An electron is emitted from a middle-mass nuclide (A=150, say) with a kinetic energy of 1.0 MeV. (a) What is its de-Broglie wavelength? (b) Calculate the radius of the emitting nucleus. (c) Can such an electron be confined as a standing wave in a “box” of such dimensions? (d) Can you use these numbers to disprove the (abandoned) argument that electrons actually exist in nuclei?
The electron can only be confined within a given structure only if its wavelength is smaller than the atomic radius of the nuclide. Now, the wavelength of an electron is associated with the momentum according to the de-Broglie concept. Thus, by comparing the wavelength with the calculated atomic radius of the nuclide, consider the strong argument for the case of confinement.
The kinetic energy of a particle in motion:
The energy and momentum relation according to relativistic concept,
The atomic radius of a nuclide using its nucleon or mass number,
The de-Broglie wavelength of a particle of smaller size:
Consider the known data, and hc = 1240 MeV.fm
Substitute the value of momentum from equation (ii) in equation (iv), consider de-Broglie wavelength of the electron as follows:
Substitute the values as:
Hence, the value of the wavelength is .
Using the given data in equation (iii), determine the radius of the emitting nucleus as follows:
Hence, the value of the radius is 6.4 fm .
Since, from parts (a) and (b) calculations the electron cannot be confined in the nuclide. Recall that at least is needed in any particular direction, to support a standing wave in an “infinite well.” A finite well is able to support slightly less than (as one can infer from the ground state wave function in Fig. 39-6), but in the present case is far too big to be supported.
A strong cas e can be made on the basis of the remarks in part (c), above.
Because of the 1986 explosion and fire in a reactor at the Chernobyl nuclear power plant in northern Ukraine, part of Ukraine is contaminated with ,which undergoes beta-minus decay with a half-life of 30.2y . In 1996, the total activity of this contamination over an area of was estimated to be . Assume that the is uniformly spread over that area and that the beta-decay electrons travel either directly upward or directly downward. How many beta-decay electrons would you intercept were you to lie on the ground in that area for (a) in 1996 and (b) today? (You need to estimate your cross-sectional area that intercepts those electrons.)
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