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She holds annual conferences and lectures at important dental and periodontology congresses and scientific meetings. The conducted research studies by Prof. Surlin have led to an extended publishing activity, comprising more than 30 Web of Science-indexed papers and over 10 published books and book chapters.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"171921",title:"Prof.",name:"Petra",middleName:null,surname:"Surlin",slug:"petra-surlin",fullName:"Petra Surlin",profilePictureURL:"https://mts.intechopen.com/storage/users/no_image.jpg",biography:"Prof. Petra Surlin, DMD, PhD, is Professor of Periodontology at the University of Medicine and Pharmacy of Craiova, Romania. Following a periodontology specialization at the Universite Paris 7, Prof. Surlin became a consultant periodontology physician in 2012. The research work of Prof. Surlin is mainly focused on the field of periodontal medicine, specifically on the immunological interactions existing between periodontal disease and certain systemic conditions, as diabetes mellitus, rheumatoid arthritis or hepatic diseases. The conducted research studies by Prof. Surlin, including research grants awarded by the Romanian Agency of Research and Innovation, have led to an extended publishing activity, comprising more than thirty Web of Science-indexed papers and over ten published books and book chapters with national and international publishing houses. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"57485",title:"Small-Angle Scattering from Mass and Surface Fractals",doi:"10.5772/intechopen.70870",slug:"small-angle-scattering-from-mass-and-surface-fractals",body:'A great number of natural systems provide us with examples of nano- and micro- structures, which appear similar under a change of scale. These structures are called fractals [1], and can be observed in various disordered materials, rough surfaces, aggregates, metals, polymers, gels, colloids, thin films, etc. Quite often, the physical properties (mechanical, optical, statistical, thermodynamical, etc.) depend on their spatial configurations, and a great deal of activity has been performed in elucidating such correlations [2–4]. To this aim, in the last decade, important steps have been performed in three directions: instrumentation [5–7], computer programs [8, 9], and the development of new methods for sample preparation [10–14]. The first two have been proved to be very useful for the physical execution and data analysis, and the third one for preparation of nano- and micro-materials with the pre-defined structure and functions.
These achievements have stimulated a great interest in the development of theoretical investigations for structural modeling of self-similar objects at nano- and micro-scale. In particular, models based on deterministic or exact self-similar fractals (i.e., fractals that are self-similar at every point, such as the Koch snowflake, Cantor set, or Mandelbrot cube) have been frequently used, since this type of fractals allows an analytical representation of various geometrical parameters (radius of gyration) or of the scattering intensity spectrum. Although for most fractals generated by natural processes, this is only an approximation, in the case of deterministic nano- and micro-materials obtained recently such as 2D Sierpinski gaskets [15] and Cantor sets [16], or 3D Menger sponge [17] and octahedral structures [18], this approximation becomes exact.
Small-angle scattering of X-rays (SAXS) and/or neutrons (SANS) are well established techniques for probing the nano/micro scale structure in disordered materials [19–21]. While in the case of X-rays, the scattering is mostly determined by the interaction of the incident radiation with electrons, in the case of neutrons, the scattering is determined by their interaction with the atomic nuclei and with the magnetic moments in magnetic materials. Since the wavelengths of X-rays (0.5–2 Å) are of the same order of magnitude as those of thermal neutrons (1–10 Å), often, the data analysis and interpretation procedures for SANS can be interchanged with SAXS, and the developed theoretical models can be applied, generally, to both techniques [22]. However, using neutrons is very important in studying magnetic properties of materials as well as in emphasizing or concealing certain features of the investigated sample [23]. In the later case, the possibility of wide variations in the neutron scattering lengths (which can be negative sometimes) is exploited, and this is a unique feature of SANS, which makes it a preferred method over SAXS in structural analysis of biological materials [24, 25].
As compared with other methods of structural investigations, SAXS/SANS have the advantage that they are noninvasive, the physical quantities of interest (specific surface, radius of gyration, volume, or the fractal dimension) are averaged over a macroscopic volume and they can be extracted with almost no approximation [26]. In particular, for self-similar objects (either exact or statistical), the most important advantage is that SAXS/SANS can distinguish between mass [27] and surface fractals [28]. Experimentally, the difference is revealed through the value of the scattering exponent τ in the region where the scattering intensity I(q) decays as a power-law, i.e., I(q) ∝ q−τ, where q = (4π/λ) sin θ is the scattering vector, λ is the wavelength of the incident radiation, and 2θ is the scattering angle. For mass fractals τ = Dm, where Dm is the mass fractal dimension with 0 < Dm < 3. For surface fractals τ = 6 − Ds, where Ds is the surface fractal dimension with 2 < Ds < 3. Thus, in practice if the absolute value of the measured scattering exponent is smaller than 3, the sample is a mass fractal with fractal dimension τ (in the measured q-range), and if the exponent is between 3 and 4, the sample is a surface fractal with fractal dimension 6 − τ.
Besides the fractal dimension (either mass or surface one), traditionally from SAXS/SANS patterns, we can also obtain the overall size of the fractal, as well as the size of the basic structural units composing the fractal. The last years have brought a breakthrough in the theoretical analysis of SAXS/SANS experimental data, allowing for the extraction of additional structural information and detailed modeling of fractals using a deterministic approach [29–40]. This progress was stimulated by recent advances in nanotechnology, which allows preparation of both mass and surface deterministic fractals at sub-micrometer scale [15–18, 41, 42], as well as by instrumentation which allows novel structural features to be recorded in experimental data [43].
This chapter focuses on the interpretation of SAXS/SANS data from deterministic mass and surface fractals. First, a brief theoretical background on the basics of SAS theory, and on description of mass and surface fractals, is presented. Here, we also include the theory of some well-known methods of generating fractals, such as iterated function system of cellular automata. Novel data analysis methods for extracting additional structural information are presented and illustrated by applications to various models of mass and surface fractals.
In this section, some important concepts for the analysis of SAXS/SANS are reviewed, and analytical and numerical procedures for calculating the scattering intensity from some basic geometrical shapes are described. As we will see in the next section, these geometrical shapes will form the “scattering units” of the fractals. Then, the basic notions of fractal theory including fractal dimension, mass, and surface fractals are presented and defined in a rigorous manner, and two general methods for generating fractal structures are presented. These concepts are then applied in calculating the SAXS/SANS patterns from several theoretical models of mass and surface fractals.
In a SAS experiment, a beam of X-rays or neutrons is emitted from a source and strikes the sample. A small fraction is scattered by the sample and is recorded by the detector. In Figure 1, the incident beam has a wave vector ki, and the scattered beam with the wave vector ks makes the angle 2θ with the direction of the incoming or transmitted beam.
Schematic representation of a small-angle scattering experiment. 2θ is the scattering angle and λ is the wavelength of incident beam. The sample shown is a two-phase system of polydisperse scatterers with the same shape and random orientations, embedded in a matrix or solution.
