Question

In the second type of helium-ion microscope, a 1.2-MeV ion passing through a cell loses 0.2 MeV per $\mu$m of cell thickness. If the energy of the ion can be measured to 6 keV, what is the smallest difference in thickness that can be discerned? (a) 0.03 $\mu$m; (b) 0.06 $\mu$m; (c) 3 $\mu$m; (d) 6 $\mu$m.

Short answer

The smallest thickness difference discernible is 0.03 $\mu \text{m}$ (option a).

Step-by-step solution

Identify the Problem

We need to find the smallest thickness difference that can be detected, given that the energy loss per micrometer ($\mu \text{m}$) is 0.2 MeV and the smallest measurable energy difference is 6 keV.

Convert Units

First, convert the smallest measurable energy difference from keV to MeV. Since 1 MeV = 1000 keV, 6 keV = 0.006 MeV.

Apply the Relation Between Energy Loss and Thickness

The energy loss per unit thickness is 0.2 MeV per $\mu \text{m}$. Therefore, the equation relating energy loss $\mathrm{\Delta }E$ to thickness $\mathrm{\Delta }x$ is $\mathrm{\Delta }E=0.2\times \mathrm{\Delta }x$.

Solve for Thickness Difference

Rearrange the equation to solve for thickness change: $\mathrm{\Delta }x=\frac{\mathrm{\Delta }E}{0.2}$. Substitute $\mathrm{\Delta }E=0.006$ MeV into the equation to get $\mathrm{\Delta }x=\frac{0.006}{0.2}=0.03\mu \text{m}$.

Key concepts

Energy Loss

In the context of helium-ion microscopy, energy loss refers to the reduction in energy of helium ions as they pass through a material. This phenomenon occurs due to interactions between the ions and atoms in the material.
These interactions can cause the ions to lose energy through processes such as electron excitation and atom displacements. In our example, a helium ion passing through a cell experiences an energy loss of 0.2 MeV per micrometer of cell thickness.
Understanding energy loss is crucial, as it impacts the resolution and effectiveness of the helium-ion microscope. This loss allows for precise measurements of material thickness and helps visualize structures at a microscopic level.

Micrometer Measurement

In scientific measurements, particularly in helium-ion microscopy, precision is key. The micrometer (μm) is a unit of length in the metric system, equal to one millionth of a meter. This tiny unit is perfect for measuring the thickness of cells or layers in various samples.
In our problem, it's crucial to determine how thick a sample must be for the energy loss to be detectable. Since the energy loss is tied closely to the thickness of the cell, micrometers serve as an ideal unit for these small-scale measurements.
This capability allows scientists to discern incredibly small differences in thickness, enhancing the accuracy and applicability of the helium-ion microscope in scientific research.

Energy Conversion

In the energy measurements of helium ions, conversion between different units is essential. An understanding of energy conversion helps bridge quantities, making the analysis accurate and meaningful.
Here, the energy is initially measured in keV (kiloelectronvolts) and is converted to MeV (megaelectronvolts) to match the units used for energy loss calculations. Recall that 1 MeV is 1000 times greater than 1 keV. Thus, when needing a precise measurement like 6 keV, conversion to 0.006 MeV is necessary.
This conversion ensures consistency across calculations, providing reliable results, and helps assure that energies are precisely compared and evaluated.

Ion Energy Measurement

Ion energy measurement is vital to exploit the full analytical power of the helium-ion microscope. Accurate measurement of energy helps detect small differences in material structures and thicknesses.
The helium-ion microscope allows measurement down to the keV scale, detecting very small changes in ion energies. This sensitivity ensures high-resolution imaging, contributing to the detailed insights available from this technology.
With such precise capability, it becomes possible to measure energy discrepancies that correlate with minute differences in thickness. This precision is imperative for applications ranging from materials science to biology, where details at the cellular and molecular level can be both subtle and significant.