PRL 110, 129501 (2013)
week ending 22 MARCH 2013
PHYSICAL REVIEW LETTERS
Comment on ‘‘Refractive Index of Silicon at Ray Energies’’ In a recent Letter, Habs et al. [1] reported a measurement of ¼ ReðnÞ-1 at photon energies near 1 MeV, where n denotes the index of refraction of silicon. They pointed out that the predicted short-wavelength limit of this quantity is ðE Þ ffi 4:8 1010 =E 2 , where E is in MeV. However, they observed a small positive value at E ¼ 0:786 MeV, and even reached ð1:48 0:13Þ 109 at 1.165 MeV. The authors attribute some of this unexpected result to the hitherto neglected contribution of Delbru¨ck scattering to the Kramers-Kronig dispersion relation for ðE Þ. Goldberger and Watson [2] write (notation of Ref. [1]), @cNc Z 1 ðEÞ P dE 2 ; (1) ðE Þ ¼ E E 2 0 where Nc denotes the number of atoms=cm3 , is the total cross section, and P denotes the principal value. The authors show in their Fig. 3 the cross sections for both inelastic and elastic (recoil taken by crystal) Delbru¨ck scattering. The
former attains a maximum of about 5 mb at 1 MeV, and falls off sharply on either side. In Jackson’s textbook [3] one finds the problem (7.14) of calculating the effect of a resonant contribution to the real part of the dielectric function. I have modified his expression, and used it to estimate the change in the ðE Þ that would be produced by a cross section of the form inelD ðEÞ ¼
2 EE0 inelD ðE0 Þ: ðE E0 2 Þ2 þ E0 2 2 2
Here, E0 is the resonant energy, is the full width at halfmaximum, and inelD ðE0 Þ is the cross section at the resonance. The cross section for elastic Delbru¨ck scattering shows a power-law increase at low energies, followed by a flat plateau at high energy. I parametrize it as follows: 0 E E1 elD ðEÞ ¼ elD ðE1 ÞðE=E1 Þ2 : E1 E 1 elD ðEÞ ¼ elD ðE1 Þ
(3)
Evaluating Eq. (1) with these expressions, I find ( denotes the Heaviside step function)
8 lnðE0 ðiE0 Þ=E 2 Þ > inelD ðE0 Þ > Im ðE 2 E 2 Þ=E þi > E0 0 0 @cNc < ðE Þ ¼ > 2 > E E þE1 E þE1 elD ðE1 Þ E1 > 2 E1 E ðE E1 Þ ln E E1 þ ðE1 E Þ ln E1 E ; : E1 where the upper (lower) line is for inelastic (elastic) scattering. I choose both E0 and E1 ¼ 1 MeV, inelD ðE0 Þ ¼ 5 mb,
(2)
(4)
elD ðE1 Þ ¼ 0:1 mb, and ¼ 0:25 MeV, as suggested by the shapes of the cross sections. Figure 1(a) shows my assumed cross sections as functions of energy. The contributions of these to ðE Þ are displayed in Fig. 1(b). The elastic Delbru¨ck result was multiplied by 10 to make it visible. It is clear that these contributions are a million times too small to cause the change in sign. Although my modeling of Delbru¨ck scattering is naive, this situation will prevail in a more refined approach. I conclude that Delbru¨ck scattering plays no role in changing the sign of , in contrast to the claims of Ref. [1]. Even taking into account the Z4 dependence on the atomic number, no significant Delbru¨ck contribution to can occur for heavy elements. J. T. Donohue Centre d’Etudes Nucle´aires de Bordeaux-Gradignan Universite´ Bordeaux 1 CNRS/IN2P3, BP 120, 33175 Gradignan, France Received 14 September 2012; published 18 March 2013 DOI: 10.1103/PhysRevLett.110.129501 PACS numbers: 41.50.+h, 12.20.m, 42.50.Xa, 78.20.e
FIG. 1 (color online). (a) The assumed inelastic (solid, red) and elastic (dotted, blue) Delbru¨ck cross sections. (b) The contributions to ðE Þ corresponding to (a). For the elastic Delbruck, the result has been multiplied by 10.
0031-9007=13=110(12)=129501(1)
[1] D. Habs, M. M. Gu¨nther, M. Jentschel, and W. Urban, Phys. Rev. Lett. 108, 184802 (2012). [2] M. L. Goldberger and K. M Watson, Collision Theory (John Wiley & Sons, New York, 1964), p. 559. [3] J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1975), 2nd ed.
