# Dependence of effective spectrum width of synchrotron radiation on particle energy

## Abstract

In the classical theory of synchrotron radiation, for the exact quantitative characterization of spectral properties, the concept of effective spectral width is introduced. In the first part of our work, published in EJPC 75 (2015), the effective spectral width as a function of the energy E of the radiating particle was obtained only in the ultra-relativistic approximation. In this article, which can be considered as a natural continuation of this work, a complete investigation is presented of the dependence of the effective width of the synchrotron radiation spectrum on energy for any values of E and for all the polarization components of the radiation. Numerical calculations were carried out for an effective width not exceeding 100 harmonics.

## 1 Introduction

As one of the major quantitative characteristics of spectral distributions for electromagnetic radiation, one commonly uses the concept of spectral half-width. For spectral distributions having a sharp maximum, the spectral half-width is the most informative physical characteristic.

However, once a spectral distribution has no pronounced maximum, the spectral half-width ceases to be an adequate quantitative characteristic. In particular, this is exactly the case of spectral distributions for synchrotron radiation (SR), and therefore the SR spectral half-width has neither been calculated theoretically, nor measured experimentally.

Instead of the spectral half-width, the present study proposes the introduction of a new precise quantitative characteristic of SR spectral distributions: the effective spectral width. We show how this quantity can be calculated theoretically, and which physically relevant information can be obtained using this quantity.

At present, the theory of synchrotron radiation is a fairly well-developed area of theoretical physics. Its main elements are described in monographs (e.g. [1, 2, 3, 4, 5, 6, 7]) and numerous articles. The high polarization degree of radiation and unique structure of spectral distributions are among the most important SR physical features. All theoretically predictable SR properties have been confirmed by experiment. The development of SR theory allows one not only to predict radiation characteristics qualitatively, but also to offer exact quantitative characteristics of physically important properties. For example, the high degree of SR polarization was qualitatively predicted by theory more than half a century ago (see, e.g., [1]); precise quantitative characteristics for linear polarization were also obtained, whereas such quantitative characteristics for circular polarization were obtained much later [8, 9, 10, 11].

In order to set up the problem, we now present some well-known expressions of the classical SR theory for the physical characteristics of synchrotron radiation, which can be found in [1, 2, 3, 4, 5, 6, 7].

*c*is the speed of light;

*W*is the total radiated power of unpolarized radiation, which can be written as

*e*is the particle charge;

*R*is the orbit radius;

*H*is the control field strength; \(m_{0}\) is the charge rest mass; \(\gamma \) is the relativistic factor. The index

*s*numbers the polarization components: \(s=2\) corresponds to the \(\sigma \)-component of linear polarization; \(s=3\) corresponds to the \(\pi \)-component of linear polarization; \(s=1\) corresponds to right-hand circular polarization; \(s=-1\) corresponds to left-hand circular polarization; \(s=0\) corresponds to the power of unpolarized radiation. The functions \(f_{s}(\beta ;\,\nu ,\,\theta )\) have the form [1, 2, 3, 4, 5, 6, 7]

## 2 Spectral distribution for polarization components of synchrotron radiation in the upper half-space

## 3 Effective spectral width for polarization components of synchrotron radiation

As one of the quantitative characteristics of physical properties for spectral distributions of SR polarization components, we propose to introduce the concept of effective spectral width \(\Lambda _{s}(\beta )\). Let us define \(\Lambda _{s}(\beta )\) as follows.

In practice, the most interesting case is the ultra-relativistic limit (\( \beta \approx 1\), equivalent to \(\gamma \gg 1\)). In this case, the analytical study of effective spectral width and other physically interesting quantitative characteristics for spectral distributions of SR polarization components can be significantly extended. This study was carried out in [13].

