Monday, April 21, 2014

Neutrons Used to Probe Dark Matter

From [1]
I'm reading up on a recent expeirment that used a neutron/gravity spectrometer to look for evidence of dark matter and dark energy.  I haven't got to review enough material to say something truly pithy here yet, but I thought I'd point you towards the stuff that's available.  First, here's a discussion of the experiment[1] from Texas A&M's own Dr. Schleich.  He's talked about this kind of thing before, (using neutrons for experiments involving gravity).  In fact, a little more than a year ago he gave a talk here on the KC interferometer and how it measured the acceleration due to gravity as opposed to the gravitational redshift as claimed by the authors[2].  His summary of the current set of experiments comes with the added bonus of a pointer to the open access version of the Physics Review Letters article on the experiment[3].  If you'd like to know how exactly you reflect a neutron from a wall without using coulombic forces which they don't particularly respond to anyway, (the answer has to do with Bragg diffraction), be sure to check out this article on an early test of a neutron interferometer.  Sadly, it's not open access, but again with the advice about local university libraries.






Wednesday, April 16, 2014

Parabolic Range/Height Elliptical Envelope Results of the Day

We're still working on figuring out a way to geometrically, or verbally, explain the fact that an ellipse is formed by by tracing through the apexes of a family of parabolas that describe the trajectories of a projectile launched with the same initial velocity but different launch angles.  Try saying that three times fast, and in the meantime, see picture 1, and refer to the excellent open access article that derives the ellipse mathematically[1].

After I wrote about the elliptical envelope a few days ago +rocktoasted asked if there was a way to describe the construction of the ellipse without using equations or algebra.  Hence, the search for a geometrical explanation was born.  +Bruce Elliott joined in on the work, and now we have a new collaborator, the only already PhD'ed physicist in the house, my wife Elaine.  If you'd like to get in on the collaboration by contributing observations that lead to explaining how to construct the ellipse in [1], please do! Maybe in the end we can right the whole thing up and submit it to one of the +Mathematical Association of America's journals.  It might just be the first ad-hoc internet math collaboration to publish there.  Who knows??? But, I digress :)

We played around with the diagram of how to make a parabola from the intersection of a plane and set of cones, and came up with the following observations.

Elaine intuited that the base circle shown in picture 2 was actually traced out by the line representing the ground for each parabolic trajectory.

Using the equations from the paper [1], we saw two mathy things immediately.  First, if you put 45 degrees, (the angle that gives the maximum range), into the x equation, and 90 degrees, (the angle that gives the maximum height) into the y equation, you'll find that the maximum range is always twice the maximum possible height of the projectile at a given fixed velocity.  Bruce saw this first and mentioned it in respect to the eccentricity of the ellipse in the paper always being equal to one half.

The second thing we found requires a bit of calculus.  The x range is equal to half of the derivative with respect to theta of the apex height.

I'm not entirely sure what the significance of this yet, but it feels kind of nice from a circular perspective.  The x coordinate of a point on a circle is equal to the derivative of the y coordinate of a circle, (picture 4).  Notice that the patch factor for the ellipse is just one over the eccentricity, (the ratio of the maximum y distance vs. the maximum x distance of the ellipse).

OK, so so far, we have a family of parabolas formed by laying a plane parallel and tangent to a cone and then pushing that plane through the cone and assuming that the ground is perpendicular to the line that drops straight down from the apex of the cone.  Here are the geometry-ish statements that go with this system.

1.  When the plane is just tangent to the cone, the parabola that's formed is actually a straight line, (line CK in the diagrams) and represents the trajectory of the projectile if you launched it straight up at a launch angle of 90 degrees.

2.  The height of the line CK is proportional to the kinetic energy of the particle at launch, or the potential energy of the particle at the apex, (they're the same for the 90 degree launch angle represented by line CK).

3.  As the launch angle sweeps from 90 degrees at CK, to 0 degrees, (the parabola has dwindled to a single point lying at B), the angle around the ground circle centered at M sweeps from 180 degrees at a launch angle of 90 degrees, (line MC) to 0 degrees at a launch angle of 0 degrees, (line MB).  When the ground line of the parabola, (line ED) lies along to the diameter of the ground circle M, the projectile has the maximum range.  This corresponds to a launch angle of 45 degrees and an angle swept through the ground circle of 90 degrees, (see picture 5).

