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## nuclear physics

### (7 points)6. Series 35. Year - 4. short half-life

What is the probability that three-quarters of the initial one mole of atoms decay during one half-life? Commonly it happens only after two half-lives. What could cause such a situation?

### (10 points)4. Series 35. Year - S. shining

1. How far from the surface of the target (suppose it is made of carbon and the laser has wavelength of $351 \mathrm{nm}$) is critical surface situated and how far does two-plasmon decay occur, if the characteristic length of plasma1)
1)
The density of plasma $n_e$ is typically expressed as a funciton $n_e = f$\frac {x}{x_c}$$, where $x$ is the distance from the target and $x_c$ is so called characteristic length of plasma, which represents scale parameter for the distance from the target.))is~$50 \mathrm{\micro m}$? Next assume
1. that the density of the plasma decreases exponentially with distance from the target,
2. that the density of the plasma decreases linearly with distance from the target.
1. What energy must electorns have in order to go through the critical surface to the real surface of the target? To calculate the distance electron travels in carbon plasma use an empirical relationship $R = 0{,}933~4 E^{1{,}756~7}$, where $E$ has units of \jd {MeV} and $R$ has units of \jd {g.cm^{-2}}.
2. What is the distance that an electron has to travel in the electric field of the plasma wave in order to reach the energies determined in second exercise?
3. Which wavelengths of scattered light are present in the case of stimulated Raman scaterring for laser with wavelength of $351 \mathrm{nm}$?

### (10 points)3. Series 35. Year - S. igniting

1. Determine the reach of helium nuclei in central hot spot (using the figure ).
2. What energy must be released in the fusion reactions in order for the fusion to spread to the closest layer of the pellet? How thick is the layer?
3. Estimate the most probable amount of energy transferred from helium nucleus to deuterium. How many collisions on average does the helium nucleus undergo in the central hot spot before it stops?

### (10 points)2. Series 35. Year - P. le bomb

What is the maximum power of a nuclear bomb?

Karel was thinking about the american presidents.

### (10 points)1. Series 35. Year - S. commencing fusion

1. Determine the energy gain of the following reactions and the kinetic energy of their products

\begin{align*} {}^{2}\mathrm {D} + {}^{3}\mathrm {T} &\rightarrow {}^{4}\mathrm {He} + \mathrm {n} ,\\ {}^{2}\mathrm {D} + {}^{2}\mathrm {D} &\rightarrow {}^{3}\mathrm {T} + \mathrm {p} ,\\ {}^{2}\mathrm {D} + {}^{2}\mathrm {D} &\rightarrow {}^{3}\mathrm {He} + \mathrm {n} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {T} &\rightarrow {}^{4}\mathrm {He} + 2\mathrm {n} ,\\ {}^{3}\mathrm {He} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + 2\mathrm {p} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + \mathrm {n} + \mathrm {p} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + {}^{2}\mathrm {D} ,\\ \mathrm {p} + {}^{11}\mathrm {B} &\rightarrow 3\;{}^{4}\mathrm {He} ,\\ {}^{2}\mathrm {D} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + \mathrm {p} . \end {align*}

1. By using the graph of fusion reaction rate (sometimes called volume rate) as a function of temperature in the Serial study text, derive the Lawson criterion for the inertial-confinement-fusion time for a temperature of your choosing, while considering the following reactions:
1. deuterium - deuterium,
2. proton - boron,
3. deuterium - helium-3.

Determine the product of the size of a fuel pellet, and the density of a compressed fuel for each case. Are there any advantages of these reactions compared to the traditional DT fusion?

1. What form would the Lawson criterion take for the non-Maxwellian velocity distribution, considering the case with the following kinetic energy of a particle
1. $E\_k = k\_B T^\alpha$,
2. $E\_k = a T^3 + b T^2 + c T$.

Could such a fusion be even possible? If so, what (the fuel) should drive the fusion reaction, what is the ideal size of the fuel pellet and what density should it be compressed to?

### (7 points)6. Series 33. Year - 4. frightened hair

Thanks to joy from the end of an exam period, Danka's hair count begun to increase by a constant rate. Later she noticed that she lost one hair, which scared her. The more hair she lost the more she feels stressed, which increases hair loss rate. More precisely, the rate of hair loss is proportional to the number of already lost hair. The rate of new hair growth remains the same. Again, we are interested, when will her last hair fall out.

Jáchym desired to calculate this for a long time.

### (8 points)3. Series 31. Year - 5. decay here, decay there

We have $A_0$ particles which decay into $B$ particles with decay constant $\lambda \_A$. $B$ particles decay into $A$ particles with decay constant $\lambda \_B$. The number of $B$ particles at the beginning is $B_0$. Find a ratio of the numbers of particles $A$ and $B$ as a function of $t$.

### (12 points)1. Series 30. Year - E. Pechschnitte

Does bread always falls on the side that has the spread on it? Explore this Murphy's law experimentally with emphasis on statistics! Does it depend on the dimensions of the slice, or the composition and the thickness of the spread? Try to explain the experimental results with a theory. Use a sandwich bread.

Terka má stůl ve špatné výšce.

### (2 points)6. Series 29. Year - 1. It's about what's inside of us

In the year 2015, a Nobel prize for Physics was given for an experimental confirmation of the oscillation of neutrinos. You have probably already heard about neutrinos and maybe you know that they interact with matter very weakly so they can pass without any deceleration through Earth and similar large objects. Try to find out, using available literature and Internet sources, how many neutrinos are at any instant moment in an average person. Don't forget to reference the sources.

### (5 points)6. Series 29. Year - 5. Particle race

Two particles, an electron with mass $m_{e}=9,1\cdot 10^{-31}\;\mathrm{kg}$ and charge $-e=-1,6\cdot 10^{-19}C$ and an alpha particle with mass $m_{He}=6,6\cdot 10^{-27}\;\mathrm{kg}$ and charge 2$e$, are following a circular trajectory in the $xy$ plane in a homogeneous magnetic field $\textbf{B}=(0,0,B_{0})$, $B_{0}=5\cdot 10^{-5}T$. The radius of the orbit of the electron is $r_{e}=2\;\mathrm{cm}$ and the radius of the orbit of the alpha particle is $r_{He}=200\;\mathrm{m}$. Suddenly, a small homogeneous electric field $\textbf{E}=(0,0,E_{0})$, $E_{0}=5\cdot 10^{-5}V\cdot \;\mathrm{m}^{-1}$ is introduced. Determine the length of trajectories of these particles during in the time $t=1\;\mathrm{s}$ after the electric field comes into action. Assume that the particles are far enough from each other and that they don't emit any radiation.

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