Boundary harmonics and respective energy of the effective spectral width for all polarization components of SR and unpolarized radiation

\(\Lambda \) | \(\nu ^{(1)}_{2}\) | \(\gamma _{2}\) | \(\nu ^{(2)}_{2}\) | \(\tilde{\gamma }_{2}\) | \(\nu ^{(1)}_{3}\) | \(\gamma _{3}\) | \(\nu ^{(2)}_{3}\) | \(\tilde{\gamma }_{3}\) | \(\nu ^{(1)}_{0}\) | \(\gamma _{0}\) | \(\nu ^{(2)}_{0}\) | \(\tilde{\gamma }_{0}\) | \(\nu ^{(1)}_{1}\) | \(\gamma _{1}\) | \(\nu ^{(2)}_{1}\) | \(\tilde{\gamma }_{1}\) | \(\nu ^{(1)}_{-1}\) | \(\gamma _{-1}\) | \(\nu ^{(2)}_{-1}\) | \(\tilde{\gamma }_{-1}\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1.1434 | 1 | 1.1434 | 1 | 1.2363 | 1 | 1.2363 | 1 | 1.1592 | 1 | 1.1592 | 1 | 1.1062 | 1 | 1.1062 | 1 | 1.1712 | 1 | 1.1712 |

2 | 1 | 1.2955 | 1 | 1.2955 | 1 | 1.4519 | 1 | 1.4519 | 1 | 1.3204 | 1 | 1.3204 | 1 | 1.2348 | 1 | 1.2348 | 1 | 1.3411 | 1 | 1.3411 |

3 | 1 | 1.4179 | 2 | 1.3237 | 1 | 1.6126 | 1 | 1.6126 | 1 | 1.4476 | 1 | 1.4476 | 1 | 1.3440 | 2 | 1.3233 | 1 | 1.4737 | 1 | 1.4737 |

4 | 1 | 1.5215 | 2 | 1.4740 | 1 | 1.7435 | 1 | 1.7435 | 1 | 1.5543 | 2 | 1.4831 | 2 | 1.4394 | 2 | 1.4394 | 1 | 1.5843 | 2 | 1.4987 |

5 | 1 | 1.6121 | 2 | 1.5827 | 1 | 1.8554 | 1 | 1.8554 | 1 | 1.6472 | 2 | 1.5990 | 2 | 1.5323 | 3 | 1.4808 | 1 | 1.6802 | 2 | 1.6210 |

6 | 1 | 1.6932 | 2 | 1.6741 | 1 | 1.9540 | 2 | 1.8592 | 1 | 1.7301 | 2 | 1.6949 | 2 | 1.6125 | 3 | 1.5854 | 1 | 1.7657 | 2 | 1.7211 |

7 | 1 | 1.7669 | 3 | 1.6998 | 1 | 2.0428 | 2 | 1.9638 | 1 | 1.8053 | 2 | 1.7786 | 2 | 1.6841 | 3 | 1.6686 | 1 | 1.8431 | 2 | 1.8082 |

8 | 1 | 1.8347 | 3 | 1.7860 | 1 | 2.1238 | 2 | 2.0562 | 1 | 1.8744 | 2 | 1.8537 | 2 | 1.7493 | 4 | 1.7015 | 1 | 1.9142 | 2 | 1.8860 |

9 | 1 | 1.8977 | 3 | 1.8610 | 1 | 2.1986 | 2 | 2.1394 | 1 | 1.9386 | 3 | 1.8770 | 2 | 1.8094 | 4 | 1.7786 | 1 | 1.9800 | 2 | 1.9570 |

10 | 1 | 1.9566 | 3 | 1.9286 | 1 | 2.2680 | 2 | 2.2161 | 1 | 1.9986 | 3 | 1.9482 | 2 | 1.8655 | 4 | 1.8455 | 1 | 2.0416 | 2 | 2.0225 |

11 | 1 | 2.0120 | 3 | 1.9906 | 1 | 2.3329 | 2 | 2.2868 | 1 | 2.0550 | 3 | 2.0131 | 3 | 1.9197 | 5 | 1.8721 | 1 | 2.0995 | 3 | 2.0448 |

12 | 2 | 2.0657 | 4 | 2.0129 | 1 | 2.3949 | 2 | 2.3537 | 1 | 2.1084 | 3 | 2.0731 | 3 | 1.9711 | 5 | 1.9353 | 1 | 2.1542 | 3 | 2.1072 |

