Describe the mysteries of atom spectra.Explain Bohr’s theory of the hydrogen atom.Explain Bohr’s planetary model of the atom.Illustrate power state making use of the energy-level diagram.Describe the triumphs and limits the Bohr’s theory.

You are watching: For a hydrogen atom in its ground state, use the bohr model to compute each of the following.

The great Danish physicist Niels Bohr (1885–1962) made prompt use that Rutherford’s planetary version of the atom. (Figure 1). Bohr ended up being convinced the its validity and also spent component of 1912 at Rutherford’s laboratory. In 1913, after return to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based upon the planetary model of the atom. For decades, many questions had been asked about atomic characteristics. From their sizes to your spectra, lot was known about atoms, but small had been described in regards to the laws of physics. Bohr’s theory defined the atomic spectrum the hydrogen and established new and generally applicable ethics in quantum mechanics.

Figure 1. Niels Bohr, Danish physicist, used the planetary design of the atom to define the atomic spectrum and also size of the hydrogen atom. His numerous contributions to the breakthrough of atomic physics and also quantum mechanics, his an individual influence on countless students and colleagues, and his an individual integrity, specifically in the challenge of Nazi oppression, earned the a significant place in history. (credit: Unknown Author, via Wikimedia Commons)

Mysteries of atom Spectra

As detailed in Quantization of Energy, the energies that some small systems are quantized. Atomic and also molecular emission and absorption spectra have been recognized for end a century to it is in discrete (or quantized). (See number 2.) Maxwell and also others had actually realized the there have to be a connection in between the spectrum of one atom and also its structure, something prefer the resonant frequencies of musical instruments. But, regardless of years of initiatives by many great minds, no one had a workable theory. (It was a running joke that any type of theory of atomic and also molecular spectra can be damaged by throwing a publication of data in ~ it, so complex were the spectra.) complying with Einstein’s proposal the photons through quantized energies straight proportional to your wavelengths, it ended up being even more evident that electrons in atoms can exist just in discrete orbits.

Figure 2. Part (a) shows, from left to right, a discharge tube, slit, and also diffraction grating producing a line spectrum. Component (b) shows the emission line spectrum because that iron. The discrete lines indicate quantized power states because that the atoms that develop them. The heat spectrum because that each aspect is unique, giving a an effective and lot used analysis tool, and many line spectra were famed for numerous years before they can be defined with physics. (credit for (b): Yttrium91, Wikimedia Commons)

In some cases, it had actually been possible to devise formulas that described the emissions spectra. As you can expect, the easiest atom—hydrogen, v its single electron—has a fairly simple spectrum. The hydrogen spectrum had actually been observed in the infrared (IR), visible, and ultraviolet (UV), and also several collection of spectral lines had actually been observed. (See number 3.) These series are named after at an early stage researchers that studied castle in certain depth.

The it was observed hydrogen-spectrum wavelengths deserve to be calculated making use of the complying with formula:

displaystylefrac1lambda=Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\,

where λ is the wavelength of the emitted EM radiation and also R is the Rydberg constant, figured out by the experiment to be R = 1.097 × 107 / m (or m−1).

The constant nf is a positive integer linked with a particular series. For the Lyman series, nf = 1; because that the Balmer series, nf = 2; for the Paschen series, nf = 3; and so on. The Lyman collection is completely in the UV, while part of the Balmer collection is visible v the remainder UV. The Paschen collection and every the remainder are completely IR. Over there are reportedly an unlimited number of series, although castle lie progressively farther into the infrared and become challenging to observe together nf increases. The constant ni is a optimistic integer, yet it have to be greater than nf. Thus, for the Balmer series, nf = 2 and ni = 3, 4, 5, 6, …. Keep in mind that ni can approach infinity. If the formula in the wavelength equation was simply a recipe designed to fit data and also was not based upon physical principles, the did indicate a depths meaning. Balmer an initial devised the formula for his collection alone, and also it was later found to describe all the other collection by using different values the nf. Bohr to be the very first to understand the depths meaning. Again, we view the interplay between experiment and also theory in physics. Experimentally, the spectra were well established, an equation was found to to the right the experimental data, but the theoretical structure was missing.

