The law of light refraction. Methodological materials

Light dispersion- this is the dependence of the refractive index n substances depending on the wavelength of light (in vacuum)

or, which is the same thing, the dependence of the phase speed of light waves on frequency:

Dispersion of a substance called the derivative of n By

Dispersion - the dependence of the refractive index of a substance on the wave frequency - manifests itself especially clearly and beautifully together with the effect of birefringence (see Video 6.6 in the previous paragraph), observed when light passes through anisotropic substances. The fact is that the refractive indices of ordinary and extraordinary waves depend differently on the frequency of the wave. As a result, the color (frequency) of light passing through an anisotropic substance placed between two polarizers depends both on the thickness of the layer of this substance and on the angle between the planes of transmission of the polarizers.

For all transparent, colorless substances in the visible part of the spectrum, as the wavelength decreases, the refractive index increases, that is, the dispersion of the substance is negative: . (Fig. 6.7, areas 1-2, 3-4)

If a substance absorbs light in a certain range of wavelengths (frequencies), then in the absorption region the dispersion

turns out to be positive and is called abnormal (Fig. 6.7, area 2–3).

Rice. 6.7. Dependence of the square of the refractive index (solid curve) and the light absorption coefficient of the substance
(dashed curve) versus wavelengthlnear one of the absorption bands()

Newton studied normal dispersion. The decomposition of white light into a spectrum when passing through a prism is a consequence of light dispersion. When a beam of white light passes through a glass prism, a multi-colored spectrum (Fig. 6.8).

Rice. 6.8. The passage of white light through a prism: due to the difference in the refractive index of glass for different
wavelengths, the beam is decomposed into monochromatic components - a spectrum appears on the screen

Red light has the longest wavelength and the smallest refractive index, so red rays are deflected less than others by the prism. Next to them will be rays of orange, then yellow, green, blue, indigo and finally violet light. The complex white light incident on the prism is decomposed into monochromatic components (spectrum).

A prime example of dispersion is a rainbow. A rainbow is observed if the sun is behind the observer. Red and violet rays are refracted by spherical water droplets and reflected from their inner surface. Red rays are refracted less and enter the observer's eye from droplets located at a higher altitude. Therefore, the top stripe of the rainbow always turns out to be red (Fig. 26.8).

Rice. 6.9. The emergence of a rainbow

Using the laws of reflection and refraction of light, it is possible to calculate the path of light rays with total reflection and dispersion in raindrops. It turns out that the rays are scattered with the greatest intensity in a direction forming an angle of about 42° with the direction of the sun's rays (Fig. 6.10).

Rice. 6.10. Rainbow location

The geometric locus of such points is a circle with center at the point 0. Part of it is hidden from the observer R below the horizon, the arc above the horizon is the visible rainbow. Double reflection of rays in raindrops is also possible, leading to a second-order rainbow, the brightness of which, naturally, is less than the brightness of the main rainbow. For her, the theory gives an angle 51 °, that is, the second-order rainbow lies outside the main one. In it, the order of colors is reversed: the outer arc is colored purple, and the lower one is painted red. Rainbows of the third and higher orders are rarely observed.

Elementary theory of dispersion. The dependence of the refractive index of a substance on the length of the electromagnetic wave (frequency) is explained on the basis of the theory of forced oscillations. Strictly speaking, the movement of electrons in an atom (molecule) obeys the laws of quantum mechanics. However, for a qualitative understanding of optical phenomena, we can limit ourselves to the idea of electrons bound in an atom (molecule) by an elastic force. When deviating from the equilibrium position, such electrons begin to oscillate, gradually losing energy to emit electromagnetic waves or transferring their energy to lattice nodes and heating the substance. As a result, the oscillations will be damped.

When passing through a substance, an electromagnetic wave acts on each electron with the Lorentz force:

Where v- speed of an oscillating electron. In an electromagnetic wave, the ratio of the magnetic and electric field strengths is equal to

Therefore, it is not difficult to estimate the ratio of the electric and magnetic forces acting on the electron:

Electrons in matter move at speeds much lower than the speed of light in a vacuum:

Where - tension amplitude electric field in a light wave, is the phase of the wave determined by the position of the electron in question. To simplify calculations, we neglect damping and write the electron motion equation in the form

where, is the natural frequency of vibrations of an electron in an atom. The solution to such a differential inhomogeneous equation we already looked at it earlier and got

Consequently, the displacement of the electron from the equilibrium position is proportional to the electric field strength. Displacements of nuclei from the equilibrium position can be neglected, since the masses of the nuclei are very large compared to the mass of the electron.

An atom with a displaced electron acquires a dipole moment

(for simplicity, let us assume for now that there is only one “optical” electron in the atom, the displacement of which makes a decisive contribution to the polarization). If a unit volume contains N atoms, then the polarization of the medium (dipole moment per unit volume) can be written in the form

Possible in real environments different types vibrations of charges (groups of electrons or ions) contributing to polarization. These types of oscillations can have different amounts of charge e i and masses t i, as well as various natural frequencies (we will denote them by the index k), in this case, the number of atoms per unit volume with a given type of vibration Nk proportional to the concentration of atoms N:

Dimensionless proportionality coefficient fk characterizes the effective contribution of each type of oscillation to the total polarization of the medium:

On the other hand, as is known,

where is the dielectric susceptibility of the substance, which is related to the dielectric constant e ratio

As a result, we obtain the expression for the square of the refractive index of a substance:

Near each of the natural frequencies, the function defined by formula (6.24) suffers a discontinuity. This behavior of the refractive index is due to the fact that we neglected attenuation. Similarly, as we saw earlier, neglecting damping leads to an infinite increase in the amplitude of forced oscillations at resonance. Taking into account attenuation saves us from infinities, and the function has the form shown in Fig. 6.11.

Rice. 6.11. Dependence of the dielectric constant of the mediumon the frequency of the electromagnetic wave

Considering the relationship between frequency and electromagnetic wavelength in vacuum

it is possible to obtain the dependence of the refractive index of a substance P on the wavelength in the region of normal dispersion (sections 1–2 And 3–4 in Fig. 6.7):

The wavelengths corresponding to the natural frequencies of oscillations are constant coefficients.

