Friday, May 16, 2014

Lab Book 2014_05_15 More Leak Detecting and Relativistic Projectile Range Comparisons

Lab Book 2014_05_15     Hamilton Carter

Summary
Almost the entire day was spent finally actually using the leak detector!  The new stopcock was attached to the glass helium Dewar early this morning.  After that, I attended a theory meeting.  After cleaning and vacuum greasing a few fittings we found out that the glass liquid helium Dewar is leak tight!!!  The next step, glass Dewar-wise, will be to modify the table that it sits in so that it can be placed between the poles of the electromagnet.  

I did some more thinking about the relativistic trajectory problem and found some possilbe symmetries in Shahin's expression for the y vs. x.  Both the expressions for hang time and the maximum range equations are interesting. Hang time is interesting because it's actually the same for the relativistic and classical cases.  Range is interesting because the expression involves the vertical component of the projectiles rapidity.


Leak Detecting Procedure
First, attach the system to be evacuated and evacuated to the top of the T-connector.  This allows the system to be vacuumed first by the secondary roughing pump and later by the leak detector’s built-in diffusion pump. 


Make sure that the leak detector to system valve is closed fully in the clockwise direction isolating the system to be detected from the leak detector’s diffusion pump.


Turn on the leak detector to begin roughing out the diffusion pump.  When the ‘< 10^-2’ indicator light turns on, begin pouring liquid nitrogen into the trap between the diffusion pump and the T joint.  Continue to pour in liquid nitrogen until the trap overflows.  One trap filling should last for up to 10 hours.  Open the gold/yellow auxiliary pump valve by turning it clockwise until it stops.  This will connect the pump to the vessel to be leak detected.


Once the trap is filled, wait 20 to 40 minutes for the roughing and diffusion pump system to completely outgas.  After this, push the filament switch directly below the green indicator lights up.  The filament active light, (the middle light), should light up.  The filament outgasses when it is first turned on causing the vacuum pressure to rise.  You will probably see the vacuum gauge on the right hand side of the control box begin to rise towards the red region.  If the needle makes it into the red region, push the filament switch down to avoid damaging the filament.  The control box should, however, automatically turn off the filament if the gauge reaches the red region.  If necessary, wait another five minutes and then repeat the filament turn on procedure.

Make sure the vacuum gauge for the system to be leaked detected reads below 10^-1 Torr. 


When it does, and when the filament can be left on stably, then slowly begin to open the lead detector to system valve in the counterclockwise direction.  There is a lot of play in the valve, so you will be able to turn it quite a bit without having too large of an effect.  Once again, keep an eye on the vacuum gauge on the right hand side of the control box.  If the needle approaches the red region, begin to close the system valve until the needle once again moves in the green direction.  Wait for the needle to stabilize and then slowly begin to further open the system valve.  Repeat this process until the system valve is completely open.  Once this is done, you can try closing the yellow/gold auxiliary pump valve.  This will usually improve the overall vacuum as the exhaust gasses from the roughing pump are now isolated from the system.

You’re now ready to begin leak detecting with helium.  Point a very low helium flow from a small tube, at various joints in the system, (beginning with the leak detector itself), and listen for the audible leak detection tone to increase in frequency.  If it does increase in frequency, you have found a leak.
To repair a leak, first tighten the leak detector to system valve all the way clockwise.  Then, loosen the small black vacuum release valve located on the side of the copper filter pie entering the T junction.  This will release the vacuum from the system under test.  Perform the necessary repairs, and then repeat the steps listed above from the auxiliary evacuation of the system onwards.  Keep in mind that the filament is still on, so it does not need to be outgassed again.  Make sure however that the system to be leak detected is down to 10^-1 Torr at least before beginning to re-open the leak detector to system valve.

If the system is leak tight, then you can move the sensitivity knob to the next lowest setting as desired.  For Dewars, a leak tight system at the 3*10^-8 setting is sufficient.

The following picture shows the bracing for the vacuum hose attached to the Dewar.  The bracing turned out to be very necessary.  The metal flex hose contracts when a vacuum is pulled and expands when it is lost.  without the bracing shown, (the bracket on the rubber hose), the ball joint would have been put under undo stress.


Theory Work:  Relativistic Projectiles
The diagram in McAllen elicited a few more thoughts today.  There’s an EJP article with a more useful form of MacColl’s relativistic range expression which is:


The first few terms amount to constants and can be left out of the expression since they will fall away when the derivative is taken to determine the launch angle that maximizes range.  We can then rewrite the expression as:



Compare this to the classical expression


and you begin to wonder if the arctanh term can be linked to the vertical component of velocity in the classical expression.  In fact, it can!  The arctanh term is the expression for the rapidity of the vertical component of the initial velocity.  This begins to beg the question: Can certain classical problems be solved in relativistic frameworks by substituting in the expression for rapidity for velocity?  I need to take a longer look at the expression for the maximum height.
For maximum the height, there is one indication that the vertical component of velocity just doesn’t play in the game.  MacColl points out that the time of flight for both the relativistic and classical particles is the same for the same absolute value of initial momentum and angle of fire, and is



 Keep in mind that MacColl’s mu, is velocity as measured in the lab frame times the relativistic gamma factor also known as the cosh of rapidity.  Also keep in mind that he uses gamma to indicate acceleration, or g, not the usual relativistic gamma factor.

One last qeustion: why are Shahin and MacColl's formulas for the trajectory different?  Is it just notational, or are they actually inequivalent?

Shahin:


Notice the possible symmetry in Shahin's case.  The second expression contains the horizontal component of momentum in the cosh along with the horizontal component of work.  It's scaled by energy over force, a notation which is feels like the distance the work was done over.  The third component contains the orthogonal projection of sinh over the same angle of work divided by horizontal momentum.  This term is scaled by the horizontal component of momentum divided by force however.  Finally, remember that E_o, and p_o c are the horizontal and vertical components of the momentum four vector and that the cosh of rapidity and the sinh of rapidity are the corresponding components of the four velocity vector.

MacColl:


One last note:  Look at the cosine like law in Shahin's velocity and acceleration solutions.

To do:  derive the above equation to determine why the gamma changing as the altitude changes doesn't matter.

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