Chapter 4: Atomic Structure

• Structure of the atom is responsible for nearly all the properties of matter that have shaped

the world around us.

• Every atom consists of a small nucleus of protons and neutrons with a number of electrons

some distance away.

• It is tempting to think that the electrons circle the nucleus as planets do the sun, but classical

electromagnetic theory denies the possibility of stable electron orbits.

• Niels Bohr applied quantum ideas to atomic structure in 1913 (inspired from Balmer’s

formula).

4.1 THE NUCLEAR ATOM

• An atom is largely empty space.

• Late nineteenth century: It was known that all atoms contain electrons.

◦ Since electrons carry negative charges whereas atoms are neutral, positively charged

matter of some kind must be present in atoms.

◦ But what kind? And arranged in what way?

• Thomson model (Plum pudding model): One suggestion, made by the

British physicist J. J. Thomson (Discovery of electron) in 1898, was that

atoms are just positively charged lumps of

matter with electrons embedded in them,

like raisins in a fruitcake (Fig. 4.1).

• Because Thomson had played an important

role in discovering the electron, his idea

was taken seriously.

• But the real atom turned out to be quite

different.

Figure 4.1 The Thomson model of the atom. The Rutherford scattering experiment showed it to

be incorrect.

Joseph John

Thomson

1856-1940

Nobel Prize in

Physics in 1906

• What is inside a fruitcake?

• Rutherford model: Hans Geiger and Ernest Marsden used as probes the

fast alpha particles in 1911 at the suggestion of Ernest Rutherford

(Discovery of alpha and beta radioactivity and discovery of atomic

nucleus)

◦ Fast alpha particles emitted by certain radioactive elements.

◦ Alpha particles are helium atoms that have lost two electrons each,

leaving them with a charge of +2e.

• Geiger and Marsden placed a sample of an alpha-emitting substance

behind a lead screen with a small hole in it, as in Fig. 4.2, so that a

narrow beam of alpha particles was produced.

◦ This beam was directed at a thin gold foil.

• It was expected that the alpha particles would go right through the foil

with hardly any deflection. This follows from the Thomson model!!

Ernest Rutherford

1871-1937

Nobel Prize in

Chemistry in

1908

• With only weak electric forces exerted on

them, alpha particles that pass through a thin

foil ought to be deflected only slightly, 1° or

less.

Figure 4.2 The Rutherford scattering experiment.

• What Geiger and Marsden actually found was that although most of the alpha particles

indeed were not deviated by much, a few were scattered through very large angles.

• Some were even scattered in the backward direction.

• Alpha particles are relatively heavy (almost 8000 electron masses) and those used in this

experiment had high speeds (typically 2x10

m/s).

• It was clear that powerful forces were needed to cause such marked deflections.

• The only way to explain the results, Rutherford found, was to picture an atom as being

composed of a tiny nucleus in which its positive charge and nearly all its mass are

concentrated, with the electrons some distance away (Fig. 4.3).

• With an atom being largely empty space, it is easy to see

why most alpha particles go right through a thin foil.

• However, when an alpha particle happens to come near a

nucleus, the intense electric field there scatters it through a

large angle.

• The atomic electrons, being so light, do not appreciably

affect the alpha particles.

Figure 4.3 The Rutherford model of the atom.

• All the atoms of any one element turned out to have the same unique nuclear charge, and

this charge increased regularly from element to element in the periodic table.

• The nuclear charges always turned out to be multiples of +e; the number Z of unit positive

charges in the nuclei of an element is called the atomic number of the element.

• We know now that protons, each with a charge +e, provide the charge on a nucleus, so the

atomic number of an element is the same as the number of protons in the nuclei of its

atoms.

Nuclear Dimensions

• Rutherford assumed that the size of a target nucleus is small compared with the minimum

distance R to which incident alpha particles approach the nucleus before being deflected

away.

