The Compton formula was published in 1923 in the Physical Review. Compton explained that the X-ray shift is caused by particle-like momentum of photons. Compton wavelength

Compton Scattering

Compton Scattering Formula

The Compton formula was published in 1923 in the Physical Review. Compton explained that the X-ray shift is caused by particle-like momentum of photons. Compton scattering formula is the mathematical relationship between the shift in wavelength and the scattering angle of the X-rays. In the case of Compton scattering the photon of frequency f collides with an electron at rest. Upon collision, the photon bounces off electron, giving up some of its initial energy (given by Planck’s formula E=hf), While the electron gains momentum (mass x velocity), the photon cannot lower its velocity. As a result of momentum conservetion law, the photon must lower its momentum given by:So the decrease in photon’s momentum must be translated into decrease in frequency (increase in wavelength Δλ = λ’ – λ). The shift of the wavelength increased with scattering angle according to the Compton formula:whereλ is the initial wavelength of photonλ’ is the wavelength after scattering,h is the Planck constant = 6.626 x 10^{-34} J.sm_{e} is the electron rest mass (0.511 MeV)c is the speed of lightΘ is the scattering angle.The minimum change in wavelength (λ′ − λ) for the photon occurs when Θ = 0° (cos(Θ)=1) and is at least zero. The maximum change in wavelength (λ′ − λ) for the photon occurs when Θ = 180° (cos(Θ)=-1). In this case the photon transfers to the electron as much momentum as possible.The maximum change in wavelength can be derived from Compton formula:The quantity h/m_{e}c is known as the Compton wavelength of the electron and is equal to 2.43×10^{−12} m.

See also:

Definition of Compton Scattering

See also:

Compton Scattering

See also:

Cross-Sections of Compton Scattering

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Definition: Compton scattering is the inelastic or nonclassical scattering of a photon by a charged particle, usually an electron. Radiation Dosimetry

Compton Scattering

Key characteristics of Compton Scattering

Compton scattering dominates at intermediate energies.

It is the scattering of photons by atomic electrons

Photons undergo a wavelength shift called the Compton shift.

The energy transferred to the recoil electron can vary from zero to a large fraction of the incident gamma ray energy

Definition of Compton Scattering

Compton scattering is the inelastic or nonclassical scattering of a photon (which may be an X-ray or gamma ray photon) by a charged particle, usually an electron. In Compton scattering, the incident gamma ray photon is deflected through an angle Θ with respect to its original direction. This deflection results in a decrease in energy (decrease in photon’s frequency) of the photon and is called the Compton effect. The photon transfers a portion of its energy to the recoil electron. The energy transferred to the recoil electron can vary from zero to a large fraction of the incident gamma ray energy, because all angles of scattering are possible. The Compton scattering was observed by A. H.Compton in 1923 at Washington University in St. Louis. Compton earned the Nobel Prize in Physics in 1927 for this new understanding about the particle-nature of photons.

See also:

Compton Scattering

See also:

Compton Formula

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At small values of gamma ray energy the photoelectric effect dominates. The cross-section curve have sharp discontinuities. Cross-Section of Photoelectric Effect

Photoelectric Effect

Cross-Sections of Photoelectric Effect

At small values of gamma ray energy the photoelectric effect dominates. The mechanism is also enhaced for materials of high atomic number Z. It is not simple to derive analytic expression for the probability of photoelectric absorption of gamma ray per atom over all ranges of gamma ray energies. The probability of photoelectric absorption per unit mass is approximately proportional to:

τ_{(photoelectric)} = constant x Z^{N}/E^{3.5}

where Z is the atomic number, the exponent n varies between 4 and 5. E is the energy of the incident photon. The proportionality to higher powers of the atomic number Z is the main reason for using of high Z materials, such as lead or depleted uranium in gamma ray shields.Although the probability of the photoelectric absorption of gamma photon decreases, in general, with increasing photon energy, there are sharp discontinuities in the cross-section curve. These are called “absoption edges” and they correspond to the binding energies of electrons from atom’s bound shells. For photons with the energy just above the edge, the photon energy is just sufficient to undergo the photoelectric interaction with electron from bound shell, let say K-shell. The probability of such interaction is just above this edge much greater than that of photons of energy slightly below this edge. For gamma photons below this edge the interaction with electron from K-shell in energetically impossible and therefore the probability drops abruptly. These edges occur also at binding energies of electrons from other shells (L, M, N …..).

