Chapter 3, Nuclear Properties Video Solutions, Introductory Nuclear Physics

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Problem 1

Show that the mean-square charge radius of a uniformly charged sphere is $\left\langle r^{2}\right\rangle=3 R^{2} / 5$.

Susan Hallstrom

Numerade Educator

01:51

Problem 2

(a) Derive Equation 3.9. (b) Fill in the missing steps in the derivation of Equation $3.13$ beginning with Equation $3.9 .$

Manik Pulyani

Numerade Educator

09:41

Problem 3

Compute the form factors $F(q)$ for the following charge distributions:
(a) $\rho(r)=\rho_{0} . \quad r<R$
(b) $\rho(r)=\rho_{0} e^{-\ln 2 r r^{2} / R^{2}}$
$=0 . \quad r>R$

Rajesh Kumar

Numerade Educator

02:18

A nuclear charge distribution more realistic than the uniformly charged distribution is the Fermi distribution, $\rho(r)=\rho_{0}\{1+\exp [(r-R) / a]\}^{-1}$.
(a) Sketch this distribution and compare with Figure 3.4. (b) Find the value of $a$ if $t=2.3 \mathrm{fm}$. (c) What is' the significance of the parameter $R$ ? (d) Evaluate $\left\langle r^{2}\right\rangle$ according to this distribution.

Chai Santi

Numerade Educator

01:31

Problem 5

Why is the electron screening correction, which is a great difficulty for analyzing electronic $\mathrm{X}$ rays, not a problem for muonic $\mathrm{X}$ rays?

Rikhil Makwana

Numerade Educator

04:45

Problem 6

(a) Using a one-electron model, evaluate the energies of the muonic $\mathrm{K} \mathrm{X}$ rays in Fe assuming a point nucleus. and compare with the energies shown in Figure 3.8. (b) Evaluate the correction $\Delta E$ due to the finite nuclear size. Compare the corrected value with the measured energies.

Narayan Hari

Numerade Educator

02:07

Problem 7

(a) From the known masses of ${ }^{15} \mathrm{O}$ and ${ }^{15} \mathrm{~N}$, compute the difference in binding energy. (b) Assuming this difference to arise from the difference in Coulomb energy, compute the nuclear radius of ${ }^{15} \mathrm{O}$ and ${ }^{15} \mathrm{~N}$.

Penny Riley

Numerade Educator

01:14

Problem 8

Given the following mass doublet values (in units of $10^{-t} \dot{u}$ ). compute the corresponding values for the atomic mass of ${ }^{37} \mathrm{Cl}$ :
$$
\begin{aligned}
m\left(\mathrm{C}_{3} \mathrm{H}\right)-m\left({ }^{37} \mathrm{Cl}\right) &=41922.2 \pm 0.3 \\
m\left(\mathrm{C}_{2} \mathrm{D}_{8}\right)-m\left({ }^{37} \mathrm{ClH}_{3}\right) &=123436.5 \pm 0.1 \\
m\left(\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}_{2}\right)-m\left({ }^{37} \mathrm{Cl}_{2}\right) &=104974.24 \pm 0.08
\end{aligned}
$$
Here $D={ }^{2} \mathrm{H}, C={ }^{12} C$, and $O={ }^{16} O$. Include in your calculation the effect of uncertainties in the H, D, O, and C masses.

Zhuxi Luo

Numerade Educator

04:22

Problem 9

Compute the total binding energy and the binding energy per nucleon for (a) ${ }^{7} \mathrm{Li} ;$ (b) ${ }^{20} \mathrm{Ne} ;$ (c) ${ }^{56} \mathrm{Fe}$; (d) ${ }^{235} \mathrm{U}$.

Mayank Tripathi

Numerade Educator

09:26

Problem 10

For each of the following nuclei, use the semiempirical mass formula to compute the total binding energy and the Coulomb energy: (a) ${ }^{21} \mathrm{Ne}$ : (b) ${ }^{57} \mathrm{Fe} ;$ (c) ${ }^{209} \mathrm{Bi} ;$ (d) ${ }^{256} \mathrm{Fm}$.

Kai Chen

Princeton University

02:05

Problem 11

Compute the mass defects of (a) ${ }^{32} \mathrm{~S} ;$ (b) ${ }^{20} \mathrm{~F}:$ (c) ${ }^{238} \mathrm{U}$.

