The labels come from the older alphabetical labeling of shells starting with rather than using the principal quantum numbers 1, 2, 3, …. A more energetic x ray is produced when an electron falls into an shell vacancy from the shell; that is, an to transition. Similarly, when an electron falls into the shell from the shell, an x ray is created. The energies of these x rays depend on the energies of electron states in the particular atom and, thus, are characteristic of that element: every element has it own set of x-ray energies. This property can be used to identify elements, for example, to find trace small amounts of an element in an environmental or biological sample.
Characteristic X-Ray EnergyCalculate the approximate energy of a x ray from a tungsten anode in an x-ray tube. Strategy How do we calculate energies in a multiple-electron atom?
In the case of characteristic x rays, the following approximate calculation is reasonable. Characteristic x rays are produced when an inner-shell vacancy is filled.
Inner-shell electrons are nearer the nucleus than others in an atom and thus feel little net effect from the others. This is similar to what happens inside a charged conductor, where its excess charge is distributed over the surface so that it produces no electric field inside. It is reasonable to assume the inner-shell electrons have hydrogen-like energies, as given by.
As noted, a x ray is produced by an to transition. Since there are two electrons in a filled shell, a vacancy would leave one electron, so that the effective charge would be rather than.
For tungsten, , so that the effective charge is Solution gives the orbital energies for hydrogen-like atoms to be , where. As noted, the effective is Now the x-ray energy is given by where and Thus, Discussion This large photon energy is typical of characteristic x rays from heavy elements. It is large compared with other atomic emissions because it is produced when an inner-shell vacancy is filled, and inner-shell electrons are tightly bound.
Characteristic x ray energies become progressively larger for heavier elements because their energy increases approximately as. Significant accelerating voltage is needed to create these inner-shell vacancies. In the case of tungsten, at least Tungsten is a common anode material in x-ray tubes; so much of the energy of the impinging electrons is absorbed, raising its temperature, that a high-melting-point material like tungsten is required. Among these are the universal dental and medical x rays that have become an essential part of medical diagnostics.
See [link] and [link]. X rays are also used to inspect our luggage at airports, as shown in [link] , and for early detection of cracks in crucial aircraft components. An x ray is not only a noun meaning high-energy photon, it also is an image produced by x rays, and it has been made into a familiar verb—to be x-rayed. The denser the material, the darker the shadow.
Since x-ray photons have high energies, they penetrate materials that are opaque to visible light. The more energy an x-ray photon has, the more material it will penetrate. So an x-ray tube may be operated at The depth of penetration is related to the density of the material as well as to the energy of the photon.
The denser the material, the fewer x-ray photons get through and the darker the shadow. Thus x rays excel at detecting breaks in bones and in imaging other physiological structures, such as some tumors, that differ in density from surrounding material. Because of their high photon energy, x rays produce significant ionization in materials and damage cells in biological organisms.
Modern uses minimize exposure to the patient and eliminate exposure to others. Biological effects of x rays will be explored in the next chapter along with other types of ionizing radiation such as those produced by nuclei.
As the x-ray energy increases, the Compton effect see Photon Momentum becomes more important in the attenuation of the x rays. Here, the x ray scatters from an outer electron shell of the atom, giving the ejected electron some kinetic energy while losing energy itself. Chemical composition of the medium, as characterized by its atomic number , is not important here. Low-energy x rays provide better contrast sharper images.
However, due to greater attenuation and less scattering, they are more absorbed by thicker materials. Greater contrast can be achieved by injecting a substance with a large atomic number, such as barium or iodine. The structure of the part of the body that contains the substance e. Breast cancer is the second-leading cause of death among women worldwide. Early detection can be very effective, hence the importance of x-ray diagnostics. This could not only prevent limitation of x-ray energy on the length of system, but also would simplify the system structure.
In this section, we will derive the optimal position of the grating in a single absorption grating incoherent x-ray dark-field imaging system shown in figure 1 a. The distance from the source G to grating G 1 was z 1 , and the distance from the grating G 1 to detector D was z 2.
