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β 1 = 26.254 ( 10.00 eV E 1 ) / eV 1 nm = 26.254 ( 10.00 7.00 ) 1 nm = 8.875 nm ,
T ( L , E 1 ) = 16 E 1 U 0 ( 1 E 1 U 0 ) e 2 β 1 L = 16 7 10 ( 1 7 10 ) e 17.75 L / nm = 3.36 e 17.75 L / nm .

For a higher-energy electron with E 2 = 9.00 eV :

β 2 = 26.254 ( 10.00 eV E 2 ) / eV 1 nm = 26.254 ( 10.00 9.00 ) 1 nm = 5.124 nm ,
T ( L , E 2 ) = 16 E 2 U 0 ( 1 E 2 U 0 ) e 2 β 2 L = 16 9 10 ( 1 9 10 ) e 5.12 L / nm = 1.44 e 5.12 L / nm .

For a broad barrier with L 1 = 5.00 nm :

T ( L 1 , E 1 ) = 3.36 e 17.75 L 1 / nm = 3.36 e 17.75 · 5.00 nm / nm = 3.36 e −88 = 3.36 ( 6.2 × 10 −39 ) = 2.1 % × 10 −36 ,
T ( L 1 , E 2 ) = 1.44 e 5.12 L 1 / nm = 1.44 e 5.12 · 5.00 nm / nm = 1.44 e 25.6 = 1.44 ( 7.62 × 10 −12 ) = 1.1 % × 10 −9 .

For a narrower barrier with L 2 = 1.00 nm :

T ( L 2 , E 1 ) = 3.36 e 17.75 L 2 / nm = 3.36 e 17.75 · 1.00 nm / nm = 3.36 e −17.75 = 3.36 ( 5.1 × 10 −7 ) = 1.7 % × 10 −4 ,
T ( L 2 , E 2 ) = 1.44 e 5.12 L 2 / nm = 1.44 e 5.12 · 1.00 nm / nm = 1.44 e 5.12 = 1.44 ( 5.98 × 10 −3 ) = 0.86 % .

Significance

We see from these estimates that the probability of tunneling is affected more by the width of the potential barrier than by the energy of an incident particle. In today’s technologies, we can manipulate individual atoms on metal surfaces to create potential barriers that are fractions of a nanometer, giving rise to measurable tunneling currents. One of many applications of this technology is the scanning tunneling microscope (STM), which we discuss later in this section.

Check Your Understanding A proton with kinetic energy 1.00 eV is incident on a square potential barrier with height 10.00 eV. If the proton is to have the same transmission probability as an electron of the same energy, what must the width of the barrier be relative to the barrier width encountered by an electron?

L proton / L electron = m e / m p = 2.3 %

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Radioactive decay

In 1928, Gamow identified quantum tunneling as the mechanism responsible for the radioactive decay of atomic nuclei. He observed that some isotopes of thorium, uranium, and bismuth disintegrate by emitting α -particles (which are doubly ionized helium atoms or, simply speaking, helium nuclei). In the process of emitting an α -particle, the original nucleus is transformed into a new nucleus that has two fewer neutrons and two fewer protons than the original nucleus. The α -particles emitted by one isotope have approximately the same kinetic energies. When we look at variations of these energies among isotopes of various elements, the lowest kinetic energy is about 4 MeV and the highest is about 9 MeV, so these energies are of the same order of magnitude. This is about where the similarities between various isotopes end.

When we inspect half-lives (a half-life is the time in which a radioactive sample loses half of its nuclei due to decay), different isotopes differ widely. For example, the half-life of polonium-214 is 160 µ s and the half-life of uranium is 4.5 billion years. Gamow explained this variation by considering a ‘spherical-box’ model of the nucleus, where α -particles can bounce back and forth between the walls as free particles. The confinement is provided by a strong nuclear potential at a spherical wall of the box. The thickness of this wall, however, is not infinite but finite, so in principle, a nuclear particle has a chance to escape this nuclear confinement. On the inside wall of the confining barrier is a high nuclear potential that keeps the α -particle in a small confinement. But when an α -particle gets out to the other side of this wall, it is subject to electrostatic Coulomb repulsion and moves away from the nucleus. This idea is illustrated in [link] . The width L of the potential barrier that separates an α -particle from the outside world depends on the particle’s kinetic energy E . This width is the distance between the point marked by the nuclear radius R and the point R 0 where an α -particle emerges on the other side of the barrier, L = R 0 R . At the distance R 0 , its kinetic energy must at least match the electrostatic energy of repulsion, E = ( 4 π ε 0 ) −1 Z e 2 / R 0 (where + Z e is the charge of the nucleus). In this way we can estimate the width of the nuclear barrier,

Questions & Answers

what is meant by Z in nuclear physic
Shubhu Reply
atomic n.o
Gyanendra
no of atoms present in nucleus
Sanjana
Note on spherical mirrors
Shamanth Reply
what is Draic equation? with explanation
M.D Reply
what is CHEMISTRY
trpathy Reply
it's a subject
Akhter
what is a Higgs Boson please?
FRANKLINE Reply
god particles is know as higgs boson, when two proton are reacted than a particles came out which is used to make a bond between than materials
M.D
bro little abit getting confuse if i am wrong than please clarify me
M.D
the law of refraction of direct current lines at the boundary between two conducting media of
BATTULA Reply
what is the black body of an ideal radiator
Areej Reply
uncertainty principles is applicable to
Areej
fermions
FRANKLINE
what is the cause of the expanding universe?
FRANKLINE
microscopic particles or gases
Areej
Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart. One explanation for dark energy is that it is a property of space.
Areej
Thanks for your contribution Areej.
FRANKLINE
no problem
Areej
what is photoelectric equation
HIMANSHU Reply
How does fringe intensity depend upon slit width in single slit diffraction?
Abhishek Reply
VOLTE
Md Reply
what is Draic equation
M.D
about nuclear angular momentum
rahul Reply
what is spin
MUKESH Reply
what concepts do you need?
Sumit Reply
l want physics concepts but I have not clarify on the main concepts of physics because I have studied 10 th class in Telugu medium .so what I will do
seshu Reply
what is electro magnet
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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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