A beam of α-particles, of kinetic energy 5.30 MeV and intensity 10^4 particle/sec, is incident n

A beam of α-particles, of kinetic energy 5.30 MeV and...

A beam of $\alpha$-particles, of kinetic energy $5.30 \mathrm{MeV}$ and intensity $10^{4}$ particle/sec, is incident normally on a gold foil of density $19.3 \mathrm{~g} / \mathrm{cm}^{3}$, atomic weight 197 , and thickness $1.0 \times 10^{-5} \mathrm{~cm}$. An $\alpha$ particle counter of area $1.0 \mathrm{~cm}^{2}$ is placed at a distance $10 \mathrm{~cm}$ from the foil. If $\Theta$ is the angle between the incident beam and a line from the center of the foil to the center of the counter, use the Rutherford scattering differential cross section, (4-9), to find the number of counts per hour for $\Theta=10^{\circ}$ and for $\Theta=45^{\circ} .$ The atomic number of gold is 79 .

Key Concepts

Solid Angle

Solid angle is a measure of the two-dimensional angle in three-dimensional space that an object subtends at a point, and it is measured in steradians. In scattering experiments, the detector's effective solid angle determines the fraction of scattered particles that are captured. By combining the differential cross section with the solid angle, one can compute the expected number of particles detected at a specific scattering angle.

Target Material Properties

The target’s properties, such as density, thickness, and atomic weight, are important for determining the number of scattering centers available for interaction with the incident particles. A thin foil target provides a controlled environment where the number density of nuclei can be accurately calculated, which is essential for predicting the number of scattering events using the Rutherford scattering formula.

Particle Beam Intensity

This term refers to the number of particles in the beam that impinge on the target per unit time. The intensity is crucial as it directly affects the rate at which scattering events occur. In experimental setups, knowing the beam intensity allows researchers to scale the theoretical scattering probabilities to actual count rates observed in detectors.

Differential Cross Section

The differential cross section is a measure of the probability per unit solid angle that a scattering event occurs at a specific angle. It quantifies how the intensity of scattered particles varies with angle and is a central quantity in scattering experiments. In the context of Rutherford scattering, the differential cross section helps predict the angular distribution of scattered particles, which is crucial for comparing theoretical predictions with experimental data.

Rutherford Scattering

This concept involves the deflection of charged particles by the Coulomb force exerted by the atomic nuclei. It was first explained by Ernest Rutherford and forms the basis of understanding how particles scatter off nuclei. The Rutherford scattering formula relates the number of particles scattered at a given angle to the properties of both the incident particle and the target nucleus, providing a way to probe the structure of the atom.

Alpha Particles

Alpha particles are helium nuclei, consisting of two protons and two neutrons. They are commonly used in scattering experiments due to their relatively high mass and charge, which make the effects of Coulomb repulsion clear in measurements. Their kinetic energy and interaction with target nuclei allow for detailed study of the nuclear structure and scattering dynamics.

Transcript

00:01 Hi there, so for this problem we are told that a beam of alpha particles of kinetic energy that is equal to 5 .3 megalletron balls and intensity of 10 to the 4 particle per second is incited normally on a gold foil of density, so the density of gold in this case, is given and that value is 19 .3 grams per cubic centimeter.

00:50 And atomic weight, in the atomic weight of this, is 197.

00:58 And thickness, the thickness is also given, and that is equal to 1 times 10 to the minus 5 centimeters.

01:08 An alpha particle counter of area 1 centimeter square is placed at a distance of 10 centimeters from the foil.

01:23 Now, if theta is the angle between the incident beam n aligned from the center of the foil to the center of the counter, we need to use rutherford scattering differential cross -ception, which is equation 4 .9 of the boot.

01:41 We define the number of counts per hour when theta is equal to 10 degrees and for theta equals to 45 degrees.

01:56 The atomic number of goal is also given and the atomic number of goal is 79.

02:02 So with this information, we need to obtain what we are asked for, which is the number of yes, the number of counts per hour.

02:16 So we start with the equation 4 .9, which is that the differential of the number is equal to 1 over 4 times pi times epsilon sub 0, this to the square times.

02:34 The atomic number times the charge of the electron square, divided by two times the mass times this speed square, all of this to the square times the neparian logarithm of 1 over the sign to the 4 of theta divided by 2.

02:57 And this times the differential in the solid angle.

03:03 Now the solid angle of the detector, that is the differential in this, is equal to the differential in the area, divided by the radius to the square, which is equal to, we know that the area in this case is given, that is one centimeter, and the radius of separation distance is 10 centimeters.

03:26 So we will elevate that to the square.

03:28 So from this, we obtain that this is equal to 10 to the minus 2 strat.

03:35 Now, also, we know that end, that we are going to call n, which is the number, of nuclei per cubic centimeter times the thickness, then what we, well, we are just to that product in here.

04:07 So n is just going to be 19 .3, and this divided by 197 times 1 .661 times 10 to the minus 24...

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