To describe the scattering from assemblies of objects with scattering length bj, we write the scattering length density SLD as ρ(r) = ∑jbj(r − rj) [21], where
where v is the total volume irradiated by the beam. In the following, we consider scattering occurring in a particulate system where particles of density ρm are dispersed in a uniform solid matrix of density ρp. Then, the excess scattering SLD is defined by Δρ = ρm − ρp. We also consider that the objects are fractals that are randomly distributed and with uncorrelated positions and orientations. Thus, the scattering intensities of each object are added, and the intensity from the entire sample can be obtained from a single object averaged over all orientations, according to [29]:
where n is the concentration of objects, V is the volume of each object, and
The symbol 〈⋯〉 stands for ensemble averaging over all orientations, and for an arbitrary function f, it is calculated according to:
where, in spherical coordinates qx = q cos ϕ sin θ, qy = q sin ϕ sin θ, qz = q cos θ.
Another useful form of the scattering intensity, as a function of the correlation function, is the following [21]:
where γ(r) ≡ 〈ρ(r) ∗ ρ(−r)〉 is the correlation function of the object, with γ(r) = 0 for r > D and D is the largest dimension in the object. The symbol ∗ denotes a convolution, and thus, the correlation function can be seen as an averaged self-convolution of density distribution.
At low values of the scattering vector (q ≲ 2π/D), the above expression can be further exploited. By considering first the MacLaurin series
and then using only the first two terms of this approximation into Eq. (5), the Guinier equation is obtained:
where
is the radius of gyration of the object. In practice, a plot of logI(q) vs. q2 is used to obtain the slope
However, as we shall see in the next sections, inside a fractal, the scattering units have defined positions and correlations, and the interference among rays scattered by different units may no longer be ignored, so that the scattering amplitudes of individual units have to be added. By considering that the fractal is composed of N balls of size R, its form factor becomes [29]:
where
where I(0) = n|Δρ|2V2, and
A physical sample almost always consists of fractals that have different sizes, which is called polydispersity. An exception to this rule is protein solutions, in which all have the same size and shape. Thus, the corresponding scattering intensity from polydisperse fractals can be regarded as the sum of each individual form factor weighted with the corresponding volume V and contrast Δρ. We consider here a continuous distribution DN(l) of fractals with different sizes l, defined in such a way that DN(l)dl gives the probability of finding a fractal with dimension l lying in the range (l, l + dl). Although any kind of broad distribution can be used, we take here, as an application, a log-normal distribution of fractal sizes, such as:
where
where the form factor F(q) is given by Eq. (9). The effect of polydispersity is to smooth the scattering curves [20, 21] (see also Figure 2a).
In the next section, we shall make use of chaos game representation (CGR) and cellular automata (CA) to generate positions of the N scattering units/points. Thus, we can start with the Debye formula [44]
where Is(q) is the intensity scattered by each fractal unit, and rij is the distance between units i and j. When the number of units exceeds few thousands, the computation of the term sin(qrij)/(qrij) is time consuming, and thus it is handled via a pair-distance histogram g(r), with a bin-width commensurate with the experimental resolution [45]. Thus, Eq. (13) becomes
where g(ri) is the pair-distance histogram at pair distance ri. For determining fractal properties, we can neglect the form factor, and consider
As a first example, we derive the scattering intensity of a ball with unit density, radius R, and volume V = (4π/3)R3. To do so, we can rewrite the normalized scattering amplitude given by Eq. (3), in spherical coordinates, such as:
Note that since balls represent here the basic units of the fractal, we have chosen the notation F0(q) instead of F(q), in spirit of Eq. (9). We can choose the polar axis to coincide with the direction of q, and therefore
By performing an integration by parts of the last expression, the normalized scattering amplitude of the ball of radius R becomes:
Thus, the total scattering intensity (see also Eq. (2)) becomes:
Figure 2a shows the scattering intensity of a ball of radius R = 10 nm. The scattering is represented on a double logarithmic scale and shows the presence of two main regions. At low values of the scattering vector (q ≲ π/R), we have the Guinier region, which is a plateau with I(q) ∝ q0, and from which one can obtain the radius of gyration, as described in the previous section. At higher values (i.e., q ≳ π/R), there is a power-law decay of the type I(q) ∝ q−4 and with many minima. This is called the Porod region and generally it gives information about the specific surface of the investigated object. The main feature here is that by increasing the relative variance σr, the scattering curve becomes smoother, and the value of the scattering exponent is preserved. Figure 2b shows that by decreasing the size of the ball, the corresponding scattering curve has the same characteristics and the Porod region is shifted to the right with the corresponding factor (here by 10).
(a) SAS intensity from a ball of radius R = 10 nm from Eq. (19) (lowest curve). The higher the relative variance, the smoother the curve. Polydisperse SAS intensity, according to Eq. (12) at various values of relative variances. (b) SAS intensities from a ball of radius R = 10 nm (left curve), and R = 1 nm (right curve).
The second example is an equilateral triangle of edge size a. This is a slightly more complicated structure since it does not have a center of symmetry as a ball, and thus an orientational averaging is required. Thus, its height is
We choose a Cartesian coordinate system where one edge is parallel to the x-axis and the opposite vertex coincides with the origin. Thus, Eq. (3) becomes a surface integral given by:
which can be calculated and transformed into:
where α = hqx and β = hqy. As we shall see in next sections, the scattering amplitude of a system of triangles can be obtained by properly taking into account their scaling, rotation, and translations.
The averaging over all orientations is performed by allowing the triangle to rotate in a 2D space, and thus the average given by Eq. (4) for 3D case, becomes now:
with qx = q cos ϕ and qy = q sin ϕ.
Similarly to scattering from a ball, the intensity curve of a triangle also shows the Guinier region at low-q (q ≲ 2π/a), and a Porod region at high-q (q ≳ 2π/a) as shown in Figure 3a. However, for a triangle, the absolute value of the scattering exponent in the Porod region is equal to 3. This is in contrast to the value of 4 obtained for the ball in the previous example (see Figure 3b). The difference arises due to the fact that the triangle is a 2D object while the ball is a 3D one. In addition, due to the lack of symmetry, scattering from a triangle does not show pronounced minima as in the case of scattering from a ball.
(a) SAS intensity from an equilateral triangle of edge size a = 1 nm (right curve), and a = 3 nm (left curve), respectively. (b) A comparison between the SAS intensity of a ball of radius R = 1 nm (left curve) and a triangle of edge size a = 1 nm (right curve).
In the previous section, we have seen that regardless of the shape and Euclidean dimension, the SAS intensity from basic geometrical structures always reproduces a Guinier region followed by a Porod one. Without loosing from generality, we will restrict in the following to calculate SAS intensity from systems of triangles. In principle, any geometrical shape can be chosen but we prefer here triangles due to the fact that the both well-known techniques for generating fractal structures: iteration function systems and cellular automata, in their basic form, involve triangles in the construction process.
