P111B Homework #2 (due Monday January 22, 2024) 1. **D-D fusion reactions. Deuterium-deuterium (D-D) fusion reactions...

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P111B Homework #2 (due Monday January 22, 2024) 1. **D-D fusion reactions. Deuterium-deuterium (D-D) fusion reactions have two "branches" that occur with approximately equal probability. The two reactions are D+DT+P and D+D He+N, where T, P, and N stand for triton (3H), proton, and neutron, respectively. The figure on the next page show data from an experiment that measured the properties of the D- D fusion reaction. The data were produced by shooting a 35 microAmp D+ beam into a 2-mm thick chamber with very cold D gas of density 1.4 1016 atoms/cm. (a) Examine the top figure on the next page. Sketch what the figure would look like if the He+N branch was half as likely as the T+P branch. (b) How many neutrons per second are produced when using a 116 keV D beam? (c) Assume that they used a standard time interval for all of their mea- surements and that they measured 800 3He counts in their 45 detector when they used a 110 keV beam. About how many proton counts did they measure with the 110 keV beam in the 22 detector? The 90 detector? (d) The authors used this parametric fit to their differential cross section data: do = a + bcos + c cos 0 where a, b, and c are constants. Once they determined a, b, and c, what formula did they use to determine the total cross section? For the 117 keV beam, their fitted coefficients in mb/sr (= millibar/sterradian) for the proton-triton branch are a = 1.243, b = 0.55, and c = 0.34. What is the total cross section for this case? 2. *NIF neutron detector. In the national ignition facility (NIF), fusion reac- tions take place in a capsule that is compressed to high densities and tem- peratures in a very small volume, like a small bomb that explodes in the center of the chamber. One of the detectors is 4.5 m from the fuel capsule and has an area of 40 20 mm. If the detector records 431 counts produced Assume that all of their detectors subtended the same solid angle. 1 c.m. Differential Cross Section (mb/sr) 1 COUNTS -600 3HE T 400 200 Ed = 90 keV lab 0 = 45 0 100 200 300 400 CHANNEL 24 110 keV nHe 20 1.8 1.6 pH Integrated Cross Section (mb) 10 H(d,n)He 0 20 40 60 80 100 120 0.8 0 20 40 60 80 100 120 140 160 180 c.m. Angle (deg) Ed (keV) Figure 1: (top) Counts measured in a silicon detector at an angle of 45 when the D gas target was bombarded by a 90 keV D beam. The x-axis is proportional to the energy of the incident particles. The nominal energies of the 3He, T, and P products in a D-D reaction are 0.82, 1.01, and 3.03 MeV. (bottom left) Measured differential cross section do/d as a function of 0. The curves are fits to the data. (bottom right) Total cross section o as a function of the energy of the incident D beam. by neutrons in a particular explosion, how many neutrons did the explosion produce? 3. **Electric breakdown. In the presence of an electric field, arcs occur when an "avalanche" leads to exponential increase in the number of free electrons. Consider a lone electron in a neutral gas. The electric field will accelerate it until it collides with a molecule. When it collides with the molecule, the electron often loses most of its energy. If, however, it has gained enough energy to ionize the molecule, then two electrons now start from nearly zero energy and are accelerated (once again) by the electric field. If they both gain enough energy to ionize molecules before they collide with them, then the two electrons create four free electrons, the four electrons create eight electrons, and so on. Under these conditions, an avalanche that creates a lightning-like electric arc occurs. (a) Find a formula for the critical electric field E in terms of the molecule's ionization energy Wion, the pressure p and temperature T of the gas, and other parameters. (b) Use your formula to estimate the critical electric field in N2 at STP. The ionization energy for a nitrogen molecule is about 15.6 eV. [Hint: consult EXAMPLE 14.3 for additional relevant parameters.]4 (c) In many physics experiments, a high voltage is applied to a pair of conductors that are separated by a partial vacuum. If the voltage ex- ceeds a certain threshold, breakdown" (i.e, an