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5. Quadrupole moments in the shell model. We will calculate an estimate of the quadrupole moment for the special case of a single proton moving in an orbital around a closed shell spherical core. So, the only contribution to the quadrupole moment is from this single proton. We will also assume that the proton moves in an orbital with j = l + 1/2. The space wave function of the proton is Yj,m = R(r)Ym,0p) where Y is the spherical harmonic, and R is the radial part of the wave function. They are normalized, i.e. I = âˆ« |uPuX.&] pue [=Up"'1x""x| (a) Since m = m + m, what must m and m be if m = j? [2 marks] (b) Show that the quadrupole moment, when m = j, is given by To do this, start with the quadrupole moment given by Q = âˆ« (3z^2 r)dV. Given that the spherical harmonic Y0, = âˆš(3/4Ï€) cos Î¸, write the quadrupole moment operator in terms of Y2o and use the integral, âˆ« Y0,Y2o,dÎ© = âˆš(5/16Ï€). Also note âˆš(2/3) = 1/âˆš3. (c) Apply this result to the ground state of Sc, which has j = 3/2. Write the configuration for this ground state and confirm that the condition of j = l + 1/2 holds. Estimate (r^2) using r = 1.2 Ã… fm. Compare your result to the measured quadrupole moment of -0.145 Â± 0.003 b. [2 marks] (d) Apply this result to the ground state of F, which has j = 2. Compare your result to the measured quadrupole moment of 7.6 Â± 0.4 fm^2 (note that this measurement did not determine the sign of the quadrupole moment, only the magnitude). [2 marks]

Name: 5 quadrupole moments in the shell model we will calculate an estimate of the quadrupole moment for the special case of a single proton moving in an orbital around a closed shell spherical co 03366
Uploaded: 2023-07-25T13:16:18-08:00
Duration: 21 s
Channel: Numerade
Description: 5 quadrupole moments in the shell model we will calculate an estimate of the quadrupole moment for the special case of a single proton moving in an orbital around a closed shell spherical co 03366

          5. Quadrupole moments in the shell model. We will calculate an estimate of the quadrupole moment for the special case of a single proton moving in an orbital around a closed shell spherical core. So, the only contribution to the quadrupole moment is from this single proton. We will also assume that the proton moves in an orbital with j = l + 1/2. The space wave function of the proton is Yj,m = R(r)Ym,0p) where Y is the spherical harmonic, and R is the radial part of the wave function. They are normalized, i.e.

I = âˆ« |uPuX.&] pue [=Up"'1x""x|

(a) Since m = m + m, what must m and m be if m = j? [2 marks]
(b) Show that the quadrupole moment, when m = j, is given by
To do this, start with the quadrupole moment given by Q = âˆ« (3z^2 r)dV. Given that the spherical harmonic
Y0, = âˆš(3/4Ï€) cos Î¸, write the quadrupole moment operator in terms of Y2o and use the integral, âˆ« Y0,Y2o,dÎ© = âˆš(5/16Ï€). Also note âˆš(2/3) = 1/âˆš3.

(c) Apply this result to the ground state of Sc, which has j = 3/2. Write the configuration for this ground state and confirm that the condition of j = l + 1/2 holds. Estimate (r^2) using r = 1.2 Ã… fm. Compare your result to the measured quadrupole moment of -0.145 Â± 0.003 b. [2 marks]
(d) Apply this result to the ground state of F, which has j = 2. Compare your result to the measured quadrupole moment of 7.6 Â± 0.4 fm^2 (note that this measurement did not determine the sign of the quadrupole moment, only the magnitude). [2 marks]

5 quadrupole moments in the shell model we will calculate an estimate of the quadrupole moment for the special case of a single proton moving in an orbital around a closed shell spherical co 03366

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