id stringlengths 10 31 | questions stringlengths 56 2.53k | solutions stringlengths 0 12.7k | final_answers listlengths 1 14 | graphs listlengths 1 5 ⌀ |
|---|---|---|---|---|
atomic/1-7 | (I) The ionization energy $E_I$ of the first three elements are
| $Z$ | Element | $E_I$ |
|-----|---------|-------|
| 1 | H | 13.6 eV |
| 2 | He | 24.6 eV |
| 3 | Li | 5.4 eV |
(a) Explain qualitatively the change in $E_I$ from H to He to Li.
(b) What is the second ionization energy of He, th... | ( a ) The table shows that the ionization energy of He is much larger than that of H. The main reason is that the nuclear charge of He is twice than that of H while all their electrons are in the first shell, which means that the
potential energy of the electrons are much lower in the case of He. The very low ionizat... | [
"\\text{The ionization energy increases from H to He due to increased nuclear charge. Li has lower ionization energy due to electron shielding.}",
"E_{II} = 54.4 \\text{ eV}",
"\\text{The energy dependence on } l \\text{ is due to atomic nucleus polarization and orbital penetration effects.}"
] | null |
atomic/4-14 | Find the threshold energy (kinetic energy) for a proton beam to produce the reaction
$$
p + p \rightarrow \pi^0 + p + p
$$
with a stationary proton target. | **Problem 4012** gives
$$
E_p = m_p + 2m_\pi + \frac{m_\pi^2}{2m_p} = 938 + 2 \times 135 + \frac{135^2}{2 \times 938} = 1218 \, \text{MeV}.
$$
Hence the threshold kinetic energy of the proton is $T_p = 1218 - 938 = 280 \, \text{MeV}$. | [
"T_p = 280 \\, \\text{MeV}"
] | null |
atomic/3-21 | (i) The decay $K \to \pi \gamma$ is absolutely forbidden by a certain conservation law, which is believed to hold exactly. Which conservation law is this?
(ii) There are no known mesons of electric charge two. Can you give a simple explanation of this?
(iii) Explain how the parity of pion can be measured by observati... | (i) The decay is forbidden by the conservation of strangeness number, which holds exactly in electromagnetic interaction.
(ii) According to the prevailing theory, a meson consists of a quark and an antiquark. The absolute value of a quark’s charge is not more than $2/3$. So it is impossible for the charge of a meson c... | [
"\\text{Strangeness number conservation}",
"\\text{A meson consists of a quark and an antiquark. The absolute value of a quark's charge is not more than } \\frac{2}{3}. \\text{ So it is impossible for the charge of a meson to be equal to 2.}",
"\\text{The parity of } \\pi^0 \\text{ is odd.}",
"\\text{The equa... | [
{
"type": "image_url",
"image_url": {
"url": "data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYYAAAC3CAYAAAAW0flZAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAEnQAABJ0Ad5mH3gAAEyeSURBVHhe7Z0HmBbV1ccPbZEOoQiKFAGxYUWsoCHWBLtJbImoIfksUTTRqDEaNRo+O8ZIEruoGKKiYsTYPqMSQREbIMWCIl16lbLzze+6l8w7zrx7L7P77q... |
atomic/4-36 | An X-ray photon of initial frequency $3 \times 10^{19} \, \text{Hz}$ collides with an electron
at rest and is scattered through $90^\circ$. Find the new frequency of the X-ray.
The electron Compton wavelength is $2.4 \times 10^{-12}$ meters. | Suppose that the target electron is free. Then the wavelength of the
scattered photon is given by (*Problem 4035(a)*)
$$\lambda' = \lambda_0 + \frac{h}{mc} (1 - \cos \theta),$$
where $\lambda_0$ is the wavelength of the incident photon, $h/(mc)$ is the electron's
Compton wavelength $\lambda_c$. At scattering angl... | [
"\\nu' = 2.42 \\times 10^{19} \\, \\text{Hz}"
] | null |
atomic/4-31 | (a) The $\Omega^-$ was discovered in the reaction $K^- + p \rightarrow \Omega^- + K^+ + K^0$. In terms of the masses of the various particles, what is the threshold kinetic energy for the reaction to occur if the proton is at rest?
