{"id":297,"date":"2018-11-12T05:51:52","date_gmt":"2018-11-12T05:51:52","guid":{"rendered":"http:\/\/phyp04.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=297"},"modified":"2019-04-29T09:43:15","modified_gmt":"2019-04-29T09:43:15","slug":"nuclear-models-7","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp04\/chapter\/nuclear-models-7\/","title":{"rendered":"Nuclear Models-7"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n\u00a0 \u00a0 Applications of Shell Model<\/strong>\r\n\r\n \r\n\r\n1.<\/strong>\u00a0Spin<\/strong>\r\n\r\n \r\n

We have seen in the last section that shell model can successfully predict ground state spin and parity of odd-A nuclei. The shell model also accounts spin of Even-A nuclei fairly well in general.<\/p>\r\n \r\n

Spin of Even \u2013A nuclei (with N = odd, Z = odd)<\/strong><\/span><\/p>\r\n \r\n

W L Nordheim in 1950 gave a rule to calculate the most probable value of spin in odd-odd nuclei which is known as Nordheim rule. According to this rule the most probable value of spin in odd-odd nuclei is given as<\/p>\r\n \r\n\r\nThe Nordheim number (NN<\/sub>) is defines as\r\n\r\n??<\/sub> = ??<\/sub> \u2013 ? ?<\/sub> + ??<\/sub> \u2212 ??<\/sub>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(1)\r\n\r\n \r\n\r\nwhere\r\n\r\n \r\n\r\nJp<\/sub> = angular momentum of odd-proton\r\n\r\n \r\n\r\nJn<\/sub> = angular momentum of odd-neutron\r\n\r\n \r\n\r\nRules for calculating spin<\/strong>\r\n\r\n \r\n\r\n(a) If NN<\/sub> = 0, then the value of j is:\u00a0\u00a0 = | jp<\/sub> \u2212 jn\u00a0<\/sub>|.<\/strong> This rule is known as \u201cStrong rule\u201d.\r\n\r\n \r\n\r\n(b) If NN<\/sub> = \u00b1 1, then the value of j is: = jp<\/sub> + jn<\/sub> or j = | jp<\/sub> \u2212 jn<\/sub> |.<\/strong> This rule is known as \u201cWeak rule\u201d.\r\n\r\n \r\n

A comparison of predicated spin (\u00a0j<\/em>????<\/sub> ) values and experimentally obtained spin ( ????\u00a0<\/sub><\/em>) values for some even-<\/span>A<\/em> nuclei is given in table 1. It can be seen from the table that <\/span>predicated values are in <\/span>well agreement with experimentally obtained values.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n

Table 1: Spin values for Even-A nuclei<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

In addition to the prediction of ground state spins and parities of nuclei, the shell model can also predict spins and parities of excited states as well, as shown in fig. 1.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

Fig. 1: Spins and parities of excited states<\/p>\r\n \r\n\r\n2. Nuclear Magnetic Moment<\/strong>\r\n\r\n \r\n

The Shell model in principle can be used to predict the magnetic moment (\u03bc<\/em>) of nuclei, which can then be compared with the experimental values. Thus, the last unpaired nucleon determines the magnetic moment of the entire nucleus.<\/p>\r\n \r\n

The magnetic moment of a nucleus is the vector sum of the spin magnetic moment\u00a0\u03bcs<\/sub><\/em> \u20d7\u20d7\u20d7\u20d7\u00a0 \u00a0and orbital magnetic moment\u00a0\u03bcL<\/sub><\/em> \u20d7\u20d7\u20d7\u20d7 :<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nWhere\u00a0\u03bcs<\/sub><\/em> \u20d7\u20d7\u20d7\u20d7 is the vector sum of the intrinsic magnetic moments of the individual nucleons in the nucleus. The intrinsic moments for proton and neutrons are given as\r\n\r\n<\/div>\r\n??<\/sub> = ??<\/sub> ??\/?<\/sub> and ??<\/sub> = ??<\/sub> ??\/?<\/sub>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(3)\r\n\r\n \r\n\r\nWhere = e\u045b\/ 2Mp, is the nuclear magneton which is analogous to the Bohr magneton in atoms but with the electron mass replaced by the proton mass, M<\/span>p<\/sub> being the proton mass, ??<\/sub> and ??<\/sub> are the gyromagnetic ratios for the proton and the neutron. It has numerical values equal to<\/span>\r\n

\r\n\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 ??<\/sub> = 2 \u00d7 2.7927 and ??<\/sub> = -2 \u00d7 1.9131\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (4)\r\n\r\n \r\n\r\nThe magnetic moment of a nucleus of spin I<\/em> (total angular momentum) can be written as\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nIn order to measure the magnetic moments, a magnetic field is applied and it is the component of\u00a0??<\/sub> in the filed direction of which determines magnetic moment.\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nWhere ??<\/sub> is the magnetic quantum number which can take values\u00a0??<\/sub>\u00a0= I, I -1, \u2026. \u2013<\/em> I<\/em>. B is the magnetic induction filed. The component of along the z direction corresponding to ??\u00a0<\/sub>= I\u00a0<\/em>usually gives the measured magnetic moment where value of ? is the largest.\r\n\r\n \r\n\r\n\"\"\r\n\r\nAs stated earlier ?N<\/sub> is the nuclear magnetron, g is the gyromagnetic factor (or g-factor) & I is the spin.\r\n