The intensity of each pixel on the detector will oscillate as G 1 moves perpendicular to the beam propagation direction over one grating period. In this paper, a single absorption grating incoherent imaging system is used to achieve x-ray dark-field imaging. Shipped from UKs. Substitution of equations 2 — 5 into 6 will yield. Citation lists with outbound citation links are available to subscribers only. They found that closer objects to phase grating resulted in a greater phase shift and higher system sensitivity.
Projection of G 1 will be represented by G 2 with a period of in the following discussion. Figure 1. The intensity of each pixel on the detector will oscillate as G 1 moves perpendicular to the beam propagation direction over one grating period. The intensity oscillation is named the phase stepping curve, which is a function of the relative position x g between the grating G 1 and detector figure 1 b.
The blue curve represents the phase stepping curve without an object, and the red curve represents the phase stepping curve with an object. The mathematical expression of the phase stepping curve can be written as. A beam of x-rays will be refracted with a refraction angle after passing through an object. The beam separation between the incident beam and refracted beam is on the detector plane figure 1 a , and then the phase stepping curve with an object will produce a phase shift with the phase stepping curve without an object figure 1 b.
Since the refraction angle of x-rays is usually in the order of micro radians, by the small angle approximation, we can obtain. When beam separation between the incident beam and refracted beam equals , the phase shift of the phase stepping curve is according to equation 3. Meanwhile, the geometric relations in figure 1 a can be expressed by. In the case of the same refraction angle caused by an object, larger phase shift will lead to easy detection of the refraction angle. Therefore, the system sensitivity can be defined as [ 22 ].
Substitution of equations 2 — 5 into 6 will yield. When the system length and period of the absorption grating are constant, the system sensitivity will reach its maximum at with value of. It can be seen from equation 8 that increased system length and reduced period of grating will increase the system sensitivity.
However, the intensity on the detector plane will reduce when the system length increases. The decrease in period will reduce the height of the absorption materials in absorption grating, thus the absorption of x-rays by grating will decrease and it will degrade the quality of images. In addition to refraction, a certain degree of scattering occurs when a beam of x-rays passes through an inhomogeneous object figure 2 a. A beam of x-rays will be split into several secondary beams after passing through an object, and each secondary beam will be refracted many times in the object due to inhomogeneity.
Thus, each secondary beam will have different diffusion paths, resulting in a certain degree of divergence in emergent light. The intensity of the scattering signal can be represented by standard deviation of angular distribution in emergent light. More inhomogeneous objects will result in greater probability of refraction and stronger scattering of x-rays.
Figure 2. X-rays are a scattered and b refracted after passing through an object. Therefore, we assume that and refraction angle are positively correlated:. For the single absorption grating incoherent imaging system, since the refraction signal reaches its maximum for absorption grating in middle of the system, the scattering signal will also reach its maximum at that position. However, the standard deviation is not easily measured directly. We use the visibility of phase stepping curve to characterize the standard deviation.
The visibility of the phase stepping curve is defined as. To eliminate the adverse effects of uneven absorption grating on visibility, the visibility should be normalized by. When a beam of x-rays is incident on a homogeneous object, it also will be split into several secondary beams figure 2 b. These secondary beams will travel in a straight line without being refracted inside the object due to the internal homogeneity of the object, and the emergent light will still be a beam of parallel light. In general, both refraction and scattering will occur simultaneously after x-rays are incident on an object.
However, the intensity of the refraction and scattering signal differ from one object to another. For homogeneous objects, the refraction signals are more dominant and characterized by the refraction angle. By comparison, the scattering signals for inhomogeneous objects are more dominant and characterized by relative visibility.
The relationship between the scattering signal and relative visibility is: the larger the scattering signal, the smaller the relative visibility. The aforementioned theory was validated by experiments using the experimental setup shown in figure 1 a. For each step, an image with the object in a beam and another image without the object in a beam were recorded. Three images were taken at each step in order to reduce the noise.
The results are shown in figure 3.