Figure 3. A schematic the the hydrogen spectrum mirrors several collection named because that those who added most to their determination. Component of the Balmer collection is in the clearly shows spectrum, if the Lyman series is entirely in the UV, and also the Paschen series and others are in the IR. Worths of nf and also ni are shown for several of the lines.

Example 1. Calculating wave Interference that a Hydrogen Line

What is the distance between the slits that a grating that produces a first-order maximum for the second Balmer line at an edge of 15º?

Strategy and Concept

For an integrated Concept problem, we must first identify the physical principles involved. In this example, we require to understand two things:

the wavelength the lightthe conditions for an interference maximum because that the pattern from a dual slit

Part 1 deals through a object of the current chapter, while part 2 considers the wave interference product of wave Optics.

Solution for Part 1

Hydrogen spectrum wavelength. The Balmer collection requires the nf = 2. The an initial line in the series is required to be for ni = 3, and also so the second would have ni = 4.

The calculation is a straightforward applications of the wavelength equation. Start the determined values because that nf and also ni yields

eginarraylllfrac1lambda&=&Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\ ext &=&left(1.097 imes10^7 ext m^-1 ight)left(frac12^2-frac14^2 ight)\ ext &=&2.057 imes10^6 ext m^-1endarray\

Inverting to find λ gives

eginarrayllllambda&=½.057 imes10^6 ext m^-1=486 imes10^-9 ext m\ ext &=&486 ext nmendarray\

Discussion for Part 1

This is without doubt the experimentally observed wavelength, corresponding to the second (blue-green) heat in the Balmer series. Much more impressive is the reality that the same simple recipe predicts all of the hydrogen spectrum lines, including brand-new ones observed in succeeding experiments. What is nature telling us?

Solution for Part 2

Double-slit interference (Wave Optics). To acquire constructive interference for a twin slit, the path length difference from 2 slits must be one integral multiple of the wavelength. This problem was expressed by the equation d sin θ =, where d is the distance between slits and also θ is the angle from the initial direction that the beam. The number m is the order of the interference; m=1 in this example. Addressing for d and entering well-known values yields

displaystyled=fracleft(1 ight)left(486 ext nm ight)sin15^circ=1.88 imes10^-6 ext m\

Discussion for part 2

This number is similar to those supplied in the interference examples of advent to Quantum Physics (and is close come the spacing between slits in frequently used diffraction glasses).

Bohr’s solution for Hydrogen

Bohr had the ability to derive the formula because that the hydrogen spectrum using straightforward physics, the planetary version of the atom, and also some really important brand-new proposals. His first proposal is that only particular orbits space allowed: we say the the orbits of electron in atoms space quantized. Each orbit has a various energy, and also electrons can move to a higher orbit by taking in energy and drop come a reduced orbit by create energy. If the orbits are quantized, the quantity of energy absorbed or emitted is likewise quantized, producing discrete spectra. Photon absorption and emission are amongst the primary methods that transferring energy into and also out that atoms. The energies of the photons space quantized, and also their power is defined as being equal to the readjust in power of the electron when it move from one orbit to another. In equation form, this is Δ= hfEi − Ef.

Figure 4. The planetary version of the atom, as modified by Bohr, has the orbits that the electron quantized. Only specific orbits room allowed, explaining why atomic spectra room discrete (quantized). The energy brought away native an atom by a photon comes from the electron dropping indigenous one enabled orbit to another and also is hence quantized. This is similarly true for atomic absorption of photons.

Here, ΔE is the change in energy between the initial and also final orbits, and also hf is the energy of the soaked up or emitted photon. That is rather logical (that is, supposed from our everyday experience) that power is involved in an altering orbits. A blast of energy is compelled for the space shuttle, because that example, to rise to a higher orbit. What is not expected is the atomic orbits have to be quantized. This is no observed for satellites or planets, which can have any type of orbit given the proper energy. (See number 4.)

Figure 5 shows an energy-level diagram, a convenient way to screen energy states. In the present discussion, us take these to be the allowed energy level of the electron. Energy is plotted vertically with the shortest or floor state in ~ the bottom and with excited says above. Offered the energies that the lines in an atom spectrum, the is possible (although sometimes an extremely difficult) to identify the energy levels of an atom. Energy-level diagrams are used for plenty of systems, consisting of molecules and nuclei. A concept of the atom or any other device must suspect its energies based upon the physics that the system.

