Lecture Notes 12

EPF 0024: Physics II 1

12.0 Atomic Physics


Outline

12.1 Introduction12.2 The Nuclear Atom12.3 Line Spectra of Atomic Hydrogen12.4 Bohr’s Hydrogen Atom Model12.5 Atomic Radiation


Objective

To explain nuclear atom model and line spectra.

To analyze Bohr model of hydrogen atom.

To calculate energies and wavelengths of photons for transitions in atomic hydrogen.

To discuss various types of atomic radiation


12.1 IntroductionSpeculations about the

microscopic structure of matter has been going on for centuries. Greek philosophers theorized about the smallest possible piece of an element and called it an atom.

Theories about nature of atom include plum pudding [Fig. 12.1 (a)] and Rutherford’s models [Fig. 12.1 (b)]. The latter model is now accepted and called the nuclear atom.

Fig.12.1: (a) Plum pudding model and (b) Rutherford’s solar system model


12.2 The Nuclear Atom

An experiment performed by Rutherford that resulted in proposing the nuclear model of the atom is shown in Fig. 12.2 (a).

According to results of experiment, Rutherford proposed that an atom consists of a small, positively charged nucleus surrounded by a number of electrons (Fig. 12.2 (b)).

Fig. 12.2: Rutherford’s scattering experiment


When low pressure gas is sealed in a tube and a large voltage applied between the ends of tube, the gas emits EM radiation characteristic of the individual gas atoms. Examples are neon lamps. When this radiation is allowed to pass through a diffraction grating, it is separated into the various wavelengths (Fig. 12.3). A series of bright line is observed.

12.3 Line spectra of atomic hydrogen

Fig. 12.3: The line spectrum of an atom


This type of spectrum, with its bright lines in different colors, is referred to as line spectrum. Fig. 12.4 (a) shows the visible part of the emission line spectrum of hydrogen atoms.

Interestingly, if light of all colors is passed through a tube of hydrogen gas, some wavelengths will be absorbed by the atoms, giving rise to absorption spectrum at same location (Fig. 12.4 (b).

Fig. 12.4: Line spectrum of hydrogen: (a) emission spectrum (b) absorption spectrum


Fig. 12.5 illustrates some of the groups or series of lines in the spectrum of the simplest atom, atomic hydrogen.

The group of lines in the visible region is known as the Balmer series after Johann J. Balmer (1825-1898) who deduced an empirical equation that gave the values for the observed wavelengths.

Fig. 12.5: Line spectrum of atomic hydrogen


The empirical equations for determining the wavelengths for the series are:

where the constant R = 1.097 107 m1 is called the Rydberg constant and n is called the principle quantum number.

... ,6, 5, 4 1311 seriesPaschen

... ,5, 4, 3 1211 seriesBalmer

... ,4, 3, 2 1111 seriesLyman

22

22

22

nn

R

nn

R

nn

R

(12.1)


Example

Find (a) the longest and (b) the shortest wavelengths of the Balmer series.


Solution

nm 656

m10524.131

21m10097.1

1211

1622

17

22

nR

nm 365

m10743.2021m10097.1

1211

162

17

22

nR

(a)

(b)


In 1913 Bohr presented a model for the hydrogen atom that led to equations such as Balmer’s.

He combines Rutherford’s nuclear atom with Plank’s theory of quantization of energy and hypothesized that:

(i) in H-atom electrons can only occupy certain discrete energy levels (Fig. 12.6).

(ii) while in these stationary orbits, electrons do not radiate energy.

