Question

How is energy deposited in the interaction with matter of each of the following? photons, electrons, alphas, neutrons. Is the full energy of the radiation deposited? If not, where does the rest go? Is the energy deposited with high linear energy transfer or low linear energy transfer? Is the radiation track sparse or dense?

Answer 1

The behavior of photons in matter is completely different from that of charged particles. In particular, the photon’s lack of an electric charge makes impossible the many inelastic collision with atomic electrons so  characteristic of charged particles. For this kind of radiation the most important mechanism of interaction are:

a)    Photoelectric effect
b)    Compton and Rayleigh scattering
c)    Pair production

As consequence of such kind of interactions a photon that interacts with the target is completely removed from the incident beam, in other words a beam of photons that cross a medium is not degraded in energy but only attenuated in intensity. Moreover, due to the smallest cross section of all this kind of reactions,x?ray or ?-ray are many times more penetrating than charged particles. The attenuation of the incident beam is exponential with the thickness of the absorbing medium and can be expressed by the following relation:

I(x)= I0 exp (-x?l)

where ?l is the linear attenuation coefficient, I0 is the incident beam intensity and x the thickness. The linear attenuation coefficient is related to the cumulative cross section by the relation:

?l =?A ?tot

where ?A is the number of atoms per unit of mass and ?tot is the total cross section. The total or cumulative cross section ?tot is the sum of all the cross sections of the interactions mentioned above. A plot of this quantity is shown in Figure 1.6 where the different components have been highlighted.

In photoelectric absorption, a photon disappears being absorbed by an atomic electron. The process results in ionization by subsequent ejection of the electron from the atom. The energy of the liberated electron is the difference between the photon energy and the energy needed to extract the electron from the atom i.e. the binding energy of the electron. The recoil momentum is absorbed by the nucleus to which the ejected electron was bound. If the resulting photoelectron has sufficiently enough of kinetic energy, it may be a source of a secondary ionization occurring along its trajectory, and in the case of the semiconductor material, it may create further e-h pairs. If the electron does not leave the detector the deposited energy corresponds to the energy possessed by the incident photon. This feature of the photoelectric effect allows calibrating the gain of the detector chained with its readout system if the energy required to create a single e-h pair is known. The range R of the electron having the kinetic energy E is of the order of some micrometers, as given by the follow equation:

R[um] = 40.8 10^(-3) x ( E[keV] )^1.5

Thus the cloud of generated charge is confined close to the photon absorption point. The clear image may be smeared by escape photons, which can leave the detector volume leading to less amount of energy deposited. These photons are actually the fluorescence photons emitted by de-exciting atoms. Photons of fluorescence radiation are emitted by atoms after the ejection of a deep shell (K, L) electron. The incident photon creates a vacancy in the shell, thus leaving an atom in an excited state. Then, the vacancy can be filled by an outer orbital electron, giving rise to the emission of the characteristic X-rays photons of the fluorescence radiation. The missing energy, which is conveyed by the escape photons leads to, so called escape peaks in the measured energy spectrum. Photon interaction coefficient for photoelectric absorption depends strongly on the atomic number of the absorbing material. The relevant cross section increases roughly as Z^3. For silicon, the photoelectric effect is a dominant process for photon energies below 100 keV.

Cross sections of photons in Carbon (a) and Lead (b) in barns/atom; 1barn=10-24 cm2.

The Compton scattering, instead of photoelectric effect, involves the free electrons. In matter of course, the electrons are bound to an atom; however, if the photon energy is high with respect to the binding energy, this latter energy can be ignored and the electrons can be treated as essentially free. When Compton scattering occurs, the electron is scattered away in conjunction with a new photon that have a lower energy than the incoming one. In Rayleigh scattering the photon interact with the whole atom and the only effect of this interaction is a deflection of the incoming photon; it does not participate to the absorption and for most purposes can be neglected.

