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25METHOD FOR INTERFEROMETRIC DETERMINATION OF X-RAY TRAIN LENGTH
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
KNYAZYAN Z. H. student master
Departament of Physics, Gyumri State Pedagogical Institute, Gyumri, Armenia
ИНТЕРФЕРОМЕТРИЧЕСКИЙ МЕТОД ИЗМЕРЕНИЯ ДЛИНЫ ЦУГА РЕНТГЕНОВСКОГО 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
Князян З. Г., студент магистор, Гюмрийский государственный педагогический институт, Армения
АННОТАЦИЯ
Разработан, изгoтовлен и опробoван особый интерферометр-цугомер. Предложен рентгеноинтерферометрический метод измерения длины цуга рентгеновского изучения. Доказано, что интерференционная картина исчезает, когда разность ходов между налагающими волнами станет больше, чем длина цуга рентгеновского монохроматического излучения. Определено предел исчезновения интерференционной картины в зависимости от величины разностей ходов. Определена длина цуга рентгеновского излучения, которая близка к значению, определяемому теоретическим путем.
ABSTRACT
It is elaborated, made and tested a special interferometer-train metering device. А method for interferometric determination of X-Ray train length is offered. It is proved that the interference pattern disappears, when the path difference between imposing waves is more than the monochromatic X-Ray train length. The limit of disappea^nce of interference pattern depending on the value of path difference is determined. The X-Ray train length is determined which is close to the meaning determined theoretically.
Ключевые слова: интерференция, интенсивность, волновой цуг, длина цуга, длина когерентности, динамическое рассеяние.
Key words:interference, intensity, wave train, train length, length of coherency, dynamic scattering.
Introduction
It is known [1-4] 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. 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. 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 about train [5].
In our opinion the interferometric method is fairly appropriate for measuring the train length of X-ray coherent radiation. This aim in view the radiation from X-ray source is split into two beams and these are made to intersect after passage of different paths and produce an interference pattern. Then the path difference of these two beams is increased till the disappearance of obtained interference pattern. It is evident that the interference pattern disappears from the moment when the path difference between these two beams exceeds the train length of monochromatic X-rays. So, by measuring the path difference at the moment of pattern disappearance one measures the train length, i.e., the duration of coherent radiation producing this interference pattern. This experiment is rather complicated and requires the employment of methods of X-ray interferometry.
It is known [6-10] that in optics the interference pattern disappears at sufficiently large path differences between the waves. This effect, however, was not observed at the interference of X-rays until X-ray interferometers appeared. X-ray interferometers appeared as a result of the discovery of anomalous passage (the Borman effect) and of following rapid development of the dynamical theory of X-ray interference. Based on the experience of making interferometers of different design and of their applications, we have designed and made an X-ray train metering device for measurement of train length (duration of coherent radiation).
The aim of the present work is to elaborate, make and test an X-ray train metering device and use that for determination of X-ray train length.
1. Experimental procedure and its substantiation
At the choice of experimental procedure it is necessary to have in mind that:
1. As was mentioned above, the interference pattern disappears as the path difference between the interfering waves is gradually increased;
2. The interference pattern disappears in case when the amplitudes of interfering waves vary considerably: the sum and difference of amplitudes of these waves insignificantly differ from one another (the visibility diminishes);
3. At first sight it may seem that the phase (path) difference between interfering waves may be easily increased by placing in the path of one of the waves a medium with refractive index different from unity. However, that may entail new difficulties.
As the refractive index of X-rays for media (other than vacuum or air) is slightly different from unity, to obtain differences in optical path lengths in excess of train length it is necessary to introduce a medium of about 10 cm extent, owing to which first, the dimensions of interferometer would extremely increase and, second, due to the absorption in medium the wave would so attenuate that the interference pattern would disappear on account of the Item 2 reasoning. Detailed discussions of this problem have shown that for measuring X-ray train length it is necessary to produce a phase difference between interfering waves by providing a difference in their geometrical paths.
For solution of this task we came down to the interferometer schemes shown in Fig. 1, that were devised to meet the following conditions:
1. The waves superimposing in these interferometers overlap exactly on the surface of third block irrespective of their points of entry from the first block, i.e., these instruments function as interferometers.
