### Beta Decay Parity Violation And A Few Fischbach Notes

And now, in the ongoing series inspired by the Fischbach article on the seasonal variation of radioactive decay rates, a few more notes on beta decays.  Today, will be about beta decay and parity, (aka mirror image symmetry), violation.  In 1957[5], C.N. Yang and T.D. Lee performed a survey of the existing experiments involving beta decay and found that there was no experimental evidence showing that beta decays maintained mirror symmetry, (parity), something that had been simply assumed until then because there was experimental evidence that electromagnetic and strong interactions did maintain mirror symmetry.  A short time after Yang and Lee's report, Wu[6] completed an experiment that showed that in fact, electrons emitted from a polarized Cobalt 60 source violated mirror image symmetry.  The article following Dr. Wu's by Garwin et al[7]. showed that muon decay, (another weak force interaction), also violated mirror image symmetry.

Vectors and Axial Vectors and What it all Means
To understand the two experiments in question, you just have to know a few new physics concepts: the difference between vectors and axial vectors and how they behave differently under reflection.  A vector describes quantities like position or momentum that have a both magnitude and a direction.  When a vector is mirror image reflected in space all its coordinates simply take negative values.  When two vectors are used to form a cross product resulting in a third vector, the third vector is a special type of vector called an axial vector.  The components of an axial vector won't change sign under reflection.  Because the normal vectors that went into the cross product both change sign when they are are reflected, the two sign changes result in an additional product of -1 times -1 which equals +1, see picture 1 [8].  Consequently, the components of the axial vector do not change sign under mirror image reflection.

What Wu et al. and Garwin et al. saw was that the electrons emitted from each of their experiments had a preferred direction of travel with respect to the spin of the Cobalt 60 nucleus in Wu's case, and with respect to the spin of the incident muon in Garwin's case.  Once that preferred direction was measured, the rest was history.  If the experiment were reflected in a mirror, the direction of the spin of the nucleus or muon would not change, but the momentum of the electron, (and hence its preferred direction of travel), would change making the mirror image imperfect.  This is shown in the figure adapted from Cottingham[8] shown below in picture 2.  The thick orange arrows labeled as cobalt 60 indicate the direction of the spin of the Cobalt nucleus.

Notes
Coming back again to how small the world of physics is, this morning as I was searching for my papers on Yang and Lee's study of parity violation in beta decays and Wu's subsequent experiment, I came across an earlier paper by Fischbach[2], who's recent paper on seasonal variations in beta decay led me to my recent foray into all things beta decay and neutrino, (now that my friends is a run-on sentence!).  In 1981, Fischbach, Freeman, and Cheng authored a paper that studied the quantum mechanical Hamiltonians of both normal and anti-matter hydrogen-like atoms suspended in a gravitational field.  They were looking for aspects of quantum gravity that could be tested experimentally.

So, why did this come up in my search for beta decay papers?  Fischbach and Freeman were actually being hosted by Yang at Stony Brook University when they wrote the paper.  In other interesting news, Freeman had gone on to work for Hughes Aircraft, either a sister or descendant company of Hughes Research, where Robert Forward[3][4] was working in the 60's when he wrote his papers on linearized general relativity and communicated with Bryce DeWitt regarding his superconductor as gravitomagnetism detector work.

References:
1.  Post-Newtonian gravity
http://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism

2.  Fischbach on hydrogenic gravity
http://dx.doi.org/10.1103%2FPhysRevD.23.2157
Fischbach E., Freeman B. & Cheng W.K. (1981). General-relativistic effects in hydrogenic systems, Physical Review D, 23 (10) 2157-2180. DOI:

3.  Forward on linearized gravity and other thigns
http://copaseticflow.blogspot.com/2013/05/gravitomagnetism-and-antigravity-for.html

4.  Test of space tethers as proposed by Forward
http://copaseticflow.blogspot.com/2013/05/a-little-news-two-approximations-and.html

5.  Lee and Yang
http://dx.doi.org/10.1103%2FPhysRev.104.254
Lee T. & Yang C. (1956). Question of Parity Conservation in Weak Interactions, Physical Review, 104 (1) 254-258. DOI:

6.  Wu's experiment
http://dx.doi.org/10.1103%2FPhysRev.105.1413
Wu C.S. (1957). Experimental Test of Parity Conservation in Beta Decay, Physical Review, 105 (4) 1413-1415. DOI:

7.  Garwin et al.'s experiment
http://dx.doi.org/10.1103%2FPhysRev.105.1415
Garwin R., Lederman L. & Weinrich M. (1957). Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: the Magnetic Moment of the Free Muon, Physical Review, 105 (4) 1415-1417. DOI:

8.  An Introduction to Nuclear Physics
http://dx.doi.org/10.1017%2FCBO9781139164405
Cottingham W.N. & Greenwood D.A. An Introduction to Nuclear Physics, DOI:

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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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 differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…