The charge density of the electron cloud of the hydrogen atom is -4-2r/a Pe(r) = (1) with q > 0 the charge of the proton, and a is the Bohr radius. Compute the polar- izability of the hydrogen atom. Use the approximation of Example 4.1. Assume the electron cloud is unchanged, but is displaced relative to the proton by -d = -dk with |d < a by an external electric field Eo = Eok. Then, in equilibrium, the force on the proton is F, = 0 = q [Eo +E.(d)] (2) where E(d) is the electric field created at position d by just the electron charge density Pe(r) (tacking origin of the spherical coordinate system still at the center of the electron charge distribution). Hence use Gauss' Law to find the electric field created by the electron charge density, and then expand your result in powers of d/a « 1.
The charge density of the electron cloud of the hydrogen atom is -4-2r/a Pe(r) = (1) with q > 0 the charge of the proton, and a is the Bohr radius. Compute the polar- izability of the hydrogen atom. Use the approximation of Example 4.1. Assume the electron cloud is unchanged, but is displaced relative to the proton by -d = -dk with |d < a by an external electric field Eo = Eok. Then, in equilibrium, the force on the proton is F, = 0 = q [Eo +E.(d)] (2) where E(d) is the electric field created at position d by just the electron charge density Pe(r) (tacking origin of the spherical coordinate system still at the center of the electron charge distribution). Hence use Gauss' Law to find the electric field created by the electron charge density, and then expand your result in powers of d/a « 1.
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![**Example 4.1**: A primitive model for an atom consists of a point nucleus (+q) surrounded by a uniformly charged spherical cloud (−q) of radius a (Fig. 4.1). Calculate the atomic polarizability of such an atom.
Figures:
- **Figure 4.1**: Displays a sphere with radius a, representing the electron cloud with charge −q surrounding a point nucleus with charge +q at the center.
- **Figure 4.2**: Shows the effect of an external electric field E. The nucleus is displaced by a small distance d to the right, while the electron cloud remains centered, indicating polarization.
**Solution**:
In the presence of an external field E, the nucleus will be shifted slightly to the right and the electron cloud to the left, as shown in Fig. 4.2. (Because the actual displacements involved are extremely small, as shown in Prob. 4.1, it is reasonable to assume that the electron cloud retains its spherical shape.) Say that equilibrium occurs when the nucleus is displaced a distance d from the center of the sphere. At that point, the external field pushing the nucleus to the right exactly balances the internal field pulling it to the left: \(E = E_e\), where \(E_e\) is the field produced by the electron cloud. Now the field at a distance d from the center of a uniformly charged sphere is
\[ E_e = \frac{1}{4\pi\epsilon_0} \frac{qd}{a^3} \]
(Prob. 2.12). At equilibrium, then,
\[ E = \frac{1}{4\pi\epsilon_0} \frac{qd}{a^3} \]
or
\[ p = qd = (4\pi\epsilon_0 a^3)E \]
**4.1 Polarization**
The atomic polarizability is therefore
\[ \alpha = 4\pi\epsilon_0 a^3 = 3\epsilon_0 v \] \hfill (4.2)
where v is the volume of the atom. Although this atomic model is extremely crude, the result (Eq. 4.2) is not too bad—it's accurate to within a factor of two or so for many simple atoms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85e4b871-9d1c-4ae2-b799-fb57954f3d49%2F1450d8ce-ba43-4e09-bc61-43dc8adb1198%2F2t2owb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example 4.1**: A primitive model for an atom consists of a point nucleus (+q) surrounded by a uniformly charged spherical cloud (−q) of radius a (Fig. 4.1). Calculate the atomic polarizability of such an atom.
Figures:
- **Figure 4.1**: Displays a sphere with radius a, representing the electron cloud with charge −q surrounding a point nucleus with charge +q at the center.
- **Figure 4.2**: Shows the effect of an external electric field E. The nucleus is displaced by a small distance d to the right, while the electron cloud remains centered, indicating polarization.