In the region of anomalous dispersion (), the frequency of the external electromagnetic field is close to one of the natural frequencies of oscillations of molecular dipoles, that is, resonance occurs. It is in these areas (for example, area 2–3 in Fig. 6.7) that significant absorption of electromagnetic waves is observed; the light absorption coefficient of the substance is shown by the dashed line in Fig. 6.7.

The concept of group velocity. The concept of group velocity is closely related to the phenomenon of dispersion. When real electromagnetic pulses propagate in a medium with dispersion, for example, wave trains known to us, emitted by individual atomic emitters, they “spread out” - an expansion of extent in space and duration in time. This is due to the fact that such pulses are not a monochromatic sine wave, but a so-called wave packet, or a group of waves - a set of harmonic components with different frequencies and different amplitudes, each of which propagates in the medium with its own phase velocity (6.13).

If a wave packet were propagating in a vacuum, then its shape and spatio-temporal extent would remain unchanged, and the speed of propagation of such a wave train would be the phase speed of light in vacuum

Due to the presence of dispersion, the dependence of the frequency of an electromagnetic wave on the wave number k becomes nonlinear, and the speed of propagation of the wave train in the medium, that is, the speed of energy transfer, is determined by the derivative

where is the wave number for the “central” wave in the train (having the greatest amplitude).

We will not derive this formula in general view, but let’s use a particular example to explain its physical meaning. As a model of a wave packet, we will take a signal consisting of two plane waves, propagating in the same direction with the same amplitudes and initial phases, but differing in frequencies, shifted relative to the “central” frequency by a small amount. The corresponding wave numbers are shifted relative to the “central” wave number by a small amount . These waves are described by expressions.

FOR LECTURE No. 24

"INSTRUMENTAL METHODS OF ANALYSIS"

REFRACTOMETRY.

Literature:

1. V.D. Ponomarev “Analytical Chemistry” 1983 246-251

2. A.A. Ishchenko “Analytical Chemistry” 2004 pp. 181-184

REFRACTOMETRY.

Refractometry is one of the simplest physical methods of analysis at a cost minimum quantity of the analyte and is carried out in a very short time.

Refractometry- a method based on the phenomenon of refraction or refraction i.e. changing the direction of light propagation when passing from one medium to another.

Refraction, as well as absorption of light, is a consequence of its interaction with the medium. The word refractometry means measurement refraction of light, which is estimated by the value of the refractive index.

Refractive index value n depends

1) on the composition of substances and systems,

2) from the fact in what concentration and what molecules the light beam encounters on its path, because Under the influence of light, molecules of different substances are polarized differently. It is on this dependence that the refractometric method is based.

This method has a number of advantages, as a result of which it has found wide application both in chemical research and in the control of technological processes.

1) Measuring refractive indexes is a very simple process that is carried out accurately and with minimal time and amount of material.

2) Typically, refractometers provide an accuracy of up to 10% in determining the refractive index of light and the content of the analyte

The refractometry method is used to control authenticity and purity, to identify individual substances, to determine the structure of organic and inorganic compounds when studying solutions. Refractometry is used to determine the composition of two-component solutions and for ternary systems.

Physical basis of the method

REFRACTIVE INDEX.

The greater the difference in the speed of light propagation in the two, the greater the deviation of a light ray from its original direction when it passes from one medium to another.

these environments.

Let us consider the refraction of a light beam at the boundary of any two transparent media I and II (See Fig.). Let us agree that medium II has a greater refractive power and, therefore, n 1 And n 2- shows the refraction of the corresponding media. If medium I is not vacuum or air, then the ratio of the sin angle of incidence of the light beam to the sin angle of refraction will give the value of the relative refractive index n rel. Value n rel. can also be defined as the ratio of the refractive indices of the media under consideration.

n rel. = ----- = ---

The value of the refractive index depends on

1) nature of substances

The nature of a substance in this case is determined by the degree of deformability of its molecules under the influence of light - the degree of polarizability. The more intense the polarizability, the stronger the refraction of light.

2)wavelength of incident light

The refractive index measurement is carried out at a light wavelength of 589.3 nm (line D of the sodium spectrum).

The dependence of the refractive index on the wavelength of light is called dispersion. The shorter the wavelength, the greater the refraction. Therefore, rays of different wavelengths are refracted differently.

3)temperature , at which the measurement is carried out. Required condition determining the refractive index is compliance with the temperature regime. Usually the determination is performed at 20±0.3 0 C.

As the temperature increases, the refractive index decreases; as the temperature decreases, it increases..

The correction for temperature effects is calculated using the following formula:

n t =n 20 + (20-t) 0.0002, where

n t – Bye refractive index at a given temperature,

n 20 - refractive index at 20 0 C

The influence of temperature on the values of the refractive indices of gases and liquids is associated with the values of their volumetric expansion coefficients. The volume of all gases and liquids increases when heated, the density decreases and, consequently, the indicator decreases

The refractive index measured at 20 0 C and a light wavelength of 589.3 nm is designated by the index n D 20

The dependence of the refractive index of a homogeneous two-component system on its state is established experimentally by determining the refractive index for a number of standard systems (for example, solutions), the content of components in which is known.

4) concentration of the substance in solution.

For many aqueous solutions of substances, refractive indices at different concentrations and temperatures are reliably measured, and in these cases reference books can be used refractometric tables. Practice shows that when the dissolved substance content does not exceed 10-20%, along with the graphical method, in many cases it is possible to use linear equation type:

n=n o +FC,

n- refractive index of the solution,

no- refractive index of a pure solvent,

C- solute concentration,%

F-empirical coefficient, the value of which is found

by determining the refractive index of solutions of known concentration.

REFRACTOMETERS.

Refractometers are instruments used to measure the refractive index. There are 2 types of these devices: Abbe type and Pulfrich type refractometer. In both cases, measurements are based on determining the maximum refraction angle. In practice, refractometers of various systems are used: laboratory-RL, universal RL, etc.

The refractive index of distilled water is n 0 = 1.33299, but practically this indicator is taken as reference as n 0 =1,333.

The operating principle of refractometers is based on determining the refractive index by the limiting angle method (the angle of total reflection of light).

Handheld refractometer

Abbe refractometer

Chapter 31

HOW DOES THE REFRACTIVE INDEX ARISE?

