Operational Tips: X-ray Coverage and Magnification Formulas and Calculator

X-ray Coverage and Magnification Formulas

Some of the first questions to answer in imaging applications are: what is the X-ray coverage on my object? What’s my maximum magnification possible? What size cone angle do I need? What formulas should I use in my calculations?

For X-ray imaging, it’s important to know how much area you need to illuminate with an X-ray beam. Once you have this information, you will be able to calculate your magnification factor and determine your enclosure size, detector requirements, and more.

Important Angles and Distances in X-ray Imaging

The diagram below illustrates the relevant distances and angles required to understand object coverage and magnification.

Geometry of a typical X-ray application showing the X-ray source, the Object being imaged, and the X-ray detector.

X-rays are generated on the target inside the tube, and exit through the X-ray window in a cone shape. This cone is defined by the cone angle (θ), originating from the X-ray spot on the target face inside the tube. There is some fixed distance between the X-ray spot on the target and the flange of the window assembly defined by the X-ray tube’s internal geometry, this is called the Spot to Window distance. This value can be obtained from Micro X-Ray for your specific tube. Typical Spot to Window distances for Microfocus tubes are under 10mm, and typical Spot to Window distances for Minifocus tubes are around 25mm.

Related to the Spot to Window distance is the Window to Object distance. This is the distance the object to be scanned is placed from the X-ray window. In cases where high geometric magnification is important, the object to analyze is often placed as close to the window as possible. The sum of these two distances is the Source to Object distance (sometimes called the Focus to Object Distance, or FOD), which is the distance between the X-ray spot inside the tube to the object being measured.

The final distance to consider is the Object to Detector distance. As the name implies, this is the distance between the object being scanned and the X-ray detector’s surface. The further away you are able to place your detector, the higher the magnification of the object will be.

X-ray Coverage Formulas

The area covered by the X-ray beam can be determined with some High School trigonometry, with just a touch of geometry thrown in. We know the angle of the radiation cone, and we can take a slice of the cone as a right angle triangle with an angle defined as θ/2, and the adjacent side equivalent to the distance from the focal spot to the object. With the angle defined by the X-ray tube, and the adjacent side defined by the distance from the X-ray spot on the target to the object, the radius of coverage of an object is defined as the length of the opposite side, and can be calculated as below (note the conversion from degrees to radians in the formula):

Calculation of the radius of the X-ray radiation coverage of an object

The coverage area on the detector may be calculated with the same formula, but swapping out the distance variables for Source to Detector distance:

Calculation of the radius of the X-ray radiation coverage of an X-ray detector

Geometric Magnification Formula

Geometric magnification is an easy concept with a (unnecessarily?) confusing name. This is simply the magnification of the object being imaged on the surface of the detector. For example, if you are measuring a 50μm feature, an 8x geometric magnification factor will blow that feature up to 400μm on the detector surface. Understanding your magnification and the size of the features you need to resolve will help you select the correct detector for your application.

Try this: hold your cell phone flashlight a fixed distance away from a flat table, the put a single finger half way between the camera LED and the table. The shadow of your finger on the table will be about 2x its real size. If you hold your phone in one spot and move your finger up and down, you can see the magnification change. As you move your finger closer to the table, the shadow shrinks; as you move your finger closer to your phone, the shadow grows. You can also change the magnification by holding your finger a constant distance from your phone, and moving the phone/finger pair closer and further from the table.

Geometric magnification of X-rays works identically to its counterpart in visible light and shadows. The magnification factor is just a simple ratio comparing the Source to Detector distance to the Source to Object distance:

Calculation of the Geometric Magnification ratio

X-ray Coverage Calculator

Try it yourself! Use the calculator below to check your coverage dimensions and magnification factor for your application or use case.

Practical Limits

It’s important to understand that your can’t increase your magnification infinitely, as there are some real-world constraints to contend with. We’ll cover these more in future articles, but for now keep in mind the following:

X-ray radiation intensity falls off with distance squared. This means the further away your source is from your detector, the fewer X-rays make it to the detector. At some point, the detector is just too far away to record any meaningful amount of X-ray flux.
The opposite is also true! If your detector is too close to the source, there may be so much flux that the detector floods and can’t produce a reliable image.
The resolution of an image is not only controlled by the magnification, but also the X-ray source. As a rule of thumb (and of course there are exceptions – there are always exceptions), the smallest image you’ll be able to resolve is around the same size as your X-ray spot. No matter how much you magnify an image, you’ll never resolve a 10μm feature with a 100μm spot size X-ray tube.

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