Figure 5. An energy-level chart plots power vertically and also is helpful in visualizing the energy states of a system and also the transitions in between them. This diagram is because that the hydrogen-atom electrons, reflecting a change between 2 orbits having actually energies E4 and E2.

Bohr to be clever enough to uncover a method to calculate the electron orbital energies in hydrogen. This was critical first step that has actually been improved upon, yet it is well worth repeating here, since it walk correctly define many features of hydrogen. Presume circular orbits, Bohr proposed that the angular inert L of an electron in that orbit is quantized, that is, it has only specific, discrete values. The worth for L is offered by the formula L=m_evr_n=nfrach2pileft(n=1,2,3,dots ight)\, where L is the angular momentum, me is the electron’s mass, rn is the radius the the n th orbit, and also h is Planck’s constant. Keep in mind that angular inert is . Because that a tiny object at a radius rImr2 and omega=fracvr\, so that L=left(mr^2 ight)fracvr=mvr\. Quantization claims that this worth of mvr have the right to only be equal to frach2,frac2h2,frac3h2\, etc. At the time, Bohr himself did not understand why angular momentum must be quantized, yet using this presumption he was able to calculate the energies in the hydrogen spectrum, something nobody else had actually done in ~ the time.

From Bohr’s assumptions, we will now derive a variety of important properties of the hydrogen atom indigenous the classic physics we have covered in the text. We begin by noting the centripetal pressure causing the electron to monitor a circular path is supplied by the Coulomb force. Come be much more general, we note that this evaluation is valid for any type of single-electron atom. So, if a nucleus has Z proton (= 1 because that hydrogen, 2 for helium, etc.) and only one electron, that atom is referred to as a hydrogen-like atom. The spectra of hydrogen-like ion are comparable to hydrogen, but shifted to higher energy through the better attractive force in between the electron and nucleus. The size of the centripetal force is fracm_ev^2r_n\, if the Coulomb pressure is kfracleft(Zq_e ight)left(q_e ight)r_n^2\. The tacit presumption here is that the cell nucleus is more massive 보다 the stationary electron, and the electron orbits around it. This is constant with the planetary design of the atom. Equating these,

kfracZq_e^2r_n^2=fracm_ev^2r_n ext (Coulomb = centripetal)\.

Angular inert quantization is proclaimed in an previously equation. We settle that equation for v, substitute it into the above, and rearrange the expression to attain the radius that the orbit. This yields:

displaystyler_n=fracn^2Za_ extB, ext for permitted orbits left(n=1,2,3dots ight)\,

where aB is defined to be the Bohr radius, due to the fact that for the shortest orbit (= 1) and for hydrogen (= 1), r1 = aB. The is left because that this chapter’s Problems and Exercises to show that the Bohr radius is

displaystylea_ extB=frach^24pi^2m_ekq_e^2=0.529 imes10^-10 ext m\.

These last 2 equations deserve to be supplied to calculation the radii that the enabled (quantized) electron orbits in any type of hydrogen-like atom. That is exceptional that the formula offers the correct size of hydrogen, which is measure up experimentally to be really close to the Bohr radius. The earlier equation additionally tells us that the orbital radius is proportional come n2, as shown in figure 6.

Figure 6. The allowed electron orbits in hydrogen have actually the radii shown. This radii were an initial calculated through Bohr and also are provided by the equation r_n=fracn^2Za_ extB\. The lowest orbit has the experimentally showed diameter the a hydrogen atom.

To gain the electron orbit energies, we begin by noting the the electron power is the amount of that is kinetic and also potential energy: En = KE + PE.