(b) Suppose the $K^0$ travels at a speed of $0.8c$. It decays in flight into two neutra... | (a) At threshold the invariant mass squared is
$$
S = (E_K + m_p)^2 - p_K^2 = (m_\Omega + 2m_K)^2 \ .
$$
With $E_K^2 = p_K^2 + m_K^2$, this gives
$$
E_K = \frac{(m_\Omega + 2m_K)^2 - m_p^2 - m_K^2}{2m_p} \ .
$$
Hence the threshold kinetic energy is
$$
T_K = E_K - m_K = \frac{(m_{\Omega} + 2m_K)^2 - (m_p + m_K)^2}{... | [
"T_K = \\frac{(m_{\\Omega} + 2m_K)^2 - (m_p + m_K)^2}{2m_p}",
"\\pi"
] | null |
atomic/3-26 | Given two angular momenta $\text{J}_1$ and $\text{J}_2$ (for example $\text{L}$ and $\text{S}$) and the corresponding wave functions.
(a) Compute the Clebsch–Gordan coefficients for the states with $\mathbf{J} = \mathbf{j}_1 + \mathbf{j}_2, M = m_1 + m_2$, where $j_1 = 1$ and $j_2 = 1/2$, $J = 3/2, M = 1/2$, for the ... | (a) First consider
$$
\left|\frac{3}{2}, \frac{3}{2}\right\rangle = |1, 1\rangle \left|\frac{1}{2}, \frac{1}{2}\right\rangle.
$$
Applying the operator
$$
L_- = J_x - i J_y = (j_{1x} - i j_{1y}) + (j_{2x} - i j_{2y}) \equiv L^{(1)} + L^{(2)}
$$
to the above:
$$
L^{-} \left|\frac{3}{2}, \frac{3}{2}\right\rangle = ... | [
"\\left|\\frac{3}{2}, \\frac{1}{2}\\right\\rangle = \\sqrt{\\frac{2}{3}} |1, 0\\rangle \\left|\\frac{1}{2}, \\frac{1}{2}\\right\\rangle + \\sqrt{\\frac{1}{3}} |1, 1\\rangle \\left|\\frac{1}{2}, -\\frac{1}{2}\\right\\rangle",
"\\sigma_1 : \\sigma_2 : \\sigma_3 = 9 : 1 : 2",
"\\sigma_1 : \\sigma_2 : \\sigma_3 = 0... | null |
atomic/3-19 | An experiment is performed to search for evidence of the reaction $pp \rightarrow HK^+ K^+$.
(a) What are the values of electric charge, strangeness and baryon number of the particle H? How many quarks must H contain?
(b) A theoretical calculation for the mass of this state H yields a predicted value of $m_H = 2150$ ... | (a) As $K^+$ has $S = 1$, $B = 0$, H is expected to have electric charge $Q = 0$, strangeness number $S = -2$, baryon number $B = 2$. To satisfy these requirements, H must contain at least six quarks (uu dd ss).
