Relativistic mass increase

As the nuclei become heavier, the strong attraction of the electrons by the very large nuclear charge causes the electrons to move very rapidly and behave relativistically, i.e. their relative mass (m) increases according to equation 1, and the effective Bohr radius (ao) for inner electrons with large average speeds decreases according to equation 230. 

The basis of these effects is found in the relativistic mass increase for a moving particle. Its mass, m, increases with its velocity, v, according to the relation... 

If we remember that the eigenvalues of the Schrodinger equation are directly proportional to the mass m of a particle while the expectation mean value scales as Mm, we expect that the eigenvalues of the Dirac equation will show both an increase in the binding energy of an electron and a contraction of the radial wave functions towards the origin because of the relativistic increase in the mass given by ... 

Relativistic effects result in contraction of the radial size of ns orbitals because they penetrate the core to the nucleus very effectively. As such, electrons in ns orbitals obtain high instantaneous velocities in the vicinity of the nucleus and thereby result in a relativistic increase in mass [m = mj - vjc)f ] and hence a decrease in Bohr radius [aQ = A ne lm X See ... 

Synchrocyclotrons are a modification (by McMillan and Veksler ) of the original cyclotron of Lawrence and Livingston . Synchrocyclotrons (also called frequency modulated cyclotrons) compensate for the relativistic increase in mass of heavy particles by appropriately modulating the radio-frequency on the dees of the cyclotron. In a cyclotron, in contrast to other high energy circular accelerators, the magnetic field is kept constant in time and the particles increase their orbits (i.e. spiral paths). Synchrocyclotrons are used to accelerate protons, deuterons, or helium ions. 

In isochronous cyclotrons, the magnetic field is increased along the radius at the rate of the relativistic increase of the mass of the accelerated particles. The time required for one revolution is constant during the acceleration, and a fixed-frequency RF system can be used. For vertical focusing, the magnetic field is appropriately shaped with radial or spiral sectors... 

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

White dwarfs are formed hot and gradually cool at nearly constant radius. With increasing mass, the star becomes squashed down until it is highly relativistic and very small. 

Diquark condensation makes the EoS harder, which leads to an increase in the maximum mass of the quark star configuration when compared to the case without diquark condensation. For finite temperatures the masses are smaller than at T = 0. For asymptotically high temperatures and densities the EoS behaves like a relativistic ideal gas, where the relation pressure versus energy density is temperature independent. In contrast to the bag model where this behavior occurs immediately after the deconfinement transition, our model EoS has a temperature dependent P(e) relation also beyond this point. 

As mentioned, most calculations we have done so far have concerned molecular systems. However, prior to development of the non-BO method for the diatomic systems, we performed some very accurate non-BO calculations of the electron affinities of H, D, and T [43]. The difference in the electron affinities of the three systems is a purely nonadiabatic effect resulting from different reduce masses of the pseudoelectron. The pseudoelectrons are the heaviest in the T/T system and the lightest in the H/H system. The calculated results and their comparison with the experimental results of Lineberger and coworkers [44] are shown in Table 1. The calculated results include the relativistic, relativistic recoil. Lamb shift, and finite nuclear size corrections labeled AEcorr calculated by Drake [45]. The agreement with the experiment for H and D is excellent. The 3.7-cm increase of the electron affinity in going from H to D is very well reproduced by the calculations. No experimental EA value is available for T. 

The values of ioni/ulion energies and atomic sizes are influenced by retain islic dlccls that, for valence electrons, increase with the value of 1 /. and become sufficiently important in the elements of the 6lh period (C s Rn) to explain largely their chemical differences from the elements of the 5lli period (Rh- Xe). The initial relativistic effect is to cause a decrease in the radius of the 1 s atomic orbital of Ihe atom. The I mass of the electron in the Is orbital becomes higher as the nuclear charge increases because the velocity of the electron increases. 

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