We start first with a simple model consisting of three triangles of edge size a/2, with a = 1 nm, as shown in Figure 4a. For this configuration, we can write [40]:
where βs is the scaling factor, F0(q) is given by Eq. (22), and the translation vectors are given by:
(a) A model of three triangles (gray) of edge size a/2, with a = 1 nm; (b) the corresponding SAS intensity (highly oscillating curve). The smoother curve is the SAS intensity corresponding to a single triangle of edge size a = 1 nm and whose center coincides with the center of white triangle in part (a).
The corresponding scattering intensity is shown in Figure 4b, and as expected, it consists of a Guinier region followed by a Porod one with scattering exponent −3. For comparison, the same figure shows the scattering intensity of a single of edge size a = 1 nm.
The second example is a system of 6 triangles of edge sizes a/3 arranged in such a way that they form a hexagon as shown in Figure 5a in black. For this configuration, the translation vectors can be written as [38]:
The corresponding scattering intensity is shown in Figure 5b and shows a superposition of maxima and minima. Excepting the distribution of minima in the Porod region, there is no significant difference between this scattering curve and the one corresponding to the system of three triangles shown in Figure 4b.
(a) Two models consisting only of equilateral triangles: hexagon (black color) and Star of David (black and gray colors); (b) the corresponding SAS intensities of the hexagon (continuous curve) and Star of David (dashed curve).
The last example is slightly more complicated and it consists of one hexagon of edge size a/3 (black) and six triangles of edge size a/3 (gray), with a = 1 nm arranged as in Figure 5a. This configuration is equivalent to a system of one triangle of edge size a and three triangles of edge size a/3. This is known in the literature, also as the Star of David. The translation vectors of the three triangles can be obtained in a similar way, as in the case of the first example. The corresponding scattering intensity is shown in Figure 5b (red). It can be seen that due to various sizes of the triangles involved in the construction, an intermediate regime emerges between the Guinier and Porod regions, approximately at 2 ≲ q ≲ 8 nm−1. As we shall see later, for even more complex structures, this intermediate region will evolve into a fractal one.
In particular, if the triangles composing the system are arranged in such a way that they form a Sierpinski gasket, the intermediate region will correspond to a mass fractal region [29–31]. When the triangles have a power-law distribution in their sizes, the intermediate region will correspond to a surface fractal one [37, 38]. The great advantage of the SAS technique consists in the possibility to differentiate these two types of fractal regimes, as discussed at the beginning of this chapter.
As it was already pointed out before, the main characteristic of fractals obtained from a SAS experiment is the fractal dimension. Mathematically, the α-dimensional Hausdorff measure is defined by [29]
where A is a subset of an n-dimensional Euclidean space, {Vi} is a covering of A with ai = diam(Vi) ≤ a, and the infimum is on all possible coverings. Then, the Hausdorff dimension D of the set A is given by:
and it represents the value of α for which the Hausdorff measure changes its value from zero to infinity.
However, in practice, this definition is quite inconvenient to be used, and here we shall use the “mass-radius” relation for calculating the Hausdorff dimension of the fractals [1]. In this approach, the total fractal measure (i.e., mass, surface area, volume) of the fractal within a ball of radius r centered on the fractal is given by:
where logA(r)/ log r → 0 for r → ∞.
As an application, for a deterministic mass fractal of length L, scaling factor βs, and k structural units in the first iteration, we can write [1]:
and using Eq. (29), one obtains a formula for calculating the fractal dimension of the mass fractal:
Thus, if a finite iteration of the fractal consists of N scattering balls of radius a, the fractal dimension is given by the asymptotic:
in the limit of large number of iterations.
If the quantity to be measured is the mass M(r) embedded in a disk of radius r, Eq. (30) becomes M(r) ∝ rDm, which leads to I(q) ∝ q−Dm. The lower the value of mass fractal dimension Dm, the less compact is the structure. In a similar way, for a surface fractal of fractal dimension Ds, its surface obeys S(r) ∝ r2 − Ds and thus the scattering intensity decays as I(q) ∝ q−(2d − Ds).
More generally, in a two-phase system where one phase is of dimension Dm and the second phase is its complement set of dimension Dp (“pores”), the “boundary” between the two phases also forms a set of dimension Ds (“surface”). Thus, for a mass fractal, we have Ds = Dm < d and Dp = d, while for a surface fractal we have Dm = Dp = d and d − 1 < Ds < d [37, 38]. The possibility of differentiating between mass and surface fractals makes SAS a very convenient technique for measuring fractal dimensions of materials at nano- and micro-scales.
As a first method of generating fractals, we consider an iterated function system (IFS). By definition, an IFS is a complete metric space (X, d) together with a finite set wn : X → X of contraction mappings and contractivity factors sn, n > 1, 2, ⋯, N. In general, a transformation
By considering a hyperbolic IFS and if we denote by (H(X), h(d)), the space of nonempty compact subsets with the Hausdorff metric h(d), then the transformations W : H(X) → H(X) defined by
The unique fix point A ∈
We generate here the attractor by using random iteration algorithm, and thus we assign the probability pn > 0 to wn for n = 1, 2, ⋯ where
Another important method to generate exact self-similar fractals is by using cellular automata (CA) [47–49]. They provide simple models for dynamical systems dealing with the emergence of collective phenomena such as chaos, turbulence, or fractals. Basically, a cellular automaton is a set of cells on a grid (rectangular, hexagonal, etc.) that evolves through a number of discrete steps according to a set of rules based on the states of neighboring cells. The rules are then applied alternatively for as many times as needed. The grid is n-dimensional but for our purposes, we will choose n = 1.
In this chapter, we shall present Rule 90 [49] together with the corresponding structure factor based on Pantos formula given by Eq. (15). For particular values of the number of steps, Rule 90 generates exact shapes of the Sierpinski gasket (SG). However, for most of the number of steps, it generates intermediate structures between two consecutive iterations of SG. Thus, for Rule 90, CA extends considerably the number of structures generated, and therefore new classes of materials consisting of a “mixture” of SG at various iterations can be investigated.
More generally, SAS from CA could be used to check whether a generated structure is a mass or surface fractal (or none of them), i.e., whether there exists a power-law distribution of some entities (collections of cells). In addition, through the oscillations of the scattering curve in the fractal region, SAS from CA can shed some light on the randomness of the generated structures. This could be of particular interest since some rules like Rule 30 generates so-called pseudo-random structures. However, this is beyond the scope of this chapter. Here, we shall restrict ourselves to calculation and interpretation of SAS intensities from basic structures, such as those based on SG. This shall facilitate a quick comparison with the theoretical model based on SG presented in the next section, and thus to support the validity of the obtained results.