(iii)The angular momentum L of the electron can assume only certain discrete values

12.4 Bohr’s Hydrogen Atom Model

Fig.12.6: The Bohr Model of the Atom


When an electron in an initial orbit with a larger energy Ei changes to a final orbit with a smaller energy Ef, the emitted photon has an energy

Fig. 12.7 shows an electron of mass m and speed v in an orbit of radius r. Total energy of electron consists of kinetic energy and electrostatic potential energy. Nucleus assumed to contain Z protons (Atomic Number).

hfEE fi (12.2)

Fig. 12.7: An electron in a uniform circular motion (Bohr model)

+Ze


The total energy E of the orbiting electron is:

The centripetal force is provided by the electrostatic force, that is

rkZemvUKE

22

21 (12.3)

2

22

rkZe

rmv

(12.4)


Using Equations (12.3) and (12.4) we obtain an expression for the total energy, that is

To determine r, Bohr made a third hypothesis that: (iii) The angular momentum L of the electron can assume only certain discrete values (similar to Planck's assumption about energy), that is

rkZe

rkZe

rkZeE

221 222

(12.5)

2

2 hnrmvrvmrIL nnn

(12.6)


Using Equations (12.4) and (12.6) we obtain an expression for rn, the nth Bohr orbit, that is

For hydrogen atom (Z = 1), the smallest Bohr orbit (n = 1) has a radius r1 = 5.29 1011m and is called the Bohr radius.

... ,3 ,2 ,1 m105.29

4

211

2

22

2

nZn

Zn

mkehrn (12.7)


From Equations (12.5) and (12.7) the corresponding expression for the total energy for the nth orbit is obtained, that is:

Fig. 12.8 shows a representation of energy level diagram for H-atom (Z = 1) using equation (12.8).

... , 3, 2, 1 ZeV 6.13 ZJ1018.2

2

2

2

2

218

2

2

2

422

nnn

nZ

hemkEn

(12.8)


Fig. 12.8: Energy level diagram for H-atom


The lowest energy level at n = 1 has a value 13.6 eV and is called the ground state of the atom. The energy levels at n = 2 and above are called the excited states.

The energy needed to remove an electron from the ground state (n = 1) to infinity (n = ) for hydrogen atom is equal to the binding energy of the electron at ground state but positive (+13.6 eV). This energy is called the ionization energy of the atom. Supplying this amount of energy removes the electron from the atom, producing positive hydrogen ion H+.


Bohr then combined his model of the atom (equation 12.8) with Einstein’s idea of the photon through the equation Ei – Ef = hf and obtained the expression

From which the wavelengths in the line spectrum of hydrogen can be predicted.

fifiif

fifi

nnnnnn

Zch

emk

nZ

nZ

hemkEEhchf

... ,3 ,2 ,1, 1121

2

222

3

422

2

2

2

2

2

422

(12.9)


The value for the expression 22mk2e4/(h3c) is 1.097 107 m1, which agrees very well with the experimental value for R, the Rydberg constant, obtained earlier.

Fig. 12.9 shows the Lyman and Balmer series for the hydrogen atom (Z = 1) calculated from Equation (12.9). Bohr’s model shows that Lyman series occurs when electrons make transitions from higher energy levels with ni = 2, 3, 4, … to the ground state (nf = 1).


Fig. 12.9: Lyman and Balmer series in H-atom


Just as an electron emits a photon when it jumps to a lower level, it can also absorb a photon and jump to a higher level.

This process occurs only if the photon has the proper energy. In particular, the photon must have an energy that precisely matches the energy difference between the lower and higher level to which the electron is raised.


Example 1

An electron in a hydrogen atom is in the initial state ni = 4. Calculate the wavelength of the photon emitted by this electron if it jumps to the final state (a) nf = 3, (b) nf = 2, or (c) nf = 1. (take the Rydberg constant R = 1.097 107 m1.)


Solution

(a)

(b)

(c)

nm 1875161

91

m 101.0971 111 1

17

1

22

if nnR

nm 2.486161

41

m 101.0971 111 1

17

1

22

if nnR

nm 23.97161

11

m 101.0971 111 1

17

1

22

if nnR

11122

if nnR


Example 2

A hydrogen atom with its electron in the initial state ni = 5 emits a photon with a wavelength of 434 nm. To which state did the electron jump?


Solution

2

m10097.1m1043455m10097.1m10434

111

1792

2179

2

2

22

RnRnn

nnR

i

if

if


12.6 Atomic Radiation

There are various types of radiation associated with multielectron atom. Examples range from X-rays that are energetic enough to pass through a human body, to the soft white light of a fluorescent light bulb.

