Friday, January 23, 2015

Lead Sample Purity and Tube Cutting: Lab Book 2015_01_22


For background on the experiment, please scroll down.

We can get a cheap piece of 3/4 inch diameter Pb from Rotometals.  Here are the details
3/4 17.14 per foot
Nuclead has the same thing.  Check for purity and price.
Also there’s Mayco.
Next question, how pure does pure have to be?

Pb purity data:
The following are reference articles about superconductors.  Each of them describes the purity of the Pb samples used.  The lower bar is set by the RMP article referenced below, as well as one other that mentions the use of ‘commercial’ grade material and the evidence of a transition to a superconducting state for this material.  The final reference from 1886 in the section defines commercial level material to have a quality not lower than 99% pure.  This information is being researched to determine what purity of sample we should use.  It would seem that a higher purity sample will produce fewer unexpected experimental results, as well as fewer alternative hypotheses to track down.  There is, however, a trade-off with the cost of the sample.
The Specific Heat of Lead in the Temperature Range 1'K to 75'K*, M. HoRowITz) A. A. SILvIDI, S. F. MALAKER) AND J. G. DAUNT  99.99% pure

Equilibrium Curve and Entropy Difference between the Supraconductive and the Normal State in Pb, Hg, Sn, Ta, and Nb, J. G. Daunt and K. Mendelssohn, Proc. R. Soc. Lond. A 1937 160, doi: 10.1098/rspa.1937.0099, published 1 May 1937  purity > 99.999
Look at 15% field lines frozen in Pb:  free version of the article:
The frozen in field lines are hypothesized to be due to the presence of more impurities in Pb, (15% frozen in lines),and Sn, (~10%), than in Hg, (0%), which “can easily be prepared in a state of extreme purity.”
This article references an article that describes the phenomenon of frozen-in field lines and how to detect them:

It was also commented that the sharpness of the transition from the normal to the superconducting state appeared to be dependent on the number of impurities.  Fewer impurities led to a sharper transition from normal to superconducting.
Section 4.i. of this article:
Section 4.i. discusses the freezing-in of flux lines again.  A dependence on the purity of the material is mentioned.  The expulsion of flux lines spontaneously without a change in flux, as often mentioned by Hirsch, is discussed in section 4.e. 
Notes on the impurity of ‘commercial tin’ used in the experiments described in the RMP article can be found below in an excerpt of a book apparently about making tin cans:
It sounds like the metal was in fact fairly impure, whatever that means quantitatively.
Here’s more information on the purity of metals.  The N numbers, of purity are discussed.  3N is 99.9 percent pure.  4N is 99.99% pure.
Here’s a quote on the quantitative quality of ‘commercial’ tin:
“In the United States, the purity levels for commercial grades of tin are defined by the American Society for Testing Materials (ASTM) Standard Classification B339. The highest grade is AAA, which contains 99.98% tin and is used for research. Grade A, which contains 99.80% tin, is used to form tinplate for food containers. Grades B, C, D, and E are lesser grades ranging down to 99% purity. They are used to make general-purpose tin alloys such as bronze and solder.”
Here’s the source
From page 625 of the following, written in 18xx, closer to the time of the RMP article in 1935
“Purity of commercial tin—The tin of commerce generally contains a very small quantity of iron and lead and traces of other metals, but rarely exceeding 1 percent of impurity.”
Random Notes
On glass Dewars

Sample Stage Preparation
Work was continued on building the sample state.  The Dewar stick was separated from its attached brass platform using a tubing cutter.  The aluminum stick was very soft and the first cut resulted in the stick becoming crimped.  A second cut was made more successfully at a position above the one crimped by the initial cut.  Care was taken not to cut any of the wires travelling through the tube.


Initial Crimp



Final Cut



Background
Hirsch's theory of hole superconductivity proposes a new BCS-compatible model of Cooper pair formation when superconducting materials phase transition from their normal to their superconducting state[1].  One of the experimentally verifiable predictions of his theory is that when a superconductor rapidly transitions, (quenches), back to its normal state, it will emit x-rays, (colloquially referred to here as H-rays because it's Hirsch's theory).

