3. Uranium-238 (92 protons) decays by alpha emission. By modelling the nucleus and forces involved, make a rough estimate of its mean life by working through the following steps: (i) Take the Uranium nucleus to have a diameter of 10-14 m, inside which the attractive strong force dominates. (ii) Model the Coulomb repulsion by a rectangular barrier¹. Assume reasonable values for the height and width, as discussed in the lectures. (iii) Assume that the alpha-particle has a kinetic energy of 4 MeV. What is its average speed? How often does it try to get through the barrier? (iv) What is the probability of the alpha-particle quantum tunneling through the barrier on any one attempt? (You may find it useful to start from the formula in the notes for the probability of transmission through a rectangular barrier of width a and height Vo, and show that for E<
3. Uranium-238 (92 protons) decays by alpha emission. By modelling the nucleus and forces involved, make a rough estimate of its mean life by working through the following steps: (i) Take the Uranium nucleus to have a diameter of 10-14 m, inside which the attractive strong force dominates. (ii) Model the Coulomb repulsion by a rectangular barrier¹. Assume reasonable values for the height and width, as discussed in the lectures. (iii) Assume that the alpha-particle has a kinetic energy of 4 MeV. What is its average speed? How often does it try to get through the barrier? (iv) What is the probability of the alpha-particle quantum tunneling through the barrier on any one attempt? (You may find it useful to start from the formula in the notes for the probability of transmission through a rectangular barrier of width a and height Vo, and show that for E<
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![3. Uranium-238 (92 protons) decays by alpha emission. By modelling the nucleus and
forces involved, make a rough estimate of its mean life by working through the following
steps:
(i) Take the Uranium nucleus to have a diameter of 10-14 m, inside which the attractive
strong force dominates.
(ii) Model the Coulomb repulsion by a rectangular barrier¹. Assume reasonable values for
the height and width, as discussed in the lectures.
(iii) Assume that the alpha-particle has a kinetic energy of 4 MeV. What is its average
speed? How often does it try to get through the barrier?
(iv) What is the probability of the alpha-particle quantum tunneling through the barrier on
any one attempt?
(You may find it useful to start from the formula in the notes for the probability of
transmission through a rectangular barrier of width a and height Vo, and
show that for E<<Vo, the probability is exp [−2a√2m(V。 – E)] ).
16E
Vo](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6713d75e-2fb3-4074-927a-cea8cea15561%2F4cf38897-6c7a-43dc-b7c1-16146dd60b8c%2F76e3k7_processed.png&w=3840&q=75)
Transcribed Image Text:3. Uranium-238 (92 protons) decays by alpha emission. By modelling the nucleus and
forces involved, make a rough estimate of its mean life by working through the following
steps:
(i) Take the Uranium nucleus to have a diameter of 10-14 m, inside which the attractive
strong force dominates.
(ii) Model the Coulomb repulsion by a rectangular barrier¹. Assume reasonable values for
the height and width, as discussed in the lectures.
(iii) Assume that the alpha-particle has a kinetic energy of 4 MeV. What is its average
speed? How often does it try to get through the barrier?
(iv) What is the probability of the alpha-particle quantum tunneling through the barrier on
any one attempt?
(You may find it useful to start from the formula in the notes for the probability of
transmission through a rectangular barrier of width a and height Vo, and
show that for E<<Vo, the probability is exp [−2a√2m(V。 – E)] ).
16E
Vo
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