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15ИНТЕРФЕРЕНЦИЯ РЕНТГЕНОВСКИХ ЛУЧЕЙ С УЧЕТОМ ДЛИТЕЛЬНОСТИ
КОГЕPЕНТНОГО ИЗЛУЧЕНИЯ
Дрмеян Г.Р.
доктор технических наук, профессор Гюмрийский государственный педагогический институт, Армения
Абоян А. О.
доктор физико-математических наук, профессор Национальный политехнический университет Армении
Григорян Г. Р.
Кандидат физико-математических наук, Гюмрийский государственный педагогический институт, Армения
INTERFERENCE OF X-RAYS WITH DUE REGARD FOR DURATION OF COHERENT RADIATION
Drmeyan H.R., doctor of science in technology Departament of Physics, Gyumri State Pedagogical Institute, Gyumri, Armenia
Aboyan A.H., doctor of science in physics and mathematics Department of Applied Mathematics and Physics, National Polytechnic University of Armenia
Grigoryan G. R., PhD of science in physics and mathematics Departament of Physics, Gyumri State Pedagogical Institute, Gyumri, Armenia
АННОТАЦИЯ
Исследованы зависимости видимости интерференционных картин от разности интенсивностей интерферирующих волн. Объяснены причины уменьшения видимости интерференционной картины с увеличением разности хода между налагающими волнами. Показано, что для точного определения ширины и интенсивности дифракционных максимумов необходимо учесть время продолжительности когерентного излучения. Дано определение цуга и группы волн. Доказано, что когда толщина кристалла порядка или меньше длины цуга, длительность времени когерентного излучения можно не учитывать, а при толщине кристалла больше длины цуга неучет (пренебрежение) длительности когерентного излучения приводит к недопустимым ошибкам.
ABSTRACT
It is investigated the dependence of the visibility of interference patterns on the difference of interfering waves intensities. The cause of decrease in interference pattern visibility with increasing path difference between the imposing waves is explained. It is shown that for correct determination of the width and intensity diffraction maxima the duration of coherent radiation must be taken into account. The definition of the wave train and wave packet is given. It is proved that when the crystal thickness is about or less than the length of train the duration of coherent radiation can be ignored, and when the crystal thickness is more than the length of train its neglect leads to unacceptable errors.
Ключевые слова: интерференция, интенсивность, цуг волны поезд, длина цуга, длина когерентности, динамическое рассеяние.
Keywords: interference, intensity, wave train, train length, length of coherency, dynamic scattering.
1. Introduction
In different problems of X-ray interference it is supposed that irrespective of the size of an irradiated crystal X- rays are coherently scattered over its whole bulk.
It is known [1-3] that the waves are coherent when they are emitted by an atom during the same emission event. Hence, the interference pattern is non-observable even when the waves emitted by the same atom are made to interfere at different times and besides with a delay, and large duration of coherent radiation (duration of one event).
Indeed, the drop in visibility of Nomarski fringes with increasing order of interference [4,5] was noted by Michelson yet. For determination of visibility he introduced a function
V =
I -1 .
max_mm
1 max + 1 min
(1)
where Imax and Imin are the intensities of bright and neighboring dark fringes respectively.
According to this definition, in case of strictly monochromatic wave the visibility for any path difference
is unity (Imin = 0, V = 1). At any slightest deviation from monochromaticity, the visibility of fringes V decreases with increasing path difference (order of interference), and at that the faster, the higher is the nonmonochromaticity. So, the visibility is characterized by the degree of radiation monochromaticity. As strictly monochromatic (i.e., exactly sinusoidal) waves do not practically feasible [1] the visibility of fringes always diminishes with increasing order of interference and, eventually, the interference patterns disappear at sufficiently large path differences.
The present survey aims at investigation and detailed analysis of the dependence of the visibility of interference patterns on the difference of interfering waves intensities. This research is especially important for practical use of poly-interferometers, in which different intensity beams are used [68], as well as for estimation of the length of train and duration of coherent radiation.
To establish the cause of decrease in interference pattern
visibility with increasing path difference between the waves, we shall investigate in what follows the nature of white light (with continuous spectrum of X-ray radiation) and examples of real monochromatic radiation.
Let us first define the wave train and wave packet [1]. The wave train is defined as the vibration of the form described at some length by a simple sinusoidal curve (with constant or slightly varying amplitude), the amplitude of which is zero outside this length. The wave packet is the resultant of a series of harmonic waves, the frequencies of which are grouped about the main frequency so that if the frequencies of some members of group differ from the main frequency by more than some small share of the latter, then their amplitudes shall be notably less than those, the frequencies of which are close to the main one, i.e., almost all energy is concentrated at frequencies near the main frequency of wave packet.