• Rutherford scattering therefore gives us a way to find an upper limit to nuclear dimensions.

• An alpha particle will have its smallest R when it approaches a nucleus head on, which will

be followed by a 180° scattering.

• At the instant of closest approach the initial kinetic energy KE of the particle is entirely

converted to electric potential energy, and so at that instant

• since the charge of the alpha particle is 2e and that of the nucleus is Ze. Hence

The maximum KE found in alpha particles of natural origin is 7.7 MeV, which is 1.2x10

Since 1/4

=9.0x10

when Z=79

: Radius of gold nucleus

4.2 ELECTRON ORBITS

• The planetary model of the atom and why it fails.

• The Rutherford model of the atom, so convincingly confirmed by experiment,

◦ pictures a tiny, massive, positively charged nucleus surrounded at a relatively great

distance by enough electrons

◦ enough electrons to render the atom electrically neutral as a whole.

• The electrons cannot be stationary in this model, because there is

nothing that can keep them in place against the electric force

pulling them to the nucleus.

• If the electrons are in motion, however, dynamically stable orbits

like those of the planets around the sun are possible (Fig. 4.5).

Figure 4.5 Force balance in the hydrogen atom.

• Let us look at the classical dynamics of the hydrogen atom, whose single electron makes it

the simplest of all atoms.

◦ We assume a circular electron orbit for convenience, though it might as reasonably be

assumed to be elliptical in shape.

◦ The centripetal force holding the electron in an orbit r from the nucleus is provided by

• The total energy of the electron is negative.

◦ This holds for every atomic electron and reflects the fact that it is bound to the nucleus.

◦ If E were greater than zero, an electron would not follow a closed orbit around the

nucleus.

Example 4.1

Experiments indicate that 13.6 eV is required to separate a hydrogen atom into a proton and an

electron; that is, its total energy is E=-13.6 eV. Find the orbital radius and velocity of the

electron in a hydrogen atom.

The Failure of Classical Physics

• The analysis above is a straightforward application of Newton’s laws

of motion and Coulomb’s law of electric force (classical physics) and

is in accord with the experimental observation that atoms are stable.

• However, it is not in accord with electromagnetic theory (another

classical physics) which predicts that accelerated electric charges

radiate energy in the form of em waves.

• An electron pursuing a curved path is accelerated and therefore

should continuously lose energy, spiraling into the nucleus in a

fraction of a second (Fig. 4.6).

Figure 4.6 An atomic electron should, classically, spiral rapidly into the nucleus as it radiates energy due to its

acceleration

• But atoms do not collapse.

• This contradiction further illustrates what we saw in the previous two chapters,

◦ The laws of physics that are valid in the macroworld do not always hold true in the

microworld of the atom.

4.3 ATOMIC SPECTRA

• Each element has a characteristic line spectrum

• The existence of spectral lines is an important aspect of the atom that finds no explanation in

classical physics.

• We saw in Chap. 2 that condensed matter (solids and liquids) at all temperatures emits em

radiation in which all wavelengths are present, though with different intensities.

◦ Witnessing the collective behavior of a great many interacting atoms rather than the

characteristic behavior of the atoms of a particular element.

◦ At the other extreme, the atoms or molecules in a rarefied gas are so far apart on the

average that they only interact during occasional collisions. Under these circumstances,

we would expect any emitted radiation to be characteristic of the particular atoms or

molecules present, which turns out to be the case.

• When an atomic gas or vapor at

somewhat less than atmospheric

pressure is suitably “excited,”

usually by passing an electric

current through it, the emitted

radiation has a spectrum which

contains certain specific

wavelengths only.

Figure 4.7 An idealized spectrometer.

• An idealized arrangement for observing such atomic spectra is shown in Fig. 4.7; actual

spectrometers use diffraction gratings.

• Figure 4.8 shows the emission

line spectra of several elements.

• Every element displays a unique

line spectrum when a sample of it

in the vapor phase is excited.

