As defined in Fig. 4.2, the angle of refraction θ t or θ 2 is given by Snell’s law:
n 1 sin θ 1 ¼ n 2 sin θ 2 or n 1 sin θ i ¼ n 2 sin θ t
ð4:2Þ
where n 1 and n 2 are the indices of refraction for both materials. In the X-ray region,
n is very close to but slightly less than 1. Thus, it is simpler to use the refractive index
decrement δ, which is the deviation of n from 1:
δ ¼ 1 À n
ð4:3Þ
In the X-ray region, the angular deviations caused by refraction are miniscule
(Fig. 4.2), and this means conventional lenses with large apertures are out of the
question. Away from grazing incidence, the reflectivity for X-rays is also negligible,
and this means that conventional mirrors with large deflection angles are also not
feasible. Below we will see how scientists have overcome these issues to make
practical X-ray optics.
4.2.1 The Complex Index of Refraction
The refractive index that we have used so far does not take into account an additional
phenomenon—absorption. By combining n with the absorption coefficient β, one
can define a complex index of refraction ñ:
e n ¼ n À iβ ¼ 1 À δ À iβ
ð4:4Þ
Fig. 4.2 Left: refraction and reflection for diamond with visible light. Right: comparison of
refraction in diamond for yellow light and soft X-rays. With this scale, the deviation of the refracted
X-ray from a straight line is not visible
4.2 X-ray Optical Constants and Equations
71
n 1 sin θ 1 ¼ n 2 sin θ 2 or n 1 sin θ i ¼ n 2 sin θ t
ð4:2Þ
where n 1 and n 2 are the indices of refraction for both materials. In the X-ray region,
n is very close to but slightly less than 1. Thus, it is simpler to use the refractive index
decrement δ, which is the deviation of n from 1:
δ ¼ 1 À n
ð4:3Þ
In the X-ray region, the angular deviations caused by refraction are miniscule
(Fig. 4.2), and this means conventional lenses with large apertures are out of the
question. Away from grazing incidence, the reflectivity for X-rays is also negligible,
and this means that conventional mirrors with large deflection angles are also not
feasible. Below we will see how scientists have overcome these issues to make
practical X-ray optics.
4.2.1 The Complex Index of Refraction
The refractive index that we have used so far does not take into account an additional
phenomenon—absorption. By combining n with the absorption coefficient β, one
can define a complex index of refraction ñ:
e n ¼ n À iβ ¼ 1 À δ À iβ
ð4:4Þ
Fig. 4.2 Left: refraction and reflection for diamond with visible light. Right: comparison of
refraction in diamond for yellow light and soft X-rays. With this scale, the deviation of the refracted
X-ray from a straight line is not visible
4.2 X-ray Optical Constants and Equations
71