So, the train of waves is determined from the waveform, and the wave packet - from the frequency distribution of energy. Corresponding to a lengthy train of waves is a narrow frequency range and vice versa, to a wave packet, the frequencies of which are confined within a narrow interval, there corresponds a lengthy train. The white light may be represented both as a group of pulses (wave packet) and a set of sinusoidal wave trains. These representations are equivalent because a pulse is mathematically expandable into series of harmonic waves using the Fourier theorem.
Since it is impossible to form a strictly monochromatic beam consisting of a single wavelength component (infinite train) by means of experimental arrangements intended for production of monochromatic radiation, real monochromatic beams represent finite frequency ranges.
The formation of infinitely long train (of strictly monochromatic radiation) would lead to the Corvallo paradox [1]: if the white light consists of a set of infinitely long trains, then these may be separated by means of a spectroscope and the spectrum would be observable before the lighting and after quenching of the source.
Proceeding from the aforesaid let us now discuss possible reasons for reduction of interference pattern visibility as the path difference increases.
1. In case of a set of large number of sinusoidal waves with slightly differing wavelengths, the maxima of all waves at zero path difference coincide and a sharp interference pattern is observed. With increasing path difference the maxima of different wavelengths begin to diverge, the visibility is reduced and at large path differences the interference maxima are not visible.
2. For a set of final length wavetrains the interference occurs between two parts of the same wavetrain formed in two channels of interferometer. In case of small path differences these parts appear in observation point almost concurrently and interfere, whereas at large path differences one part appears in the observation point after arrival of the other, i.e., the parts meeting in observation point are from different emission events and the interference pattern are invisible.
So, irrespective of whether the light is described as a set of final length wavetrains or a set of large number of sinusoidal waves with slightly differing wavelengths, at sufficiently large path differences the interference does not occur, and the
interference fringes are invisible.
In X-ray case (multibeam interference) one may assume that separate sinusoidal components of pulse simultaneously excite all atomic planes of the crystal. However, the excitation of some sinusoidal waves is suppressed by the others (the summary excitation is zero) in all parts of crystal except for the part excited by the packet maximum (wavetrain). Thus, in reality a single emission event excites only the part of planes determined by the length of train or the spatial extension of the packet maximum (the duration of coherent radiation). Hence, separate parts of crystal sufficiently distant one from the other (large path differences) are excited by radiation from different emission events and for this reason are scattering incoherently.
2. Estimation of wavetrain length and of the duration of coherent radiation
Now let us estimate the length of wavetrain and duration of coherent radiation. As is evident from the aforesaid, since coherent are only the waves that are parts of the same train, the duration of coherent radiation is the duration of one emission event (duration of wavetrain emission).
The finiteness of wavetrain and, hence, the finiteness of the duration of coherent radiation as well as of the natural width of spectral line are stipulated in the classical theory by the gradual decrease in oscillator amplitude (damping due to the emission of radiation) or in the quantum theory by decreasing probability of staying in the initial state [9 -11].
According to classical concepts, each transition of an atom from one level to the other may be associated with one classical oscillator [9], which, from the point of view of classical electrodynamics, produces damped oscillations during the emission event.
The damping of oscillator due to the emission proceeds
according to law Wt = W0e Y , where W0 and Wt are the oscillator energies (of the oscillating electron) at instants t = 0 and t respectively and y (the damping factor) is determined by the relation [11]
Y =
8n2e2 1 3mc A2
i.e., by the value of radiation intensity. Hence, we have for the effective lifetime or the decay time of the oscillator
1
t = — = ■
3mc
8n2 e2
a.
Y e (2)
It is seen that the energy of oscillator (electron) and, hence, the radiation intensity decreases exponentially, and, hence, the time t may be accepted as the duration of a separate emission event or the duration of coherent radiation.
From the quantum theory viewpoint [9,10] in the energy-time uncertainty relation one has
AW At = h (3)
In the case under consideration AW is the width of the level and At is the lifetime of level or the duration of emission event (duration of coherent radiation).
For duration of coherent radiation we obtain from (3)
t = -
AW
(4)
that is related to the transition probability per second as 1
t = —.
Y (5)
If several transitions are possible from the excited to the low-lying levels, then the total probability of transition will be expressed as
Y
(6)
where k is the index of the low-lying level to which the transition is performed.
