Erratum to: W. Helfrich: Possibility of electrical conduction in filled bands

Abstract

It has been proposed in the discussion that the energy balance in an electric field is satisfied simply by momentum changes of the neutral particle causing the electron drag. Such an interpretation is not possible, as may be readily seen in the ring model. Here an electron keeps its wave vector when accelerated by a field, thus the total wave vector of the system consisting of the neutral particle and the electrons will also be conserved. Therefore, a change in the wave vector of the neutral particle requires that at least one electron alters its wave vector. Because of the exclusion principle this electron has to be excited into an empty band.As to the calculations we found that the result for {(0] Nv [ 2)+ (2] Nv [0)} has to be corrected to give approximatelyWe wish to emphasize, however, that repeated checks and further simplifications of the model confirmed that (Nv~ 2 can be made nonzero. Only two possible explanations for this admittedly strange result appear to be left: (1) Divergence of the perturbation calculation in higher than second order. (2) Excitation of electrons into the conduction band by the combined action of the neutral particle and the electric field.

Steric Interaction of Fluid Membranes in Multilayer Systems

“…The derivation of the entrop} 7 ASq per mode is not so easy and, perhaps, preferably done in terms of real sinusoidal rather than complex modes. We may express the new modes by the old aq cos q r = uq e iqr -u_qe 1 bq sin q r = i(uq e iqr -u_qe~ (8) recalling that uq = u_q. Denoting the amplitude of some sinusoidal mode simply by a, we may write for the Gaussian…”

“…If the deformations are not too strong, the associated curvature-elastic energy per unit area of plane and, equally, of membrane can be expressed by [8] JCr d 2 The thermal undulations of the membrane are now restricted by two parallel rigid plates of separation 2d positioned at z = ±d. The membrane thickness is regarded as negligible.…”

“…The reduction of the mode amplitudes being caused by the steric effect, we re-use the arguments and the formalism laid down in Eqs. (8) to (17) to obtain a formula for the free energy of steric interaction. In analogy to (17) …”

Effect of ionic surfactants on the iridescent color in lamellar liquid crystalline phase of a nonionic surfactant

“…For the lamellar bilayer membrane systems of a common nonionic surfactant C 12 E 5 , the periodic structure of bilayers is assumed to be stabilized by the Helfrich steric undulations [15][16][17] . Small amounts of ionic surfactants give a significant influence on the behavior of the dilute bilayer solutions of the nonionic agent [18] .…”

On Hydration Repulsive Forces between Various Phospholipid Bilayers

“…On the other hand, two additional repulsive forces must be taken into account to understand the interaction between neutral membranes, such as phospholipid bilayers in water. The one is a force due to thermal fluctuations [2,3]. This force acts mainly when the lipid bilayers are in a fluid (L  ) phase.…”