Nearly free electron models assume that electrons are free except for at the boundary of the Brillouin zone, where they interact with a potential. As an example, the Kronig-Penney model assumes that there are a periodic array of delta-function potentials.This interaction creates discontinuities and forbidden energies which qualitatively explain the behavior of insulators. Another class of theories of electronic conduction are bound electron models. Tight binding models are a major class of bound electron models. Following Slater, electronic conduction in insulators or near-insulators may be quantitatively modeled by an electron hopping between atomic orbitals to which it is bound. Therefore the energy band structure is a “Linear Combination of Atomic Orbitals,” and electrons move between neighboring atoms. Tight binding models offer superior tunablility relative to free electron models, plastic pots 30 liters and may model topological quantum properties such as the Quantum Hall Effect. A fundamental experiment of electromagnetism is the Hall Effect. A strip with a current running through it under the force of an electric field, in the presence of a normal magnetic field acquires a transverse voltage.
This is expected on the basis of the Lorentz Force Law. With division it may be shown that the transverse conductivity, or “Hall conductivity” is linear in magnetic field strength.These plateaus in resistivity are · h/e2 , for c and integer, to within one part in one trillion. These plateaus are used to define the standard for resistivity. The centers of the plateaus correspond the the Landau Level energy, which is the only energy at which electrons can move at when disorder is present. Electrons at all other energies are localized. This experimental result motivated the introduction of topology into condensed matter to describe such quantized behaviors. Within a magnetic field, charged particles can only occupy certain energy levels. These energy levels are the Landau levels, which for a magnetic field perpendicular to a two dimensional plane describe the cyclotron orbits of electrons. Here, we follow the discussion Electrons in a tight-binding chain of atoms may be modeled in terms of a linear combination of the atomic orbitals of each atom in the chain, modified by additional potentials from the Coulomb interaction of atoms. Here, a model is developed where all orbitals are considered the same at each site, and that “hopping” is only possible between adjacent sites. Here, numerical and analytical solutions are developed to solve for the spectrum of the Hamiltonian of this finite one dimensional lattice. First, the lattice is defined, and the model clarified in terms of base kets corresponding to states on each site, and this formalism is presented with the time-evolution of states in the Schrödinger equation.
Then a model for the energies on each lattice site, and corresponding to “hops” is developed. From this, the Hamiltonian is determined. From the Hamiltonian, the spectrum of energies may either be determined numerically using a numerical diagonalization algorithm, or analytically by developing the “translation operator,” and observing that the Hamiltonian is diagonal in a basis of eigenkets of the translation operator.Electrons may be modeled using the Tight-Binding model. This model assumes a probability to stay on a lattice site , and a probability to hop to a new site Here, a model is developed where all orbitals are identical, and the orbitals are arranged in a 2D square lattice with equal spacing between sites. It is assumed that hopping is only possible between nearest-neighboring sites. An analytical solution is presented, and an expression for the Hamiltonian is developed and numerically solved. The numerical solution agrees within machine precision to the analytical solution. The methods for the two dimensional tight binding model are very similar to methods for the one dimensional tight binding model. The only differences are that the system is a two dimensional square lattice, and that a formalism for the representation of the matrix Hamiltonian is developed which allocates one row per site, and moves through rows and then columns.The detachment of a grape berry from its pedicel generally damages the berry because the vascular tissues and associated parenchyma, collectively known as “the brush”, remain attached to the pedicel and are pulled out of the berry on detachment, leaving an open wound sometimes called a “wet” stem scar on the berry’s stem-end. Berry detachment may also remove pieces of skin or cause the whole berry to rupture. Such mechanical damage can reduce the yield and quality of machine-harvested grapes for wine or raisins. Stem-end picking damage also limits the quality and storage life of stemless table grapes. Certain plant growth regulators known as “abscission agents” activate an abscission zone at the pedicel-fruit boundary .
The activation of this abscission zone reduces fruit detachment force and promotes the development of dry stem scars . Abscission agents could reduce picking damage and thereby serve as harvest aides if treated grapes can be harvested after the abscission zone is activated, but before the fruit abscises. However, once the abscission zone has been activated, development proceeds quickly and may lead to excessive preharvest fruit drop.The first compound tested as an abscission agent for grape was ethephon, a phosphonic-acid compound that decomposes to release the gaseous plant hormone ethylene. Ethephon can induce the abscission of mature grape berries within 7 to 14 days after treatment, but high dosages are needed. The use of ethephon as an abscission agent for grapes would require an application dosage which is higher, and a preharvest interval that is shorter, than those for the current registered use of ethephon on grapes in order to enhance berry color. Such changes could be expected to increase ethephon residues on treated fruit, and it seems unlikely that regulatory agencies would approve a use that could increase ethephon residues on grapes since existing residues are already a concern. However, it should be noted that Ferrara et al. found that effective dosages of ethephon did not result in excessive residues. Jasmonates, including methyl jasmonate, a natural product, have also been shown to induce the abscission of various fleshy fruits, including blueberry , orange , and tomato. Moreover, the Environmental Protection Agency of the United States ruled that MeJA was exempted from the requirement of a tolerance for residues in or on all food commodities when applied pre-harvest, a ruling that could facilitate the development of jasmonates as active ingredients in agrichemicals. A screening trial confirmed that MeJA and coronatine, a jasmonate mimic, also induce abscission in grape. The exogenous application of jasmonates stimulates ethylene production in several fruits, including apple , orange, grape, strawberry , and tomato. The work carried out on orange and grape showed that the application of MeJA stimulated ethylene production by the fruit which was followed by fruit abscission. Malladi et al. provided indirect evidence that MeJA stimulates the abscission of blueberry fruits at least partly via ethylene action, as the co-application of MeJA with aminoethoxyvinylglycine , an ethylene biosynthesis inhibitor, round plastic pots attenuated MeJA effects on abscission. However, MeJA still induced some abscission in blueberry, even when co-applied with AVG which suggests that MeJA may initiate some abscission processes independently of ethylene. Moreover,grape berries treated with MeJA or 1-aminocyclopropane-1-carboxylic acid , the direct precursor of ethylene, produced similar levels of ethylene in the first 2 days after treatment , but thereafter, berries treated with MeJA produced less ethylene than berries treated with ACC, even though the MeJA-treated grapes generally had lower FDF, greater abscission, and a higher proportion of dry stem scars than the grapes treated with ACC. Together, these findings suggest that ethylene and jasmonic acid can promote fruit abscission via independent pathways and interact to promote abscission.
