Techniques: LAPW Band Structure Calculations, Muffin-tin Potential.
For this month's Practical Quantum Mechanics project we investigate poor performance of an LED torch. We'd bought the new Deben 'Huntmaster' torch from Amazon, and were disappointed to find that its beam was only just reaching 5m, when the product literature claimed 75m or better (See Fig 1). Given the significant disparity between the actual and expected beam-length, we decided some DIY diagnostics were in order.
The first step is to determine the theoretical brightness of the galenium-arseninde LED that features in the 'Huntmaster'. That depends on the density of states in the p-band of galenium, since the beam is formed when excited electrons drop back from the p-band to the k-band. For this you'll need an ab initio self-consistent band-structure calculation. We went for the tried and trusted LAPW method of Anderson, with an 'N=3' implementation. If you don't have time to write your own code, the University of Gotlingen has a suite of LAPW programs to download free.
The LAPW method considers a one-electron Schrodinger equation, with the effects of the lattice ions and valence electrons modelled as a background potential V(r). The 'self consistent' part just means that the output from one run of the LAPW calculations is used to generate the background potential as input for the next, and you keep repeating that until the calculations converge. But to kick things off you need a starting potential, and that's straightforward to produce using the 'muffin tin' model, where the potential is assumed to be flat in the interstices between adjacent spherical potentials of the lattice nuclei, just like in a real muffin tin (see Fig 2).
Again you can download a program to calculate a muffin tin potential. For galenium-arsenide use an HCP (hexagonal close-packed) lattice setting, with a spacing of 0.238 (using natural units).
Running the calculations will take about 2 solid years on an I-Phone 5, but you can speed things up by booking time on a supercomputer (check it's got floating-point acceleration capabilities). We used the cloud service provided by the Berkeley IT group, and finished all the calculations in an overnight run. Note that to start the LAPW program you might need to click the 'yes' option in a pop-up dialogue box as shown below...
If you've got things right the calculated energy density of states ought to look something like Fig 3.
To calculate the brightness of the beam, just integrate the number of states over the width of the P-band, and multiply by Avogadro's number and the mass of the LED in Kilograms. The result in our case was around 6,283,000,000,000,000,000,000,000,000,000,000,000. Allowing for dispersion, that should easily give a 75m beam with an excitation current of around 50mA, which should be well within the theoretical capabilities of the three AAA batteries that power the torch. And therein lay the clue to the poor performance of our Huntmaster. A quick check with a multi-meter confirmed that its batteries were almost flat, so we inserted new replacements and the beam was every bit as bright as Deben's claims in the literature.
So the lesson here is that so-called 'new' batteries supplied with torches need to be checked.
New month we investigate annoying crackles in MP3 playback on an iPhone 5s. A loose headphone jack, perhaps, or could it be something more sinister? We'll show you how to run a self-consistent Hartree-Fock calculation to rule out the possibility of Block-wave scattering by evanescent phonons arising from interstitial impurities in the silicon lattice of the i-phone chip-set.