particle range through material and particle lifetimes dan claes crop fall 2006 workshop saturday,...
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Particle RANGE Through Materialand
Particle Lifetimes
Dan ClaesCROP Fall 2006 Workshop
Saturday, October 14, 2006
Earth Moon
While the size of the moon represented is roughly to scale for the earth as shown…
…the distance separating them is about 12 too small!
Earth Moon
Scale of the atomic orbital is about a factor of 5,000 too small!
The nucleus would be invisibly small if corrected in scale to the size of depicted orbital.
Atoms really are mostly empty space!
V
V
Muon in head-on collision with a nucleus
Muon in head-on collision with an electron
Actually if an incoming “sees”
n nuclei per unit area “in its way”
there will be Zn electrons per unit area!
V
Carbon 6COxygen 8OAluminum 13AlIron 26FeCopper 29CuLead 82Pb
What about a single, high energy, charged particle?
While the mass of matter is due primarily to it’s nuclei
The volume of matter is due primarily to it’s electron clouds
Imagine a narrow, well-collimated beam of mono-energetic particles
passing through a slab of matter
EoE
Eo
EoE
Energy loss
E1
E
EoE
Energyloss
E1
E
If the target is thick, this implies that the overall mean energy loss thickness
For sufficiently high initial E0
(or thin enough targets)all particles get through.
EoE0
but if the target is thick enough
thickness
)(xN
Occasionally might justcatch a muon
stopped within thethe thickness ofour detector!
Such an occurrencewould be “signaled” by a coincidence betweenthe top two counters
with NO SIGNAL in the bottom.
n p + ee
ee +
Ne* Ne +
N C + e e
Pu U +
20 10
20 10
13 7
13 6
236 94
232 92
Fundamental particle decays
Nuclear decays
Some observed decays
+ DECAY MODES
It almost seems a self-evident statement:
Any decay that’s possible will happen!
What makes it possible?What sort of conditions must be satisfied?
initialtotal mm Total charge q conserved.
J (angular momentum) conserved.
probability of decaying(at any time - now or later)
= constant
???? What’s this mean equally likely at any instant ????
must be expressed as a probability per unit time
If we observe one, isolated nucleusit is equally likely it decays
this moment tas any other moment t (even years from now)
It either decays or it doesn’t.
A quantum mechanically model for this random behavior suggests
Suppose a given particle has a 0.01 probability of decaying in any given sec.
Does this mean if we wait 100 sec it will definitely have decayed?
If we observe a large sample N of such particles,
for 1 sechow many can we expect to have decayed?
Even a tiny speck of material can include well over trillions and trillions of atoms!
0.01N
Imagine flipping a coin until you get a head.
Is the probability of needing to just one flip the same as the probability of needing to flip
10 times?
Probability of a head on your 1st try,P(1) =
Probability of 1st head on your 2nd try,P(2) =
Probability of 1st head on your 3rd try,P(3) =
Probability of 1st head on your 10th try,P(10) =
1/2
1/4
1/8
(1/2)10 = 1/1024
What is the total probability of ALL OCCURRENCES?
P(1) + P(2) + P(3) + P(4) + P(5) + •••=1/2+ 1/4 + 1/8 + 1/16 + 1/32 + •••
A six-sided die is rolled
repeatedly until it gives a 6.
What is the probability that one roll is enough?What is the probability that one roll is enough?1/6
What is the probability that it will take exactly 2 rolls?
(probability of miss,1st try)(probability of hit)=
36
5
6
1
6
5
What is the probability that exactly 3 rolls will be needed?
# decays N (counted by a geiger counter)
the size of the sample studied
t time interval ofthe measurement
NN each decay represents a loss in theoriginal number of radioactive particles
NN / fraction of particles lost
Note: for 1 particle this must be interpreted as the probability of decaying.
This argues that:
t
NN /constant
This is the decay constant
0.10.09
0.0810.0729
0.065610.059049
0.05314410.04782969
0.043046721
imagine the probability of decaying within any single second is
p = 0.10
the probability of surviving instead during that same single second is
P(1) = 0.10 =P(2) = 0.90 0.10 =P(3) = 0.902 0.10 = P(4) = 0.903 0.10 =P(5) = 0.904 0.10 = P(6) = 0.905 0.10 = P(7) = 0.906 0.10 = P(8) = 0.907 0.10 = P(9) = 0.908 0.10 =
P(N)probability
that it decaysin the Nth
second(but not thepreceeding
N-1seconds)
1 p = 0.90
Probability of Decaying in the Nth Second
0
0.02
0.04
0.06
0.08
0.1
0.12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Number of Seconds
Pro
bab
ilit
y
Series1
tey
Probability of living to time t=N sec, but decaying in the next second
(1-p)Np
Probability of decaying instantly (t=0) is?
Probability of living forever (t ) is?
0
0
0.10.18
0.2430.2916
0.328050.3542940.3720090.3826380.387420.38742
0.3835460.3765730.3671580.3558610.3431520.3294260.3150130.3001890.285180.27017
0.2553110.2407220.2264970.2127110.1994160.1866530.1744490.1628190.1517710.1413040.1314130.1220870.1133120.1050710.0973450.0901140.0833550.0770470.0711670.065693
0.0606020.0558720.0514820.0474110.04364
0.0401490.0369190.0339340.0311770.0286320.0262840.02412
0.0221250.0202880.0185980.0170420.0156120.0142970.0130890.01198
We can calculated an “average” lifetime from (N sec)×P(N)
(1 sec)×P(1)=(2 sec)×P(2)=(3 sec)×P(3)=(4 sec)×P(4)=(5 sec)×P(5)=
N=1
sum=3.026431
sum=6.082530
sum=8.3043 sum=9.690773
sum=9.260956 sum=9.874209
the probability of decaying within any single second
p = 0.10 = 1/10
= 1/
where of course is the average lifetime(which in this example was 10, remember?)
This exponential behavior can be summarized by the rules for our imagined sample of particles
fraction still surviving by time t = et
where = 1/ (and is the average lifetime)
teNtN 0)(N
um
ber
su
rviv
ing
Rad
ioac
tive
ato
ms
time
tNN 0logloglogN
NdtdN
teNtN )0()(
tet )(Pprobability of surviving
through to time t then decaying that moment
(within t and t)
dt
NdN / or
)(
)0( 0
tN
N
tdt
N
dN
t
N
tN )0(
)(log
…and for the calculus savvy…
Some Backgrounds to this estimate:
Accidentals: 2 stray cosmic rays, each passing individually through just one detector, but bysheer chance coincident, can set off the trigger.Another accidental passing through the 2nd counter within 10 sec, will fake a muon decay.
accidentals rate bottom counter’s singles rate × gate width
Inefficiencies: Can trigger when a cosmic track is seen by the top two counters, but missed (since its not 100% efficient) by the bottom. If anaccidental passes through the bottom within 10 sec it gets counted as a muon decay.
2-fold trgr rate(top 2) × (1efficiency of bottom) bottom’s singles rate × gate width