A superconductor can be rapidly transitioned back to its normal state by placing it in a strong magnetic field.  My experiment will look for H-rays emitted by both a Pb and a YBCO superconductor when it is quenched by a strong magnetic field.
This series of articles chronicles both the experimental lab work and the theory work that’s going into completing the experiment.

The lab book entries in this series detail the preparation and execution of this experiment… mostly.  I also have a few theory projects involving special relativity and quantum field theory.  Occasionally, they appear in these pages.

Call for Input
If you have any ideas, questions, or comments, they're very welcome!

References
1.  Hirsch, J. E., “Pair production and ionizing radiation from superconductors”, http://arxiv.org/abs/cond-mat/0508529

Wednesday, January 21, 2015

Mechanics I: Distance Traveled along a 1-D path

I ran into a rather abstruse question in today's first mechanics recitation.  The question gave the one dimensional position of a particle with respect to time as

$x = 10 - 4t + 2t^3$

It then asked for the distance traveled by the particle between t = 0 and t = 2.  The suggested answer, (from the prof in charge of TAs), was to plot the trajectory of the particle, thereby demonstrating the distance and displacement were different.  Here's the plot:


The idea is that you can see that the particle travelled form 10 to 8 and then back to 18, so the total distance is more than the displacement from 10 to 18 i.e. 8.

The question came up as to how to do this to get the exact answer.  Here goes

What we want to do is add up all the small, (read infinitesimal), distances travelled by the particle between time 0 and time 2.  The phrases, 'adding up', and infinitesimal provide the tip off that we'll do an integral, so:

Getting to the Integral the better way with the chain rule
Since we want to add up all the little pieces of distance that the particle traveled, we want something like:

$distance = \int_{t = 0}^{t = 2} dx$

which is all fine and jolly except we have $dx$ and integration limits in time.  I'd like my integration variable to be the same as the variable that describes the limits of the integral.  To get an expression for dx in terms of dt, we can use the chain rule, taking the total derivatives of both sides of

$x = 10 - 4t + 2t^3$

We arrive at:

$1dx = \left(-4+6t^2\right)dt$

Notice that I took the derivative of each side using the chain rule.  The derivative of $x$ is the derivative of the outside with respect to x, $1$, times the derivative of the variable inside, $dx$. The same things works out for each expression that contains $t$.

The not nearly as good way of looking at getting to dx in terms of dt
Let's do a substitution of variables.  From the original position expression we can get, (by taking a derivative with respet to time),

$$ \dfrac{dx}{dt} = -4 + 6t^2$$

Being the lazy physicists that we are, we multiply both sides by $dt$ to get:

$$ dx = \left(-4 + 6t^2\right) dt$$

Now, On with the derivative
The next big thing to keep in mind is that we need to add up all distances the same regardless of their directions, (the negative vs. the positive x direction).  To achieve this, we can either take the absolute value of dx, or we can use a trick I like better which is to square dx and then take its square root.  I prefer this because when I start dealing with two or more dimensions, I'll already have the Pythagorean theorem.  In other words when I say, 

$$\sqrt{dx^2}$$

I'm thinking

$$\sqrt{dx^2 + dy^2}$$

but in this problem, since we're in one dimension, $$dy = 0$$

To illustrate why we have to use either the absolute value, or the square root of the square, here's our expression for $dx$ vs time without the absolute value/square root:




With the absolute value or the square root/square combination we get the following:


Notice in the first plot, dx goes negative for a while.  This will be counted as negative distance traveled when we integrate.  Since we want to know the total distance traveled, no matter which direction, we need to use the expression shown in the second plot where the 'negative direction' distance has been flipped positive by the absolute value.

Here's a diagram that might help make more sense of the how we're going to sum distance. Notice that the red arrow moves in the backwards direction, but we still count it as posititive distance covered.



In any event, we wind up with the integral

$$distance = \int_{t = 0}^{t = 2}  \sqrt{\left(4 + 6t^2\right)^2} dt$$

Performing this integral in Sage gives:

$12.35$ which is larger than the displacement of $8$ and slightly larger than our initial estimate from the plot of about $12.$

Coming soon, the Sage code here on the page...

