GPhO 2019, Теория — Задача 2. Controlled fusion

The 3nd Gulf Physics Olympiad — Theoretical Competition · 7 октомври 2019 г. · 11 т.

Условие

In all your subsequent calculations, you may use the following physical constants and their numerical values.

Boltzmann constant $k_B = 1.381\times10^{-23}\ \mathrm{J\cdot K^{-1}}$

Elementary charge $e = 1.602\times10^{-19}\ \mathrm{C}$

Electron’s mass $m_e = 9.109\times10^{-31}\ \mathrm{kg}$

Planck’s constant $h = 6.626\times10^{-34}\ \mathrm{m^2\cdot kg\cdot s^{-1}}$

Permittivity of free space $\epsilon_0 = 8.854\times10^{-12}\ \mathrm{F\cdot m^{-1}}$

Nuclear fusion is a reaction where light atomic nuclei merge to form a larger nucleus. The difference in the rest energies of the fusing nuclei and the fusion product is released as heat. For instance, if deuterium (consisting of one neutron and one proton, denoted as D) and tritium (consisting of two neutrons and one proton, denoted as T) merge, they will form an $\alpha$-particle, a neutron, and $14\ \mathrm{MeV}$ of energy. While humans have learned how to ignite fusion reaction explosively in hydrogen bombs, they are still struggling to succeed in controlled fusion, i.e. to control fusion reaction so that the released heat could be used for operating power plants. The most feasible reaction for a controlled fusion is the above mentioned D-T reaction which will be addressed by this Problem.

Part A. General considerations (0.5 points)

In what follows, we shall express the temperature in electron volts; this is a common practice for so high temperatures. $1\ \mathrm{eV}$ corresponds to such a temperature $T$ by which the characteristic thermal energy $k_BT$ equals to the potential energy of an electron in electrostatic potential of $V = 1\ \mathrm{V}$.

For a power plant, the released fusion energy must be larger than the total energy loss. It can be shown that for an optimally designed D-T reactor (device in which the controlled fusion takes place), the temperature of the deuterium and tritium nuclei should be $T_0 = 14\ \mathrm{keV}$ while the product of the number density of particles $n$ (the number of particles per volume) and the confinement time $\tau$ (the time during which density $n$ remains roughly constant) should not be less than $2\times10^{20}\ \mathrm{s/m^3}$; this requirement is know as the Lawson criterion. The main technological challenge is to achieve a long enough confinement of the hot plasma.

A.i. Express the fusion temperature $T_0$ in Kelvins. [0,5 т.]

Part B. Tokamak (2.5 points)

The most popular design of fusion reactors is tokamak. In a tokamak, charged particles move along magnetic field lines and are confined because the field lines are confined into a finite volume of space. Qualitatively, the magnetic field lines have the same shape as in the case of an infinitely long straight current passing coaxially through a circular current loop. In the following subtasks, you’re expected to provide the sketches in a 3d projection as shown in the figure.

B.i. Sketch magnetic field lines of a infinitely long straight current. [0,5 т.]

B.ii. Sketch magnetic field lines of a circular current loop. [0,5 т.]

B.iii. Sketch a magnetic field line of an infinitely long straight current passing coaxially through a circular current loop which starts from a small distance from the circular current. [0,75 т.]

B.iv. For the same current configuration as before, sketch a magnetic field line starting from a small distance from the straight current. [0,75 т.]

Part C. Cold fusion (3.5 points)

“Cold fusion” refers to a muon-catalytic fusion process by which an electron in a hydrogen molecule (which can include one deuterium and one tritium nucleus) is substituted by a muon. Muon, having a 207 times larger mass than an electron, brings the nuclei in the molecule closer to each other, thereby increasing the probability of their fusion. The idea of such a catalytic fusion was suggested in 1947-48 by A. Sakharov and F.C. Frank, and lead to a short-lived research boom in 1989 after an erroneous report of a successful fusion at room temperatures by M. Fleischmann and S. Pons. The problem with muon-catalytic fusion is that the energetic cost of producing one muon is larger than the total energy released by fusion reaction mediated by one muon; the possible solutions are either decreasing the energetic cost of a muon, or increasing the number of fusions mediated by a single muon. In what follows, we consider a simple approach to understand why substituting electrons with muons will decrease the size of an atom.

