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Coherent bremsstrahlung and e±-pair production in periodic nanostructures
Nuclear Instruments and Methods in Physics Research B 234 (2005) 87–98 www.elsevier.com/locate/nimb
Coherent bremsstrahlung and e±-pair production in periodic nanostructures S. Bellucci a, V.A. Maisheev a b
b,*
INFN – Laboratori Nazionali di Frascati, P.O. Box 13, 00044 Frascati, Italy Institute for High Energy Physics, 142281 Protvino, Moscow Region, Russia Received 9 September 2004 Available online 19 November 2004
Abstract Coherent bremsstrahlung and electron–positron pair production of high energy electrons and c-quanta moving in a three-dimensional periodic lattice consisting of a complicated system of atoms are considered. The peculiarities of formation and possibilities of utilization of these processes are discussed. 2004 Elsevier B.V. All rights reserved.
1. Introduction Considerable advances have been made in recent years in the creation of various nanostructures and particularly in the synthesis of the so-called carbon nanotubes. In [1] a superlattice of singlewall nanotubes was described. According to this paper single-wall nanotubes are uniform in diameter and self-organize into ropes, which consist of 100–500 nanotubes in a two-dimensional triangu˚ . Then, in lar lattice with a lattice constant of 17 A a number of papers [2–6] the channeling mechanism and channeling radiation in similar lattices were considered. As expected, these effects can be *
Corresponding author. E-mail address: [email protected] (V.A. Maisheev).
utilized for high energy physics purposes, such as beam deflection, photon sources creation, etc. In this paper we consider the coherent processes of bremsstrahlung and e±-pair production on the periodic superlattice of nanostructures. These processes are the result of the interaction of an electron (or a c-quantum) moving in such a lattice with its periodical electric field. For the first time, the theory of this phenomenon in single crystals was published in [7,8]. Since this time the phenomenon was investigated experimentally [9,10] and it was utilized for obtaining linearly polarized cbeams [11–13], in polarization measurements [14] and for increasing the e±-beam intensity [15]. The processes of coherent bremsstrahlung and pair production were also described in detail in the literature (see for example [16–18]).
0168-583X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.09.016
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It should be noted that the coherent processes as a rule were considered for simple crystallographic structures, such as monoatomic single crystals. In this paper we investigate the abovementioned processes for complicated enough periodic structures, consisting of a large number of atoms. As we will show below, the new peculiarities take place at the formation of coherent processes on the lattice with the basic sizes in tens of angstroms.
2. Electric fields of single crystals and nanostructures The continuum three-dimensional potential (averaged over thermal fluctuations of the lattice) of a single crystal or some periodic nanostructure consisting of atoms, has the form [18,19] 4peZ X uðrÞ ¼ U ðgÞeigr ; ð1Þ V g where e is the elementary charge, Z is the atomic number of the material, V is the fundamental cell
volume, g is a reciprocal lattice vector [7,16], r is the radius vector drawn from the origin of the fundamental cell, U ðgÞ ¼ SðgÞ
ð1 F ðgÞÞ expðAg2 =2Þ; g2
ð2Þ
S(g) is the structure factor [7], F(g) is the form factor of an atom in the structure and A is the meansquare amplitude of the thermal vibrations of atoms. Thus, the potential of the periodic nanostructure is determined, if we know the quantity U(g), which can be measured (strictly speaking the quantity jF(g)j2 is measured) from the diffraction of X-rays or electrons. The structure factor S(g) is determined with the help of the following relation [7]: SðgÞ ¼
N X
expðigqj Þ;
ð3Þ
j¼1
where the qj is the radius-vector of the jth atom in the fundamental cell of the lattice, N is the total number of atoms in the fundamental cell. Let us consider the triangle superlattice consisting of single nanotubes (see Fig. 1). In this case we
Fig. 1. Three-dimensional superlattice of (10, 10) armchair single-wall nanotubes and Cartesian coordinate system (xyz). The fundamental cell of the structure is presented with the help of the thick lines. The black points are atoms of the nanotubes. ˚ , b = 2.4 A ˚ , radius of circles R = 6.8 A ˚. OA = AB = BC = CO = 17 A
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can write for qj = xji + yjj + zjk (i, j, k are the unit vectors along x, y, z-axes) the following relations: 4pj1 þ u1 ; xj1 ¼ R cos N ð4Þ 4pj2 þ u2 ; xj2 ¼ R cos N 4pj1 þ u1 ; y j1 ¼ R sin N ð5Þ 4pj2 þ u2 ; y j2 ¼ R sin N zj1 ¼ 0;
