Bully Metric Bohr Model

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

Figure 1. 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 (see Figure 1). 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

Calculation of the orbits requires two assumptions, classical electromagnetism and a quantum rule.

• 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. This classical equation determines that the product of the orbital radius (r) with the square of the electron's momentum (p), is constant:

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

• a quantum rule

The magnitude of angular momentum is an integer (n) multiple of ħ:

$$L=rmv_{\perp }=rp_{\perp }=n\hbar .$$

For a circular orbit, the electron's total momentum (p) will always be perpendicular to the orbital radius (r), thus:

$rp=n\hbar .$

• a couple known limitations

Bohr's model assumes that the mass of the nucleus is much larger than the electron mass, allowing the nucleus to sit mostly stationary while the electron orbits around it. This limitation can be explicitly stated as:

$(Z+N)>>{\frac {m_{\mathrm {e} }}{u}}.$

where m_e is the electron's mass, Z is the number of protons, N is the number of neutrons, and u is the unified atomic mass unit.

Bohr's model does not include relativistic corrections that become necessary for a system where two charged points orbit each other at speeds approaching that of light. This limitation can be stated as follows:

$p<<m_{\mathrm {e} }c.$

where m_e is the electron's mass and c is the speed of light.

Conversion to Bully Metric Units

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 model becomes:

$rp^{2}=m_{\mathrm {e} }\,Z\alpha \hbar c=Z(\alpha )(m_{\mathrm {e} }c\hbar ).$

$rp=n\hbar .$

$(Z+N)>>{\frac {m_{\mathrm {e} }}{u}}.$

$p<<m_{\mathrm {e} }c.$

In Bully Metric units, the speed of light (c = 1 la / ta), the reduced Planck constant (ħ = 1 An), and the elementary charge (1 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 Quantization Rule in Bully Units

Figure 2. 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).

The quantization rule:

$rp=n\hbar .$

Can be written in Bully units as:

$rp=n\,An$

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 (see Figure 2). 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 (9.159 picometers), then the quantization of angular momentum would require the electron's perpendicular momentum to be at least 1 actionat per micropan, or in other words, a million actionats per length apan (1.151 × 10^-23 kg * (m / s)). The slope of negative one indicates that momentum in Bully units is proportional to the inverse of the radius in Bully units.

Bohr's Classical Electromagnetism in Bully Units

Figure 3. Bohr's model of the hydrogen atom on a log-log plot in Bully Metric units. The black line represents allowed radius-momentum value combinations according to Bohr's classical electromagnetism equation. The other lines represents allowed radius-momentum value combinations according to quantization of angular momentum. The points where the black line intersects with other lines are solutions (energy levels) of Bohr's model

Bohr's classical electromagnetism equation:

$rp^{2}=Z(\alpha )(m_{\mathrm {e} }c\hbar ).$

Can be written in Bully units as shown below (note that 137.035999177 is the inverse fine-structure constant and the value 23717311.411 is obtained from table 1 above):

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

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

$rp^{2}={\frac {23717311.411}{137.035999177}}{\frac {An^{2}}{la}}.$

When momentum and radius are plotted on a log-log graph using Bully units (see Figure 3), Bohr's classical electromagnetism 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).

Bohr Model Limitations in Bully Units

The identified limitations:

$(Z+N)>>{\frac {m_{\mathrm {e} }}{u}}.$

$p<<m_{\mathrm {e} }c.$

Can be written in Bully Units as:

$(Z+N)>>0.00054858.$

$p<<23717311.411{\frac {An}{la}}.$

Bohr Model Solutions in Bully Units

Table 1: Bohr Model Hydrogen Solutions

n	Momentum $\left({\frac {An}{la}}\right)$	Radius $\left(la\right)$
∞	0.000	∞
1000	173.074	5.777889273
100	1730.736	0.057778893
10	17307.358	0.000577789
9	19230.398	0.000468009
8	21634.198	0.000369785
7	24724.798	0.000283117
6	28845.597	0.000208004
5	34614.717	0.000144447
4	43268.396	0.000092446
3	57691.194	0.000052001
2	86536.792	0.000023112
1	173073.583	0.000005778