The potential for synergistic effects has sustained interest in research on the coapplication of jasmonates and ethylene-promoting compounds. This is especially important because of the relatively high dosages of MeJA needed for consistent efficacy when applied alone and because MeJA is much more expensive than ethephon. Because of the high rate of ethephon needed for abscission activity, the short time between abscission zone activation and fruit drop, and concerns about ethephon residues, alternatives to ethephon are desired. 1-Aminocyclopropane-1-carboxylic acid is not particularly effective at stimulating abscission on its own, but the co-application of MeJA with ACC improved efficacy in such a way that lower dosages of MeJA could be used. Recent improvements in jasmonic acid biosynthesis have the potential to make JA and its metabolite MeJA more available and affordable than they are now. The effects of JA on the abscission of fleshy fruits has not been tested, and the relative efficacy of JA versus MeJA with respect to grape berry abscission is unknown. Therefore, two studies were conducted to compare the efficacy of JA and MeJA at inducing abscission of Thompson Seedless grapes and to determine if JA interacts with ACC to promote abscission.Methyl jasmonate at 2 mM was ineffective, whereas 4 mM and 8 mM MeJA were equally effective at inducing preharvest abscission, reducing fruit detachment force , and increasing the proportion of detached berries with dry stem scars . Jasmonic acid was as effective or more effective than MeJA at inducing the abscission of Thompson Seedless grapes. Compared with MeJA, JA induced a similar or higher preharvest berry abscission, a similar or lower fruit detachment force, and a similar or higher percentage of detached berries with dry stem scars after treatment with JA, versus MeJA . The most effective treatment overall was 4 mM JA, which induced the highest level of preharvest abscission and percentage of detached berries with dry stem scars . Grapes that were treated with 4 mM JA also measured among the lowest FDF values .MeJA can stimulate the abscission of many fruits and endogenous JA is known to promote abscission of floral organs and fruits, but the data presented here may be the first report of an exogenous application of JA stimulating the abscission of a mature fruit. Moreover, JA appears to be at least as effective, and possibly more effective, than MeJA at stimulating grape berry abscission. Interestingly, this is in contrast with a recent study that showed MeJA was more effective than JA at inducing abscission in lupine flowers. Improved efficacy at lower dosages could facilitate the commercial development of jasmonate-based abscission agents for grapes and other fleshy fruits because these natural products are expensive. In the first study, 4 mM JA was more effective than 8 mM JA; however, the opposite appeared to be the case in the second study. Therefore, additional research is needed to clarify the lowest dosage of JA that is consistently effective. The general range of effective dosages agrees with previous research that used MeJA as the active ingredient. Abscission zone activation reduces FDF and promotes the development of dry stem scars, both of which could help minimize picking damage and possibly improve the quality of destemmed table grapes. However, the final stage of AZ activation is undesirable unless catchment systems can be employed. Previous studies with MeJA suggested that harvest should occur within 3 days after treatment . Jasmonic acid also stimulates rapid abscission zone activation, with 14 to 25% abscission observed 2 days after treatment, and 52% to 60% by 3 DAT. Without catchment systems, it may be necessary to harvest treated fruit within 2 days to avoid excess crop loss. This could be logistically difficult, and data are lacking to determine whether the abscission zone would be sufficiently developed by 1 or 2 DAT to provide the potential quality benefits that are desired from abscission agents. Another outcome that needs to be determined is whether the abscission zone could be activated preharvest and develop during postharvest storage. If so, this could be a way to achieve fruit quality benefits for stemless table grapes while minimizing the risk of preharvest fruit drop. Lavee demonstrated that preharvest applications of plant growth regulators can affect the postharvest abscission of grapes. Specifically, the preharvest application of 1- naphthaleneacetic acid and some other synthetic auxins reduced the postharvest abscission of “Muscat of Alexandria” that were held at room temperature for three days before entering cold storage. However, the abscission of grapes that were placed into cold storage immediately after picking was greatly suppressed, regardless of whether the grapes were pretreated with auxins or not. This suggests that the effect of plant growth regulators on postharvest abscission may depend on postharvest storage conditions.