Dear Avinash Ingle ,

the most crucial parameter in the calculation of the interaction volume is the mean penetration depth (or better the mean information depth) of the sample at GIXRD conditions.

When going to grazing incidence we have to consider two cases:

a) incidence angle phii is about or smaller than the critical angle phic of total reflection,

and b) phii is significantly larger than phic.

phic is mainly related to the density rho of the material of interest. For small phic we have sin(phic) ~ phic ~ sqrt(2*delta) with delta as the deviation of the real part of the refractive index from 1 (real part re(n) of refractive index n in the x-ray regime is described by re(n)=1-delta).

Making some simplifications from Parratt*) we have:

phic[mrad] ~ (20/E[keV] * sqrt(rho[g/cm³])

So at Cu K alpha (E~8keV) and for rho=1g/cm³ we have phic = 2,5mrad. For rho= 4g/cm³ we have phic = 5mrad, and for rho=9g/cm³ we have 7,5mrad.

Thus within that range of sample densities your incident angle phii of 1° ~ 17mrad is much larger than the critical angle of total reflection and condition b) will be satisfied.

So we can go for a simple ray tracing (please see the attachement). For estimation of the mean penetration/information depth we have to set the total length of the x-ray path in the sample equal to the inverse of the attenuation coefficient µ of the sample material.

A special simple case shows up when chosing the exit glancing angle phie almost equal to the incoming glancing angle phii.

We will then have: d1/e = sin(phi)/(2*µ).

For phie significantly different from phii you should apply the formula presented in the attachment.

You can calculate the x-ray linear attenuation coefficient µ from the x-ray mass-attenuation coefficient µ/rho by multiplying by the density rho.

µ/rho values are easily derived from the NIST/XCOM data base.

https://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html

For the condition a) you have to calculate the penetration depth according to formalism presented by L.G. Parratt.

*) see:

L. G. Parratt; Surface Studies of Solids by Total Reflection of X-Rays;

Phys. Rev. 95 (2), pp. 359-369 (1954)

Article Role of x-ray diffraction analysis in surface engineering: I...

Dear Avinash Ingle ,

the most crucial parameter in the calculation of the interaction volume is the mean penetration depth (or better the mean information depth) of the sample at GIXRD conditions.

When going to grazing incidence we have to consider two cases:

a) incidence angle phii is about or smaller than the critical angle phic of total reflection,

and b) phii is significantly larger than phic.

phic is mainly related to the density rho of the material of interest. For small phic we have sin(phic) ~ phic ~ sqrt(2*delta) with delta as the deviation of the real part of the refractive index from 1 (real part re(n) of refractive index n in the x-ray regime is described by re(n)=1-delta).

Making some simplifications from Parratt*) we have:

phic[mrad] ~ (20/E[keV] * sqrt(rho[g/cm³])

So at Cu K alpha (E~8keV) and for rho=1g/cm³ we have phic = 2,5mrad. For rho= 4g/cm³ we have phic = 5mrad, and for rho=9g/cm³ we have 7,5mrad.

Thus within that range of sample densities your incident angle phii of 1° ~ 17mrad is much larger than the critical angle of total reflection and condition b) will be satisfied.

So we can go for a simple ray tracing (please see the attachement). For estimation of the mean penetration/information depth we have to set the total length of the x-ray path in the sample equal to the inverse of the attenuation coefficient µ of the sample material.

A special simple case shows up when chosing the exit glancing angle phie almost equal to the incoming glancing angle phii.

We will then have: d1/e = sin(phi)/(2*µ).

For phie significantly different from phii you should apply the formula presented in the attachment.

You can calculate the x-ray linear attenuation coefficient µ from the x-ray mass-attenuation coefficient µ/rho by multiplying by the density rho.

µ/rho values are easily derived from the NIST/XCOM data base.

https://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html

For the condition a) you have to calculate the penetration depth according to formalism presented by L.G. Parratt.

*) see:

L. G. Parratt; Surface Studies of Solids by Total Reflection of X-Rays;

Phys. Rev. 95 (2), pp. 359-369 (1954)