Cross section of photoelectric effect.

See also:

Definition of Photoelectric Effect

See also:

Photoelectric Effect

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Definotion of Photoelectric Effect. In the photoelectric effect, a photon undergoes an interaction with an electron which is bound in an atom. Radiation Dosimetry

Photoelectric Effect

The photoelectric effect dominates at low-energies of gamma rays.

The photoelectric effect leads to the emission of photoelectrons from matter when light (photons) shines upon them.

The maximum energy an electron can receive in any one interaction is hν.

Electrons are only emitted by the photoelectric effect if photon reaches or exceeds a threshold energy.

A free electron (e.g. from atomic cloud) cannot absorb entire energy of the incident photon. This is a result of the need to conserve both momentum and energy.

The cross-section for the emission of n=1 (K-shell) photoelectrons is higher than that of n=2 (L-shell) photoelectrons. This is a result of the need to conserve momentum and energy.

Definition of Photoelectric effect

In the photoelectric effect, a photon undergoes an interaction with an electron which is bound in an atom. In this interaction the incident photon completely disappears and an energetic photoelectron is ejected by the atom from one of its bound shells. The kinetic energy of the ejected photoelectron (E_{e}) is equal to the incident photon energy (hν) minus the binding energy of the photoelectron in its original shell (E_{b}).

E_{e}=hν-E_{b}

Therefore photoelectrons are only emitted by the photoelectric effect if photon reaches or exceeds a threshold energy – the binding energy of the electron – the work function of the material. For gamma rays with energies of more than hundreds keV, the photoelectron carries off the majority of the incident photon energy – hν.Following a photoelectric interaction, an ionized absorber atom is created with a vacancy in one of its bound shells. This vacancy is will be quickly filled by an electron from a shell with a lower binding energy (other shells) or through capture of a free electron from the material. The rearrangement of electrons from other shells creates another vacancy, which, in turn, is filled by an electron from an even lower binding energy shell. Therefore a cascade of more characteristic X-rays can be also generated. The probability of characteristic x-ray emission decreases as the atomic number of the absorber decreases. Sometimes , the emission of an Auger electron occurs.

See also:

Albert Einstein and Photoelectric Effect

See also:

Photoelectric Effect

See also:

Cross-Section of Photoelectric Effect

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In the paper on the photoelectric effect Albert Einstein solved the paradox by describing light as composed of discrete quanta. Albert Einstein and Photoelectric Effect. Radiation Dosimetry

Discovery of Photoelectric Effect

The phenomenon, that a surface (typically alkali metals) when exposed to electromagnetic radiation (visible light) emits electrons, was discovered by Hertz and Hallwachs in 1887 during experiments with a spark-gap generator. Hertz found that the sensitivity of his spark-gap device can be increased by exposition to visible or ultraviolet light and that light obviously had some electrical effect. He did not further pursue investigation of this effect.Shortly after Hertz’s discovery in 1899, English physicist J.J.Thomson showed that UV light, which fall onto metal surface, trigger the emission of electrons from the surface. In 1902, Hungarian physicist Philipp Lenard made the first quantitative measurements of the photoelectric effect. He observed that the energy of individual emitted electrons increased with the frequency of the light (which is related to the color).

While this is interesting, it is hardly explainable by classical theory of electromagnetic radiation which assumed the existence of a stationary medium (the luminiferous aether) through which light propagated. Subsequent investigations into the photoelectric effect results in the fact that these explorations did not fit with the classical theory of electromagnetic radiation.In 1905, Albert Einstein published four groundbreaking papers on the photoelectric effect, Brownian motion, special relativity, and the equivalence of mass and energy. These papers were published in the Annalen der Physik journal and contributed significantly to the foundation of modern physics. In the paper on the photoelectric effect (“On a Heuristic Viewpoint Concerning the Production and Transformation of Light”) he solved the paradox by describing light as composed of discrete quanta (German: das Lichtquant), rather than continuous waves.This theory was builded on Max Planck’s blackbody radiation theory, which assumes that luminous energy can be absorbed or emitted only in discrete amounts, called quanta. The photon’s energy in each quantum of light is equal to its frequency (ν) multiplied by a constant known as Planck’s constant (h), or alternately, using the wavelength (λ) and the speed of light (c):

E=hc/λ=hν

Each photon above a threshold frequency (specific for each material) has the needed energy to eject a single electron, creating the observed effect. Einstein’s theory predicts that the maximum kinetic energy of emitted electron is dependent only on the frequency of the incident light and not on its intensity. Shining twice as much light (high-intensity) results in twice as many photons, and more electrons releasing, but the maximum kinetic energy of those individual electrons remains the same. Experimentation in the photoelectric effect was carried out extensively by Robert Millikan in 1915, Robert Millikan showed that Einstein’s prediction was correct. This discovery contributed to the quantum revolution in physics and earned Einstein the Nobel Prize in Physics in 1921.

