for first part see unitary_limit.gnumeric

at low energy the total cross section = 4 pi a^2 where 
a=scattering length = -d delta0/d k (-Derivative of delta0 wrt k)
one can crudely estimate the slope by eye

U	sigma	slope = -a
18	4	-.5
21	6	+.5
24	54	-2
27	24	-1
30	18	-1
33	16	-1
36	14	-1
39	12	-1
42	12	-1
45	11	-1
48	10	-1
51	9	-1
54	7	-.8
57	4	-.4
59	1	0
60	.5	+.4
61	46	+2
61.5	1200	+4
62	680	-2.5
63	80	-2

big a^2 -> big sigma