Hi Sweetluv,
To answer your question, I am adding some excerpts from a document I found early on during my research.
"A human being may be killed in many ways but sureness is often overlooked by those who may be emotionally unstrung by the seriousness of this act they intend to commit. The specific technique employed will depend upon a large number of variables, but should be constant in one point: Death must be absolutely certain."
"The simplest local tools are often much the most efficient."
"A hammer, axe, wrench, screw driver, fire poker, kitchen knife, lamp stand, or anything hard, heavy and handy will suffice."
"All such improvised weapons have the important advantage of availability and apparent innocence."
"Blunt weapons require some anatomical knowledge for effective use. Their main advantage is their universal availability. A hammer may be picked up almost anywhere in the world. Baseball and [illeg] bats are very widely distributed. Even a rock or a heavy stick will do, and nothing resembling a weapon need be procured, carried or subsequently disposed of."
"Blows should be directed to the temple, the area just below and behind the ear, and the lower, rear portion of the skull. Of course, if the blow is very heavy, any portion of the upper skull will do. The lower frontal portion of the head, from the eyes to the throat, can withstand enormous blows without fatal consequences."
The simple answer is that an improvised weapon may not be identifiable. It goes along with other known facts in Joan's case, removal of clothing and identifiable jewelry, scattered belongings, concealed leads. Everything I have found points to a cover-up. Four people promoted a false explanation even though they had exculpatory evidence in their possession: Tim Burke, Andrew Palombo, Carmen Tammaro, and George Webster.
The blow to Joan's skull was done with tremendous force. She would have bled out quickly and died instantly. Unless you found the weapon with Joan's blood or hair on it, you might not ever know what was used, complicating investigation. It was probably something easily disposed.
Based on the known facts in Joan's case, it's easy to understand why this document stood out to me. This was contained in documents declassified under an FOIA on May 31, 1997. The estimated date of the original CIA document was December 31, 1953.