See also:

Photoelectric Effect

See also:

Definition of Photoelectric Effect

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Gamma Ray Attenuation describes attenuation of monoenergetic gamma rays collimated into a narrow beam and their passage through the material. Radiation Dosimetry

The total cross-section of interaction of a gamma rays with an atom is equal to the sum of all three mentioned partial cross-sections:σ = σ_{f} + σ_{C} + σ_{p }

σ_{f} – Photoelectric effect

σ_{C} – Compton scattering

σ_{p} – Pair production

Depending on the gamma ray energy and the absorber material, one of the three partial cross-sections may become much larger than the other two. At small values of gamma ray energy the photoelectric effect dominates. Compton scattering dominates at intermediate energies. The compton scattering also increases with decreasing atomic number of matter, therefore the interval of domination is wider for light nuclei. Finally, electron-positron pair production dominates at high energies.Based on the definition of interaction cross-section, the dependence of gamma rays intensity on thickness of absorber material can be derived. If monoenergetic gamma rays are collimated into a narrow beam and if the detector behind the material only detects the gamma rays that passed through that material without any kind of interaction with this material, then the dependence should be simple exponential attenuation of gamma rays. Each of these interactions removes the photon from the beam either by absorbtion or by scattering away from the detector direction. Therefore the interactions can be characterized by a fixed probability of occurance per unit path length in the absorber. The sum of these probabilities is called the linear attenuation coefficient:

The attenuation of gamma radiation can be then described by the following equation.

I=I_{0}.e^{-μx}

, where I is intensity after attenuation, I_{o} is incident intensity, μ is the linear attenuation coefficient (cm^{-1}), and physical thickness of absorber (cm).

The materials listed in the table beside are air, water and a different elements from carbon (Z=6) through to lead (Z=82) and their linear attenuation coefficients are given for three gamma ray energies. There are two main features of the linear attenuation coefficient:

The linear attenuation coefficient increases as the atomic number of the absorber increases.

The linear attenuation coefficient for all materials decreases with the energy of the gamma rays.

Half Value Layer

The half value layer expresses the thickness of absorbing material needed for reduction of the incident radiation intensity by a factor of two. There are two main features of the half value layer:

The half value layer decreases as the atomic number of the absorber increases. For example 35 m of air is needed to reduce the intensity of a 100 keV gamma ray beam by a factor of two whereas just 0.12 mm of lead can do the same thing.

The half value layer for all materials increases with the energy of the gamma rays. For example from 0.26 cm for iron at 100 keV to about 1.06 cm at 500 keV.

Mass Attenuation Coefficient

When characterizing an absorbing material, we can use sometimes the mass attenuation coefficient. The mass attenuation coefficient is defined as the ratio of the linear attenuation coefficient and absorber density (μ/ρ). The attenuation of gamma radiation can be then described by the following equation:

I=I_{0}.e^{-(μ/ρ).ρl}

, where ρ is the material density, (μ/ρ) is the mass attenuation coefficient and ρ.l is the mass thickness. The measurement unit used for the mass attenuation coefficient cm^{2}g^{-1}.For intermediate energies the Compton scattering dominates and different absorbers have approximately equal mass attenuation coefficients. This is due to the fact that cross section of Compton scattering is proportional to the Z (atomic number) and therefore the coefficient is proportional to the material density ρ. At small values of gamma ray energy or at high values of gamma ray energy, where the coefficient is proportional to higher powers of the atomic number Z (for photoelectric effect σ_{f} ~ Z^{5}; for pair production σ_{p} ~ Z^{2}), the attenuation coefficient μ is not a constant.

