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The Bohr AtomA guide

Bruce Cameron Reed

Chapter 1

Preliminaries

This chapter summarizes background information relevant to the historical contextof the Bohr model. Specifically, we look at what was known of the sizes of atoms, theintroduction of quantum concepts to explain phenomena in thermal physics,regularities in spectroscopic data, and the problem of the predicted collapse ofatoms in pre-Bohr atomic models. This chapter also sets up some expressions forforces and energies involved in the physics of charged particles orbiting each other.These are useful in analyzing the collapse problem, and set the stage for thederivation of the Bohr atom in chapter 2.

1.1 On the sizes of atomsAn idea of the effective sizes of atoms can be obtained via the following simpleargument. Suppose that some element (what is had in mind is a metallic solid) hasdensity ρ grams per cubic centimeter and atomic mass A grams per mole. Thenumber of cubic centimeters per mole of atoms will then be A/ρ. One mole of anykind of entity is Avogadro’s number NA of them, so the volume per atom will beA/(ρ NA). If the atoms are imagined to be packed together tightly as spheres, eachwill effectively occupy a cube of side length L and volume L3, as sketched infigure 1.1.

The effective diameter of each atom will be

ρ=L

AN

.A

1/3⎛⎝⎜

⎞⎠⎟

Table 1.1 shows results for a few metallic elements drawn from across the periodictable; atomic masses are rounded to the nearest whole number, and values of L aregiven in Ångstroms (Å). Recall that 1 Å = 10−10 m, one ten-billionth of a meter, aunit named after Swedish spectroscopist Anders Ångstrom (1814–74) and conven-ient as it corresponds closely to the effective sizes of atoms. Despite a range of a

doi:10.1088/978-0-7503-3611-6ch1 1-1 ª IOP Publishing Ltd 2020

factor of over 30 in atomic masses and densities, the effective sizes are remarkablysimilar: bulk densities and atomic weights indicate that all atoms act as if they areabout 2–3 Å in diameter, or ∼1–2 Å in radius.

1.2 Spectroscopy, the quantum concept, and the Balmer formulaThe history of spectroscopy is extremely rich; only a very brief summary is givenhere. This section is a bit lengthy, but important for understanding variousphenomena and theories which underlay Bohr’s atomic model.

That prism-shaped pieces of glass could be used to split beams of sunlight into theconstituent colors of the visible spectrum was known in antiquity. Systematic experi-ments with the solar spectrum began in the 1600s. In particular, Isaac Newton foundthat upon passing a narrow pinhole-created ray of sunlight through a prism, thespectrum so created could be recombined into white light by passing it through asecond prism; flames and stars also served as convenient light sources. Newton believedlight to be composed of particulate ‘corpuscles’, but this idea was superseded in theearly 1800s by a wave theory developed by Thomas Young (1773–1829) in order toexplain interference effects.

In 1802, English chemist/physicist William Wollaston (1766–1828) built a formalspectrometer that included a slit and lens to focus spectra on a screen for finerexamination. This revealed that the colors in the solar spectrum were not distributedcontinuously, but rather that some colors were missing. This phenomenoncame to be called ‘dark lines’. This work was picked up by German scientist

Figure 1.1. Closely-packed atoms occupying cubes of side length L. Imagine extending the structure to threedimensions.

Table 1.1. Effective diameters for various elements.

Element ρ (g cm−3) A (g mol−1) L (Å)

Li 0.534 7 2.79Al 2.7 27 2.55Fe 7.87 56 2.28Cu 8.96 64 2.28Pb 11.35 207 3.12Pt 21.45 195 2.47U 18.95 238 2.75

The Bohr Atom

1-2

Joseph von Fraunhofer, who is considered to have invented the modern form of thespectroscope around 1814; student laboratory spectrometers still follow his funda-mental design. Fraunhofer also refined the development of diffraction gratings, whichare essentially a fence of very closely-spaced slits. By knowing the spacing of the slits,it is possible to relate colors to wavelengths via Young’s wave theory, and Fraunhoferthus became the founder of quantitative spectroscopy. Fraunhofer cataloged over500 of Wollaston’s dark lines in the solar spectrum; these are now also known asFraunhofer lines. Today, millions of such lines are known. Figure 1.2 shows some ofthe Fraunhofer lines in the visible part of the solar spectrum; it was common to referto particular lines with letters. The scale along the bottom of the figure giveswavelengths in Ångstroms (Å). Modern usage is tending toward using nanometers(nm; 1 nm = 10−9 meters = 10 Å), but in this book Ångstroms will be used. Humanresponse to light is a matter of individual physiology, but in general for the visible partof the electromagnetic spectrum, wavelengths run from about 3500–4000 Å for thedeepest visible bluish-violet colors to 6500–7000 Å for the reddest.