(b) At minimum incident energy, the particles are produced at rest in the center-of-mass frame. As $(\Sigma... | [
"Q = 0, \\, S = -2, \\, B = 2",
"6",
"4.208 \\, \\text{GeV}/c",
"H \\to \\Lambda + n, \\, \\Sigma^0 + n, \\, \\Sigma^- + p",
"H \\to \\Lambda + p + e^- + \\bar{\\nu}, \\, \\Sigma^0 + p + e^- + \\bar{\\nu}"
] | null |
atomic/3-10 | The lifetime of the muon is $10^9$, $10^2$, $10^{-2}$, $10^{-6}$ second. | $10^{-6}$ s (more precisely $\tau_\mu = 2.2 \times 10^{-6}$ s). | [
"10^{-6}"
] | null |
atomic/4-38 | (a) Two photons energy $\varepsilon $ and $E$ respectively collide head-on. Show that the velocity of the coordinate system in which the momentum is zero is given by
$$
\beta = \frac{E - \varepsilon }{E + \varepsilon }
$$
(b) If the colliding photons are to produce an electron-positron pair and $\varepsilon $ is 1 eV... | (a) Let **P**, **p** be the momenta of the photons, where $P = E, \ p = \varepsilon$. The total momentum of the system is $|\mathbf{P} + \mathbf{p}|$, and the total energy is $E + \varepsilon$. Hence the system as a whole has velocity
$$
\beta = \frac{|\mathbf{P} + \mathbf{p}|}{E + \varepsilon} = \frac{E - \varepsil... | [
"\\beta = \\frac{E - \\varepsilon}{E + \\varepsilon}",
"E = \\frac{m_{e}^2}{\\varepsilon} = 261 \\text{ GeV}"
] | null |
atomic/3-17 | (a) Explain why the following reactions are not observed, even if the kinetic energy of the first proton is several BeV:
1. $p + p \rightarrow K^+ + \Sigma^+$
2. $p + n \rightarrow \Lambda^0 + \Sigma^+$
3. $p + n \rightarrow \Xi^0 + p$
4. $p + n \rightarrow \Xi^- + K^+ + \Sigma^+$
(b) Explain why the following deca... | (a) The reactions involve only strongly interacting particles and should obey all the conservation laws. If some are violated then the process is forbidden and not observed. Some of the relevant data and quantum numbers are given in Table 3.3.
1. $p + p \rightarrow K^+ + \Sigma^+$, the baryon number, the isospin and i... | [
"\\text{Baryon number, isospin, and its third component are not conserved.}",
"\\text{Strangeness number ($\\Delta S = -2$) and third component of isospin are not conserved.}",
"\\text{Strangeness number ($\\Delta S = -2$) and third component of isospin are not conserved.}",
"\\text{Strangeness number ($\\Del... | null |
atomic/3-28 | The $\Sigma^*$ is an unstable hyperon with mass $m = 1385 \text{ MeV}$ and decay width $\Gamma = 35 \text{ MeV}$, with a branching ratio into the channel $\Sigma^* \rightarrow \pi^+\Lambda$ of 88%. It is produced in the reaction $K^-p \rightarrow \pi^-\Sigma^{*+}$, but the reaction $K^+p \rightarrow \pi^+\Sigma^{*+}$ d... | (a) As $\Sigma^{*+}$ is produced in the strong interaction $K^-p \rightarrow \pi\Sigma^{*+}$, which conserves strangeness number, the strangeness number of $\Sigma^{*+}$ is equal to that of $K^-$, namely, $-1$. As $S(K^+) = +1$, the reaction $K^+p \rightarrow \pi^+\Sigma^{*+}$ violates the conservation of strangeness n... | [
"-1",
"\\text{strong decay}",
"1"
] | null |
atomic/3-43 | (a) Describe the experiments that prove
(1) there are two kinds of neutrino,
(2) the interaction cross section is very small.
(b) Write down the reactions in which an energetic neutrino may produce a single pion with
(1) a proton, and with
(2) a neutron.
(c) Define helicity and what are its values for neutri... | (a) (1) For two-neutrino experiment see **Problem 3009(3)**.
(2) The first observation of the interaction of free neutrinos was made by Reines and Cowan during 1953–1959, who employed $\bar{\nu}_e$ from a nuclear reactor, which have a broad spectrum centered around 1 MeV, as projectiles and cadmium chloride ($\text{C... | [
"\\sigma_\\nu \\sim 10^{-44} \\, \\text{cm}^2",
"\\nu_\\mu + p \\rightarrow \\mu^- + p + \\pi^+",
"\\begin{align*}\n\\nu_\\mu + n &\\rightarrow \\mu^- + n + \\pi^+ \\\\\n&\\rightarrow \\mu^- + p + \\pi^0\n\\end{align*}",
"H = -1",
"H = +1",
"\\mu^+ \\rightarrow e^+ + \\gamma",
"\\mu^+ \\rightarrow e^+ +... | null |
atomic/3-44 | A sensitive way to measure the mass of the electron neutrino is to measure
(a) the angular distribution in electron-neutrino scattering.