As an example of a deterministic mas fractal, we calculate the scattering from a two dimensional Sierpinski gasket (SG), generated by three different methods. In order to calculate the scattering intensities of an ensemble of triangles, we use the following properties:
When the size of a triangle is scaled as a → βsa, then the form factor scales as F(q) → F(βsq);
When the triangle is translated by a vector b such as
Zero-th iteration of SG consists of a single triangle of edge size a (here, a = 1 nm). First iteration (m = 1) consists of four smaller triangles, each of the edge length a/2 as shown in Figure 3a. At second iteration (m = 2), the same operation is repeated for each of the triangles of edge length a/2. In the limit of large number of iterations m, the total number of triangles of edge size am = a/2m is:
Therefore using Eq. (31), one obtains the fractal dimension of SG as:
At m-th iteration, the positions of the triangles forming the SG are given by:
where
where the translation vectors are given by Eq. (25),
By using the property given in Eq. (37), the fractal structure factor can be written as [29]:
Thus, by introducing Eqs. (39) and (22) into Eq. (10), we obtain an analytical expression for the scattering intensity:
By ignoring the form factor F0 in Eq. (40), an analytical expression of the structure factor is obtained. This case is discussed in [40].
The corresponding scattering intensities are shown in Figure 5 right part, for a triangle (m = 0) and for the first three iterations of SG (m = 1, 2, 3). At low q-values (q ≲ 2 nm−1), all the scattering curves are characterized by a Guinier region. A main feature of scattering from deterministic mass fractals is that after the Guinier region, it follows a fractal regime in which the absolute value of the scattering exponent equals the fractal dimension of the fractal. The length of the fractal regime increases with increasing the iteration number since the distances between the scattering units of the fractals (here triangles) decrease. In Figure 5 right part, the fractal regime is clearly seen within the range 2 ≲ q ≲ 14 nm−1 for m = 3. It is characterized by a succession of maxima and minima superimposed on a power-law decay (also known as a generalized power-law decay [29]). The number of minima in the fractal regime is equal with the fractal iteration number and from their periodicity, we can extract the value of the scaling factor [29]. Beyond the fractal regime, one obtain as expected, the Porod region where the exponent of the power-law decay is −3 (or −4 for three-dimensional objects). In Figure 5 right part, at m = 3 the Porod regime begins near q ≳ 14 nm−1.
By taking into account, the effect of polydispersity or the random distribution of the scattering units, the scattering curve is smoothed and thus the maxima and minima are smeared out. The “smoothness” of the curve increases with increasing the relative variance of the distribution function, and they can be completely smeared out when a threshold is reached. Experimentally, most of the times, SAS experiments give this type of behavior, when the curve is completely smeared out. Please note that if we neglect the contribution of the form factor F0 in Eq. (40), then the Porod region is replaced by an asymptotic region, from which the number of scattering units inside the fractal can be obtained (see Figure 6). This case is presented and discussed in the following, for the SG generated using CGR and IFS, respectively.
Left part: first four iterations of the Sierpinski gasket generated using a deterministic algorithm; right part: The corresponding SAS intensities of the Sierpinski gasket. The curves are shifted vertically by a factor 10m, for clarity. The lowest curve corresponds to m = 0, and the highest one corresponds to m = 3.
CGR representation gives directly the positions of the scattering units in the fractal. Figure 6 left part, shows the SG generated from CGR for N = 30, 130, 230, and for 430 points, respectively. The figure clearly shows that by increasing the number of points, the obtained structure approaches better the structure of SG. From the same figure, we can see that for N = 430 points, the second iteration of SG can be quite clearly distinguished. Thus, a convenient way to calculate the scattering intensity is to use Pantos formula given by Eq. (15), since we neglect the shape of the scattering units.
Figure 6 right part shows the scattering structure factor of SG built from the CGR, for the four structures in the left part. Generally, all scattering curves are characterized by the presence of the three main regions specific to SG, obtained using the analytic representation (Figure 5) right part: Guinier region at low q, fractal region at intermediate q, and here an asymptotic region instead of a Porod one, since we neglect the form factor. The same figure shows that by increasing the number of particles, the length of the fractal region also increases. This is to be expected, since increasing the number of points leads to a better approximation of the deterministic SG. However, for SG generated using CGR, a transition region appears (at 40 ≲ q ≲ 200), since in this region, the pair distance distribution function does not follow a power-law decay distribution of the number of distances. Finally, in the asymptotic region, the curves are proportional to 1/N, and thus, the number of points can be recovered. The asymptotic values of the curves are marked by horizontal dashes lines in Figure 5 right part.
Rule 90 (Figure 7 left part) is an elementary cellular automaton rule, and it produces SG for particular values of the number of steps. Although the system is generated on a rectangular grid with cells of finite size, we consider here (as in the previous example) that the scattering units are points centered in the cells. Figure 7 left part, shows the structure generated by Rule 90 for p = 31, 41, 51, and for p = 63 steps, respectively. Note that p = 31 and p = 63 correspond to SG at iterations m = 3, and m = 4, respectively, while p = 41 and p = 52 correspond to some intermediate structures between the two consecutive iterations.
Left part: Sierpinski gasket generated using CGR, for different values of the number of scattering units; right part: the corresponding SAS intensities of the Sierpinski gasket from CGR. The highest curve corresponds to N = 30 and the lowest one corresponds to N = 430.
The corresponding structure factors are shown in Figure 7 right part. In the scattering curve, all the three main regions are present: Guinier, fractal, and asymptotic regions. However, as opposed to the Guinier region of the SG generated deterministically, or through CGR, here its length decreases with increasing the number of steps. This is a consequence of the construction algorithm used, namely the structure increases its size at every step. Decreasing the length of the Guinier region leads to an increase of the fractal one, since the smallest distances between scattering points remain the same. This is indicated in the scattering curve, by approximately equal positions (q ≃ 360 nm−1) of the first minima, for each of the four steps. As expected, the absolute value of the scattering exponent in the fractal region coincides with the fractal dimension of the structure, and the asymptotic behavior at large q (shown by horizontal dotted lines in Figure 7 right part) tens to 1/N, where N is the total number of scattering points.
Recently, it has been shown that surface fractals can be built as a sum of mass fractals [37, 38]. As an example, we consider here a generalized version of the Koch snowflake (KS). The construction algorithm starts from an equilateral triangle with edge length a, which is the zero-th mas fractal iteration (black triangle in Figure 8 left part). At the second step, each side is divided into three segments, each of length a/3, and a new triangle pointing outward is added. The base of the new triangle coincides with the central segment. This is the first mass fractal iteration (three orange triangles in Figure 8 left part). By repeating the same procedure for each of the new triangles, one obtains the KS having the fractal dimension:
where am is the side length at mth iteration. In Figure 8, each of the triangle is scaled down by a factor of 0.6 ⋅ am. The case when the triangles are not scaled down corresponds exactly to the well-known KS and its scattering properties have been extensively studied in [38].
