Abstract
Atoms, molecules, and crystalline solids are examined with regard to energy quantization. The formation of energy bands in crystals is explained.
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Notes
- 1.
Diamond is a notable exception. Theoretically, its melting point is at 5000 Κ; it is never reached in practice, however, since diamond transforms to graphite at about 3400 Κ! (See [1], p. 406.)
- 2.
By “solid” we will henceforth always mean crystalline solid.
- 3.
Again, diamond is a notable exception given that its thermal conductivity exceeds that of metals at room temperature [1]. This conductivity is, of course, exclusively due to lattice vibrations.
- 4.
The name “line spectrum” is related to the fact that each frequency appears as a line in a spectroscope.
- 5.
Exceptions to this rule are hydrogen (H, 1) and helium (He, 2).
- 6.
An exception is hydrogen, where the energy of its single electron depends only on n (En = − κ/n2), thus is a property related to shells rather than to subshells.
References
Turton, R.: The physics of solids. Oxford University Press, Oxford (2000)
Bransden, B.H., Joachain, C.J.: Physics of atoms and molecules, 2nd edn. Prentice –Hall, Harlow/Munich (2003)
Schiff, L.I.: Quantum mechanics, 3rd edn. McGraw-Hill, New York (1968)
Millman, J., Halkias, C.C.: Integrated electronics. McGraw-Hill, New York (1972)
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Questions
Questions
-
1.
Show that the energy distance between successive energy levels of the hydrogen atom decreases as we move up the energy-level diagram.
-
2.
Evaluate the frequency and the wavelength of the radiation emitted by a hydrogen atom during a transition of its electron from the orbit n to the orbit (n–1), where n > 1.
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3.
(a) Show that the radius 25a0 (where a0 is the Bohr radius) corresponds to an allowable orbit for the electron in a hydrogen atom, and evaluate the angular momentum and the energy of the electron in that orbit. (b) At what distance from the nucleus does the electron have an angular momentum equal to 3 h/π?
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4.
Find the possible frequencies of radiation emitted by an excited hydrogen atom when the electron, initially moving on the third Bohr orbit, finally – albeit not necessarily directly – returns to the fundamental first orbit.
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5.
A hydrogen atom is excited by absorbing a photon; it then returns in two steps to its fundamental state, emitting two photons of wavelengths λ1 and λ2. Find the wavelength λ of the photon that was absorbed by the atom.
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6.
A moving electron hits a hydrogen atom and excites it from the fundamental to the third energy level. The atom then returns in two steps to its initial state, emitting two photons. (a) Find the minimum speed of the electron that hit the atom. (b) Find the frequencies of the two photons.
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7.
Ionization of a hydrogen atom can be produced by collision with a moving electron or by absorption of a photon. (a) What must be the minimum speed of the electron? (b) What must be the maximum wavelength of the photon?
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8.
A photon hits a hydrogen atom and causes ionization to the latter. The liberated electron then falls onto another hydrogen atom and excites it from the fundamental to the immediately higher energy level. Find the minimum frequency of the photon that hit the first hydrogen atom.
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9.
As we know, the lowest energy level of a many-electron atom is the 1s level (i.e., the energy level corresponding to the 1s subshell). Why then doesn’t the totality of electrons in the atom occupy this particular level?
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10.
Draw a comparative energy-level diagram (not necessarily an exact one!) for the electrons of the oxygen (Ο) atom and those of the ozone (Ο3) molecule.
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11.
Why do crystals have energy bands instead of energy levels like atoms and molecules? Why are higher-energy bands wider than lower-energy bands?
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12.
A band in a crystal has its full quota of electrons, while another band of the same crystal is partly filled. Which band is wider? Explain.
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13.
Consider two diamond crystals, a small one and a big one. Compare the widths of corresponding energy bands of the two crystals.
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14.
In what ways do the free electrons affect the stability and the physical properties of metals?
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15.
Give examples that demonstrate that the correspondence between the energy levels in each of a set of identical atoms, and the energy bands of a crystal composed from these atoms, is not perfect.
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Papachristou, C.J. (2020). Atoms, Molecules, and Crystals. In: Introduction to Electromagnetic Theory and the Physics of Conducting Solids. Springer, Cham. https://doi.org/10.1007/978-3-030-30996-1_1
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