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A Generalization of the Bohr model for Hydrogen-like
Atoms and Ions for Application to the Structure of Nuclei

The Bohr model of a hydrogen-like ion predicts that the total energy E of an electron is given by

E = −Z²R/n²

where Z is the net charge experienced by the electron, n is the principal quantum number and R is a constant equal to approximately 13.6 electron volts (eV). This formula is the result of the total energy being equal to

E = − Ze²/(2rn)

where e is the charge of the electron and rn is the orbit radius when the principal quantum number is n. The orbit radius is given by

rn = n²h/(Zmee²)

where h is Planck's constant divided by 2π and me is the mass of the electron.

This model gives strikingly good results for the ionization energies of the electrons in atoms and ions. See Ionization. It works as well or better for multiple electrons in the outer shell as for single electrons there.

What follows is a generalization that may help explain the structure of nuclei.

Generalization

Consider a particle of mass m moving in a circular orbit of radius r around a particle of charge Z. The angular rate of rotation of the particle is ω. Let the force on the particle be represented as

F = −HZf(r)/r²

where H is a constant and the minus sign indicates that it is an attractive force. The function f(r) is for now completely arbitrary, and thus this formula is completely general for a central force.

Potential Energy

For the above force the potential energy V(r) is

V(r) = ∫rF(s)ds
= ∫r[−HZ(f(s)/s²)]ds
= −HZ∫r(f(s)/s²)ds

Quantization

The angular momentum is quantized. The tangential velocity is ωr and thus angular momentum is mωr². Quantization means

mωr² = nh
and hence
ω = nh/(mr²)
and for later use
ω² = n²h²/m²r4

where n is an integer, called the principal quantum number for the system, and h is Planck's constant divided by 2π.

Orbital Balance

A balance of centrifugal force with the attractive force on the particle requires

mω²r = HZf(r)/r²
which means that
ω² = HZf(r)/(mr³)

The two expression derived above for ω² can be equated to give

HZf(r)/(mr³) = n²h ²/(m²r4)
which reduces to
rf(r) = n²h²/(HZm)

This is the quantization condition for r. Only a discrete set of value of r satisfy this relation. Therefore the quantization condition should be expressed as

rnf(rn) = n²h²/(HZm)

to indicate that it holds only for discrete values. The corresponding quantization condition for ω is

ωn = nh/(mrn²)

If the quantization condition for r is divided on both sides by rn³ the result is

f(rn)/rn² = (n²h²/(mHZ))/rn³
or, equivalently
HZf(rn)/rn² = (h²/m)n²/rn³

The Left-hand-side (LHS) of the above equation is the magnitude of the force on the particle. This means at discrete points the force is equal to

− (h²/m)n²/rn³

If this were true at all values of r then this expression could be substituted for the force formula in the equation for potential energy. This would mean that

*V(r) = −HZ∫r(f(s)/s²)ds
= −HZ(n²h²/(HZm))∫r(1/s³)ds
which reduces to
*V(r) = −(n²h²/m)[−½(1/s²)]r

which evaluates to
*V(r) = −½(n²h²/m)/r²

The asterisk preceding the above formulas is to indicate that the formula is not strictly true because the quantization condition holds only at discrete points. However there is an approximation that exists. The potential energy is the area under the curve for force, such as the one shown below, using a positive force rather than a negative force which would be appropriate for the case being considered. The red vertical lines indicate the partition of the range of distances from the center of attraction. If the tops of the red lines are connected a set of trapezoids would be formed whose areas would closely approximate the area under the force curve.

There is a trapezoid formula for the approximation of an integral which can be applied for the set of points {rn for n=1, 2, …} where the force is equal to −((h²/m))(n²/r³). Thus an approximation for the potential energy is thus derived.

In particular when the particle moves from an orbit for quantum number n to (n+1) the change in potential energy ΔV is approximately equal to the area of the trapezoid; i.e.,

(h²/m)½[n²/rn² + (n+1)²/rn+1²](rn+1−rn)

Bounds for the Potential Energy

It was previously found that for any q (h²/m)q²/r³ matches the magnitude of the force F(r) at rq. As the following graph illustrates, for values of r such that rn<r<rn+1

kn²/rn³ < F(r) < k(n+1)²/rn+1³

where for convenience (h²/m) is denoted as k.

Therefore the integrals of these functions over the interval are ordered in the same way. The integral of F(r) is the change in the potential energy ΔV over the interal. The integral

∫(1/r³)dr
evaluates to
−½(1/r²)

Thus

½kn²[1/rn²−1/rn+1²] < ΔV
and
ΔV < ½k(n+1)²[1/rn² − 1/rn+1²]

Kinetic Energy

The kinetic energy K of the system is given by

K = ½mω²r²
= ½(n²h²/m)/r²

Thus for a change in the quantum number from n to (n+1)

ΔK = ½(h²/m)[(n+1)²/rn+1²−n²/rn²]

A comparison of the formula for ΔK with the trapezpidal approximation for ΔV(r) gives an approximation of the spectrum because any discrepancy between ΔV and ΔK is compensated for by the emission or adsorption of a photon.

(To be continued.)


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