At very high energy another effect starts to be relevant: the pair production. In this process the photon interacts with an electron or a nucleus producing a positron-electron pair. In order to produce the pair the photon must have at least an energy of 1.022 MeV. In Figure 1, with knuc and ke, are shown the two components of the pair production cross section, respectively for the interaction with nuclei or electrons. Another possible interaction, but usually negligible compared to the previous ones is the Photonuclear reaction, in this case the photon interact directly with the nucleus. The related cross section is shown in Figure 1 in dotted line (?g.d.r.). The above cross section in barns/atom (1barn = 10-24 cm2, approximately the section of an uranium nucleus) expresses the probability of an interaction. A more suitable quantity, often used to characterize the absorption of a photon shower, is the mass attenuation coefficient. The mass attenuation coefficient is defined as:

?m=?A ?tot/?

where ? is the density of the material. Figure 2 shows the mass attenuation coefficient of the silicon with the indication of its different components.

The neutron has a mass of 1.675 × 10-27 kg and a half-life of 885.7 ± 0.8 second with no charge. Since a neutron has no charge and does not interact with orbital electrons, it can easily pass through the target nucleus to cause various reactions. However, not all the possible reactions contribute to the dose distribution equally. The degree of importance of the reactions depends on the target nucleus and the energy of neutron. The hydrogen, carbon, oxygen and nitrogen atoms which compose the most of human soft tissue (H,10.5%; C, 22.6%; O, 63.7%; N, 2.34%) interact differently with neutron [8]. For instance, the elastic collision is related to the atomic weight, and thus hydrogen does not have elastic scattering interaction. Therefore, with the higher hydrogen content, the dose absorbed in fat exposed to a neutron beam is about 20% higher than in muscle, because the dose deposited in tissue from a high energy neutron beam is contributed predominantly by recoil process [4]. The neutron beam is divided into three categories by energy: below 0.5 eV as thermal neutrons, above 0.5 eV up to 10 keV as intermediate and above 10 keV as fast neutrons. Low energy neutron results neutron capture interactions to cause ?-rays while high energy neutron interacts elastic scattering in the matter. In a dosimetrical aspect, the dose in the human body is dominated by the contribution of recoil protons resulted from elastic scattering of hydrogen nuclei above 10 keV below which ?-rays resulted from thermal neutron capture interaction dominates.

1) Elastic scattering (n,n): Elastic scattering occurs when a neutron strikes a nucleus and rebounds elastically (Fig. 2). The amount of kinetic energy transferred depends upon the angle of impact and hence the direction of motion of the neutron and nucleus after the collision. When a neutron hits a nucleus, it may rebound completely or bounce off in different directions. In the former interaction, the kinetic energy is conserved completely. In the latter interaction, the large amount of kinetic energy is transferred to the nucleus so that the recoiling nucleus becomes ion pairs to lose energy through the excitation and ionization. This interaction is important at lower energy region up to 10 MeV and not effective above 150 MeV. The fast neutrons are thermalized by elastic scattering interactions. The elastic scattering interaction is related closely with the atomic weight of the target. The relationship of a neutron mass 1 with the initial kinetic energy E0 hits a nucleus mass A to the final kinetic energy E1 is: [(A - 1)/(A + 1)] ? (E1/E0) ? 1. When a neutron hits the nucleus of hydrogen (A = 1), the energy spectrum of scattering neutron varies from 0 to E0whereas the spectrum varies from 0.176E0 to E0 if it hits the carbon. In average, the kinetic energy of a neutron E encountering a nucleus of atomic weight A, the energy loss is 2EA/(A + 1)2. Thus to reduce the energy of neutrons with the fewest number of elastic collisions target nuclei with small A should be used. For example, to reach to the thermal equilibrium (0.025 eV: most probable energy for neutrons at 293°K), hydrogen atom needs only 18 collisions whereas carbon atom needs 110 collisions.

2) Inelastic scattering (n,n'): The interactions of fast neutrons of the therapeutic range energy are dominated by this interaction, which in turn causes the emission of photons, neutrons, and charged particles [4]. Non-elastic scattering differs from inelastic scattering by only that the secondary particles are not neutrons (e.g., 12C(n,?)9Be E? = 1.75 MeV). A neutron may strike a nucleus and be absorbed momentarily, which is forming an excited state nucleus releasing radiation eventually. When a neutron hits and enters into a nucleus, the nucleus is excited into an unstable condition. After then, the excited nucleus returns to the stable state by emitting ?-ray (e.g., 14N(n,n')14N E?? = ~10 MeV). Thus, the average energy loss depends on the energy levels within the nucleus. If all the excited states of the nucleus are too high, inelastic scattering does not occur so that this interaction happens only when high energy neutrons interact with heavy nuclei. For the hydrogen, its nucleus does not have the excited state thus only elastic scattering happens. A variety of emissions, however, may follow if the energy of neutron and the atomic mass are high enough. If more than one neutron is emitted, nuclear fission will occur.

