Weighted Variance

I had to review some basic statistics, so I figured I'd post a quick summary.  I tend to forget how to treat error when averaging, so this is a good reminder.

If we have a series of measurements  with variances , we might like to average them.  But, how should we weight them?

The standard way to weight such points is by inverse variance, which is the part I tend to forget:


We recall that a weighted average is:



The weighted variance of these points is much as you'd expect, but with a correction for the effective number of points.  That's the tricky part.



where the effective number of points is



Source: Lyons' Statistics for Nuclear and Particle Physicists

Gnuplot Fitting

If you aren't using gnuplot, you should check it out.  For those that do, I just figured out a fun tidbit on how to print fit errors that I couldn't find on the Intarwebs for the life of me.

The "fit" command in gnuplot works much like the "plot" command; you give it a data file and a "using" specification, and also the function you want to fit with and its parameters like so:

f(x) = a*x + b
a = -0.01
b = 0
fit f(x) 'somefile.txt' using 1:2:3 via a,b

You do need to give a first guess for parameters, as you see above.  The "using" specification here tells it to use the first column for x, second for y and the third for error in y (which you want to include if you can).  I've had plenty of trouble in the past getting a fit to converge, but generally you need to keep playing with your initial parameters and things will work out.  Once the fit is done you get something like this:

After 4 iterations the fit converged.
final sum of squares of residuals : 8.19305
rel. change during last iteration : -6.6513e-10


degrees of freedom    (FIT_NDF)                        : 9
rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.954117
variance of residuals (reduced chisquare) = WSSR/ndf   : 0.910339


Final set of parameters            Asymptotic Standard Error
=======================            ==========================


a               = -0.00123217      +/- 0.0002571    (20.87%)
b               = 0.0347759        +/- 0.001622     (4.665%)


correlation matrix of the fit parameters:
               a      b      
a               1.000 
b              -0.872  1.000 

Huzzah!  You get your final fit parameters with error; you get degrees of freedom and stdfit and chisquare.  This is fun and all, but how to I put that in my plot?  First, you can get at some environment variables to use in your gnuplot script, like "FIT_WSSR" and "FIT_STDFIT."  As far as the final parameters and errors, here's the best way I've found to put them on your plot:

set fit errorvariables

fit f(x) 'group_asym_out.txt' index 1 u 1:2:3 via a,b 

set label sprintf("a = %2.6f +- %2.6f",a,a_err) at graph 0.7,0.15

set label sprintf("b = %2.6f +- %2.6f",b,b_err) at graph 0.7,0.1

plot 'somefile.txt' u 1:2:3 w yerr, f(x) title 'f(x)=a*x+b'

Here I've put the tricky part in bold.  We have to set labels on the graph somewhere; mercifully we can use the "graph" coordinates instead of the coordinates of the plot itself, so that top right is graph (1,1) and bottom left is graph (0,0).  Now we access our parameters "a" and "b" using "sprintf" in the label.

To get the errors on these parameters, we need to include "set fit errorvariables" which enables the error variables "_err," for example "a_err" and "b_err."  The graph still needs some stylistic attention, but at this point it looks this:

Sources: Janert's Gnuplot in Action

Čerenkov Radiation Application

Last week I gave a quick run-down of the justification of Cherenkov radiation from classical electrodynamics.  In practice, this effect is the centerpiece of many experiments, and I'll run through how and what we need to know in this post.

We recall the velocity of the charge particle must exceed the phase velocity of the E&M field to emit Cherenkov radiation, or with index of refraction n:



Cherenkov radiation comes in the form of an electromagnetic shock wave, a conical wavefront formed following the particle, emitted at angle:



Since we'd like to pick up this radiation in photomultiplier tubes to observe the effect and therewith the incident charge particle, it's good to know how many photons to expect.  Following the math of the last post on Cherenkov radiation (or just following along in Jackson), we take the energy differential over frequency and divide by de Broglie's h bar omega and then by L we get the number of photons per unit length:



Cherenkov radiation contributes to the mechanism of several types of particle detectors, including electromagnetic calorimeters and non-scintillating hodoscopes; there are two types of detectors specifically designed to directly take advantage of the Cherenkov radition emitted by high energy charge particles: threshold Cherenkov detectors and differential Cherenkov detectors.

