In STM, an elevation of the tip above the surface being scanned can be determined with a great precision, because the tunneling-electron current between surface atoms and the atoms of the tip is extremely sensitive to the variation of the separation gap between them from point to point along the surface. Assuming that the tunneling-electron current is in direct proportion to the tunneling probability and that the tunneling probability is to a good approximation expressed by the exponential function \(e^{-2 \beta L}\) with \(\beta=10.0 / \mathrm{nm}\) determine the ratio of the tunneling current when the tip is \(0.500 \mathrm{nm}\) above the surface to the current when the tip is \(0.515 \mathrm{nm}\) above the surface.

Step by step solution

01

Calculate Tunneling Current for L = 0.500 nm

Calculate the tunneling current for L = 0.500 nm by using the given expression for the tunneling probability: \(I_1 \propto e^{-2 \beta L_1 }\) Plug in the given values, \(\beta = 10.0 \, \mathrm{nm^{-1}}\) and \(L_1 = 0.500 \, \mathrm{nm}\), into the expression: \(I_1 \propto e^{-2(10.0 \, \mathrm{nm^{-1}})(0.500 \, \mathrm{nm})}\)

02

Calculate Tunneling Current for L = 0.515 nm

Calculate the tunneling current for L = 0.515 nm using the same expression for the tunneling probability: \(I_2 \propto e^{-2 \beta L_2 }\) Plug in the given values, \(\beta = 10.0 \, \mathrm{nm^{-1}}\) and \(L_2 = 0.515 \, \mathrm{nm}\), into the expression: \(I_2 \propto e^{-2(10.0 \, \mathrm{nm^{-1}})(0.515 \, \mathrm{nm})}\)

03

Calculate the Ratio of Tunneling Currents

Now we have to find the ratio between the calculated tunneling currents: \(\frac{I_1}{I_2} = \frac{e^{-2(10.0 \, \mathrm{nm^{-1}})(0.500 \, \mathrm{nm})}}{e^{-2(10.0 \, \mathrm{nm^{-1}})(0.515 \, \mathrm{nm})}}\) To simplify, we can use the properties of exponentials: \(\frac{I_1}{I_2} = e^{\left[-2(10.0 \, \mathrm{nm^{-1}})(0.500 \, \mathrm{nm}) + 2(10.0 \, \mathrm{nm^{-1}})(0.515 \, \mathrm{nm})\right]}\)

04

Calculate the Final Result

We can now calculate the final result: \(\frac{I_1}{I_2} = e^{\left[-2(10.0 \, \mathrm{nm^{-1}})[(0.500 \, \mathrm{nm}) - (0.515 \, \mathrm{nm})]\right]}\) \(\frac{I_1}{I_2} = e^{\left[-2(10.0 \, \mathrm{nm^{-1}})(-0.015 \, \mathrm{nm})\right]}\) \(\frac{I_1}{I_2} = e^{\left[0.3\right]}\) Using a calculator, we get: \(\frac{I_1}{I_2} \approx 1.35\) Thus, the ratio of the tunneling current when the tip is 0.500 nm above the surface to the current when the tip is 0.515 nm above the surface is approximately 1.35.

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