CALCULATIONS: Pion Decay

Pion Decay into Muon; Muon into Electron

Ricardo L. Carezani

Commentary

An interesting comment was made from a student at CERN about this kind of decay:

"2) We measure the momentum of the particles using the standart magnetic field technique. One interesting fact about the above decay is that the absolute value of the outgoing muon momentum differs (as well as direction as pointed in 1) from that of the incoming kaon. Not only that, but interestingly enough, it is always < the original kaon's momentum."

As you can see in this example, the muon's momentum of 95.101 *is less than* the pion's momentum equal to 134.164 (in MeV). Adding these three momenta, AD can calculate the incoming Pion energy value when it enters the device.

I keep asking for experimental values to perform AD calculations in this manner. I want to calculate what NATURE does.

Basic Listing

10 REM A Pion decays into a Muon and this into an Electron at 90 degees,
20 REM supposing that they lose energy by ionization
30 REM and are trapped by a nucleus.
40 PRINT "PION DECAY TO MUON"
50 U=931 :REM Mass unit in MeV.
60 MP=140 :REM Pion rest mass in MeV.
70 MM=106 :REM Muon rest mass in MeV.
80 ME=.511 :REM Electron rest mass in MeV.
90 MH2O=18*U :REM  Mass of water molecule in MeV.
100 DPM=MP-MM :REM Pion-Muon rest mass energy in MeV.
110 ECM=33.85746 :REM Muon Kinetic Energy in MeV.
120 BETAM#=SQR(1-(1-(ECM/MP))^2) :REM Beta Muon.
130 PRINT "Betam=";BETAM#;" Beta Muon."
140 PM#=MM*BETAM# :REM Muon momentum in MeV/c.
150 PRINT "pm=";PM# ;" Muon momentum in Mev/c."
160 REM H2O (water molecule) momentum equal to Muon momentum.
170 VH2O#=PM#/MH2O :REM H2O velocity.
180 PRINT "vH2O=";VH2O#;" Velocity of water molecule."
190 KE#=((MH2O*(VH2O#^2))/2) :REM  Kinetic Energy of water molecule in MeV.  
200 PRINT "KE=";KE#;" Kinetic Energy of water molecule in MeV."
210 KEM#=MP*(1-SQR(1-BETAM#^2)) :REM Muon KE test.
220 PRINT "KEM=";KEM#;" Muon Kinetic Energy test."
230 TEST#=ECM+KE# :REM Muon plus H2O KE equal total energy(34)
240 PRINT "Test=";TEST#;" Muon plus H2O KE, equal to 34 MeV."
250 PRINT
260 PRINT "MUON DECAY TO ELECTRON"
270 ECE=105.481 :REM Electron Kinetic Energy.
280 BETAE#=SQR(1-(1-(ECE/MM))^2) :REM Beta electron.
290 PRINT "Betae=";BETAE#;" Beta Electron."
300 PE#=ME*BETAE# :REM Electron momentum.
310 PRINT "pe=";PE#;" Electron momentum in MeV/c."
320 VW#=PE#/MH2O :REM Water molecule velocity.
330 PRINT "Vw=";VW#;" Velocity of water molecule."
340 KEW#=(MH2O*(VW#^2))/2 :REM KE of water molecule.
350 PRINT "KEw=";KEW#;" KE of water molecule in MeV."
360 PRINT
370 PRINT "The neutral water molecule will not produce a track, for two"
380 PRINT "reasons: it can't produce ionization since it is electrically"
390 PRINT "neutral and its KE is small. Probably its recoiling energy"
400 PRINT "will be distributed to many molecules."
410 REM Here we suppose that the PION and MUON decay when in motion.
420 PRINT
430 PRINT "              PION AND MUON DECAY IN MOTION"
440 PRINT "PION DECAY TO MUON"
450 REM We suppose that the residual Pion KE is KEPM=100 MeV.
460 KEPM=100 :REM Pion KE in MeV when the Muon decay happens.
470 PRINT "KEPM=";KEPM;" Pion KE in MeV when Muon decay happens." 
480 BP#=SQR(1-(1-(KEPM/MP))^2) :REM Beta Pion.
490 PRINT "Bp=";BP#;" Beta Pion."
510 PP#=MP*BP# :REM Pion momentum in MeV/c.
520 PRINT "pP=";PP#;" Pion momentum in MeV/c."
530 TKEMM=KEPM+MP :REM Total energy of Pion rest mass plus its KE in MeV.
540 PRINT "TKEMM=";TKEMM;" Total Energy: Pion rest mass plus its KE in MeV."
550 KEMM=TKEMM-MM :REM Muon KE in MeV.
560 PRINT "KEMM=";KEMM;" Muon KE in MeV."
570 BETAMM#=SQR(1-(1-(KEMM/TKEMM))^2) :REM Beta Muon.
580 PRINT "BetaMM=";BETAMM#;" Beta Muon."
590 PMM#=MM*BETAMM# :REM Muon momentum.
600 PRINT "pmm=";PMM#;" Muon momentum in MeV/c."
610 KEPMM#=TKEMM*(1-SQR(1-(BETAMM#^2))) :REM Muon KE in MeV. (TEST).
620 PRINT "KEPMM=";KEPMM#;" Muon KE in MeV. (TEST)."
630 PRINT
640 PRINT "MUON DECAY TO ELECTRON"
650 :REM We suppose that the Muon loses 114 MeV in the ionization path.
660 ML=114 :REM Energy lost by Muon.
670 PRINT "ML=";ML;" Energy lost by Muon."
680 MRE=KEMM-ML :REM Muon residual energy in MeV.
690 PRINT "MRE=";MRE;" Muon residual energy."
700 S=MM+MRE :REM Muon rest mass plus KE in MeV.
710 PRINT "S=";S;" Muon total energy."
720 E=S-.511 :REM Electron KE in MeV.
730 PRINT "E=";E;" Electron KE."
740 BE#=SQR(1-(1-(E/S))^2) :REM Beta electron.
750 PRINT "Be=";BE#;" Beta electron."
760 PRINT
770 PRINT "In this case, the Muon and Electron vector momentum"
780 PRINT "will be in the same direction as the Pion and Muon"
790 PRINT "vector momentum, respectively (Probabilistically)."

