### LENR and Electron Effective Mass

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:

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

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

5.  Physics mill on band-gap structure

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…