This feels nice and intuitive, and the three mentioned points certainly match up, but if you'd like a proof of the relations between the launch angle and the angle swept out on the ground circle, see picture 6.

There's one last thing about the distance along the diameter of the ground circle, (line BC), how it's related to the angle theta in the picture above, and how that's related to the maximum height of the projectile in each parabola, but I'll save that for later.

1.  Open Access paper on the elliptical envelope of parabolic apexes.

2.  Original post on article

2.a. permalink to this post

3.  post on Da Vinci's Parabolic Compass

Tuesday, April 15, 2014

Superconductors, the London Moment, and Spin Currents

In addition to the Meissner effect pictured to the left, superconductors have other odd properties.  One of them is something known as the London moment.  It's named after the London brothers who did some of the earliest work on the properties of superconductors.  If I had to take a guess, I'd say it was specifically named after  Fritz London who authored the two volume set of books, "Superfluids"[1].  So much for the naming of the thing, here's what happens.  When a superconductor is rotated, it produces a magnetic field aligned with the axis of rotation.  The magnetic field is linearly related to the angular velocity of the superconductor by the following equation, (picture 2)[2].

where m_e is the mass of the electron, c is the speed of light, and e is the charge of an electron.

London theorized that the  magnetic field which bears his name was a result of the superconducting cooper pairs lagging behind the initial rotation of the superconductor bulk.  Thereafter, they never quite catch up and there's a relative current between the superconducting electrons and the body of the superconductor.  It's this current he reasoned that is responsible for the magnetic field.

Dr. Jorge Hirsch of UCSD has posited a different mechanism for the production of the London moment in his paper titled "Spin currents in superconductors[2]".  Hirsch theorizes that the current responsible for the London  moment is not due to the lag of superconducting electrons spinning up and trying to catch  up with the rest of the superconductor, but instead is due to counterposed currents of electrons whose spins are opposite to one another travelling around the surface of the superconductor.  The currents come in pairs.  A current of spin up electrons moving in the clockwise direction around a superconductor is always accompanied by a current of spin down electrons moving in the opposite direction.  In this way, in equilibrium, there is no detectable charge or spin currents since each pair of counter-spinning counter-traversing currents effectively cancels each other out.  According to Hirsch, these hypothetical counterposed spin currents are setup in the superconducting material when it enters the superconducting state.  A diagram of two pairs of the spin currents in a spherical superconductor is shown in picture 3[2].

Hirsch presents a very nice macroscopic explanation of how the spin currents create the London moment currents when the superconductor is rotated.  At the most basic level, the London moment current due to the spin currents compensates for the extra centripetal force on the superconducting electrons.  I'll have to leave that for another time though.

1.  London's book on superconductors

2.  Hirsch's paper on superconductors and spin currents (open access)

Sunday, April 13, 2014

Projectile Motion: Pushing the Envelope

Think everything that's publishable for say an old classical topic like projectile motions has already been published?  Turns out the old 'lob the projectile at a constant velocity in a constant gravitational field' problem is still producing.  Check out this paper from J. L. Fernandez-Chapou, A. L. Salas-Brito, and C. A. Vargas published in 2004.  It eventually made its way into the American Journal of Physics.  In the paper, the authors show that if you write down the trajectory of a projectile in terms of its launch angle and then solve for the x and y position when the projectile has reached it's maximum height, the solutions will trace out a nice little ellipse like the figure below excerpted from the arxiv version.

1.  Elliptic envelope of parabolic trajectories paper

1.a.  AJP version of the paper

Friday, April 11, 2014

Adventures in Hosing

We're trying a new type of roughing pump vacuum hosing.  We're moving away from the traditional classic, red rubber tubing and trying out Kuritech K7130 Polywire Hose!  The kids and I went on an adventure a week and a half ago to check the hosing out at Bryan Hose and Gasket!  Everyone was super nice, and it looked like if you turned up later in the day you could have free popcorn.  It was Sam's first science trip.  Jr. used to go on these all the time when were at Brookhaven National Laboratory.  The best part, besides the obvious of getting to hang out in a hose and gasket store?  On the way out, Sam and Jr. saw their first real bulldozer, (see the right hand side of the building in the picture).  After weeks of seeing them in Mater cartoons, they thought it was awesome!

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