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An Analytical Study on the Compton Wavelength Transactions of the Korean Nuclear Society Autumn Meeting PyeongChang, Korea, October 30-31, 2008 An Analytical Study on the Compton Wavelength Sun-Tae Hwang and Dong-Kwan Shin TEKCIVIL Corporation 301 Songjeon B/D, 1182 Dunsan-dong, Seogu, Daejeon 302-121, Korea 1. Introduction p of a particle are related to its rest mass m 2 2 2 4 Mathematically, the Compton wavelength λc of by the invariant relation, p ∙pc - E = -m c . a particle is given by λc = h/mc, where h is the Now, by conservation of energy, Planck constant, m is the particle's rest mass 2 2 2 1/2 and c is the speed of light. For the electron as pγ +mec = p'γ +(me c +p'e ) (3) -12 a reference constant, it is about 2.426x10 m, intermediate between the size of an atomic nu- Rearranging and squaring both sides, Eq.(4) is cleus and an atom. The λc means a critical dis- obtained. tance below which certain quantum mechanical 2 2 2 2 2 2 2 effects take place for any particle. The energy pγ +p'γ -2pγp'γ+2mec(pγ-p'γ)+me c =me c +p'e of a photon of this wavelength is equal to the (4) 2 rest mass energy mec of an electron[1]. Now, Subtraction of Eq.(2) from Eq.(4) yields the λc is analytically reviewed. mec(pγ-p'γ) = pγp'γ - pγ․p'γ (5) 2. Compton Scattering Effect Finally, dividing both sides by mecpγp'γ, Eq.(6) Compton scattering effect is the result of a for the relation between the energies of the in- high energy photon colliding with an electron. cident and the scattered photons is obtained. First, consider the interaction between a pho- 2 ton with momentum vector pγ and an electron 1/E'γ - 1/Eγ = (1/mec )․(1 - cosθ) (6) initiallly at rest with mass me. Next call p'γ and p'e the momenta of the photon and the electron where θ is the angle between their final momen- after the interaction, respectively. By conserva- tum vectors. In the meantime, considering the tion of momentum, the relation between the in- wavelike character of photons, their energy E = itial and final momenta is hυ = hc/λ and momentum p are related by the relativistic equation for electron of rest mass of pγ = p'γ +p'e (1) zero, namely p = hυ/c = E/c. Accordingly, from Eq.(6) the formula for "wavelength shift" in the Reqaranging and squaring both sides, Eq.(2) wavelength of incoherently scattered photons is can be obtained. obtained as follows: 2 2 2 pγ +p'γ - 2pγ․p'γ = p'e (2) △λ = (h/mec)․(1 - cosθ) (7) The total relativistic energy E and momentum where θ is the angle between the trajectories 119 Transactions of the Korean Nuclear Society Autumn Meeting PyeongChang, Korea, October 30-31, 2008 of the incident and scattered photon. When the 4. Conclusions electron moves in free space with a momentum p, its wavelength is given by λe = h/p which is The Compton wavelength λc can be thought known as the de Broglie wavelength[2]. Mean- of as a fundamental limitation on measuring the time, even though the value h/mec is called the position of a particle, taking quantum mechanics Compton wavelength of the electron, this is not and special relativity into accouint. In quantum an actual wavelength but really a proportionality mechanics, this is a convenient unit of length constant for the wavelength shift △λ. Looking that is characteristic of an elementary particle at Eq.(7), it become clear that the entire shift and equals to Planck's constant h divided by can be measured purely in terms of the angle the product of the particle's mass m and the at which the photon gets scattered. Everything speed of light c. Hence, λc depends on m. else on the right side of Eq.(7) is a constant. The quantum field theory of light postulates that photons behave like particles except for 3. Compton Wavelength λc the absence of any rest mass. Using just kine- matics, the change in the wavelength of X-ray Normally, the Compton wavelength λc is as- photon △λ = λ‘-λ can be expressed in the sociated with an incident photon colliding with form Eq.(7). The Compton wavelength λc is ac- an electron at rest. The incident photon imparts tually the de Broglie wavelength for the elec- some momentum to an electron and scatters tron moving at the speed of light. with less energy. By just using the kinematics The Compton scattering shows how such a in Compton scattering event, the change in the collision in Section 3 might be represented, wavelength of photon can be expressed in the with an X-ray photon striking an electron at form Eq.(7). Generally, the Compton wavelength rest and being scattered away from its origin- of a particle is equal to h/mc. In particular, the al path while the electron receives an impulse λc of the electron is the characteristic length and is recoiled. scale of QED(quantum electrodynamics). The CODATA 2002 value for λc of the elec- References tron is 2.4263102175×10-12m with a standard uncertainty of 0.0000000033×10-12m[3]. Other [1] http://www.britannica.com/eb/topic-130389/ particles have different values. CODATA(Com- Compton-wavelength mittee on Data for Science and Technoloy) was [2] Clayton, Roderick K., Light and Living established in 1966 as an interdisciplinary com- Matter Volume 1: The Physical Part, Mc- mittee of the International Council of Science Graw Hill Book Company, New York, NY, (ICSU), formerly the International Council of p.14n, 1970 (http://hyperphysics.phy-astr. Scientific Unions. It seeks to improve the com- gsu.edu/hbase/debrog.html) pilation, critical evaluation, storage, and retriev- [3] CODATA 2002 value for Compton wave- al of data of importance to science and tech- length for the electron from NIST. nology. The CODATA recommended values of [4] CODATA Recommended Values of the fundamental physical constants are published at Fundamental Physical Constants: 2006 the NIST Reference on Constants, Units, and (http://physics.nist.gov/cuu/Constants/codata. Uncertainty[4]. pdf), NIST. 120.
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