129501-1
Ó 2013 American Physical Society
week ending 22 MARCH 2013
PHYSICAL REVIEW LETTERS
Comment on ‘‘Refractive Index of Silicon at Ray Energies’’ In a recent Letter, Habs et al. [1] reported a measurement of ¼ ReðnÞ-1 at photon energies near 1 MeV, where n denotes the index of refraction of silicon. They pointed out that the predicted short-wavelength limit of this quantity is ðE Þ ffi 4:8 1010 =E 2 , where E is in MeV. However, they observed a small positive value at E ¼ 0:786 MeV, and even reached ð1:48 0:13Þ 109 at 1.165 MeV. The authors attribute some of this unexpected result to the hitherto neglected contribution of Delbru¨ck scattering to the Kramers-Kronig dispersion relation for ðE Þ. Goldberger and Watson [2] write (notation of Ref. [1]), @cNc Z 1 ðEÞ P dE 2 ; (1) ðE Þ ¼ E E 2 0 where Nc denotes the number of atoms=cm3 , is the total cross section, and P denotes the principal value. The authors show in their Fig. 3 the cross sections for both inelastic and elastic (recoil taken by crystal) Delbru¨ck scattering. The
former attains a maximum of about 5 mb at 1 MeV, and falls off sharply on either side. In Jackson’s textbook [3] one finds the problem (7.14) of calculating the effect of a resonant contribution to the real part of the dielectric function. I have modified his expression, and used it to estimate the change in the ðE Þ that would be produced by a cross section of the form inelD ðEÞ ¼
2 EE0 inelD ðE0 Þ: ðE E0 2 Þ2 þ E0 2 2 2
Here, E0 is the resonant energy, is the full width at halfmaximum, and inelD ðE0 Þ is the cross section at the resonance. The cross section for elastic Delbru¨ck scattering shows a power-law increase at low energies, followed by a flat plateau at high energy. I parametrize it as follows: 0 E E1 elD ðEÞ ¼ elD ðE1 ÞðE=E1 Þ2 : E1 E 1 elD ðEÞ ¼ elD ðE1 Þ
(3)
Evaluating Eq. (1) with these expressions, I find ( denotes the Heaviside step function)
8 lnðE0 ðiE0 Þ=E 2 Þ > inelD ðE0 Þ > Im ðE 2 E 2 Þ=E þi > E0 0 0 @cNc < ðE Þ ¼ > 2 > E E þE1 E þE1 elD ðE1 Þ E1 > 2 E1 E ðE E1 Þ ln E E1 þ ðE1 E Þ ln E1 E ; : E1 where the upper (lower) line is for inelastic (elastic) scattering. I choose both E0 and E1 ¼ 1 MeV, inelD ðE0 Þ ¼ 5 mb,
(2)
(4)
elD ðE1 Þ ¼ 0:1 mb, and ¼ 0:25 MeV, as suggested by the shapes of the cross sections. Figure 1(a) shows my assumed cross sections as functions of energy. The contributions of these to ðE Þ are displayed in Fig. 1(b). The elastic Delbru¨ck result was multiplied by 10 to make it visible. It is clear that these contributions are a million times too small to cause the change in sign. Although my modeling of Delbru¨ck scattering is naive, this situation will prevail in a more refined approach. I conclude that Delbru¨ck scattering plays no role in changing the sign of , in contrast to the claims of Ref. [1]. Even taking into account the Z4 dependence on the atomic number, no significant Delbru¨ck contribution to can occur for heavy elements. J. T. Donohue Centre d’Etudes Nucle´aires de Bordeaux-Gradignan Universite´ Bordeaux 1 CNRS/IN2P3, BP 120, 33175 Gradignan, France Received 14 September 2012; published 18 March 2013 DOI: 10.1103/PhysRevLett.110.129501 PACS numbers: 41.50.+h, 12.20.m, 42.50.Xa, 78.20.e
FIG. 1 (color online). (a) The assumed inelastic (solid, red) and elastic (dotted, blue) Delbru¨ck cross sections. (b) The contributions to ðE Þ corresponding to (a). For the elastic Delbruck, the result has been multiplied by 10.
0031-9007=13=110(12)=129501(1)
[1] D. Habs, M. M. Gu¨nther, M. Jentschel, and W. Urban, Phys. Rev. Lett. 108, 184802 (2012). [2] M. L. Goldberger and K. M Watson, Collision Theory (John Wiley & Sons, New York, 1964), p. 559. [3] J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1975), 2nd ed.
129501-1
Ó 2013 American Physical Society