(b) the electron energy spectrum in beta-decay.
(c) the neutrino flux from the sun. | In the Kurie plot of a $ \beta $ spectrum, the shape at the tail end depends on the neutrino mass. So the answer is (b). | [
"(b)"
] | null |
atomic/3-16 | Which of the following reactions violate a conservation law?
Where there is a violation, state the law that is violated.
- $\mu^+ \rightarrow e^+ + \gamma$
- $e^- \rightarrow \nu_e + \gamma$
- $p + p \rightarrow p + \Sigma^+ + K^-$
- $p \rightarrow e^+ + \nu_e$
- $p \rightarrow e^+ + n + \nu_e$
- $n \rightarrow p + ... | - $\mu^+ \rightarrow e^+ + \gamma$ is forbidden because it violates the conservation of lepton number, which must hold for any interaction.
- $e^- \rightarrow \nu_e + \gamma$, $p + p \rightarrow p + \Sigma^+ + K^-$ are forbidden because they violate electric charge conservation.
- $p \rightarrow e^+ + \nu_e$ is for... | [
"\\mu^+ \\rightarrow e^+ + \\gamma \\text{ violates lepton number conservation}",
"e^- \\rightarrow \\nu_e + \\gamma \\text{ violates charge conservation}",
"p + p \\rightarrow p + \\Sigma^+ + K^- \\text{ violates charge conservation}",
"p \\rightarrow e^+ + \\nu_e \\text{ violates baryon number conservation}... | null |
atomic/3-29 | A particle $X$ has two decay modes with partial decay rates $\gamma_1 (\text{sec}^{-1})$ and $\gamma_2 (\text{sec}^{-1})$.
(a) What is the inherent uncertainty in the mass of $X$?
(b) One of the decay modes of $X$ is the strong interaction decay
$$
X \rightarrow \pi^+ + \pi^+.
$$
What can you conclude about the is... | (a) The total decay rate of particle $X$ is
$$
\lambda = \gamma_1 + \gamma_2.
$$
So the mean lifetime of the particle is
$$
\tau = \frac{1}{\lambda} = \frac{1}{\gamma_1 + \gamma_2}.
$$
The inherent uncertainty in the mass of the particle, $\Gamma$, is given by the uncertainty principle $\Gamma \tau \sim \hbar$. Hen... | [
"\\Gamma \\sim \\hbar(\\gamma_1 + \\gamma_2)",
"I = 2"
] | null |
atomic/3-11 | List all of the known leptons. How does $\mu^+$ decay? Considering this decay and the fact that $\nu_\mu + n \rightarrow e^- + p$ is found to be forbidden, discuss possible lepton quantum number assignments that satisfy additive quantum number conservation laws. How could $\nu_\mu$ produce a new charged "heavy lepton"? | Up to now, 10 kinds of leptons have been found. These are $e^-$, $\nu_e$, $\mu^-$, $\nu_\mu$, $\tau^-$ and their antiparticles $e^+$, $\bar{\nu}_e$, $\mu^+$, $\bar{\nu}_\mu$, $\tau^+$. $\nu_\tau$ and $\bar{\nu}_\tau$ have been predicted theoretically, but not yet directly observed.