Chemical spectroscopy flourished with the work of Robert Bunsen (1811–99; ofburner fame) and Gustav Kirchhoff (1824–87) in the 1860s. If you have studiedelectrical circuits, you may have come across ‘Kirchhoff’s rules’ for currentconservation at nodes; this is the same Kirchhoff. In the 1860s, this work culminatedin what are now known as Kirchhoff’s three laws of spectroscopy. These aresketched in figure 1.3, and can be summarized as

(1) An incandescent solid, liquid, or gas under high pressure emits a continuousspectrum: light of all colors appears.

(2) A heated gas under low pressure emits a ‘bright line’ spectrum: spectral linesat particular wavelengths appear. This is also known as an ‘emission’spectrum. The specific lines that appear depend on the gas involved.

(3) When viewed through a cool, low-density gas, a continuous spectrumcreated by a source as in law (1) produces an ‘absorption-line’ spectrum:a continuous spectrum with particular wavelengths removed.

The especially striking fact here is that if the same gas is used in (2) and (3), thewavelengths that appear in emission in (2) are identical to those which are absorbedout of the spectrum in (3). Also particularly striking in (2) and (3) is that eachelement has its own unique pattern of lines. This gave rise to the science of chemicalspectroscopy: the ability to identify the presence of elements by their spectra. It was

Figure 1.2. Solar spectrum showing prominent Fraunhofer absorption lines. Source: Public domain from nl:Gebruiker:MaureenV https://commons.wikimedia.org/wiki/File:Fraunhofer_lines.jpg.

The Bohr Atom

1-3

not understood why these patterns were present, but they acted like identifyingfingerprints of their corresponding elements.

Figure 1.4 shows spectra of a few common elements, with prominent spectral linesindicated by their wavelengths. Figure 1.5 shows the spectrum of hydrogen, whichplayed a key role in Bohr’s atomic theory. At visible wavelengths, the hydrogenspectrum comprises a series of lines which get closer and closer together withdecreasing wavelength. The reddest line occurs at 6563 Å; the others, strictly aninfinite number of them, occur at decreasing wavelengths down to about 3647 Å in thedeep violet, where they ‘pile up’. (For some qualifications regarding exact wave-lengths, see comments associated with table 2.1 in chapter 2.) In practice, spectral lineshave finite widths due to phenomena such as the Doppler effect and the effects of theforces that atoms in a gas exert on each other (‘pressure broadening’); you cannotseparate all of the lines, even with a spectrometer of infinitely great resolution.

An enormous step forward in understanding the nature of light came in 1865 withthe publication of James Clerk Maxwell’s theory of electromagnetism. For studentsof physics and electrical engineering, if you have not yet come to Maxwell’sequations in your curriculum, you surely will. One of the results of Maxwell’stheory was that if an electrical charge is oscillated back-and-forth at some frequency,it will emit energy in the form of an ‘electromagnetic wave’ which will travel throughspace at the speed of light, c. This was a quite remarkable development: Maxwellsucceeded in marrying all electrical, magnetic, and optical phenomena within oneunified theory. All light is a form of electromagnetic radiation.

hot solid,liquid, or

high-pressure gas

emission spectrum

absorption spectrum

continuous spectrum

diffraction grating,spectrometer

cool gas

heatedgas

same gas samewavelengths

Figure 1.3. Sketch of Kirchhoff’s laws of spectroscopy. See also figures 1.4 and 1.5. Colors are approximate.Source: Adapted by the author from Talos https://commons.wikimedia.org/wiki/File:Kirchhoff-spectroscopy-law.svg.

The Bohr Atom

1-4

Hydrogen

656348614340

Helium

6678587650154471

Carbon

65785889 60135145/5151

Magnesium

5172/5183

Sodium

5890/5896

Neon

5882

Silicon

5041/5055

Figure 1.4. Optical spectra of several elements, with wavelengths of some prominent lines indicated inÅngstroms. Sources: Public domain from

https://commons.wikimedia.org/wiki/File:Carbon_Spectra.jpghttps://commons.wikimedia.org/wiki/File:Helium_spectra.jpghttps://commons.wikimedia.org/wiki/File:Hydrogen_Spectra.jpghttps://commons.wikimedia.org/wiki/File:Magnesium_Spectra.jpghttps://commons.wikimedia.org/wiki/File:Neon_spectra.jpghttps://commons.wikimedia.org/wiki/File:Silicon_Spectra.jpghttps://commons.wikimedia.org/wiki/File:Sodium_Spectra.jpg.

