Nuclear monopole moment

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 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... 

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 ... 

Quadrupole moment - A coefficient of the third term (after monopole and dipole) in the power series expansion of the electric potential of an array of charges. A nucleus of spin greater than 1/2 has a non-vanishing nuclear quadrupole moment which can interact with the electric field gradient of the surrounding electrons. Molecular quadrupole moments have an influence on intermolecular forces. 

In considering the electric monopole interaction and the resulting isomer shift it is implicitly assumed that the nuclear charge distribution is spherical. However, nuclei in states with a nuclear angular momentum quantum number /> have non-spherical charge distributions which are characterised by a nuclear quadrupole moment. When the nuclear... 

In a spherical nucleus we assume the charge distribution to be spherical and the nucleus acts as a monopole. In the deformed nuclei, the nuclear charge has a non-spherical distribution. The potoitial at a point x,y,z (Fig. 11.6) will be found to vary depending on the charge distribution and mode of rotation of the nucleus. The nuclear charge may be distributed to form a dipole, a quadrupole, etc. Nuclei are therefore divided into differoit classes dq>ending on their electrical moments monopoles, dipoles, quadrupoles, octupoles etc. 

The experiments show that in the ground states of nuclei the electric moments of 2 order with odd I and all 2-pole magnetic moments with even I are missing. Only electric monopole, magnetic dipole, electric quadrupole, magnetic octupole, etc., moments are observed in experiments. This means that the 2-pole character unambiguously determines whether the moment is electric or magnetic, and these latter attributes may be omitted. This rule can be viewed as a consequence of a symmetry property (parity) of the nuclear states. 

According to O Eq. (25.56), one obtains that the energy shift due to electric monopole interaction is proportional to the product of the s electronic density and the second moment of the nuclear charge distribution also called the mean square nuclear charge radius (see also O Sect. 2.2.3.1 in Chap. 2, Vol. 1). When the nucleus is considered to be a homogeneously charged sphere with a radius R (often called charge equivalent nuclear radius), then... 

Atomic nuclei can be stretched like cigars (prolate shape) or compressed like discs (oblate shape). The deformation is described by the electric quadrii-pole moment Q (prolate Q > 0 oblate Q < 0). The principal interaction is, of course, the normal electrostatic (Coulomb) force on the charged nucleus monopole interaction). The differential interaction, which depends on the structure of the nucleus and on the valuation of the field across its finite extension, is of course very much smaller quadrupole interaction). It gives rise to an electric hyperfine structure. The energy contribution depends on the direction of the nuclear spin in relation to the electric field gradient. For the electric hyperfine interaction one obtains... 

In a recent work (26,27) we have clearly shown the importance of higher moments than the first nonzero moments in molecular chemical physics. The situation is similar in nuclear physics. The presence of a charge (monopole) does not preclude the presence of higher multipole moments (28). 

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