3) Neutron capture (n,?): The neutron may be captured by the nucleus of absorbing matter, and only the absorbing atom emits ?-ray. The interaction is the same as nonelastic scatter, but this occurs only at the low energy levels. This process leads to disappearance of neutron. The result of this interaction is an isotope of the same element as the original nucleus with the increased mass number. The probability of this interaction is inversely proportional to the energy of the neutron. The scattered neutron lost its energy is captured by a specific nucleus so that the probability is called as "Capture Cross Section." The probability of a specific nucleus capture is differed from target nucleus and its energy. It varies from almost 0 for 4He, 0.0035 for 12C, 0.33 for 1H, and 1.70 for 14N Barns (10-28 m2). The neutron capture interaction accounts for a significant fraction of the energy transferred to tissue by neutrons in the low energy ranges (e.g., 1H(n,?)2H Q = 2.2 MeV E? = 2.2 MeV and 14N(n,p)14C Q = 0.626 MeV Ep = 0.58 MeV). The hydrogen capture reaction is the major contributor to dose in the tissue from thermal neutrons.

2. Interactions of proton

The proton has a mass of 1.67-27 kg with a positive charge having a half-life of 1035 years. The advantage of proton beam over the photon comes from the both of the high energy and low energy interactions. For the protons with a higher energy, the several processes of energy transfer such as direct inelastic collisions by proton, inelastic collisions by delta rays, and elastic and non-elastic nuclear reactions may occur. The secondary particles are also important, because they can be scattered to a considerable range. By these interactions, it shows characteristic depth dose curve of low dose at the entry region and high dose at a specific depth. Unlike photons or neutron, it has a short build up region followed by a maximum energy deposition region near the end (the Bragg peak). As they move through target material, they interact by electronic or nuclear reaction. The electronic interactions are ionization and excitation of atomic electrons whereas the nuclear reaction interactions are Coulomb scattering, elastic collision and non-elastic nuclear collision.

1) Coulomb scattering: At the entry region, the primary protons lose their energy mainly by Coulomb interactions with the outer shell electrons to cause excitation or ionization (Fig. 3). Inelastic collision may occur without loss of energy in this region. But the energy loss per interaction is small so that there is no significant deflection of proton at this area. The range of secondary electrons is less than 1 mm and the most of the dose is absorbed locally. As the protons travel through tissue, the energy get lowered so that the number of ionization events rapidly increases and reaching its apex known as a Bragg peak. Shortly after the Bragg peak, the number of ionizations quickly diminishes to zero. The energy loss of the proton beam is constantly related with the elementary component of the body which is known as the ping power.

the target density, Z2,, me, v, Z1, and L are referred as target atomic number, the electron the mass, the velocity, the charge and the stopping function. As the energy lowers, the velocity lowers to 0, causing a peak (the Bragg peak) to occur. The width of peak depends on range straggling in medium and initial energy spectrum while the peak to plateau ratio depends on the width of energy spectrum. However, the Bethe-Bloch model becomes invalid at the low energies (<10 MeV/A).

Usually, the values of stopping powers are obtained from experiments and simulations and are similar to cross-sections in the sense that they are natural properties of the materials [9]. The stopping powers for various materials are given in International Commission on Radiation Units and Measurements report 49 [10]. Mass stopping power is energy loss per unit path length in g/cm2. Therefore, the low atomic number (Z) materials have the greater mass stopping power than high-Z materials. For example, the stopping power for a 1 MeV proton is 25.4 MeV/kg/m2 in water and 6.39 MeV/kg/m2 in lead. High Z materials scatter the proton at a larger angle without much energy loss so that those materials are used to spread out the beam.