Kinetic power is the acquainted KE=frac12m_ev^2\, assuming the electron is not moving at relativistic speeds. Potential power for the electron is electrical, or PEqeV, where V is the potential as result of the nucleus, which looks choose a point charge. The nucleus has actually a positive charge Zqe ; thus, V=frackZq_er_n\, recalling an previously equation for the potential as result of a point charge. Since the electron’s fee is negative, we check out that PE=-frackZq_er_n\. Entering the expressions for KE and PE, we find


Now us substitute rn and also v from earlier equations right into the over expression because that energy. Algebraic manipulation yields

displaystyleE_n=-fracZ^2n^2E_0left(n=1,2,3,dots ight)\

for the orbital energies that hydrogen-like atoms. Here, E0 is the ground-state energy (= 1) for hydrogen (= 1) and is offered by

displaystyleE_0=frac2piq_e^4m_ek^2h^2=13.6 ext eV\

Thus, because that hydrogen,

displaystyleE_n=-frac13.6 ext eVn^2left(n=1,2,3dots ight)\

Figure 7. Energy-level diagram because that hydrogen showing the Lyman, Balmer, and also Paschen series of transitions. The orbit energies room calculated utilizing the above equation, an initial derived through Bohr.

Figure 7 shows an energy-level diagram for hydrogen that additionally illustrates exactly how the assorted spectral collection for hydrogen are concerned transitions in between energy levels.

Electron complete energies space negative, due to the fact that the electron is bound come the nucleus, analogous to gift in a feet without sufficient kinetic power to escape. Together n ideologies infinity, the complete energy becomes zero. This synchronizes to a cost-free electron with no kinetic energy, because rn it s okay very huge for large n, and the electrical potential energy thus becomes zero. Thus, 13.6 eV is necessary to ionize hydrogen (to walk from –13.6 eV to 0, or unbound), one experimentally proved number. Given an ext energy, the electron becomes unbound with some kinetic energy. For example, giving 15.0 eV come an electron in the soil state of hydrogen strips that from the atom and also leaves it v 1.4 eV of kinetic energy.

Finally, let us think about the energy of a photon emitted in a downward transition, offered by the equation to be ∆E = hfEi − Ef.

Substituting En = (–13.6 eV/n2), we view that

displaystylehf=left(13.6 ext eV ight)left(frac1n_ extf^2-frac1n_ exti^2 ight)\

Dividing both sides of this equation by hc gives an expression for frac1lambda\:

displaystylefrachfhc=fracfc=frac1lambda=fracleft(13.6 ext eV ight)hcleft(frac1n_ extf^2-frac1n_ exti^2 ight)\

It can be displayed that

displaystyleleft(frac13.6 ext eVhc ight)=fracleft(13.6 ext eV ight)left(1.602 imes10^-19 ext J/eV ight)left(6.626 imes10^-34 ext J cdot ext s ight)left(2.998 imes10^8 ext m/s ight)=1.097 imes10^7 ext m^-1=R\

is the Rydberg constant. Thus, we have actually used Bohr’s assumptions to derive the formula very first proposed by Balmer years earlier as a recipe to fit experimental data.

displaystylefrac1lambda=Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\

We watch that Bohr’s concept of the hydrogen atom answers the question regarding why this previously well-known formula describes the hydrogen spectrum. That is due to the fact that the energy levels space proportional to frac1n^2\, wherein n is a non-negative integer. A downward shift releases energy, and also so ni need to be better than nf. The various series are those whereby the transitions end on a certain level. Because that the Lyman series, nf = 1—that is, all the transitions end in the floor state (see likewise Figure 7). Because that the Balmer series, nf = 2, or all the transitions end in the first excited state; and so on. What was as soon as a cooking recipes is now based in physics, and something brand-new is emerging—angular momentum is quantized.

Triumphs and also Limits the the Bohr Theory

Bohr did what no one had been may be to perform before. Not just did he explain the spectrum that hydrogen, he appropriately calculated the size of the atom from basic physics. Few of his concepts are generally applicable. Electron orbit energies room quantized in every atoms and also molecules. Angular momentum is quantized. The electrons do not spiral right into the nucleus, as supposed classically (accelerated charges radiate, so that the electron orbits classically would decay quickly, and also the electrons would certainly sit top top the nucleus—matter would certainly collapse). This are major triumphs.

But there are borders to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s design is what we speak to semiclassical. The orbits space quantized (nonclassical) however are assumed come be straightforward circular (classical). As quantum mechanics to be developed, it became clear that there space no well-defined orbits; rather, there space clouds that probability. Bohr’s theory likewise did not define that some spectral lines space doublets (split right into two) as soon as examined closely. Us shall examine plenty of of these elements of quantum mechanics in more detail, but it have to be retained in mind the Bohr did no fail. Rather, that made very important actions along the course to better knowledge and also laid the structure for all of atomic physics the has due to the fact that evolved.