§ 1. Refractive index

§ 2. Field radiated by the medium

§ 3. Dispersion

§ 4. Absorption

§ 5. Light wave energy

§ 1. Refractive index

We have already said that light moves slower in water than in air, and in air a little slower than in a vacuum. This fact is taken into account by introducing the refractive index n. Let us now try to understand how the decrease in the speed of light occurs. In particular, it is especially important to trace the connection of this fact with some physical assumptions or laws that were previously expressed and boil down to the following:

a) the total electric field under any physical conditions can be represented as the sum of fields from all charges in the Universe;

b) the radiation field of each individual charge is determined by its acceleration; the acceleration is taken taking into account the delay arising from the finite speed of propagation, always equal to c. But you will probably immediately cite a piece of glass as an example and exclaim: “Nonsense, this position is not suitable here. It must be said that the delay corresponds to the speed c/n.” However, this is wrong; Let's try to figure out why this is wrong. It seems to the observer that light or any other electrical wave propagates through a substance with a refractive index n at a speed c/n. And this is true with some accuracy. But in fact, the field is created by the movement of all charges, including charges moving in the medium, and all the components of the field, all its components, propagate with a maximum speed c. Our task is to understand how the apparent slower speed occurs.

Fig. 31.1. Passage of electric waves through a layer of transparent substance.

Let's try to understand this phenomenon using a very simple example. Let a source (let's call it an “external source”) be placed at a large distance from a thin transparent plate, say glass. We are interested in the field on the other side of the plate and quite far from it. All this is shown schematically in Fig. 31.1; points S and P here are assumed to be located at a large distance from the plane. According to the principles we have formulated, the electric field far from the plate is represented by the (vector) sum of the fields of the external source (at point S) and the fields of all charges in the glass plate, each field being taken with a delay at speed c. Recall that the field of each charge does not change from the presence of other charges. These are our basic principles. Thus, the field at point P

can be written as

where E s is the field of the external source; it would coincide with the desired field at point P if there were no plate. We expect that in the presence of any moving charges, the field at point P will be different from E r

Where do moving charges in glass come from? It is known that any object consists of atoms containing electrons. An electric field from an external source acts on these atoms and rocks the electrons back and forth. The electrons in turn create a field; they can be considered as new emitters. The new emitters are connected to the source S, since it is the source field that causes them to oscillate. The total field contains a contribution not only from the source S, but also additional contributions from the radiation of all moving charges. This means that the field in the presence of glass changes, and in such a way that inside the glass its propagation speed appears different. It is this idea that we use in quantitative considerations.

However, an accurate calculation is very difficult, because our statement that the charges experience only the action of the source is not entirely correct. Each given charge “feels” not only the source, but, like any object in the Universe, it also feels all other moving charges, in particular the charges oscillating in the glass. Therefore, the total field acting on a given charge is a combination of fields from all other charges, the movement of which in turn depends on the movement of this charge! You see that deriving an exact formula requires solving a complex system of equations. This system is very complex and you will learn it much later.

And now let's turn to completely simple example to clearly understand the manifestation of all physical principles. Let us assume that the action of all other atoms on a given atom is small compared to the action of the source. In other words, we are studying a medium in which the total field changes little due to the movement of the charges in it. This situation is typical for materials with a refractive index very close to unity, for example, for rarefied media. Our formulas will be valid for all materials with a refractive index close to unity. In this way we can avoid the difficulties associated with solving the complete system of equations.

You may have noticed along the way that the movement of charges in the plate causes another effect. This motion creates a wave that propagates back in the direction of the source S. This backward moving wave is nothing more than a ray of light reflected by a transparent material. It comes not only from the surface. Reflected radiation is generated at all points within the material, but the overall effect is equivalent to reflection from the surface. Taking into account reflection lies beyond the limits of applicability of the present approximation, in which the refractive index is considered so close to unity that reflected radiation can be neglected.

Before moving on to the study of the refractive index, it should be emphasized that the basis of the phenomenon of refraction is the fact that the apparent speed of wave propagation is different in different materials. The deflection of a light beam is a consequence of changes in effective speed in different materials.

Fig. 31.2. Relationship between refraction and change in speed.

To clarify this fact, we have noted in FIG. 31.2 a series of successive maxima in the amplitude of a wave falling from a vacuum onto glass. An arrow perpendicular to the indicated maxima marks the direction of wave propagation. Everywhere in the wave oscillations occur at the same frequency. (We have seen that forced oscillations have the same frequency as the oscillations of the source.) It follows that the distances between the maxima of the waves on both sides of the surface coincide along the surface itself, since the waves here must be consistent and the charge on the surface oscillates at the same frequency. The shortest distance between wave crests is the wavelength equal to the speed divided by the frequency. In a vacuum, the wavelength is l 0 =2pс/w, and in glass l=2pv/w or 2pс/wn, where v=c/n is the wave speed. As can be seen from Fig. 31.2, the only way to “stitch” the waves at the boundary is to change the direction of movement of the wave in the material. Simple geometric reasoning shows that the “matching” condition reduces to the equality l 0 /sin q 0 =l/sinq, or sinq 0 /sinq=n, and this is Snell’s law. Let the bending of the light no longer concern you now; you just need to find out why, in fact, the effective speed of light in a material with refractive index n is equal to c/n?

Let's return again to FIG. 31.1. From the above it is clear that it is necessary to calculate the field at point P from the oscillating charges of the glass plate. Let us denote this part of the field, which is represented by the second term in equality (31.2), by E a. Adding the source field E s to it, we obtain the total field at point P.

The task before us here is perhaps the most difficult of those that we will tackle this year, but its difficulty lies only in large quantities foldable members; each member in itself is very simple. Unlike other times when we used to say: “Forget the conclusion and look only at the result!”, now for us the conclusion is much more important than the result. In other words, you need to understand the entire physical “kitchen” with which the refractive index is calculated.

To understand what we are dealing with, let’s find what the “correction field” E a should be so that the total field at point P looks like the source field slowed down when passing through a glass plate. If the plate had no influence on the field, the wave would propagate to the right (along the axis

2) by law

or, using exponential notation,

What would happen if the wave passed through the plate at a lower speed? Let the thickness of the plate be Dz. If there were no plate, the wave would travel the distance Dz in the time Dz/c. And since the apparent speed of propagation is c/n, then time nDz/c will be required, i.e. more by some additional time equal to Dt=(n-l) Dz/c. Behind the plate the wave again moves with speed c. Let's take into account the additional time for passing through the plate, replacing t in equation (31.4) with (t-Dt), i.e. Thus, if you put on a record, then the formula for the wave should become

This formula can also be rewritten in another way:

from which we conclude that the field behind the plate is obtained by multiplying the field that would exist in the absence of the plate (i.e. E s) by exp[-iw(n-1)Dz/c]. As we know, multiplying an oscillating function of the type e i w t by e i q means a change in the oscillation phase by an angle q, resulting from a delay in the passage of the plate. The phase is delayed by the amount w(n-1)Dz/c (precisely delayed, since there is a minus sign in the exponent).

