Barrier Penetration and Quantum Tunneling (Qualitative)

BARRIER PENETRATION AND QUANTUM TUNNELING (Qualitative)

• According to classical ideas, a particle striking a hard wall has no chance of leaking through it. But, the behaviour of a quantum particle is different due to the wave nature associated with it.

• We know that when an electromagnetic wave strikes at the interface of two media, it is partly reflected and partly transmitted through the interface and enters the second medium.

• In a similar way the de Broglie wave also has a possibility of getting partly reflected from the boundary of the potential well and partly penetrating through the barrier.

• Fig. 7.4 shows a particle with energy E < V approaching potential barrier of height V.

• An electron of total energy E approaches the barrier from the left. From the view-point of classical physics, the electron would be reflected from the barrier because its energy E is less than V.

• For the particle to overcome the potential barrier, it must have an energy equal to or greater than V.

Quantum mechanics leads to an entirely new result. It shows that there is a finite chance for the electron to leak to the other side of the barrier.

It is noted that the electron tunneled through the potential barrier and hence in quantum mechanics, this phenomenon is called tunneling.

The transmission of electrons through the barrier is known as barrier penetration.

Expression for Transmission Probability

• Now let us consider the case of a particle of energy E<V approaching a potential barrier of finite height and width as shown in fig. 7.5.

• The particle in region I has certain probability of passing through the barrier to reach region II and then emerge out on the other side in region III.

• The particle lacks the energy to go over the top of the barrier, but tunnels through it. Higher the barrier and wider it is, the lesser is the probability of the particle tunneling through it.

Fig. 7.5 When a particle of energy E < V, approaches a potential barrier, the de Broglie waves that correspond to the particle are partly reflected and partly transmitted. That is the particle has a finite chance of penetrating the barrier

• Let us now consider a beam of identical particles, all having kinetic energy E. The beam is incident on the potential barrier of height V and width a from region I.

• On both sides of the barrier V = 0. This means that no forces act on particles in regions I and III.

As shown in fig. 7.5, the wave function ΨI, represents the particle moving towards the barrier from region-I while ΨI - represents the particle reflected moving away from the barrier.

The wave function ΨII represents the particle inside the barrier. Some of the particles end up in region III while the others return to region I.

Quantum mechanics shows that the transmission probability T for a particle to pass through the barrier is given by

This probability is approximately given by

T0 is a constant close to unity. It shows that the probability of particle penetration through a potential barrier depends on the height and width of the barrier.

Significance of the study of barrier penetration problems

1. Tunnelling is a very important physical phenomena which occurs in certain semiconductor diodes. In such diodes electrons pass through potential barriers even though their kinetic energies are smaller than the barrier heights.

2. The tunneling effect also occurs in the case of the alpha particles. The kinetic energy of alpha particle is only a few MeV but it is able to escape from a nucleus whose potential wall is perhaps 25 MeV high. 

3. The ability of electrons to tunnel through a potential barrier is used in the Scanning Tunneling Microscope (STM) to study surfaces on an atomic scale of size.

Terminology related to microscope 

(a) Microscope

A microscope is an instrument which is used to view the magnified image of a smaller object which cannot be clearly seen with a naked eye.

(b) Optical microscope

It is a microscope which uses light radiation to illuminate the object.

(c) Resolving power

It is the ability of the microscope to show two closer objects as separated ones.

The resolving power is inversely proportional to wavelength of light used. In an electron microscope, beam of electrons are used to illuminate the specimen.

The wave length a associated with these electrons is about 0.1 Å or less. Hence, its resolving power is very high. The minimum distance that can be resolved in the electron microscope is about 10 Å.

(d) Magnification Power

It is the ability of the microscope to show the image of an object in an enlarged manner.

In an optical microscope, 

F→ Focal length of objective lens in mm 

f→ Focal length of eye piece in mm 

Δ - Length of microscope (16 cm) 

D- Least distance of distinct vision (25 cm)

Thus, the magnification is about 1000 X (one thousand times).

In the case of electron microscope, Δ is very large ( >1 m) F and f can be reduced to less than a millimetre. So, the magnification power of electron microscope is about 105 X. 

(e) Depth of focus

It is defined as the ability of the objective of microscope to produce a sharp focussed image when the surface of the object is not truly plane.

The deviation from plane surface occurs when the specimen is severely etched or when certain constituents of the structure are depressed or elevated from the etched surface.

Electron Microscope

Definition

It is a microscope which uses electron beam to illuminate a specimen and it produces an enlarged image of the specimen.

It has very high magnification power and resolving power when compared to optical microscope.

Principle

Like an optical microscope, its purpose is to magnify extremely minute objects. The resolving power of microscope is inversely proportional to the wavelength of the radiation used for illuminating the object under study.

Higher magnification as well as resolving power can be obtained by utilizing waves of shorter wavelength (λ).

Electron microscope uses electron waves whose wavelength is given by the formula

For V = 10,000 V, λ = 0.1225 Å which is extremely short. Electron microscopes giving magnification more than 2,00,000 X are common in Science & Technology Medical Research Laboratories.

An electron microscope consists of the following essential parts:

(i) Electron Gun. Its function is to provide a narrow beam of electrons of uniform velocity.

(ii) Electrostatic and magnetic lenses. Their function is to refract and properly focus the electron beam.

(iii) Fluorescent screen or photographic plate. They are used to receive the highly magnified image of the extremely small object being studied.

Types of Electron Microscopes

There are four types of electron microscopes. They are

1. Transmission Electron Microscope (TEM)

2. Scanning Electron Microscope (SEM)

3. Scanning Transmission Electron Microscope (STEM)

4. Scanning Tunneling Microscope (STM).

Fig. 7.6 gives comparison of an optical and electron microscope.