Skip to content

Commit

Permalink
updated Ch 11
Browse files Browse the repository at this point in the history
  • Loading branch information
@saturnaxis
saturnaxis committed Jun 14, 2024
1 parent 9793a22 commit 800b82e
Show file tree
Hide file tree
Showing 4 changed files with 537 additions and 121 deletions.
147 changes: 146 additions & 1 deletion Chapter_11/nuclear-physics.ipynb
View file
Original file line number Diff line number Diff line change
Expand Up @@ -1525,7 +1525,152 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Fusion"
"## Fusion\n",
"The only primary souce in widespread use that is *not* derived from the Sun is nuclear power. Energy emitted by stars arises from *nuclear fusion* reations in which light nuclei fuse together. This process contrasts with nuclear fission, in which large nuclei divide.\n",
"\n",
"When $\\rm ^{236}U^*$ fissions, it divides into nuclei with a larger binding energy per nucleon, thereby releasing energy. If two light nuclei fuse together, they alos form a nucleus with a larger binding energy per nucleon and energy is released.\n",
"\n",
"The most energy is released if two isotopes of hydrogen (deuteron and triton) fuse toghether in the reaction\n",
"\n",
"\\begin{align}\n",
"{\\rm ^2H} + {\\rm ^3H} &\\rightarrow n + {\\rm ^4He}. \\qquad (Q=17.6\\ {\\rm MeV})\n",
"\\end{align}\n",
"\n",
"About $3.5\\ {\\rm MeV}$ per nucleon is released because of the stron binding of $\\rm ^4He$. Less than $1\\ {\\rm MeV}$ per nucleon is released in fission.\n",
"\n",
"\n",
"### Formation of Elements\n",
"\n",
"In the first few minutes of the universe, the light elements of hydrogen and helium were formed. Millions of years later, the heavier elements were formed in stars through nuclear fusion. Thre are two main cycles for producing energy in stars.\n",
"\n",
"The first is the **proton-proton chain**, which converts 4 protons into an $\\alpha$ particle. As stars form due to gravitational attraction, the heat (average speed of the protons) increases substantially so they can overcome their Coulomb repulsion (with the help of quantum tunneling) and fuse by the following reactions:\n",
"\n",
"\\begin{align}\n",
"{\\rm ^1H} + {\\rm ^1H} &\\rightarrow {\\rm ^2H} + \\beta^+ + \\nu. \\qquad (Q=0.42\\ {\\rm MeV})\n",
"\\end{align}\n",
"\n",
"This reaction produces a deuteron $(\\rm ^2H)$ and is a special kind of weak-interaction beta decay process. It is extremely slow, because only 1 collision in about $10^{26}$ produces a reaction. This is good, otherwise the Sun would explode!\n",
"\n",
"The deuterons that accumulate can combine with a proton $\\rm ^1H$ to produce $\\rm ^{3}He$:\n",
"\n",
"```{math}\n",
":label: pp_chain_I\n",
"{\\rm ^2H} + {\\rm ^1H} &\\rightarrow {\\rm ^3He} + \\gamma. \\qquad (Q=5.49\\ {\\rm MeV})\n",
"```\n",
"\n",
"The ${\\rm ^3He}$ can then combine to produce ${\\rm ^4He}$ through:\n",
"\n",
"\\begin{align}\n",
"{\\rm ^3He} + {\\rm ^3He} &\\rightarrow {\\rm ^4He} + {\\rm ^1H} + {\\rm ^1H}. \\qquad (Q=12.85\\ {\\rm MeV}).\n",
"\\end{align}\n",
"\n",
"```{note}\n",
"The first two reactions which create the deuteron and helium-3 must occur more than once to produce the enough matter for the final reaction that produces helium-4. \n",
"```\n",
"\n",
"A total of six ${\\rm ^1H}$ are requred to produce ${\\rm ^4He}$ and two ${\\rm ^1H}$. This process consumes four protons. The total $Q$ for 6 ${\\rm ^1H}$ produce ${\\rm ^4H}$ is $24.7\\ {\\rm MeV}$, where an additional $2\\ {\\rm MeV}$ come from teh annihilation of two electron-positiron pairs for a total of $26.7\\ {\\rm MeV}$.\n",
"\n",
"The proton-proton chain is slow because Eq. {eq}`pp_chain_I` limits the entire process. As the reaction proceeds, the star's temperature increases and eventually carbon-12 nuclei are formed through by the **triple-alpha process** (i.e., converting 3 ${\\rm ^4He}$ into a ${\\rm ^{12}C}$).\n",
"\n",
"Another cycle can produce the ${\\rm ^4He}$, if enough **carbon** is produced (or already present) near the stellar core. The series of reactions responsible are called the **CNO cycle**:\n",
"\n",
"\\begin{align}\n",
"{\\rm ^1H} + {\\rm ^{12}C} &\\rightarrow {\\rm ^{13}N} + \\gamma, \\\\\n",
"{\\rm ^{13}N} &\\rightarrow {\\rm ^{13}C} + \\beta^+ + \\nu, \\qquad (t_{1/2} = 9.96\\ {\\rm min}) \\\\\n",
