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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 % .


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

total binding energy of ionic crystal at equilibrium is
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How does, ray of light coming form focus, behaves in concave mirror after refraction?
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Refraction does not occur in concave mirror. If refraction occurs then I don't know about this.
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Izevbogie Reply
Anything which changes itself with respect to time or surrounding
and what's time? is time everywhere same
how can u say that
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Radioactive substance
These substance create harmful radiation like alpha particle radiation, beta particle radiation, gamma particle radiation
But ask anything changes itself with respect to time or surrounding A Not any harmful radiation
explain cavendish experiment to determine the value of gravitational concept.
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of course g is constant
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normal distribution of errors
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What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
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give any fix value to wave length
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
this msg is out of mistake. sorry friends​.
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
physics is the study of non living things if we added it with biology it becomes biophysics and bio is the study of living things tell me please what is this?
physics is the study of matter,energy and their interactions
all living things are matter
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thanks buvanas
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