zj2 ¼ b=2;
ð6Þ
where j1, j2 = 1, 2, . . ., N/2 are the indices corresponding to atoms placed in two parallel planes, R is the radius of the ring, b is the period (i.e. the size of the fundamental cell) in the z-direction, u1, u2 are the angle shifts. The reciprocal lattice vector g with the l, m, n-indices in Cartesian coordinates (see Fig. 1) have the form pffiffiffi 2p 2p ðl mÞi þ ðl þ mÞj= 3 þ nk; ð7Þ g¼ a b where a is the size of the fundamental cell in the xy-plane (a = OA = AB, see Fig. 1). It is well-known [7,16] that there are some numbers (very often they are integer numbers) for the structure factors in monoatomic single crystals. In the case under consideration it is necessary to calculate the structure factor according to Eqs. (3)–(6) for every reciprocal vector. In the general case these factors are complex numbers. The equations presented here describe the three-dimensional potential of the lattice of nanotubes, as shown in Fig. 1. This figure illustrates that the ideal lattice consists of uniformly oriented (10, 10) armchair single-wall nanotubes (the angle shifts are selected in the following way: u1 = 0, u2 = 6). For a real nanotube structure we expect that the angle shifts will be different for every nanotube in the lattice. (Notice that the problem of a nanotube orientation in the superlattice is not simple, see [20].) With the aim to take into account this situation we introduce the continuum three-dimensional potential aver-
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aged over angle shifts. This potential has the form described by Eqs. (1)–(3), where the structure factors read Sðl; m; nÞ ¼ cos2 ðpn=2ÞNJ 0 ðRgðl; m; 0ÞÞ:
ð8Þ
Here J0(x) is the Bessel function of the zeroth order and g(l, m, 0) is the value of the (l, m, n)-reciprocal vector projection on the xy-plane. The physical meaning of this potential is the following: it is the most likely value of the potential at any space point, if the angle shifts of nanotubes are distributed randomly. It should be noted that our consideration of potentials has a model independent form [19], or, in other words, it is valid for any atomic potentials. The last fact finds a reflection in that the dependence on the atomic potential takes place by means of atomic form factors. Besides, it is easy to obtain two-dimensional potentials from Eq. (1) by integration over the z-coordinate, as it was considered in [19]. As a result the two-dimensional potentials are also described by Eq. (1), in which the sum is calculated of only over (l, m, 0)-vectors of the reciprocal lattice. Fig. 2 illustrates the calculations of the twodimensional lattice potentials for (10, 10) armchair single-wall nanotubes in both of the following cases: (a) for a lattice with fixed angle shifts (u1 = 0, u2 = 6), see Fig. 1; (b) for a lattice with averaging over angle shifts. In all calculations in the paper the Molier atomic form factor was employed. The amplitude of the thermal vibrations ˚ (as in a diamond single cryswas equal to 0.039 A tal). The increase of this value by a factor 2 is insignificant for all computed results of the coherent processes in this paper. However, the behavior of potentials noticeably depends on the amplitude of the thermal vibrations, but it takes place in a small volume (A3/2) in the vicinity of the atomic centers. The lattice potentials shown in Fig. 2 are close to similar ones, which were calculated in [4] on the basis of the Doyle–Turner approximation for atomic potentials. Notice that in the cited paper the method of calculation is different from that used here.
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dr 2 2 a a x ¼ r0 ð1þ ð1 xÞ Þðw1 þ w1 Þ ð1 xÞðw2 þ w2 Þ ; dx 3 ð9Þ dr 2 ¼ r0 ðx2 þ ð1 xÞ2 Þðwa1 þ w1 Þ þ xð1 xÞðwa2 þ w2 Þ ; dx 3 ð10Þ
where r0 ¼ aQED Z 2 r2e , aQED = 1/137.04, re is the E classical electron radius, x ¼ Ec is the ratio of emitted photon energy Ec to the initial energy E of the electron (or the ratio of the electron energy to the energy of the c-quantum for pair production), wa1 , wa2 are well-known functions in the theory of coherent processes [7,16] (for high energies of the interacting particles they are almost constants), w1, w2-functions are w1 ¼ 4
2 ð2pÞ X dg2 jU ðgÞj2 2? ; NV gk g
w2 ¼ 24
2 2 ð2pÞ2 X 2 d g? ðg k dÞ jU ðgÞj ; NV g4k g
ð11Þ
ð12Þ
where gk is the projection of the g-vector on the direction of the particle motion, g2? ¼ g2 g2k , dvalue is equal to d¼ Fig. 2. Two-dimensional periodic potential of the superlattice (see Fig. 1). The upper and lower half-figures are the potentials described by Eqs. (4)–(6) and (8), respectively. The numbers near equipotential curves are the values of the potential in eV.