We said earlier that the plate adds the field E a to the original field E S = E 0 exp, but instead we found that the action of the plate is reduced to multiplying the field by a factor that shifts the phase of the oscillations. However, there is no contradiction here, since the same result can be obtained by adding a suitable complex number. This number is especially easy to find for small Dz, since e x for small x is equal to (1+x) with great accuracy.

Fig. 31.3. Construction of the field vector of a wave passing through the material at certain values of t and z.

Then we can write

Substituting this equality into (31 6), we obtain

The first term in this expression is simply the source field, and the second should be equated to E a - the field created by the oscillating charges of the plate to the right of it. The field E a is expressed here through the refractive index n; it, of course, depends on the source field strength.

The meaning of the transformations made is most easily understood using a diagram complex numbers(see Fig. 31.3). Let us first plot E s (z and t are chosen in the figure such that E s lies on the real axis, but this is not necessary). The delay during the passage of the plate leads to a delay in the phase of E s, i.e., it turns E s by a negative angle. This is the same as adding a small vector E a, directed almost at right angles to E s. This is precisely the meaning of the factor (-i) in the second term (31.8). It means that for real E s the quantity E a is negative and imaginary, and in the general case E s and E a form a right angle.

§ 2. Field radiated by the medium

We must now find out whether the field of oscillating charges in the plate has the same form as the field E a in the second term of (31.8). If this is so, then we will thereby find the refractive index n [since n is the only factor in (31.8) that is not expressed in terms of fundamental quantities]. Let us now return to the calculation of the field E a created by the charges of the plate. (For convenience, we have written down in Table 31.1 the notations that we have already used and those that we will need in the future.)

WHEN CALCULATING _______

E s field created by the source

E a field created by plate charges

Dz plate thickness

z distance normal to the plate

n refractive index

w frequency (angular) radiation

N is the number of charges per unit volume of the plate

h number of charges per unit area of the plate

q is the electron charge

m electron mass

w 0 resonant frequency of an electron bound in an atom

If the source S (in Fig. 31.1) is located on the left at a sufficiently large distance, then the field E s has the same phase along the entire length of the plate, and near the plate it can be written in the form

On the plate itself at point z=0 we have

This electric field affects every electron in the atom, and they are influenced electric force qE will fluctuate up and down (if e0 is directed vertically). To find the nature of the movement of electrons, let us imagine atoms in the form of small oscillators, that is, let the electrons be elastically connected to the atom; this means that the displacement of electrons from their normal position under the influence of a force is proportional to the magnitude of the force.

If you've heard of a model of an atom in which electrons orbit around a nucleus, then this model of an atom will seem simply funny to you. But this is just a simplified model. An accurate theory of the atom, based on quantum mechanics, states that in processes involving light, electrons behave as if they were attached to springs. So, suppose “that the electrons are subject to a linear restoring force and therefore behave like oscillators with mass m and resonant frequency w 0 . We have already studied such oscillators and know the equation of motion to which they obey:

(here F is the external force).

In our case, the external force is created by the electric field of the source wave, so we can write

where q e is the charge of the electron, and as E S we took the value E S = E 0 e i w t from equation (31.10). The equation of electron motion takes the form

The solution to this equation, which we found earlier, looks like this:

We found what we wanted - the movement of electrons in the plate. It is the same for all electrons, and only the average position (“zero” of motion) is different for each electron.

Now we are able to determine the field E a created by the atoms at point P, since the field of the charged plane was found even earlier (at the end of Chapter 30). Turning to equation (30.19), we see that the field E a at point P is the velocity of the charge, delayed in time by the value z/c, multiplied by a negative constant. Differentiating x from (31.16), we obtain the speed and, introducing delay [or simply substituting x 0 from (31.15) into (30.18)], we arrive at the formula

As one would expect, the forced oscillation of the electrons led to a new wave propagating to the right (this is indicated by the exp factor); the amplitude of the wave is proportional to the number of atoms per unit area of the plate (multiplier h), as well as the amplitude of the source field (E 0). In addition, other quantities arise that depend on the properties of the atoms (q e, m, w 0).

Most important point, however, is that formula (31.17) for E a is very similar to the expression E a in (31.8), which we obtained by introducing a delay in a medium with a refractive index n. Both expressions coincide if we put

Note that both sides of this equation are proportional to Dz, since h, the number of atoms per unit area, is equal to NDz, where N is the number of atoms per unit volume of the plate. Substituting NDz instead of h and reducing by Dz, we get our main result - the formula for the refractive index, expressed in terms of constants depending on the properties of the atoms and the frequency of light:

This formula “explains” the refractive index, which is what we were striving for.

§ 3. Dispersion

The result we obtained is very interesting. It gives not only the refractive index expressed in terms of atomic constants, but indicates how the refractive index changes with the frequency of light w. With the simple statement "light travels at a slower speed in a transparent medium" we could never arrive at this important property. It is, of course, also necessary to know the number of atoms per unit volume and the natural frequency of the atoms w 0 . We do not yet know how to determine these quantities, since they are different for different materials, and we cannot present a general theory on this issue. General theory properties of various substances - their natural frequencies and

etc. - formulated on the basis of quantum mechanics. In addition, the properties of various materials and the value of the refractive index vary greatly from material to material, and therefore one can hardly hope that it will be possible to obtain a general formula suitable for all substances.

Nevertheless, let's try to apply our formula to different environments. First of all, for most gases (for example, air, most colorless gases, hydrogen, helium, etc.), the natural frequencies of electron vibration correspond to ultraviolet light. These frequencies are much higher than the frequencies of visible light, that is, w 0 is much greater than w, and as a first approximation, w 2 can be neglected compared to w 0 2. Then the refractive index is almost constant. So, for gases the refractive index can be considered a constant. This conclusion is also true for most other transparent media, such as glass. Taking a closer look at our expression, we can see that as c increases, the denominator decreases, and, therefore, the refractive index increases. Thus, n increases slowly with increasing frequency. Blue light has a higher refractive index than red light. This is why blue rays are deflected more strongly by a prism than red rays.

