Nuclear spin statistics

These limitations lead to electron spin multiplicity restrictions and to differing nuclear spin statistical weights for the rotational levels. Writing the electronic wavefunction as the product of an orbital fiinction and a spin fiinction there are restrictions on how these functions can be combined. The restrictions are imposed by the fact that the complete function has to be of synnnetry... 

Figure 5.18 Nuclear spin statistical weights (ns stat wts) of rotational states of various diatomic molecules a, antisymmetric s, symmetric o, ortho p, para and rotational, nuclear spin...

All other homonuclear diatomic molecules with / = for each nucleus, such as F2, also have ortho and para forms with odd and even J and nuclear spin statistical weights of 3 and 1, respectively, as shown in Figure 5.18. 

The rotational temperature of H2O in the molecular beam is quite low, about 10 K. As in the hydrogen molecule, the water molecule has para and ortho rotational levels with nuclear spin-statistics of 1 3 respectively. Since the para and ortho rotational levels have different nuclear wavefunc-tions, the conversion between the para and ortho levels is extremely slow, as in the hydrogen molecule. In H2O, the nuclear spin-statistics for the lowest rotational levels are as follows ... 

The nuclear spin statistical weight factors for " NH3 are determined in Section 8.4.1 of Ref. [3] and we do not repeat the derivation here. The results are summarized in Table 2. The 24 nuclear spin functions (see Chapter 8 in Ref. [3])... 

Each listed type of randomization (orientional, hydrogen-bonding network, nuclear spin statistics, isotopes, impurities, defects, and others that could be cited) makes independent contributions to S0 0. Hence, it seems safe to conclude that no macroscopic sample of real substance that ever appeared on Earth satisfies S0 = 0, i.e., that every real substance represents an imperfect exception to the third law as commonly stated. 

M. E. KeUman The idea by Prof. Quack of using molecular experiments to test fundamental symmetries is very interesting. In connection with the Na3 pseudorotation experiments we heard about yesterday, have you considered testing permutation symmetry of identical particles, that is, the Pauli principle and nuclear spin statistics ... 

Nuclear spin statistical weights were discussed in Section 5.3.4 and the effects on the populations of the rotational levels in the v = 0 states of H2, 19F2, 2H2, 14N2 and 1602 illustrated as examples in Figure 5.18. The effect of these statistical weights in the vibration-rotation Raman spectra is to cause a J" even odd intensity alternation of 1 3 for H2 and 19F2 and 6 3 for 2H2 and 14N2 for 1602, all transitions with J" even are absent. It is for the... 

Nuclear Spin Effects on Rotation. There is an interesting effect on the rotational partition function, even for the hydrogen molecule, due to nuclear spin statistics. The Fermi postulate mandates that the overall wavefunction (including all sources of spin) be antisymmetric to all two-particle interchanges. A simple molecule like (1H1)2, made of two electrons (S = 1/2) and two protons (spin 7=1/2), will have two kinds of molecule ... 

Clearly the symmetric It = 2 and 0 functions satisfy (6.258) and are therefore associated with rotational levels having odd N. Conversely, the antisymmetric / / = I functions satisfy (6.257) and must be associated with even N levels. These nuclear spin statistical results are very important, because they determine the nature of the nuclear hyperfine structure if the different nuclear spin states remain degenerate, the spin statistics then help to determine rotational level populations and hence spectroscopic intensities. 

In order to account for the nuclear spin statistics of O2, the representation just defined should be thiimed out by the extraction of spectroscopically relevant matrix from the matrix turned to eigenvectors of the representation at R o° (in the limit of a free O2, seee.g. [11]). 

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