The Bohr Atom

1-5

If the frequency of the oscillation is f back-and-forth cycles per second, then thewavelength λ of the emitted wave must be given by the universal wave equation,λ f = c. In principle, there is no limit to f, so λ is likewise unlimited: it is possible tohave an electromagnetic (EM) wave of any wavelength. The full run of wavelengthsis now known as the electromagnetic spectrum. Since atoms within any object that isat any temperature above absolute zero are constantly in motion, they constantlyemit EM energy while absorbing it from other emitters. The range of visible wave-lengths is only a small segment of the entire EM spectrum: we see only a tiny fractionof the ‘light’ that is emitted by any object. Figure 1.6 shows a depiction of the EMspectrum and some of the phenomena associated with different wavelengths.Wavelengths are listed along the top of the diagram; the second row gives energiesin electronvolts, which we will come to soon. Human beings sense infrared light(IR; beyond red) as heat (think of an open oven door), while shorter-wavelengthultraviolet light (UV) is energetic enough to ionize atoms and molecules in yourbody and give you a sunburn.

While all objects at temperatures above absolute zero emit energy across the EMspectrum, what figure 1.6 does not depict is that the intensity of the emission is notuniform across all wavelengths. There proves to be a characteristic shape to theintensity/wavelength distribution, which for historical reasons is called a blackbody

3500 Å 7000 Å4000 Å

65634861434141023646

Balmerserieslimit

Infinitenumber of lines

in here

Balmer lines

Continuousspectrum

6500 Å6000 Å5500 Å5000 Å4500 Å

Figure 1.5. Sketch of visible-wavelength continuous spectrum and the first four lines of the Balmer series.Wavelengths are in Ångstroms and are as in air (see section 2.3). Colors are approximate.

Figure 1.6. The electromagnetic spectrum. The red line at the end of the infrared (IR) part of the spectrumcorresponds to room-temperature thermal energy. Source: KristianMolhave https://commons.wikimedia.org/wiki/File:TheElectromagneticSpectrum.jpg. This file is licensed under the Creative Commons Attribution 2.5 Genericlicense. This illustration was made for theOpensource Handbook of Nanoscience and Nanotechnology.

The Bohr Atom

1-6

spectrum. This is somewhat far afield from the Bohr atom (although there is animportant connection), but there is a point here worth mentioning in order toaddress a question that might have occurred to you: why do we ‘see’ in the visiblepart of the spectrum? It turns out that for objects whose surface temperatures are likethat of the Sun (∼5800 K), their EM radiation is most intense at greenish-yellowwavelengths. Human beings evolved to take advantage of this; it would not be ofmuch use for us to see at, say, x-ray wavelengths. We might speculate that ifintelligent beings have evolved on a planet orbiting a cooler, reddish star, their eyeswill involve different chemistry than ours.

Figure 1.7 shows the spectrum of the Sun (jagged white curve) as a function ofwavelength in nanometers; the abbreviation AM0 means ‘air mass zero’, whichcorresponds to the spectrum as measured outside Earth’s atmosphere. The yellowcurve corresponds to a ‘best-fit’ theoretical blackbody spectrum for a temperature of5778 K. The scale on the y-axis gives the number of Watts of solar energy per squaremeter per nanometer wavelength interval at the location of the Earth. The intensitypeaks at about 500 nm = 5000 Å. Hotter objects have the peak intensity of theirblackbody spectra shifted to bluer wavelengths, and cooler ones toward the redder;think of a hot burner on your stove.

A side story on infrared radiation. In 1800, German–British astronomer WilliamHerschel (1738–1822) was testing filters to enable him to observe sunspots, andnoticed that a lot of heat seemed to be produced when he was using a filter which letred light pass through. He then directed sunlight to pass through a prism, and held athermometer up to the projected spectrum, discovering that it read a highertemperature when held off the red end of the spectrum than when placed insidethe visible part. This led him to the conclusion that there exist forms of light outsidethe visible spectrum; he is now considered to have discovered infrared radiation.Sometimes a simple experiment can reveal remarkable information.

Here is a key point that will come up again. Maxwell’s theory made it clear that ifa charged particle is oscillating at some frequency, it will emit EM radiation of thatsame frequency, which means a particular wavelength. If light of particular wave-lengths appear in an emission spectrum or do not appear in an absorption spectrum, it

Figure 1.7. Solar spectrum intensity versus wavelength. Source: Public domain, Danmichaelo https://commons.wikimedia.org/wiki/File:Solar_AM0_spectrum_with_visible_spectrum_background_(en).png.

The Bohr Atom

1-7

means that particular energies are somehow either being emitted or absorbed. This wasa crucial clue for Bohr.