arc) is likely to occur. The dependence of this critical voltage on the gas pressure has an in- teresting behavior that can be understood in terms of the electron's mean-free path. When the chamber is pumped down, as the pressure is lowered from atmospheric pressure, the breakdown voltage decreases, owing to the effect considered in part (a). However, at some point, the electron's mean-free path exceeds the distance between the metal conductors. When this happens, a free electron becomes less likely to collide with a molecule, so it becomes less likely to contribute to an avalanche. (i) Sketch the expected dependence of the critical voltage as a function of pressure in the chamber. (ii) Use the parameters from 2 Assume an isotropic reaction. 3 Assume that an experiment is being conducted in pure nitrogen without any other gas such as 4The actual value differs from this estimate. 02. 3 part (b) to estimate the pressure (in atmospheres) where the minimum of the voltage-pressure curve occurs for metal plates that are separated by 10 cm.5 4. ***Inelastic bounce off a sphere. In an actual bounce off a surface, two quantities matter: the magnitude of the normal force (which is determined by the coefficient of restitution) and the magnitude of the tangential force (which is determined by the coefficient of friction). Reconsider Exam- ple 14.5 but change the nature of the bounce. Assume that the coefficient of friction is zero (as in Example 14.5) but that the coefficient of restitution is v, final = EV1, initial with 0 < 1. (Example 14.5 assumes = 1.) (a) Make a sketch that shows that the relationship between the incident angle d'incident and the final angle a final is tan incident = tan a final [Hint: Decompose the velocity vectors.] (b) Write a program to solve for the differential cross section do/d nu- merically. [Hint: My program follows the following steps. (i) Given b, find Cincident. (ii) Given Cincident, find . (iii) Use 0(b) to find | db/d0|. (iv) For convenience, I normalized the impact parameter 6 by the ra- dius R.] (c) Show that your program reproduces Eq. (14.25) when = 1. (d) Plot the differential cross section vs. 0 for = 0.3, 0.7 and 1.0. (e) Interpret your graph. Why does do/d shift the way it does? 5. *Greenhouse effect. In general, cross sections are energy dependent. An example of great practical importance is the absorption of photons by the "greenhouse gases" water, carbon dioxide, and methane. Their cross sec- tions are relatively small at visible wavelengths, allowing much of the in- cident solar radiation to reach the surface of the earth. However, since the earth is much colder than the sun, it radiates longer wavelength infrared light (primarily 8-30 m) that is often absorbed by atmospheric greenhouse gases, warming the atmosphere and ultimately the earth. (a) Use the graphs that appear at the end of this file to make an order-of- magnitude estimate of the mean-free path of visible light for absorp- 5As for part (b), your answer will differ from the true value. "Include a copy of the commands you used when you turn in your solution. Mathematica is the preferred programming language but you may use another language if you prefer. 4 tion by carbon dioxide, verifying that an increase in CO2 concentra- tion has a negligible effect on the incident radiation from the sun. The concentration of CO2 in the upper atmosphere is around 0.04%. (b) Water molecules, carbon dioxide, and methane in Earth's atmosphere absorb about 40% of the energy that the Earth tries to radiate at in- frared wavelengths. If the concentration of greenhouse gases increases by 20%, what is the absorbed fraction of energy now? "Because water vapor is an important greenhouse gas, in reality, the effect of an increase in methane and CO2 is amplified or diminished by its effect on the H2O concentration. Ignore this subtlety and lump all three major constituents together for your estimate. 