3. Cross sections of coherent bremsstrahlung and pair production From Eq. (1) one can see that the potential of the lattice consisting of nanotubes represents a three-dimensional periodic structure. A charged particle or c-quantum moving in this medium will interact periodically with the lattice. The results of this interaction are the following well-known processes: coherent bremsstrahlung (for a charged particle) and e±-pair production (for a c-quantum). If the expansion of the potential in Fourier series is known, the differential cross sections per atom of these processes are, correspondingly
mc2 x 2E 1 x
ð13Þ
for bremsstrahlung and mc2 x ð14Þ d¼ 2E xð1 xÞ for e±-pair production m is the electron mass). The summation in Eqs. (11) and (12) takes place under the following condition: gk P d:
ð15Þ
The d-value has an important physical meaning, i.e. it is the minimal transferred momentum to the medium. The inverse value (1/d) determines the formation length of the radiation process. Eqs. (9) and (10) for cross sections are valid at conditions that the thickness of medium where the particle propagates is more than the formation length and approximately 10 periods (or more) must be to exist in the medium along the particle path [7,18].
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It is significant that in this section we employ the system units in which the value of the reciprocal (direct) lattice vector is measured in units of k1 c ðkc Þ, where kc is the electron Compton length. This system is adopted in the theory of coherent radiation [7,16] and is convenient for performing specific calculations numerically. The equations presented here for cross sections have the same form as for single crystals. The basic difference between single crystals and nanostructures is determined by the difference in the size of the fundamental cells and disposition of atoms, or, in other words, by the values of the structure factors. From Eqs. (11) and (12) one can see that every nonzero term in the sum is proportional to the following factor: 2
M¼
2
jSðgÞj ð1 F ðgÞÞ : NV g2
ð16Þ
We compare the intensities of coherent processes in the nanotube lattice with the intensities of these processes in a single crystal of diamond, for the case when the electron beam moves under a small angle relative to the h0 0 1i axis of the single crystal. Also, from Eqs. (9)–(12) one can see that the intensities of both coherent processes considered
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here are proportional to the inverse volume of the elementary cell. So the volume of such a cell for a (10, 10) armchair single-wall nanotube is ˚ 3. For a diamond crystal the same equal to 600 A ˚ 3. At first sight, from here volume is equal to 46 A it follows that the intensity of the coherent radiation in the nanotube lattice should be approximately 13 times less than in the diamond crystal (at equivalent conditions). However, this conclusion is not true. From Eqs. (9) and (10) follow that the intensity is the sum of various terms. It turns out that in the case of a nanotube lattice the number of effective terms grows with increasing of time. This fact is illustrated in Fig. 3. Here the solid curves present calculations of the MV value the jSðgÞj2 for diamond ¼ 8 at amplitudes of thermal N ˚ as continuous functions vibrations 0 and 0.039 A of the g-value. However, in both cases of lattices the set of g-values is discrete. The values of the MV-factor for discrete g-values are shown in Fig. 3 (the solid intercepts are for a nanotube lattice and dashed ones are for diamond). One can see that only some values for diamond and a lot of values for the lattice of (10, 10) armchair singlewall nanotubes belong to the most effective range. In the latter case small values of the M-factors are partially compensated by the large number of them.
Fig. 3. The factor MV as a function of the g-value for a nanotube lattice and a diamond single crystal. The numbers near the arrows are indices of some points of the diamond reciprocal lattice. Incident electrons move under a small angle relative to the h0 0 1i diamond axis (see the explanation in the text).
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It should be noted that coherent radiation has a linear polarization and coherent pair production has a dependence of the cross sections on the linear polarization of the c-quanta. One can also write the relations describing these phenomena. Besides, at very small angles of motion for electrons, positrons or c-quanta with respect to the nanotube axis, the coherent mechanism is violated. Partially this takes place due to channeling in the nanotubes. On the other hand, at angles of motion < Umax/mc2 (where Umax is the potential barrier) the deflection angle of the electron is larger than the inverse Lorentz factor. In this case, the radiation processes have a magneto bremsstrahlung character [18]. For a (10, 10) singlewall nanotube this angle is about 0.3 mrad. Note that at high energies of the particles a similar effect takes place for some strong planar orientations (the corresponding planar angle is less than 0.05 mrad). For calculations of the coherent bremsstrahlung and pair production processes we have found the periodic averaged potential for the nanotube superlattice, which, generally speaking is not periodic in a strict sense. Some explanations and discussion of the coherent radiation in imperfect periodic structures can be found in Appendix A.
