Hyperfine levels

The early MBER spectra were of two types first, pure rotational (microwave) transitions and, secondly, radiofrequency transitions between different hyperfine levels the latter were observable only for Ar-HCl, because... 

Time second s Duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. 

Time. The unit of time in the International System of units is the second "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental state of the atom of cesium-133" (25). This definition is experimentally indistinguishable from the ephemetis-second which is based on the earth s motion. 

The rate of spontaneous decay increases with v3 so that for higher frequency transitions, such as in the visible region of the spectrum, spontaneous decay is fast (of the order of nanoseconds) whereas rotational transitions or transitions between hyperfine levels within an atom are very slow. 

Fig. 2. Sample of hfs data recorded in present experiment of I93g,m7i The upper portion shows the atomic hyperfine levels for the two spins.

Here is yet another bizarre result of quantum mechanics for you to ponder. The lx wavefunction for a hydrogen atom is unequal to zero at the origin. This means that there is a small, but nonzero probability that the electron is inside the proton. Calculation of this probability leads to the so-called hyperfine splitting —the magnetic dipoles on the proton and electron interact. This splitting is experimentally measurable. Transitions between the hyperfine levels in the lx state of hydrogen are induced by radiation at 1420.406 MHz. Since this frequency is determined by... 

We have calculated exactly the Zeeman effect for the levels IS, 3S and 3P. Indeed it is necessary to know the shift for all the hyperfine levels very well. These calculations are very classical and we just present the results in a Zeeman diagram (see Fig. 5). The most important part in the diagram is the crossing between the 38 2 (F=l, mp=-l) and 3P1/2(F=1, mj =0) levels, because the quadratic Stark effect is proportional to the square of the induced electric field and inversely proportional to the difference of energy between the two considered levels. Moreover the selection rules for the quadratic Stark effect in our case (E perpendicular to B) impose Am.F= l. So it is near this crossing that the motional Stark shift is large enough to be measured. In our calculations the Stark effect is introduced by the formalism of the density matrix [4] where the width of the levels are taken into account. The result of the calculation presented on... 

The spin-averaged 2p-level broadening T2p in pH was determined indirectly from the intensity balance between the total Balmer series and the Lyman a transition [10,12], a method which is, however, only strictly valid in the limit of equal widths for all 2p hyperfine levels [8,12,14], No hadronic effects were observed in antiprotonic deuterium because of the weakness of the K transitions due to the enhanced absorption from the pD 1 = 1 states. 

Figure 8.13. Hyperfine level structure of the J — 28 rotational level in the v = 0, X state of Na2. The quadrupole coupling constant, eqoQ> is f°und to have a negative value [22]. The levels are labelled with their F values.

Figure 8.14. Hyperfine levels for the J = 1 rotational level of Na2 in its ground electronic state, and the observed transitions [23]. Each level is labelled by its F value.

The lowest level studied experimentally was. 7=1, for which seven hyperfine transitions were observed and the molecular constants determined [23]. For an odd J value the allowed values of / are 1 and 3, so that the following hyperfine levels exist ... 

We now list the results, which are summarised in the hyperfine level diagram shown in figure 8.14, where the observed transitions are also indicated. The constants determined from the experiments [23] were (in kHz) ... 

We are now in a position to calculate the first-order g-factors for the hyperfine levels in ortho-H2, making the same numerical substitutions as before, but additionally with 7 = 1. The result is... 

Given values of the constants appearing in the expressions for the matrix elements listed in (8.238), we can calculate the energies of the hyperfine levels for N = 1 and hence calculate the transition frequencies. This we shall now do, but note that this is the easy direction in which to proceed Brooks, Lichten and Reno [42] had the more difficult problem of determining the constants from the experimental data. Their values [42] are as follows (MHz) ... 

It is instructive to consider in quantitative detail the analysis of a particular hy-perfine transition a simple example would seem to be the F = 2 <+ 1 transition in the level It = 1, N = 0, J = 1, which is observed at a frequency of 20.846 MHz for the v = 0 level. The expressions for the matrix elements of the magnetic and electric hy-perfine terms, (8.258), (8.259) and (8.270), show that for N =0 only the Fermi contact interaction is non-zero and the energies of the hyperfine levels are... 

All of the magnetic and electric hyperfine matrix elements were derived in our discussion of the LiO spectrum. We now use the symbol m to denote nuclear spin-free terms, as listed above in equations (8.406), and denote the hyperfine terms, previously given in equations (8.384) and (8.385), by the symbol hf. Since the 14N nucleus has spin 7=1, each J level is split into three hyperfine levels, characterised by / =./,. / 1, except for the J = 1 /2 level which has only two hyperfine components, with F = 1 /2 and 3 /2. Consequently if we neglect matrix elements off-diagonal in J for the moment, each characteristic set of J, F levels is described by a 4 x 4 matrix, or two 2x2 matrices, as follows. 

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