"{\\rm ^1H} + {\\rm ^{13}C} &\\rightarrow {\\rm ^{14}N} + \\gamma, \\\\\n",
"{\\rm ^1H} + {\\rm ^{14}N} &\\rightarrow {\\rm ^{15}O} + \\gamma, \\\\\n",
"{\\rm ^{15}O} &\\rightarrow {\\rm ^{15}N} + \\beta^+ + \\nu, \\qquad (t_{1/2} = 2.04\\ {\\rm min}) \\\\\n",
"{\\rm ^1H} + {\\rm ^{15}N} &\\rightarrow {\\rm ^{12}C} + {\\rm ^4He}.\n",
"\\end{align}\n",
"\n",
"```{note}\n",
"Four ${\\rm ^1H}$ and one ${\\rm ^{12}C}$ are required to produce ${\\rm ^4He}$ and ${\\rm ^{12}C}$. The {\\rm ^{12}C} nucleus serves as a catlyst.\n",
"```\n",
"\n",
"The proton-proton chain is probably responsible for most of our Sun's energy production, but the CNO cycle is a much more rapid fusion reaction. It requires higher temperatures (${\\sim}2\\times 10^7\\ {\\rm K}$) than are present in the Sun, because of the higher Coulomb barrier of ${\\rm ^{12}C}$ compared to the ${\\rm ^1H}$ for the protons.\n",
"\n",
"A hydrostatic equilibrium exists in the Sun between the gravitational attraction that contracts a star and a gas presssure that pushes out due to the fusion process. As the lighter nuclides are consumed to produce heavier nuclides, the gravitatioanl attraction succeeds in contracting the star's mass into a smaller volume and the temperature increaes.\n",
"\n",
"A higher temperature allows the nulcides with higher $Z$ to fuse. This process continues until a large part of the Sun's mass is converted to iron. Then the Sun collapses under its own graviational attraction to become a white dwarf. For more massive stars, they can contract more which allows them to become a neutron or black hole, but this is sensitive to the star's initial mass.\n",
"\n",
"### Nuclear Fusion on Earth\n",
"\n",
"Some scientists beleive that controlled nuclear fusion is ultimatlely our best source of terrestrial energy. Among the several possible fusion reactions, three of the simplest involve the three isotopes of hydrogen:\n",
"\n",
"\\begin{align}\n",
"{\\rm ^2H} + {\\rm ^2H} &\\rightarrow n + {\\rm ^3He}, \\qquad (Q=3.3\\ {\\rm MeV}) \\\\\n",
"{\\rm ^2H} + {\\rm ^2H} &\\rightarrow p + {\\rm ^3He}, \\qquad (Q=4.0\\ {\\rm MeV}) \\\\\n",
"{\\rm ^2H} + {\\rm ^3H} &\\rightarrow n + {\\rm ^4He}. \\qquad (Q=17.6\\ {\\rm MeV})\n",
"\\end{align}\n",
"\n",
"Deuterium $({\\rm ^2H})$ exists in vast quantities in seawater. Assuming that there are $10^{21}\\ {L}$ of water on Earth, the natural abundance of deuterium $(0.015\\%)$ gives $10^{43}$ deuterons. These deuterons (when fused together to make $\\rm ^{3}He$) would produce over $10^{30}\\ {\\rm J}$ of energy. This is enough to support present world energy consumption for a few billion years.\n",
"\n",
"Three main conditions are necessary for controlled nuclear fusion:\n",
"\n",
"1. The temperature must be hot enough to allow the ions (e.g., $d$ and $t$) to overcome the Coulomb barrier and fuse thier nuclei together. This requires a temperature of $100-200$ million $\\rm K$.\n",
"2. The ions must be confined to a small volume to allow the ions to fuse. A suitable ion density is $2-3 \\times 10^{20}\\ {\\rm ions/m^3}$.\n",
"3. The ions must be held together in close proximity at high temperature long enough to avoid plasma cooling, where a suitable time is $1-2\\ {\\rm s}$.\n",
"\n",
"The suitable values given above assume magnitic confinement. The product of the plasma density $n$ and the containment time $\\tau$ must have a minimum value at a sufficiently hgih temperature to initiate fusion and produce as much energy as it consumes. The minimum value is\n",
"\n",
"\\begin{align}\n",
"n\\tau \\geq 3 \\times 10^{20}\\ {\\rm s/m^3}.\n",
"\\end{align}\n",
"\n",
"The above relation is calle dthe [Lawson criterion](https://en.wikipedia.org/wiki/Lawson_criterion). A triple product of $n\\tau T$ called the *fusion product* is also sometimes use:\n",
"\n",
"\\begin{align}\n",
"n\\tau T \\geq 5\\times 10^{21}\\ {\\rm s\\cdot keV/m^3}.\n",
"\\end{align}\n",
"\n",
"The factor $Q$ is used to represent the ratio fo the power produced in the fusion reation to the power required to produce the fusion (heat). **This $Q$ factor is not to be confused with the $Q$ value for the release of binding energy.**\n",
"\n",
