{"id":833,"date":"2021-11-21T05:19:57","date_gmt":"2021-11-21T05:19:57","guid":{"rendered":"http:\/\/www.calm-ebsd.com\/?page_id=833"},"modified":"2021-11-25T11:06:02","modified_gmt":"2021-11-25T11:06:02","slug":"braggs-equation","status":"publish","type":"page","link":"https:\/\/www.calm-ebsd.com\/index.php\/introduction-2\/braggs-equation\/","title":{"rendered":"Bragg’s equation"},"content":{"rendered":"\n

In any diffraction technique Bragg’s law<\/strong> is one of the fundamentals:<\/p>\n\n\n\n

n\\cdot\\lambda = 2\\cdot d_{(hkl)}\\cdot\\sin\\theta_{(hkl)}<\/pre><\/div>\n\n\n\n
For the determination of d<\/em>(hkl)<\/sub> which is used to describe the lattice parameters, the Bragg angle<\/strong> has to be measured<\/strong> and the wavelength must be known<\/strong>.<\/p>\n\n\n\n
Electron wavelength<\/p>\n\n\n\n
The wavelength of electron  \u03bbe<\/em><\/sub>  is a function of the acceleration voltage E<\/em>0<\/sub> used to controll the electron energy. 
The classic approach<\/strong> <\/p>\n\n\n\n
\\lambda_e [\\textnormal{\\AA}]= \\dfrac{h}{\\sqrt{2m_0eE_0\\rule{0em}{1.8ex}}}=\\dfrac{12.2653}\n{\\sqrt{E_0\\rule{0em}{1.8ex}} \\;[\\textrm{V}] }\n\n<\/pre><\/div>\n\n\n\n
of deBroglie is often assumed as sufficient for applications in SEM.<\/p>\n\n\n\n
For higher acceleration energies – commonly used in TEM – the relativistic approach<\/strong> <\/p>\n\n\n\n
\\begin{align*}\n\\lambda_e [\\textnormal{\\AA}]&= \\dfrac{h}{\\sqrt{2m_0eE_0\\left(1+\\dfrac{e}{2m_0c^2}E_0\\right)}}\\\\\n        & = \\dfrac{12.2653}{\\sqrt{E_0+\\rule{0em}{2.25ex}0.97845\\cdot 10^{-6}{E_0}^2}}\n\\end{align*} \n<\/pre><\/div>\n\n\n\n
has been developed. The left diagram (below) shows that the differences between the two approaches are quite small. However, the relevant relative difference (right diagram) proves a practically linearly increasing offset, which is 1% at 20kV and 1.5% at 30kV. This would not be negligible for the desired accuracy in CALM so that it uses the relativistic approach.<\/p>\n\n\n\n