Moment monopole

The first term in (4.6), Jp (r)r dr, depends only on the radial distribution of the nuclear charge. This term represents the so-called nuclear monopole moment, note that it is related to the extended finite size of the nucleus. ... 

The simplest moment to evaluate is the monopole moment, which has only the component k = I = m = Q, so that the operator becomes 1 and, independent of coordinate system, we have... 

The first term in the square bracket in this equation is the electric monopole moment, which is equal to the nuclear charge, Ze. The second term in the square bracket is the electric dipole moment while the third term in the square bracket is the electric quadmpole moment. For a quantum mechanical system in a well-defined quantum state, the charge density p is an even function, and because the dipole moment involves the product of an even and an odd function, the corresponding integral is identically zero. Therefore, there should be no electric dipole moment or any other odd electric moment for nuclei. For spherical nuclei, the charge density p does not depend on 0, and thus the quadmpole moment Q is given by... 

Here Dz is the electric dipole moment and M0 is the molecular monopole moment. If A 0 =0 (neutral molecule) and Dz - 0, Qzz = Qzz. If the molecule is neutral and if the molecule has a non-zero electric dipole moment,... 

Since the density is normalized within the unit cell, and the integral in Eq.(4.48) is over the spherical cell, the monopole moments have to be renormalized, and this is done by adding/subtracting the constant SSCA in accordance with charge neutrality of the spherical cell. 

In the BGFM, the ASA+M technique [57] is used to improve on the ASA energies. In the ASA+M, the multipole moments which are lost in the ASA (that is, all but the spherical l = 0 monopole moment) are added to the Madelung potential and energy. This significantly increases the accuracy [81], and makes the BGFM a reliable method in many cases [76, 82]. 

Tho critical baryon density corresponds to pmax = 0. This leads to the ( (juations Uv + Us = G or Uv — Us — 2Mg = 0 or both. The first condition Uv + Us = 0 is fulfilled earlier than the second one. When the first condition is reached the nucleus becomes unbound, i.e., unstable with respect to emission of nucleons. So it is impossible to compress the nucleus more than the critical density in a self-consistent manner such densities should occur only as shortlived intermediate stages in a heavy-ion collision. Wc performed a constraint calculation with the monopole moment [23, 24, 25], which produced self-consistent solutions up to w 3po for the case of Pb. A chart for the critical densities of nuclear matter with different parameter sets is given in ref. [26]. We found that the critical densities are much larger for nuclear matter c ompared to finite nuclei in all available parameter sets. The TMl parameter set was chosen for our calculations, because it gives a larger critical density of about 3po ... 

Thus the only multipole components which can appear for a homonuclear diatomic are Qqq (the charge or monopole moment), Q2Q (the quadrupole moment), Q q (the hexadecapole moment), and so on. 

To carry out the calculations it is customary to use a multipole expansion and to separate electric and magnetic couplings. A nucleus can only possess even electric moments, i.e. a monopole moment (/ = 0), namely its protonic charge and a quadrupole moment (/ = 2) if the charge distribution is not spherical. Higher moments are neglected. Magnetically only odd moments exist. It suffices to consider the dipole moment (/ = 1). 

The absence of a zeroth-order moment in Eq. (5.9) reflects the fact that magnetic monopole moments do not exist. The magnetic dipole moments can thus be shown to be independent of the gauge origin Rgo (see Exercise 5.3] as a direct consequence of the absence of magnetic monopole moments. 

This quantity is proportional to the total number of molecules with J= 2, that is, thepopulationofthe7= 2 system summed over all different Wvalues. All systems must possess some overall population hence every distribution has a monopole moment. If the system has only a monopole moment, then the J distribution is uniformly distributed in space and the system is said to be unpolarized. If higher-order multipole moments exist, however, the system is polarized. 

By inspection of (9.13.39), we see that the translation matrix W is a lower triangular matrix with unit diagonal elements. This structure implies that the monopole moment at P makes contributions to all multipoles at P, that the dipole moment at P makes contributions to the dipole and all higher moments at P, and so on. In particular, we note that a charge distribution that is represented exactly by a finite number of multipoles centred at P, will require an infinite expansion at a different position P for an exact evaluation. 

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