Since spectral lines always appear at particular wavelengths, it was natural toassume that something within the structure of atoms which was electrically chargedmust jiggle back-and-forth at just the right frequencies to emit light detectable in theoutside world. With Joseph John ‘JJ’ Thomson’s 1897 discovery of electrons as auniversal constituent of matter and the fact that they proved to weigh about 1/2000as much as individual hydrogen atoms, it was equally natural to presume that atomscomprised clouds of positive electrical material within which were embeddednegatively-charged electrons which for some reason were doing the requisite jiggling.The positive cloud was needed to give atoms overall electrical neutrality, and to givethem their correct 1–2 Å radii. This Tomson atomic model has often been likened toa plum pudding, with electrons akin to embedded raisins.

The next major step toward the Bohr model occurred in 1900. German physicistMax Planck (1858–1947), whose research area was thermodynamics and statisticalmechanics, was attempting to develop a general mathematical expression forblackbody radiation intensity as a function of temperature. At the time, there wasno detailed understanding of the internal structure of atoms, so Planck and othersidealized them as abstracted ‘oscillators’ which could vibrate at any chosenfrequency and hence emit EM radiation. However, predicted intensity/wavelengthrelationships based on this scheme never agreed properly with measured values, andthe situation was becoming a crisis for an otherwise very successful and well-established branch of physics. Planck found a way around the impasse, but in a waythat was regarded as very radical. In classical physics, there is no relationshipbetween the frequency of an oscillator and the amount of energy it possesses. As arough example, think of the common laboratory experiment where you tie one endof a string to a vibrator which is usually arranged to shake the string up-and-down ina vertical plane, while the other end of the string is fixed, usually by running it over apulley and tying it to a heavy mass. You can adjust the energy of the vibrator, whichdictates its up-and-down amplitude, quite independently of adjusting the frequencyof the vibration. This is a mechanical analogy, but the same independence waspresumed to hold for atomic-level oscillators and the EM waves which they created.Planck’s radical departure was that he found that if he assumed that the energy of anoscillator was in fact restricted to an integer multiple of its frequency, he couldreproduce blackbody intensity curves with great accuracy. To make the units workout, frequency had multiplied by a suitable constant, which he designated as h, andwhich we now know as Planck’s constant. Remarkably, one universal value of hworked for all temperatures. Planck’s constant is now regarded as a fundamentalconstant of physics.

Expressed mathematically, Planck’s assumption was that the energy E of anoscillator of frequency f is given by

= = …E nhf n( 1, 2, 3, ). (1.1)

The Bohr Atom

1-8

Frequency has units of ‘per second’ or s−1, and energy has units of Joules (J), so hmust have units of J/(s−1) = J s, read as ‘Joule-seconds’ and usually abbreviated to J s.The numerical value, purely empirically, is h = 6.626 × 10−34 J s, a very tiny number.The minuteness of this value means that, on a macroscopic scale where you aredealing with Joules of energy, the possible oscillatory ‘energy states’ of a system areindistinguishable from each other.

Because E is restricted to multiples of hf through the integer multiplier n, E is saidto be ‘quantized’, that is, it is restricted to particular values for a given f. Frequencyitself is still regarded as a continuously-valued variable. Planck is considered to bethe father of ‘quantum physics’ or ‘quantum mechanics’, a designation he waspersonally never too comfortable with given his very classical training. Planck wasawarded the 1918 Nobel Prize for Physics for his discovery of ‘energy quanta’. Anexcellent (and technically involved) history of the development of the quantumconcept can be found in Kuhn (1987).

Planck considered the energy radiated by his oscillators to be classical in that itwas still in the form of an EM wave; the idea of electromagnetic radiation ascomprising a stream of photons or ‘quanta’ would not begin to be taken seriouslyuntil Einstein’s analysis of the photoelectric effect in 1905. None of Planck, Einstein,or Bohr used the term ‘photon’ in their groundbreaking publications; that terminol-ogy would not be coined until 1926, when it was introduced by American chemistGilbert Lewis of Lewis dot diagram fame (Lewis 1926). However, we can play fast-and-loose with history here to get to a valuable expression that relates energy towavelength, as in figure 1.6. From the universal wave equation, we can write f = c/λ.Taking a single photon to correspond to n = 1, we can write Planck’s formula as

λ= =E hf

hc. (1.2)

When dealing with atomic-scale phenomena, it is customary to quote energies inelectronvolts (eV); recall that one eV is the magnitude of the electron chargeexpressed as a number of Joules of energy: 1 eV = 1.602 … × 10−19 J. Chemicalreactions, which involve exchanges of electrons, are typically of energies of a few eV.Equation (1.2) can be put into a handy form for estimating the energy of a photon ineV if its wavelength is specified in Ångstroms. Designate the latter by λÅ. Thewavelength in meters will be (10−10 λÅ). In SI units and rounding h and c to threedecimal places, we get

λ λ

λ

= = × ×

= ×

−

− −

−

−

Ehc

(10 m)(6.626 10 J s)(2.998 10 m s )

(10 m)

1.986 10Joules.