2) Non-elastic scattering (p,d,p',n,?): Non-elastic interactions with protons occur at higher energies and produce secondary particles which usually stop in the vicinity of the interaction and have a relatively high biological effectiveness. Primary protons are lost in non-elastic nuclear interactions. The contributions to the absorbed dose by non-elastic scattering are about 5% in 100 MeV, 10% in 150 MeV, and 20% in 250 MeV [11]. With 250 MeV energy, about 20% of the incident protons show a non-elastic nuclear interaction with the target nuclei to generate charged particles such as proton (p,p), deuteron (p,d), alpha particles (p,?) or recoil protons (p,p'). These secondary products are absorbed locally. The interaction may generate non-charged particle such as neutron (p,n) or ?-rays (p,?) (i.g., 12C(p,?)13N). These non-charged particles can pass a relatively longer ranged to be absorbed by surroundings (Fig. 3). Numerous neutrons are produced by nuclear interaction of protons, thus the neutron induced interactions should be examined in detail for the actual proton therapy. Behind the distal endpoint of the Bragg peak, the absorbed dose by the n-secondary particles due to neutrons went up to about 70-80% of the total absorbed dose, the contributions to which were the n-secondary protons produced by the (n,p) reaction and the n-secondary alpha particles (n,?). Therefore, though the dose contribution of low energy protons less than a few hundred keV is about 5%, the biological implications of this interaction are not that simple. The endpoint processes transfer electrons mainly through ionization collisions to generate many ions and radicals.

3) Electron exchange: As the proton slows down, it causes increased interaction with orbital electrons to make maximum interaction at the end of range. Finally, at the end of interactions, the energy is lowered below the proton's stopping power so that they exchange electrons with hydrogen atoms of the target. This is called charge-changing process [12,13]. Both electron capture process p + H2O ? H + H2O+ and electron loss process H + H2O ? p+e- + H2O occur. The ions and radicals induce biological damage in the bio-cells effectively despite of the slight dose contribution [14]. These electrons can move only a few micrometers at the most, which is almost the same scale as a chromosome in the cell nucleus [14]. The advantage of low energy proton beams are from the spatial distribution of ions and radicals which may form clusters and attack bio-molecules such as DNA [15]. Thus, the RBE increases in a depth beyond the endpoint of the Bragg peak [16]. Matsuzaki et al. [14] reported that the effective dose for each secondary particle by multiplying the absorbed dose by the factor, and the ratio of the effective dose was found to be increased to 20:140:180 for electrons, protons and alpha particles, respectively. According to the radiation weighting factors defined in the International Commission on Radiological Protection (ICRP) 2007 recommendation [17], these factors for electrons, protons, and alpha particles are 1, 2, and 20, respectively. This result suggests that the proton dosimetry beyond the Bragg peak is difficult to evaluate accurately.

3. Interactions of heavy charged particles

1) Electron collision: Though the charged particles are still losing their energy by the interactions with atomic electrons at the entry region, the angular and energy straggling is much lower than protons as the heavy particles have much larger mass [18]. In the energy interval of therapeutic interest, heavy charged particles show diverse interactions from pure fragmentation at high energy levels to Rutherford scattering and inelastic scattering interactions in low energy levels depending on the nuclear structure of target matter (Fig. 4). Except at low velocities, the heavy charged particles lose a negligible amount of energy in nuclear collisions. Also, heavy charged particles colliding with electrons will lose only a small fraction of their energy per collision (usually about 25 eV, but on the average 100 eV and at most >> 4 mE/M) [19]. Thus, the heavy particles have much larger relative dose in the Bragg peak and small lateral scattering than protons and they offer an improved dose conformation as compared with photon and proton beam. Another characteristic of heavy charged particle beam is that by the lower-charge fragments, they produce considerable dose tails after the Bragg peak.

