Skip to main content
SpringerLink
Log in

Application of the Donkin Formula in the Theory of Reflecting and Turning Devices

  • ELECTROPHYSICS
  • Published:
Technical Physics Aims and scope Submit manuscript

Abstract

Electrostatic turning devices are electron- and ion-optical elements changing the direction of movement of a parallel monochromatic beam of charged particles by a given angle without affecting a beam’s parallelism. The trajectory similarity principle for electric fields homogeneous in Euler terms ensures the fulfillment of this property for the fields with a homogeneity of a zero power. The Donkin formula for 3D homogeneous harmonic functions produces extremely wide class of analytic expressions describing homogeneous electric potentials of a zero power. This paper considers the algorithm of synthesis of electrostatic turning devices that transform input parallel monochromatic beams into output parallel monochromatic beams. The algorithm is based on the Donkin formula and ensures beam stability for small deviations from the electric field’s symmetry plane.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

Similar content being viewed by others

Notes

  1. In this device, the input beam diverging with a small cone angle is not strictly parallel so that the telescopic turning device with field U(x, y, z) ~ arctan(x/z) produces an intermediate imaginary source with large input angles directly at the inlet of the energy analyzing region.

  2. Accurate analysis shows that the continuity of the electric potential at boundaries is formally not required for the similarity principle either. The potential is a Euler homogeneous function on the right and on the left of the boundary. Therefore, after recalculation of the velocities of charged particles obeying the required proportion, the velocities and coordinates of particles crossing the infinitely thin boundary with an electric potential jump (see [25]) turn out to be in the same proportion that is required for similarity of the text fragments of trajectories.

  3. Degenerate case f  '(0) = 0 can also be of certain interest, but its analysis is beyond the scope of this article.

REFERENCES

  1. P. G. Gabdullin, Yu. K. Golikov, N. K. Krasnova, and S. N. Davydov, Tech. Phys. 45, 232 (2000).

    Article  Google Scholar 

  2. P. G. Gabdullin, Yu. K. Golikov, N. K. Krasnova, and S. N. Davydov, Tech. Phys. 45, 330 (2000).

    Article  Google Scholar 

  3. G. M. Fikhtengol’ts, A Course of Differential and Integral Calculus (Fizmatlit, Moscow, 2001), Vol. 1.

    Google Scholar 

  4. V. I. Smirnov, Advanced Mathematics Course (BKhV-Peterburg, St. Petersburg, 2008), Vol. 1.

  5. A. S. Berdnikov, I. A. Averin, N. K. Krasnova, and K. V. Solov’ev, Vestn. Aktyubinskogo Reg. Gos. Univ. Fiz.-Mat. Nauki, No. 2, 147 (2016).

    Google Scholar 

  6. A. S. Berdnikov, I. A. Averin, N. K. Krasnova, and K. V. Solov’ev, Usp. Prikl. Fiz. 5 (1), 10 (2017).

    Google Scholar 

  7. A. S. Berdnikov, I. A. Averin, N. K. Krasnova, and K. V. Solov’ev, Vestn. Aktyubinskogo Reg. Gos. Univ. Fiz.-Mat. Nauki, No. 2, 17 (2016).

    Google Scholar 

  8. W. F. Donkin, Philos. Trans. R. Soc. London 147, 43 (1857).

    Article  ADS  Google Scholar 

  9. W. F. Donkin, Proc. R. Soc. London 8, 307 (1856–1857).

  10. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea Publ. Co., 1931).

    MATH  Google Scholar 

  11. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (AMS Press, 1944), Part II.

  12. Yu. K. Golikov, Vestn. Aktyubinskogo Reg. Gos. Univ. Fiz.-Mat. Nauki, No. 2, 165 (2016).

    Google Scholar 

  13. Yu. K. Golikov, Vestn. Aktyubinskogo Reg. Gos. Univ. Fiz.-Mat. Nauki, No. 2, 59 (2016).

    Google Scholar 

  14. Yu. K. Golikov, K. G. Utkin, and V. V. Cheparukhin, Calculation of Elements of Electrostatic Electron-Optical Systems: Study Guide (Leningr. Politekh. Inst., Leningrad, 1984).

    Google Scholar 

  15. L. N. Gall’ and Yu. K. Golikov, Nauchn. Priborostr. 24 (1), 11 (1987).

  16. Yu. K. Golikov and N. K. Krasnova, Prikl. Fiz., No. 2, 5 (2007).

  17. I. A. Averin, A. S. Berdnikov, and N. R. Gall, Tech. Phys. Lett. 43, 156 (2017).

    Article  ADS  Google Scholar 

  18. Yu. K. Golikov and N. K. Krasnova, Theory of the Synthesis of Electrostatic Energy Analyzers (Politekh. Univ., St. Petersburg, 2010).

    Google Scholar 

  19. N. K. Krasnova, Doctoral Dissertation in Mathematics and Physics (St. Petersburg Polytechnic Univ., St. Peterburg, 2013).

