Il Nuovo Cimento B (1965-1970)

, Volume 53, Issue 1, pp 24–33 | Cite as

Frequency mixing in scattering of electromagnetic waves by free electrons

I. — Classical theory
  • H. Prakash
  • Vachaspati
Article
First Online:
Received:
Revised:

Summary

It is shown that if the incident electromagnetic wave contains two frequencies, the wave scattered from free electrons contains a complete spectrum of the combination frequencies in addition to the fundamentals, second harmonics and summation and beat frequencies. The expressions for the scattering cross-sections of the summation and beat frequencies differ from those of Fried and Frank but are in agreement with our quantum-theoretical results. It is also shown that the scattering cross-section increases in the presence of a very-low-frequency wave of high intensity because, in this case, the summation and beat frequencies are the same as the greater of the two frequencies and therefore the energy scattered into these combination frequencies should be added to that into the greater of the two fundamental frequencies. This increase is enormous when the ratio of the two frequencies approaches zero. This may make it possible to reduce very appreciably the time of exposure of sensitive films in X-ray experiments by irradiating the scatterer simultaneously by a maser beam also.

Keywords

Free Electron Suffix Beat Frequency Combination Frequency Complete Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Смещивание частот при рассеянии злектромагнитных волн свободными злектронами

I: Классическая теория

Реэюме

Покаэано, что если падаюшая злектромагнитная волна содержит две чаьтоты, то рассеянная свободными злектронами волна содержит полный спектр комбинированных частот в дополнение к основным, вторым гармоникам и суммарной частоте и частоте биений. Выражение для поперечных сечений рассеяния суммарных частот и частот биений отличаются от поперечных сечений Фрайда и Франка, но согласуются с нащими квантовыми теоретическими реэультатами. Также покаэывается, что поперечное сечение рассеяния увеличивается в присутствии волны очень ниэкой частоты высокой интенсивности, потому что в зтом случае суммарная частота и частота биений являются такими же, как больщая иэ зтих двух частот и, следовательно, знергия, рассеянная в зти комбинационные частоты, должна складываться с больщей частотой. Это увеличение является очень больщим, когда отнощение двух частот приближается к нулю. Это делает воэможным сохранить весьма эаметно время зкспоэиции чувствительных пленок в зкспериментах с рентгеновскими лучами, при одновременном облучении рассеивателя лучом маэера.

Riassunto

Si dimostra che se l’onda elettromagnetica incidente contiene due frequenze, l’onda diffusa dagli elettroni liberi contiene uno spettro completo delle frequenze di combinazione oltre alle fondamentali, le seconde armonche e le frequenze di somma e di battimento. Le espressioni della sezione trasversale delle frequenze di somma e di battimento differiscono da quelle di Fried e Frank ma sono in accordo con i nostri risultati in meccanica quantistica. Si dimostra anche che la sezione d’urto di diffusione cresce in presenza di un’onda di frequenza bassissima di alta intensità perchè, in questo caso, le frequenze di somma e di battimento sono uguali alla maggiore delle due frequenze e quindi l’energia diffusa in queste due frequenze di combinazione deve essere aggiunta a quella diffusa nella maggiore delle due frequenze fondamentali. Questo aumento è enorme quando il rapporto fra le due frequenze tende a zero. Con questo metodo si può ridurre apprezzabilmente il tempo di esposizione di pellicole sensibili in esperimenti con raggi X irradiando contemporaneamente il diffusore anche con il fascio di un maser.

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. (1).
    Vachaspati andS. L. Punhani:Proc. Nat. Inst. Sci. India,29, 129 (1963).MathSciNetGoogle Scholar
  2. (2).
    Vachaspati:Phys. Rev.,128, 664 (1962);130, 2598 (1963).ADSCrossRefGoogle Scholar
  3. (3).
    Z. Fried:Nuovo Cimento,22, 1303 (1961).CrossRefGoogle Scholar
  4. (4).
    L. S. Brown andT. W. B. Kibble:Phys. Rev.,133, A 705 (1964).Google Scholar
  5. (5).
    Vachaspati:Indian Journ. Pure Appl. Phys.,2, 373 (1964).Google Scholar
  6. (6).
    H. Prakash andVachaspati:Indian Journ. Pure Appl. Phys. 5, 21 (1967).Google Scholar
  7. (7).
    Z. Fried andJ. H. Eberly:Phys. Rev.,136, B 871 (1964).Google Scholar
  8. (8).
    H. Prakash andVachaspati:Indian Journ. Pure Appl. Phys. (to be published).Google Scholar
  9. (9).
    H. Prakash andVachaspati:Indian Journ. Pure Appl. Phys. (to be published).Google Scholar
  10. (10).
    H. Prakash:Harmonics in Elementary-Particle Physics: Nonlinear Electromagnetic Interactions, Ph. D. Thesis (unpublished), University of Roorkee, 1966, Ch. III, V and VII.Google Scholar
  11. (11).
    Z. Fried andF. M. Frank:Nuovo Cimento,27, 218 (1963).MathSciNetCrossRefGoogle Scholar
  12. (12).
    H. Prakash andVachaspati:Nuovo Cimento,53 B, 34 (1968).ADSCrossRefGoogle Scholar
  13. (13).
    We shall take the velocity of light as unity and follow the summation convention for the Greek suffixes which run from 0 to 3. We shall use the metricg 00=1,g 11=g 22=g 33=−1. Suffixesa, b, c will take the values 1 and 2.Google Scholar
  14. (14).
    We exclude the casek (1)=k (2). Ifk (1)=k (2), terms involving (2k (1)−k (2)) etc. will give coherent contributions to scattering into the fundamental frequency; this makes the calculations complicated.Google Scholar
  15. (*).
    Fried andFrank have given the correct matrix elements (eq. (13)) but incorrect expressions for the scattering cross-sections (eqs. (14) and (15)). They may have made some mistake in their calculations which are not given in the paper.Google Scholar

Copyright information

© Società Italiana di Fisica 1968

Authors and Affiliations

  • H. Prakash
    • 1
  • Vachaspati
    • 1
  1. 1.Department of PhysicsUniversity of AllahabadAllahabad

Personalised recommendations