$\mu^+$ decays according to $\mu^+ \... | [
"\\{e^-, \\nu_e, \\mu^-, \\nu_\\mu, \\tau^-, e^+, \\bar{\\nu}_e, \\mu^+, \\bar{\\nu}_\\mu, \\tau^+\\}",
"\\mu^+ \\rightarrow e^+ + \\nu_e + \\bar{\\nu}_\\mu",
"L_e = 1 \\quad \\text{for} \\quad e^-, \\nu_e",
"L_\\mu = 1 \\quad \\text{for} \\quad \\mu^-, \\nu_\\mu",
"\\nu_\\mu + n \\rightarrow A^+ + \\nu_A +... | null |
atomic/4-39 | The universe is filled with black-body microwave radiation. The average photon energy is $E \sim 10^{-3} \text{ eV}$. The number density of the photons is $\sim 300 \text{ cm}^{-3}$. Very high energy $\gamma$-rays make electron-positron-producing collisions with these photons. This pair-production cross section is $\si... | (a) Let the energies and momenta of the high energy photon and a microwave photon be $E_1$, $\text{p}_1$, $E_2$, $\text{p}_2$ respectively. For $e^{+}e^{-}$ production we require
$$(E_1 + E_2)^2 - (\mathbf{p_1} + \mathbf{p_2})^2 \ge (2m)^2,$$
where $m$ is the electron mass. As $E_1 = p_1, E_2 = p_2$, this becomes
$... | [
"E_{\\min} = 2.6 \\times 10^{14} \\text{ eV}",
"l = 1.6 \\times 10^{4} \\text{ light years}",
"l \\ll R",
"\\gamma p \\rightarrow \\pi^0 p, \\quad \\gamma p \\rightarrow \\pi^+ n"
] | null |
atomic/3-27 | Estimate the ratios of decay rates given below, stating clearly the selection rules ("fundamental" or phenomenological) which are operating. Also state whether each decay (regardless of the ratio) is strong, electromagnetic or weak. If at all possible, express your answer in terms of the fundamental constants $G, \alph... | (a) Consider $K^+ \rightarrow \pi^+\pi^0$. For nonleptonic weak decays $\Delta I = 1/2$. As $I(K) = 1/2$, the isospin of the $2\pi$ system must be 0 or 1. The generalized Pauli's principle requires the total wave function of the $2\pi$ system to be symmetric. As the spin of $K$ is 0, conservation of the total angular m... | [
"\\frac{K^+ \\to \\pi^+ \\pi^0}{K^0_s \\to \\pi^+\\pi^-} \\approx 1.5 \\times 10^{-3}",
"\\frac{\\rho^0 \\to \\pi^0 \\pi^0}{\\rho^0 \\to \\pi^+\\pi^-} \\approx 0",
"\\frac{K^0_L \\to \\mu^+ \\mu^-}{K^0_L \\to \\pi^0 \\pi^0} \\approx 10^{-5}",
"\\frac{K^+ \\to \\pi^+ \\pi^+ e^- \\nu}{K^- \\to \\pi^+ \\pi^- e^-... | null |
atomic/4-30 | The $\rho^-$ meson is a meson resonance with mass 769 MeV and width 154 MeV. It can be produced experimentally by bombarding a hydrogen target with a $\pi^-$-meson beam,
$$
\pi^- + p \rightarrow \rho^0 + n.
$$
(a) What is the lifetime and mean decay distance for a 5 GeV $\rho^0$?
(b) What is the $\pi^-$ threshold en... | (a) The $\rho^{0}$ has Lorentz factor
$$
\gamma_{0} = \frac{E_{p}}{m_{\rho}} = \frac{5}{0.769} = 6.50.
$$
Its proper lifetime is
$$
\tau_{0} = \hbar/\Gamma = \frac{6.58 \times 10^{-22}}{154} = 4.27 \times 10^{-24} \text{ s}.
$$
In laboratory frame the lifetime is
$$
\tau = \gamma_{0} \tau_{0} = 2.78 \times 10^{-23... | [
"\\tau = 2.78 \\times 10^{-23} \\, \\text{s}",
"d = 8.23 \\times 10^{-13} \\, \\text{cm}",
"E_{\\pi} = 1077 \\, \\text{MeV}",
"N = 1.3 \\times 10^{-3}",
"0"
] | null |
atomic/3-18 | Listed below are a number of decay processes.
(a) Which do not occur in nature? For each of these specify the conservation law which forbids its occurrence.
(b) Order the remaining decays in order of increasing lifetime. For each case name the interaction responsible for the decay and give an order-of-magnitude estim... | (a) $p \to e^+ + \pi^0$, forbidden as the lepton number and the baryon number are not conserved.