10Å

34 8 1

10Å

15

Å

Now convert Joules to eV: 1 eV = 1.602 × 10−19 J, or 1 J = 6.242 × 1018 eV. Hence

λ λ= × × =

−E

(1.986 10 )(6.242 10 )eV

12 400eV. (1.3)

15 18

Å Å

The Bohr Atom

1-9

The actual value of the numerator is 12 398.4 Å, but it has been rounded off veryslightly for convenience. This is known as the ‘12 400’ formula. For a photon ofλ = 5000 Å, its energy must be ∼12 400/5000 ∼ 2.48 eV. If we take the range ofvisible photons to be from 3500 to 7000 Å, the corresponding energy range is from3.54 down to 1.77 eV. These are much more convenient numbers to deal with thantheir Joule-enumerated equivalents. Common multiples of the electronvolt are thekilo-electronvolt (1 keV = 1000 eV) and the mega-electronvolt (1 MeV = 106 eV); thelatter appears frequently in nuclear physics.

On the topic of unit conversions, another one you may run into is thatspectroscopists sometimes quote wavelengths and frequencies in reciprocal centi-meters (cm−1), which means the number of wavelengths that fit within onecentimeter. This quantity is also known as the wave number. We can arrange thePlanck formula to relate energies in eVs and wave numbers as follows. Reciprocalwavelengths are given by

λ= E

hc1

.

If energy is in Joules, then 1/λ will emerge in reciprocal meters. For energies in eV,

λ= ×

× ×= ×

−

−−E

E1 (1.602 10 )

(6.626 10 )(2.998 10 )(8.065 10 ) (m ).eV

19

34 85

eV1

Since 1 m = 100 cm, then one reciprocal meter = 10−2 reciprocal cm, so we canconvert this to give a result in cm−1 by multiplying by 10−2:

λ= −E

18065 (cm ). (1.4)eV

1

An energy of 1 eV corresponds to a wave number of 8065 cm−1. Conversely, awave number of 1 cm−1 corresponds to EeV = (1/8065) eV = 1.24 × 10−4 eV. Wavenumber is often designated by the symbol ν̃, the Greek letter ‘nu’ (as in number) witha tilde (∼) over it.

If you know a wavelength directly in Ångstroms, then the wave number in cm−1 isjust 108/λÅ because 1 Å = 10−8 cm. For example, if λÅ = 5000 Å, the wave number is20 000 cm−1; you should get the same result by taking the corresponding energy,2.48 eV, and using equation (1.4).

We now leave Max Planck, but a few further comments on the difficulties of hislater life are appropriate. In 1944, his house in Berlin was destroyed in a bombingraid; he was living in the country at the time, but lost decades of papers andcorrespondence. His son Erwin, to whom he was very close, was part of a plot toassassinate Adolf Hitler in July, 1944; the plot failed, and Erwin was captured by theGestapo and executed by hanging the following January. Planck is now memorial-ized by several ‘Max Planck Institutes’ which carry out research in fields across thenatural and social sciences.

Stepping back in history, we come to the last piece in the foundations of theBohr model, a formula published in 1885, the year of Bohr’s birth, by

The Bohr Atom

1-10

Johann Jakob Balmer (1825–1898; figure 1.8), an otherwise obscure Swissmathematician (Balmer 1885). Balmer earned a PhD at the University of Baselin 1849, and made his living as a teacher at a secondary school for girls as well aslecturing at his alma mater. His main field of interest was geometry, but he isremembered for a contribution to physics.

Balmer found that, purely empirically, the wavelengths of the lines in the visiblepart of the hydrogen spectrum could be described by the simple formula

λ =−

= …jj

j(3646 Å)4

( 3, 4, 5, ). (1.5)j

2

2

⎛⎝⎜

⎞⎠⎟

Balmer had no physical argument behind this result, it was purely a matter of theformula reproducing measured wavelengths. At the time of his work, over a dozenhydrogen lines had been detected in the spectra of some stars, and his formula fit allof them to within the experimental errors. In the rarefied outer atmospheres of stars,pressure-broadening effects are not as dramatic as in Earth-bound laboratories, andmore lines can be photographed and clearly distinguished from each other.