• Spectroscopy is therefore a

useful tool for analyzing the

composition of an unknown

substance.

Figure 4.8 Some of the principal lines in the

emission spectra of hydrogen, helium, and

mercury

• When white light is passed through a gas, the gas is found to absorb light of certain of the

wavelengths present in its emission spectrum.

• The resulting absorption line spectrum consists of a bright background crossed by dark

lines that correspond to the missing wavelengths (Fig. 4.9); emission spectra consist of

bright lines on a dark background.

Figure 4.9 The dark lines in the absorption spectrum of an element correspond to bright lines in its

• The number, intensity, and exact wavelengths of the lines in the spectrum of an element

depend upon temperature, pressure, the presence of electric and magnetic fields, and the

motion of the source.

• It is possible to tell by examining its spectrum not only what elements are present in a light

source but much about their physical state.

Spectral Series

• A century ago the wavelengths in the spectrum of an element were found to fall into sets

called spectral series.

• The first such series was discovered by J. J. Balmer in 1885 in the course of a study of the

visible part of the hydrogen spectrum.

• Figure 4.10 shows the Balmer series.

• As the wave-length decreases, the lines are

found closer together and weaker in

intensity until the series limit at 364.6 nm is

reached,

• beyond which there are no further separate

lines but only a faint continuous spectrum.

Figure 4.10 The Balmer series of hydrogen. The H

line is

red, the H line is blue, the H

and H lines are violet, and the other lines are in the near ultraviolet.

• Balmer’s formula for the wavelengths of this series is

• What is n for H ? What happens for n= ?

• The Balmer series contains wavelengths in the visible portion of the hydrogen spectrum.

• The spectral lines of hydrogen in the ultraviolet and infrared regions fall into several other

series. In the ultraviolet the Lyman series contains the wavelengths given by the formula

Figure 4.11 The spectral series of hydrogen. The wavelengths in each series are related by simple formulas.

4.4 THE BOHR ATOM

• Electron waves in the atom.

• The first theory of the atom to meet with any success was put forward in

1913 by Niels Bohr.

• The concept of matter waves leads in a natural way to this theory, as de

Broglie found.

• Bohr himself used a different approach, since de Broglie’s work came a

decade later.

• Start by examining the wave behavior of an electron in orbit around a

hydrogen nucleus.

• Since the electron velocities are much smaller than c (v<<c), we will

assume that γ=1 and for simplicity omit γ from the various equations.

• The de Broglie wavelength of this electron is

Niels Henrik

David Bohr

1885-1962

Nobel Prize in

Physics in 1922

By substituting 5.3x10

-11

m for the radius r of the electron

orbit (see Example 4.1), we find the electron wavelength to be

• This wavelength is exactly the same as the

circumference of the electron orbit, 2r=33x10

-11

• The orbit of the electron in a hydrogen atom

corresponds to one complete electron wave joined on

itself (Fig. 4.12).

Figure 4.12 The orbit of the electron in a hydrogen atom corresponds to a complete electron

de Broglie wave joined on itself.

• The fact that the electron orbit in a hydrogen atom is one electron wavelength

in circumference provides the clue we need to construct a theory of the atom.

• If we consider the vibrations of a wire loop (Fig. 4.13), we find that their

wavelengths always fit an integral number of times into the loop’s

circumference so that each wave joins smoothly with the next.

• If the wire were perfectly elastic, these vibrations would continue

indefinitely.

• Why are these the only vibrations possible in a wire loop?

Figure 4.13 Some modes of vibration of a wire loop. In each case a whole number of wavelengths fit into the

circumference of the loop.

• If a fractional number of wavelengths is placed around the loop, as in

Fig. 4.14, destructive interference will occur as the waves travel

around the loop, and the vibrations will die out rapidly.

Figure 4.14 A fractional number of wavelengths cannot persist because destructive

interference will occur.

