Mirror nuclei

Example Problem Consider the mirror nuclei 25Mg and 25Al. What is the energy difference between their ground states ... 

The observation of the masses of mirror nuclei suggests the strong or nuclear force between a neutron and a proton is the same. This equivalence leads naturally to considering the neutron and the proton as corresponding to two states of the same particle, the nucleon. (A similar simation holds for the it meson, where the it0, tt+,... 

Consider the A = 14 isobars, 14C, 14N, and 140.14C and 140 are mirror nuclei and have ground states with T3 = + 1. As such they must be part of an isospin triplet with T = 1 (T3 = 0, +1). Thus, in the 7) = 0 nucleus, 14N, there must be a state with T= 1, r3 = 0 that is the analog of the 7) = 0 ground states of 14C and 140. (See Problems section for further details.) We expect the three members of this multiplet to have approximately the same energy levels after correction for the Coulomb effect and the neutron-proton mass difference. 

The decay of the neutron into the proton is an important example of decay between mirror nuclei. In the (3 decay of mirror nuclei, the transformed nucleons (neutron —> proton or proton neutron) must be in the same shell and have very similar wave functions. This gives rise to a large matrix element Mif 2 and a very small log ft value. For the (3 decay of mirror nuclei to their partners, log ft values are about 3, which is unusually small. Such transitions are called superallowed transitions. 

The separation of states of different T implies that one of a set of isobaric nuclei is stable and the others unstable against beta decay. The Wigner theory predicts superallowed decay between the states of a given supermultiplet because no change of spatial wave function is needed. This is found (a) for the positron decay of odd mirror nuclei (b) for transitions between the low states of nuclei with mass number 4 + 2 in which both T= and P = 0 states are found in the (1,0,0) supermultiplet. 

The beta decay between the isobars is superallowed, as expected for mirror nuclei. 

A strong overlap of initial and final wave functions is expected for mirror nuclei, in which the mass numbers are identical, but the atomic number in one nucleus is equal to the neutron number of the other. The large overlap integral results in a very low fi value (superallowed transitions). Most of the known P emitters decay by allowed or superallowed transitions. Among the light nuclides there are many mirror pairs. 

A similar tool that is useful for studying mirror nuclei and isobaric analog states is the charge-transfer reaction of the type (p,n) or (n,p), the nuclear reaction equivalent of beta-decay. Mirror nuclei are pairs of isobars that can be interconverted by exchanging a neutron and a proton, e.g., and An isobaric analog state of nucleus is also a state in the... 

A is a normalization constant and T/.m are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. 

The water molecule possesses two mirror planes of symmetry, as shown in Fig. 6-3. One mirror plane lies in the plane of the diagram through which the whole molecule reflects into itself across the plane. The other, through the oxygen nucleus in the yz plane of the figure, and shown by the dotted line, reflects Ha into Hb and vice versa. 

The nucleus has cylindrical symmetry around an axis normal to the boundary and mirror symmetry across the grain-boundary plane. Figure 19.27 shows a cross section of the nucleus centered in a patch of boundary of constant circular area, Ac. The area of the nucleus projected on the boundary is indicated by A. The total interfacial energy of this configuration is then... 

An example of the energy level matching in the mirror pair 17F, 170 is shown in Figure 6.7. The agreement of the levels is quite remarkable and can be taken as strong evidence for the charge independence of the nuclear force, that is, the protons and neutrons move in essentially identical but separate orbitals in the nucleus. 

The frequency exaltation of the Kekule mode is mirrored by the structural manifestations in the twin states, discussed with reference to Figures 16 and 17. Thus, the repulsive jr-curve in the ground state softens the potential and thereby enables the ground-state molecule to distort along the Kekule mode when angular strain is exerted. In contrast, the attractive jr-curve in the twin excited state stiffens the potential and restores the local Deh symmetry of the benzene nucleus. The two physical effects are in perfect harmony and find a natural reflection in the VB model. 

In the first example, the models are superposable (i.e., homomers) in the second, nonsuperposable mirror images (i.e., enantiomers) in the third, not mirror images (i.e., diastereomers). Note that the tags are permanent i.e., H and (H)are different kinds of atoms. Alternatively, one proton in each structure may be replaced by Z, representing any nucleus not present in the molecule. 

Orbitals (GTO). Slater type orbitals have the functional form e, if) = NYi, d, e- -- (5.1) is a normalization constant and T are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. centre of a bond. 5.2 Classification of Basis Sets Having decided on the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions... 

Messenger RNA makes a mirror image copy of a stretch of the DNA molecule and then moves RNA out of the nucleus through the nuclear pores into the cytoplasm. There the RNA locates the ribosomes where it consumes the protein products with the help of transfer RNA molecules. 

Fig. 7.14. Relief plots of the negative of the Laplacian distributions for triplet and singlet states of CFj. The lower diagrams are for the plane containing the nuclei, the upper ones for the perpendicular symmetry plane containing the C nucleus, the plane containing the non-bonded charge maxima. There are two non-bonded maxima in the triplet, one in the singlet. The point labelled a is a (3, — 1) critical point in the VSCC of triplet carbon. There is no radial maximum or lip at the point labelled h and its mirror image and the VSCC of singlet carbon exhibits holes at these two points. The maxima present in the VSCCs of the F atoms are not shown as they are larger by a factor of 10 than those on the carbons.

Suboptimal erythropoiesis can be classified by changes in the size of RBCs noted on examination of the peripheral blood. Because the excretory and endocrine functions of the kidney usually mirror each other, renal dysfunction can lead to anemia by reduction in EPO production, resulting in a normochromic, normocytic pattern. Other causes of insufficient erythropoiesis include replacement of bone marrow by fibrosis, solid tumors, or leukemia, as well as defects in erythroid maturation. Relative deficiencies in the cofactors required for heme-RBC synthesis such as iron, folate, and vitamin B may also be important contributors. Structurally, RBC macrocytosis denotes defects in the maturation of the nucleus, whereas microcytosis is indicative of cytoplasmic defects (reduced hemoglobin synthesis). (A detailed description regarding the pathogenesis and treatment of anemic disorders is found in Chap. 99.)... 

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