Question

X-rays with a wavelength of \(1.00 \mathrm{nm}\) are scattered off two small tumors in a human body. If the two tumors are a distance of \(10.0 \mathrm{~cm}\) away from the X-ray detector, which has an entrance aperture of diameter \(1.00 \mathrm{~mm}\), what is the minimum separation between the two tumors that will allow the X-ray detector to determine that there are two tumors instead of one?

Short answer

Answer: The minimum separation between the two tumors that will allow the X-ray detector to determine that there are two tumors instead of one is approximately \(1.22\times10^{-4}\,\mathrm{mm}\) or \(0.122\,\mathrm{\mu m}\).

Step-by-step solution

Using the Rayleigh Criterion

The Rayleigh criterion for the resolution of two sources is given by $$sin(\theta) = 1.22\frac{\lambda}{D}$$ where \(\theta\) is the angular separation between the two sources, \(\lambda\) is the wavelength of the light, and \(D\) is the diameter of the entrance aperture of the detector. In this problem, the values are given as: \(\lambda = 1.00\,\mathrm{nm}\), and \(D=1.00\,\mathrm{mm}\). Now let's plug these values into the formula and solve for the angular separation \(\theta\).

Calculating the Angular Separation

$$sin(\theta) = 1.22\frac{\lambda}{D} = 1.22\frac{1.00\,\mathrm{nm}}{1.00\,\mathrm{mm}}$$ Before solving this equation, we need to convert nm to mm. There are 1,000,000 nm in 1 mm, so: $$1.00\,\mathrm{nm} = \frac{1.00}{1000000}\,\mathrm{mm}$$ Substituting this back into our equation, we get: $$sin(\theta) = 1.22\frac{\frac{1.00}{1000000}\,\mathrm{mm}}{1.00\,\mathrm{mm}} = 1.22\frac{1.00}{1000000}$$ Now, we compute the angular separation by computing the inverse sin of the value. $$\theta = sin^{-1}(1.22\frac{1.00}{1000000}) \approx 1.22\times10^{-6}$$ The angular separation \(\theta\) has been calculated. Now we need to find the minimum separation between the tumors.

Finding the Minimum Separation Between the Tumors

The minimum separation between the two tumors can be found using the given distance \(R\) and the calculated angular separation \(\theta\): $$S=R\tan(\theta)$$ Here, \(R\) is the distance between the tumors and the detector, which is given as \(10.0\,cm = 100\,mm\). Plugging in the given values and solving for the separation: $$S = 100\,\mathrm{mm}\times\tan(1.22\times10^{-6}) \approx 1.22\times10^{-4}\,\mathrm{mm}$$ So the minimum separation between the two tumors that will allow the X-ray detector to determine that there are two tumors instead of one is approximately \(1.22\times10^{-4}\,\mathrm{mm}\) or \(0.122\,\mathrm{\mu m}\).

Key concepts

Rayleigh Criterion

The Rayleigh Criterion is fundamental to our understanding of resolution in imaging systems. Named after the British physicist Lord Rayleigh, this criterion scientifically establishes the conditions under which two light sources can be considered distinct from one another by an imaging system.

The equation canonically represented is: \[\sin(\theta) = 1.22\frac{\lambda}{D}\]
Here, \(\theta\) represents the angular separation of the two sources as perceived by the imaging system. The wavelength of light, represented by \(\lambda\), and the diameter of the aperture through which light enters, signified by \(D\), are the other critical components in this formula. When the equation is satisfied, two objects can be resolved as separate. This concept is essential not just in optics but also in technologies like X-ray diffraction, essential for the field of medicine and materials science.

Angular Separation

Angular separation describes the angle between lines of sight to two points from a given perspective. In the context of X-ray diffraction resolution, it typically refers to the angle that can be resolved by the detection equipment. The smaller the angle, the closer the two points or sources, such as tumors, can be while still being detected as distinct.

To deduce the angular separation using the Rayleigh Criterion, we compute: \[\sin(\theta) = 1.22\frac{\lambda}{D}\]
The angular separation is not simply the physical distance between two objects but rather how that physical distance is perceived from a specific viewpoint, which in our exercise is the X-ray detector's location. Understanding angular separation is key for professionals using imaging techniques to diagnose or inspect distinct objects or features.

Wavelength Conversion

Wavelength conversion is crucial in X-ray diffraction calculations and involves changing the units of wavelength to match other units in the equation. The Rayleigh Criterion requires consistent units for an accurate calculation.

In the given exercise, we convert the X-ray's wavelength from nanometers to millimeters for it to match the aperture's diameter's unit: \[1.00\,\mathrm{nm} = \frac{1.00}{1000000}\,\mathrm{mm}\]
This step is not to be overlooked, as using mismatched units can result in significant errors. In practical applications—be it physics, medicine, or engineering—such conversions enable precise calculations and are integral to advancements in imaging and measurement technologies.

Minimum Separation Calculation

The minimum separation calculation is a pivotal step in resolving the fine details of an image and differentiating between closely located objects. It gives us the smallest distance required between two entities to perceive them as separate under specific observational conditions.

In our exercise, this separation is found by capitalizing on the known distance from the tumors to the detector and the previously computed angular separation: \[S = R\tan(\theta)\]
Where \(S\) is the minimum separation needed between the tumors, \(R\) is their distance to the detector, and \(\theta\) is their angular separation. Calculating minimum separation is not just a theoretical exercise but a practical necessity, especially in sensitive applications such as medical diagnostics wherein the finest details can be life-critical.