The Promised Sage Code

Plot the position vs. time

Show the graph of dx/dt vs. time with all distance positive

More to come...

Monday, January 19, 2015

Cosine Laws, Polyhedra, and Legendre Functions



I didn't make it into the lab today what with the holiday and all, but I did have time to read one of my favorite journals, American Mathematical Monthly from the +Mathematical Association of America .  The journal features a very interesting article[1] by Marshall Hampton[3] about cosine identities.  The article got me back to musing about solving for potentials with spherical symmetries and Legendre polynomials again[5].  I don't have time to work through this now, so I'm just recording my meandering thoughts here for future self, and anyone else that would like to take a look.

Hampton writes down the generalization of the law of cosines for polyhedra rather than just the plane, (pun intended), old triangle.  Here it is

$$0 = \sum_j\vec{n}_i\cdot\vec{n}_j\Delta_j = \Delta\left(i\right) - \sum_j c_{ij}\Delta_j$$

Where, $$c_{ij}$$ is the cosine between two faces of the polyhedra i and j, and $$\vec{n}_i$$ is a vector field normal to the i'th face.

Dr. Hampton states that this expression is arrived at through the divergence theorem for polyhedra.

Here's the question, sketchy as it may be.  Since the Legendre polynomials are a solution of Laplace's equation under certain boundary conditions, and since the polynomials can be generated by the law of cosines, in light of the series of cosines above, if we extend the sum out to an infinite number of identical faces for the polyhedra, can we arrive back at the Legendre polynomials?

In addition to his recent article in AMM, Dr. Hampton has produced quite a bit of other material worth checking out [2][3][4].

References:
1.  http://www.jstor.org/stable/10.4169/amer.math.monthly.121.10.937
sadly, behind a pay wall, but see [3]
2.  Marshall Hampton's mathematical coloring book
http://www.d.umn.edu/~mhampton/mathcolor17b.pdf
3.  Marshall Hampton's home page
http://www.d.umn.edu/~mhampton/
4.  Marshall Hampton on Diff Eqs and Sage
https://books.google.com/books?id=78cSvjHo1zUC&lpg=PA233&ots=LZjUB_iA-x&dq=Marshall%20Hampton%20cosines&pg=PP1#v=onepage&q&f=false
5.  Legendre polynomials on Copasetic Flow
http://copaseticflow.blogspot.com/2013/01/law-of-cosines-and-legendre-polynomials.html

Sunday, January 18, 2015

Experimental Estimates and Deconstruction: Lab Book 2015_01_16


Took apart the apparatus at the bottom of the proposed Dewar stick.  This is stick that will eventually support all the required apparatus in liquid helium.  Pictures follow.  Per normal, if this is your first day on the site, scroll to the bottom for the experimental background.



The inside of the Dewar measure out at 1 and 1/8 inches.  That works out to about 1.25 cm.  Then, plugging that into an expression for the size of cylinder we can get to fit


Working with square cylinders
Inscribe a square inside a circle
circle radius
0.5625
1.42875
cm
square side
0.795495129
1.010279
cm
Square radius
1.010278814

The above distances are to the wall of the Dewar.  If we back off of this a little bit and give ourselves  an 1/8 of an inch clearance at all the corners, we get
Working with square cylinders
Inscribe a square inside a circle
circle radius
0.5
1.27
cm
square side
0.707106781
0.898026
cm
Square radius
0.898025612

a radius of 0.89 cm.  That gives a maximum energy of about 290 keV and a total flux of 10,000 events per quench.  How does this jive with the sensitivity of the NaI detector?  This fits well within the range of the signals the detector is sensitive to:
Source
Peak Channel
Energy eV
Cd109


am241
110
26344
Cs 137
121
32000
Am241
221
59541
Cd109


Cs 137
2118
662000

For reference, here’s what the Cs137 spectrum looked like in the fiberglass Dewar

As it turns out, the brass disc at the bottom of the stick is too wide to fit in the Dewar.  It will b removed soon.