2. When the primary wave is incident on the first blocks of
b2 and (Fig.1) in points A and
interferometers a
A3 B
B3 C
C
A A B'
1 and 3, 1 and
this part of interferometer are equal, while at the incidence on
points A , B and C2 they are not.
3. In interferometers a , b and (Fig.1) the phase differences between interfering waves at first preserve their zero
A B C
values as 3, 3 and 3 points are approached respectively
A B C
from 1, 1 and 1 points in the direction shown in the figure by arrows, then acquire some non-zero values and, eventually, abruptly go to zero.
al bl
4. As the points of incidence in interferometers
t A3
to 3
B,
^ and , ^ and 3, the path differences in points
B' C' C
3, 1 and 3 of the third blocks are zero. In the same interferometers at the incidence of primary
A B C
beam in points 2 , 2 and 2 , there arise path differences
A B' C'
in 2 , 2 and 2 points of the third blocks. It is easy to see
AA
that this is because in case of incidence on points 1 and 3
B1 and B3 C
C
1 and 3 the distances between blocks in
c ABC
and 1 (Fig. 1) shift from 1, 1 and 1 C
and 3 , the phase differences between the interfering waves first preserve zero values, then gradually increase and finally acquire their maximum and constant value. It is easy to see that the above devices function as interferometers, i.e., the waves superimpose exactly on the surface of the third block only when the following conditions are observed: the angles of wage-shaped second blocks in interferometers a1, b1 shall be almost twice as large as the angles of wage-shaped third blocks in the same interferometers; in interferometer b1 the
pp EE F a b
surfaces and 1 1 slope symmetrically. In 2 and 2
EF = E F
interferometers the steps satisfy the conditions 11 22 and EF1 + EF EF , and in C2 interferometers the
EF = EF
steps are equal, 1 1.
P 1
dj
Fig. 1. Some versions of interferometers for production of phase changes of superimposing waves
The proof of these assertions for ai type interferometer is given below in Section 2.
Thus, the following observations may be made: • The paths passed in the second block by waves that interfere in the third block are different and, hence, these waves are differently attenuated and there arises a difference in their amplitudes. However, the contrast of interference patterns shall not be visibly affected as, first, the paths in the second block differ one from another by shares of millimeter, and, second,
as the passage is anomalous (especially in thicker parts of the second block), the absorptions of interfering waves in the second block differ insignificantly.
At last, if in separate cases the differences in amplitudes of differently absorbed waves that emerge from the second block and interfere in the third block are remarkable, one may reduce these differences to zero by placing additional absorbers in their
dj
and 1 d2 ). In
paths (Fig. 1 the absorber is a wedge
a,
Pj
bj dj
and
1 interferometers (Fig.1 d1 ), while in a2 , b2 and
c P d
2 ones the absorber is a plane-parallel plate 2 (Fig.1 2 ). These absorbers insignificantly change the phase of waves, but equalize the absorptions. If needed, the shifts of beams due to the presence of these absorbers may be also taken into account.
The advantage of
a b
a,
and C
interferometers
over 2 , 2 and 2 ones is that in former interferometers the path difference between interfering waves is continuously changeable, owing to which it will be possible to detect the moment when the path difference exceeds the train length.
• At first sight the disadvantage of these interferometers consists in the fact that owing to the inclination of incidence surfaces the Bragg angles may change. But the inclination of surfaces is very small (1-2 angular degrees) and for this reason the values of reflection angles are preserved with high accuracy and the blocks retain the reflecting positions.
2. Calculation of a wedge-shaped block interferometer (the train-length meter)
Assume now that the second block of interferometer is wedge-shaped, or rather its lower part is a parallelepiped and the upper part is a wedge (Fig.2), the reflecting planes being normal both to the large surfaces of blocks and the surface of base.
Fig. 2. A schematic drawing for calculation of interferometer - train-length meter with wedge-shaped blocks
Let the monochromatic X-ray beam be incident at Bragg angle on the first block of interferometer. The beam incident A
on 1 point of the first block passes two first blocks and is
focused on O point, and the beam incident on A2 point is focused on M point. When the incidence point on the front
A A
surface of the first block travels from 1 to 2 , the focal
A
points on the front surface of the third block travel from
O
to M , i.e., OM line is the locus of focal points, and like the second block the third one has a parallelepipedic bottom and wedge-shaped top.