$\Omega^- \rightarrow \Xi^0 + K^-$, forbidden because the energy is not conserved as $m_{Ω} < ( m_\Xi + m_{K})$.
(b) The allowed decays are arranged below in increasing order of lifetime:
- $\rho^0 \rightarrow \pi^+ + \pi... | [
"p \\to e^+ + \\pi^0, \\text{ forbidden by lepton number and baryon number conservation}",
"\\Omega^- \\to \\Xi^0 + K^-, \\text{ forbidden by energy conservation}",
"\\rho^0 \\to \\pi^+ + \\pi^-, \\text{ strong decay, } \\approx 10^{-24} \\text{ s}",
"\\pi^0 \\to \\gamma + \\gamma, \\text{ electromagnetic dec... | [
{
"type": "image_url",
"image_url": {
"url": "data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAApoAAAGECAYAAACWOICJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAEnQAABJ0Ad5mH3gAAOyoSURBVHhe7N0HYBzXfeD/7/a+wGLRG0EA7L13dVFdstxkx3Eu3UnuLndOcezL2XGKndwlufwTx4njXmRbltUpiUViE3sFCBb03oHFLrb3nf8bEJJIibLlmC... |
atomic/4-37 | Consider Compton scattering of photons colliding head-on with moving
electrons. Find the energy of the back-scattered photons $(\theta = 180^\circ)$ if the
incident photons have an energy $h\nu = 2 \, \text{eV}$ and the electrons have a kinetic
energy of $1 \, \text{GeV}$. | Denote the energies and momenta of the electron and photon before and
after collision by $E_e, \, p_e, \, E_\gamma, \, p_\gamma, \, E'_e, \, p'_e, \, E'_\gamma, \, p'_\gamma$ respectively. Conservation
of energy and of momentum give
$$E_\gamma + E_e = E'_\gamma + E'_e ,$$
or
$$
p_{\gamma} + E_e = p'_{\gamma} + E... | [
"29.7 \\, \\text{MeV}"
] | null |
atomic/3-20 | Having 4.5 GeV free energy, what is the most massive isotope one could theoretically produce from nothing?
(a) $^2\text{D}$.
(b) $^3\text{He}$.
(c) $^3\text{T}$. | With a free energy of 4.5 GeV, one could create baryons with energy below 2.25 GeV (To conserve baryon number, the same number of baryons and antibaryons must be produced together. Thus only half the energy is available for baryon creation). Of the three particles only $^2\text{D}$ has rest energy below this. Hence the... | [
"^2\\text{D}"
] | null |
atomic/3-45 | How many of one million 1-GeV neutrinos interact when traversing the earth? ($ \sigma = 0.7 \times 10^{-38} \text{ cm}^2/n $, where $ n $ means a nucleon, $ R = 6000 \text{ km}, \rho \approx 5 \text{ g/cm}^2 $, $\langle A \rangle = 20$)
(a) all.
(b) $\approx 25$.
(c) none. | Each nucleon can be represented by an area $ \sigma $. The number of nucleons encountered by a neutrino traversing earth is then
$$
N_n = \frac{2R\sigma \rho N_A}{\langle A \rangle } \langle A \rangle,
$$
where $ N_A $ = Avogadro's number. The total number of encounters (collisions) is
$$
N = N_\nu N_n = 2R\sigma \r... | [
"25"
] | null |
atomic/3-42 | (a) How many neutrino types are known to exist? What is the spin of a neutrino?
(b) What properties of neutrinos are conserved in scattering processes? What is the difference between a neutrino and an antineutrino? Illustrate this by filling in the missing particle:
$$ \nu_\mu + e^- \rightarrow \mu^- + ? . $$
(c) As... | (a) Two kinds of neutrino have been found so far. These are electron-neutrinos and muon-neutrinos and their antiparticles. Theory predicts the existence of a third kind of neutrino, $\tau$-neutrino and its antiparticle. The neutrino spin is 1/2.
