Tuesday, September 9, 2014

Proper Velocity!!! and Getting Index Notation Worked Out: EM II Notes 2014_09_09

Summary:  It looks like I'll finally get a good understanding of the gamma notation for moving proper velocities to lab velocities and back.  It'll be nice to know it inside and out, but a little irksome given all that can be done with the hyperbolic notation we're not using.  I want to maintain my fluency in both.

There may be a subtle second notation for inverted Lorentz transforms.  As it turns out, the subtle notation difference of moving around indices in the top and the bottom with spaces is meant to keep track of which index comes first when you go back to side by side notation.

First, we cover Lorentz transforms, (which are not in fact tensors), and contractions and arrive at the interesting result in equation 1.99:

$\Lambda^\mu_\rho \Lambda^\sigma_\mu T^\rho_\sigma = \delta^\sigma_\rho T^\rho_\sigma$

Which indicates the transpose of the Lorentz transform times itself follows a sort of orthogonality rule making use of contravariant indices.

Q:  Does this obviate the need for the $\eta$ metric?

A:  But wait!  There's so much more!  This is a way to write things without the $\eta$ cruising through everywhere, but it also explains the oddball spacing of the indices.  Maybe I just don't remember this from MacConnel?  First, the Lorentz transform is not a tensor.  Now, we hve that down.  The next bit is how to arrive at the above expression.

$\eta_{\mu\nu} \Lambda^\mu_{\;\rho} \Lambda^\nu_{\;\sigma} = \eta_{\rho\sigma}$

The next step is to plow the $\eta$ on the l.h.s. in and lower the $\nu$ on the second transform.

$\Lambda^\mu_{\;\rho} \Lambda_{\mu\sigma} = \eta_{\rho\sigma}$

We then raise the sigma

$\eta^{\sigma\lambda} \Lambda^\mu_{\;\rho} \Lambda_{\mu\sigma} = \eta^{\sigma\lambda}\eta_{\rho\sigma}$

$\eta^{\sigma\lambda} \Lambda^\mu_{\;\rho} \Lambda_{\mu\sigma} = \eta_\rho^\lambda = \delta_\rho^\lambda$

$\Lambda^\mu_{\;\rho} \Lambda^{\;\lambda}_\mu = \eta_\rho^\lambda = \delta_\rho^\lambda$

Now, since we can see that the first two indices are $\mu$s, we can call the above statement a transpose.  The index locations matter to get the transpose to be a transpose, making sure the rows and columns are handled properly.

NOTE:  This doe arise in MacConnel.  His notation is slightly different.  Instaed of $\Lambda^\mu_{\;\rho}$, he writes $\Lambda^\mu_{.\rho}$

We're going to need the delLambertian soon and it's important to note that it is

$\Box = -\partial_0\partial_0 + \partial_i\partial_i = \partial^\mu \partial_\mu$

A few notes follow on why the D'Alambertian is written with one index up and one down.  It has to do with the negative sign in the first entry of the Minkowski metric.  As it turns out, the index up version of the Kronecker delta is the same as the index down version, but not so for $\eta$ because of the $-1$ $00$ entry.  This is all much simpler if you never start writing your indices in the 'wrong' location in the first place.

OK, now for the interesting stuff.  First, with the choice of signature for the Minkowski metric here, we wind up having to write down $d\tau$ as \

$d\tau = -ds^2 = dt^2 - dx^2 - dy^2 - dz^2$

Given $d\tau$ we define the four velocity to be

$U^\mu = \dfrac{dx^\mu}{d\tau}$

Which is technically mixing frames, but then, that's what proper velocity does.  Intersting that we're starting with four velocity, proper velocity here.  It's really nice to get this out of the way early on and benefit from it.

Picture of the Day
Presenting the Takeno metric line element.  Hopefully one of the payoffs of all this will be understanding this more fully, and possibly using it to explain the circular Unruh effect.  However, a far more productive use is in explaining the Thomas Precession.

Saturday, September 6, 2014

Stepping Back Up With Classes: Lab Book 2014_09_06

Summary: Classes started this week.  They're a lot of fun, but they take time.  Consequently, the lab work is moving a little more slowly.  I'm looking into what we can accomplish with a YBCO superconductor sample.  The advantage is that we can test our experimental techniques using relatively cheap liquid nitroogen intead of iquid helium.  The downside is that with the size of YBCO sample we have, the expected maximum energy is only 3 deV which is kind of low without a specialized detector.

For more background on the experiment, please scroll to the bottom of the post.

The percolator peak does not appear when the detector is initially turned on.  The attenuator does however appear to create a rather copious amount of noise.

Suppose we used YBCO as a sample.  The energy we could expect doing a back of the napkin calculation is 3.8 keV.  The flux is approximately 230 photons for our sample size.  For a 25 square mm detector that may be available, this gives a total flux of 57 particles if we’re only 0.5 cm away.  This may work.