5 CO2 Extinction deCross Section (cm) Density, kg/m 10 10 -25 -26 Shardanand and Rao, 1997 Sneep and Ubachs, 2005 He et al. 2018 Jordan et al. 2019 Wilmouth et al.2019 This study (BBCES) (a) O This study (CRD) n-based calculation -1.0 350 400 450 90 550 600 650 700 10 -6 10 10-8 10-10 wavelength (nm) Density vs Altitude NRLMSISE-00 Model - F10.7 = F10.7 = 140, Ap = 15 avg 12 80 100 120 140 160 180 200 Altitude, km 220 240 260 280 6 300 P111B Homework #2 (due Monday January 22, 2024) 1. **D-D fusion reactions. Deuterium-deuterium (D-D) fusion reactions have two "branches" that occur with approximately equal probability. The two reactions are D+DT+P and D+D He+N, where T, P, and N stand for triton (3H), proton, and neutron, respectively. The figure on the next page show data from an experiment that measured the properties of the D- D fusion reaction. The data were produced by shooting a 35 microAmp D+ beam into a 2-mm thick chamber with very cold D gas of density 1.4 1016 atoms/cm. (a) Examine the top figure on the next page. Sketch what the figure would look like if the He+N branch was half as likely as the T+P branch. (b) How many neutrons per second are produced when using a 116 keV D beam? (c) Assume that they used a standard time interval for all of their mea- surements and that they measured 800 3He counts in their 45 detector when they used a 110 keV beam. About how many proton counts did they measure with the 110 keV beam in the 22 detector? The 90 detector? (d) The authors used this parametric fit to their differential cross section data: do = a + bcos + c cos 0 where a, b, and c are constants. Once they determined a, b, and c, what formula did they use to determine the total cross section? For the 117 keV beam, their fitted coefficients in mb/sr (= millibar/sterradian) for the proton-triton branch are a = 1.243, b = 0.55, and c = 0.34. What is the total cross section for this case? 2. *NIF neutron detector. In the national ignition facility (NIF), fusion reac- tions take place in a capsule that is compressed to high densities and tem- peratures in a very small volume, like a small bomb that explodes in the center of the chamber. One of the detectors is 4.5 m from the fuel capsule and has an area of 40 20 mm. If the detector records 431 counts produced Assume that all of their detectors subtended the same solid angle. 1 c.m. Differential Cross Section (mb/sr) 1 COUNTS -600 3HE T 400 200 Ed = 90 keV lab 0 = 45 0 100 200 300 400 CHANNEL 24 110 keV nHe 20 1.8 1.6 pH Integrated Cross Section (mb) 10 H(d,n)He 0 20 40 60 80 100 120 0.8 0 20 40 60 80 100 120 140 160 180 c.m. Angle (deg) Ed (keV) Figure 1: (top) Counts measured in a silicon detector at an angle of 45 when the D gas target was bombarded by a 90 keV D beam. The x-axis is proportional to the energy of the incident particles. The nominal energies of the 3He, T, and P products in a D-D reaction are 0.82, 1.01, and 3.03 MeV. (bottom left) Measured differential cross section do/d as a function of 0. The curves are fits to the data. (bottom right) Total cross section o as a function of the energy of the incident D beam. by neutrons in a particular explosion, how many neutrons did the explosion produce? 3. **Electric breakdown. In the presence of an electric field, arcs occur when an "avalanche" leads to exponential increase in the number of free electrons. Consider a lone electron in a neutral gas. The electric field will accelerate it until it collides with a molecule. When it collides with the molecule, the electron often loses most of its energy. If, however, it has gained enough energy to ionize the molecule, then two electrons now start from nearly zero energy and are accelerated (once again) by the electric field. If they both gain enough energy to ionize molecules before they collide with them, then the two electrons create four free electrons, the four electrons create eight electrons, and so on. Under these conditions, an avalanche that creates a lightning-like electric arc occurs. (a) Find a formula for the critical electric field E in terms of the molecule's ionization energy Wion, the pressure p and temperature T of the gas, and other parameters. (b) Use your formula to estimate the critical electric field in N2 at STP. The ionization energy for a nitrogen molecule is about 15.6 eV. [Hint: consult EXAMPLE 14.3 for additional relevant parameters.]4 (c) In many physics experiments, a high voltage is applied to a pair of conductors that are separated by a partial vacuum. If the voltage ex- ceeds a certain threshold, breakdown" (i.e, an arc) is likely to occur. The dependence of this critical voltage on the