### Qualitative Superconductor Hysteresis Monitoring: Lab Book 2014_12_30

Lab Book 2014_12_30

Today I worked on getting a qualitative susceptometer up and running.  By qualitative I mean that we won't be attempting to analyze the returned data to accurately determine any characteristics of the superconducting materials that are to be studied, only the state of the materials, and the existence, or not of a hysteresis curve for the material.

The why of it all
I’m experimenting with a quick and dirty susceptometer that can be used with the hray experiment to determine when we’ve quenched the superconductor.  Today’s work is using a ferromagnetic core, not a superconductor.  The responses are similar and I don’t have to worry about cooling a superconductor.

Experimental Setup
Something old and something new
I’m pressing a General Radio 1311-A audio oscillator and a Tektronix TDS 210 into service together.

The general radio signal generator is being used to drive the coil surrounding the iron core.  The oscilloscope is measuring both the signal from the General Radio source and the pickup coil on the iron core as shown below

Voltage vs. time vs. frequency
In the following waveforms, the driving voltage is on channel 1 and the response voltage from the pickup coil is shown on channel two.  Channel one is a cleaner signal than channel two.  There’s a 180 degree phase difference between the signal just as you’d expect from Lenz’s law and the output on the pickup coil increases with frequency as you’d expect from Faraday’s law.
 400 Hz 500 Hz 1000 Hz 2000 Hz 5000 Hz 10000 Hz

X-Y data vs. frequency
The expected hysteresis loop began to appear once the frequency of the generator was brought sufficiently high.  The following data table illustrates this.
 400 Hz 500 Hz 1000 Hz 2000 Hz 5000 Hz 10000 Hz

The final data point at 10,000 Hz is shown below with bandwidth limiting on the scope enabled to cut down on the signal noise:

What we’re seeing
As the frequency of the oscillator is increased, the loop is opening up because the ferromagnetic response of the core can’t track quickly enough with the driving current.  Consequently there’s a phase difference between the two signals and this appears in the x-y display mode as a gradually opening loop.
It should be noted that the signal isn’t large enough to cause the iron core to saturate.  If it was, we’d see horizontal flat extensions at either corner of the loop.  A power amplifier may be added to the arrangement tomorrow to cause saturation.

Here’s a set of hysteresis curves that depend on frequency[1].  Notice that the enclosed loop becomes wider as the frequency increases, just as in the data shown above.

How Ferromagnets and Superconductors are Alike, (well, one way anyway)
I mentioned that ferromagnets behaved in a similar manner to superconductors when subjected to a magnetic field.  It’s because their hysteresis curves don’t look too much different.  Here’s a hysteresis curve[2] measured for superconducting Pb.  The square is the theoretically ideal response.  The curved lines are the actual response of the superconductor.

References:

1.  Jiles, J.C., “Frequency dependence of hysteresis curves in conducting magnetic materials
”, Journal of Applied Physics 76, 5849 (1994) http://dx.doi.org.lib-ezproxy.tamu.edu:2048/10.1063/1.358399

2. Rjabinin, Shubnikow, “Dependence of Magnetic Induction on the Magnetic Field in Supraconducting Lead”, Nature, 134, (1934), 286-287

### 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…