Lesson_2a_formulae.pdf

Lesson 2a

Following to B.G. Streetman

2.1 Atoms and Electrons, introduction to physical models

In the 1920s it became necessary to develop a new theory to describe phenomena on the atomic

scale. A long series of careful observations had been made that clearly indicated that many events

involving electrons and atoms did not obey the classical laws of mechanics. It was necessary,

therefore, to develop a new kind of mechanics to describe the behavior of particles on this small

scale. This new approach, called quantum mechanics, describes atomic phenomena very well and also

properly predicts the way in which electrons behave in solids — our primary interest here. Through

the years, quantum mechanics has been so successful that now it stands beside the classical laws as a

valid description of nature.

A special problem arises when students first encounter the theory of quantum mechanics.

The problem is that quantum concepts are largely mathematical in nature and do not involve the

“common sense” quality associated with classical mechanics. At first, many students find quantum

concepts difficult, not so much because of the mathematics involved, but because they feel the

concepts are somehow divorced from “reality.” This is a reasonable reaction, since ideas that we

consider to be real or intuitively satisfying are usually based on our own observation. Thus the

classical laws of motion are easy to understand because we observe bodies in motion every day.

On the other hand, we observe the effects of atoms and electrons only indirectly , and naturally we

have very little feeling for what is happening on the atomic scale. It is necessary, therefore, to

depend on the facility of the theory to predict experimental results rather than to attempt to force

classical analogues onto the non-classical phenomena of atoms and electrons.

Our approach in this chapter will be to investigate the important experimental observations

that led to the quantum theory, and then to indicate how the theory accounts for these

observations. Discussions of quantum theory must necessarily be largely qualitative in such a brief

presentation, and those topics that are most important to solid state theory will be emphasized

here.

2.2 Experimental observations that led to the development of quantum theory

The experiments that led to the development of quantum theory were concerned with the nature of

light and the relation of optical energy to the energies of electrons within atoms. These

experiments supplied only indirect evidence of the nature of phenomena on the atomic scale;

however, the cumulative results of a number of careful experiments showed clearly that a new theory

was needed.

2.2.1 The Photoelectric Effect

An important hypothesis of Planck indicated that radiation from a heated sample is emitted in

discrete units of energy, called quanta; the energy units were described by hv, where v is the

frequency of the radiation, and h is a quantity now called Planck’s constant

(

−34

⋅

). Soon after Planck developed this hypothesis, Einstein interpreted an

important experiment that clearly demonstrated the discrete nature (quantization) of light. This ex-

periment involved absorption of optical energy by the electrons in a metal and the relationship

between the amount of energy absorbed and the frequency of the light (Fig.2- 1). Let us suppose

that monochromatic light is incident on the surface of a metal plate in a vacuum.

The electrons in the metal absorb energy from the light, and some of the electrons receive

enough energy to be ejected from the metal surface into the vacuum. This phenomenon is called the

photoelectric effect. If the energy of the escaping electrons is measured, a plot can be made of the

maximum energy as a function of the frequency v of the incident light (Fig. 2-lb).

One simple way of finding the maximum energy of the ejected electrons is to place another

plate above the one shown in Fig. 2-la and then create an electric field between the two plates. The

potential necessary to retard all electron flow between the plates gives the energy . For a

particular frequency of light incident on the sample, a maximum energy is observed for the

emitted electrons. The resulting plot of vs. ν is linear, with a slope equal to Planck’s constant.

The equation of the line shown in Fig. 2-lb is

E m

= ν , (2-1)

−

where q is the magnitude of the electronic charge. The quantity V (volts) is a characteristic of the

particular metal used. When V is multiplied by the electronic charge, an energy (joules) is obtained

which represents the minimum energy required for an electron to escape from the metal into a

vacuum. The energy qV is called the work function of the metal. These results indicate that the

electrons receive an energy hv from the light, and lose an amount of energy qV in escaping from the

surface of the metal.

This experiment demonstrates clearly that Planck's hypothesis was correct — light energy is

contained in discrete units rather than in a continuous distribution of energies. Other experiments also

indicate that, in addition to the wave nature of light, the quantized units of light energy can be

considered as localized packets of energy, called photons. Some experiments emphasize the wave

nature of light, while other experiments reveal the discrete nature of photons. This duality is

fundamental to quantum processes and does not imply an ambiguity in the theory.

2.2.2 Atomic Spectra

One of the most valuable experiments of modern physics is the analysis of absorption and emission

of light by atoms. For example, an electric discharge can be created in a gas, so that the atoms

begin to emit light with wavelengths characteristic of the gas. We see this effect in a neon sign,

which is typically a glass tube filled with neon or a gas mixture, with electrodes for creating a dis-

charge. If the intensity of the emitted light is measured as a function of wavelength, one finds a

series of sharp lines rather than a continuous distribution of wavelengths. By the early 1900s the

characteristic spectra for several atoms were well known. A portion of the measured emission

spectrum for hydrogen is shown in Fig. 2-2, in which the vertical lines represent the positions of

observed emission peaks on the wavelength scale. Wavelength ( λ) is usually measured in

angstroms (

−

) and is related (in meters) to frequency by

= , where c is the speed

of light (

10 8

m /

). Photon energy hv is then related to wavelength by

(2-2)

λ /

The lines in Fig. 2-2 appear in several groups labeled the Lyman, Balmer, and Paschen series

after their early investigators. Once the hydrogen spectrum was established, scientists noticed

several interesting relationships among the lines. The various series in the spectrum were observed

to follow certain empirical forms:

Lyman:

⎝

−

⎠

n =

432

...

(2-3a)

Balmer:

⎝

−

⎠

n =

543

...

(2-3b)

Paschen:

⎝

−

⎠

n =

...

(2-3c)

where R is a constant called the Rydberg constant (R = 109,678 cm -1 ). If the photon energies hv are

plotted for successive values of the integer n , we notice that each energy can be obtained by taking

sums and differences of other photon energies in the spectrum (Fig. 2-3). For example, E 42 in the

Balmer series

is the difference between E 4l and E 21 in the Lyman series. This relationship among the various

series is called the Ritz combination principle. Naturally, these empirical observations stirred a great

deal of interest in constructing a comprehensive theory for the origin of the photons given off by

atoms.

⎛

⎞

⎛

⎞

⎛

⎞

2.3 The Bohr model

The results of emission spectra experiments led Niels Bohr to construct a model for the hydrogen

atom, based on the mathematics of planetary systems. If the electron in the hydrogen atom has a

series of planetary-type orbits available to it, it can be excited to an outer orbit and then can fall to

any one of the inner orbits, giving off energy corresponding to one of the lines of Fig. 2-3. To

develop the model, Bohr made several postulates:

1. Electrons exist in certain stable, circular orbits about the nucleus. This assumption implies

that the orbiting electron does not give off radiation, as classical electromagnetic theory

would normally require of a charge experiencing angular acceleration; otherwise, the electron

would not be stable in the orbit but would spiral into the nucleus as it lost energy by

radiation.

2. The electron may shift to an orbit of higher or lower energy, thereby gaining or losing

energy equal to the difference in the energy levels (by absorption or emission of a photon of

energy hv ) .

3. The angular momentum of the electron in an orbit is always an integral multiple of Planck's

constant divided by

2π ( 2

is often abbreviated for convenience). This assumption,

P =

, n = 1,2,3,4,... (2-5)

is necessary to obtain the observed results of Fig. 2-3.

If we visualize the electron in a stable orbit of radius r about the proton of the hydrogen atom,

we can equate the electrostatic force between the charges to the centripetal force :

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