gas pressure has an in- teresting behavior that can be understood in terms of the electron's mean-free path. When the chamber is pumped down, as the pressure is lowered from atmospheric pressure, the breakdown voltage decreases, owing to the effect considered in part (a). However, at some point, the electron's mean-free path exceeds the distance between the metal conductors. When this happens, a free electron becomes less likely to collide with a molecule, so it becomes less likely to contribute to an avalanche. (i) Sketch the expected dependence of the critical voltage as a function of pressure in the chamber. (ii) Use the parameters from 2 Assume an isotropic reaction. 3 Assume that an experiment is being conducted in pure nitrogen without any other gas such as 4The actual value differs from this estimate. 02. 3 part (b) to estimate the pressure (in atmospheres) where the minimum of the voltage-pressure curve occurs for metal plates that are separated by 10 cm.5 4. ***Inelastic bounce off a sphere. In an actual bounce off a surface, two quantities matter: the magnitude of the normal force (which is determined by the coefficient of restitution) and the magnitude of the tangential force (which is determined by the coefficient of friction). Reconsider Exam- ple 14.5 but change the nature of the bounce. Assume that the coefficient of friction is zero (as in Example 14.5) but that the coefficient of restitution is v, final = EV1, initial with 0 < 1. (Example 14.5 assumes = 1.) (a) Make a sketch that shows that the relationship between the incident angle d'incident and the final angle a final is tan incident = tan a final [Hint: Decompose the velocity vectors.] (b) Write a program to solve for the differential cross section do/d nu- merically. [Hint: My program follows the following steps. (i) Given b, find Cincident. (ii) Given Cincident, find . (iii) Use 0(b) to find | db/d0|. (iv) For convenience, I normalized the impact parameter 6 by the ra- dius R.] (c) Show that your program reproduces Eq. (14.25) when = 1. (d) Plot the differential cross section vs. 0 for = 0.3, 0.7 and 1.0. (e) Interpret your graph. Why does do/d shift the way it does? 5. *Greenhouse effect. In general, cross sections are energy dependent. An example of great practical importance is the absorption of photons by the "greenhouse gases" water, carbon dioxide, and methane. Their cross sec- tions are relatively small at visible wavelengths, allowing much of the in- cident solar radiation to reach the surface of the earth. However, since the earth is much colder than the sun, it radiates longer wavelength infrared light (primarily 8-30 m) that is often absorbed by atmospheric greenhouse gases, warming the atmosphere and ultimately the earth. (a) Use the graphs that appear at the end of this file to make an order-of- magnitude estimate of the mean-free path of visible light for absorp- 5As for part (b), your answer will differ from the true value. "Include a copy of the commands you used when you turn in your solution. Mathematica is the preferred programming language but you may use another language if you prefer. 4 tion by carbon dioxide, verifying that an increase in CO2 concentra- tion has a negligible effect on the incident radiation from the sun. The concentration of CO2 in the upper atmosphere is around 0.04%. (b) Water molecules, carbon dioxide, and methane in Earth's atmosphere absorb about 40% of the energy that the Earth tries to radiate at in- frared wavelengths. If the concentration of greenhouse gases increases by 20%, what is the absorbed fraction of energy now? "Because water vapor is an important greenhouse gas, in reality, the effect of an increase in methane and CO2 is amplified or diminished by its effect on the H2O concentration. Ignore this subtlety and lump all three major constituents together for your estimate. 5 CO2 Extinction deCross Section (cm) Density, kg/m 10 10 -25 -26 Shardanand and Rao, 1997 Sneep and Ubachs, 2005 He et al. 2018 Jordan et al. 2019 Wilmouth et al.2019 This study (BBCES) (a) O This study (CRD) n-based calculation -1.0 350 400 450 90 550 600 650 700 10 -6 10 10-8 10-10 wavelength (nm) Density vs Altitude NRLMSISE-00 Model - F10.7 = F10.7 = 140, Ap = 15 avg 12 80 100 120 140 160 180 200 Altitude, km 220 240 260 280 6 300