It will be convenient to express Balmer’s formula as a prescription for inversewavelengths, that is, wave numbers. Inverting the formula gives

λ= − = −j

jj

j1 1

(3646 Å)

4 4

(3646 Å)

/4 1,

j

2

2

2

2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

or

λ= − = × −−

j j1 4

(3646 Å)

14

1(1.097 10 m )

14

1. (1.6)

j2

7 12

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Figure 1.8. Left: Johann Balmer. Source: https://upload.wikimedia.org/wikipedia/commons/5/51/Balmer.jpeg.Right: Johannes Rydberg. Source https://upload.wikimedia.org/wikipedia/commons/7/7e/JanneRydberg.jpg.

The Bohr Atom

1-11

The factor of 1.097 × 107 m−1 is now known as the Rydberg constant forhydrogen, abbreviated RH; a more precise value can be found in equation (2.11).This terminology refers to Swedish spectroscopist Johannes Rydberg (1854–1919;figure 1.8), who in 1888 found that a generalized version of Balmer’s formula of theform

λ= − >R

n jj n

1 1 1( ) (1.7)

n j,2 2

⎛⎝⎜

⎞⎠⎟

described many, although not all, of the spectral lines of many other elements, witheach element characterized by its own value of R (Rydberg 1890). The Balmer–Rydberg formula hinted that there was some regularity to atomic spectra, and hencepresumably to atomic structure itself. But it was by no means clear what thatstructure might be.

1.3 Circular orbit forces and energeticsBohr’s atomic model can be described as a sort of forced marriage between theclassical mechanics of circular orbits with electrical forces and some arbitraryadaptations of Planck’s quantum hypothesis. In this section, the classical mechanicspart of the derivation is set up.

Bohr’s model is commonly described as an electron of charge −e in orbit about anucleus containing Z protons and hence having charge +Ze. Z is the commonsymbol for the atomic number of an element. In his original derivation, Bohr did notworry about the motion of the electron and the nucleus about their common centerof mass, that is, he imagined the nucleus to be fixed while the electron orbited it.However, I account for this motion for sake of completeness. Neglecting this effect isequivalent to assuming that the nucleus is infinitely heavy compared to the orbitingelectron, and for even the case of hydrogen this is not at all a bad approximationbecause the lone proton that comprises the nucleus is over 1800 times heavier thanthe orbiting electron. But there are some cases, such as when dealing with acomparison of transition wavelengths between ordinary hydrogen and so-called‘heavy hydrogen’, where the effects of non-infinite-mass nuclei are crucial to thecalculations. In many cases I will assume that the nucleus is infinitely heavycompared to the orbiting electron for sake of simplifying a calculation. But it isalways easier to start with a general formula and simplify as necessary, rather thanto try to go the other way.

Figure 1.9 shows the electron/nucleus system. The electron has mass me andexecutes a circular orbit of radius re about the center of mass (CM). The nucleus hasmass mnuc and orbital radius rnuc about the CM. The orbital speeds are ve and vnuc,and are presumed to be such that the electron and nucleus are always directlyopposite each other; if they were not, the CM would move.

If we take the CM to define the coordinate origin, then basic mechanics tells usthat the masses and orbital radii must be related according as

The Bohr Atom

1-12

= ⇒ =m r m rrr

mm

. (1.8)e ee

enuc nuc

nuc

nuc

The manipulation to form rnuc/re will be used in a moment.For the electron and nucleus to remain opposite each other, they must have the

same orbital period. In a circular orbit, the period is given by vπr2 / , so this demands

v vv v= ⇒ =r r r

r. (1.9)

e

e ee

nuc

nucnuc

nuc

Now, the total kinetic energy of the system will be the sum of the individualkinetic energies:

v v= +KE m m12

12

.e e2

nuc nuc2

Replace vnuc with (1.9), and use equation (1.8) for rnuc/re. This gives

v= +KE mm

m12

1 . (1.10)e ee2

nuc

⎡⎣⎢

⎤⎦⎥

This expression has the advantage of expressing the total kinetic energy of thesystem in terms of that of the electron alone and the ratio of the two masses. If thenucleus is infinitely heavy, then the KE of the system will be that of the electronalone.

It is customary to express this kinetic energy in a modified form by the followingmanipulation:

Figure 1.9. Nucleus of mass mnuc and charge +Ze and an electron of mass me and charge −e in circular orbitsabout their mutual center of mass CM. Colors are for illustrative purposes only.