2) Nuclear collision: Nuclear interactions of heavy particles are occurred by either gazing or head-on collisions. Unlike gazing collisions, the head-on collisions occurs less frequently but these interactions transfer large energy to cause projectile breaks into many small pieces, and no high-velocity fragment survives. Heavy ions having gazing interactions with nuclei may result in fragmentation of the incident ions or target nucleus. The resultant charged fragments of the target nuclei that interpenetrate undergo significant interactions. In the interaction, evaporated nucleons (changing the characteristics of the nucleons) and light clusters are produced. The importance of the fragments depends upon how it affects the absorbed dose distribution in linear energy transfer (LET) which in turn depends upon the nature of the medium, ion type and its energy [20]. Further, these effects increase as a function of the beam energy. For example, for a 12C beam at 200 MeV/n about 30% of the primary carbon ions are involved in nuclear reactions and do not reach the Bragg maximum at about 8.6 cm depth in water, whereas at 400 MeV/n only the 30% of the primary particles reach the Bragg peak at about 27.5 cm depth in water since 70% of 12C are absorbed by nuclear reactions [21]. The interactions may serve advantages for the therapy verification with the similar mechanisms as positron emission tomography (PET) imaging. Verification of dose delivery to the tumor is possible by taking advantage of the property of positrons in producing 511 keV annihilation gamma photons [22,23]. These isotopes travel almost the same velocity as the main beam and stop in almost the same place and they emit gamma rays to be detected in a conventional PET scanner. As a consequence the location of the spread out Bragg peak and therefore the high dose treatment volume is visualized [24]. The production yield of some positron emitter nuclide such as 11C and 10C has been studied using GEANT4 (GEometry ANd Tracking-4) computer code [25]. The secondary positron emission is exploited for visualizing the dose distribution during irradiation in hadron therapy and consequently allowing a safer irradiation of the tumor volume by supplying sufficient quality for monitoring in head and neck cancer treatments [26].

3) Nuclear fusion: While the fragments lighter than 12C, such as for example 11B, are mainly produced by projectile and target fragmentation, the occurrence of fragments heavier than 12C is also very significant because they are found in a considerable amount with rather low energies which may contribute to the increase of the RBE of the carbon beam (Fig. 5) [21]. The interactions such as complete fusion and/or break-up fusion produce these fragments as evaporation residues. For example, at the low energy threshold of about 5 MeV/n, a complete fusion interaction of 12C + 12C ? 24Mg may occur. As a result, many radioactive and stable isotopes may be produced in interaction between heavy ion beam and the elements of soft tissues [27]. In a calculation model for the carbon therapy, the 12C interact with the elements 16O, 12C, 14N and 1H, which are the most abundant nucleus in fat and muscle tissue, produce isotopes such as: 12C + 12C ? 6Li + 18F, 12C + 12C ? 5Li + 19F, 12C + 12C ? 4Li + 20F, 12C + 14N ? 24Na + (?,d, p), and 12C + 16O ? 24Na + (?,d, p). The 18F and 24Na have a half-life of 110 minutes and 15 hours respectively. Thus it is inferred that after carbon ion therapy the patient must be quarantined [27]. Still, thorough studies of the nuclear interactions of the heavy particles in the therapeutic energy range are needed before their clinical applications.

2. Heavy charged particles

The heavy charged particle beams show very diverse RBEs as they travel through the matter. While the protons are producing relatively localized biological damage according to the dose, the heavy particles are producing many ion tracks around the path so that they cause locally multiplied damages [44]. Thus simple dose scaling or plotting of dose from a reference depth dose is not appropriate. To understand the models of heavy charged particles, it is needed to understand ion track and track structures. Actually, it is the track structure rather than LET which implicate radiobiologic effect for the heavy particle beams.

The expression 'track structure' refers to an 'event-by-event' description of the physical processes following irradiation, represented as a matrix Sn (i,X,E), where i is the interaction type, X is its position, and E is the deposited energy [44]. In comparison with the photon beam, which is sparsely ionizing radiation, the particle beams produce dense ionization tracks with more 'clustered damage' [45]. The mechanism of producing clustered damage is that the particle beams have very complex track structures characterized by energy depositions along with the primary particle path and radially projecting secondary beams so called 'delta-rays'. A delta ray is characterized by very fast electrons produced in quantity by alpha particles or other fast energetic charged particles knocking orbiting electrons out of atoms. Collectively, these electrons are defined as delta radiation when they have sufficient energy to ionize further atoms through subsequent interactions on their own. Delta rays appear as branches in the main track. These branches will appear nearer the start of the track of a heavy charged particle, where more energy is imparted to the ionized electrons. With the assumption that biological damage is determined by locally ejected ?-electrons, increased effectiveness of particle radiation can be described by a combination of the photon dose response and microscopic dose distribution. For this purpose, a local effect model (LEM) has been suggested which showed that the predictions of the LEM are in good agreement with clinical data [46,47]. The input parameters of this model are radial dose distribution, size of cell nucleus and X-ray sensitivity (?/? ratio) [48] to determine local damage probability of microscopic track structure by calculating the dose in small compartments by the reference of X-rays.