Another step would be to cycle the permanent magnet at a few Hertz to quench the superconductor repeatedly and increase the total flux.  The inside of the power supply control box is shown below.

The two potentiometers that control the supply are shown in the table below.  The coarse control is on the left and the fine control is on the right.

One technique might be to replace these potentiometers with a voltage controlled resistor.  The control circuit is shown below.

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, September 4, 2014

Showing that SpaceTime Intervals are invariant: EM II notes 2014_09_03

Summary:  Continuing notes on the tensor version of the Lorentz tranform.  It's time to start on the second set of examples.

The interval in four space is invariant under Lorentz transforms and is called the Lorentz scalar.

The Lorentz transform also applies to differential distances as,

$dx^{\prime\mu} = \Lambda^\mu_\nu x^\mu$

We were asked in class to work out $x^2+y^2+z^2-t^2 = x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-t^{\prime 2}$

The transforms we'll use are:

$x = \gamma\left(x^\prime + vt^\prime\right)$

$t = \gamma\left(t^\prime + vx^\prime\right)$

Substituting these into the l.h.s. gives

$\gamma^2\left(x^\prime + vt^\prime\right)^2 - \gamma^2\left(t^\prime + vx^\prime\right)^2 = x^{\prime 2} - t^{\prime 2}$

$= \gamma^2\left(x^{\prime 2} +2vtx + v^2t^{\prime 2}\right) - \gamma^2\left(t^{\prime 2} + 2vxt+v^2x^{\prime 2}\right)= x^{\prime 2} - t^{\prime 2}$

$= \gamma^2\left(x^{\prime 2} + v^2t^{\prime 2}\right) - \gamma^2\left(t^{\prime 2} + v^2x^{\prime 2}\right) = x^{\prime 2} - t^{\prime 2}$

$\gamma^2 x^{\prime 2} - \gamma^2v^2x^{\prime 2} = \gamma^2 x^{\prime 2}\left(1 - v^2\right) = x^{\prime 2}$

Similarly

$\gamma^2 v^{\prime 2}t^{\prime 2} - \gamma^2 t^{\prime 2} = -\gamma^2 t^{\prime 2}\left(-v^2 + 1\right) = -t^{\prime 2}$

So

$x^{\prime 2} - t^{\prime 2} = x^{\prime 2} - t^{\prime 2}$

Done

Picture of the Day:
Here's a throwback Thursday pic from 1946.  These are the infamous Carter Boys.  My dad is the twin on the right.

Wednesday, September 3, 2014

Tensor Based Special Relativity Begins! EM II Notes 2014_08_25

Summary:  This one took awhile.  I got busy in the lab  These notes start with rotation matrix properties and the transpose products of matrices.  special relativity via tensors also begins.  Specifically, the Lorentz transformation tensor components are reviewed and the number of independent parameters are counted.

Did a few concrete checks that a matrix times its transpose is symmetric.  Sure enough, it is.
$\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\begin{pmatrix} a & c \\ b & d \\ \end{pmatrix} = \begin{pmatrix} a^2 + b^2 & ca + bd \\ ca+bd & c^2+d^2 \\ \end{pmatrix}$

$\begin{pmatrix}a & b & c\\ d & e & f \\ g & h & i \\ \end{pmatrix}\begin{pmatrix} a & d & g\\ b & e & h \\ c & f & i \\ \end{pmatrix} = \begin{pmatrix} a^2 + b^2 + c^2 & ad + be + cf & ag + bh + ci\\ ad + be + cf & d^2 + e^2 + f^2 & gd + eh + fi \\ ga + hb + ic & gd + he + if & g^2 + h^2 +i^2 \\ \end{pmatrix}$

Sure enough, symmetric!

We cover the Minkowski metric, written as

$\eta_{\mu\nu} = \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\\ \end{pmatrix}$

Using the Minkowski metric shown above, we can right the invariant interval as:\\
$x^2 + y+^2 + z^2 - t^2 = \eta_{\mu\nu}x^\mu x^\nu$

Lorentz transforms will now be denoted as\\
$x^{\prime \mu} = \Lambda_\nu^\mu x^\nu$

Here's an interesting bit.  The Lorentz transform tensor's components can be written as\\
$\Lambda_0^0 = \gamma$

$\Lambda_i^0 =\gamma v_i$

$\Lambda_0^i =\gamma v_i$

$\Lambda_i^j = \delta_{ij} + \dfrac{\gamma - 1}{v^2} v_i v_j$

The $i$, $j$ terms contain the same $\gamma - 1$ term that was in the earlier notes about the vector Lorentz transform.  See

http://goo.gl/Z9MGjK for a complete explanation
The pretext to Eqn. 1.70 is re-specifying the invariant time-space interval in terms of tensor notation

