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Para esta lectura:
Summary from last time:
- Coocer ley de desintegracin radiactiva, vida media y periodo de semidesintegrac.
-Ser capaz de calcular los factores de Q de la desintegracin alfa y comprender-las implicaciones para la estabilidad nuclear
Una mirada de la mecnica cuntica en la desintegracin alfa
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By 1928, George Gamouw (brilliant Russian born theorist)
developed a quasi-classical model for alpha decay
Good friend ofLev Landau and another
Russian physicist Dmitri Ivanenkothe Three Musketeers
Worked in several different fields
Early 1900s: radioactive decay known to have characteristic rates and energies.
However, no clear explanation as to why......
Attempted twice to defect in 1932, trying to kayak ~250 km over the Arctic Sea toNorway. Both attempts failed.......
In 1933, tried a less dramatic approach: disappeared when attending a conference in
Brussels. Turned up In 1934 in the United States.
One of the most significant physicists never to have won NP.
contributed to big bang theory and theoretical biology
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Cmo algunas partculas se forman en el ncleo?
Consider 2 protons + 2 neutrons in heavy nucleus:Liquid drop modelBE~ 8 MeV/nucleon for all nucleons
However, shell model BE~ 6 MeV/nucleon forouter nucleons
Total BE of 4 outer nucleons ~ 4 x 6 MeV = 24 MeV
Imagine 4 nucleones amalgamate to form -particleDetails not really clear.... (QM plays a large role)
As que por qu no acaba de salir??
Why dont heavy nuclei release -particles and decompose instantaneously?
BE of an a particle ~ 28.3 MeV
PotentialEnergy
4 nucleons in nucleus
a particle outside nucleus
BEis depth of the potential energy well in which the nucleons sit,i.e. Piense of as a negative potential energy:
Distance
Barrier at edge of nucleus due to coulomb potential
i.e. -particle has (+) kinetic energy and en principio podria dejer el nucleo
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Pictorial representation of potentials:
Potential energy of nucleons:
Difference in potential
energy of 4 nucleons and a:
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Entonces, cmo la partcula alfa nunca abandona ?
QM tunel a traves de la barrera:
Probability of tunnelling out =
Square of ratio of wavefunctions (Y)
inside and outside barrier
Tunnelling through 1/r Coulombpotential difficult to analyze
The solution: approximate to
sequence of square barriers:
By classical physics, there is no possibility for alpha particle a subir la barrier
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Reminder of square barrier tunnelling:
Transmission (T) probability given by:
U0 = barrier height
t = barrier thicknessm = particle mass
E = particle energy
By multiplying transmission probabilities:
1 2 32 ( )2 2 2 ..........
t dtt t tT e e e e
(where integral is over thickness for which U(t) > E)
Through sequence of barriers:
Simply rewriting this byreplacing t with radial coordinate r:
GT ewith Gamow factor:
R is radius of strong potential (i.e. nucleus)
b is distance at which U(r) = E
Note: sqrt (barrier height)
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Tunnelling through a Coulomb barrier:
For r > R, Coulomb barrier potential is: (z = 2 foraparticle)
Substitute into Gamow factor:
Integrate:
( do this with change of variable k using r = b cos2(k) )
Typically, coulomb barrier potential is much larger than particle kinetic energy
Then b >> R, and one can make the approximations: 1)
2)
1
cos 2R Rb b
2
2
R R Rb bb
Gamow factor becomes:
Z ofdaughternucleus
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gives the transmission probability ofa penetrating theCoulomb barrier when it approaches it
However, to know decay rate we must also know how often particle approaches
barrier:
Recall that b is the distance r at which U(r) = E, i.e.
Also, since particles are non-relativistic:
Substituting these obtains
the simplest form of G:
(fine structure constant)
T = Exp(-G)
Assume -particle has velocity V0wi th in nucleus, radius R
Will then make V0/2R approaches to barrier per second
Thus probability of leaving nucleus per unit time:
Have their normal meaning
=
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Kinetic energy ofa particle
Shorter t1/2 for lower Z and higher Q
1) Z
2) Q
Lifetime behaviour within alpha decay groups:
1) Z dependence:
=
Potenti
al
Energy
r
Lower potential barrier for lower Z nuclei
- G factor lower
- Higher probability of escape: lower t1/2In nucleus
Outside nucleus
2) Q dependence Higher Q higher V0 inside nucleus
More attempts to escape, lower t1/2
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Functional form of tunnelling probability function:
Where the variables f and g are given by:
Approximately constant: ln(R) and RZ vary only slightlybetween common -emitters.V0 assumed to be constant.
Constant
Recall the empirical Geiger-Nuttall relation:
Take the logarithm of both sides:
functionally identical to tunnelling equation above (assuming KE of
liberated -particle takes lions share of Q released in decay)
Log dependence ofl and E get enormous range in decay constant for relatively
small changes in KE of -particle due to 1/r dependence of coulomb barrier
=
b1
=
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Only one unknown left: V0.
For simplicity, assume V0 = V:
If the mass of daughter nucleus is >> a particle, then E ~ Q
Finalizing the tunnelling rate equation:
Get finalized equation:
=
fits well to experimental decay rates
best fits yield empirical relation for radius:
R = 1.53 x A1/3 fm.
Compare this to empirical relation found from
scattering measurements: R = 1.2 x A1/3 fm.
Empirical radius from decay fits are aconvolution of nuclear and -particle radii
ln(l)Z (m/Q)1/2
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Worked example:
2Gv e
Rl
24 8Zc mc RZGv c
a a
Calculate the half life for the emission, , which has Q = 6 MeV235 231 493 91 2Np Pa He
(Neptunium protactinium)
Aparent = 235
Zdaughter= 91
= 0.8 x 10-5 s-1
Rparent = 1.53 x A1/3 fm = 9.44 fm
Determines height of coulomb barrier that
alpha particle has to tunnel through
a) Calculate energy of emitted alpha particle:Q KE of alpha = m
av2
ma= 6.64 10-27 kg
v = 1.7 x 107 m/s
b) Calculate emission probability from nucleus:
Fine structure
constant = 1/137
= 147.3 87.3 = 60
Careful to convert
R from fm to m
T1/2 = ln(2) / l = 86,600 s = 24 hours
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works well in broad terms:
Explains Geiger-Nuttall relation and other caracteristicas claves
predicts most emission lifetimes fairly accurately
find that Q
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For next lecture:
A closer look at beta decay
Summary:
- explain alpha emission probability in terms of barrier tunnelling
- be able to apply tunnelling probability function to determine decay rates
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2. Potencial pozo infinito
3. Potencial pozo finito cuadrado
4. Potencial pozo redondeado (Wood-Saxon)
5.Potencial exponencial
6. Potencial de Yukawa
1. Potencial tipo oscilador armnico:
7. Potencial con centro repulsivo impenetrable.
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