Question

) Define the momentum transfer vector q = kr − ki - What are its direction and magnitude, q? Give an expression for the latter in terms of the incident wavelength λ and angle θ1. (5M) - For the present case where α1 ≪ 1 derive an approximated relation between q and α1 and show that the reflection coefficients depend on the 4-th power of the magnitude of the momentum transfer, q as: Rs(q) = Rp(q) ∼ C q 4 . Give the proportionality factor, C, in terms of the incident wavelength λ and the angle α1. Question 3:Total external reflection almost) [Total Marks: 33.5] A monochromatic plane wave, propagating in air, hits a flat surface of a non-magnetic sample with an incident angle ; ~ 90o degrees [see Fig.(1)]. For x-rays fields the index of refraction may be safely taken very slightly less than one; So, assume that n2 =1, =10-5 (3a) With reference to Fig.(1): - Make two sketches with the two possible configurations for the reflection/refraction of the EM wave ('s' and 'p' polarization incidence). Sketch clearly the(E;,B;,k),Er,B,,k,),(Et,B,k) vectors and identify the plane of incidence and the interface plane (3M) - Using the general Fresnel equations for the s-polarization reflection and transmission amplitude coefficients, r's,ts provide an expression for the reflection and transmission power coefficients, Rs,Ts in terms of the incident and the transmission angles. (5M) - Provide the same expression for the 'p'-polarization incidence. (5M) 3b)At this point,set;=1and=2and=1 andt=a2 -Using the Snell's law show that for <1 it results that 2<1 if 1. - In addition show that, (5M) (0.5M) a? - a2 ~ 28. - Given that 1,2 < 1 and the help of the above expression show that the approximated reflection power coefficients, Rs,Rp become equal; give an expression in terms of and 1 (5M)) 3cDefine the momentum transfer vector q=kr-k, - What are its direction and magnitude, q? Give an expression for the latter in terms of the incident wavelength and angle 1. (5M) -For the present case where 1 < 1 derive an approximated relation between q and 1 and show that the reflection coefficients depend on the 4-th power of the magnitude of the momentum transfer, q as: Rs(q)=Rp(q)~ 04 Give the proportionality factor, C, in terms of the incident wavelength A and the angle 1. (5M)

          ) Define the momentum transfer vector q = kr − ki - What are its direction and magnitude, q? Give an expression for the latter in terms of the incident wavelength λ and angle θ1. (5M) - For the present case where α1 ≪ 1 derive an approximated relation between q and α1 and show that the reflection coefficients depend on the 4-th power of the magnitude of the momentum transfer, q as: Rs(q) = Rp(q) ∼ C q 4 . Give the proportionality factor, C, in terms of the incident wavelength λ and the angle α1.
Question 3:Total external reflection almost) [Total Marks: 33.5] A monochromatic plane wave, propagating in air, hits a flat surface of a non-magnetic sample with an incident angle ; ~ 90o degrees [see Fig.(1)]. For x-rays fields the index of refraction may be safely taken very slightly less than one; So, assume that
n2 =1,
=10-5
(3a) With reference to Fig.(1): - Make two sketches with the two possible configurations for the reflection/refraction of the EM wave ('s' and 'p' polarization incidence). Sketch clearly the(E;,B;,k),Er,B,,k,),(Et,B,k) vectors and identify the plane of incidence and the interface plane (3M) - Using the general Fresnel equations for the s-polarization reflection and transmission amplitude coefficients, r's,ts provide an expression for the reflection and transmission power coefficients, Rs,Ts in terms of the incident and the transmission angles. (5M) - Provide the same expression for the 'p'-polarization incidence. (5M)
3b)At this point,set;=1and=2and=1 andt=a2
-Using the Snell's law show that for <1 it results that 2<1 if 1. - In addition show that,
(5M) (0.5M)
a? - a2 ~ 28.
- Given that 1,2 < 1 and the help of the above expression show that the approximated reflection power coefficients, Rs,Rp become equal; give an expression in terms of  and 1 (5M))
3cDefine the momentum transfer vector q=kr-k,
- What are its direction and magnitude, q? Give an expression for the latter in terms of the incident wavelength  and angle 1. (5M) -For the present case where 1 < 1 derive an approximated relation between q and 1 and show that the reflection coefficients depend on the 4-th power of the magnitude of the momentum transfer, q as:
Rs(q)=Rp(q)~ 04
Give the proportionality factor, C, in terms of the incident wavelength A and the angle 1.
(5M)

define the momentum transfer vector q kr ki what are its direction and magnitude q give an expression for the latter in terms of the incident wavelength and angle 1 5m for the present case w 57143

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