Left part: Rule 90 for p = 31, 41, 52 and p = 63 number of steps, respectively; right part: the corresponding scattering intensities. The leftmost curve corresponds to p = 63 and the rightmost one corresponds to p = 31.
Since a surface fractal can be constructed as a sum of mass fractals, the normalized scattering amplitude is written as a sum of the scattering amplitudes of mass fractals. If we denote
Here, the scaling factor is βs = 3/10, and the generative functions are given by:
and respectively by:
The translation vectors are defined as:
and the vectors
Figures 8 and 9 left parts show the construction of the generalized KS when the ratio between the sizes of the triangles and the distances between them are 0.6 and 0.4, respectively. Figures 8 and 9 right parts show the corresponding scattering intensities. The general feature is that when the ratio is 0.4, in the fractal region the overall agreement between the total scattering intensity and the intensity corresponding to uncorrelated triangles is slightly better. The curve also shows that the absolute value of the scattering exponent is now 6 − Ds, with Ds = 1.26 given by Eq. (41), which is a “signature” of scattering from surface fractals. The reason for such behavior is that at a given iteration, the surface fractal consists of scattering units of different sizes following a power-law distribution (see Figure 8 left part), while mass fractals consist of scattering units of the same size (see Figure 5 left part). A mathematical derivation of the value of the scattering exponent in the case of scattering from surface fractals can be found in [37, 38].
Left part: a generalized Koch snowflake model. Right part: highly oscillating curve—the corresponding scattering intensity, smoother curve—the scattering intensity from a system containing the same number of triangles but whose positions are randomly.
Left part: a generalized Koch snowflake model. Right part: highly oscillating curve—the total scattering intensity, smoother curve—the scattering intensity from a system containing the same number of triangles but whose positions are random.
The centers of triangles in Figure 8 left part coincide with the centers of triangles of the regular KS. From another hand, it has been recently shown that the scattering intensity corresponding to a system of triangles whose sizes follow a power-law distribution and whose positions are uncorrelated, approximates under certain conditions the scattering intensity of the same system but when the positional correlations are not considered [38]. Figure 8 right part, shows also the scattering intensity of a system of triangles whose positions are uncorrelated (red curve). We can observe that in the fractal region (20 ≲ q ≲ 400 nm−1), the approximation is not quite satisfactory. The reason of this behavior is that here the scaling factor is βs, and thus the distances between the centers of the triangles is of the same order as their size. However, when the distances between the scattering units are much bigger than the sizes of the units, the approximation of the uncorrelated positions of the triangles works fairy well in the fractal region [38] (Figure 10).
In this chapter, we have presented and discussed some general concepts in small-angle scattering (SAS; neutrons, X-ray, light) from deterministic mass and surface fractals. To do so, we have considered the Sierpinski gasket (SG) as a model for deterministic mass fractals, and a generalized version of Koch snowflake (KS) as a model for deterministic surface fractals. The model for SG has been introduced through three main algorithms: deterministic and random iteration function system (IFS), and through cellular automata (CA). KS has been constructed in the framework of deterministic IFS.
The SAS intensities (fractal and structure factor) has been calculated from a system of 2D, monodisperse diluted (i.e., spatial correlation can be neglected) and randomly oriented fractals. They are characterized by the presence of three main regions: Guinier (at low q), fractal (at intermediate q), and Porod/asymptotic (at high q). We have shown that in the case of mass fractals, we can extract structural information about the fractal dimension (from the exponent of the SAS curve in the fractal region), scaling factor (from the periodicity of minima in the fractal region), iteration number (from the number of minima in the fractal region), and the total number of scattering units inside a fractal (from the value of the structure factor in the asymptotic region). In addition, mass fractals generated using IFS are able to reproduce fairly well the scattering curve of deterministic IFS under the proper conditions. Mass fractals generated from CA increase their size with increasing the number of steps. This growing process is also reflected by a decrease of the length of Guinier region. We have also shown that a surface fractal can be considered as a superposition of mass fractals at various iterations and the range of structural information which can be extracted is similar to the case of scattering from mass fractals.
The IFS and CA algorithms used to generate the mass fractals models can be easily extended to more general structures and can be used to address various questions. For example, in the case of CA, SAS could be used to determine the fractal dimension of an arbitrary structure generated using one dimensional rules, it can shed some light on the randomness of some structures, or it can reveal the existence of a power-law distribution of some entities (of arbitrarily shape) generated by a specific rule.
During the last decades, an impressive technological development has been achieved permitting the manipulation of single photons with a high degree of statistical accuracy. However, despite the significant experimental advances, we still do not have a clear physical picture of a single photon state universally accepted by the scientific community, especially involved in quantum electrodynamics. In this chapter, based on the present state of knowledge, we make a synthesis of the physical characteristics of a single photon put in evidence by the experiments, and we advance theoretical developments for its representation. Accordingly, the concept of the wave-particle nature of a single photon becomes physically comprehensive and in agreement with the experimental evidence.
\nHowever, before advancing in the theoretical developments, we consider that it is important starting with a brief historical review on the efforts carried out previously for understanding the nature of light while simultaneously making a synthesis of the main experimental results which are of crucial importance for the comprehension of the birth of the photon concept.
\nThe very first scientific publications on the nature of light are due to ancient Greeks who believed light is composed of corpuscles [1, 2]. Around 300 BC Euclid published the book Optica in which he developed the laws of reflection based on the rectilinear propagation of light. Two centuries later, Ptolemy of Alexandria published the book Optics, in which he included extensively all the previous knowledge on light. In this book, colours as well as refraction of the moonlight and sunlight by the earth’s atmosphere were analysed. After Ptolemy of Alexandria, almost no progress has been reported until the seventeenth century.
\nIn the year of 1670, Newton revived the ideas of ancient Greeks and advanced the theory following that light is composed of corpuscles that travel rectilinearly [3]. Ten years later, Huygens developed the principles of the wave theory of light [1, 4, 5]. Huygens’ wave theory was a hard opponent to Newton’s corpuscle concept. In the beginning of the nineteenth century, Young obtained experimentally interference patterns using different sources of light and explained some polarisation observations by assuming that light oscillations are perpendicular to the propagation axis [1, 6]. Euler and Fresnel explained the diffraction patterns observed experimentally by applying the wave theory [6]. In 1865, Maxwell published his theory on the electromagnetic waves establishing the relations between the electric and magnetic fields and showing that light is composed of electromagnetic waves [7]. A few years later, Hertz confirmed Maxwell’s theory by discovering the long-wavelength electromagnetic radiation [1, 7]. Thus, at the end of the nineteenth century, the scientific community started to accept officially the wave nature of light replacing Newton’s theory.