When a charged particle penetrates in matter, it will interact with the electrons and nuclei present in the material through the electromagnetic force. If the charged particle is a proton, an alpha particle or any other charged hadron (discussed in Chap. 1), it can also undergo a nuclear interaction and this will be discussed in Sect. 2.5. In the present section we ignore this possibility. If the particle has 1 MeV or more as energy, as is typical in nuclear phenomena, the energy is large compared to the binding energy of the electrons in the atom. To a first approximation, matter can be seen as a mixture of free electrons and nuclei at rest. The charged particle will feel the electromagnetic fields of the electrons and the nuclei and in this way undergo elastic collisions with these objects. The interactions with the electrons and with the nuclei present in matter will give rise to very different effects. Let us assume for the sake of definiteness that the charged particle is a proton. If the proton collides with a nucleus, it will transfer some of its energy to the nucleus and its direction will be changed. The proton is much lighter than most nuclei and the collision with a nucleus will cause little energy loss. It is easy to show, using non-relativistic kinematics and energy– momentum conservation, that the maximum energy transfer in the elastic collision of a proton of mass ‘m’ with nucleus of mass ‘M’ is given by (see Sect. 7.3, Eq. 7.1): Emax = 1 2 mv2 4 mM (m + M) 2 If the mass of the proton m is much smaller than the mass of the nucleus M, we therefore have Emax ? 1 2 mv2 4 m M (m << M) In the limit that the mass of the nucleus goes to infinity, no energy transfer is possible. In a collision with a nucleus the proton will lose little energy, but its direction can be changed completely; it can even bounce backwards. In collisions with electrons, on the other hand, a large amount of energy can be transferred to the electrons, but the direction of the proton can only be slightly changed. Indeed, there is a maximum possible kinematical angle of deviation in such collisions. It needs a relativistic calculation to derive this angle. As a result, most of the energy loss of the proton is due to the collisions with the electrons, and most of the change of direction is due to the collisions with the nuclei. A proton, and more generally any charged particle, penetrating in matter leaves behind a trail of excited atoms and free electrons that have acquired some energy in the collision. The energy distribution of these electrons is dn dE ? 1 E2 Most of these electrons have only received a very small amount of energy. However, some of the electrons acquire sufficient energy to travel macroscopic distances in matter. These high-energy electrons are sometimes called ?-electrons. These have sufficient energy themselves to excite or ionise atoms in the medium. This type of energy loss due to the interaction of the charged particle with electrons is often referred to as ‘energy loss due to ionisation’. This is strictly speaking not correct since many atoms are only brought to an excites state, illustrates the passage of a charged particle in matter and shows some of the ionisation electrons.trajectory When discussing the biological effects of radiation the term ‘Linear Energy Transfer’ (LET) is often used to refer to the energy loss of charged particles. The linear energy transfer is defined as the amount of energy transferred, per unit track length, to the immediate vicinity of the trajectory of the charged particle. For heavyand low-velocity particles, the energy loss per unit track length and the LET are the same. For light and fast particles, however, the two quantities differ considerably. Part of the energy loss of an electron of several MeV is used to eject energetic ?-electrons from the atoms in the medium. These energetic electrons do not deposit their energy in the immediate vicinity of the track and therefore do not contribute to the LET. The energy loss of a high-energy charged particle in matter due to its interactions with the electrons present in the matter is given by the Bethe-Bloch equation: dE dx = ? Znucl Ar (0.307 MeVcm2/g) Z2 ?2 1 2 ln2mec2?2? 2Tmax I2 ? ?2 ? ?(?) 2

Diffusion cloud chamber with tracks of ionizing radiation (alpha particles) that are made visible as strings of droplets

In dosimetry, linear energy transfer (LET) is the amount of energy that an ionizing particle transfers to the material traversed per unit distance. It describes the action of radiation into matter.