**Solution**:
In the presence of an external field E, the nucleus will be shifted slightly to the right and the electron cloud to the left, as shown in Fig. 4.2. (Because the actual displacements involved are extremely small, as shown in Prob. 4.1, it is reasonable to assume that the electron cloud retains its spherical shape.) Say that equilibrium occurs when the nucleus is displaced a distance d from the center of the sphere. At that point, the external field pushing the nucleus to the right exactly balances the internal field pulling it to the left: \(E = E_e\), where \(E_e\) is the field produced by the electron cloud. Now the field at a distance d from the center of a uniformly charged sphere is
\[ E_e = \frac{1}{4\pi\epsilon_0} \frac{qd}{a^3} \]
(Prob. 2.12). At equilibrium, then,
\[ E = \frac{1}{4\pi\epsilon_0} \frac{qd}{a^3} \]
or
\[ p = qd = (4\pi\epsilon_0 a^3)E \]
**4.1 Polarization**
The atomic polarizability is therefore
\[ \alpha = 4\pi\epsilon_0 a^3 = 3\epsilon_0 v \] \hfill (4.2)
where v is the volume of the atom. Although this atomic model is extremely crude, the result (Eq. 4.2) is not too bad—it's accurate to within a factor of two or so for many simple atoms.
![**Charge Density of the Electron Cloud in Hydrogen Atom**
The charge density of the electron cloud of the hydrogen atom is given by:
\[
\rho_e(r) = \frac{-q}{\pi a^3} e^{-2r/a},
\]
where \( q > 0 \) is the charge of the proton, and \( a \) is the Bohr radius.
---
**Problem Statement:**
Compute the polarizability of the hydrogen atom. Use the approximation of Example 4.1.
**Assumptions:**
- The electron cloud remains unchanged but is displaced relative to the proton by \(-\mathbf{d} = -d\hat{k}\) with \( |d| \ll a \) due to an external electric field \(\mathbf{E}_0 = E_0 \hat{k}\).
**Equilibrium Condition:**
In equilibrium, the force on the proton is zero:
\[
F_p = 0 = q \left[\mathbf{E}_0 + \mathbf{E}_e(\mathbf{d})\right]
\]
where \(\mathbf{E}_e(\mathbf{d})\) is the electric field at position \(\mathbf{d}\) created by the electron charge density \(\rho_e(r)\), with the origin of the spherical coordinate system still at the center of the electron charge distribution.
**Task:**
Apply Gauss’s Law to find the electric field created by the electron charge density, then expand your result in powers of \(d/a \ll 1\).
---
The text includes mathematical expressions for the charge density and the equilibrium condition of a hydrogen atom under an applied electric field, explaining how to compute its polarizability using established approximations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85e4b871-9d1c-4ae2-b799-fb57954f3d49%2F1450d8ce-ba43-4e09-bc61-43dc8adb1198%2Fskcig0u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Charge Density of the Electron Cloud in Hydrogen Atom**
The charge density of the electron cloud of the hydrogen atom is given by:
\[
\rho_e(r) = \frac{-q}{\pi a^3} e^{-2r/a},
\]
where \( q > 0 \) is the charge of the proton, and \( a \) is the Bohr radius.
---
**Problem Statement:**
Compute the polarizability of the hydrogen atom. Use the approximation of Example 4.1.
**Assumptions:**
- The electron cloud remains unchanged but is displaced relative to the proton by \(-\mathbf{d} = -d\hat{k}\) with \( |d| \ll a \) due to an external electric field \(\mathbf{E}_0 = E_0 \hat{k}\).
**Equilibrium Condition:**
In equilibrium, the force on the proton is zero:
\[
F_p = 0 = q \left[\mathbf{E}_0 + \mathbf{E}_e(\mathbf{d})\right]
\]
where \(\mathbf{E}_e(\mathbf{d})\) is the electric field at position \(\mathbf{d}\) created by the electron charge density \(\rho_e(r)\), with the origin of the spherical coordinate system still at the center of the electron charge distribution.
**Task:**
Apply Gauss’s Law to find the electric field created by the electron charge density, then expand your result in powers of \(d/a \ll 1\).
---
The text includes mathematical expressions for the charge density and the equilibrium condition of a hydrogen atom under an applied electric field, explaining how to compute its polarizability using established approximations.
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