Alice .

Numerade Educator

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Problem 12

Given the following mass defects, find the corresponding atomic mass: (a) ${ }^{-24} \mathrm{Na}:-8.418 \mathrm{MeV} ;$ (b) ${ }^{144} \mathrm{Sm}:-81.964 \mathrm{MeV}:$ (c) ${ }^{240} \mathrm{Pu}:+50.123 \mathrm{MeV}$.

Susan Hallstrom

Numerade Educator

04:52

Problem 13

Evaluate (a) the neutron separation energies of ${ }^{7} \mathrm{Li} .{ }^{91} \mathrm{Zr}$, and ${ }^{236} \mathrm{U}$ : $(b)$ the proton separation energies of ${ }^{30} \mathrm{Ne} .{ }^{55} \mathrm{Mn}$. and ${ }^{187} \mathrm{Au}$.

Mayank Tripathi

Numerade Educator

12:22

Problem 14

Examine carefully the $S_{\mathrm{n}}$ and $S_{\mathrm{p}}$ values given in Table $3.1$ and draw conclusions about the strength of the binding of the last proton or neutron in the mirror pairs $\left({ }^{17} \mathrm{O} .{ }^{17} \mathrm{~F}\right)$ and $\left({ }^{41} \mathrm{Ca} .{ }^{4 \mathrm{t}} \mathrm{Sc}\right)$. Try to account for general or systematic behavior. Compare the nucleon separation energies in nuclei with identical numbers of protons or neutrons (for example. $S_{\mathrm{n}}$ in ${ }^{16} \mathrm{O}$ and ${ }^{17} \mathrm{~F}$ or $S_{p}$ in ${ }^{16} \mathrm{O}$ and ${ }^{17} \mathrm{O}$ ). Extend these systematics by evaluating and tabulating the $S_{\mathrm{n}}$ and $S_{\mathrm{p}}$ for ${ }^{4} \mathrm{He}$. ${ }^{3} \mathrm{He}$. ${ }^{5} \mathrm{Li}$ and for ${ }^{56} \mathrm{Ni}$. ${ }^{57} \mathrm{Ni}$, and ${ }^{57} \mathrm{Cu}$. $($ Note: Nuclei with $Z$ or $N$ equal to $2,8.20$, or 28 have unusual stability. We explore the reasons for this behavior in Chapter 5.)

Susan Hallstrom

Numerade Educator

04:52

Problem 15

Use the semiempirical mass formula to obtain an expression for the two-neutron separation energy when $A \gg 1$. (Hint: A differential method is far easier than an algebraic one for this problem.) Estimate the size of the various terms and discuss the $A$ dependence. Compare with the following data for $\mathrm{Al}$ and Te:
\begin{tabular}{ll|llll}
${ }^{25} \mathrm{Al}$ & $31.82 \mathrm{MeV}$ & ${ }^{117} \mathrm{Te}$ & $18.89 \mathrm{MeV}$ & ${ }^{124} \mathrm{Te}$ & $16.36 \mathrm{MeV}$ \\
${ }^{26} \mathrm{Al}$ & $28.30 \mathrm{MeV}$ & ${ }^{118} \mathrm{Te}$ & $18.45 \mathrm{MeV}$ & ${ }^{125} \mathrm{Te}$ & $16.00 \mathrm{MeV}$ \\
${ }^{27} \mathrm{Al}$ & $24.42 \mathrm{MeV}$ & ${ }^{119} \mathrm{Te}$ & $18.17 \mathrm{MeV}$ & ${ }^{136} \mathrm{Te}$ & $15.69 \mathrm{MeV}$ \\
${ }^{2 \mathrm{~A}} \mathrm{Al}$ & $20.78 \mathrm{MeV}$ & ${ }^{120} \mathrm{Te}$ & $17.88 \mathrm{MeV}$ & ${ }^{127} \mathrm{Te}$ & $15.41 \mathrm{MeV}$ \\
${ }^{29} \mathrm{Al}$ & $17.16 \mathrm{MeV}$ & ${ }^{121} \mathrm{Te}$ & $17.46 \mathrm{MeV}$ & ${ }^{128} \mathrm{Te}$ & $15.07 \mathrm{MeV}$ \\
${ }^{30} \mathrm{Al}$ & $15.19 \mathrm{MeV}$ & ${ }^{123} \mathrm{Te}$ & $17.04 \mathrm{MeV}$ & ${ }^{129} \mathrm{Te}$ & $14.86 \mathrm{MeV}$ \\
${ }^{31} \mathrm{Al}$ & $13.03 \mathrm{MeV}$ & ${ }^{133} \mathrm{Te}$ & $16.80 \mathrm{MeV}$ & ${ }^{130} \mathrm{Te}$ & $14.50 \mathrm{MeV}$
\end{tabular}
Why do we choose two-neutron. rather than one-neutron, separation energies for this comparison?