13 | 2 | 2.1167 | 4 | 2.0723 | 1 | 2.4532 | 2 | 2.4165 | 1 | 2.1591 | 3 | 2.1292 | 3 | 2.0196 | 5 | 1.9922 | 1 | 2.2061 | 3 | 2.1654 |

14 | 2 | 2.1652 | 4 | 2.1273 | 1 | 2.5097 | 2 | 2.4751 | 1 | 2.2074 | 3 | 2.1819 | 3 | 2.0656 | 5 | 2.0445 | 1 | 2.2557 | 3 | 2.2200 |

15 | 2 | 2.2114 | 4 | 2.1788 | 1 | 2.5628 | 2 | 2.5314 | 1 | 2.2537 | 3 | 2.2318 | 3 | 2.1094 | 6 | 2.0690 | 1 | 2.3030 | 3 | 2.2716 |

16 | 2 | 2.2559 | 4 | 2.2275 | 1 | 2.6140 | 2 | 2.5851 | 1 | 2.2980 | 3 | 2.2793 | 3 | 2.1512 | 6 | 2.1190 | 1 | 2.3485 | 3 | 2.3207 |

17 | 2 | 2.2984 | 4 | 2.2737 | 1 | 2.6623 | 2 | 2.6363 | 1 | 2.3407 | 4 | 2.2999 | 3 | 2.1913 | 6 | 2.1656 | 1 | 2.3921 | 3 | 2.3675 |

18 | 2 | 2.3392 | 4 | 2.3177 | 1 | 2.7107 | 2 | 2.6846 | 1 | 2.3818 | 4 | 2.3457 | 4 | 2.2302 | 6 | 2.2094 | 1 | 2.4342 | 3 | 2.4123 |

19 | 2 | 2.3787 | 4 | 2.3598 | 1 | 2.7557 | 2 | 2.7324 | 1 | 2.4215 | 4 | 2.3894 | 4 | 2.2682 | 7 | 2.2315 | 1 | 2.4748 | 3 | 2.4553 |

20 | 2 | 2.4168 | 5 | 2.3807 | 1 | 2.7998 | 2 | 2.7779 | 2 | 2.4603 | 4 | 2.4313 | 4 | 2.3049 | 7 | 2.2738 | 1 | 2.5141 | 3 | 2.4967 |

21 | 2 | 2.4538 | 5 | 2.4215 | 1 | 2.8422 | 2 | 2.8217 | 2 | 2.4980 | 4 | 2.4716 | 4 | 2.3402 | 7 | 2.3139 | 1 | 2.5522 | 4 | 2.5154 |

22 | 2 | 2.4895 | 5 | 2.4608 | 1 | 2.8831 | 2 | 2.8636 | 2 | 2.5345 | 4 | 2.5104 | 4 | 2.3744 | 7 | 2.3521 | 1 | 2.5891 | 4 | 2.5556 |

23 | 2 | 2.5243 | 5 | 2.4985 | 1 | 2.9225 | 2 | 2.9035 | 2 | 2.5700 | 4 | 2.5479 | 4 | 2.4076 | 7 | 2.3886 | 1 | 2.6250 | 4 | 2.5943 |

24 | 2 | 2.5581 | 5 | 2.5350 | 1 | 2.9619 | 2 | 2.9440 | 2 | 2.6045 | 4 | 2.5842 | 4 | 2.4397 | 8 | 2.4091 | 1 | 2.6599 | 4 | 2.6318 |

25 | 2 | 2.5910 | 5 | 2.5702 | 1 | 2.9991 | 2 | 2.9827 | 2 | 2.6380 | 4 | 2.6194 | 4 | 2.4710 | 8 | 2.4446 | 1 | 2.6938 | 4 | 2.6681 |

26 | 2 | 2.6231 | 5 | 2.6043 | 1 | 3.0356 | 2 | 3.0201 | 2 | 2.6707 | 4 | 2.6536 | 5 | 2.5017 | 8 | 2.4787 | 1 | 2.7269 | 4 | 2.7033 |

27 | 2 | 2.6545 | 5 | 2.6374 | 1 | 3.0715 | 2 | 3.0566 | 2 | 2.7025 | 5 | 2.6712 | 5 | 2.5318 | 8 | 2.5116 | 2 | 2.7596 | 4 | 2.7376 |

