We describe the construction, validation, and testing of a concept inventory for an "Introduction to Physics of Semiconductors" course offered by the department of physics to undergraduate engineering students. By design, this inventory addresses both content knowledge and the ability to interpret content via different cognitive processes outlined in Bloom's revised taxonomy. The primary challenge comes from the low number of test takers. We describe the "Rasch modeling" analysis for this concept inventory, and the results of the calibration on a small sample size, with the intention of providing a useful blueprint to other instructors. Our study involved 101 students from Oklahoma State University and fourteen faculty teaching or doing research in the field of semiconductors at seven universities. The items were written in four-option multiple-choice format. It was possible to calibrate a 30-item unidimensional scale precisely enough to characterize the student population enrolled each semester and, therefore, to allow the tailoring of the learning activities of each class. We show that this scale can be employed as an item bank from which instructors could extract short testlets and where we can add new items fitting the existing calibration.