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:

File:Bohr atom model.svg

The Bohr model of the hydrogen atom (Template:Nowrap) or a hydrogen-like ion (Template:Nowrap), 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 Template:Nowrap transition depicted here produces the first line of the Balmer series, and for hydrogen (Template:Nowrap) it results in a photon of wavelength 656 nm (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 5.77788928 µla (0.0529 nm) for hydrogen. Once an electron is in this lowest orbit, it can get no closer to the nucleus. Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper.
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.

Additional point:

1. Like Einstein's theory of the photoelectric effect, Bohr's formula assumes that during a quantum jump a discrete amount of energy is radiated. However, unlike Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels; Bohr did not believe in the existence of photons.

Calculation of the orbits

• classical electromagnetism

The electron is held in a circular orbit by electrostatic attraction. The centripetal force is 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 equation determines the electron's speed at any radius:

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

It also determines the electron's total energy at any radius:

$E=-{\frac {1}{2}}m_{\mathrm {e} }v^{2}.$

The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton.

• A quantum rule

The angular momentum Template:Nowrap is an integer multiple of ħ:

$m_{\mathrm {e} }vr=n\hbar .$

Conversion to Bully Metric Units

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

Table 1: Electron Properties

Electron Mass		Rest Energy (mc2)		Angular Frequency (${\frac {mc^{2}}{\hbar }}$)
Template:Val	An ta / la2	Template:Val	An / ta	Template:Val	ta-1

Bully Quantum Rule

Bohr's quantum rule:

$m_{\mathrm {e} }vr=n\hbar .$

Can be written in Bully Units as:

$vr={\frac {n}{23717311.411}}\,{\frac {la^{2}}{ta}}$

Bully Classical Electromagnetism

Bohr's classical velocity equation:

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

Can be written in Bully Units as:

$v^{2}r={\frac {Zk_{\mathrm {e} }e^{2}}{23717311.411{\frac {An\,ta}{la^{2}}}}}.$

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:

$v^{2}r={\frac {Z{\alpha }}{23717311.411}}{\frac {la^{3}}{ta^{2}}}.$

Substituting in the value for the inverse fine-structure constant:

$v^{2}r={\frac {Z}{23717311.411\times {137.035999177}}}{\frac {la^{3}}{ta^{2}}}.$

Bully Bohr Atom

Bohr's two equations written in Bully Metric format are:

$|v|r={\frac {n}{23717311.411}}\,{\frac {la^{2}}{ta}}$

$v^{2}r={\frac {Z}{23717311.411\times {137.035999177}}}{\frac {la^{3}}{ta^{2}}}.$

Where r is the radius of the Bohr orbital, v is the electron velocity as it orbits the nucleus, n is the principal quantum number and Z is the atom's atomic number. From these two equations, one can determine the radius (r), velocity (v), and energy (E) of each of the Bohr electron orbits.

$r={\frac {(vr)^{2}}{v^{2}r}}={\frac {137.035999177\,({\frac {n^{2}}{Z}})}{23717311.411}}la.$

$|v|={\frac {v^{2}r}{|v|r}}={\frac {\frac {Z}{n}}{137.035999177}}{\frac {la}{ta}}.$

$E={\frac {1}{2}}mv^{2}={\frac {23717311.411\,({\frac {Z}{n}})^{2}}{(137.035999177)^{2}}}{\frac {An}{ta}}.$

Table 2: Bohr Model Energy Levels

n	Velocity $\left({\frac {la}{ta}}\right)$	Energy $\left({\frac {An}{ta}}\right)$	E0 + E	Momentum $\left({\frac {An}{la}}\right)$	Radius $\left(la\right)$
∞	0.000000	0.000	23717311.411	0.000	∞
1000	0.000007	-0.001	23717311.410	173.074	5.777889
100	0.000073	-0.063	23717311.348	1730.736	0.057779
10	0.000730	-6.315	23717305.096	17307.358	0.000578
9	0.000811	-7.796	23717303.615	19230.398	0.000468
8	0.000912	-9.867	23717301.544	21634.198	0.000370
7	0.001042	-12.888	23717298.523	24724.798	0.000283
6	0.001216	-17.541	23717293.870	28845.597	0.000208
5	0.001459	-25.260	23717286.151	34614.717	0.000144
4	0.001824	-39.468	23717271.943	43268.396	0.000092
3	0.002432	-70.165	23717241.246	57691.194	0.000052
2	0.003649	-157.872	23717153.539	86536.792	0.000023
1	0.007297	-631.489	23716679.922	173073.583	0.000006