\nNevertheless, new events supporting the particle nature of light occurred in the beginning of the twentieth century. Stefan and Wien discovered the direct relationship between the thermal radiation energy and the temperature of a black body [8, 9]. However, the emitted radiation energy density as a function of the temperature calculated by Rayleigh failed to describe the experimental results at short wavelengths. Scientists had given the name “UV catastrophe” to this problem revealing the necessity of a new theoretical approach. Planck managed to establish the correct energy density expression for the radiation emitted by a black body with respect to temperature, in excellent agreement with the experiment [8]. For that purpose, he assumed that the bodies are composed of “oscillators” which have the particularity of emitting the electromagnetic energy in “packets” of hν, where ν is the frequency and h is a constant that was later called Planck’s constant. During the same period, the experiments carried out by Michelson et al. [10] demonstrated that the speed of light in vacuum is a universal physical constant corresponding to the product of the frequency ν times the wavelength λ, that is, c = λ ν.
\nIn 1902, Lenard pointed out that the photoelectric effect, discovered by Hertz 15 years earlier [11], occurs beyond a threshold frequency of light and the kinetic energy of the emitted electrons does not depend on the incident light intensity. Based on Planck’s works, Einstein proposed a simple interpretation of the photoelectric effect assuming that the electromagnetic radiation is composed of quanta with energy hν [12]. He advanced that the energy of a light ray when spreading from a point consists of a finite number of energy quanta localised in points in space, which move without dividing and are only absorbed and emitted as a whole. Although that was a decisive step towards the particle theory of light, the concept of the light quanta was still not generally accepted, and Bohr, who was strongly opposed to the particle concept of light [13], announced in his Nobel lecture (1922) that the light quantum hypothesis is not compatible with the interference phenomena and consequently it cannot throw light in the nature of radiation. Bohr’s statement was rather surprising because Taylor’s experiments, consisting of repeating Young’s double slit diffraction at extremely low light intensities, had already demonstrated since 1909 that light rays are composed of discrete parts whose spots compose the diffraction patterns by gradual accumulation on the detection screen [14]. Compton published his studies on X-rays scattered by free electrons in 1923 advancing that the experimental results could only be interpreted based on the light quanta model [15].
\nThus, the photoelectric effect and Compton scattering have been initially considered as the undoubtable demonstrations of the particle nature of light and historically were the strongest arguments in favour of the light quanta concept, which started to be universally accepted, and Lewis introduced the word “photon”, from the Greek word phos (Φωs, which means light) [1, 4].
\nTherein, it is extremely important to mention that Wentzel in 1926 [16] and Beck in 1927 [17], as well as much later Lamb and Scully in the 1960s [18], demonstrated that the photoelectric effect can be interpreted remarkably well by only considering the wave nature of light, without referring to photons at all [19]. Furthermore, the Compton scattering has been fully interpreted by Klein and Nishina in 1929 [20] also by considering the electromagnetic wave nature of light without invoking the photon concept. On the other hand, Young’s experiment, initially presented as the most convincing argument for the wave nature of light, was applied by Taylor at very low intensities to demonstrate the particle concept of light [14]. Indeed, much later Jin et al. [21] published an excellent theoretical interpretation of Young’s diffraction experiments based only on the particle representation of light.
\nThus, the picture on the nature of light in the 1930s was rather confusing since both opposite sides defending the wave or the particle nature advanced equally strong arguments. Hence, Bohr, inspired by de Broglie’s thesis on the simultaneous wave character of particles, announced the complementarity principle according to which light has both wave and particle natures appearing mutually exclusively in each specific experimental condition [1, 2, 19].
\nThe development of lasers [22] in the 1960s and the revolutionary parametric down-convertion techniques [23, 24] in the 1970s, have made it possible to realise conditions in which, with a convenient statistical confidence, only a single photon may be present in the experimental apparatus. In this way, the double-prism experiment [25] realised in the 1990s contradicted for the first time Bohr’s mutual exclusiveness demonstrating that a single photon exhibits both the wave and particle natures in the same experimental conditions.
\nAccording to the experimental investigations, it has been always stated that a photon has circular, left or right, polarisation with spin \n
The lateral expansion of a single photon, considered locally as an indivisible entity, was always an intriguing part of physics. With the purpose of studying the lateral expansion of the electromagnetic rays, Robinson in 1953 [32] and Hadlock in 1958 [33] carried out experiments using microwaves crossing small apertures and deduced that no energy is transmitted through apertures whose dimensions are smaller than roughly ∼λ/4. In 1986, for the same purpose, Hunter and Wadlinger [34, 35] used X-band microwaves with λ = 28.5 mm and measured the transmitted power through rectangular or circular apertures of different dimensions. They concluded that no energy is transmitted when the apertures are smaller than ∼λ/π confirming that the lateral expansion of the photons is a fraction of the wavelength.
\nThus, the experiments have shown that the single photon is not a point and cannot be localised at a coordinate, as stated by Einstein, while it can exhibit both the wave and particle natures in the same experimental conditions contradicting Bohr’s mutual exclusiveness. However, quantum electrodynamics (QED) has been developed during the 1930s to 1960s based upon the point particle model for the photon [36, 37, 38, 39]. In fact, the point photon concept has permitted to establish an efficient mathematical approach for describing states before and after an interaction processes [19, 39, 40, 41], but it is naturally inappropriate for the description of the real nature of a single photon.
\nFinally, what we can essentially draw out by summing up the experimental evidence is that a single photon is a minimum, local, indivisible part of the electromagnetic field with precise energy hν and momentum hν/c, having circular left or right polarisation with spin \n
In what follows, we present first the standard theoretical representation of the electromagnetic field quantization resulting in photons, and next we proceed to recent advances based on the vector potential quantization enhanced to a single photon state.
\nSince the formulation of Maxwell’s equations, the vector potential \n
where \n
In 1949, Ehrenberg and Siday were the first to put in evidence the influence of the vector potential on charged particles [42] deducing that it is a real physical field. Ten years later, Aharonov and Bohm re-infirmed the influence of the vector potential on electrons in complete absence of electric and magnetic fields [43]. That was confirmed experimentally by Chambers [44], Tonomura et al. [45], and Osakabe et al. [46] demonstrating without any doubt the reality of the vector potential field end its direct influence on charges.
\nFrom a theoretical point of view [43], the behaviour of a particle with charge q and mass m in the vicinity of a solenoid where the vector potential is present is described by the Hamiltonian:
\nwith \n
If the solenoid is extremely long along the z axis, then the magnetic field is uniform in the region inside and zero outside. The scalar potential Φ can be put to zero by assuming that the solenoid is not charged. In this case, in the outside region, the electric and magnetic fields are zero, but the vector potential is not zero and depends on the magnetic field flux in the solenoid:
\nwhere r is the radial distance from the z axis of the solenoid, S is the surface of the circle with radius r perpendicular to z, and \n
The Schrödinger equation for a charged particle outside the solenoid, where the vector potential is not zero, writes in complete absence of any other external potential:
\nwith \n
where \n
The exponential part of the wave function of Eq. (6) entails that two particles have equal charge and mass moving both outside the solenoid at the same distance from the axis, but the first in the same direction with the vector potential \n
Interference patterns for electrons in analogue conditions have been observed experimentally [44, 45, 46] demonstrating that the vector potential is a real physical field and interacts directly with charged particles in complete absence of magnetic and electric fields and of any other potential.
