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## nuclear physics

### 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.