4. Calculation of coherent processes The direction of the particle motion relative to the Cartesian coordinate system shown in Fig. 1 can be determined with the help of the a and h angles. We denote by h the angle with the z-axis and a is the azimuthal angle around this axis (a = 0, when the momentum of the incoming particle lies in the yz-plane). Sometimes it is convenient to use an alternative pair of angles, i.e. hx = hcos a and hy = hsin a. Taking into account these angles, we can write (at h 1) 2 2p ðl mÞ2 þ ðl þ mÞ2 =3 ; ð17Þ g2? ¼ a gk ¼
pffiffiffi 2p 2p h ðl mÞ sin a þ ðl þ mÞ cos a= 3 þ n: a b ð18Þ
In the case when g? d (this condition is valid when the particle energy is high enough) the terms corresponding to vectors of the reciprocal lattice with n = 0 yield the main contribution to the cross sections. Further, we carry out a calculation of the radiation coherent processes in this approximation. It means that at high energies the coherent processes depend predominantly on the transversal electric fields in nanostructures. It is well-known [7], that the maximum of the coherent bremsstrahlung spectrum corresponds to the condition gk = d or, in an expanded form, pffiffiffi 2p h ðl mÞ sin a þ ðl þ mÞ cos a= 3 ¼ d: ð19Þ a Let us assume that the particle move in the xzplane. It means that a = p/2. Then Eq. (10) can be rewritten as follows: 2p hðl mÞ ¼ d: a
ð20Þ
The choice of points of the reciprocal lattice under the condition l m = 1 corresponds to selecting a row as shown in Fig. 4. The dashed region corresponds to points of the reciprocal lattice, whose contribution in intensity is large enough. If the initial energy E and the relative energy x of the emitted photon are fixed, then from Eq. (20) we can find the corresponding angle h. The result of a calculation of the intensity of the coherent radiation is presented in Fig. 5 for energy E = 10 GeV and x = 0.5. In this case the angle takes the value h 0.018 radian. Before discussing this result we can draw the following important conclusion: if one knows the intensity (cross section) for some energy E of the particle, then one can find the corresponding intensity for another energy with the help of the simple relation drcnew drc ðEn ; x; hx =k; hy =jÞ ¼ j ðE; x; hx ; hy Þ; dx dx
ð21Þ
where En is the new energy, j = En/E, rc is the coherent part of the cross section. As already noted, the amorphous part of the cross section is practically constant at high energy of the particles. Thus, it is very easy to get the energy behavior of the intensity of the coherent processes.
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Fig. 4. Reciprocal lattice for a structure of nanotubes (see Fig. 1). The numbers near the points are (l, m)-indices of the reciprocal lattice vectors. The arrow Ph is the projection of the particle momentum on the xy-plane.
Fig. 5. The bremsstrahlung spectrum x dr of 10-GeV electrons dx moving at angles h = 0.018 and a = p/2 radian (curves 1 and 2). Curves 1 and 2 correspond to potentials, which are described by Eqs. (4)–(6) and (8). Curve 3 is calculated for hx = 0.006, hy = 0.002 radian and for a potential, which is described by Eq. (8). Curve 4 is the amorphous part of the cross section (see the explanation in the text).
The calculations presented in Fig. 5 were carried out for both potentials (see Eqs. (4)–(8)). Fig. 5 shows that the intensity of the first potential Eqs. (4)–(7) is 1.5 times higher than for the other. However, we think that the second potential is more realistic. We have also compared these calculations with similar ones for the (1 0 0) plane of the reciprocal lattice of a diamond single crystal. For a diamond single crystal the maximum intensity takes place at h = 0.00133 radian (for x = 0.5). In this case the electron momentum lies in the (0 1 1) plane of the single crystal. The absolute value of the coherent cross section is 4.3 and 6.5 times larger in comparison with the nanotube lattice for the first and second potentials. Thus, the decrease in intensity is less than it may be expected. The purpose and mechanism of this effect were discussed already (see also Fig. 3). We call attention to the fact that the coherent processes in nanostructures have a tens of times wider angle sensitivity. This result
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can be seen from Eqs. (19) and (20): the angle h is proportional to the characteristic minimal period of a structure. The curves 1 and 2 (see Fig. 5) present a special orientation of the electron beam when the momentum of electron is perpendicular to rows of reciprocal lattice points (see also Fig. 4). In the general case the direction of the electron beam is arbitrary relative to the reciprocal lattice and the quasimonochromaticity of the emitted photons is lost. Curve 3 presents a sample of a similar behavior. It should be noted that curves 1 and 2 (see Fig. 5) are shown for the purposes of comparison of the obtained data with the analogous ones for the diamond single crystal. Besides, similar orientations are widely covered in the literature. The authors of the paper [16,21] proposed to name such a kind of spectra as ideal ones. Due to the specific motion of electrons, the form of the real spectra differ from ideal ones as it is illustrated by the dashed curve, which corresponds to curve 1.
