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## nuclear physics

### 5. Series 21. Year - S. sequence, hot orifice and white dwarf

- Derive Taylor expansion of exponential and for $x=1$ graphically show sequence of partial sums of series \sum_{$k=1}^{∞}1⁄k!$ with series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$.

Using the same method compare series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$ and series of partial sums of series \sum_{$k=1}^{∞}x^{k}⁄k!$, therefore series {\sum_{$k=1}^{n}x^{k}⁄k!}_{n=1,2,\ldots}$, now for $x=-1$.

- The second task is to find concentration of electrons and positrons on temperature with total charge $Q=0$ in empty and closed cavity (you can choose value of $Q.)$ Further calculate dependence of ration of internal energy $U_{e}$ of electrons and positrons to the total internal energy of the system $U$ (e.g. the sum of energy of electromagnetic radiation and particles) on temperature and find value of temperature related to some prominent temperature and ratios (e.g. 3 ⁄ 4, 1 ⁄ 2, 1 ⁄ 4, …; can this ratio be of all values?).

Put your results into a graph – you can try also in 3-dimensions.

To get the calculation simplified, it could help to take some unit-less entity (e.g. $βE_{0}$ instead of $β$ etc.).

- Solve the system of differential equations for $M(r)$ and $ρ(r)$ in model of white dwarf for several well chosen values of $ρ(0)$ and for every value find the value which it get close $M(r)$ at

$r→∞$. This is probably equal to the mass of the whole star. Try to find the dependence of total weight on $ρ(0)$ and find its upper limit. Compare the result with the upper limit of mass for white dwarf (you will find it in literature or internet). Assume, that the star consists from helium.

Zadali autoři seriálu Marek Pechal a Lukáš Stříteský.

### 4. Series 21. Year - S. quantum harmonic oscillator

Calculate time dependence of wave function of particle, which is located in potential $V(x)=\frac{1}{2}kx$ and which is at time $τ=0$ described by wave function

$ψ_{R}(X,0)=\exp(-((X-X_{0}))⁄4)$,

ψ$_{I}(X,0)=0$.

It is wave packet with the center not in the origin. We can tell you, that this is so called *coherent stat of harmonic oscillator* and wave packet should oscillate around origin with angular frequency
√( $k⁄m)$ same as classical particle.

If you can calculate the previous, then you can try what will be the behaviour of wave packet of different width (e.g. denominator in exponential is different from 4) of how the behaviour will look like with different potential.

Zadal autor seriálu Marek Pechal.

### 2. Series 21. Year - S. cutting of wild plains

### Uranium storage

Very important question is storing of radioactive waste. Usually it is stored in cylindrical containers immersed in water, which keeps the surface at constant temperature 20 °C. Your task is to find the temperature distribution inside containers of square base of edge length 20 cm. Container is relatively long, therefore just temperature distribution in horizontal cross section is of interest. Uranium will be in block of square base of edge 5 cm. From the experience with cylindrical capsules we know, that it will have constant temperature of about 200 °C.

### Heating wire

Lets have a long wire of circular cross section and radius $r$ from a material of heat conductivity $λ$ and specific conductivity $σ$. Then a electric field is applied. Lets the electric field inside the wire is constant and parallel with the axis of the wire and the strength is $E$. Then the current through wire will be $j=σE$ and will create Joule's heat with volume wattage $p=σE$.

Because the material of the wire has non-zero temperature conductivity, some equilibrium gradient of temperature will form. The gradient fulfills Poisson's equation $λΔT=-p$. Assume, that the end of wire is kept at temperature $T_{0}$. This gives a border condition needed to solve the equation. Due to symmetry we can take into account only two dimensions: on cross section of wire (temperature will be independent of shift along the axis of symmetry). Now it is easy to solve the problem with methods described in text.

However, we will make our situation little bit more complex and will assume, that specific electrical conductivity $σ$ is function of temperature. So we will have a equation of type Δ$T=f(T)$.

Try to solve this equation numerically and solve it for some dependency of conductivity on temperature (find it on internet, in literature of just pick some nice function) and found temperature profile in wire profile. Try to change intensity of electric field $E$ and plot volt-amper characteristics, you can try more than one temperature dependency. $σ(T)$ (e.g. semiconductor which conductivity increase with temperature, or metal, where conductivity is decreasing) etc.

Do not limit your borders, we would be glad for any good idea.

### Capacity of a cube

Calculate capacity of ideally conductive cube of edge length 2$a$ (2Ax2Ax2A). If you think, it is simple, try to calculate for cuboid (AxBxC) or other geometrical shapes.

**Hint:**
Capacity is a ration of the charge on the cube to the potential on the surface of cube (assuming that the potential in infinity is zero). Problem can be solved by selecting arbitrary potential of cube and solving Laplace equation Δ$φ=0$ outside of the cube and calculating total charge in cube using Gauss law. E.g. calculating intensity of electrical field and derivating potential and calculation of flow through nicely selected surface around the cube.