\nThe vector potential, being a real field, is considered as the fundamental link between the electromagnetic wave theory issued from Maxwell’s equations and the particle concept in quantum electrodynamics (QED) [19, 36, 39]. We will show analytically how this link is established.
\nIn the classical theory [5, 7], the energy density of a mode k of the electromagnetic wave writes:
\nwhere ε0\n and μ0\n are the electric permittivity and magnetic permeability of the vacuum, respectively, related to the speed of light in vacuum c by \n
In the case of a monochromatic plane wave with angular frequency \n
where \n
Introducing Eqs. (10) and (11) in Eq. (9), the energy density now depends on the square of the vector potential amplitude:
\nThe mean value over a period, thus over a wavelength, is time independent:
\nNote that the last equation expressing the mean energy density over a period of the mode k of the electromagnetic wave is independent on any external volume yielding that in the classical description, a free of cavity electromagnetic radiation mode expands naturally within a minimum volume. In a given cavity, this volume corresponds roughly to that imposed by the boundary conditions and the cut-off wave vectors [4, 5, 7].
\nOn the other hand, in the quantum description, the energy density for a number \n
In order to link the classical to the quantum description [4, 9, 19], the classical mean energy density over a period, expressed by Eq. (13), is imposed to be equivalent to the quantum mechanics expression of Eq. (14) for \n
The last relation is the fundamental link between the classical and quantum theory of light which is used to define in QED the vector potential amplitude operators for a single photon [19, 26, 29, 36, 37, 38, 39, 40, 41]:
\nwhere \n
Therein, it is worth noting that an external arbitrary volume parameter V appears in the vector potential amplitude of the single photon, expressed by Eq. (15), which is supposed to be an intrinsic physical property. This could entail the unphysical interpretation that a single photon in an infinite cavity has zero vector potential, thus zero electric and magnetic fields and consequently zero energy. This ambiguity, which is scarcely quoted in the literature, is lifted by considering that, in the case of a single photon, the volume V in Eq. (15) is equivalent to that defined by the boundary conditions in a cavity for the single radiation mode k.
\nThe energy of the electromagnetic field in a volume V considered as a superposition of different k-modes and λ-polarisations is obtained directly from Eq. (13):
\nwhere the summation over the λ-polarisations takes only two values corresponding to circular left and right [19, 36, 37, 38, 39, 40, 41].
\nReplacing in Eq. (17) the vector potential amplitude and its conjugate by the relations of the vector potential amplitude operators defined in Eq. (16), we get the “normal ordering” radiation Hamiltonian corresponding to the order \n
and the “anti-normal ordering” Hamiltonian corresponding to the order \n
where we have used the fundamental commutation relation in quantum electrodynamics:
\nIn Dirac’s representation the eigenfunctions take the simple expression \n
The successive action of both operators in the normal order corresponds to the photon number Hermitian operator \n
In this representation the normal and anti-normal ordering radiation Hamiltonians write, respectively:
\nWe obtain a harmonic oscillator Hamiltonian for the electromagnetic field by considering the mean value of the normal ordering and anti-normal ordering Hamiltonians:
\nThus, in QED the electromagnetic field is considered to be an ensemble of harmonic oscillators each represented by a point particle, the photon, whose eigenfunction is denoted simply by \n
Although we have no experimental facts showing the harmonic oscillator nature of a single photon, this representation has been adopted since the 1930s [37].
\nIn a different way, a harmonic oscillator representation for the electromagnetic field can be obtained by the intermediate of the canonical variables of position \n
Introducing the last expressions in Eq. (17), we get the electromagnetic field energy:
\nwhere the (+) sign is obtained when Eq. (17) is considered initially to be in the “normal order”, \n
With the purpose of establishing a harmonic oscillator representation for the electromagnetic field, it is generally considered that \n
Replacing in the last equation the classical canonical variables of position and momentum with the corresponding Hermitian operators [19, 29, 41]:
\nand putting \n
At that level it is important to note that, for a harmonic oscillator of a particle with mass \n
to the quantum mechanics Hamiltonian:
\nwhere \n
Consequently, the harmonic oscillator Hamiltonian for a particle of mass m expressed by Eq. (31) is a quite physical result (e.g., phonons in solid-state physics) obtained with a perfect correspondence between the classical canonical variables of momentum and position \n
Conversely, this is not the case for the electromagnetic field [19, 29, 39] because commutations between the canonical variables \n
Obviously, as frequently quoted [2, 19, 39], the fundamental mathematical ambiguity consisting of cancelling the commuting classical variable term \n
In fact, since no experiment has yet demonstrated that a single photon is a harmonic oscillator, the main reason for considering the electromagnetic field as an ensemble of harmonic oscillators lies in the importance of the zero-point energy (ZPE) issued in absence of photons from the eigenvalue \n
The summation of the last expression over all modes and polarisations is infinite and represents the principal singularity in the QED formalism [19, 26, 29, 36, 39].
\nNevertheless, the zero-point energy is very important because it is considered to be the basis for the explanation of the vacuum effects such as the spontaneous emission, the Lamb shift and the Casimir effect. However, as pointed out by many authors [19, 26, 39, 41], it is important to underline that the explanation of the spontaneous emission and the Lamb shift in QED is not due to Eq. (33) but precisely to the commutation properties of the photon creation and annihilation operators, \n
Conversely, the zero-point energy expressed by Eq. (33) is useful for the explanation of the spontaneous emission and the Lamb shift in the classical description of radiation [2, 39, 47].
\nRegarding the Casimir effect, it is often commented that caution has to be taken concerning the interpretation of its physical origin because it has been demonstrated by different methods [48, 49, 50] that it can be easily explained using classical electrodynamics without invoking at all the zero-point energy.
\nHence, in view of the above, the normal ordering Hamiltonian is the one mainly used in QED, casting aside the vacuum singularity issued from the harmonic oscillator formalism, while the zero-point energy issued from the harmonic oscillator Hamiltonian is principally useful in the classical formalism for the interpretation of the vacuum effects [2, 19, 39, 47].
\nWe have analysed in Section 3.1 the electromagnetic field energy quantization according to the harmonic oscillator representation. Now, we will analyse the vector potential field quantization following the second quantisation process.
