Here's an excerpt from
this paper:
"The second type of response is a
weak response: frequent approaching and retreating from the
predator and inspecting it. The third type is an average response:
it is a real alarm when the birds tend to be close to the predator,
they move restlessly around the object of alarm by bowing, pivoting
and tail-flicking. Chaffinches often raise their crest and
their neck is usually somewhat extended. The fourth type is a
strong response: chaotic movements and intense display, perhaps
including dive attacks on the predator. While mobbing
predators, chaffinches use ‘chink’ and ‘rain’ calls (Hinde 1954;
Marler 1956; Korbut 1989). During the second type of alarm
chaffinches use mainly ‘rain’ calls and only some ‘chink’ calls.
During the third type of alarm both call types are used in
approximately equal proportions. During the fourth type of
alarm almost all calls given by chaffinches are ‘chink’ calls."
Here's some more info about the rain call:
http://www.springerlink.com/content/k521m47123548353/
http://books.google.fi/books?id=2iFmsVSyV4gC&pg=PA134&lpg=PA134#v=onepage&q=&f=false
According to this, the rain call type recorded by Chris is also common in the UK:
http://www.garden-birds.co.uk/birds/chaffinch.htm