$\eta_{\mu\nu}x^\mu x^\nu = \eta_{\mu\nu}x^{\mu\prime} x^{\nu\prime}$

Moving to all unprimed variables we get

$\eta_{\mu\nu}x^\mu x^\nu = \eta_{\mu\nu}\Lambda^\mu_\rho \Lambda^\nu_\sigma x^\rho x^\sigma$

Then we have a good example of replacing dummy indices by any name we please including the dummy indices on the other side of the equation to get

$\left(\eta_{\rho\sigma} - \eta_{\mu\nu}\Lambda^\mu_\rho \Lambda^\nu_\sigma \right)x^\rho x^\sigma = 0$

and finally
$\eta_{\mu\nu}\Lambda^\mu_\rho \Lambda^\nu_\sigma= \eta_{\rho\sigma}$

Independent Lorentz Parameters

There are $4 \times 4$ parameters to begin with.  The transform is symmetric, so that eliminates the bottom triangle as redundant.  That leaves the upper triangle and the diagonal which gives 10 constraints, also known as equations.  That leaves 6 unspecified parameters.  Three of these are the velocities in each direction.  The other three are the spatial rotation parameters.

Question
How does the Lorentz group relate to the Conformal group?  I think it's a subgroup.

Questions

Does the Minkowski metric correspond to the identity tensor?\\
If so, then does the expression

$\eta_{\mu\nu}\Lambda^\mu_\rho \Lambda^\nu_\sigma = \eta_{\rho\sigma}$
Correspond to the transpose identity in the $O\left(3\right)$ group?

A This does correspond to the transpose identity, but the Minkowski metric doesn't have anything to do with it.  See \url{http://goo.gl/7zTnJV} for more details on how transposes look in index notation.

Q Why are the transformation in 1.68 only examples of pure boosts?  What extra parameters do they need to be completely general?

A: They need off diagonal terms in the three by three that's in the $i$, $j$, indices.  Two boosts in different directions will multiply to create terms in these positions giving a Thomas-Wigner rotation.  Any general boost can be decomposed to a pure boost and a rotation, but the terms do not commute.  Depending on the order of the decomposition, boost then rotation, or rotation then boost, you get different terms.

Q: Why do we need to take the transpose of the l.h.s. of 1.75 to show that it's symmetric?  Wouldn't showing the product of the transform and its transpose is symmetric work just as well?  I ask because the Minkowski metric that is subtracted out is diagonal and won't/can't affect the symmetry.

A: This was a bit of overkill in the notes.  Symmetric matrices are their own transposes.

Q: How does the expression $\left(4 \times 5\right)/2$ enter into the discussion?

A: This is a combinatoric expression for the number of independent parameters in a symmetric matrix.  The anti-symmetric counterpart is

$n\left(n-1\right)/2$

One way to look at it for the two dimensional case is to consider the antisymmetric case and keep in mind that the number of parameters for a completely general $n \times n$ matrix is $n^2$, and just subtract the anti-symmetric portion to arrive at the result above.

The above result also generalizes to higher dimensional tensors.  For example, in a three index tensor, we get

$\dfrac{\left(n+2\right)\left(n+1\right)n}{3!}$

Things to memorize

The rotation matrices with $det\left(M\right)$ do not form a group because the product of any two members will take you back to the space of $SO\left(3\right)$, i.e. matrices with a positive determinant.

The number of independent parameters is found by subtracting the number of independent equations, (which can be solved for parameters, making them therefore not independent), from the total number of parameters.

Matrix identity

$\left(AB\right)^T = B^T A^T$

Symmetric matrices by definition have a lower triangle's worth of redundant equations.

Picture of the Day:
The Tunnel Top Bar in San Francisco

Tuesday, September 2, 2014

Testing the Scintillator Near the Magnetic Field: Lab Book 2014_09_01

Summary:  The x-ray detector will be very near a rather large pulsed magnetic field in the experiment.  Tests were run today to determine how the scintillator reacts, if at all to the field. There were no visually available indications that the detector had behaved differently at all.  There is one channel that has a consistently higher count when the data is analyzed, however, this doesn't appear to be statistically significant though.

If you're new to the experiment, please scroll to the bottom for background material.

