The relationship of the Rutherfor differential cross section between laboratory and center of mass systems at relativistic energies is treated in Section 1.6 of Chapter 1 in [Leroy and P.G. Rancoita (2016)]. Chapter 1 is availble for downloading at https://www.worldscientific.com/worldscibooks/10.1142/9167#t=aboutBook.
Leroy and P.G. Rancoita (2016), Principles of Radiation Interaction in Matter and Detection - 4th Edition -, World Scientific. Singapore, .https://www.worldscientific.com/worldscibooks/10.1142/9167#t=aboutBook, ISBN-978-981-4603-18-8 (printed), ISBN.978-981-4603-19-5 (ebook); it is also partially accessible via google books.