Example:

How much water schielding do you require, if you want to reduce the intensity of a 500 keV monoenergetic gamma ray beam (narrow beam) to 1% of its incident intensity? The half value layer for 500 keV gamma rays in water is 7.15 cm and the linear attenuation coefficient for 500 keV gamma rays in water is 0.097 cm^{-1}.The question is quite simple and can be described by following equation:If the half value layer for water is 7.15 cm, the linear attenuation coefficient is:Now we can use the exponential attenuation equation:thereforeSo the required thickness of water is about 47.5 cm. This is relatively large thickness and it is caused by small atomic numbers of hydrogen and oxygen. If we calculate the same problem for lead (Pb), we obtain the thickness x=2.8cm.

Linear Attenuation Coefficients

Table of Linear Attenuation Coefficients (in cm-1) for a different materials at gamma ray energies of 100, 200 and 500 keV.

The pair production is a phenomenon of nature where energy is direct converted to matter. At high energies of gamma rays pair production dominates. Radiation Dosimetry

Positron-Electron Pair Production

In general, pair production is a phenomenon of nature where energy is direct converted to matter. The phenomenon of pair production can be view two different ways. One way is as a particle and antiparticle and the other is as a particle and a hole. The first way can be represented by formation of electron and positron, from a packet of electromagnetic energy (high energy photon – gamma ray) traveling through matter. It is one of the possible ways in which gamma rays interact with matter. At high energies this interaction dominates. In order for electron-positron pair production to occur, the electromagnetic energy of the photon must be above a threshold energy, which is equivalent to the rest mass of two electrons. The threshold energy (the total rest mass of produced particles) for electron-positron pair production is equal to 1.02MeV (2 x 0.511MeV) because the rest mass of a single electron is equivalent to 0.511MeV of energy.If the original photon’s energy is greater than 1.02MeV, any energy above 1.02MeV is according to the conservation law split between the kinetic energy of motion of the two particles.The presence of an electric field of a heavy atom such as lead or uranium is essential in order to satisfy conservation of momentum and energy. In order to satisfy both conservation of momentum and energy, the atomic nucleus must receive some momentum. Therefore a photon pair production in free space cannot occur.Moreover, the positron is the anti-particle of the electron, so when a positron comes to rest, it interacts with another electron, resulting in the annihilation of the both particles and the complete conversion of their rest mass back to pure energy (according to the E=mc^{2} formula) in the form of two oppositely directed 0.511 MeV gamma rays (photons). The pair production phenomenon is therefore connected with creation and destruction of matter in one reaction.

Positron-Electron Pair Production – Cross-Section

The probability of pair production, characterized by cross section, is a very complicated function based on quantum mechanics. In general the cross section increases approximately with the square of atomic number (σ_{p} ~ Z^{2}) and increases with photon energy, but this dependence is much more complex.

Cross section of pair production in nuclear field and electron field.

See also:

Compton Scattering

See also:

Interaction of Gamma Radiation with Matter

See also:

Attenuation

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In the photoelectric effect, a photon undergoes an interaction with an electron which is bound in an atom. The photoelectric effect dominates at low-energies of gamma rays. Radiation Dosimetry

Photoelectric Effect

Albert Einstein and Photoelectric Effect / Discovery

The phenomenon, that a surface (typically alkali metals) when exposed to electromagnetic radiation (visible light) emits electrons, was discovered by Hertz and Hallwachs in 1887 during experiments with a spark-gap generator. Hertz found that the sensitivity of his spark-gap device can be increased by exposition to visible or ultraviolet light and that light obviously had some electrical effect. He did not further pursue investigation of this effect.Shortly after Hertz’s discovery in 1899, English physicist J.J.Thomson showed that UV light, which fall onto metal surface, trigger the emission of electrons from the surface. In 1902, Hungarian physicist Philipp Lenard made the first quantitative measurements of the photoelectric effect. He observed that the energy of individual emitted electrons increased with the frequency of the light (which is related to the color).