PhET Explorations: Models that the Hydrogen Atom

How go scientists number out the framework of atoms without looking in ~ them? shot out different models by shooting irradiate at the atom. Check how the forecast of the model matches the speculative results.


Click to download the simulation. Run making use of Java.

Section Summary

The planetary design of the atom photos electrons orbiting the cell nucleus in the way that planets orbit the sun. Bohr offered the planetary design to develop the an initial reasonable concept of hydrogen, the simplest atom. Atomic and also molecular spectra are quantized, through hydrogen spectrum wavelengths provided by the formulafrac1lambda =Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\,where λ is the wavelength of the emitted EM radiation and also R is the Rydberg constant, which has the value = 1.097 × 107 m−1.The constants ni and also nf are positive integers, and also ni need to be better than nf.Bohr appropriately proposed the the energy and radii that the orbits of electrons in atoms room quantized, with energy for transitions between orbits provided by ∆= hf = Ei − Ef, where ∆E is the change in energy in between the initial and final orbits and hf is the energy of an took in or emitted photon. That is advantageous to plot orbital energies top top a upright graph referred to as an energy-level diagram.Bohr proposed that the permitted orbits are circular and must have actually quantized orbit angular momentum given by L=m_e extvr_n=nfrach2pi left(n=1, 2, 3 dots ight)\, where L is the angular momentum, rn is the radius the the nth orbit, and h is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of one orbit is provided by r_n=fracn^2Za_ extBleft( extallowed orbits n=1, 2, 3, … ight)\Z is the atomic variety of an aspect (the number of electrons is has when neutral) and also aB is defined to be the Bohr radius, which is a_ extB=frach^24pi ^2m_e extkq_e^2= ext0.529 imes ext10^- ext10 ext m\.Furthermore, the energies that hydrogen-like atoms are provided by E_n=-fracZ^2n^2E_0left(n=1, 2, 3 … ight)\, where E0 is the ground-state energy and is provided by E_0=frac2pi ^2q_e^4m_ek^2h^2= ext13.6 eV\.Thus, because that hydrogen, E_n=-frac ext13.6 eVn^2left(n,=,1, 2, 3 … ight)\.The Bohr Theory provides accurate worths for the energy levels in hydrogen-like atoms, but it has actually been boosted upon in numerous respects.

Conceptual Questions

How perform the enabled orbits for electrons in atoms differ indigenous the allowed orbits because that planets around the sun? define how the post principle applies here.Explain how Bohr’s rule for the quantization of electron orbital angular momentum different from the actual rule.What is a hydrogen-like atom, and how are the energies and also radii of its electron orbits concerned those in hydrogen?

