Quantum mechanics makes some rather astonishing predictions about how particles behave. One of the most astonishing to me is that an electron's wave function can interact with a periodic potential, (say from the lattice sites of a crystal), and an applied force,(from a constant electric field for example), to make it behave as though its mass is vastly different, (sometimes even negative), compared to its rest mass in free space. Semiconductor physicists make use of this property all the time. It is also this property that Widom and Larsen[3] utilize in their theory of LENR paper. The energy of an electron within a crystal depends on its quasi-momentum as shown in picture 1. The quasi-momentum multiplied by the distance between crystal lattice sites is shown on the x axis and the electron's energy is shown on the y axis. Notice that the graph includes regions of energy called gaps that the electron does not occupy. The resgions of energy that are allowed are called bands. For a much more complete explanation of band-gap theory, see +Jonah Miller's excellent post on the subject[5].

It can be shown mathematically, although I won't go into the details here, that this dependence of energy on quasi-momentum leads to an equation relating the electron's effective mass to the curvature, (second derivative , of the E vs. q graph shown above. When a force is applied to an electron in a crystal, as the electron's momentum, (q) increases, its effective mass changes. This can lead to very small effective masses in the conductance bands of a semiconductor and very large effective masses in the valence bands. Interestingly, when the second derivative of the E vs. q graph is negative the electron can be shown to have a negative effective mass, and it will move in the opposite direction to the force applied. Electrical engineers call these negative effective mass electrons 'holes'. It is this motion in the opposite direction that leads to the phenomenon known as Bloch oscillations.

The effective mass variation led Widom and Larsen to hypothesize that if there was a large enough gain in the effective mass of the electron, then it could behave like the muons described in yesterday's post. The key argument made by Tennfors[4] against the Widom and Larsen[3] theory is that the effective mass gain is not high enough.

**Historical Aside**

For a kinder, gentler introduction to electrons behaving as waves, check out these two open access articles from one of the first experiments that showed electrons could diffract in the same manner as light[1][2].

**References:**

**1. Davisson's report on electron diffraction, (open access)**

http://www3.alcatel-lucent.com/bstj/vol07-1928/articles/bstj7-1-90.pdf

2. Davisson and Germer in Physical Review (open access)

http://dx.doi.org/10.1103%2FPhysRev.30.705

Davisson C. & Germer L. (1927). Diffraction of Electrons by a Crystal of Nickel, Physical Review, 30 (6) 705-740. DOI: 10.1103/PhysRev.30.705

3. Widom and Larsen on low energy nuclear reactions. This appears to be open access.

http://link.springer.com/article/10.1140/epjc/s2006-02479-8

4. Tennfors commenting on Widom and Larsen's article. This also appears to be open access.

http://link.springer.com/article/10.1140/epjp/i2013-13015-3

5. Physics mill on band-gap structure

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