PION DECAY TO MUON
Betam= .6520674 Beta muon.
pm= 69.1191406 Muon momentum in MeV/c.
vH2O= 0.0041245 Velocity of Water molecule.
KE= 0.1425425 Kinetic Energy of water molecule in MeV.
KEM= 33.8574600 Muon Kinetic Energy test.
Test= 34.0000038 Muon plus H2O KE, equal to 34 MeV.

MUON DECAY TO ELECTRON.
Betae= 0.9999880 Beta Electron.
pe= 0.5109938 Electron momentum in MeV/c.
Vw= 0.0000305 Velocity of water molecule in MeV.
KEw= 0.0000078 KE of water molecule in MeV.

              PION AND MUON DECAY IN MOTION
PION DECAY TO MUON
KEPM= 100 Pion KE when Muon decay happens.
Bp= 0.958315 Beta Pion.
pP= 134.1640930 Pion momentum in MeV/c.
TKEMM= 240 Total Energy: Pion rest mass plus its KE in MeV.
EMM= 134 Muon KE in MeV.
BetaMM=.8971792 Beta Muon.
pmm= 95.1009979 Muon momentum in MeV/c.
KEPMM= 134.0000153 Muon KE in MeV.(TEST).

MUON DECAY TO ELECTRON
ML= 114 Energy lose by Muon.
MRE= 20 Muon residual energy.
S= 126 Muon total energy.
E= 125.4889984 Electron KE.
Be= 0.9999918 Beta electron.

In this case the Muon and Electron vector momentum will be in the same
direction as the Pion and Muon vector momentum, respectively.
(Probabilistically).