(b) In a scattering process, the lepton number of each kind of neutrino i... | [
"3",
"\\nu_e",
"0",
"v = 2.6 \\times 10^7 \\, \\text{m/s}"
] | null |
atomic/1-31 | Spin-orbit splitting of the hydrogen $2p$ state is $10^{-6}$, $10^{-4}$, $10^{-2}$, $10^0$ eV. | For the $2p$ state of hydrogen atom, $n = 2$, $l = 1$, $s = 1/2$, $j_1 = 3/2$, $j_2 = 1/2$. The energy splitting caused by spin-orbit coupling is given by
$$
\Delta E_{ls} = \frac{hcR\alpha^2}{n^3 l \left( l + \frac{1}{2} \right) (l + 1)} \left[ \frac{j_1(j_1 + 1) - j_2(j_2 + 1)}{2} \right],
$$
where $R$ is Rydberg's... | [
"10^{-4} \\, \\text{eV}"
] | null |
atomic/2-1 | Discuss 4 independent arguments against electrons existing inside the nucleus.
| *First argument - Statistics.* The statistical nature of nuclei can be deduced from the rotational spectra of diatomic molecules. If a nucleus (A,Z) were to consist of A protons and (A-Z) electrons, the spin of an odd-odd nucleus or an odd-even nucleus would not agree with experimental results. Take the odd-odd nucleus... | [
"\\text{Nuclei cannot be composed of protons and electrons}",
"\\text{Electron cannot be bound in the nucleus}",
"\\text{Nuclear magnetic moment does not support electrons in the nucleus}",
"\\text{Continuous spectrum contradicts electrons being inside nucleus}"
] | null |
atomic/2-16 | Hard sphere scattering:
Show that the classical cross section for elastic scattering of point particles from an infinitely massive sphere of radius $R$ is isotropic. | Solution:
In classical mechanics, in elastic scattering of a point particle from a fixed surface, the emergent angle equals the incident angle. Thus if a particle moving along the $-z$ direction impinges on a hard sphere of radius $R$ at a surface point of polar angle $\theta$, it is deflected by an angle $\alpha = 2\t... | [
"\\frac{d\\sigma}{d\\Omega} = \\frac{R^2}{4}"
] | null |
atomic/2-29 | Figure 2.5 shows a plot of the average binding energy per nucleon $E$ vs. the mass number $A$. In the fission of a nucleus of mass number $A_0$ (mass $M_0$) into two nuclei $A_1$ and $A_2$ (masses $M_1$ and $M_2$), the energy released is
$$
Q = M_0 c^2 - M_1 c^2 - M_2 c^2.
$$
Express $Q$ in terms of $\varepsilon(A)$... | The mass of a nucleus of mass number $A$ is
$$
M = Z m_p + (A - Z) m_n - B / c^2,
$$
where $Z$ is its charge number, $m_p$ and $m_n$ are the proton and neutron masses respectively, $B$ is the binding energy. As $Z_0 = Z_1 + Z_2$, $A_0 = A_1 + A_2$, and so $M_0 = M_1 + M_2 + (B_1 + B_2)/c^2 - B_0/c^2$, we have
$$
Q =... | [
"Q \\approx 216\\ \\text{MeV}"
] | null |
atomic/2-11 | The binding energy of $^{90}_{40}\text{Zr}_{50}$ is 783.916 MeV. The binding energy of $^{90}_{39}\text{Y}_{51}$ is 782.410 MeV. Estimate the excitation energy of the lowest $T = 6$ isospin state in $^{90}\text{Zr}$. | The energy difference between two members of the same isospin multiplet is determined by the Coulomb energies and the neutron-proton mass difference. Thus *(Problem 2009)*
$$ \Delta E = E(A, Z + 1) - E(A, Z) = \Delta E_e - (m_n - m_p)c^2 $$
$$ = \frac{3e^2}{5R}(2Z + 1) - 0.78 = \frac{3(2Z + 1) c\hbar \alpha}{5R} - 0.... | [
"E = 13.40 \\, \\text{MeV}"
] | null |
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