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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. d +9 E FIGURE 4.1 FIGURE 4.2 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 you'll see in Prob. 4.1, it is rea- sonable 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 = Ee, where E, is the field pro- duced by the electron cloud. Now the field at a distance d from the center of a uniformly charged sphere is 1 qd E. 47T €0 a3 (Prob. 2.12). At equilibrium, then, E = 1 qd or p= qd = (4r€ga³)E. 47€0 a3 4.1 Polarization 169 The atomic polarizability is therefore a = 47 €ga = 3eOv, (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 four or so for many simple atoms.
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. d +9 E FIGURE 4.1 FIGURE 4.2 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 you'll see in Prob. 4.1, it is rea- sonable 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 = Ee, where E, is the field pro- duced by the electron cloud. Now the field at a distance d from the center of a uniformly charged sphere is 1 qd E. 47T €0 a3 (Prob. 2.12). At equilibrium, then, E = 1 qd or p= qd = (4r€ga³)E. 47€0 a3 4.1 Polarization 169 The atomic polarizability is therefore a = 47 €ga = 3eOv, (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 four or so for many simple atoms.
The charge density of the electron cloud of the hydrogen atom is -4-2r/a Pe(r) (1) Ta3 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 [E, + E.(d)] (2) where Ee(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.
Transcribed Image Text:The charge density of the electron cloud of the hydrogen atom is -4-2r/a Pe(r) (1) Ta3 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 [E, + E.(d)] (2) where Ee(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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