(a) The Coulomb energy of a uniformly charged sphere of radius r^', i.e., the energy required to

(a) The Coulomb energy of a uniformly charged sphere of...

(a) The Coulomb energy of a uniformly charged sphere of radius $r^{\prime}$, i.e., the energy required to assemble the charge, is $$ V=\frac{3}{5} \frac{Z^{2} e^{2}}{4 \pi \epsilon_{0} r^{\prime}} $$ Take $r^{\prime}=1.1 A^{1 / 3} \mathrm{~F}$, which is consistent with the electron scattering measurements, and show that $V$ then assumes the form of the Coulomb term of the semiempirical mass formula. (b) Evaluate, in mass units, the coefficient of $Z^{2} / A^{1 / 3}$ in the expression obtained for $V$, and compare with the empirical value of the coefficient $a_{3}$ given in (15-31).

(a) The Coulomb energy of a uniformly charged sphere of radius $r^{\prime}$, i.e., the energy required to assemble the charge, is
$$
V=\frac{3}{5} \frac{Z^{2} e^{2}}{4 \pi \epsilon_{0} r^{\prime}}
$$
Take $r^{\prime}=1.1 A^{1 / 3} \mathrm{~F}$, which is consistent with the electron scattering measurements, and show that $V$ then assumes the form of the Coulomb term of the semiempirical mass formula. (b) Evaluate, in mass units, the coefficient of $Z^{2} / A^{1 / 3}$ in the expression obtained for $V$, and compare with the empirical value of the coefficient $a_{3}$ given in (15-31).

Key Concepts

Coulomb Energy

Coulomb energy refers to the electrostatic energy stored in a distribution of electric charge. In the context of nuclei, it is the energy required to assemble a uniformly charged sphere from an initially dispersed distribution of protons. This energy is calculated by integrating the repulsive force between infinitesimal elements of charge over the entire sphere and plays a significant role in determining the binding energy of the nucleus.

Uniform Charge Distribution

The concept of a uniformly charged sphere assumes that the positive charge (due to protons) is spread evenly throughout the volume of the nucleus. This simplification allows for the derivation of analytical expressions for the Coulomb energy and leads to results that are consistent with experimental observations. It is critical for understanding various nuclear properties, as it provides a manageable model for the complex distribution of charge inside the nucleus.

Nuclear Radius Scaling

Nuclear radius scaling involves the empirical relationship r = r?A^(1/3), where A is the mass number and r? is a constant approximately equal to 1.1 fm. This scaling law reflects the roughly constant density of nuclear matter across different nuclei and is central to many models in nuclear physics, including the derivation of the Coulomb term in the semiempirical mass formula.

Semiempirical Mass Formula

The semiempirical mass formula (SEMF) or liquid drop model provides an approximate description of the binding energy of nuclei by incorporating various macroscopic terms, such as volume, surface, Coulomb, asymmetry, and pairing energies. The Coulomb term, derived from the uniformly charged sphere model, represents the energy cost associated with the electrostatic repulsion among protons and helps explain trends in nuclear stability and binding energy.

Dimensional Analysis and Unit Conversion

Dimensional analysis in nuclear physics involves ensuring that the derived expressions, like the Coulomb energy term, are consistent in terms of physical units when comparing theoretical predictions with experimental data. Converting quantities such as fm (femtometers) to mass units and using appropriate constants (like the Coulomb constant and the elementary charge) is crucial for accurately evaluating coefficients, such as the one appearing in the Coulomb term of the SEMF.

Transcript

00:01 Hi there.

00:01 So for this problem, for part a, we are told that the column energy of an uniformly charged sphere of radius r prime, and that is the energy required to assemble the charge, is the following expression, that is, the potential is equal to 3 divided by 5 times the atomic number to the square, times the charge of an electron to the square and this divided by 4 times pi times epsilon sub 0 times r prime we need to take that r prime is equal to 1 .1 times a elevated to 1 divided by 3 and then which is consistent with the electron scattering measurements and show that the potential then assumes the form of the column term of the semi -empirical mass formula.

01:08 So in this case, what we need to do is to simply evaluate the function that we are given for the potential.

01:18 So we will have 3 divided by 5.

01:22 In the denominator, we will have 4 times pymptons epsilon sub 0, and this times 1 .1 times 10 to the minus 15.

01:33 For the value that we are given.

01:38 And now in the numerator we have the charge of an electron square and then this times the atomic number to the square times a elevated to minus 1 divided by 3.

01:51 So from this, we can simplify this as just a times the atomic number elevated to the square times a elevated to minus 1 divided by 3.

02:05 And this has the same form as the column term of the mass formula.

02:11 So that's a solution for part a of this problem.

02:16 Now for part b, we are asked about to evaluate in mass units the coefficient of the atomic number to the square divided by a, elevated to 1 divided by 3 in the expression that we obtained from before, and compared with the empirical value of the coefficient a3 that is given in 15.

02:36 Now, the energy coefficient in the potential above is the following...

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