### Calculating x-ray Attenuation

I received detailed plans for the liquid helium Dewar that's going to be used in the h-ray experiment yesterday and spent some time calculating the amount of attenuation we can expect to the x-rays theoretically produced by our quenched Pb superconductor.  I'm making the assumption that most of the attenuation of x-ray flux will come from the aluminum walls and aluminum backed insulation, not from the plastic walls.  A cross-section of the walls is shown to the left, (picture 1).  The wall on the left of the cross-section is made of .125 inch thick aluminum.  The five sheets of insulation between the inner and outer walls are Mylar coated with .003 inch thick aluminum.The inner wall on the right hand side of the cross section is constructed of .1 inch of 'low thermal conductivity plastic'.  This gives me a grand total of 0.35 cm of aluminum between the source and the   detector.  The graph shown below (picture 2) is the amount of flux transmitted through the aluminum with respect to energy.  The attenuation values were taken from an online attenuation calculator[1]. The attenuation value without coherent scattering was used and was multiplied by the density of aluminum to arrive at a linear attenuation number mu.  The percent of transmitted flux was then calculated as

exp(-mu*thickness of material)

In addition to the attenuation due to the Dewar walls, I did two more calculations to get a ball park figure for how much attenuation I should expect from the superconducting solenoid magnet that will surround the sample and that is made from copper clad niobium alloy wire that is .045 cm thick.  The first graph is for copper of this thickness and the second graph is for Niobium, (pictures 3 and 4).

The theory predicts x-rays of about 382 keV energy.  If x-rays are produced at a much lower energy, say around 50 keV, it's conceivable that they could be missed with the detector outside the liquid helium Dewar.

Any thoughts, pointers, or questions are always welcome!

References:
1.  Aluminum attenuation
http://atom.kaeri.re.kr/cgi-bin/w3xcom?m=13

2.  Copper attenuation
http://atom.kaeri.re.kr/cgi-bin/w3xcom?m=29

3.  Niobium attenuation
http://atom.kaeri.re.kr/cgi-bin/w3xcom?m=41

4.  x-ray attenuation documentation
http://en.wikibooks.org/wiki/Basic_Physics_of_Nuclear_Medicine/Attenuation_of_Gamma-Rays

### 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…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

### 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…