Problems & Exercises

By calculating that wavelength, present that the very first line in the Lyman collection is UV radiation.Find the wavelength the the third line in the Lyman series, and identify the kind of EM radiation.Look increase the worths of the quantities in a_ extB=frach^24pi ^2m_e extkq_e^2\ , and also verify that the Bohr radius aB is 0.529 × 10−10 m.Verify the the soil state energy E0 is 13.6 eV by using E_0=frac2pi ^2q_e^4m_ek^2h^2\.If a hydrogen atom has actually its electron in the = 4 state, how much power in eV is required to ionize it?A hydrogen atom in an excited state have the right to be ionized v less energy than as soon as it is in its floor state. What is n for a hydrogen atom if 0.850 eV of power can ionize it?Find the radius that a hydrogen atom in the = 2 state follow to Bohr’s theory.Show the fracleft(13.6 exteV ight)hc=1.097 imes10^7 ext m=R\ (Rydberg’s constant), as discussed in the text.What is the smallest-wavelength heat in the Balmer series? Is it in the visible component of the spectrum?Show the the whole Paschen collection is in the infrared component of the spectrum. To perform this, friend only have to calculate the shortest wavelength in the series.Do the Balmer and Lyman collection overlap? to answer this, calculate the shortest-wavelength Balmer line and also the longest-wavelength Lyman line.(a) Which heat in the Balmer collection is the very first one in the UV part of the spectrum? (b) How many Balmer collection lines are in the visible component of the spectrum? (c) How many are in the UV?A wavelength that 4.653 µm is it was observed in a hydrogen spectrum for a shift that ends in the nf = 5 level. What was ni because that the early stage level that the electron?A singly ionized helium ion has actually only one electron and also is denoted He+. What is the ion’s radius in the floor state contrasted to the Bohr radius that hydrogen atom?A beryllium ion with a single electron (denoted Be3+) is in an excited state v radius the very same as that of the ground state that hydrogen. (a) What is n for the Be3+ ion? (b) how much power in eV is necessary to ionize the ion indigenous this excited state?Atoms can be ionized by heat collisions, such together at the high temperatures found in the solar corona. One together ion is C+5, a carbon atom with only a solitary electron. (a) by what variable are the energies the its hydrogen-like levels better than those of hydrogen? (b) What is the wavelength of the an initial line in this ion’s Paschen series? (c) What type of EM radiation is this?Verify Equations r_n=fracn^2Za_ extB\ and a_B=frach^24pi ^2m_ekq_e^2=0.529 imes10^-10 ext m\ using the approach stated in the text. The is, equate the Coulomb and centripetal forces and also then insert an expression for velocity native the problem for angular momentum quantization.The wavelength the the 4 Balmer collection lines for hydrogen are uncovered to it is in 410.3, 434.2, 486.3, and also 656.5 nm. What mean percentage distinction is found between these wavelength numbers and also those guess by frac1lambda=Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\? the is amazing how well a basic formula (disconnected initially from theory) could duplicate this phenomenon.


hydrogen spectrum wavelengths: the wavelengths of visible light from hydrogen; deserve to be calculation by

displaystylefrac1lambda =Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)\

Rydberg constant: a physical continuous related come the atomic spectra with an created value of 1.097 × 107 m−1

double-slit interference: an experiment in which waves or corpuscle from a solitary source impinge upon 2 slits so that the result interference pattern might be observed

energy-level diagram: a diagram provided to analyze the power level of electrons in the orbits of an atom

Bohr radius: the typical radius that the orbit of one electron about the cell core of a hydrogen atom in its ground state

hydrogen-like atom: any atom with just a single electron

energies the hydrogen-like atoms: Bohr formula because that energies the electron states in hydrogen-like atoms: E_n=-fracZ^2n^2E_0left(n= ext1, 2, 3,dots ight)\

Selected solutions to Problems & Exercises

1. displaystylefrac1lambda=Rleft(frac1n_ extf^2-frac1n_ exti^2 ight)Rightarrow lambda =frac1Rleft;n_ exti=2,n_ extf=1\, therefore that

displaystylelambda =left(fracm1.097 imes ext10^7 ight)left=1 ext. ext22 imes ext10^-7 extm= ext122 nm\ , i m sorry is UV radiation.

3. eginarrayllla_ extB&=&frach^24pi ^2m_e extkZq_e^2\ ext &=&fracleft( ext6.626 imes ext10^- ext34 extJ cdot ext s ight)^24pi ^2left(9.109 imes ext10^- ext31 extkg ight)left(8.988 imes ext10^9 extNcdot extm^2/C^2 ight)left(1 ight)left(1.602 imes ext10^- ext19 extC ight)^2\ ext &=& ext0.529 imes ext10^- ext10 extmendarray\

5. 0.850 eV

7. 2.12 × 10−10 m

9. 365 nm; that is in the ultraviolet.

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11. No overlap; 365 nm; 122 nm

13. 7

15. (a) 2; (b) 54.4 eV

17. displaystylefrac extkZq_e^2r_n^2=fracm_eV^2r_n\, so that displaystyler_n=frac extkZq_e^2m_eV^2=frac extkZq_e^2m_efrac1V^2\. From the equation displaystylem_evr_n=nfrach2pi\, we can substitute for the velocity, giving:

displaystyler_n=frac extkZq_e^2m_ecdot frac4pi ^2m_e^2r_n^2n^2h^2\

so that

displaystyler_n=fracn^2Zfrach^24pi ^2m_e extkq_e^2=fracn^2Za_ extB\,


displaystylea_ extB=frach^24pi ^2m_e extkq_e^2\.