In the threshold Cherenkov detector, the medium in the tank is chosen carefully for a refractive index that indicates the passage of charge particles that exceed a given velocity threshold.  Usually this is done with a gas such as hydrogen, nitrogen or carbon dioxide, and further control of the refractive index comes through the pressure of the gas in the tank.

The differential Cherenkov detector allows the measurement of a particle's velocity while rejecting particles outside a given mass range.  This is accomplished by accepting a small annulus around the track of incident particles at some angle theta.  This corresponds, at a given refractive index, to a velocity resolution:



The minimum velocity resolution is often constrained by the minimum angular resolution, which tends to be limited by dispersion in the slit.

Sources:  Leo's Techniques and Fernow's Intro to Experimental Particle Physics

Čerenkov Radiation

This phenomena, aside from providing the eerie lighting in this picture from Idaho National Lab, is a central tool for many particle physics experiments.  Actually, it's so important, it won Chernkov the Noble prize in 1958.  We recall that it's analogous to a sonic boom, but for light, and we can explain it in purely in terms of classical electrodynamics.

The derivation is in full in Jackson, but I'll sketch it here.  We consider a charged particle moving through a medium, where all the distant interactions of the particle with the atoms in the material are represented by a macroscopic dielectric constant .   Taking the Fourier transforms of the E&M wave equations, we get the E and B fields as follows:

Then, after integrating in k and recognizing the modified Bessel function, we get three field components in terms of  , in each direction, where here 1 is parallel to the particle's velocity.  I'll skip straight to the limit where    (a and b are impact parameters, and lambda is  ), which gives:

We integrate over frequency to get energy over distance , which has Jackson explains is done rather elegantly by finding the energy flow through a cylinder radius a.  After this integration and applying our limit, we get an expression which is multiplied by:

If lambda has a positive real part, this vanishes at large distance as the energy is deposited near the path.  If lambda is purely imaginary, the exponential is unity and the expression is independent of scattering distance a, so some of the energy escapes to infinity as radiation!  When is lambda imaginary?  When epsilon is real (there is little absorption) and  

 .  

Or, the speed of the particle is greater than the phase velocity of the E&M fields in the material.  That's it!  

We've now seen that with the right circumstances, namely the speed of the particle is greater than the phase velocity in the material, some of the energy of the particle escapes as "Cherenkov" radiation.  You can go on to calculate the angle of emission and the photon yields, but I'll leave that to another post in which I'll describe the practical applications of this effect.

Source: Jackson's Classical Electrodynamics

Stern-Gerlach

Everyone remembers the Stern-Gerlach experiment; we usually recall it in the context of the discovery of quantized intrinsic spin.  However, I like very much the treatment of Sakurai in his Quantum Mechanics text, which uses Stern-Gerlach as the introduction to shock us into a quantum mechanical way of thinking.

The setup is familiar: Heat up silver atoms which escape through a hole in the oven.  Collimate the beam and send it though an inhomogeneous magnetic field; the force in the z direction on the atom is .  Since the atoms in the oven were randomly oriented, there's no preferred orientation of , so classically we expect to see a vertical splay of particles coming out of the magnetic field.  Of course, we instead see two distinct components, scattered up and down, which leads us to the idea of quantized spin angular momentum.

The fun part begins with sequential Stern-Gerlach setups, adding the ability to rotate the subsequent apparatus on its side to act in the x direction instead of the z. The picture below, stolen from Wikipedia which obviously it stole from Sakurai, shows the results.

The first result makes sense; we removed the minus z portion of the beam, so it doesn't recur.  The second, in which the plus z portion is now split 50-50 into plus and minus x portions, is interesting; maybe 50% of the split beam was z plus and x plus while the other 50% was z plus and x minus?  The third result is the doozy.  We get z minus out at the end, but didn't we remove it in the first splitting?  Apparently the x measurement in the middle destroys our previous information on the z direction of the spin!  We can't know both the spin in z and in x simultaneously; this is obviously not a spinning top.