Hence, for the lifetime of level we obtain 1 1 t = - = -
Y
Therefore, the effective lifetime of the i-th level is expressed in terms of the widths of initial and possible final k levels as follows:
1
h
h
t v. aw yaw., '
' I I / j IK
where AW., is the width of the k-th level.
ik
(7)
tc1 = 0.146 -10 c, t
quant
= 0.114-10-
width of spectral line based on the energy conservation law. Hence, the duration of coherent radiation may be determined in the framework of quantum mechanics also with the help of total damping of radiations with analogous results. But if the lifetime of excited state (duration of coherent radiation) is determined from the total damping of radiation, it is clear that the durations of separate acts should not exceed the total duration.
The length of train l is determined by the relation
l = OT, (9)
where c is the propagation speed of electromagnetic waves. The dimensions of coherently scattering domains of the irradiated volume are determined precisely by the length of train. If the dimensions of irradiated volume exceed the length of train, then the irradiated volume does not scatter the rays coherently as a whole and for this reason the methods for calculation of the intensity and width of reflexes are not correct. As is seen from (9), the length of X-ray wavetrain is of the order
of 10-5 ... 10-6 cm. Indeed, the values of t for e.g., CuKa and
MoKa , are respectively 1,077x10-15c and 2,29x10-16c, whence we obtain for sizes of coherently scattering volumes
W, = 3.23 -10-
and
L
= 6.87 -10-5 cm.
As is seen from (2), in the classical electrodynamics the effective lifetime of oscillator depends only on the wavelength, whereas as is seen in (7) in the quantum theory the lifetime of level depends on transition probabilities (on the widths of levels).
It is easy to ascertain that the durations of coherent radiation both in classical (the effective lifetime of the oscillator (2)) and quantum mechanics (the lifetime of level (7)) are of the same order of magnitude. We can confirm that based on a case study.
Substituting the numerical values of e, m and c in (2), we obtain
t = 4.53A2 c. (8)
For gold Ka (K -LJ [9] the width of line is 58eV (the width of the K -th level is 54eV and that of LIII level is -4 eV),
the wavelength Ak = 0.17982 A. For duration of cohe-rent radiation we obtain respectively from (8) and (4) with due regard for the above values
As the dimensions of irradiated volume for gases, liquids and amorphous solids are determined by cross sections of the incident beam that practically could not be less than 10-2 cm, it is evident that in these cases the dimensions of irradiated volume much exceed the size of train, and for such specimens it is impossible to calculate the intensities and widths of diffraction maxima without taking the duration of coherent radiation into account.
In case of crystalline specimens one could disregard the duration of coherent radiation if the sizes of small crystals are less than those of trains, i.e., if the crystals are smaller than cm.
As there is no strict monochromaticity and it is impossible in nature, we arrive at the following essential conclusion: irrespective of the fact whether the light (X-rays) is represented by a set of wavetrains or, vice versa, by sinusoidal waves with different wavelengths (wave packet), in case of sufficiently large irradiated volumes not all the waves scattered in different parts of this volume will be coherent one with respect to the other. For this reason it seems necessary to revise the existing theory of X-ray interference from the viewpoint of finiteness of coherent radiation duration.
3. The influence of the finiteness of X-ray train length on the scattering intensity
It is seen that the classical and quantum values of coherent radiation duration agree sufficiently well.
In the quantum-mechanical case the duration of coherent radiation was determined based on the width of line produced by a collection (ensemble) of atoms. For this reason it may seem that the durations of radiation from separate atoms are much larger than that determined from (7).
However, the formalism of quantum mechanics allows for radiation damping in the same full measure as the classical theory, and the quantum theory also readily accounts for natural
time
Fig. 1.
For the first time the incoherence of X-ray waves pertaining
n
16
0
T
to different trains was noticed by Professor P. Bezirganyan [12].
In his early works [13-18] the research was conducted within the framework of kinematical theory of X-ray interference. In these works the diffraction of X- rays in gases, liquids, amorphous solids and crystals has been investigated on assumption that the train of X-rays has a form of step shown in Fig.1, i.e., in the period of time from zero to t the amplitude of wave is constant and different from zero, and outside that it is zero, where t is the duration of coherent radiation.
The basic conclusions drawn during these research efforts
are:
1.The average intensity of waves scattered in gases is a function of the duration of radiation,- the intensity of scattering increases with duration of coherent radiation.
2.For not too small scattering angles in gases, the total intensity of scattered waves increases with decreasing scattering angle on account of the increase in the number of motifs simultaneously sharing in the interference at the observation point.
