Transmission and scattering problems involving complex potentials are important in physics, in particular in describing nuclear collisions. We describe many pedagogical features of transmission in one dimension and scattering and absorption cross sections in three dimensions for a rectangular absorptive potential and compare the results with scattering from real barriers and a hard sphere. For a given energy in one dimension we show that the absorption reaches a maximum for a critical value of the absorption strength and then monotonically decreases with an increase in the absorption strength. An infinitely absorptive well becomes almost fully reflective and thus is similar to an infinitely high barrier. Similar results are found in three dimensions. We show that the absorption cross section in three dimensions diverges at a threshold energy, which is of critical importance for exothermic reactions.

Topics
Thermal neutrons, Optical absorption, Threshold energy, Educational aids, Elastic scattering, Wentzel-Kramers-Brillouin approximation, Scattering problem, Scattering theory, Scattering matrix, Statistical mechanics models
1.
B.
 
Sahu
,
I.
 
Jamir
,
E. P. F.
 
Lyngdoh
, and
C. S.
 
Shastry
, “
Fusion under a complex barrier
,”
Phys. Rev. C
https://doi.org/10.1103/PhysRevC.57.1853
 
57
,
1853
–
1864
(
1998
).
Google Scholar
Crossref
Search ADS
 
2.
M.
 
Razavy
,
Quantum Theory of Tunneling
(
World Scientific
, Singapore,
2003
), Chaps. VI and VII.
Google Scholar
Crossref
Search ADS
 
3.
E. P. F.
 
Lyngdoh
,
I.
 
Jamir
, and
C. S.
 
Shastry
, “
Schrödinger equation with complex potentials: The phenomenon of absorption
,”
Phys. Educ.
 
15
,
145
–
153
(
1998
).
Google Scholar
 
4.
S.
 
Mahadevan
,
A.
 
Uma Maheshwari
,
P.
 
Prema
, and
C. S.
 
Shastry
, “
Quantum mechanical transmission with absorption
,”
Phys. Educ.
 23 (to be published).
5.
L. I.
 
Schiff
,
Quantum Mechanics
, 3rd ed. (
McGraw-Hill
, New York,
1968
), Chap. V.
Google Scholar
 
6.
A. E.
 
Green
,
T.
 
Sawada
, and
D. S.
 
Saxon
,
The Nuclear Independent Particle Model
(
Academic
, New York,
1968
), Chap. III.
Google Scholar
 
7.
X. Z.
 
Li
,
J.
 
Tian
,
M. Y.
 
Mei
, and
C. X.
 
Li
, “
Sub-barrier fusion and Selective resonant tunneling
,”
Phys. Rev. C
https://doi.org/10.1103/PhysRevC.61.024610
 
61
,
024610
–
1
024610
–
5
(
2000
).
Google Scholar
Crossref
Search ADS
 
8.
R. G.
 
Newton
,
Scattering Theory of Wave and Particles
(
McGraw-Hill
, New York,
1996
), Chap. XII.
Google Scholar
 
9.
M.
 
Abramowitz
 and
I. A.
 
Stegun
,
Handbook of Mathematical Functions
(
Dover
, New York,
1965
), p.
4377
.
Google Scholar
 
10.
See for example,
M. A.
 
Preston
,
Physics of the Nucleus
(
Addison-Wesley
, Reading, MA, 1962), p.
555
.
11.
P.
 
Frobrich
 and
R.
 
Lipperheide
,
Theory of Nuclear Reactions
(
Clarendon
, Oxford,
1996
), Chap. IX.
Google Scholar
 
12.
B.
 
Sahu
,
S. K.
 
Agarwalla
, and
C. S.
 
Shastry
, “
Fusion, resonances and scattering in C+12C12 reaction
,”
Pramana, J. Phys.
 
61
,
51
–
66
(
2003
).
Google Scholar
Crossref
Search ADS
 
13.
B.
 
Sahu
,
G. S.
 
Mallick
, and
S. K.
 
Agarwalla
, “
Soluble complex potential model for heavy-ion collision: Resonance and fusion in C+12C12 reaction
,”
Nucl. Phys. A
https://doi.org/10.1016/j.nuclphysa.2003.08.010
 
727
,
299
–
318
(
2003
).
Google Scholar
Crossref
Search ADS
 
© 2007 American Association of Physics Teachers.
2007
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.