"The breakeven point is $Q=1$, and ignition occurs for $Q\\gg 1$. For controlled fusion produced in the laboratory, temperatures equivalent to $kT = 20\\ {\\rm keV}$ are satisfactory. For uncontrolled fusion (i.e., a H-bomb), high temperatures and densities are acheived over a very brief time by using a fission bomb. Fusion bombs do not produce radiation effects nearly as severe as those of fission bombs, because the primary products are not dangerously radioactive.\n",
"\n",
"### Controlled Thermonuclear Reactions\n",
"\n",
"A controlled thermonuclear reaction of nuclear fusion in the laboratory is one of the primary goals of science and engineering. Scientists do not expect to reach this goal for several more decades. The first fusion reaction will likely be the $D + T$ reaction. The tritium will be derived from two possible reactions:\n",
"\n",
"\\begin{align}\n",
"n + {\\rm ^6Li} &\\rightarrow {\\rm ^3H} + {\\rm ^4He}, \\\\\n",
"n + {\\rm ^7Li} &\\rightarrow {\\rm ^3H} + {\\rm ^4He} + n. \n",
"\\end{align}\n",
"\n",
"The lithium is required to generate the tritium. It is also used as the heat transfer medium and a neutron radiation shield. \n",
"\n",
"For $Q=1$, a product of a few times $10^{21}\\ {\\rm s\\cdot keV/m^3}$ will be required for a commercial reactor using $D+T$. There are two major schemes to control thermonuclear reactions:\n",
"\n",
"1. Magnetic confinement fusion (MCF), and\n",
"2. Inertial confinement fusion (ICF).\n",
"\n",
"```{margin}\n",
"**Magnetic Confinement of Plasma**\n",
"```\n",
"\n",
"The primary effort of research laboratories around the world for several years has been a device called the [tokamak](https://en.wikipedia.org/wiki/Tokamak). As many as six separate magnetic fields can be used to contain and heat the plasma.\n",
"\n",
"In a schematic cross section of a typical magnetic containment vessel, \n",
"\n",
"- The center is the location of the hot plasma where the $D+T$ reaction takes place. \n",
"- The plasma is surrounded by vaccum to keep out impurities that would poison the reaction.\n",
"- A wall surrounds the plasma and vacuum region, which would be subjected to intense radiation. The plasma must be kept from touching the enclosure.\n",
"- A lithium blanket is the next layer, which absorbs neutrons to breed more tritium.\n",
"- Next is a radiation shield to prevent radiation from reaching the magnets, which may be superconducting in a commerical reactor.\n",
"\n",
"The heating of the plasma to sufficiently high temperatures begins with the resistive heating from the electric current flowing in the plasma. This is insufficient to attain the high ignition temperature. As a result, there are two other schemes to add additional heat: (1) injection of high-energy $(40-120\\ {\\rm keV})$ neutral fuel atoms that interact with the plasma, and (2) radio-frequency (RF) induction heating of the plasma (similar to a microwave oven).\n",
"\n",
"The most significant fusion project in the near future is a large tokamak fusion reactor called the [**ITER**](https://en.wikipedia.org/wiki/ITER) (International Thermonuclear Experimental Reactor). It is currently under construction in France and is expected to generate self-sustained fusion power of $500\\ {\\rm MW}$ for $1000\\ {\\rm s}$.\n",
"\n",
"```{margin}\n",
"**Inertial Confinement**\n",
"```\n",
"The concept of intertial confinement fusion is to use an intense high-powered beam of heavy ions or a laser called a *driver* to implode a pea-sized target composed of $D+T$ to a density and temperature high enough to cause fusion ignition. \n",
"\n",
"Several US-based institutions are doing research and development in laser fusion. \n",
"\n",
"The [National Ignitition Facility](https://en.wikipedia.org/wiki/National_Ignition_Facility) (NIF) at Lawrence Livermore National Lab fires 192 lasers in to a high-Z cylinder, which produces x-rays. These x-rays heat the small fuel pellet containing the $D+T$. It produced $1.3\\ \\rm MJ$ of UV laser energy in 2010, just short of the $1.5\\ {\\rm MJ}$ needed for ignition. \n",
"\n",
"Sandia National Lab used a device called a [*Z-pinch*](https://en.wikipedia.org/wiki/Z-pinch) that uses a huge jolt of current to produe a powerful magnetic field that squeezes ions inot implosion and heats the plasma. Fracne is building a device called [Laser Megajoule](https://en.wikipedia.org/wiki/Laser_M%C3%A9gajoule) with similar objectives as the NIF.\n"
]
},
{
Expand Down
Loading

0 comments on commit 800b82e

Please sign in to comment.