3.In case of gases under high pressure and liquids with due regard for the finiteness of coherent radiation time, the radiated volume seems as though to be decreased: the intensity of diffraction halo decreases but the scattering angle at which it was observed is unchanged.
4.At incidence of plane wave on a plane crystal the intensity of scattered waves with due regard for the duration of coherent radiation decreases; in case of point source the intensity of scattered waves increases with the incidence angle.
5.In case when the length of wavetrain much exceeds the crystal thickness, the finiteness of the duration of coherent radiation is of no importance.
The case when during one emission event the source damping, i.e., the value of primary wave amplitude exponentially decreases, as exp |-y t}, where y is the damping coefficient, has been investigated in [18]. It was also shown that when the length of train is less than the crystal thickness, the damping strongly affects the intensity and width of reflexes, and almost does not affect these when the train length much exceeds the crystal size.
In [18] the primary radiation is represented by the integral of monochromatic waves of the following type:
1 X
E (t ) = — [ a(o)exp{iat}da,
where
with those obtained in [18].
In [19, 20] the dynamical scattering of X- rays in Bragg case was investigated by means of Darwin method for an infinitely thick crystal and a crystal plate with due regard for primary wave damping. It was shown that the total reflection does not occur even in case of infinitely thick crystal. The relative intensity of reflected wave is much less than unity at small durations of coherent radiation, but increases with that going to unity at infinitely large durations of coherent radiation.
The dynamical scattering of X- rays with due regard for primary wave damping in transparent and absorbing crystals has been investigated in [21,22] by means of Darwin method using more rigorous approach to the allowance for source damping. It was shown that with allowance for the source damping the total reflection does not occur, as it was in the previous case, and the more is the damping, the less is the relative intensity. The allowance for source damping does not distort the reflection symmetry and the center of reflection curve is inclined at the same angle, at which it was obtained in the absence of damping.
In absorbing crystals due to the allowance for damping the asymmetry of reflection is retained. With increasing of the role of damping of primary radiation source, the reflection curves grow more and more symmetric and the maximum of relative intensity decreases.
In [23] a dynamical theory ofX-ray interference allowing for source damping in Laue and Bragg cases has been developed in spherical wave approximation. It was shown that the allowance for source damping leads to a shift of pendulum fringes, the maxima of fringes being displaced to the front surface of crystal (Laue case). In the Bragg case, due to the damping the distribution of intensity on the front surface is displaced in direction opposite to the absorption-initiated displacement.
Both in Laue and in Bragg cases the allowance for damping leads to some reduction in the contrast of pendulum fringes.
Based on Section 1 data it is concluded that for calculation of scattered wave intensity and diffraction widths of spectral lines it is necessary first to find out the relation of X-ray train length to dimensions of perfection ranges of single crystals. For this reason the determination of X-ray train length is important not only from the view point of interference theory, but also of X-ray structural analysis in general. It is especially important for X-ray interferometry and holography, as well as for verification of theoretical ideas set forth in Section 1.
The next article will be devoted to problems of experimental determination of the length of train of X-ray waves with the help of a three-block interferometer (called by us a trein meter).
a(œ) =-E-^exp{-iœ0 r }:
y + i(a-a0 ) c
E0 = e^4^exp
c2 R
y-
A0 is the amplitude of initial oscillations, r is the distance from observation point to the source, 9 is the angle made by the electron acceleration direction to the radius-vector r.
The results obtained in [17] coincide almost completely
Refernces
[1] Ditchburn, R.W. (1965). Light. London
[2] Franson, M., Slawskij, S. (1967). Coherence in Optics (in Russian).
[3] Born, M. & Wolf, E. (1968). Principles of Optics. Paris
[4] Michelson, A.A. (1891). Phil. Mag. 31, 388.
[5] Michelson, M. (1934). Light Waves and their Application.
[6] Aboyan, A.H. (1996). Crys.Res.Technol. 31,513-519.
[7] Aboyan, A.H., Khzardzhyan, A.A.(2003). Izv. NAN Armenia, ser. fiz., 38, 326 - 334 (in Russian)
r
c
[8] Aboyan, A.H. (2007). Izv. NAN Armenia i SEU Armenia, ser. tech. nauk. 60, 252 - 260 (in Russian).
[9] Heitler, M. (1956). Quantum Theory of Radiation.
[10] Sobelman, I.I. (1963). Introduction to the Theory of Atomic Spectra. Moscow (in Russian).
