You are given a new unknown element of atomic weight 124 (For time being, assume that this element has not been discovered) and you wish to find out the radius of the nucleus of the atoms present in it. You do the following steps:

###### This problem is original, and is inspired from MIT 5.111 Principles of Chemical Science Lecture-2, where the students perform an in-class imitation of the scattering experiment and calculate the radius of the "ping-pong ball" nuclei.

###### Since this problem is very different from conventional ones, people are invited to take part in discussions, and also suitable corrections will be made, if necessary.

**Step 1** : You carefully cut out a \( 3 \) cm by \( 3 \) cm mono-layer of atoms from it, and weigh it accurately and precisely to get \( 0.6 \) nano grams. You can safely assume that the mono-layer looks similar to the figure above.

Note that this is **not** the actual mono-layer you have, but its lattice is similar to the above showed picture. We also assume that the atoms are spherical. The yellow dots represent the nuclei of the atoms.

**Step 2**: You put aside the mono-layer, and take a sample of radioactive radium bromide, which emits \( \alpha \) radiation. With the help of a Geiger counter ( or simply, a Detector ), you measure the count rate of this beam of \( \alpha \) particles to be \( 132000 \) hits min\(^{-1}\). The setup is something like this :

**Step 3**: Without further delay, you subject the mono-layer you have with the same radiation uniformly throughout its surface. You also place a Geiger counter beside the emitter to measure the count rate of rebounded \( \alpha \) particles, which you found out to be \( 2 \) hits min\(^{-1}\). The setup is something like this:

Now, given this information, you calculate the radius of nuclei of the atoms in the mono-layer to be some \( \lambda \times 10^{\beta} \) metres, where \( 1 < \lambda < 10 \). Enter \( \beta \) as your answer.

**Details and Assumptions**

Avogadro Number is taken as \( 6.023 \times 10^{23} \).

Pi is taken as \( \frac{22}{7} \).

\( \alpha \) particles are assumed to be point sized, when compared to large sized atoms in the mono-layer.

Step 2 and Step 3 are performed for the same amount of time.

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