28 | 3 | 2.6849 | 6 | 2.6558 | 1 | 3.1063 | 2 | 3.0922 | 2 | 2.7336 | 5 | 2.7045 | 5 | 2.5611 | 8 | 2.5434 | 2 | 2.7914 | 4 | 2.7709 |

29 | 3 | 2.7148 | 6 | 2.6880 | 1 | 3.1401 | 2 | 3.1266 | 2 | 2.7640 | 5 | 2.7369 | 5 | 2.5897 | 9 | 2.5623 | 2 | 2.8226 | 4 | 2.8033 |

30 | 3 | 2.7443 | 6 | 2.7193 | 1 | 3.1728 | 3 | 3.1407 | 2 | 2.7937 | 5 | 2.7685 | 5 | 2.6176 | 9 | 2.5933 | 2 | 2.8530 | 4 | 2.8350 |

31 | 3 | 2.7731 | 6 | 2.7499 | 1 | 3.2053 | 3 | 3.1740 | 2 | 2.8227 | 5 | 2.7992 | 5 | 2.6449 | 9 | 2.6233 | 2 | 2.8828 | 4 | 2.8659 |

32 | 3 | 2.8012 | 6 | 2.7797 | 1 | 3.2374 | 3 | 3.2078 | 2 | 2.8512 | 5 | 2.8292 | 5 | 2.6716 | 9 | 2.6525 | 2 | 2.9119 | 4 | 2.8960 |

33 | 3 | 2.8288 | 6 | 2.8088 | 1 | 3.2690 | 3 | 3.2415 | 2 | 2.8790 | 5 | 2.8585 | 5 | 2.6977 | 9 | 2.6808 | 2 | 2.9404 | 4 | 2.9256 |

34 | 3 | 2.8559 | 6 | 2.8373 | 1 | 3.2995 | 3 | 3.2728 | 2 | 2.9064 | 5 | 2.8872 | 6 | 2.7234 | 10 | 2.6983 | 2 | 2.9684 | 5 | 2.9414 |

35 | 3 | 2.8824 | 6 | 2.8652 | 1 | 3.3297 | 3 | 3.3039 | 2 | 2.9331 | 5 | 2.9152 | 6 | 2.7487 | 10 | 2.7260 | 2 | 2.9959 | 5 | 2.9703 |

36 | 3 | 2.9085 | 6 | 2.8926 | 1 | 3.3592 | 3 | 3.3346 | 2 | 2.9594 | 5 | 2.9427 | 6 | 2.7736 | 10 | 2.7530 | 2 | 3.0228 | 5 | 2.9986 |

37 | 3 | 2.9339 | 7 | 2.9086 | 1 | 3.3885 | 3 | 3.3646 | 2 | 2.9853 | 5 | 2.9696 | 6 | 2.7979 | 10 | 2.7793 | 2 | 3.0492 | 5 | 3.0263 |

38 | 3 | 2.9589 | 7 | 2.9352 | 1 | 3.4186 | 3 | 3.3946 | 2 | 3.0106 | 5 | 2.9960 | 6 | 2.8219 | 10 | 2.8050 | 2 | 3.0752 | 5 | 3.0535 |

39 | 3 | 2.9835 | 7 | 2.9612 | 1 | 3.4461 | 3 | 3.4241 | 2 | 3.0356 | 6 | 3.0112 | 6 | 2.8453 | 10 | 2.8302 | 2 | 3.1007 | 5 | 3.0801 |

40 | 3 | 3.0078 | 7 | 2.9869 | 1 | 3.4723 | 3 | 3.4511 | 2 | 3.0601 | 6 | 3.0371 | 6 | 2.8684 | 11 | 2.8465 | 2 | 3.1258 | 5 | 3.1063 |

41 | 3 | 3.0316 | 7 | 3.0121 | 1 | 3.4998 | 3 | 3.4793 | 2 | 3.0842 | 6 | 3.0625 | 6 | 2.8911 | 11 | 2.8712 | 2 | 3.1505 | 5 | 3.1319 |