[11] Frish, C.E. (1963). Optical Spectra of Atoms. Moscow (in Russian).
[12] Bezirganyan, P.A. (1963). Dokl.Akad.Nauk Arm.SSR, 37, 197- 201
[13] Bezirganyan, P.A. (1964). JTP, 34, 1895 - 1900.
[14] Bezirganyan, P.A. (1964). Izv.Akad.Nauk SSSR, 28, 882 - 884 (in Russian).
[15] Bezirganyan, P.A. (1965). JTP, 35, 359 - 367.
[16] Bezirganyan, P.A. (1965). JTP, 35, 1701 - 1706.
[17] Bezirganyan, P.A., Gasparyan, L.G. (1970). JTP, 40, 2427-2433.
[18] Bezirganyan, P.A., Gasparyan, L.G. (1971). Izv. Akad. Nauk Arm. SSR, ser. fiz., 6, 106 - 115.
[19] Bezirganyan, P.A. (1966). JTP, 36, 514 - 520.
[20] Bezirganyan, P.A. (1966). Izv.Akad.Nauk Arm.SSR, ser. fiz., 1, 25 - 30.
[21] Bezirganyan, P.A., Azizyan, S.L. (1973). Molodoi nauchni rabotnik, YSU, 2, 18 - 25 (in Russian).
[22] Azizyan, S.L., Bezirganyan, P.A. (1971). JTP, 41, 2186 - 2190.
[23] Bezirganyan, P.A., Truni, K.G., Vardanyan, D.M., Levonyan, L.V. (1973). Molodoi nauchni
rabotnik, YSU, 6, 15-20 (in Russian).
НАПРЯЖЁННО-ДЕФОРМИРОВАННОЕ СОСТОЯНИЕ ЖЕЛЕЗОБЕТОННЫХ ТРАПЕЦИЕВИДНЫХ И ЭЛЛИПТИЧЕСКИХ БАЛОК
Немировский Юрий Владимирович
докт. физ.-мат. наук, профессор, Институт теоретической и прикладной механики им. С.А. Христиановича СО РАН Болтаев Артем Иванович аспирант,
Институт теоретической и прикладной механики им. С.А. Христиановича СО РАН
STRESS-STRAIN STATE OF REINFORCED CONCRETE TRAPEZOIDAL AND ELLIPTICAL BEAMS
Nemirovskii Y. V., DSc, Professor Siberian branch of Russian academy of sciences Khristianovich institute of theoretical and applied mechanics
Boltaev A.I., Research Assistant, Siberian branch of Russian academy of sciences Khristianovich institute of theoretical and applied mechanics
АННОТАЦИЯ
Поставлена и решена задача определения напряжений, деформаций и перемещений железобетонных балок до момента образования трещин, с учётом реальных диаграмм деформирования арматуры и бетона. Используемый в работе подход позволяет рас-сматривать балки с поперечным сечением состоящим из нескольких слоёв. Все слои сечения могут иметь различную зависимость изменения ширины по высоте сечения, могут быть вы-полнены из различных марок бетона и иметь различную структуру армирования. Увеличение количества слоёв не вносит каких либо серьёзных усложнений в расчёт и что самое главное, не изменяет алгоритма решения задачи. Следовательно, наряду с эллиптическими и трапециевидными сечениями, можно также рассматривать широко распространённые в практике железобетонного проектирования тавровые, двутавровые и другие виды поперечных сечений. Приведены численные эксперименты, которые показывают влияние изменения класса арматуры, марки бетона, структуры армирования и формы сечения на напряжённо-деформированное состояние балок.
Предложенный подход позволяет, с единых позиций, определять напряженно дефор-мированное состояние стержневых конструкций, выполненных как из железобетона, так и из бетона, древесины и других материалов, не подчиняющихся классическому закону Гука.
ABSTRACT
Posed and solved the problem of determining the stress, strain and displacement concrete beams until cracking, taking into account the real strain diagrams of concrete and reinforcement. As used in the approach allows us to consider the beam with a cross section consisting of several layers. All layers may have different sectional dependence of the width adjustment section may be made of different grades of concrete and have a different structure reinforcement. Increasing the number of layers does not introduce any major complications in the calculation, and most importantly, does not change the algorithm for solving the problem. Therefore, along with the elliptical, and trapezoidal, it can also be seen in the widespread practice of reinforced concrete design tees, H-sections and other types of cross-sections. Numerical experiments show that the effect of changing the class of reinforcement, concrete grade, reinforcement of the structure and shape of the cross section in the tensely deformed state of the beams.