Took a background spectrum with the Dewar in place.  The percolator peak was not present when the spectrum was started, but had appeared by the time the spectrum was finished.
 Bias 1500 V Gate Window 0.5 Us Threshold 1.5mV Attenuation 3 dB Data set HBC_00029 Source background Start Time 8:35 AM Stop Time 12:43 PM Date 2014_09_01 x-y scope V/div 1, 0.5 Shielded? Yes Tube Harshaw B-

There are two hypothesis regarding the cause of the percolating peak.  One is that the attenuator has a time dependent flaw, the other is that the detector itself has a time dependent flaw, (perhaps running the tube for too long on maximum bias?)
The objective of the background spectrum was to make sure that a background peak could be distinguished from the Cs 137 peak by determining where each of them resides.
The first background peak resides at channel 113.  The second peak is at channel 289.  The complete peak map so far with the Dewar present and 3 dB of attenuation is
 Source Peak Channel Count Rate Cd109 am241 110 3.526 Background 113 1.177 Cs 137 121 3.77 Am241 221 7.2717 Background 289 3.210 Cd109

Yet another Cd 109 spectrum will be taken with the Dewar in place for comparison to the background count rate.

Pulsed Current Source Detector Testing
A five minute background spectrum was taken with the NaI detector located immediately next to the pulsing coil as shown below.  Arcing was occurring in the pulsed supply.  The exact location of the arcing has not been isolated yet.

No visually noticeable change in spectrum was observed.  The data will be compared with a spectrum without the pulsing supply firing.  It appears we’re in the clear with regard to x-rays from the switch firing.  The detector was sitting directly on top of the case holding the switch and no change in background was detected.
 Bias 1500 V Gate Window 0.5 Us Threshold 1.5mV Attenuation 3 dB Data set HBC_00030 Source background Start Time 2:10 PM Stop Time 2:15 PM Date 2014_09_01 x-y scope V/div 1, 0.5 Shielded? Yes Tube Harshaw B-

 Bias 1500 V Gate Window 0.5 Us Threshold 1.5mV Attenuation 3 dB Data set HBC_00031 Source background Start Time 2:22 PM Stop Time 2:28 PM Date 2014_09_01 x-y scope V/div 1, 0.5 Shielded? Yes Tube Harshaw B-

There may be two points at channels 147 and 220 that are statistically significant in the following.  Retaking a pulsed spectrum over a shorter window to see if these peaks remain.

 Bias 1400 V Gate Window 0.5 Us Threshold 1.5mV Attenuation 3 dB Data set HBC_00032 Source background Start Time 29.7 seconds Stop Time Date 2014_09_01 x-y scope V/div 1, 0.5 Shielded? Yes Tube Harshaw B-

 Bias 1400 V Gate Window 0.5 Us Threshold 1.5mV Attenuation 3 dB Data set HBC_00033 Source background Start Time 29.9 seconds Stop Time Date 2014_09_01 x-y scope V/div 1, 0.5 Shielded? Yes Tube Harshaw B-

To rule out random background, two more runs will be taken with shorter sampling windows around the pulsed supply firing.  One run will be taken with a pulse firing.  The other will be taken with background only.  The results are shown below.
 Bias 1400 V Gate Window 0.5 Us Threshold 1.5mV Attenuation 3 dB Data set HBC_00034 Source Pulsed once Start Time 5.7 seconds Stop Time Date 2014_09_01 x-y scope V/div 1, 0.5 Shielded? Yes Tube Harshaw B-

 Bias 1400 V Gate Window 0.5 Us Threshold 1.5mV Attenuation 3 dB Data set HBC_00035 Source Background Start Time 5.6 seconds Stop Time Date 2014_09_01 x-y scope V/div 1, 0.5 Shielded? Yes Tube Harshaw B-

Is the following significant?

It’s unclear whether or not the increased count in the last few channels above is significant.
Here are the spectra with all the available data entered

There’s an outlier at channel 220 which is where the other outlier at channel 224 was recorded.
I checked for a difference in the overflow channel of the detector.  Here’s the available data so far:
 Run Overflow Channel Adjusted to 5.7 second run Duration Poisson Uncertainty VeryShortPulse 256 256 5.7 16 VeryShortNoPulse 239 243.2678571 5.6 15.45962 Difference 17 12.73214286

The difference in the counts from the overflow channel for the two runs is within the Poisson uncertainty.

Percolating Peak
The high count peak in the low channels range is still somewhat of a mystery.  It is probably not due to the detector/PMT combination however.  During one run today, I forgot to attach the input cable from the PMT to the QVT and recorded the following spectrum:

The peak has the following observed properties:
1.  The peak only appears when attenuation is not equal to 0
2.  The peak does not appear immediately after the detector has been turned on, but only after a greater than one minute delay, (I'm working on characterizing the delay).
3.  The peak moves in channel number to the right as attenuation is increased.  This is independent of which attenuator switches are activated.

Plans
The pulse data isn’t significant enough to warrant more runs at this time.  The detector will be located further away from the pulser switch during the actual experiment.
To Do:  Take similar data with the actual experimental setup.
The data taking today did highlight the need for an automated data acquisition system.  The system should have the following features:
1.  Read all channel data into a file automatically.
2.  Activate and deactivate the QVT sampling based on an external signal.

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