1.2.1 Appendix
Let us integrate an arbitrary function
from
to
.
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula/appendix/img04.gif)
(A1)
Defining
as
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula/appendix/img06.gif)
(A2)
where
– average value of a function
on the interval from
to
,
. Then equation (A1) can be rewritten as
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula/appendix/img09.gif)
(A3)
Considering a Taylor series expansion of the integrand (A3) in and neglecting
and higher order members, we get
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula/appendix/img11.gif)
(A4)
The second term in (A4) vanishes upon integration, therefore (A4) can be expressed as
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula/appendix/img12.gif)
(A5)
where the correction factor is
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula/appendix/img13.gif)
(A6)