• By considering the behavior of electron waves in the hydrogen atom as analogous to the

vibrations of a wire loop, then, we can say that

• An electron can circle a nucleus only if its orbit contains an integral number of de Broglie

wavelengths.

• This statement combines both the particle and wave characters of the electron since the

electron wavelength depends upon the orbital velocity needed to balance the pull of the

nucleus.

• It is easy to express the condition that an electron orbit contain an integral number of de

Broglie wavelengths. The circumference of a circular orbit of radius r is 2r, and so the

condition for orbit stability is

where r

designates the radius of the orbit that contain n wavelengths. The integer n is called the

quantum number of the orbit. Substituting for , the electron wavelength given by Eq. (4.11),

yields

and so the possible electron orbits are those whose radii are given by

The radius of the innermost orbit is customarily called the Bohr radius of the hydrogen atom and is

denoted by the symbol a

The other radii are given in terms of a

by the formula

4.5 ENERGY LEVELS AND SPECTRA

• A photon is emitted when an electron jumps from one energy level to a lower level

• The various permitted orbits involve different electron energies. The electron energy E

given in terms of the orbit radius r

by Eq. (4.5) as

Substituting for r

from Eq (4.13), we see that

• The energies specified by Eq.

(4.15) are called the energy levels

of the hydrogen atom and are

plotted in Fig. 4.15.

• An atomic electron can have only

these energies and no others.

• The lowest energy level E

called the ground state of the

atom, and the higher levels E

, E

, . . . are called excited

states.

• As the quantum number n

increases, the corresponding

energy E

approaches closer to 0.

• The work needed to remove an

electron from an atom in its

ground state is called its

ionization energy.

• The ionization energy is

accordingly equal to -E

, the

energy that must be provided to

raise an electron from its ground

state to an energy of E=0, when it

is free.

• For hydrogen, it is 13.6 eV.

Figure 4.15 Energy levels of the hydrogen atom.

Example 4.2

An electron collides with a hydrogen atom in its ground state and excites it to a state of n=3. How

much energy was given to the hydrogen atom in this inelastic (KE not conserved) collision?

Example 4.3

Hydrogen atoms in states of high quantum number have been created in the laboratory and

observed in space. They are called Rydberg atoms. (a) Find the quantum number of the Bohr

orbit in a hydrogen atom whose radius is 0.0100 mm. (b) What is the energy of a hydrogen

atom in this state?

Origin of Line Spectra

• The presence of discrete energy levels in the hydrogen atom suggests the connection.

• According to our model, electrons cannot exist in an atom except in certain specific energy

levels.

• The jump of an electron from one level to another, with the difference in energy between the

levels being given off all at once in a photon rather than in some more gradual manner, fits

in well with this model.

• Recall that E

is a negative quantity (-13.6 eV), so -E

is a positive quantity. The frequency

of the photon released in this transition is therefore

Since =c/ν 

• Equation (4.18)

states that the

radiation emitted by

excited hydrogen

atoms should

contain certain

wavelengths only.

• These wavelengths,

furthermore, fall

into definite

sequences that

depend upon the

quantum number n

f the final energy

level of the electron

(Fig. 4.16).

Figure 4.16 Spectral lines

originate in transitions between energy levels. Shown are the spectral series of hydrogen. When n=, the electron is

free.

• Since n

> n

in each case, in order that there be an excess of energy to be given off as a

photon, the calculated formulas for the first five series are

Example 4.4

Find the longest wavelength present in the Balmer series of hydrogen, corresponding to the H



line.

4.8 ATOMIC EXCITATION

• How atoms absorb and emit energy.

• There are two main ways in which an atom can be excited to an

energy above its ground state and thereby become able to radiate.

