Chapter 4

Atomic Structure

4 Atomic Structure

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4 Atomic Structure

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• Structure of the atom is responsible for nearly all the properties of

matter (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: Thomson model

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• 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?

Joseph John

Thomson

1856-1940

Nobel Prize in

Physics in 1906

• Thomson 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.

4. 1 The Nuclear Atom: Rutherford model

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• Rutherford model: Hans Geiger and Ernest Marsden used the

fast alpha particles as probes in 1911 at the suggestion of

Ernest Rutherford (Discovery of alpha and beta radioactivity

and discovery of atomic nucleus)

• Alpha particles are helium atoms that have lost two

electrons each, leaving them with a charge of +2e.

• Fast alpha particles emitted by certain radioactive elements.

Ernest

Rutherford

1871-1937

Nobel Prize in

Chemistry in

1908

• 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.

Figure 4.2 The Rutherford scattering experiment.

• 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!!

4. 1 The Nuclear Atom: Rutherford model

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• With only weak electric forces exerted on them, alpha particles that

pass through a thin foil should be deflected only slightly, 1° or less.

• 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)

Figure 4.3 The Rutherford model of the atom.

4. 1 The Nuclear Atom

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• 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.

• 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.

4. 1 The Nuclear Atom: Nuclear Dimensions

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• 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

(Distance of closest approach )

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

J. Since 1/4

=9.0x10

and Z=79,

4. 2 Electron Orbit

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• 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.

Figure 4 5 Force balance in the

hydrogen atom.

• 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).

4. 2 Electron Orbit: Hydrogen atom

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• Let us look at the classical dynamics of the hydrogen atom, whose

single electron makes it the simplest of all atoms. The centripetal force

holding the electron in an orbit r from the nucleus is provided by:

• 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 between them. The condition for a dynamically stable orbit is

The electron velocity is therefore related to its orbit radius r by the formula

(4.4 Electron velocity)

• The total energy E of the electron in a hydrogen atom is

the sum of its kinetic and potential energies, which are

Substituting for v from Eq. (4.4) gives

• 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.

(4.5 Total energy of hydrogen atom)

4. 2 Electron Orbit

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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.

4. 2 Electron Orbit: The Failure of Classical Physics

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Figure 4.6 An atomic electron

should, classically, spiral rapidly

into the nucleus as it radiates

energy due to its acceleration

• 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 following a curved path is accelerated and therefore

should continuously lose energy, spiraling into the nucleus in a

fraction of a second (Fig. 4.6).

• But atoms do not collapse.

• This contradiction further illustrates what we have discussed before;

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

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• 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 previously that condensed matter (solids and liquids) at all

temperatures emits EM radiation in which all wavelengths are present,

though with different intensities.

• At one extreme, 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.

4. 3 Atomic Spectra: Emission Line Spectra

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Figure 4.7 An idealized spectrometer

• 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 4.8 Some of the principal lines in the emission spectra

of hydrogen, helium, and mercury.

• 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.

4. 3 AtomicSpectra: Absorption Line Spectra

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• 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 emission spectrum.

• 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.

4. 3 Atomic Spectra: Spectral Series

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• 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.

Figure 4.10 The Balmer series of hydrogen. The H

line is red, the

line is blue, the H

and H

lines are violet, and the other lines are

in the near ultraviolet.

• 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.

(Balmer)

Balmer’s formula for the wavelengths of this series is

The quantity R, known as the Rydberg constant, has the

value R= 1.097x10

-1

=0.01097 nm

-1

• What is n for H



? What happens for n=?

4. 3 Atomic Spectra: Spectral Series

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• 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.

In the infrared, three spectral series have been found whose

lines have the wavelengths specified by the formulas

These spectral series of hydrogen are plotted in terms of

wavelength in Fig. 4.11.

4.4 Bohr Atom

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• 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.

Niels Henrik

David Bohr

1885-1962

Nobel Prize in

Physics in 1922

• 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 equations.

• The de Broglie wavelength of this electron is where the

electron velocity v is that given by Eq. (4.4)

(4.11 Orbital electron

wavelength)

4.4 Bohr Atom

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• By substituting 5.3x10

-11

m for the radius r of the electron orbit, we find

the electron wavelength to be

Figure 4.12 The orbit of the

electron in a hydrogen atom

corresponds to a complete electron

de Broglie wave joined on itself

• 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).

• 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.

4.4 Bohr Atom

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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 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.

• Why are these the only vibrations possible in a wire 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 ocur.

4.4 Bohr Atom

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• 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

(Condition for orbit stability)

where r

designates the radius of the orbit that contain n wavelengths.

The integer n is called the quantum number of the orbit.

4.4 Bohr Atom

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• Substituting for , the electron wavelength given by Eq. (4.11), yields

(4.13 Orbital radii in Bohr atom)

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

(Bohr radius)

The other radii are given in terms of a

by the formula

4.5 Energy Levels and Spectra

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• 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

is given in terms of the orbit radius r

by Eq. (4.5) as

Substituting for r

from Eq (4.13), we see that

(4.15)

4.5 Energy Levels and Spectra

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Figure 4.15 Energy levels of the hydrogen atom.

 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.

 The lowest energy level E

is 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.

4.5 Energy Levels and Spectra

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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?

4.5 Energy Levels and Spectra

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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?

4.5 Energy Levels and Spectra: Origin of Line Spectra

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 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

(4.18 Hydrogen spectrum)

4.5 Energy Levels and Spectra: Hydrogen Spectrum

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 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

of 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 Lyman, Balmer, Paschen, Brackett, Pfund series.

4.5 Energy Levels and Spectra : Hydrogen Spectrum

4.5 Energy Levels and Spectra

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 :

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.

• 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).

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..

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).

4.8 Atomic Excitation

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: Light Amplification by Stimulated Emission of Radiation

Figure 4.223A 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.

• 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.

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.

4.9 The LASER: Light Amplification by Stimulated Emission of Radiation

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.

• 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.

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.

4.9 The LASER: Light Amplification by Stimulated Emission of Radiation

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

, it

can be raised to E

by 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

by 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

• 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.

• 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

4.9 The LASER: Light Amplification by Stimulated Emission of Radiation

• 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.

4.9 The LASER: Light Amplification by Stimulated Emission of Radiation

• 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.

4 Solved Problems

1. The mysterious lines observed in 1896 in the spectrum of the star ζ-

Puppis fit the empirical formula

where R is the Rydberg constant. Show that these lines can be

explained by the Bohr theory as originating from He

4 Solved Problems

2. In the n = 3 state of hydrogen (for Bohr model), find the electron’s

radius, velocity, kinetic energy, and potential energy.

4 Solved Problems

3. Find the wavelengths of the transitions from n

= 3 to n

= 2 and from

= 4 to n

= 2 in atomic hydrogen.

4 Solved Problems

4. The shortest wavelength of the hydrogen Lyman series is 91.13 nm.

Find the three longest wavelengths in this series. (Hint: The shortest

wavelength is the series limit)

4 Solved Problems

5. Calculate the two longest wavelengths of the Balmer series of triply

ionized beryllium (Z = 4).

Chapter 3 Wave Properties of Particles

Additional Materials