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 71 mla (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 5.7778 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

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 Template:Nowrap is an integer multiple of ħ:

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

This equation determines the electron's momentum at any radius, 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 equation determines the square of the electron's momentum at any radius, for each integer n:

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

For each integer n, we now have two equations with two unknowns, r and p.

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 turned 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 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 our 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

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 la-2	Template:Val	An ta-1	Template:Val	ta-1

The Quantization Rule

Bohr's quantum rule:

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

Can be written in Bully units as:

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

File:Bully Metric Quantization of Angular Momentum.png

Quantization of angular momentum demands an integer value for the product of orbital radius and 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).

The lowest energy level (n = 1), when represented in Bully Metric units, has the property that the momentum perpendicular to the radius is always, of necessity, equal to the numerical inverse of the radius. This is not a special property of the Bohr atom, but rather, is a universal property of quantum mechanics called quantization of angular momentum. For example, if an electron were to orbit a nucleus at 1 micropan (0.000001 la), then the electron's perpendicular momentum would be, of necessity, 1 actionat per micropan (1000000 An / la). When 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 (negative one indicating that the momentum is proportional to the inverse of the radius), each line representing an integer value of the principle quantum number n.

Bully Classical Electromagnetism

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:

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

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

File:Bully Metric Bohr Model Hydrogen Atom.png

Bohr Model of a Hydrogen atom on a log-log plot in the Bully Metric system of measurement units

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 plotted on a log-log graph using Bully units, the equation for momentum squared, of a hydrogen atom, appears as a straight line with a slope of negative two (negative two indicating that the momentum squared is proportional to the inverse of the radius). The point on the line where the radius equals one length apan (r = 1 la), at that point the momentum is equal to the square root of the ratio of the electron mass and inverse fine-structure constant, or in other words 416 (416 = sqrt(23717311.411 / 137.035999177)). Each point where the electromagnetism equation line intersects with an angular momentum quantization line is a solution for a specific value of n. See Table 2 for a list of Bohr hydrogen atom energy level solutions in Bully Metric units. See Table 3 for 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
4	0.001824	-39.468	43268.396	0.000092446
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

File:Hydrogen transitions.svg

Electron transitions and their resulting wavelengths for hydrogen. Energy levels are not to scale.

Table 2: Bohr Model Energy Levels

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	561.025966 -561.323981 =0.298015	87.684591 -87.706872 =0.022281
n→2	473.364938 -473.617109 =0.252171