The very fact that the refractive index depends on frequency is called dispersion, since it is because of dispersion that light is “dispersed” and decomposed into a spectrum by a prism. The formula that expresses the refractive index as a function of frequency is called the dispersion formula. So, we have found the dispersion formula. (Over the past few years, "dispersion formulas" have come into use in particle theory.)

Our dispersion formula predicts a number of interesting new effects. If the frequency w 0 lies in the region of visible light, or if we measure the refractive index of a substance, such as glass, for ultraviolet rays (where w is close to w 0), then the denominator tends to zero and the refractive index becomes very large. Let, further, w be greater than w 0 . This case occurs, for example, if substances such as glass are irradiated with X-rays. In addition, many substances that are opaque to ordinary light (say, coal) are transparent to X-rays, so we can talk about the refractive index of these substances for X-rays. The natural frequencies of carbon atoms are much lower than the frequency of X-rays. The refractive index in this case is given by our dispersion formula if we set w 0 =0 (that is, we neglect w 0 2 compared to w 2).

A similar result is obtained when a gas of free electrons is irradiated with radio waves (or light). In the upper atmosphere, ultraviolet radiation from the sun knocks electrons out of atoms, resulting in a gas of free electrons. For free electrons w 0 =0 (there is no elastic restoring force). Assuming w 0 =0 in our dispersion formula, we obtain a reasonable formula for the refractive index of radio waves in the stratosphere, where N now means the density of free electrons (number per unit volume) in the stratosphere. But, as can be seen from the formula, when a substance is irradiated with X-rays or an electron gas with radio waves, the term (w02-w2) becomes negative, which means that n is less than one. This means that the effective speed of electromagnetic waves in matter is greater than c! Could this be true?

Maybe. Although we said that signals cannot travel faster than the speed of light, nevertheless, the refractive index at a certain frequency can be either greater or less than unity. This simply means that the phase shift due to light scattering is either positive or negative. In addition, it can be shown that the speed of the signal is determined by the refractive index not at one frequency value, but at many frequencies. The refractive index indicates the speed at which the wave crest moves. But the crest of the wave does not yet constitute a signal. A pure wave without any modulation, that is, consisting of endlessly repeating regular oscillations, has no “beginning” and cannot be used to send time signals. To send a signal, the wave must be modified, a mark must be made on it, that is, it must be made thicker or thinner in some places. Then the wave will contain not one frequency, but whole line frequencies, and it can be shown that the speed of signal propagation depends not on one value of the refractive index, but on the nature of the change in the index with frequency. We will put this question aside for now. In ch. 48 (issue 4) we will calculate the speed of propagation of signals in glass and make sure that it does not exceed the speed of light, although the wave crests (purely mathematical concepts) move faster than the speed of light.

A few words about the mechanism of this phenomenon. The main difficulty here is related to the fact that the forced movement of charges is opposite in sign to the direction of the field. Indeed, in expression (31.16) for the charge displacement x, the factor (w 0 -w 2) is negative for small w 0 and the displacement has the opposite sign with respect to the external field. It turns out that when the field acts with some force in one direction, the charge moves in the opposite direction.

How did it happen that the charge began to move in the direction opposite to the force? In fact, when the field is turned on, the charge does not move opposite to the force. Immediately after turning on the field, a transition mode occurs, then oscillations are established and only after this oscillation the charges are directed opposite to the external field. At the same time, the resulting field begins to move ahead in phase with the source field. When we say that the “phase speed”, or the speed of the wave crests, is greater than c, we mean precisely the phase advance.

In fig. Figure 31.4 shows an approximate appearance of the waves that arise when the source wave is suddenly turned on (i.e., when a signal is sent).

Fig. 31.4. Wave "signals".

Fig. 31.5. Refractive index as a function of frequency.

The figure shows that for a wave passing through a medium with a phase advance, the signal (i.e., the beginning of the wave) does not advance in time the source signal.

Let us now turn again to the dispersion formula. It should be remembered that the result we obtained somewhat simplifies the true picture of the phenomenon. To be accurate, some adjustments need to be made to the formula. First of all, damping must be introduced into our model of an atomic oscillator (otherwise the oscillator, once started, will oscillate indefinitely, which is implausible). We have already studied the movement of a damped oscillator in one of the previous chapters [see. equation (23.8)]. Taking damping into account leads to the fact that in formulas (31.16), and therefore

in (31.19), instead of (w 0 2 -w 2) appears (w 0 2 -w 2 +igw)" where g is the attenuation coefficient.

The second amendment to our formula arises because each atom usually has several resonant frequencies. Then, instead of one type of oscillator, you need to take into account the action of several oscillators with different resonant frequencies, the oscillations of which occur independently of each other, and add up the contributions from all oscillators.

Let a unit volume contain N k electrons with a natural frequency (w k and attenuation coefficient g k. As a result, our dispersion formula will take the form

This final expression for the refractive index is valid for a large number of substances. An approximate variation of the refractive index with frequency, given by formula (31.20), is shown in Fig. 31.5.

You can see that everywhere except in the region where w is very close to one of the resonant frequencies, the slope of the curve is positive. This dependence is called “normal” variance (because this case occurs most often). Near resonant frequencies the curve has a negative slope, in which case one speaks of "anomalous" dispersion (meaning "abnormal" dispersion) because it was observed long before electrons were known and seemed unusual at the time, C From our point of view, both inclinations are quite “normal”!

§ 4 Absorption

You've probably already noticed something strange in the last form (31.20) of our dispersion formula. Because of the attenuation term ig, the refractive index has become a complex quantity! What does this mean? Let's express n through the real and imaginary parts:

and n" and n" are real. (In" is preceded by a minus sign, and n" itself, as you can easily see, is positive.)

The meaning of the complex refractive index is most easily understood by returning to equation (31.6) for a wave passing through a plate of refractive index n. Substituting complex n here and rearranging the terms, we get

The multipliers, designated by the letter B, have the same form and, as before, describe a wave, the phase of which, after passing through the plate, is delayed by an angle w (n"-1)Dz/c. Multiplier A (exponent with actual indicator) represents something new. The exponent is negative, therefore A is real and less than one. Multiplier A reduces the amplitude of the field; as Dz increases, the value of A, and therefore the entire amplitude, decreases. When passing through a medium, an electromagnetic wave attenuates. The medium “absorbs” part of the wave. The wave leaves the medium having lost part of its energy. This should not be surprising, because the damping of the oscillators we introduced is due to the force of friction and certainly leads to a loss of energy. We see that the imaginary part of the complex refractive index n" describes the absorption (or "attenuation") of an electromagnetic wave. Sometimes n" is also called the "absorption coefficient".

