in reciprocal space. When the promolecule density, rather than the superposition of rc-modified non-neutral spherical-atom densities advocated by Hansen (1993), is evaluated in direct space, the pertinent expressions are given by (Destro et al. 1989)... 

Molecules consist of electrons and nuclei the principal difference between a molecule and an atom is that the latter has only one particle of the nuclear sort. Classical potential theory, which in this case works for quantirm mechanics, says that Coulomb s law operates between charged particles. This asserts that the potential energy of a pair of spherical, charged objects is... 

These values for J R) and K R) may be rationalized in pirrely electrostatic terms involving charge distributions of various sizes and shapes. From the point of view of electrostatics, J R) is the interaction of points and spherical charge distributions. The well-known effect, where the interaction of a point and spherical charge at a distance R is due only to the portion of the charge inside a sphere of radius R, leads to an exponential fall-off J R), as R increases. 

Electrostatic interactions between a spherical charged protein particle and an oppositely charged, deformable interface can be estimated by evaluating the electrostatic force on a small segment of the interface as that produced by an adjacent flaf section on the protein surface. The strength of this interaction is dependent on the separation distance (b) between those two segments, and so will be a function of the position of the interfacial segment ... 

Radius-time diagram of the formation of shock waves by the detonation of a spherical charge, log (1 / x) scale Fig 37... 

L, is seen going out into the air (from both ends of the charge. The left side of the plot represents the center of a symmetrical spherical charge). The first shock is then succeeded by "a rarefaction shooting forward , and then by other shocks and rarefactions, exactly the same as in the case of jetting... 

Experimental Verification of Geometrical Similarity far Shock Waves from Spherical Charges of 50/50 Pentalite... 

Saint Louis Research Laboratory Report No 8/50(1950) (Detonation of spherical charges... 

H.L. Brode, "A Calculation of the Blast Wave from a Spherical Charge ,... 

---