1) One of these ways is by a collision with another particle in which

part of their joint kinetic energy is absorbed by the atom.

o Such an excited atom will return to its ground state in an

average of 10

-8

s by emitting one or more photons (Fig. 4.18).

o Energy transfer is a maximum when the colliding particles

have the same mass: the electrons in such a discharge are

more effective than the ions in providing energy to atomic

electrons.

2) Another excitation mechanism is involved when an atom absorbs a

photon of light whose energy is just the right amount to raise the

atom to a higher energy level. This process explains the origin of

absorption spectra (Fig. 4.19).

Figure 4.18 Excitation by collision. Some of the available energy is absorbed by one of the atoms, which goes

into an excited energy state. The atom then emits a photon in returning to its ground (normal) state.

Figure 4.19 How emission and

absorption spectral lines originate.

• When white light, which contains all wavelengths, is passed through hydrogen gas, photons

of those wavelengths that correspond to

transitions between energy levels are

absorbed.

• The resulting excited hydrogen atoms

reradiate their excitation energy almost at

once, but these photons come off in random

directions with only a few in the same

direction as the original beam of white light

(Fig. 4.20).

Figure 4.20 The dark lines in an absorption spectrum are never totally dark

4.9 THE LASER

• How to produce light waves all in step.

• The laser is a device that produces a light beam with some remarkable properties:

1. The light is very nearly monochromatic.

2. The light is coherent, with the waves all exactly in phase with one

another (Fig.4.23).

3. A laser beam diverges hardly at all. Such a beam sent from the

earth to a mirror left on the moon remained narrow enough to be

detected on its return to the earth, a total distance of over three-

quarters of a million kilometers.

4. The beam is extremely intense, more intense by far than the light

from any other source

• The last two of these properties follow from the second of them.

• The term laser stands for Light Amplification by Stimulated

Emission of Radiation.

Figure 4.23 A laser produces a beam of light whose waves all have the same frequency

(monochromatic) and are in phase with one another (coherent). The beam is also well

collimated and so spreads out very little, even over long distances.

• The key to the laser is the presence in many atoms of one or more

excited energy levels whose lifetimes may be 10

-3

s or more

instead of the usual 10

-8

• Such relatively long-lived states are called metastable (temporarily stable); see Fig. 4.24.

Figure 4.24 An atom can exist in a metastable

energy level for a longer time before

radiating than it can in an ordinary energy

level.

• Three kinds of transition involving electromagnetic radiation are possible between two

energy levels, E

and E

, in an

atom (Fig. 4.25).

Figure 4.25 Transitions between two energy

levels in an atom can occur by stimulated

absorption, spontaneous emission, and stimulated

emission.

1. Stimulated absorption. If the atom is initially in the lower state E

, it can be raised to E

absorbing a photon of energy E

- E

= hν.

2. Spontaneous emission. If the atom is initially in the upper state E

, it can drop to E

emitting a photon of energy hν.

3. Stimulated emission (Einstein, in 1917, was the first to point out a third possibility). An

incident photon of energy hν causes a transition from E

to E

o In stimulated emission, the radiated light waves are exactly in phase with the

incident ones, so the result is an enhanced beam of coherent light.

o A photon of energy hν incident on an atom in the upper state E

has the same

likelihood of causing the emission of another photon of energy hν as its likelihood of

being absorbed if it is incident on an atom in the lower state E

• A three-level laser, the simplest kind, uses an assembly of atoms (or molecules) that have a

metastable state hν in energy above the ground state and a still higher excited state that

decays to the metastable state (Fig. 4.26). Figure 4.26 The principle of the laser.

• What we want is more atoms in the metastable state than in the ground state.

• If we can arrange this and then shine light of frequency ν on the assembly, there will be

more stimulated emissions from atoms in the metastable state than stimulated absorptions by

atoms in the ground state.

• The result will be an amplification of the original light. This is the concept that underlies the

operation of the laser.

• The term population inversion describes an assembly of atoms in which the majority are in

energy levels above the ground state; normally the ground state is occupied to the greatest

extent.