Laser is also radiation whose origin is the controlled energy transitions in an atom as we shall see here.


12.6.1 X-RaysX-rays are produced when

electrons, accelerated through a large potential difference, collide with a metal target made from molybdenum or platinum.

The target is contained within an evacuated glass tube (Fig. 12.11(a)). A plot of X-ray intensity per unit wavelength versus the wavelength is shown in Fig. 12.11 (b) and consists of sharp peaks or lines superimposed on a broad continuous spectrum.

Fig. 12.11: (a) X-ray tube and (b) the X-ray spectrum for molybdenum


The broad continuous spectrum is referred to as Bremsstrahlung (breaking radiation) and is emitted when the electrons decelerate upon hitting the target.

The sharp peaks are called characteristic X-rays. They are characteristic of the target material. They are marked K and K as n = 1 or K shell is involved. When energetic electron strikes the target a K-shell electrons may be knocked off. An electron in one of the outer shells fall into the K shell emitting X-ray photon in the process.


12.6.2 Lasers

The laser is an invention of the twentieth century. The operating principle of many types of lasers depend directly on the quantum mechanical structure of the atom.

It has already been shown that a photon is emitted when an electron makes a transition from a higher energy state to a lower one. This is called spontaneous emission. Normally, in spontaneous emission, an electron in an atom eventually drops to a lower level in a time that is about 108 s giving off a photon in the process.


The operation of a laser depends on stimulated emission where an incoming photon induces or stimulates an electron to change energy levels.

In stimulated emission one photon goes in and two photons come out (Fig. 12.12). This process amplifies the number of photons and therefore is the origin of the word laser = “light amplification by the stimulated emission of radiation”. The emitted photon travels in the same direction as the incoming photon and are coherent.

Fig. 12.12: Stimulated emission


Stimulated emission depends on an external source of energy to excite electrons to higher energy levels and produce what is known as a population inversion of electrons.

Fig. 12.13 compares a normal energy level population with a population inversion. The energy state in a population inversion must be metastable, in the sense that electrons remain in it for a much longer period of time than they do in an ordinary excited state.

Fig. 12.13: Population inversion


A specific example of a laser is the helium-neon laser in which the neon atoms produce the laser light. In this arrangement the helium and neon gas is enclosed in an evacuated tube and a high voltage applied. Fig. 12.14 shows the schematic diagram.

Fig. 12.14: Schematic of a helium-neon laser


Fig. 12.15: Energy level diagram.

The appropriate energy level diagram for helium and neon are shown in Fig. 12.15. The excited state E3 is metastable. Electrons are excited to this level by application of a high potential of 8000 V to the tube containing the helium-neon mixture.


12.6.3 Fluorescence and PhosphorescenceAn atom in an excited state can

emit photons of various energies as it falls to the ground state. The emission of light of lower frequency after illumination by a higher frequency as shown in Fig. 12.16 is referred to as fluorescence.

An example of fluorescence is the fluorescent light bulb. The inside of such a bulb, with a filament at one end, is filled with mercury vapor.

Fig. 12.16: The mechanism of fluorescence


When a voltage is applied to the ends of the bulb, the filament is heated and produces electrons which are accelerated through the tube and collide with the mercury atoms to produce ultraviolet light.

The inside of the tube is coated with phosphor that absorbs the ultraviolet light and then emits a visible lower-frequency light.


Another application of fluorescence is in forensics. The analysis of a crime scene is enhanced by the fact that human bones and teeth are fluorescent. Therefore, illuminating a crime scene with ultraviolet light can make items of interest stand out for easy identification.

Alternatively, the use of a fluorescent dye can make fingerprints visible thus enabling identification of a culprit at a crime scene.


Phosphorescence is similar to fluorescence, but in phosphorescent materials continue to give off secondary glow long after the initial illumination that excited the atoms. Phosphorescence may persist for periods of time ranging from a few seconds to several hours.

An example Phosphorescent materials are those used in watches and clocks so that the writing on the deice continue to be seen in the dark even though the light is switched off.

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Lecture Notes 12

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