Note also that the appearance of the imaginary part n deflects the arrow depicting E a in Fig. 31.3, to the origin.

This makes it clear why the field weakens when passing through a medium.

Typically (like glass) the absorption of light is very low. This is exactly what happens according to our formula (31.20), because the imaginary part of the denominator ig k w is much smaller than the real part (w 2 k -w 2). However, when the frequency w is close to w k, the resonant term (w 2 k -w 2) turns out to be small compared to ig k w and the refractive index becomes almost purely imaginary. Absorption in this case determines the main effect. It is absorption that produces dark lines in the solar spectrum. Light emitted from the surface of the Sun passes through the solar atmosphere (as well as the Earth's atmosphere), and frequencies equal to the resonant frequencies of atoms in the Sun's atmosphere are strongly absorbed.

Observation of such spectral lines of sunlight makes it possible to establish the resonant frequencies of atoms, and therefore chemical composition solar atmosphere. In the same way, the composition of stellar matter is determined from the spectrum of stars. Using these methods it was discovered that chemical elements on the Sun and stars are no different from those on earth.

§ 5. Light wave energy

As we have seen, the imaginary part of the refractive index characterizes absorption. Let's now try to calculate the energy transferred by a light wave. We have expressed considerations in favor of the fact that the energy of a light wave is proportional to E 2, the time average of the square of the electric field of the wave. The weakening of the electric field due to wave absorption should lead to a loss of energy, which turns into some kind of electron friction and ultimately, as you might guess, into heat.

Taking the part of the light wave incident on a single area, for example, on a square centimeter of the surface of our plate in Fig. 31.1, we can write the energy balance in the following form (we assume that energy is conserved!):

Incoming energy in 1 sec = Outgoing energy in 1 sec + Work done in 1 sec. (31.23)

Instead of the first term, you can write aE2s, where a is the proportionality coefficient connecting the average value of E2 with the energy transferred by the wave. In the second term it is necessary to include the radiation field of the atoms of the medium, i.e. we must write

a (Es+E a) 2 or (expanding the square of the sum) a (E2s+2E s E a + -E2a).

All our calculations were carried out under the assumption that

the thickness of the material layer is small and its refractive index

differs slightly from unity, then E a turns out to be much less than E s (this was done for the sole purpose of simplifying calculations). As part of our approach, member

E2a should be omitted, neglecting it in comparison with E s E a . You can object to this: “Then you need to discard E s E a, because this term is much less than El.” Indeed, E s E a

much less than E2s, but if we drop this term, we get an approximation in which environmental effects are not taken into account at all! The correctness of our calculations within the framework of the approximation made is verified by the fact that we everywhere left terms proportional to -NDz (the density of atoms in the medium), but threw out terms of order (NDz) 2 and higher degrees in NDz. Our approximation could be called the “low-density approximation.”

Note, by the way, that our energy balance equation does not contain the energy of the reflected wave. But this is how it should be, because the amplitude of the reflected wave is proportional to NDz, and the energy is proportional to (NDz) 2.

To find the last term in (31.23), you need to calculate the work done by the incident wave on the electrons in 1 second. Work, as we know, is equal to force times distance; hence work per unit time (also called power) is given by the product of force and speed. More precisely, it is equal to F·v, but in our case the force and velocity have the same direction, so the product of vectors is reduced to the usual one (up to sign). So, the work done on each atom in 1 second is equal to q e E s v. Since there are NDz atoms per unit area, the last term in equation (31.23) turns out to be equal to NDzq e E s v. The energy balance equation takes the form

The terms aE 2 S cancel and we get

Returning to equation (30.19), we find E a for large z:

(remember that h=NDz). Substituting (31.26) into the left side of equality (31.25), we obtain

Ho E s (at point z) is equal to E s (at atomic point) with a delay of z/c. Since the average value does not depend on time, it will not change if the time argument is delayed by z/c, i.e. it is equal to E s (at the atomic point) v, but exactly the same average value is on the right side (31.25 ). Both parts of (31.25) will be equal if the relation is satisfied

Thus, if the law of conservation of energy is valid, then the amount of energy of an electric wave per unit area per unit time (what we call intensity) should be equal to e 0 cE 2. Denoting the intensity by S, we get

where the bar means the average over time. Our refractive index theory produced a remarkable result!

§ 6. Diffraction of light on an opaque screen

Now is an opportune moment to apply the methods of this chapter to a different kind of problem. In ch. 30 we said that the distribution of light intensity - the diffraction pattern that appears when light passes through holes in an opaque screen - can be found by evenly distributing sources (oscillators) over the area of the holes. In other words, the diffracted wave appears as if the source is a hole in the screen. We must find out the reason for this phenomenon, because in fact there are no sources in the hole, there are no charges moving with acceleration.

Let's first answer the question: what is an opaque screen? Let there be a completely opaque screen between the source S and the observer P, as shown in Fig. 31.6, a. Since the screen is “opaque,” there is no field at point P. Why? According to general principles, the field at point P is equal to the field E s, taken with some delay, plus the field of all other charges. But, as has been shown, the E s field sets the screen charges in motion, and they in turn create a new field, and if the screen is opaque, this charge field should exactly extinguish the E s field from the back wall of the screen. Here you can object: “What a miracle they will be extinguished exactly! What if the repayment is incomplete?” If the fields were not completely suppressed (recall that the screen has a certain thickness), the field in the screen near the rear wall would be different from zero.

Fig. 31.6. Diffraction on an opaque screen.

But then it would set in motion other electrons of the screen, thereby creating a new field that tends to compensate for the original field. If the screen is thick, it has plenty of options to reduce the residual field to zero. Using our terminology, we can say that an opaque screen has a large and purely imaginary refractive index and therefore the wave in it decays exponentially. You probably know that thin layers of most opaque materials, even gold, are transparent.