The secondary electrons produced during the process of ionization by the primary charged particle are conventionally called delta rays, if their energy is large enough so that they themselves can ionize.^[3] Many studies focus upon the energy transferred in the vicinity of the primary particle track and therefore exclude interactions that produce delta rays with energies larger than a certain value ?.^[1] This energy limit is meant to exclude secondary electrons that carry energy far from the primary particle track, since a larger energy implies a larger range. This approximation neglects the directional distribution of secondary radiation and the non-linear path of delta rays, but simplifies analytic evaluation.^[4]

In mathematical terms, Restricted linear energy transfer is defined by

{\displaystyle L_{\Delta }={\frac {{\text{d}}E_{\Delta }}{{\text{d}}x}}},

where {\displaystyle {\text{d}}E_{\Delta }} is the energy loss of the charged particle due to electronic collisions while traversing a distance {\displaystyle {{\text{d}}x}}, excluding all secondary electrons with kinetic energies larger than ?. If ? tends toward infinity, then there are no electrons with larger energy, and the linear energy transfer becomes the unrestricted linear energy transfer which is identical to the linear electronic stopping power.^[1] Here, the use of the term "infinity" is not to be taken literally; it simply means that no energy transfers, however large, are excluded.

Linear energy transfer is best defined for monoenergetic ions, i.e. protons, alpha particles, and the heavier nuclei called HZE ions found in cosmic rays or produced by particle accelerators. These particles cause frequent direct ionizations within a narrow diameter around a relatively straight track, thus approximating continuous deceleration. As they slow down, the changing particle cross section modifies their LET, generally increasing it to a Bragg peak just before achieving thermal equilibrium with the absorber, i.e., before the end of range. At equilibrium, the incident particle essentially comes to rest or is absorbed, at which point LET is undefined.

Since the LET varies over the particle track, an average value is often used to represent the spread. Averages weighted by track length or weighted by absorbed dose are present in the literature, with the later being more common in dosimetry. These averages are not widely separated for heavy particles with high LET, but the difference becomes more important in the other type of radiations discussed below.^[4]

Beta particles[edit]

Electrons produced in nuclear decay are called beta particles. Because of their low mass relative to atoms, they are strongly scattered by nuclei (Coulomb or Rutherford scattering), much more so than heavier particles. Beta particle tracks are therefore crooked. In addition to producing secondary electrons (delta rays) while ionizing atoms, they also produce bremsstrahlungphotons. A maximum range of beta radiation can be defined experimentally^[5] which is smaller than the range that would be measured along the particle path.

Gamma Rays[edit]

Gamma rays are photons, whose absorption cannot be described by LET. When a gamma quantum passes through matter, it may be absorbed in a single process (photoelectric effect, Compton effect or pair production), or it continues unchanged on its path. (Only in the case of the Compton effect, another gamma quantum of lower energy proceeds). Gamma ray absorption therefore obeys an exponential law (see Gamma rays); the absorption is described by the absorption coefficient or by the half-value thickness.

LET has therefore no meaning when applied to photons. However, many authors speak of "gamma LET" anyway,^[6] where they are actually referring to the LET of the secondary electrons, i.e., mainly Compton electrons, produced by the gamma radiation.^[7] The secondary electrons will ionize far more atoms than the primary photon. This gamma LET has little relation to the attenuation rate of the beam, but it may have some correlation to the microscopic defects produced in the absorber. It must be noted that even a monoenergetic gamma beam will produce a spectrum of electrons, and each secondary electron will have a variable LET as it slows down, as discussed above. The "gamma LET" is therefore an average.

It is identical to the retarding force acting on a charged ionizing particle travelling through the matter.^[1] By definition, LET is a positive quantity. LET depends on the nature of the radiation as well as on the material traversed.

A high LET will attenuate the radiation more quickly, generally making shielding more effective and preventing deep penetration. On the other hand, the higher concentration of deposited energy can cause more severe damage to any microscopic structures near the particle track. If a microscopic defect can cause larger-scale failure, as is the case in biological cells and microelectronics, the LET helps explain why radiation damage is sometimes disproportionate to the absorbed dose. Dosimetry attempts to factor in this effect with radiation weighting factors.

Linear energy transfer is closely related to stopping power, since both equal the retarding force. The unrestricted linear energy transfer is identical to linear electronic stopping power, as discussed below. But the stopping power and LET concepts are different in the respect that total stopping power has the nuclear stopping power component,^[2] and this component does not cause electronic excitations. Hence nuclear stopping power is not contained in LET.

The appropriate SI unit for LET is the newton, but it is most typically expressed in units of kiloelectronvolts per micrometre (keV/?m) or megaelectronvolts per centimetre (MeV/cm). While medical physicists and radiobiologists usually speak of linear energy transfer, most non-medical physicists talk about stopping power.