The mean momentum p of a nucleon in a nucleus of the mass number A and atomic number Z depends on A as P ∝ A − 1 3 .All that I know is that the radius of the nucleus is proportional to mass number as R ∝ A 1 3 .From here how can one relate momentum to the mass number?

Question

The mean momentum   p of a nucleon in a nucleus of the mass number   A and atomic number   Z depends on   A as  P  ∝            A              −              1        3              .All that I know is that the radius of the nucleus is proportional to mass number as  R  ∝      A                  1        3              .From here how can one relate momentum to the mass number?

canhaulatlt · Accepted Answer

You have an answer about nuclei with angular momentum. But the result also holds for spinless nuclei, because of the uncertainty principle. A particle that is somewhere within a box of size   R has position uncertainty       σ    x    ∼  R, and therefore momentum uncertainty      σ    p    ≳  ℏ      /    RYou can mess around with the factors of two if you like, but there&#039;s your one-squiggle proportionality relationship.

Chloe Arnold · Accepted Answer

You can get a very rough idea of the momentum scales that you would expect in the nucleus simply by using the Uncertainty Principle. Essentially, when you localize a particle such that its uncertainty in position is   Δ  x, then its uncertainty in momentum   Δ  p is given by:  Δ  x  Δ  p  ≳      ℏ    2  (We use   ≳ rather than   ≥ here because we&#039;re not quite localizing a particle to a particular 1-D box, or even to a 3-D cube; the nucleus is closer to a spherical well, and the operator structure is somewhat different. That said, it shouldn&#039;t be too different if we only want a very rough estimate. And really, all that matters for the purposes of our question is that the right-hand side is a constant.)In the rest frame of the nucleus, the average momentum vector of a nucleon in the nucleus is zero, so the uncertainty in the momentum gives you a very rough measure of the magnitude of the momentum of the average nucleon. For   Δ  x, we use the nuclear radius, which is roughly proportional to       A          1              /            3      . So, rearranging with   Δ  x  ∝      A          1              /            3        ,, we have that  Δ  p  ∝      1          A              1                  /                3              =      A          −      1              /            3

The mean momentum p of a nucleon in a nucleus of the mass number A and atomic number Z depends on A as P∝A^(−1/3).

Answered question

Answer & Explanation

New Questions in Nuclear physics