Let us now see what picture will arise if we take such an opaque screen with a hole as shown in Fig. 31.6, b. What will the field be at point P? The field at point P is composed of two parts - the source field S and the screen field, i.e. the field from the movement of charges in the screen. The movement of charges in the screen is apparently very complex, but the field they create is quite simple.

Let's take the same screen, but close the holes with covers, as shown in Fig. 31.6, c. Let the covers be made of the same material as the screen. Note that the covers are placed in the places where in Fig. 31.6, b shows the holes. Let us now calculate the field at point P. The field at point P in the case shown in FIG. 31.6, in, of course, is equal to zero, but, on the other hand, it is also equal to the field of the source plus the field of the electrons of the screen and covers. We can write the following equality:

The strokes refer to the case when the holes are closed with covers; the value of E s in both cases is, of course, the same. Subtracting one equality from the other, we get

If the holes are not too small (for example, many wavelengths wide), then the presence of the covers should not affect the field at the screen, except perhaps in a narrow region near the edges of the holes. Neglecting this small effect, we can write

E walls = E" walls and, therefore,

We come to the conclusion that the field at point P with open holes (case b) is equal (up to sign) to the field created by that part of the solid screen that is located in the place of the holes! (We are not interested in the sign, since we usually deal with an intensity proportional to the square of the field.) This result is not only valid (in the approximation of not very small holes), but also important; among other things, it confirms the validity of the usual theory of diffraction:

The field E of the cover is calculated under the condition that the movement of charges everywhere in the screen creates exactly such a field that extinguishes the field E s on the back surface of the screen. Having determined the movement of the charges, we add up the radiation fields of the charges in the covers and find the field at point P.

Let us recall once again that our theory of diffraction is approximate and is valid in the case of not too small holes. If the size of the holes is small, the term E"of the lid is also small and the difference E" of the wall -E of the wall (which we assumed to be equal to zero) can be comparable and even much larger than e"of the lid. Therefore, our approximation turns out to be unsuitable.

* The same formula is obtained using quantum mechanics, but its interpretation in this case is different. In quantum mechanics, even a single-electron atom, such as hydrogen, has several resonant frequencies. Therefore, instead of the number of electrons N k with frequency w k Nf multiplier appears k where N is the number of atoms per unit volume, and the number f k (called oscillator strength) indicates with what weight a given resonant frequency is included w k .

Processes that are associated with light are an important component of physics and surround us everywhere in our everyday life. The most important in this situation are the laws of reflection and refraction of light, on which modern optics is based. The refraction of light is an important part of modern science.

Distortion effect

This article will tell you what the phenomenon of light refraction is, as well as what the law of refraction looks like and what follows from it.

Basics of a physical phenomenon

When a beam falls on a surface that is separated by two transparent substances that have different optical densities (for example, different glasses or in water), some of the rays will be reflected, and some will penetrate into the second structure (for example, they will propagate in water or glass). When moving from one medium to another, a ray typically changes its direction. This is the phenomenon of light refraction.
The reflection and refraction of light is especially visible in water.

Distortion effect in water

Looking at things in water, they appear distorted. This is especially noticeable at the boundary between air and water. Visually, underwater objects appear to be slightly deflected. The described physical phenomenon is precisely the reason why all objects appear distorted in water. When the rays hit the glass, this effect is less noticeable.
Refraction of light is a physical phenomenon that is characterized by a change in direction of movement sunbeam at the moment of moving from one environment (structure) to another.
To improve our understanding of this process, consider an example of a beam hitting water from air (similarly for glass). By drawing a perpendicular line along the interface, the angle of refraction and return of the light beam can be measured. This index (angle of refraction) will change as the flow penetrates the water (inside the glass).
Note! This parameter is understood as the angle formed by a perpendicular drawn to the separation of two substances when a beam penetrates from the first structure to the second.

Beam Passage

The same indicator is typical for other environments. It has been established that this indicator depends on the density of the substance. If the beam falls from a less dense to a denser structure, then the angle of distortion created will be greater. And if it’s the other way around, then it’s less.
At the same time, a change in the slope of the decline will also affect this indicator. But the relationship between them does not remain constant. At the same time, the ratio of their sines will remain constant value, which is reflected by the following formula: sinα / sinγ = n, where:

n is a constant value that is described for each specific substance (air, glass, water, etc.). Therefore, what this value will be can be determined using special tables;
α – angle of incidence;
γ – angle of refraction.

To determine this physical phenomenon and the law of refraction was created.

Physical law

The law of refraction of light fluxes allows us to determine the characteristics of transparent substances. The law itself consists of two provisions:

First part. The beam (incident, modified) and the perpendicular, which was restored at the point of incidence on the boundary, for example, of air and water (glass, etc.), will be located in the same plane;
The second part. The ratio of the sine of the angle of incidence to the sine of the same angle formed when crossing the boundary will be a constant value.

Description of the law

In this case, at the moment the beam exits the second structure into the first (for example, when the light flux passes from the air, through the glass and back into the air), a distortion effect will also occur.

An important parameter for different objects

The main indicator in this situation is the ratio of the sine of the angle of incidence to a similar parameter, but for distortion. As follows from the law described above, this indicator is a constant value.
Moreover, when the value of the decline slope changes, the same situation will be typical for a similar indicator. This parameter has great importance, since it is an integral characteristic of transparent substances.

Indicators for different objects

Thanks to this parameter, you can quite effectively distinguish between types of glass, as well as various precious stones. It is also important for determining the speed of light in various environments.

Note! The highest speed of light flow is in a vacuum.

When moving from one substance to another, its speed will decrease. For example, in diamond, which has the highest refractive index, the speed of photon propagation will be 2.42 times higher than that of air. In water, they will spread 1.33 times slower. For different types glass this parameter ranges from 1.4 to 2.2.

Note! Some glasses have a refractive index of 2.2, which is very close to diamond (2.4). Therefore, it is not always possible to distinguish a piece of glass from a real diamond.

Optical density of substances

Light can penetrate through different substances, which are characterized by different optical densities. As we said earlier, using this law you can determine the density characteristic of the medium (structure). The denser it is, the slower the speed at which light will propagate through it. For example, glass or water will be more optically dense than air.
In addition to the fact that this parameter is a constant value, it also reflects the ratio of the speed of light in two substances. The physical meaning can be displayed as the following formula:

This indicator tells how the speed of propagation of photons changes when moving from one substance to another.

