Bully Metric Bohr Model

The following text was copied from the Wikipedia article about the Bohr model and was adapted to use Bully Metric Units:

The Bohr model of the hydrogen atom (Z = 1) or a hydrogen-like ion (Z > 1), where the negatively charged electron confined to an atomic shell encircles a small, positively charged atomic nucleus and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount of electromagnetic energy (hν). The orbits in which the electron may travel are shown as grey circles; their radius increases as n², where n is the principal quantum number. The 3 → 2 transition depicted here produces the first line of the Balmer series, and for hydrogen (Z = 1) it results in a photon of wavelength 71 millapan (656 nanometer red light).

In atomic physics, the Bohr model or Rutherford–Bohr model was the first successful model of the atom. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's nuclear model. It supplanted the plum pudding model of J J Thomson only to be replaced by the quantum atomic model in the 1920s. It consists of a small, dense nucleus surrounded by orbiting electrons. It is analogous to the structure of the Solar System, but with attraction provided by electrostatic force rather than gravity, and with the electron energies quantized (assuming only discrete values).

Development

In 1913 Niels Bohr put forth three postulates to provide an electron model consistent with Rutherford's nuclear model:

1. The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what classical electromagnetism suggests. These stable orbits are called stationary orbits and are attained at certain discrete distances from the nucleus. The electron cannot have any other orbit in between the discrete ones.
2. The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant: $m_{\mathrm {e} }vr=n\hbar$, where $n=1,2,3,...$ is called the principal quantum number, and $\hbar =h/2\pi$. The lowest value of $n$ is 1; this gives the smallest possible orbital radius, known as the Bohr radius, of 5.777 889 micropan (52.917 721 picometers) for hydrogen. Once an electron is in this lowest orbit, it can get no closer to the nucleus.
3. Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency $\nu$ determined by the energy difference of the levels according to the Planck relation: $\Delta E=E_{2}-E_{1}=h\nu$, where $h$ is the Planck constant.

Calculation of the orbits

The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. Calculation of the orbits requires two assumptions, a quantum rule and classical electromagnetism.

• A quantum rule

The magnitude of angular momentum L = m_evr is an integer multiple of ħ:

$L=rm_{\mathrm {e} }v=n\hbar .$

This quantum rule determines the electron's momentum (p) at any radius (r), for each integer n:

$p=m_{\mathrm {e} }v={\frac {n\hbar }{r}}.$

• classical electromagnetism

The electron is held in a circular orbit by electrostatic attraction. The centripetal force is therefore equal to the Coulomb force.

${\frac {m_{\mathrm {e} }v^{2}}{r}}={\frac {Zk_{\mathrm {e} }e^{2}}{r^{2}}},$

where m_e is the electron's mass, e is the elementary charge, k_e is the Coulomb constant and Z is the atom's atomic number. It is assumed here that the mass of the nucleus is much larger than the electron mass (which is a good assumption). This classical equation determines the square of the electron's momentum (p) at any radius (r), for each integer n:

$p^{2}=(m_{\mathrm {e} }v)^{2}=m_{\mathrm {e} }\,{\frac {Zk_{\mathrm {e} }e^{2}}{r}}.$

Calculation of energy levels

Classical energy is the sum of kinetic and potential energy. Classical kinetic energy is equal to one half of the mass multiplied by the velocity squared. And from the previous section, the momentum squared turns out to be equal to the Coulomb potential multiplied by the electron mass.

$E=K+P={\frac {m_{\mathrm {e} }v^{2}}{2}}-{\frac {Zk_{\mathrm {e} }e^{2}}{r}}={\frac {p^{2}}{2m_{\mathrm {e} }}}-{\frac {p^{2}}{m_{\mathrm {e} }}}=-{\frac {p^{2}}{2m_{\mathrm {e} }}}$

The total energy here is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the atom. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. It will be advantageous to represent the Coulomb constant k_e in terms of the Reduced Planck constant ħ, the speed of light c, the elementary charge e, and the fine-structure constant α.

$k_{\mathrm {e} }={\frac {1}{4\pi \varepsilon _{0}}}={\frac {\alpha \hbar c}{e^{2}}}$

From whence Bohr's three equations become:

$p={\frac {n\hbar }{r}}.$

$p^{2}=m_{\mathrm {e} }\,{\frac {Z\alpha \hbar c}{r}}.$

$E=-{\frac {p^{2}}{2m_{\mathrm {e} }}}$

Conversion to Bully Metric Units

The Quantization Rule

Quantization of angular momentum demands an integer value for the product of orbital radius with the momentum perpendicular to the radius. This appears as a series of parallel straight lines on a log-log plot. The above graphic includes plots for principle quantum numbers one through ten (n = 1 .. 10), and for various powers of ten (n = 100, 1000, 10000, and 100000).

Bohr's quantization rule:

$p={\frac {n\hbar }{r}}.$

Can be written in Bully units as:

$p={\frac {n}{r}}\,An$

This rule is not a special property of the Bohr atom, but rather, is a universal property of quantum mechanics called quantization of angular momentum. This rule has an extremely simple form when momentum and radius are plotted on a log-log graph using Bully units. The quantization of angular momentum appears as a series of parallel straight lines with a slope of negative one, each line representing an integer value of the principle quantum number n. The lowest energy level (n = 1) has the property that the momentum is always equal to the numerical inverse of the radius. For example, if an electron were to orbit a nucleus at 1 micropan (0.000001 la), then the quantization of angular momentum would require the electron's perpendicular momentum to be 1 actionat per micropan, or in other words a million actionats per length apan (1000000 An / la). The slope of negative one indicates that momentum is proportional to the inverse of the radius.