A solution (or energy level) of the Bohr model, is a point on the momentum-radius graph that satisfies both the classical electromagnetism equation and the quantization rule. Solutions of the Bohr model can be found algebraically through simple manipulation of Bohr's two equations:

$rp^{2}={\frac {23717311.411}{137.035999177}}{\frac {An^{2}}{la}}.$

$rp=n\,An$

From whence:

$r={\frac {(rp)^{2}}{rp^{2}}}={\frac {137.035999177\,n^{2}}{23717311.411}}la$

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

Figure 3 illustrates and Table 1 lists Bohr model solutions for the Hydrogen atom with principle quantum numbers one through ten (n = 1 .. 10), and for various powers of ten (n = 100, 1000, 10000, and 100000), and for infinity (solutions are marked with an asterisk(*) and labeled as "Energy Levels" in Figure 3).

Note that for the hydrogen atom (Z = 1, N = 0), the electron/nucleon mass ratio limitation is satisfied, but the situation improves with an increased number of nucleons.

$(1+0)>>0.00054858.$

The relativistic limitation is satisfied when n=1, but improves as n increases.

$173073.583{\frac {An}{la}}<<23717311.411{\frac {An}{la}}.$

Related constants

A trio of related constants are marked in Figure 3. These include the Bohr radius ($a_{0}$), the reduced Compton wavelength ($\lambda _{\mathrm {e} }/2\pi$), and the classical electron radius ($r_{\mathrm {e} }$). Any one of these constants can be written in terms of any of the others using the fine-structure constant $\alpha$:

$r_{\mathrm {e} }=\alpha {\frac {\lambda _{\mathrm {e} }}{2\pi }}=\alpha ^{2}a_{0}.$

The Bohr radius of 5.777 889 micropan is the smallest possible orbital radius for an electron in the Bohr hydrogen atom. Once an electron is in this lowest orbit, it can get no closer to the nucleus without violating one of Bohr's criteria.

$rp^{2}={\frac {23717311.411}{137.035999177}}{\frac {An^{2}}{la}}.$

$rp=An$

$p<<23717311.411{\frac {An}{la}}.$

$(Z+N)>>0.00054858.$

However, if one were to imagine a counterfactual universe where the electron is subject to Bohr's quantization rule, but is not subject to the classical electromagnetism equation, then the electron's orbit might slide down closer to the nucleus, to the reduced Compton wavelength of 42.163 295 nanopan as shown in Figure 3. The reduced Compton wavelength is a solution of the following equations:

$rp=An$

$p=23717311.411{\frac {An}{la}}.$

Or, if one were to imagine a counterfactual universe where the electron is subject Bohr's classical electromagnetism equation, but not subject to Bohr's quantization rule, then the electron's orbit might slide down even further to the classical electron radius of 0.307 680 nanopan. The classical electron radius is a solution of the following equations:

$rp^{2}={\frac {23717311.411}{137.035999177}}{\frac {An^{2}}{la}}.$

$p=23717311.411{\frac {An}{la}}.$

Calculation of energy levels

Potential energy (P) is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. In the Bohr model, the pertinent form of potential energy is electric potential ${\textstyle -{\frac {Zk_{\mathrm {e} }e^{2}}{r}}}$. The kinetic energy (K) of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is ${\textstyle {\frac {1}{2}}mv^{2}}$. The total energy (E) of the Bohr model atom is:

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

Multiplying both sides by the radius (r) and mass (m_e):

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

Note from a previous section that Bohr's classical electromagnetism equation requires:

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

From whence:

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

Thus:

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

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 and momentum are both zero, corresponding to a motionless electron infinitely far from the proton. Table 3 lists the same solutions as Table 2 (Bohr hydrogen atom energy level solutions in Bully Metric units), but Table 3 also lists the energy and electron velocity for each solution.

Table 3: 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 4 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 4: 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