Final solution is finding a physical model, its numerical solution and realization on computer. More points you will get for deeper physical analysis and detailed commentar. For algorithm you can also get extra points.

Zadal spoluautor seriálu Lukáš Stříteský.

### 4. Series 20. Year - 3. finishing of Temelin (nuclear power station)

Estimate thickness of water needed to shield radiation from nuclear reactor of power 980 MW in proposed new block of nuclear power station Temelin. The total energy released by fission from Uranium is split following: 82% goes to kinetic energy of fragments, 6% goes to neutrinos, 6% goes to neutrons and remaining 6% goes to energy of gamma photons.

**Hint:** Probability of particle passing through a material into the depth $d$ is probably equal to e^{$-σnd}$, where $n=N/V$ is density of molecules of material (in our case number of water molecules in 1 m3 ) and $σ$ is effective cross section for absorption of particle on molecule of water. Unit of cross section is surface, therefore m^{2} but often is used unit barn = 100 fm2) and depends on energy of particles. The values of cross section can be found on Internets or in tables.

Úloha řešil Karel Tůma na zkoušce z jaderné fyziky.

### 5. Series 19. Year - 4. natural nuclear reactor

The typical composition of natural uranium is 0.72 % of isotope ^{235}U of halftime of 704 million years and the rest is isotope ^{238}U with half life of 4 468 million years.

While mining uranium in 1970's in Okla in equatorial Gabon, the uranium ore with relative content of isotope ^{235}U 0.44 % . This discrepancy can be explained by assuming existence of 'natural nuclear reactor' at the uranium deposits million years ago.

Calculate, how long the nuclear reaction was going on if the decay of ^{235}U was started by slow neutrons. The collision of slow neutron with the nucleus happens every 352 thousand years.

Úloha ze zápočtové písemky z jaderné fyziky.

### 6. Series 18. Year - 3. space probe from NASA

In Jet Propulsion Laboratory in California, U.S.A. in NASA laboratory the new rocket engine is under development. It uses momentum of $α-particles$ created during radioactive decay of fermium $^{257}_{100}Fm_{157}$, which mass is $m_{Fm}$ and half-life $T$. The second product is californium $^{253}_{98}Cf_{155}$. The mass of $α-particle$ is $m_{α}$, the mass of californium is $m_{Cf}$, and during the decay the energy $E$ is released. Assume, that each $α-particle$ leaves rocket in the same direction.

The space probe with above engine is in rest at the beginning and its mass is $M$, the mass of 'fuel' is also $M$. Calculate the speed of the probe $v$ after half of the fermium decays. Resulting speed calculate also for the following numerical values $E=1,106\cdot 10^{-12}J$, $M=4\;\mathrm{kg}$ a $T=100,5days$, for other values consult your table-book.

SR olympiáda.

### 5. Series 18. Year - 3. beta decay

When measuring decay of neutron to electron and proton the energy of the electron was detected. How can be detected, that another particle was not created? Assume the neutron to be at rest at the beginning.

Pavel Augustinský

### 4. Series 18. Year - 4. Mössbauer effect

The frequency of photon emitted by the nucleus of radioactive iron is not always the same, but is slightly different (it is true also for other elements). To make thinks simpler, assume that the energy of photon in the frame connected with the resting nucleus of iron is randomly in interval ( $E_{0}-ΔE,E_{0}+ΔE)$, where $E_{0}=14,4\;\mathrm{keV}$ (keV = kiloelektronVolt), $ΔE≈10^{-8}\;\mathrm{eV}$ (1 eV = 1,602 \cdot 10^{-19} J ).

- When the photon is emitted from a stationary nucleus the nucleus acquires opposite momentum to the photon. Calculate kinetic energy of the atom and compare it with the $ΔE$.
- So called Mössbauer effect is the transfer of momentum of recoil to the crystal (whose the atom is part of). Calculate kinetic energy of the crystal (the shift in photon energy) assuming that the crystal consist of 10^{23} atoms.

Same as the emission of photon also excitation occurs. The photon can be absorbed only if its energy in the rest frame of atom is in interval ( $E_{0}-ΔE,E_{0}+ΔE)$.

- Decide if the resting atom of iron can absorb photon emitted by another resting atom of iron.
- Calculate the relative speed of two pieces of iron needed for Doppler effect to forbid the absorption of the photon in the second piece of iron. The Doppler effect is the change in the frequency $f$, of the photon when the source is coming closer to the observer at the speed $v$. The frequency is changed to

$f′=(1+v⁄c)f$.

Assume that the Mössbauer effect takes place at the emission.

Find all the needed constants in tables.

Navrhl Pavel Augustinský.