The Bohr Atom

1-13

v v v= + = + = +KE m

mm

mm m

mm

m mm m

12

112

12

.e ee

e ee

e ee

e

2

nuc

2 nuc

nuc

2 2 nuc

nuc

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

The inverse of the term in square brackets is known as the reduced mass of thesystem. This is always given the symbol μ:

μ =+

m mm m

. (1.11)e

e

nuc

nuc

⎡⎣⎢

⎤⎦⎥

If mnuc ≫ me, then the denominator of μ reduces to approximately just mnuc, μreduces to ∼me, and the kinetic energy again reduces to that of the electron alone. Ingeneral, however,

vμ

=KEm1

2. (1.12)e e

2 2

In atomic systems, the reduced mass is always very slightly less than the mass ofthe electron (hence the ‘reduced’ terminology). For the specific case of hydrogenwhere the nucleus is a proton (mass mp), the reduced mass is designated with thesymbol μH:

μ =+

= × ∼−m m

m mm9.104425 10 kg 0.999456 . (1.13)H

e p

p ee

31⎡⎣⎢

⎤⎦⎥

An exactly analogous quantity arises in celestial mechanics, where the Sun playsthe role of the nucleus and a planet that of the electron. Reduced mass was a well-known quantity in classical mechanics long before Bohr’s time.

Now, if the electron is moving in a circular orbit, we further know fromNewtonian mechanics that it must be subject to a centripetal force of magnitude

vm r/e e e2 . What provides this force? The only force acting in the system is that of

electrical attraction between the electron and the nucleus, which is given byCoulomb’s law. Bohr sensibly assumed that this was the cause of the mutualmotions, so we put

vπε

=+

mr

Zer r4 ( )

. (1.14)e e

e o e

2 2

nuc2

In writing this expression, we are dealing with magnitudes, so the charge of theelectron goes into the algebra as ∣−e∣. In the denominator on the right side,the inverse-square force acts over the total distance between the electron and thenucleus, re + rnuc, but on the left side only the radius of the electron’s orbit appears inthe centripetal force. Also, it is customary in electrodynamics texts to write thecombination πε1/4 0 as just k, but I will leave everything in terms of fundamentalphysical constants. Also, the resulting formulae look very impressive with all theirGreek letters.

The Bohr Atom

1-14

Solve (1.14) for the square of the electron’s speed:

vπε πε

=+

=+

Ze rm r r

Zem r r r4 ( ) 4 (1 / )

.ee

o e e o e e e

22

nuc2

2

nuc2

In the last step, a factor of re was extracted from within the bracket in thedenominator to give re

2 in the denominator, and then one factor of re in this re2 is

canceled by the re in the numerator. The remaining factor of rnuc/re within thebracket can be put in terms of masses via equation (1.8), and the result can beexpressed in reduced-mass form as

vμ

πε= Ze

m r4. (1.15)e

o e e

22 2

3

Substitute this back into equation (1.12) to express the kinetic energy of thesystem as

vμ μ

μπε

μπε

= = =KEm m Ze

m rZe

m r12

12 4 8

. (1.16)e e e

o e e o e e

2 2 2 2 2

3

2

Now, electromagnetic theory also tells us that a system of two electrical chargesQ1 and Q2 separated by a distance d possesses potential energy given by πεQ Q d/41 2 0 .In applying this to our nucleus/electron system, the charges are +Ze and −e. Herethe negative sign of the electron’s charge is built into the algebra explicitly: this is anenergy calculation, and energy can be positive or negative. Equation (1.14) is amagnitude-of-force argument, so the sign of the electron charge disappeared whentaking absolute values. For the potential energy,

πεμ

πε= −

+= −PE

Zer r

Zem r4 ( ) 4

, (1.17)o e o e e

2

nuc

2

where the orbital radii were again replaced with masses by extracting a factor of re inthe denominator and putting things in terms of the reduced mass.

The total energy of the system is the sum of the kinetic and potential energies,which from (1.16) and (1.17) emerges as

μπε

= + = −E KE PEZe

m r8. (1.18)

o e e

2

The negative sign indicates that the system is in a ‘bound’ energy state. Electronsand protons attract each other, and if they are in some stable configuration, then aninput of a positive amount of energy will be required to pull them apart. This energyexpression will play an enormous role in much of the remainder of this chapter.

So far, there has been absolutely nothing ‘quantal’ about what has been derived;that will come in chapter 2. For the following section, we return to an issue raised inthe Preface.

The Bohr Atom

1-15

1.4 The radiative collapse problemIt was remarked in the Preface that one of the great dilemmas of early atomictheories was that they predicted that atoms should collapse almost instantaneously.This section develops an analysis of this issue.