  20. Yu. K. Golikov, A. S. Berdnikov, A. S. Antonov, N. K. Krasnova, and K. V. Solov’ev, Tech. Phys. 63, 1659 (2018).

    Article  Google Scholar 

  21. Yu. K. Golikov, N. A. Kholin, and T. A. Shorina, Nauchn. Priborostr. 19 (2), 13 (2009).

    Google Scholar 

  22. D. V. Grigor’ev, Candidate’s Dissertation in Mathematics and Physics (St. Petersburg State Technical Univ., St. Petersburg, 2000).

  23. K. Siegbahn, N. Kholine, and G. Golikov, Nucl. Instrum. Methods Phys. Res., Sect. A 384, 563 (1997).

    Google Scholar 

  24. A. S. Berdnikov and N. K. Krasnova, Nauchn. Priborostr. 25 (2), 69 (2015).

    Article  Google Scholar 

  25. A. S. Berdnikov, Nauchn. Priborostr. 25 (1), 48 (2015).

    Article  Google Scholar 

  26. N. K. Krasnova, Tech. Phys. 56, 843 (2011).

    Article  Google Scholar 

  27. Yu. K. Golikov and N. K. Krasnova, Tech. Phys. 56, 164 (2011).

    Article  Google Scholar 

  28. I. A. Averin and A. S. Berdnikov, Usp. Prikl. Fiz. 4 (1), 5 (2016).

    Google Scholar 

  29. Yu. K. Golikov, A. S. Berdnikov, A. S. Antonov, N. K. Krasnova, and K. V. Solov’ev, Tech. Phys. 63, 593 (2018).

    Article  Google Scholar 

  30. W. von Koppenfels and F. Stallmann, Praxis der Konformen Abbildung (Springer, 1959).

  31. M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable (Nauka, Moscow, 1965).

    Google Scholar 

  32. V. I. Lavrik and V. N. Savenkov, Handbook on Conformal Mappings (Naukova Dumka, Kiev, 1970).

    Google Scholar 

  33. P. F. Fil’chakov, Approximate Conformal Mapping Methods (Naukova Dumka, Kiev, 1964).

    Google Scholar 

  34. V. M. Kel’man, S. P. Karetskaya, L. V. Fedulina, and E. M. Yakushev, Electron-Optical Elements of Prism Spectrometers for Charged Particles (Nauka, Alma-Ata, 1979).

  35. V. M. Kel’man, I. V. Rodnikova, and L. M. Sekunova, Static Mass Spectrometers (Nauka, Alma-Ata, 1985).

  36. http://simion.com.

  37. http://wolfram.com/mathematica/.

  38. http://www.getpaint.net.

Download references

ACKNOWLEDGMENTS

The authors are grateful to the developers, researchers, and sponsors of site rspl.royalsocietypublishing.org (Proceedings of the Royal Society of London) whose selfless work has made it possible to freely operate with unique and rare references (in particular, publications [8, 9]).

Numerical calculations were performed using the SIMON 8.1 code [36]. In analytic calculations and verification of analytic expressions, as well as in preparing figures, we used Wolfram Mathematica program version 11 [37]. In editing figures, freely accessible program Paint.NET version 4 was used [38].

This research was completed using the ideas and rough manuscript copies of Prof. Yu.K. Golikov (without his direct participation). Therefore, responsibility for all inaccuracies and errors should be born exclusively by his coworkers.

The authors thank the reviewer for meticulous and profound work with great esteem to papers of Yu.K. Golikov (deceased). Thanks are also due to the reviewer for numerous useful remarks and for the wide scientific discussion in the form of anonymous exchange of letters (used in part in this study), which allowed us to considerably revise and improve the text.

Funding

This work was financed in part by state assignment no. 075-00780-19-00 for the Institute of Analytic Instrumentation, Russian Academy of Sciences.

Author information

Authors and Affiliations

  1. Peter the Great St. Petersburg Polytechnic University, 195251, St. Petersburg, Russia

    Yu. K. Golikov, N. K. Krasnova & K. V. Solov’ev

  2. Institute of Analytical Instrumentation, Russian Academy of Sciences, 190103, St. Petersburg, Russia

    A. S. Berdnikov, A. S. Antonov & K. V. Solov’ev

  3. Ioffe Institute, 194021, St. Petersburg, Russia

    A. S. Antonov

Authors
  1. Yu. K. Golikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. S. Berdnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. S. Antonov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. K. Krasnova
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. K. V. Solov’ev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. S. Berdnikov.

Ethics declarations

The authors claim that there are no conflicts of interests.

Additional information

Translated by N. Wadhwa

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Golikov, Y.K., Berdnikov, A.S., Antonov, A.S. et al. Application of the Donkin Formula in the Theory of Reflecting and Turning Devices. Tech. Phys. 64, 1850–1865 (2019). https://doi.org/10.1134/S1063784219120089

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063784219120089

Search

Navigation