Mayank Tripathi

Numerade Educator

02:02

Problem 16

In analogy with the previous problem, use the semiempirical mass formula to find approximate expressions for the variation of $S_{p}$ with $A$ holding $Z$ constant. Obtain data for several sets of isotopes, plot the data, and compare with the predictions of the semiempirical mass formula.

Amy Jiang

Numerade Educator

06:06

Problem 17

The spin-parity of ${ }^{9} \mathrm{Be}$ and ${ }^{9} \mathrm{~B}$ are both $\frac{1}{2}^{-}$. Assuming in both cases that the spin and parity are characteristic only of the odd nucieon, show how it is possible to obtain the observed spin-parity of ${ }^{10} \mathrm{~B}\left(3^{+}\right)$. What other spinparity combinations could also appear? (These are observed as excited states of ${ }^{10} \mathrm{~B}$.)

Abdul Vahid M

Numerade Educator

03:38

Problem 18

Let's suppose we can form ${ }^{3} \mathrm{He}$ or ${ }^{3} \mathrm{H}$ by adding a proton or a neutron to ${ }^{2} \mathrm{H}$, which has spin equal to 1 and even parity. Let $\ell$ be the orbital angular momentum of the added nucleon relative to the ${ }^{2} \mathrm{H}$ center of mass. What are the possible values of the total angular momentum of ${ }^{3} \mathrm{H}$ or ${ }^{3} \mathrm{He}$ ? Given that the ground-state parity of ${ }^{3} \mathrm{H}$ and ${ }^{3} \mathrm{He}$ is even, which of these can be eliminated? What is the most likely value of the ground-state angular momentum of ${ }^{3} \mathrm{H}$ or ${ }^{3} \mathrm{He}$ ? Can you make a similar argument based on removing a proton or a neutron from ${ }^{4} \mathrm{He}$ ? (What is the ground-state spin-parity of ${ }^{4} \mathrm{He}$ ?) How would you account for the spin-parity of ${ }^{3} \mathrm{Li}$ and ${ }^{5} \mathrm{He}\left(\frac{2}{2}{ }^{-}\right)$?

Penny Riley

Numerade Educator

01:36

Problem 19

(a) Consider a neutron as consisting of a proton plus a negative $\pi$ meson in an $\ell=1$ orbital state. What would be the orbital magnetic dipole moment of such a configuration? Express your result as a multiple of the proton's magnetic moment. (b) Is it possible to account for the observed theutron magnetic moment from such a model? Suppose the neutron wave function consisted of two pieces, one corresponding to a $g=0$ "Dirac" neutron and the other to proton-plus-\pi meson. What would be the relative sizes of the two pieces of the wave function? (Assume the proton also to behave like an ideal Dirac particle.) (c) Repeat the previous analysis for the proton magnetic moment: that is. consider the proton as part pure Dirac proton. plus part Dirac neutron with orbiting positive $\pi$ meson in $\ell=1$ state.

Suzanne W.

Numerade Educator

07:33

Problem 20

Suppose the proton magnetic moment were to be interpreted as due to the rotational motion of a positive spherical uniform charge distribution of radius $R$ spinning about its axis with angular speed $\omega$. (a) Show that $\mu=e \omega R^{\div} / 5$ by integrating over the charge distribution. (b) Using the classical relationship between angular momentum and rotational speed. show that $\omega R^{2}=5 / 0.4 m$. (c) Finally, obtain $\mu=(e / 2 m) s$, which is analogous to Equation $3.32 .$

Zachary Warner

Numerade Educator

01:41

Problem 21

Calculate the electric quadrupole moment of a uniformly charged ellipsoid of revolution of semimajor axis $b$ and semiminor axis $u$.

Arun Bana

Numerade Educator

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