The 22-year-old, who joined the Robins after winning the Samsung Win A Pro Contract competition, will remain at Sixfields until New Year's Day.
Boothroyd said the acquisition of Smith was with an eye on helping the Cobblers through a busy December.
He said: "We have seven games coming up in a period with the loan window closed and it is about making sure we are well equipped to deal with that period, not leaving ourselves short.
"With the loan window closing and injuries to John Johnson and David Artell, I felt we needed another defender in the squad to see us through a hectic run of games.
"Chris is here to give us the option of an extra defender through a busy period."