Barrier Tunneling — Question 3

Question 3

A thickness change can be inferred from a transmission ratio

Two otherwise identical opaque barriers have widths a and $a+\Delta a$, with $\Delta a>0$. Their decay constant is $\kappa=8.00\,\mathrm{nm}^{-1}$. Use $T(a)\simeq Ce^{-2\kappa a}$ with the same width-independent prefactor C. Transmission decreases by a factor of 100.

1. Derive the relation between the transmission ratio and $\Delta a$.

2. Find the added width and the logarithmic sensitivity to thickness.

Question 3 - Solution

A thickness change can be inferred from a transmission ratio

(a) Method and calculation

$$\frac{T(a+\Delta a)}{T(a)}=e^{-2\kappa\Delta a}=1/100.$$$$\boxed{\Delta a=\frac{\ln100}{2\kappa}=.287823\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\ln T=\ln C-2\kappa a\quad\Rightarrow\quad\boxed{\frac{d\ln T}{da}=-2\kappa=-16.0\,\mathrm{nm}^{-1}}.$$$$\boxed{\delta T/T\simeq-2\kappa\,\delta a}\qquad(|2\kappa\delta a|\ll1).$$

Physical check

The ratio removes the unknown common prefactor. The differential formula is for small changes and must not replace the logarithm for a hundredfold change.