Sakurai then calms our confusion with an analogy to polarized light, with the x and z spin directions above corresponding to zero and 45 degree polarized light.  In that context, this result would be right at home.  In E&M we write the 45 degree polarization in terms of a linear combination of the 0 degree coordinate vectors, x and y.  For spin, we must turn our thinking to an abstract spin space, where the base vectors Sz+ and Sz- take the place of base vectors x and y in physical space. The Sx spin directions can then be swapped into the idea of the 45 degree polarized light, so should be expressed in terms of our Sz+ and Sz- base vectors:



The missing piece is Sy, the last spin direction, which we correspond to circularly polarized light in this analogy.  Circularly polarized light is expressed as the same combination of base x and y vectors, but with the y portion 45 degrees out of phase.  This now brings imaginary numbers easily into play, as we express a light wave in exponential notation instead of cosines and pull out .  In our corresponding spin vector space we have now been lead to:



So, in a few paragraphs we've seen not only the weirdness of quantum mechanical phenomena, but we've constructed a complex vector space directly out of our observations in Stern-Gerlach and an analogy with our understanding of polarized light.

Source: Sakurai's Modern Quantum Mechanics.

Asymptotic Freedom

This is a scary name for a relatively simple, but crucial effect that demonstrates part of the vast difference between the QED behind electromagnetic phenomena and the QCD behind the strong phenomena of the nucleon.

In QED, we learn that the sum total of all Feynman diagrams that can describe a process contribute to the end result.  This means that all the crazy diagrams you can draw with the same external lines, no matter how many vertices creating and destroying any number of virtual particles, must go into the calculation of the process.  The saving grace is the fine structure constant , which gives the coupling strength of the vertex, is .  This means that each additional vertex of higher and higher order terms contributes less and less to the final result by orders of magnitude, and can be ignored.

Things get more complicated when we apply this method to QCD.  The coupling constant here  is determined by experiment to be greater than 1.  This means each successive order contributes more and more!  Now the solution for calculability instead comes from the dependence of the coupling on distance, the "reach" of the strong force, which is mediated by a massless gauge boson that carries charge and can thus couple to itself.

The key to understanding this puzzle lies within charge screening.  In E&M, a charge q in a dielectric medium is screened by the presence of molecular dipoles whose negative end is attracted towards the positive charge, effectively lowering the charge.  The closer to the charge, the fewer molecules in the way to decrease the observed charge strength.  It turns out the vacuum acts as a dielectric as it sprouts particle/anti-particle pairs, so this always applies.  For example the effective electron charge we know and love is really lower than the value very close to an "unscreened" electron.

In QCD, the vacuum sprouts not only particle/anti-particle pairs as in QED, but also gluon loops, as the gluon can couple to itself; it has "charge."  Thus a red charged quark begets more red charged particles in its neighborhood, which creates an anti-screening.  The further away, the more particles, the larger the charge looks!  Here we have asymptotic freedom: the QCD coupling increases at distance, but decreases close together.  Quarks are free to rattle around when near each other, but can't stray too far.

Sources: Griffiths Intro to Elementary Particles p64 and Halzen and Martin's Quarks & Leptons p10

Introduction

Welcome to our Physics topic review blog!  Jonathan, Stache and I created this blog to be a centralized and easy to access place to collect summaries of review materials.  As a Physics student, whether preparing to write a thesis or not, periodic review of our field is crucial.  As we read through papers and textbooks keeping good notes strengthens the understanding and reinforces memory.  We decided a common place to share our review notes would be an excellent resource for others in the group.

Thus, our goal is to provide regular updates on our Physics readings, here shared for our fellow graduate students as a motivation for review and further learning in our field for ourselves and others.  We hope more graduate students will join us and share their review summaries.  No area of Physics will be too simple or too complicated; the simplest of topics is still useful to recall and obtuse posts will challenge and lead to deeper reading.  The comments could be an excellent place for others to correct or ask questions.

For contributors:  
We'd like this blog to provide interesting and easy to digest snippets for those following the RSS feed.  To this end, we'll try to keep these entries short; just a handful of paragraphs should be enough to inform and jog the memory.  Links and cites to your source are crucial to allow these little bite sized entries to lead to deeper reading for those interested.

To keep things well organized, tags or "labels" as they are called in blogger, are key.  Required tags for each post should be your last name, and the general subject area; more specific keys are encouraged at your discretion.

This handy LaTeX equation maker will no doubt be useful to you all in your posts:

http://www.codecogs.com/latex/eqneditor.php

It should be self explanatory, and it is easy enough to get the original LaTeX back out (mouse over).