Serial of year 35

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(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?

(10 points)2. Series 35. Year - S. compressing

What energy must a laser impulse lasting $10 \mathrm{ns}$ have in order for the shock wave generated by it to be able to heat the plasma to a temperature at which a thermonuclear fusion reaction can occur? What will be the density of the compressed fuel? Note: Assume that the initial plasma is a monatomic ideal gas.

(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)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)
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 {^{-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)5. Series 35. Year - S. stabilizing

  1. What intensity must a laser with a wavelength of $351 \mathrm{nm}$ have in order to stabilize a Rayleigh-Taylor (RT) instability using the surface ablation of a fuel pellet? Suppose the boundary between the ablator and DT ice is corrugated with a wavelength of
  1. $0,2 \mathrm{\micro m}$,
  2. $5 \mathrm{\micro m}$.
  1. How will the intensity of the laser change if we also apply a magnetic field with magnitude $5 \mathrm{T}$?
  2. What else can help us minimize the RT instability?

(10 points)6. Series 35. Year - S. lasering

  1. How big must an aperture in a spatial filter be if we created it from a lens with a diameter of $40 \mathrm{cm}$ and its focal length is $4 \mathrm{m}$? Our Gaussian laser beam has an input diameter $30 \mathrm{cm}$ and a wavelength $1~053 nm$. The radius of the focus (parameter $\sigma $) of the Gaussian beam can be obtained using

\[\begin{equation*} r = \frac {2}{\pi }\lambda \frac {f}{D} \end {equation*}\] where $D$ is the diameter of the beam, $f$ is the focal length of the lens and $\lambda $ is the wavelength of the laser.

  1. The laser beam is focused on a surface of a nuclear fuel pellet of a $1 \mathrm{mm}$ diameter. What energy should it have in order for the intensity in its focus to reach $10^{14}^{-2}$? The radius of the focus is $25 \mathrm{\micro m}$ and a pulse lasts $10 \mathrm{ns}$. How many beams do we need to equally cover the surface of a pellet? What is their total energy?
    What energy must the laser beam have if it is not focused on a surface of a nuclear fuel pellet, but the beam diameter matches exactly the diameter of the pellet and the density is its focus reaches $10^{14}^{-2}$? Assume that we have one such beam and it shines homogenously on the pellet „from all directions“.
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