R. 5. BRADLEY The University of Leeds, Leeds, England
ITIS the purpose of this note to draw attention to certain features of the wave-mechanical internretation of atomic structure which appear to have a certain novelty although they are implicit in standard treatments. In order to prepare the ground the analogues of these features in Bohr's theory of the hydrogen atom will first he discussed, relativity corrections beina omitted. As is well known; the energies are correctly :educed if the electronic mass is replaced by the reduced mass and the latter is imagined t o be attracted t o a fixed center placed at a distance equal to the interparticle distance. This treatment obscures what is actually imagined t o happen according t o Bohr's theory, since both the nucleus and electron rotate about the mass center, as shown in the figure for circular orbits, the two particles preserving the distance between them as if joined by and M~~~the masses of electron a ~ l d a r -i ~ rod. d ~f nucleus, r, and r2 their distances from the mass center and r the interparticle distance, -,
iY1 -
r, = m + M r
m==r
m
?=-
mr,%
+ Mr,%
= p
7%
nh
= -
zr
H~~~~the ratio angular momentum orbital magnetic moment
- 2rv2 - 2rdm f M)z = - e(rlz - r?) - (Ma - mP)e 2mMc e(M - m)
This ratio is usually quoted as 2mc/e and if m were replaced by p we should obtain an incorrect result. Likexvise the orhital magnetic moment is nha (M - m) &me
M
instead of the usually quoted value,
It follows that the ratio of spin magnetic moment to the orbital magnetic moment is not 2, but 2 M / ( M - m). In a similar way the diamagnetic susceptibility due to an electron and a proton, both of which describe cFcular orbits,.is . not -e2r2/(4me3, as applies to an but
4rahzwa
Although these corrections are small they could give rise to observable effects and they bring out the adwhere is the reduced mass and the principal quanvantage of treating the two particles in an equivalent tum number. Hence fashion, a principle which it is the purpose of this note n'h2 nzh' to emphasize, and which may be carried over to the r l = T Z~= - 4rZMe9 ~ wave-mechanical treatment. The wave equation for a Since the Bohr radius, ha/(4a2me3,is equal to 0.53 i., hydrogen atom may be separated into an equation deit follows that for large values of n, rl uill be appreciable. scribing the motion of the mass center and one involvThus if n = 8, which corresponds to the experimentally ing the reduced mass. As an example of the solution known term value 1714 om.-:, r2 is approximately 0.02 of the latter equation we may quote A. and, if n = 50, rz is *0.7 A. + (ground stsite) = ( 1)" -,/ao ' On the emission of light both the electronic and Taa nuclear orbit must simultaneously contract, a rather curious but inescapable result. Asimilar treatment may where aa = h2/(4a\e2); W J d r gives the probability of be given for elliptical orbits, both particles describing finding an electron in an element of volume dr distant such orbits round the mass center, so that the maximum r from the nucleus where +* is the complex conjugated All this is of course correct, but the usual treatment value of r2 may be appreciable for high eccentricity. In the calculation of magnetic moments it is worth then goes on to consider electron distributions round a noting that the replacement of electronic by reduced fixed nucleus, instead of treating the two particles in an mass gives incorrect results. This arises from the fact equivalent manner. This may be done for a fixed that both particles contribute to the orbital magnetic mass center by the use of the consideration that W * d r moment, the total moment being ew/2c (rI2 - r 2 3 gives the probability of finding an electron in an elefor circular orbits, where w is the angular velocity and ment of volume dr distant rl from the mass center and where the negative sign arises from the fact that elec- at the same time the probability of finding the nucleus tron and nucleus are oppositely charged. The orbital in the element of volume dr distant r2 from the mass center, the two particles and the mass center lying on angular momentum is
+.
359
360
JOURNAL OF CHEMICAL EDUCATION
the same line. If this were not so, the mass center would not remain fixed. The wave functions describing the behavior of the electron and the nucleus are then obtained by replacing, r by the equal values
and
giving
.+
(L) ' - 7, xaa (M+m)
(electron) =
rl
and
These two functions are equal when rl and r, are given their appropriate values, but they may be considered separately as giving the J. values as functions of all values of 5 and r,, and they are of course not equal a t equal distances from the mass center. The nuclear
function decays much more rapidly than the electronic. The nuclear "cloud pattern" is thus a kind of shrunken version of that due to the electron. Similar results are obtained for excited states, for which the wave functions contain as factors polynomials in r. We thus have the result that the nuclear cloud is spread over the whole atom, just as is the electron cloud, and at first sight the interpenetration of these two clouds seems unlikely. Such a view overlooks the statistical nature of the cloud patterns, and it is clear that the proton and electron always lie on opposite sides of the mass center. We thus have the idea of proton waves permeating the atom, and it follows that any discussion of the combination of two hydrogen atoms in terms of atomic electron-wave functions should also allow for the proton waves. This is not likely to lead t o any serious correction to energy values. The way is opened up, however, for the concept of proton waves spreading throughout a molecule, albeit with very great attenuation, and we have here a possible factor in prototropic changes in chemistry. In principle the effects will apply to other nuclei hut will be much smaller than for hydrogen.