Fig. 6. The behavior of the bremsstrahlung cross section for a 10 GeV electron beam, as a function of the hx and hy orientation angles. The numbers near the curves are the values of the cross section in comparison with the amorphous one. Some insignificant details in the figure were deleted for the sake of clarity of the picture. Calculations are made for the potential described by Eq. (8). The black points are narrow maxima of the distribution, with the peak value being approximately equal to 4.5.
Fig. 7. The behavior of the pair production cross section for 10 GeV unpolarized c-quanta, as a function of the hx and hy orientation angles. The numbers near the curves are the values of the cross section in comparison with the amorphous one. Some insignificant details in the figure were deleted for the sake of clarity of the picture. Calculations are made for the potential described by Eq. (8).
Calculations were carried out for thickness equal to 0.01 cm. One can see that at x > 0.1 the dashed curve and curve 1 practically coincide. Figs. 6 and 7 illustrate the behavior of cross sections of both latter processes considered, for 10 GeV incoming particles at x = 0.5 as a function of the hH, hV-angles. From here one can see that the intensities have a complicated enough angle dependence and a wide angle range, where coherent processes exceed the noncoherent ones. By using Eq. (20) it is easy to recalculate the data in Fig. 7 (or in Fig. 6) for c-quanta energies higher than 10 GeV. For example, for an energy of 100 GeV we must reduce both angle ranges by 10 times (from ±0.1 to ±0.01 radian). Besides, there is a need to recalculate the numbers near the curves; so, the quantity 1.8 is replaced by 10(1.8 1) + 1 = 9 and so on. 5. Conclusion We consider the processes of the coherent bremsstrahlung of electrons and coherent pair pro-
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duction of c-quanta moving in a three-dimensional lattice of nanotubes. At present, these structures are created in small amounts, however the development of nanotechnology allows one to assume a significant progress in manufacturing much larger quantities of them in the near future. The main result of our consideration for a lattice based on (10, 10)-armchair single-wall nanotubes is that both coherent processes take place in the large angle range with high intensity. A more detailed analysis of coherent processes in a nanotube lattice, including the polarization characteristics of cross sections and the behavior of processes at low energies of the moving particles, will be published elsewhere. However, based on our consideration of the specific mechanism for the formation of coherent radiation processes, we can claim that these processes exist in different periodic nanostructures and their intensity may be very high at high energies of incident particles in a wide angle range. Besides, we think that the results of our investigations may be also useful for the X-ray analysis of periodic nanostructures: for example, the jU(g)j2-values in Eqs. (11) and (12) are similar to scattering strengths for X-rays. It should be noted that the results obtained here (at the condition of solving the problem of synthesizing nanotube lattices in large volumes) bear important consequences for creating various sources of positrons and c-radiation, polarimeters and so on. Besides, presently one can begin carrying out some experimental investigations of existing samples in electron beams.
Acknowledgements This work is supported in part from the Italian Ministry MIUR under grant COFIN 2002022534.
Appendix A. Coherent bremsstruhlung and pair production in the periodic imperfect structures The process of coherent bremsstrahlung was investigated and utilized mainly in simple single crystals. Such crystals as a diamond and silicon have the negligible small degree of mozaicity and
95
admixture. Because of this, the problem of calculations of the bremsstrahlung in imperfect structures did not practically appear with one exception. It is taking into account of the influence of atomic thermal motion on the intensity of radiation. In [7] (where this process was investigated) point out the fundamental analogy between coherent bremsstrahlung and diffraction of X-ray in single crystals. Starting from the famous article [22] the process of the X-ray diffraction gain worldwide recognition as a powerful tool for investigations of various atomic structures, due to real impetus was given in development of the theory of the phenomenon, in particularly, for imperfect periodic structures. Thus, we can use the results of this theory for calculations of the process of coherent bremsstrahlung in the periodic imperfect structures, for example, such as the nanotube superlattices. The main result of the theory [23], which we apply to coherent bremsstrahlung, is the following: the differential cross section of the