C.i. Using classical mechanics and considering an electron on a circular orbit of radius $R$ around a point-like nucleus of charge $+2e$, relate the momentum $p$ of the electron to the orbit’s radius $R$. [1 т.]

C.ii. At the ground state, the total energy is as small as possible; meanwhile, the state of the electron (muon) cannot violate the uncertainty principle. From these considerations, find an estimate for the radius $R$ at the ground state. [1 т.]

C.iii. By the subtask i, we neglected the distance between the two nuclei in the molecule which was permissible for estimating the orbital radius $R$. Now, however, we want also to get an estimate for the distance between the nuclei. To that end, consider another simple model. The two electrons (muons) on their orbit form a ball-like cloud: let us assume that there is a spherical ball of radius $R$ carrying a total charge $-2e$, homogeneously distributed over the entire volume of the ball. Inside the charged ball, there are two nuclei which can be considered as point masses and point charges (of charge $+e$ each), able to move frictionlessly inside the ball. Find the equilibrium distance $d$ between the nuclei. [1 т.]

C.iv. Based on the model suggested above, by how many times is the distance between the deuterium and tritium atoms reduced when orbital electrons are substituted by muons? [0,5 т.]

Part D. Inertial confinement fusion (4.5 points)

Third approach to controlled fusion is based on the idea that due to mass and inertia, it takes some time, although short, for any hot blob of matter to explode and scatter. In order to satisfy the Lawson criterion one can increase the confinement time, but one can also increase the number density $n$. In the inertial confinement fusion devices, powerful beams are used to create highly compressed balls of gas of densities exceeding the density of lead by hundreds of times. In what follows, we consider this approach by adopting a simple model: a liquid spherical shell of total mass $M$ and radius $r$ is surrounding a ball of gas of number density $n_0$, temperature $T_0$, and pressure $p_0 = k_Bn_0T_0$ (in reality, the shell is solid, but at really high pressures, solids essentially liquify); see the figure. Each gas molecule consists of a deuterium nucleus, tritium nucleus, and two electrons. The thickness of the walls of the spherical liquid shell $\delta$ is much smaller than $r$ ($\delta \ll r$).

D.i. Consider a small piece of shell of surface area $\Delta A$. Express its mass in terms of the quantities introduced above. [0,5 т.]

D.ii. External pressure $p_e$ ($p_e \gg p_0$) is applied to the shell. Express the initial acceleration of a small piece of the shell in terms of the quantities introduced until now. [1 т.]

D.iii. While the shell contracts due to external pressure, the pressure inside grows, and at a certain moment, it becomes larger than the external pressure. Express the minimal radius of the shell $r_m$ and the maximal temperature inside the shell $T_m$ (which are achieved when the surrounding shell stops for a moment before reversing its direction of motion) in terms of the quantities introduced above. Keep in mind that the inside temperature becomes so high that the gas is converted into a completely ionized plasma made of nuclei and electrons. You may assume that $p_e$ remains constant during the entire process (this might not be entirely true, but under this assumption, we shall still be able to get a correct order of magnitude for the answer), the shell contracts while retaining its spherical shape, and the thermal energy transferred to the shell can be neglected. [1,5 т.]

D.iv. The huge external pressure $p_e$ is created by irradiating the shell from outside, isotropically from all sides, with a laser of total output power $P$. As a result, the outer layers of the shell are evaporated, and the evaporated atomic nuclei flow away at the average speed of $u$. Estimate the pressure $p_e$ in terms of $P$, $r$, and $u$; you may assume that $u$ is much smaller than the speed of light. [1,5 т.]

Отговори

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