# Muon Lifetime and time dilation effect ```MUON LIFETIME AND TIME
DILATION EFFECT
Measuring the stopping rate of muons, as a function of depth in the
atmosphere to demonstrate the time dilation effect of special
relativity.
Melinda Jolley
THE MUON WAS SO UNEXPECTED THAT, REGARDING
ITS DISCOVERY, NOBEL LAUREATE ISIDOR ISAAC RABI
FAMOUSLY QUIPPED, “WHO ORDERED THAT?”
Where do muons come
from?
•
Not completely sure
•
Somewhere in deep space?
•
Very recent NASA research
Muons were originally
thought to be meson
It’s name is a combination of mu
and meson
•
•
Femi Coupling Constant (𝐺𝑓 )
The Muon
•
•
Nope, it is a Lepton
Nicknamed ‘The Heavy Electron’
•
𝜇&plusmn; →e&plusmn; +2 ν
THE GOALS OF MY EXPERIMENT
Determine the
the Muon
Muon Physics
•
•
•
•
Moun.exe
𝑁 𝑡 = 𝑁0 𝑒
•
Use the measured τ
•
𝑡
𝜏
−
Accepted τ = 2.20 &micro;s
Is there no Time
Dilation?
The Fermi Coupling
Constant
𝐺𝑓 =
•
ℏ 192𝜋3
𝜏𝜇 𝑚5
•
•
Rates of Muon decays at one
elevation
Approximation for rate at second
elevation
Correction factors for Energy Loss
over distance and Variations in the
shape of the muon energy
spectrum
CORRECTING FOR
ENERGY LOSS
•
•
∆𝐸 = 𝐶𝑜 ∗ ∆𝐻 ∗ 𝜌𝑎𝑖𝑟
𝜌𝑎𝑖𝑟
•
𝑎𝑣𝑒
=
1 ℎ2
𝜌
∆𝐻 ℎ1 0
𝑎𝑣𝑒
&times;𝑒
𝜌0 = 1.28 &times; 10−3
−ℎ
8400 𝑚
𝑔
.
𝑐𝑚3
THE MUON
ENERGY
SPECTRUM
•
𝑅𝑟𝑎𝑤 =
𝑆𝑡𝑜𝑝𝑝𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 𝑎𝑡 ℎ1
𝑆𝑡𝑜𝑝𝑝𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 𝑎𝑡 ℎ2
•
𝑅(𝑖𝑔𝑛𝑜𝑟𝑒 𝑠ℎ𝑎𝑝𝑒) = 𝑒
•
𝑡′ =
𝑚𝑐
𝜌𝑎𝑖𝑟(𝑎𝑣𝑒) 𝐶𝑜
•
•
𝑅𝑜 =
−𝑡′
𝜏
𝛾2
𝛾1
𝛾2 −1
𝛾1 =
𝐸1
𝑚𝑐 2
𝑅𝑟𝑎𝑤
𝑅(𝑖𝑔𝑛𝑜𝑟𝑒 𝑠ℎ𝑎𝑝𝑒)
𝑑𝛾
THE PREDICTIONS
• 𝑅𝑃𝑁𝑇𝐷 = 𝑅𝑜 ∗
𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑠𝑜𝑚𝑒 𝑜𝑡ℎ𝑒𝑟 ℎ (𝑛𝑜 𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛)
• =𝑅0 &times; 𝑒
−𝑡
𝜏
• 𝑅𝑃𝑇𝐷 = 𝑅𝑜 ∗
𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑠𝑜𝑚𝑒 𝑜𝑡ℎ𝑒𝑟 ℎ (𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛)
• =𝑅0 &times; 𝑒
(
−𝑡′
)
𝜏
THE EQUIPMENT
USED
•
Scintillator is placed at the bottom of
a black anodized aluminum alloy
tube
•
Transparent Material
HOW THE DATA LOOKS
MY DATA
• High Voltage set to -1154 Volts
• Threshold Voltage set to 206 MeV
• First location:
Pueblo, CO, Elevation 1420 m
• Second location:
Monarch Mountain Base Summit, CO, Elevation 3290 m
• ΔH= 1870 m
• Much more Pueblo Data than Monarch Data
FERMI COUPLING
CONSTANT
•
•
Mass of the Muon = 106
𝑀𝑒𝑉
𝑐2
Reduced Planck’s Constant = 6.58 &times;
10−25 𝐺𝑒𝑉 ∗ 𝑠
• Accepted value: 1.17 &times; 10−5 𝐺𝑒𝑉 −2
• Pueblo value : 1.18 &times; 10−5 𝐺𝑒𝑉 −2
• Monarch value : 1.19 &times; 10−5 𝐺𝑒𝑉 −2
STOPPING RATES
•
Pueblo’s Stopping Rate =
0.0284
•
𝑚𝑢𝑜𝑛𝑠
𝑠𝑒𝑐
Monarch’s Stopping Rate =
0.0644
𝑚𝑢𝑜𝑛𝑠
𝑠𝑒𝑐
• t=6.28 &micro;sec or 𝑡 = 2.85 𝜏
RATES
ACCOUNTING FOR
ENERGY LOSS
•
•
Remember t is the transit time
assuming no Time Dilation
t’ assumes Time Dilation Effects
• 𝜌𝑎𝑖𝑟 = 972
𝑔
.
𝑚3
• ∆𝐸 = 364 𝑀𝑒𝑉, 𝐸1 = 508 𝑀𝑒𝑉, 𝛾1 = 4.9
• 𝑡 ′ = 2.38 𝜇 sec 𝑜𝑟 𝑡 = 1.08 𝜏
• 𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑛𝑜 𝑡𝑖𝑚𝑒 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑐𝑒) = 𝑅0 &times; 0.0578
• 𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛) = 𝑅0 &times; 0.339
MUON ENERGY
SPECTRUM
CORRECTION
AND PREDICTIONS
•
𝑅(𝑖𝑔𝑛𝑜𝑟𝑒 𝑠ℎ𝑎𝑝𝑒) = 0.339
•
𝑅𝑟𝑎𝑤 =
•
𝑅𝑜 =
0.0284
0.0644
0.441
0.339
= 0.441
= 1.30
• 𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑛𝑜 𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛) = 1.30 &times; 0.056 = 0.0749
• 𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛) = 1.30 &times; 0.339 = 0.441
GREAT INCONSISTENCY OF THE
HYPOTHESIS THAT THERE IS NO
RELATIVISTIC TIME DILATION
EFFECTS ON THE MUON
The data was consistent with the Relativistic Time Dilation
Hypothesis.
“
IF WE KNEW WHAT IT WAS WE WERE
DOING, IT WOULD NOT BE CALLED
RESEARCH, WOULD IT
Einstein
Thanks to:
Dr. Brown
Dr. Wallin
My family
Special thanks to Annika, Thomas, and Olivia
”
```