process is calculated according to Eqs. (9)–(12), in which U(g) is the Fourier component of the averaged potential of a structure. Besides, in the general case the noncoherent contribution in the cross section is possible. The theory gives some methods for obtaining of averaged potentials. However, in the general case they are enough complicated. In this paper we have introduced the averaged potential on the basis of the physical representations for the case of superlattice when the angle shifts are distributed randomly. One can see that this potential have the clear and transparent physical sense. Further we suggest the method for determination of the averaged potential in more general cases. For simplicity we consider our method for a two-dimensional square supperlattice. Besides, we assume that all Na atoms (see Eqs. (3)–(6) and Fig. 1) are distributed with the equal step around the nanotube circle. Now we can build the square superlattice consisting of N2 nanotubes. Then we can enumerate every nanotube in the superlattice as ij (i, j = 1,2,3, . . ., N), where i and j are the horizontal and vertical numbers of the square cell of side a, correspondingly. And after we enumerate sequentially the atoms in every nanotube as k where k = 1, 2, 3, . . ., Na. Thus, every atom in the
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r = (i 1)aix + (j 1)ajy in the basic coordinate system. In Eq. (A.3)P the value S ij ð~ gÞ is the followN ing structure factor: 1 a exp i~gqijk , where in the local coordinate system vector qijk is
superlattice get own ijk-number. We can also determinate the angle shifts uij1 for every atom with the 1st number in the nanotube. Then the other atoms obtain the angle shifts uijk according to relation: uijk ¼ uij1 þ 2pðk 1Þ=N a :
qijkx ¼ R cos uijk þ a=2;
ðA:1Þ
ðA:4Þ
By this means we determinate all the geometrical parameters of the superlattice. Let be specific, we select the basic Cartesian coordinate system in the left and bottom corner of the superlattice with x and y axes in the horizontal and vertical directions, correspondingly. In a similar manner, we introduce the local Cartesian coordinate systems for every cell with i and j numbers. With this lattice (of Na · Na sizes) as the basic element we can build (by using of parallel translations in x and y directions with the period of Na) the twodimensional infinite periodic system and calculate its potential with the help of Eq. (1). For this pur~ gÞ for every pose, we find the structure factors Sð~ reciprocal vector ~ g: S~ ¼
N X N X i¼1
exp i~ grij S ij ð~ gÞ;
Recall that the averaged over angle shifts potential of the superlattice (in limit N ! 1) are determined by (1)l+mNaJ0(gR) factor, where (1)l+mmultiplay is due to our choice of the coordinate (the shift of coordinate system on (a/2, a/2)-vector corresponds to Eq. (8)). First we consider the superlattice where the orientation of all the nanotubes is the same, i.e. the angle shifts of the different nanotubes are equal in between. It is obvious that in this case our periodic structure is one which based on the square fundamental cell with the period of a. The all structure factors (see Eq. (A.2)) for which l/N and m/N are nonintegral numbers are equal to zero, and the others structure factors are equal to N2S, where S is structure factor for cell with the a-size. One can see that in this case the reciprocal space consists of the following vectors (corresponding to nonzero structure factors): g ¼ ~gN . Taking into account that S~ ¼ N 2 S and the corresponding volumes of both the cells are connected by relation V~ ¼ N 2 V we get the expected result: at our choice of the angles shifts the real funda-
ðA:2Þ
j¼1
where the reciprocal vector ~ g is ~ g ¼ G0 lix þ G0 mjy ;
l; m ¼ 0; 1; 2; . . . ;
qijky ¼ R sin uijk þ a=2:
ðA:3Þ
where G0 = 2p/(Na), ix and jy are the unit vectors in x and y directions and the translation vector
Table 1 Results of simulation of averaged potential. Normalized structure factors in 2–5 columns are calculated for reciprocal vectors e g ¼ 2plix =ðaN Þ l/N
STh
S1
S2
S3
S4
10 20 30 40 50 60 70 80 90 100 200
4.478713 3.175097 2.594649 2.247977 2.011157 1.836234 1.700225 1.590557 1.499696 1.422815 1.006334
4.478714 3.066866 2.604257 2.268821 2.183495 1.771523 1.702413 1.640649 1.460587 1.442049 1.068331
4.478713 3.160334 2.605523 2.246414 2.048165 1.817817 1.662717 1.560811 1.492919 1.438173 1.001786
4.478713 3.174532 2.591840 2.249317 2.018196 1.845523 1.700150 1.584478 1.499754 1.416810 1.014237
4.478713 3.175091 2.594654 2.247972 2.011253 1.836263 1.700272 1.590533 1.499752 1.422829 1.006339
STh is the theoretical value of the factor. S2 is the factor calculated at N = 100 and averaged over number of tests NT = 10. S1, S3, S4 are calculated factors for condition NT = 1 and N = 100, 1000, 10000, correspondingly. We do not illustrate factors less than 10 due to excellent agreement with the theory.