Bully Classical Electromagnetism

In Bully Metric units, the speed of light (c = 1.0 la / ta), the reduced Planck constant (ħ = 1.0 An), and the elementary charge (1.0 e) are all normalized, which means that many of the electron's properties carry the same numeric value but with differing units as shown in Table 1.

Table 1: Electron Properties

Electron Mass (m)		Rest Energy (mc2)		(mcħ)
23717311.411	An ta la-2	23717311.411	An ta-1	23717311.411	An^2 la-1

Bohr's model of the hydrogen atom on a log-log plot in the Bully Metric units. The black line represents allowed radius-momentum value combinations according to Bohr's classical electromagnetism equations. The other lines represents allowed radius-momentum value combinations according to quantization of angular momentum.

Bohr's classical electromagnetism equations:

$p^{2}=m_{\mathrm {e} }\,{\frac {Z\alpha \hbar c}{r}}.$

$E=-{\frac {p^{2}}{2m_{\mathrm {e} }}}$

Can be written in Bully units as shown below (note that 137.035999177 is the inverse fine-structure constant):

$p^{2}={\frac {Z}{r}}{\frac {23717311.411}{137.035999177}}{\frac {An^{2}}{la}}.$

$E=-{\frac {Z}{r}}{\frac {1}{2\times 137.035999177}}{\frac {An\,la}{ta}}.$

For a hydrogen atom with one proton (Z = 1), this becomes:

$p^{2}={\frac {1}{r}}{\frac {23717311.411}{137.035999177}}{\frac {An^{2}}{la}}.$

$E=-{\frac {1}{r}}{\frac {1}{2\times 137.035999177}}{\frac {An\,la}{ta}}.$

When momentum and radius are plotted on a log-log graph using Bully units, Bohr's classical electromagnetism momentum equation appears as a straight line with a slope of negative two (negative two indicating that momentum squared is proportional to the inverse of the radius).

Bully Metric Solutions for Bohr's Hydrogen Atom

$p={\frac {n}{r}}\,An$

$p^{2}={\frac {1}{r}}{\frac {23717311.411}{137.035999177}}{\frac {An^{2}}{la}}.$

$E=-{\frac {1}{r}}{\frac {1}{2\times 137.035999177}}{\frac {An\,la}{ta}}.$

• The solution

The above two equations are sufficient to find exact r and p values (two equations in two unknowns) for each given integer n. Dividing momentum squared by momentum, the radius dependence drops out:

$p={\frac {p^{2}}{p}}={\frac {1}{n}}{\frac {23717311.411}{137.035999177}}{\frac {An}{la}}.$

$v={\frac {-2E}{p}}=-{\frac {1}{n}}{\frac {1}{137.035999177}}{\frac {la}{ta}}.$

See Table 2 for the list of Bohr hydrogen atom energy level solutions in Bully Metric units. Table 3 provides a list of photons that are emitted or absorbed when an electron transitions to a different energy level within the Bohr hydrogen atom.

Table 2: Bohr Model Hydrogen Energy Levels

n	Velocity $\left({\frac {la}{ta}}\right)$	Energy $\left({\frac {An}{ta}}\right)$	Momentum $\left({\frac {An}{la}}\right)$	Radius $\left(la\right)$
∞	0.000000	0.000	0.000	∞
1000	0.000007	-0.001	173.074	5.777889273
100	0.000073	-0.063	1730.736	0.057778893
10	0.000730	-6.315	17307.358	0.000577789
9	0.000811	-7.796	19230.398	0.000468009
8	0.000912	-9.867	21634.198	0.000369785
7	0.001042	-12.888	24724.798	0.000283117
6	0.001216	-17.541	28845.597	0.000208004
5	0.001459	-25.260	34614.717	0.000144447
4	0.001824	-39.468	43268.396	0.000092446
3	0.002432	-70.165	57691.194	0.000052001
2	0.003649	-157.872	86536.792	0.000023112
1	0.007297	-631.489	173073.583	0.000005778

Table

Table 3: Photon

Transition	Lyman series (n=1)	Balmer series (n=2)	Paschen series (n=3)	Brackett series (n=4)
n→∞	631.152904 631.489478 0.336574	157.875323 157.872370 -0.002954	70.143290 70.165498 0.022207	39.468831 39.468092 -0.000738
n→9	623.360648 623.693312 0.332664	150.038067 150.076203 0.038136	62.346214 62.369331 0.023117	31.670641 31.671926 0.001285
n→8	621.290915 621.622455 0.331540	147.967622 148.005346 0.037724	60.282375 60.298474 0.016099	29.601623 29.601069 -0.000554
n→7	618.272041 618.601938 0.329896	144.948283 144.984829 0.036546	57.259259 57.277957 0.018698	26.567662 26.580552 0.012890
n→6	613.620732 613.948104 0.327372	140.295678 140.330995 0.035317	52.601056 52.624123 0.023067	21.922116 21.926718 0.004602
n→5	605.906685 606.229899 0.323214	132.579027 132.612790 0.033764	44.887329 44.905918 0.018590	14.205272 14.208513 0.003242
n→4	591.705868 592.021386 0.315518	118.373611 118.404277 0.030666	30.690963 30.697405 0.006442
n→3	561.024872 561.323981 0.299109	87.684591 87.706872 0.022281
n→2	473.364899 473.617109 0.252210