42 | 3 | 3.0550 | 7 | 3.0367 | 1 | 3.5270 | 3 | 3.5074 | 2 | 3.1079 | 6 | 3.0875 | 6 | 2.9133 | 11 | 2.8953 | 2 | 3.1748 | 5 | 3.1572 |

43 | 3 | 3.0781 | 7 | 3.0609 | 1 | 3.5534 | 3 | 3.5344 | 2 | 3.1313 | 6 | 3.1120 | 7 | 2.9354 | 11 | 2.9189 | 2 | 3.1987 | 5 | 3.1820 |

44 | 3 | 3.1008 | 7 | 3.0847 | 1 | 3.5787 | 3 | 3.5614 | 2 | 3.1543 | 6 | 3.1361 | 7 | 2.9572 | 11 | 2.9420 | 2 | 3.2223 | 5 | 3.2064 |

45 | 3 | 3.1231 | 7 | 3.1081 | 1 | 3.6046 | 3 | 3.5868 | 3 | 3.1771 | 6 | 3.1598 | 7 | 2.9786 | 12 | 2.9573 | 2 | 3.2455 | 5 | 3.2304 |

46 | 3 | 3.1453 | 7 | 3.1311 | 1 | 3.6303 | 3 | 3.6129 | 3 | 3.1996 | 6 | 3.1831 | 7 | 2.9997 | 12 | 2.9801 | 2 | 3.2684 | 5 | 3.2540 |

47 | 4 | 3.1671 | 8 | 3.1457 | 1 | 3.6550 | 3 | 3.6388 | 3 | 3.2217 | 6 | 3.2060 | 7 | 3.0205 | 12 | 3.0024 | 2 | 3.2909 | 5 | 3.2773 |

48 | 4 | 3.1887 | 8 | 3.1684 | 1 | 3.6798 | 3 | 3.6637 | 3 | 3.2436 | 6 | 3.2286 | 7 | 3.0410 | 12 | 3.0243 | 2 | 3.3131 | 5 | 3.3002 |

49 | 4 | 3.2100 | 8 | 3.1906 | 1 | 3.7032 | 3 | 3.6887 | 3 | 3.2651 | 6 | 3.2509 | 7 | 3.0612 | 12 | 3.0458 | 2 | 3.3351 | 6 | 3.3137 |

50 | 4 | 3.2309 | 8 | 3.2126 | 1 | 3.7286 | 3 | 3.7123 | 3 | 3.2864 | 6 | 3.2728 | 7 | 3.0811 | 12 | 3.0670 | 2 | 3.3567 | 6 | 3.3363 |

## 4 Analysis of numerical results for effective spectral width of synchrotron radiation

The main results of a numerical study for effective spectral width of SR polarization components are given by Table 1. Calculation results for larger \(\Lambda \) can be found in [14].

*s*, we examine the sequences of integers \(\Lambda _{s}=1,2,3\,\ldots \) and \( \nu _{s}^{(1)}=1,2,3\,\ldots \) (it is evident that \(\nu _{s}^{(2)}=\nu _{s}^{(1)}+\Lambda _{s}-1\)) and determine the regions of values \(\gamma _{s}\) for which the condition (16) (equivalent to (12)) is satisfied. It is clear that the boundary points of possible regions for \(\gamma _{s}\) can be found, according to (16), as solutions of the equations

Let us consider in more detail the calculation method and the results pertaining to the \(\sigma \)- component of linear polarization. These results are given in the column for \(s=2\) of our Table. We denote \(\nu _{2}^{(1)}=k=1,2,3\,\ldots \).

The next step concerns the region \(\Lambda _{2}=12\div 27\). Here, it is essential that the smallest possible value is \(\nu _{2}^{(1)}=2\); otherwise, the results are identical to the case \(\Lambda _{2}=7\div 11\).

In general, the column for \(s=2\) of the table indicates for each \(\Lambda _{2}\) the smallest possible value \(\nu _{2}^{(1)}=\nu _{2}^{(1)}(\Lambda _{2})\) and the corresponding largest value \(\gamma _{2}=\gamma _{2}(\Lambda _{2},\nu _{2}^{(1)}(\Lambda _{2}))\), as well as the largest possible value \( \tilde{\nu }_{2}\) for this \(\Lambda _{2}\) and the corresponding smallest value \(\tilde{\gamma }_{2}\). Possible intermediate values \(\nu _{2}^{(1)}\) between the smallest \(\nu _{2}^{(1)}(\Lambda _{2})\) and the largest \(\tilde{ \nu }_{2}\), as well as the respective intermediate values \(\gamma _{2}\), are not specified. The intermediate values of \(\gamma _{2}\) always satisfy the relations of Eq. (29).

