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Дата изменения: Thu Aug 24 09:48:48 2000 Дата индексирования: Tue Oct 2 03:36:44 2012 Кодировка: |
The scattering of electromagnetic waves off electric charges is a standard problem of classical electrodynamics where it may be thought off as the absorption of an incoming electromagnetic wave by the charge with subsequent emission of an outgoing electromagnetic wave. In quantum physics the electromagnetic wave is considered as a flux of photons which carry energy and momentum. When an electron is struck by a photon the electron recoils and some of the photon's momentum is transferred to it. At the beginning of the quantum era this effect, the Compton effect, was one of the decisive experiments that confirmed the quantum nature of photons.
Now let us calculate the Compton effect within the framework of quantum
electrodynamics by means of CompHEP package.
Start a CompHEP session and choose option QED in the menu of physical
models provided in the package. Then with the help of menu function
Enter process you have to enter the reaction equation of Compton
scattering
.
To do this you must use the particle labels from
the table displayed at the top of the screen and write the equation
in CompHEP style:
Enter Process: A,e1 -> A,e1
Next you are prompted by CompHEP to define the collision energy in the center-of-mass system (CMS). Let us set energy equal to 0.529 MeV, which corresponds to the Compton experiment and is only slightly greater than the rest energy of the electron (0.511 MeV)
Enter CMS Energy in GeV: 0.000529
Two Feynman diagrams are generated and you can inspect them on the screen by selecting option View diagrams. Here they are presented on (Fig. 1a).
The next step in the calculation is Squaring which creates diagrams for avaluation of the modulus-squared of amplitude. These, so called "squared diagrams" may be inspected via View squared diagrams menu function. See Fig. 1b.
The next step is Symbolic calculations. To start this step one should activate the corresponding menu function. On this step CompHEP performs summation and the contraction of the Lorentz labels and creates symbolic expressions for squared diagrams. CompHEP also The analytical answers for squared diagrams can be obtained in the form of REDUCE or MATHEMATICA code. To generate, for example, the MATHEMATICA code you should subsequently choose options Write results and MATHEMATICA code. As a result the symbolic code should be written in the file symb1.m of results directory. Here we present this code with some comments: (Click, please).
CompHEP user is provided with
a possibility to perform various symbolic manipulations with the output.
Say, one may sum all diagrams contributions and express the sum in terms of
the Mandelstam variables
| s=(p1+p2)2 | - CMS energy squared |
| t=(p1-p3)2 | - transverse momenta squared |
| (1) |
To make contact with experimental results we must next calculate the
invariant differential cross section which we get by applying
the formula
is the modulus of the CMS momentum.
This equation is the basis for getting the formulæ for the differential cross sections in any reference frame.
As an example consider
the formula for the differential cross section in the electron rest frame
in terms of energies of the incident and scattered
photons: wi and ws.
This expression is known as the Klein-Nishina formula1 and can be found from Eq. (1,2).
We use MATHEMATICA to perform the evaluation. See the corresponding
MATHEMATICA sessiion. The result of
evaluation is
The formula for the total cross section of Compton scattering can be
obtained by integration (1) over t from tmin=2Me2
to tmax=Me4/s. As above we use MATHEMATICA 
The result of
evaluation is