Another important indicator

When a light flux moves through transparent objects, its polarization is possible. It is observed during the passage of a light flux from dielectric isotropic media. Polarization occurs when photons pass through glass.

Polarization effect

Partial polarization is observed when the angle of incidence of the light flux at the boundary of two dielectrics differs from zero. The degree of polarization depends on what the angles of incidence were (Brewster's law).

Full internal reflection

Concluding our short excursion, it is still necessary to consider such an effect as full internal reflection.

The phenomenon of full display

For this effect to appear, it is necessary to increase the angle of incidence of the light flux at the moment of its transition from a more dense to a less dense medium at the interface between substances. In a situation where this parameter exceeds a certain limiting value, then photons incident on the boundary of this section will be completely reflected. Actually, this will be our desired phenomenon. Without it, it was impossible to make fiber optics.

Conclusion

The practical application of the behavior of light flux has given a lot, creating a variety of technical devices to improve our lives. At the same time, light has not yet revealed all its possibilities to humanity and its practical potential has not yet been fully realized.

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For some substances, the refractive index changes quite strongly when the frequency of electromagnetic waves changes from low frequencies to optical frequencies and beyond, and can also change even more sharply in certain regions of the frequency scale. The default usually refers to the optical range or the range determined by the context.

The ratio of the refractive index of one medium to the refractive index of the second is called relative refractive index the first environment in relation to the second. For executed:

where and are the phase speeds of light in the first and second media, respectively. Obviously, the relative refractive index of the second medium with respect to the first is a value equal to .

This value, other things being equal, is usually less than unity when a beam passes from a more dense medium to a less dense medium, and more than unity when a beam passes from a less dense medium to a denser medium (for example, from a gas or from a vacuum to a liquid or solid ). There are exceptions to this rule, and therefore it is customary to call the environment optically more or less dense than another (not to be confused with optical density as a measure of the opacity of a medium).

A ray falling from airless space onto the surface of some medium is refracted more strongly than when falling on it from another medium; the refractive index of a ray incident on a medium from airless space is called its absolute refractive index or simply the refractive index of a given medium, this is the refractive index, the definition of which is given at the beginning of the article. The refractive index of any gas, including air, under normal conditions is much less than the refractive index of liquids or solids, therefore, approximately (and with relatively good accuracy) the absolute refractive index can be judged from the refractive index relative to air.

Examples

The refractive indices of some media are given in the table.

Refractive indices for wavelength 589.3 nm

Environment type	Wednesday	Temperature, °C	Meaning
Crystals	LiF	20	1,3920
	NaCl	20	1,5442
	KCl	20	1,4870
	KBr	20	1,5552
Optical glasses	LK3 (Light Crown)	20	1,4874
	K8 (Cron)	20	1,5163
	TK4 (Heavy crown)	20	1,6111
	STK9 (Super heavy crown)	20	1,7424
	F1 (Flint)	20	1,6128
	TF10 (Heavy flint)	20	1,8060
	STF3 (Super heavy flint)	20	2,1862
Gems	Diamond white	-	2,417
	Beryl	-	1,571 - 1,599
	Emerald	-	1,588 - 1,595
	Sapphire white	-	1,768 - 1,771
	Sapphire green	-	1,770 - 1,779
Liquids	Distilled water	20	1,3330
	Benzene	20-25	1,5014
	Glycerol	20-25	1,4370
	Sulfuric acid	20-25	1,4290
	Hydrochloric acid	20-25	1,2540
	Anise oil	20-25	1,560
	Sunflower oil	20-25	1,470
	Olive oil	20-25	1,467
	Ethanol	20-25	1,3612

Negative index materials

phase and group velocities of waves have different directions;
it is possible to overcome the diffraction limit when creating optical systems (“superlenses”), increasing the resolution of microscopes with their help, creating nanoscale microcircuits, increasing the recording density on optical storage media).

Notes

Links

RefractiveIndex.INFO refractive index database

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See what “Refractive Index” is in other dictionaries:

REFRACTIVE INDEX- the ratio of the speed of light in a vacuum to the speed of light in a medium (absolute refractive index). The relative refractive index of 2 media is the ratio of the speed of light in the medium from which light falls on the interface to the speed of light in the second... ... Big Encyclopedic Dictionary

REFRACTIVE INDEX Modern encyclopedia

Refractive index- REFRACTIVE INDEX, a quantity characterizing the medium and equal to the ratio of the speed of light in a vacuum to the speed of light in the medium (absolute refractive index). The refractive index n depends on the dielectric e and magnetic permeability m... ... Illustrated encyclopedic Dictionary

REFRACTIVE INDEX- (see REFRACTION INDEX). Physical encyclopedic dictionary. M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1983 ... Physical encyclopedia

refractive index- 1. The ratio of the speed of the incident wave to the speed of the refracted wave. 2. Ratio of sound speeds in two media. [System non-destructive testing.… … Technical Translator's Guide

refractive index- the ratio of the speed of light in a vacuum to the speed of light in a medium (absolute refractive index). The relative refractive index of two media is the ratio of the speed of light in the medium from which light falls on the interface to the speed of light in ... ... encyclopedic Dictionary

refractive index- lūžio rodiklis statusas T sritis automatika atitikmenys: engl. index of refraction; refraction index; refractive index vok. Brechungsindex, m; Brechungsverhältnis, n; Brechungszahl, f; Brechzahl, f; Refractionsindex, m rus. refractive index, m; … Automatikos terminų žodynas

refractive index- lūžio rodiklis statusas T sritis chemija apibrėžtis Medžiagos konstanta, apibūdinanti jos savybę laužti šviesos bangas. atitikmenys: engl. index of refraction; refraction index; refractive index rus. refractive index; refractive index;… … Chemijos terminų aiškinamasis žodynas

refractive index- lūžio rodiklis statusas T sritis Standartizacija ir metrologija apibrėžtis Esant nesugeriančiai terpei, tai elektromagnetinės spinduliuotės sklidimo greičio vakuume ir tam tikro dažnio elektromagnetinės spinduliuotės fazinio gr eičio terpėje… …

refractive index- lūžio rodiklis statusas T sritis Standartizacija ir metrologija apibrėžtis Medžiagos parametras, apibūdinantis jos savybę laužti šviesos bangas. atitikmenys: engl. refraction index; refractive index vok. Brechungsindex, m rus. index… … Penkiakalbis aiškinamasis metrologijos terminų žodynas

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