Now find the coordinates of point in Fig.2. The origin of
frame is in O point, OY axis is parallel to the first block, and
OX axis is normal to that. The coordinates of
OY
point are
x=a..
tgt
2 1 - tgd-tgf
Y = a . 2 - tgd-tgj
(1)
2 1 - tgd- tg0 (2)
where a is the distance between Al and A points on the front surface of the first block, 0 is the Bragg angle, 9 is the wedge angle of the second block.
From triangle we obtain for the wedge angle a of the
third block having in mind (1) and (2)
tga = — =-àZ-
Y 2 - tgG- tgf (3)
The angle a is seen from (3) to be independent of a
distance between incidence points A and Ar2 , i.e., OM line
is the locus of focal points between O and M points. It is
A
also easy to see that when the incidence point 1 moves down
to A2 , the focus O also moves down to O point.
Thus, the shape of the third and second blocks is the same except for the fact that, in general, the angles at their wedges are not equal and related as (3). As a and 9 angles are small, formula (3) is reduced to the form
t
O '
a =
2tgd
(4)
n
whence it is seen that when 0 < 2 , angle 9 exceeds a. 3. Calculation of the path difference between superimposed waves
Now calculate the path difference that arises between the interfering waves in interferometer (Fig. 3). As is seen in Fig. 2,
A A
when the incidence points are between 1 and 2 , the path differences of interfering waves in points located between O
and
O '
are zero.
A A
When the incidence point is between 1 and 2 , the phase
the surface of the second block (Fig. 3). A simple relation is obtained for them:
A = L2tg$\ secO-1),
L
(5)
differences of interfering waves in points located between O
a M t tu a-o- * i where is the distance of incidence point on the sloping
and 1V1 differ from zero. These differences A arise only on r r &
surface of second block from the wedge base.
Fig. 3. The scanning scheme of the train length meter
Fig.4 The interference (Moire)
pattern, obtained by scanning of the train length meter
In case of
(p = 0
the phase differences are seen from (5) to
be zero and increase with distance.
At the derivation of (5) a difference was formed
A = X1-X2(1 -S) = X1 -X + X2S
Keeping in mind the smallness of X2S this difference was replaced by expression
CuK„
A = X
■X 2.
So, the path differences of interfering waves increase as
X — X
1 2 difference, that is achieved by means of scanning.
For measurement of X-ray train length, a type interferometer was prepared from dislocation-free silicon single crystal. In it (110) planes were normal to large surfaces and the base of interferometer. The interferometer geometry was as follows: the distance between the interferometer blocks was 12 . 5 mm, the blocks had 0.6 mm thickness, and the width and height were respectively 17.5 mm and 11.7 mm; L1= 4.84
mm, angles q>=2°20'20", a=1o10'50'' in 220 reflection of radiation. The scanning patterns (Moire patterns) from such an interferometer were obtained using X-ray diffraction cameras with KPC and A-3 scanning devices (Japanese make). The diagram of interferometer scanning and the corresponding interference pattern are seen in Figs. 3 and 4.
The value L3= 0.098 mm of parameter corresponding to a disappearance of Moire pattern was determined using CuK
(5), the value of a radiation train length being
l=3.76-10-5 cm, whereas that
l = CT
(where c is the
propagation speed of electromagnetic waves, t is the duration of coherent radiation), and with due regard for T = 4-53^ s
[5] was l = 3,23-10-5 cm•
Thus, as a result of pilot research one can state that:
1. A special type interferometer - the train length meter, was calculated, fabricated and tested;
2. Gradually increasing the path difference between the interfering waves, an interference pattern was obtained
-m-
by means of scanning and the point of disappearance of this pattern was determined depending on the value of these path differences.
CuK
3. The obtained length of X-ray train of a
radiation proved to be close to the calculated one.
REFERENCES
[1] Bezirganyan, P.H. (1965). JTP, 35, 359 - 367.
[2] Bezirganyan, P.H., Gasparyan, L.G. (1970). JTP, 40, 2427-2433.
[3] Bezirganyan, P.H., Gasparyan, L.G. (1971). Izv. Akad. Nauk Arm. SSR, ser. fiz., 6, 106 - 115.