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mental cell is square with the side is equal to a and the potential of this structure is described by Eq. (1). Note the presence of volume V in Eq. (1) for
Fig. 8. Simulation of the potential in nanotube superlattice. The calculated absolute values of the normalized structure factors for lattice with N = 10 are shown as functions of l, mnumbers in the following cases: for one test (top) and averaged over 100 tests (bottom). The area of the circles are proportional to their quantity. The quantity of point with l, m = 0 is equal to 40. The points with the quantities less than 10-4 are invisible. Figure illustrates the different behavior of the structure factors at averaging and gives some insight into the diffusion noise properties.
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a two-dimensional structure one can to explain that this structure are disposed in the three-dimensional space. Let us consider the second case when the angle shifts of the nanotubes are distributed randomly in the interval (0, 2p). This problem we can solve by computer simulation. The angle shifts one can define with the help of the generator of the pseudorandom numbers. Using the above-mentioned ~ gÞ=N 2 -values (normalized relations we calculate Sð~ structure factors) for enough large numbers of the first reciprocal vectors ~g. For calculations we ˚ , the number of select the a = 17 and R = 6.8 A atoms Na = 40. The number N we select in the range from some tens to some thousands. Our calculations show that the normalized structure factors have valuable quantity (as a rule from 0.1 to 5) when l/N and m/N-values are the integer numbers. These quantities are practically independent of N. The values of others normalized structure factors depend on N-number: when N arise the values of the pointed structure factors decrease. Taking the different sets of random numbers one can obtain the corresponding sets of structure factors. Obtained in such a manner, the sets may be averaged over the number of tests. The values of averaged factors of the 1st kind with high accuracy lþm equal to values ð1Þ N a J 0 ðR~ gÞ and approximately to zero for factors of the 2nd kind. Recall previous example it is easy to see that the structure factors of the 1st kind represent the averaged potential of the cell with a-size and the factors of the 2nd kind represent the diffusion noise. Table 1 and Fig. 8 illustrate some results of calculations. One can see that the calculated values of structure factors are really close to theoretical ones and the agreement became better with increase N or number of tests. Besides, this agreement is practically conserved down to very large numbers l and m. Now we can make conclusions that potential (8) introducing with the help of physical reasons is really averaged potential. It should be noted, the experimental investigations of the nanotube superlattice by X-ray diffraction support the random character of angle shifts [1] and the experimental electron density is in agreement with our calculations of the averaged potential.
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It is obvious that diffusion structure factors one can use for determination of the additional noncoherent contribution in the cross section. In the case of random angle shifts we found that diffusion factors are enough small when l, m-numbers less than 10N. From here we have estimated that noise contribution is less than some percents of incoherent one. In a whole we think that the method considered here may be utilized for determination of averaged potentials in more complicated cases. For this purpose, the known dependence of the angle shifts should be simulated. Analyzing the calculated values of structure factors we can find ones independent of N, NT-numbers and depending on these numbers. The first factors are relative to the averaged potential and others correspond to diffusion noise. References [1] A. Thess, R. Lee, P. Nikolaev, et al., Science 273 (1996) 484. [2] V.V. Klimov, V.S. Letokhov, Phys. Lett. A 222 (1996) 424. [3] L.A. Gevorgian, K.A. Ispirian, R.K. Ispirian, Nucl. Instr. and Meth. B 145 (1998) 155. [4] N.K. Zhevago, V.I. Glebov, Phys. Lett. A 250 (1998) 360. [5] V.M. Biryukov, S. Bellucci, Phys. Lett. B 542 (2002) 111. [6] S. Bellucci et al., Phys. Rev. ST Accel. Beams 6 (2003) 033502.
[7] M.L. Ter-Mikaelian, High Energy Electromagnetic Processes in Condensed Media, Wiley, New York, 1972, see here references on early authorÕs papers. [8] H. Uberall, Phys. Rev. 103 (1956) 1055; H. Uberall, Phys. Rev. 107 (1957) 223. [9] G. Barbiellini, G. Bologna, G. Diambrini, G.P. Murtas, Phys. Rev. Lett. 8 (1962) 454. [10] L. Criegee, G. Lutz, H.D. Schulz, U. Timm, W. Zimmermann, Phys. Rev. Lett. 16 (1966) 1031. [11] C. Berger, G. McClellan, N. Mistry, et al., Phys. Rev. Lett. 25 (1970) 1366. [12] V.A. Maisheev, A.M. Frolov, R. Avakian, et al., Sov. Phys. JETP 50 (1979) 856; V.A. Maisheev, A.M. Frolov, R. Avakian, et al., Nucl. Instr. and Meth. 178 (1980) 319. [13] P.J. Bassy et al., Nucl. Instr. and Meth. 211 (1983) 301. [14] A. Apyan, R.O. Avakian, B. Badelek, et al., hep-ex/ 0306028. [15] A.M. Frolov et al., Nucl. Instr. and Meth. 216 (1983) 93. [16] U. Timm, Fort. d. Phys. 17 (1969) 765. [17] H. Bilokon, G. Bologna, F. Celani, et al., Nucl. Instr. and Meth. 204 (1983) 299. [18] V.N. Baier, V.M. Katkov, V.M. Strakhovenko, Electromagnetic Processes at High Energies in Oriented Single Crystals, Singapore World Scientific, Singapore, 1998. [19] V.A. Maisheev, Nucl. Instr. and Meth. B 119 (1996) 42. [20] Y.-K. Kwon, D. Tomanek, Phys. Rev. Lett. 84 (2000) 1483. [21] G. Bologna, G. Lutz, H.D. Schulz, U. Timm, W. Zimmermann, Nuovo Cim. 42 (1966) 844. [22] W. Friedrich, P. Knipping, M. von Laue, Annal. Phys. 41 (1913) 971. [23] J.M. Cowly, Diffraction Physics, North-Holland Publishing Company, Amsterdam, Oxford, 1973.