In the ultra-relativistic case, the corresponding results have been obtained in [13].

## Notes

### Acknowledgements

The work of Bagrov and Gitman is partially supported by RFBR research project No. 15- 02-00293a and by Tomsk State University Competitiveness Improvement Program. Gitman thanks CNPq and FAPESP for their permanent support. Levin thanks CNPq for permanent support.

## References

- 1.A.A. Sokolov, I.M. Ternov,
*Synchrotron Radiation*(Akademie, Berlin, 1968)Google Scholar - 2.H. Winick, S. Doniach,
*Synchrotron Radiation Research*(Plenum Press, New York, 1980)CrossRefGoogle Scholar - 3.A.A. Sokolov, I.M. Ternov, C.W. Kilmister,
*Radiation from Relativistic Electrons*(American Institute of Physics, New York, 1986)Google Scholar - 4.V.G. Bagrov et al.,
*Synchrotron Radiation Theory and Its Development*(World Scientific, Singapore, 1999)Google Scholar - 5.P.J. Duke,
*Synchrotron Radiation: Production and Properties*(Oxford University Press, Oxford, 2000)Google Scholar - 6.H. Wiedemann,
*Synchrotron Radiation*(Springer, Berlin, 2003)Google Scholar - 7.A. Hofmann,
*The Physics of Synchrotron Radiation*(Cambridge University Press, Cambridge, 2004)CrossRefGoogle Scholar - 8.V.G. Bagrov, M.V. Dolzhin, V.B. Tlyachev, A.T. Jarovoi, Evolution of the angular distribution of circularly polarized synchrotron radiation with charge energy. Russ. Phys. J.
**47**(4), 414–423 (2004). doi: 10.1023/B:RUPJ.0000042770.97166.87 CrossRefMATHGoogle Scholar - 9.V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, A.T. Jarovoi, Evolution of angular distribution of polarization components for synchrotron radiation under changes of particle energy, in
*International Summer School: Recent Problems in Field Theory*, vol. 4, ed. by A.V. Aminovain, K. Heter (Kazan, Petrov School, 2004), pp. 9–24. (in Russian)Google Scholar - 10.V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, A.T. Jarovoi, Evolution of angular distribution of polarization components for synchrotron radiation under changes of particle energy, in
*International Conference: Particle Physics in Laboratory, Space and Universe. Moscow State University, Faculty Physics*, ed. by A.I. Studenikin (World Scientific, New Jersey, 2005), pp. 355–362CrossRefGoogle Scholar - 11.V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, A.T. Jarovoi, New theoretical results in synchrotron radiation. Nucl. Instrum. Methods Phys. Res. B
**240**(3), 638–645 (2005). doi: 10.1016/j.nimb.2005.03.286 ADSCrossRefGoogle Scholar - 12.V.G. Bagrov, M.V. Dolzhin, K.G. Seravkin, V.M. Shakhmatov, Partial contributions of individual harmonics to the power of synchrotron radiation. Russ. Phys. J.
**49**(7), 681–689 (2006). doi: 10.1007/s11182-006-0162-1 CrossRefGoogle Scholar - 13.V.G. Bagrov, D.M. Gitman, A.D. Levin, A.S. Loginov, A.D. Saprykin, Effective spectrum width of the synchrotron radiation. Eur. Phys. J. C
**75**(11), 555 (2015). doi: 10.1140/epjc/s10052-015-3798-6 ADSCrossRefGoogle Scholar - 14.V. G. Bagrov, D. M. Gitman, A. D. Levin, A. S. Loginov, A. D. Saprykin, Dependence of Effective spectrum width of synchrotron radiation on particle energy. arXiv:1703.02695v1 [physics.class-ph] (2017)

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}