[4] Bezirganyan, P.H. (1966). JTP, 36, 514 - 520.
[5] Michelson M. (1934) Light Waves and their Application.
[6] Bonse, U.& Hart, M. (1965). Appl.Phys.Letters, 6, 155 - 56.
[7] Bonse, U.& Hart, M. (1965). Z.Physik. 190, 455 - 467.
[8] Bezirganyan, P.H., Eiramdzhyan, F.O., Truni, K.G. (1973). Phys. Status Solidi (a), 20, 611- 618.
[9] Aboyan, A.H. (1996). Crys.Res.Technol. 31,513-519.
[10] Aboyan, A.H., Khzardzhyan, A.A .(2003). Izv. NAN Armenia, ser. fiz., 38, 326 - 334 (in Russian)
This work was supported by the RA MES State Committee of Science, in frame of the research project 15SH-008
CLASSIFICATION OF CONVEX POLYHEDRONS
Tatiana Puolokainen,
Petrozavodsk State University, docent, candidate of physical and mathematical sciences,
the Faculty of Mathematics
ABSTRACT
This paper is devoted to decomposition of convex polyhedrons into classes. The principle of classification is the presence or absence on the polyhedron boundary of the faces which are parallel to one direction in space. This problem is associated with the Hadwigers problem of covering convex polyhedrons with body images at homothety with coefficients less than unity. Key words: Convex polyhedrons, classification, covering, homothety.
In the paper [1, 121] Hadwiger formulated the next hypothesis: to covering of any convex body in n-dimensional Euclidean space En enough 2n bodies with smaller sizes if they are homothetic copies for this body.
In the author's papers [2, 330-334], [3, 236-239] problem of covering for convex polyhedrons of some classes with body images at homothety with coefficients less than unity is formulated and solved. In the paper [4, 62-65] author considered the influence of polyhedrons rearrangements to the number of geometric bodies, if these bodies are polyhedrons images at homotheties and if they are sufficient for a covering of polyhedron.
The problem of covering of convex polyhedrons with bodies images at homothety and problem of classification of convex polyhedrons are associated very closely. Classification of convex polyhedrons is necessary to consideration of problem of whole class of convex polyhedrons and we don't speak about a single polyhedron. The proposed lower classification of convex polyhedrons is associated with the number of polyhedrons with smaller sizes if this number is sufficient to covering of polyhedron of a certain class and with the methods of covering of polyhedrons when they are related to a certain class.
1.1. Some definitions. Introduction of classes for convex polyhedrons.
Let M - convex polyhedron and q - some straight line in space. Let we have n-faces of polyhedron M and these faces are parallel to straight line q (n>2). Two cases are possible.
Case 1. All n-faces which are parallel to the direction q, are form a single connected component.
Case 2. All n-faces which are parallel to the direction q, are form two or more connected components.
Let's consider in detail the first case. Let all faces of polyhedrons which are parallel to one direction q in space, are
form a single connected component. Then the next 5 variants are possible.
1. The connected component is homeomorphic to the ring. The boundary of connected component consists of two disjoint closed broken lines, each of which is topologically equivalent to the circle. In this case we shall say that the boundary of convex polyhedron contains prismatic part. The prismatic part - is a surface with a border. The border of the surface consists of two closed broken lines and lines do not have common points. A convex polyhedron which is containing a prismatic part, we shall relate to the class B.
Note 1. It may happen that a convex polyhedron contains not a single prismatic part which is parallel to straight line q. May be it contains a several prismatic parts and each of them is parallel to some straight line in space. This polyhedron also we will relate to class B.
2. The connected component is topologically equivalent to the set W, this set may be prepared in the following way. Let two circles w and w1 with different centers and radii are tangent internally in the point L. Part of the plane, which lies between the two circles w and w1 will be denoted as W.
W W \ W1, where W - is a ring which is bounded by
circle w, and Wl - is open ring, which is bounded by circle w1.
Connected component of the polyhedron border which is topologically equivalent to the set W, will be named a multi-faceted surface of transition type. Such surface of transition type consists of a single segment.
The multi-faceted surface of transition type consists of n -faces and is a surface with a border. This border is the union of two closed broken lines which have a one common point.
If the convex polyhedron M contains a surface of transition
-1(3-