\nConsidering the natural units \n
with
\nwhere \n
Using Eq. (36) in Eq. (34), the vector potential becomes:
\nwith \n
For \n
Suppressing the natural units (i.e., introducing c and \n
On the basis of the density of state theory, the quantization of a field in a cavity of volume V permits to transform the continuous summation over the modes to a discrete one [19, 51]:
\nThe last transformation is only valid for an ensemble of modes k whose wavelengths \n
Switching now to Heisenberg’s representation:
\nGeneralizing the coordinate system, adapting the phase and using Eq. (40), the vector potential of the electromagnetic field writes in QED [19, 29, 39, 41, 51]:
\nConsidering the scalar potential to be constant, the electric field is:
\nThe last expressions represent in a given volume V the quantized vector potential and the electric field of the electromagnetic radiation composed of a large number of modes k each with angular frequency \n
The amplitudes in Eqs. (42) and (43) have been obtained using the density of state theory and are valid only on the condition of Eq. (44). Furthermore, the boundary conditions of the electromagnetic waves considered in cavities and waveguides impose the wave vectors k of the modes to be higher than a characteristic cut-off value \n
The last equations represent the vector potential and the electric field of a large number of modes k of the quantized electromagnetic field in a finite volume V with \n
We have seen in Section 3.1 that according to the energy quantization procedure, a k-mode and λ-polarisation photon is considered to be a point harmonic oscillator represented by the simplified eigenfunction \n
As mentioned in Section 2.2, the classical expression of the mean energy density over a period for a single electromagnetic mode k, represented by Eq. (13), can be considered equivalent to that for a single photon in the quantum representation, given by Eq. (14) for \n
From a theoretical point of view, this is also compatible with the density of state theory according to which the spatial volume corresponding to a single state of the quantized field is proportional to \n
On the other hand, the dimension analysis of the vector potential issued from the general solution of Maxwell’s equations yields that it is proportional to an angular frequency [5, 7, 9]:
\nwhere J is the current density (C m−2 s\n−1) and μ the magnetic permeability.
\nIndeed, it is well established experimentally that the energy density radiated by a dipole is proportional to \n
This result is gauge independent since it concerns the natural units of the vector potential.
\nAccording to the previous considerations, for a free single k-mode photon with λ-polarisation (left or right circular), the vector potential can be written in quantum and classical formalism:
\nwhere, following to the above analysis, the amplitude writes:
\nwith \n
We can evaluate \n
where \n
Thus, the characteristic volume of a free single photon writes in agreement with Eq. (47):\n
\nReplacing \n
\nEquations (50) and (53) express the quantized vector potential amplitude and the spatial extension of a single photon with the constant \n
For a free k-mode photon, the volume Vk\n corresponds to the space in which the quantized vector potential oscillates at the angular frequency \n
which are independent on any external arbitrary volume parameter and are directly proportional to the square of the angular frequency [2, 54, 55].
\nWe can now express the quantum properties of the photon, energy, momentum, and spin by integrating the classical electromagnetic expressions over the volume \n
With the same token considering circular polarisation [4, 5, 7, 9] for the amplitudes of the electric and magnetic fields in Eq. (55), the momentum is:
\nAccording to the classical electromagnetic theory, the spin can be written through the electric and magnetic field components; hence, using again the circular polarisation, we get:
\nwhere we have taken the mean value \n
The fact that the quantum properties, energy, momentum, and spin, of the photon can be expressed through the classical electromagnetic fields integrated over the volume Vk\n signifies that the photon has naturally a spatial extension, and consequently when employing the term “wave-particle”, one must have in mind that a single photon is a “three-dimensional particle”.
\nWe can now obtain Heisenberg’s uncertainty relation for position and momentum using Vk\n. Indeed, replacing V in Eq. (16) by Vk\n, we get the photon vector potential amplitude operators:
\nThe corresponding position \n
Thus, introducing Eq. (59) in Eq. (60) and using Eq. (20) with Eq. (53), Heisenberg’s commutation relation, a fundamental concept in quantum theory, results directly [2]:
\nThe fundamental properties of the photon, energy \n
Considering Heisenberg’s energy-time uncertainty principle:
\nwe directly deduce from Eq. (62) the vector potential-time uncertainty:
\nThe energy and vector potential uncertainties with respect to time are intrinsic physical properties of the wave-particle nature of the photon.
\nObviously, the photon vector potential function \n
as well as the vector potential energy (wave-particle) equation for the photon [2, 54]:
\nwhere the vector potential operator \n
It is worth remarking the symmetry between the pairs \n
Now, when considering the propagation of a k-mode photon with wavelength \n
In fact, from a theoretical point of view, for a photon propagating in the z direction, Heisenberg’s uncertainty for the position z and momentum \n
Notice that the momentum uncertainty along the propagation axis is expressed through the uncertainty over the inverse of the wavelength.
\nConsidering now the vector potential function with the quantized amplitude \n
Obviously, the shorter the wavelength of the photon, the higher the localization probability in agreement with Heisenberg’s uncertainty and the experimental evidence.
\nThe photon vector potential is composed of a fundamental function \n
In this way, the general equation for the vector potential of the electromagnetic wave considered as a superposition of plane wave modes writes:
\nand that of a large number of cavity-free photons in quantum electrodynamics is:
\nAccording to Eqs. (55) and (62), for \n
The field \n
Combination of the expression \n
Using again Eq. (51) and recalling that the electron mass may be written as \n
entailing that the mass derives also from the EFGS and is proportional to the charge square.
\n\nEquations (50), (74), and (75) show the strong physical relationship between photons and electrons-positrons which are all related directly to the EFGS through the amplitude ξ. Obviously, photons and electrons-positrons, also probably leptons-antileptons, are issued from the same quantum vacuum field. This may be at the origin of the physical mechanism governing the photon generation during the electron-positron (and probably lepton-antilepton) annihilation and that of the electron-positron (lepton-antilepton) pair creation during the annihilation of high-energy gamma photons in the vicinity of very heavy nucleus.
\nIn this chapter we have presented recent theoretical developments complementing the standard formalism with the purpose of describing a single photon state in conformity with the experiments. We resume below the principal features.
\nThe quantization of the vector potential amplitude \n
A single photon, as a local three-dimensional entity of the electromagnetic field, is absorbed and emitted as a whole and propagates guided by the non-local vector potential function (Eq. (49)), which appears to be a natural wave function for the photon satisfying both the propagation equation (Eq. (65)) and the vector potential - energy equation (Eq. (66)). The probability for detecting a photon around a given point on the propagation axis is obtained by the square modulus of the vector potential and is proportional to the square of the angular frequency \n
Finally, the electromagnetic field ground state (EFGS) at zero frequency, a real quantum vacuum component, issues naturally from the vector potential wave function putting in evidence that photons are oscillations of the vacuum field. Furthermore, the electron-positron charge and mass are directly proportional to the vector potential amplitude quantization constant showing the strong physical relationship with the photons. Obviously, the origin of the mechanisms governing the transformations of photons to electrons-positrons and inversely lies in the nature of the electromagnetic field ground state.
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