Bare mass

Mass renormalization requires [15] that an additional term vj/yvv /5m be added where 5m is the difference between the physical and bare masses [77]. 

Then, given the bare mass of the electron as mo, we have the mass of the electron as m = mo — bm. By the Dirac equation this also contributes a counter term into the Lagrangian 8m j/ /. 

Bare mass (body, chassis) Propulsion system mass... 

As is well known, the mass counter term in equation (6) accounts for the use of the observable instead of the bare mass of the electron. The latter is not observable due to the interaction of an electrical charge with its own radiation field which is always present. The magnitude of 5m is fixed by... 

For instance, if we assume that we live in a multidimensional world with a changing compactification radius, we may expect that electron s mass and charge should vary. The effects depend on the model of origin of bare masses and coupling constants. The bare values can change or, alternatively, the bare values would stand unchanged, while the renormalization term would change. 

The effective field-theoretical Hamiltonian of the model (14), obtained via special Stratonovich-Hubbard transformation, passing from the discrete S3 tem to the continuous field theory, is the same as (13). The relevant global properties of a microscopic model is represented by the structure of the effective Hamiltonian. In this case, is a squared bare mass proportional to the temperature distance to the critical point, o > 0 is a bare coupling constant. Note that the effective Hamiltonian (13) preserves the 0(m) symmetry of the Stanley model (14). As far as the Stanley model is in this sense equivalent to the 0(m) symmetric theory, the analytic continuation m —> 0 of this model again leads to the polymer limit. 

The calculation of the correlation functions in the field theory is well documented in the literature. For d = 4, the calculation of the various quantities encounter divergence problems (due to the presence of the cut-off A of equation (42) which are removed by the proper renormalizations of the bare mass Eq and the bare coupling constant w. The analogs of the renormalization group equations (such as those of Gellman and Low, tHooft and Veltman, and Callan and Symanzik) which describe the dependence of the correlation functions (and the vertex functions) on arbitrary scale factors can be readily written for the polymer case also. " " A study of the Wilson function, j8( ), which describes the dependence of the renormalized coupling constant g on an... 

The determination of critical si2e or mass of nuclear fuel is important for safety reasons. In the design of the atom bombs at Los Alamos, it was cmcial to know the critical mass, ie, that amount of highly enriched uranium or plutonium that would permit a chain reaction. A variety of assembhes were constmcted. Eor example, a bare metal sphere was found to have a critical mass of approximately 50 kg, whereas a natural uranium reflected 235u sphere had a critical mass of only 16 kg. 

In specialized cases, a treatment known as canonization sometimes is tried to improve the amount of molecular (chemical) information made available. If Ag or Na are deliberately introduced into the sample, they will ofren combine with the molecular species present to create Ag or Na molecular ions. These ions are more stable to fragmentation than the bare molecular ions, and can therefore be observed more easily in the mass spectrum. The identification of parent ion peaks in this manner aids in detailed chemical identification. 

SERS substrates with bare metal surfaces irreversibly adsorb thioorganics (Eig. 4.59) and other compounds and can thus serve for the detection and identification of very low gas or solution concentrations of these substances [4.303]. SERS is especially well suited for the analysis of traces of gases, because it combines measurement of surface concentration with extremely high sensitivity. A monolayer in a typical focus of a laser with a diameter of 10 pm has a mass in the range of 10 femtograms even smaller amounts of substance are easily detectable, because 1% of a monolayer in a region 1-pm in diameter results in SERS of sufficient intensity. 

The total energy in ab initio theory is given relative to the separated particles, i.e. bare nuclei and electrons. The experimental value for an atom is the sum of all the ionization potentials for a molecule there are additional contributions from the molecular bonds and associated zero-point energies. The experimental value for the total energy of H2O is —76.480 a.u., and the estimated contribution from relativistic effects is —0.045 a.u. Including a mass correction of 0.0028 a.u. (a non-Bom-Oppenheimer effect which accounts for the difference between finite and infinite nuclear masses) allows the experimental non-relativistic energy to be estimated at —76.438 0.003 a.u. ... 

If the film is removed above a certain mass transfer rate there will be a sudden increase in the erosion corrosion rate which will tend to rise to the rate the bare metal can dissolve. 

The most commonly observed effect of current flow is the development of alkaline conditions at the cathode. On bare metal this alkaline zone may exist only at the metal surface and may often reach pH values of 10 to 12. When the soil solution contains appreciable calcium or magnesium these cations usually form a layer of carbonate or hydroxide at the cathodic area. On coated lines the cations usually move to holidays or breaks in the coating. On failing asphalt or asphalt mastic type coatings, masses of precipitated calcium and magnesium often form nodules or tubercles several centimetres in diameter. 

Additionally, the surfactant properties of filmers reduce the potential for stagnant, heat-transfer-resisting films, which typically develop in a filmwise condensation process, by promoting the formation of condensate drops (dropwise condensation process) that reach critical mass and fall away to leave a bare metal surface (see Figure 11.2). This function, together with the well-known scouring effect on unwanted deposits keeps internal surfaces clean and thus improves heat-transfer efficiencies (often by 5-10%). 

When calculating A/ we use the mass of an atom of H instead of the mass of a proton. This strategy allows us to use readily available isotope masses instead of the masses of bare atomic nuclei to calculate Am, because the number of electrons in the isotope will be the same as the total number of electrons in the hydrogen atoms on the other side of the equation and the masses of the electrons cancel. The electron-nucleus binding energy, which contributes to the mass of an atom, is only about 1(Th mu per proton, and so it can be ignored in elementary calculations. 

The electric monopole interaction between a nucleus (with mean square radius k) and its environment is a product of the nuclear charge distribution ZeR and the electronic charge density e il/ 0) at the nucleus, SE = const (4.11). However, nuclei of the same mass and charge but different nuclear states isomers) have different charge distributions ZeR eR ), because the nuclear volume and the mean square radius depend on the state of nuclear excitation R R ). Therefore, the energies of a Mossbauer nucleus in the ground state (g) and in the excited state (e) are shifted by different amounts (5 )e and (5 )g relative to those of a bare nucleus. It was recognized very early that this effect, which is schematically shown in Fig. 4.1, is responsible for the occurrence of the Mossbauer isomer shift [7]. 

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