While this is interesting, it is hardly explainable by classical theory of electromagnetic radiation which assumed the existence of a stationary medium (the luminiferous aether) through which light propagated. Subsequent investigations into the photoelectric effect results in the fact that these explorations did not fit with the classical theory of electromagnetic radiation.In 1905, Albert Einstein published four groundbreaking papers on the photoelectric effect, Brownian motion, special relativity, and the equivalence of mass and energy. These papers were published in the Annalen der Physik journal and contributed significantly to the foundation of modern physics. In the paper on the photoelectric effect (“On a Heuristic Viewpoint Concerning the Production and Transformation of Light”) he solved the paradox by describing light as composed of discrete quanta (German: das Lichtquant), rather than continuous waves.This theory was builded on Max Planck’s blackbody radiation theory, which assumes that luminous energy can be absorbed or emitted only in discrete amounts, called quanta. The photon’s energy in each quantum of light is equal to its frequency (ν) multiplied by a constant known as Planck’s constant (h), or alternately, using the wavelength (λ) and the speed of light (c):

E=hc/λ=hν

Each photon above a threshold frequency (specific for each material) has the needed energy to eject a single electron, creating the observed effect. Einstein’s theory predicts that the maximum kinetic energy of emitted electron is dependent only on the frequency of the incident light and not on its intensity. Shining twice as much light (high-intensity) results in twice as many photons, and more electrons releasing, but the maximum kinetic energy of those individual electrons remains the same. Experimentation in the photoelectric effect was carried out extensively by Robert Millikan in 1915, Robert Millikan showed that Einstein’s prediction was correct. This discovery contributed to the quantum revolution in physics and earned Einstein the Nobel Prize in Physics in 1921.

The photoelectric effect dominates at low-energies of gamma rays.

The photoelectric effect leads to the emission of photoelectrons from matter when light (photons) shines upon them.

The maximum energy an electron can receive in any one interaction is hν.

Electrons are only emitted by the photoelectric effect if photon reaches or exceeds a threshold energy.

A free electron (e.g. from atomic cloud) cannot absorb entire energy of the incident photon. This is a result of the need to conserve both momentum and energy.

The cross-section for the emission of n=1 (K-shell) photoelectrons is higher than that of n=2 (L-shell) photoelectrons. This is a result of the need to conserve momentum and energy.

Definition of Photoelectric effect

In the photoelectric effect, a photon undergoes an interaction with an electron which is bound in an atom. In this interaction the incident photon completely disappears and an energetic photoelectron is ejected by the atom from one of its bound shells. The kinetic energy of the ejected photoelectron (E_{e}) is equal to the incident photon energy (hν) minus the binding energy of the photoelectron in its original shell (E_{b}).

E_{e}=hν-E_{b}

Therefore photoelectrons are only emitted by the photoelectric effect if photon reaches or exceeds a threshold energy – the binding energy of the electron – the work function of the material. For gamma rays with energies of more than hundreds keV, the photoelectron carries off the majority of the incident photon energy – hν.Following a photoelectric interaction, an ionized absorber atom is created with a vacancy in one of its bound shells. This vacancy is will be quickly filled by an electron from a shell with a lower binding energy (other shells) or through capture of a free electron from the material. The rearrangement of electrons from other shells creates another vacancy, which, in turn, is filled by an electron from an even lower binding energy shell. Therefore a cascade of more characteristic X-rays can be also generated. The probability of characteristic x-ray emission decreases as the atomic number of the absorber decreases. Sometimes , the emission of an Auger electron occurs.

Cross-Sections of Photoelectric Effect

At small values of gamma ray energy the photoelectric effect dominates. The mechanism is also enhaced for materials of high atomic number Z. It is not simple to derive analytic expression for the probability of photoelectric absorption of gamma ray per atom over all ranges of gamma ray energies. The probability of photoelectric absorption per unit mass is approximately proportional to:

τ_{(photoelectric)} = constant x Z^{N}/E^{3.5}

where Z is the atomic number, the exponent n varies between 4 and 5. E is the energy of the incident photon. The proportionality to higher powers of the atomic number Z is the main reason for using of high Z materials, such as lead or depleted uranium in gamma ray shields.Although the probability of the photoelectric absorption of gamma photon decreases, in general, with increasing photon energy, there are sharp discontinuities in the cross-section curve. These are called “absoption edges” and they correspond to the binding energies of electrons from atom’s bound shells. For photons with the energy just above the edge, the photon energy is just sufficient to undergo the photoelectric interaction with electron from bound shell, let say K-shell. The probability of such interaction is just above this edge much greater than that of photons of energy slightly below this edge. For gamma photons below this edge the interaction with electron from K-shell in energetically impossible and therefore the probability drops abruptly. These edges occur also at binding energies of electrons from other shells (L, M, N …..).

Cross section of photoelectric effect.

See also:

Characteristics of Gamma Rays

See also:

Interaction of Gamma Radiation with Matter

See also:

Compton Scattering

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