To Do: 
·         Characterize the detector response with the glass Dewar
·         Calculate the solid angle with the detector much closer to the source
·         Characterize the background response in the basement
Updates for the experimental plan:
Attempt to do a 90 degree rotation of the sample between runs.  This should help to account for any directionality issues due to the sample being a cylinder and not a sphere.

A possible source for the sample

Back to spheres?
The extruded version of this might work fine


Background
Hirsch's theory of hole superconductivity proposes a new BCS-compatible model of Cooper pair formation when superconducting materials phase transition from their normal to their superconducting state[1].  One of the experimentally verifiable predictions of his theory is that when a superconductor rapidly transitions, (quenches), back to its normal state, it will emit x-rays, (colloquially referred to here as H-rays because it's Hirsch's theory).

A superconductor can be rapidly transitioned back to its normal state by placing it in a strong magnetic field.  My experiment will look for H-rays emitted by both a Pb and a YBCO superconductor when it is quenched by a strong magnetic field.
This series of articles chronicles both the experimental lab work and the theory work that’s going into completing the experiment.

The lab book entries in this series detail the preparation and execution of this experiment… mostly.  I also have a few theory projects involving special relativity and quantum field theory.  Occasionally, they appear in these pages.

Call for Input
If you have any ideas, questions, or comments, they're very welcome!

References
1.  Hirsch, J. E., “Pair production and ionizing radiation from superconductors”, http://arxiv.org/abs/cond-mat/0508529

Thursday, January 15, 2015

The Day of Inconclusive Data: Lab Book 2015_01_15

In all likelihood, we’ll go with a resistive measurement, (to detect the superconducting state or lack thereof), using a small lead wire.  The following setup is from an Alfred Leitner video.  We’ll use something even simpler, probably just a lead wire with a four point probe attached.  By the way, if you're new to the game, scroll to the bottom for background information.

We could sputter a line of lead onto a glass slide, but I don’t see the benefit yet. Working on finding out what it is.

Check this out later in the day per shaping samples:

I’m performing the check on the resistance of the primary coil as its cooled by liquid nitrogen.  The idea is that yesterday’s increase in output ignal with the superconductor ooled may have noly been do to the primary pulling more current as its resistance ramped down.  Here are a few pictures of the assembly being taken apart for the present work:






A one ohm resistor has been placed in series with the primary coil.  The readout across the resistor is proportional to the current through the circuit.  The resistor signal is the smaller signal in the following picture.

The basic lesson of the day was not to build coils that can move in any possible way.  The second attempt to check for reduced resistance, (which should happen), had contradictory results.  The first and final attempts returned the expected results, but they’re not reliable given the first two runs.  The available data follows.



First attempt:
Room temp:

Post-cooling with liquid nitrogen:


Second Experiment:
Pre-cooling:




Post-cooling:

This is also where it was observed that the oscillator has a significant phase shift.


Third experiment:
pre-cooling:


post-cooling:





Background
Hirsch's theory of hole superconductivity proposes a new BCS-compatible model of Cooper pair formation when superconducting materials phase transition from their normal to their superconducting state[1].  One of the experimentally verifiable predictions of his theory is that when a superconductor rapidly transitions, (quenches), back to its normal state, it will emit x-rays, (colloquially referred to here as H-rays because it's Hirsch's theory).

A superconductor can be rapidly transitioned back to its normal state by placing it in a strong magnetic field.  My experiment will look for H-rays emitted by both a Pb and a YBCO superconductor when it is quenched by a strong magnetic field.
This series of articles chronicles both the experimental lab work and the theory work that’s going into completing the experiment.

The lab book entries in this series detail the preparation and execution of this experiment… mostly.  I also have a few theory projects involving special relativity and quantum field theory.  Occasionally, they appear in these pages.

Call for Input
If you have any ideas, questions, or comments, they're very welcome!

References
1.  Hirsch, J. E., “Pair production and ionizing radiation from superconductors”, http://arxiv.org/abs/cond-mat/0508529