One of the results of James Clerk Maxwell’s electromagnetic theory was that anaccelerated electrical charge would radiate away energy in the form of electro-magnetic radiation. The rate of energy loss depends on the charge and themagnitude of the acceleration a. For an electron, the theory predicted a rate ofenergy loss given by

πε= −dE

dte a

c6, (1.19)

o

2 2

3

where c is the speed of light. This formula was first developed by Irish physicistJoseph Larmor in 1897.

Figure 1.10 sketches the scenario had in mind here: an electron begins with orbitalradius R around a nucleus, and then spirals inward due to energy loss, eventuallymerging with the nucleus in a catastrophic matter/anti-matter explosion. Refer toequation (1.18) for the total energy of such a system: if there is energy loss, E willbecome progressively more negative, which corresponds to the orbital radiusbecoming smaller and smaller. The essential question is: if R has a value character-istic of atomic sizes, say ∼1 Å, how long will the spiral-in take?

If the spiral-in takes much longer than the characteristic time for one orbit, wecan model the electron as being in a circular orbit at any time, with the accelerationbeing the corresponding centripetal acceleration. The validity of this assumption isinvestigated a posteriori.

Figure 1.10. An electron spirals into its nucleus, emitting electromagnetic radiation and losing energy as itgoes. Colors are for illustrative purposes only.

The Bohr Atom

1-16

This is a case where we treat the nucleus as immobile, and write the reduced massμ asme. With equation (1.15), we can then write the acceleration when the electron isat radius r as

vπε

= =ar

Zem r4

.e

e o e e

2 2

2

The rate of energy loss is then

πε πε π ε= − = −dE

dte

cZe

m rZ e

m c r6 4 961

. (1.20)o o e o e

2

3

2

2

2 2 6

3 3 2 3 4

⎛⎝⎜

⎞⎠⎟

Now, at any moment, the total energy of the system will be, from equation (1.18),

πε= −E

Zer8

.o

2

Taking the derivative of this with respect to time gives

πε=dE

dtZe

rdrdt8

. (1.21)o

2

2

⎛⎝⎜

⎞⎠⎟

Putting together equations (1.20) and (1.21) gives

πε π ε= −Ze

rdrdt

Z em c r8 96

1,

o o e

2

2

2 6

3 3 2 3 4

⎛⎝⎜

⎞⎠⎟

or

π ε= −dr

dtZe

m c r121

.o e

4

2 2 2 3 2

⎛⎝⎜

⎞⎠⎟

Separating variables and integrating from r = R at t = 0 to r = 0 at t = tcollapsegives

∫ ∫π ε= −r dr

Zem c

dt12

.R o e

t02

4

2 2 2 3 0

collapse⎛⎝⎜

⎞⎠⎟

This gives

π ε= ∼ × −t

m cZe

RZ

R4 (1.05 10 m )

.o ecollapse

2 2 2 3

43

20 33

⎛⎝⎜

⎞⎠⎟

For R = 1 Å and Z = 1 (hydrogen), the collapse time is ∼10−10 s: about one-tenthof a nanosecond. It will probably take you much more than one-tenth of ananosecond to read this sentence, during which time you have hopefully notcollapsed. Clearly, there must be something terribly wrong with this model. Theprospect of this incredibly brief collapse time is what prompted Bohr to arbitrarily

The Bohr Atom

1-17

assume that electrons had to be in some sort of non-radiating states wherein they didnot emit electromagnetic radiation even as they orbited the nucleus.

Is the assumption that the spiral-in will occur over many orbits valid? We cancheck this by comparing the typical orbital period to the spiral-in period. For anorbit of radius R and speed v, the orbital period will be 2πR/v. Using equation (1.15)for the speed (with μ = me) gives

vπ π πε π ε= = =tR

Rm R

Zem R

Z e2

24 4

.o e o eorbit 2

3/2 1/2 1/2 3/2

1/2

Hence

π επ ε

π ε= =t

tm c

ZeR

Z e

m Rm c

Z eR

4

4

1.o e

o e

o ecollapse

orbit

2 2 2 3

43

1/2

3/2 1/2 1/2 3/2

1/2 3/2 3/2 3

1/2 33/2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

The term in brackets evaluates to 2.66 × 1020 m−3/2; for R = 1 Å, tcollapse/torbit ∼266 000. The spiral-in time is much longer than the orbital time, so treating thespiral-in as a succession of circular orbits is plausible.

ReferencesBalmer J J 1885 Ann. Phys. Lpz. 261 80–7Kuhn T S 1987 Black-Body Theory and the Quantum Discontinuity 1894–1912 (Chicago, IL:

University of Chicago Press)Lewis G 1926 Nature 118 874–5Rydberg J R 1890 Philos. Mag. Ser. 5 29B 331–7

The Bohr Atom

1-18