Coherent bremsstrahlung and e±-pair production in periodic nanostructures S. Bellucci a, V.A. Maisheev a b
b,*
INFN – Laboratori Nazionali di Frascati, P.O. Box 13, 00044 Frascati, Italy Institute for High Energy Physics, 142281 Protvino, Moscow Region, Russia Received 9 September 2004 Available online 19 November 2004
Abstract Coherent bremsstrahlung and electron–positron pair production of high energy electrons and c-quanta moving in a three-dimensional periodic lattice consisting of a complicated system of atoms are considered. The peculiarities of formation and possibilities of utilization of these processes are discussed. 2004 Elsevier B.V. All rights reserved.
1. Introduction Considerable advances have been made in recent years in the creation of various nanostructures and particularly in the synthesis of the so-called carbon nanotubes. In [1] a superlattice of singlewall nanotubes was described. According to this paper single-wall nanotubes are uniform in diameter and self-organize into ropes, which consist of 100–500 nanotubes in a two-dimensional triangu˚ . Then, in lar lattice with a lattice constant of 17 A a number of papers [2–6] the channeling mechanism and channeling radiation in similar lattices were considered. As expected, these effects can be *
Corresponding author. E-mail address: [email protected] (V.A. Maisheev).
utilized for high energy physics purposes, such as beam deflection, photon sources creation, etc. In this paper we consider the coherent processes of bremsstrahlung and e±-pair production on the periodic superlattice of nanostructures. These processes are the result of the interaction of an electron (or a c-quantum) moving in such a lattice with its periodical electric field. For the first time, the theory of this phenomenon in single crystals was published in [7,8]. Since this time the phenomenon was investigated experimentally [9,10] and it was utilized for obtaining linearly polarized cbeams [11–13], in polarization measurements [14] and for increasing the e±-beam intensity [15]. The processes of coherent bremsstrahlung and pair production were also described in detail in the literature (see for example [16–18]).
0168-583X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.09.016
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It should be noted that the coherent processes as a rule were considered for simple crystallographic structures, such as monoatomic single crystals. In this paper we investigate the abovementioned processes for complicated enough periodic structures, consisting of a large number of atoms. As we will show below, the new peculiarities take place at the formation of coherent processes on the lattice with the basic sizes in tens of angstroms.
2. Electric fields of single crystals and nanostructures The continuum three-dimensional potential (averaged over thermal fluctuations of the lattice) of a single crystal or some periodic nanostructure consisting of atoms, has the form [18,19] 4peZ X uðrÞ ¼ U ðgÞeigr ; ð1Þ V g where e is the elementary charge, Z is the atomic number of the material, V is the fundamental cell
volume, g is a reciprocal lattice vector [7,16], r is the radius vector drawn from the origin of the fundamental cell, U ðgÞ ¼ SðgÞ
ð1 F ðgÞÞ expðAg2 =2Þ; g2
ð2Þ
S(g) is the structure factor [7], F(g) is the form factor of an atom in the structure and A is the meansquare amplitude of the thermal vibrations of atoms. Thus, the potential of the periodic nanostructure is determined, if we know the quantity U(g), which can be measured (strictly speaking the quantity jF(g)j2 is measured) from the diffraction of X-rays or electrons. The structure factor S(g) is determined with the help of the following relation [7]: SðgÞ ¼
N X
expðigqj Þ;
ð3Þ
j¼1
where the qj is the radius-vector of the jth atom in the fundamental cell of the lattice, N is the total number of atoms in the fundamental cell. Let us consider the triangle superlattice consisting of single nanotubes (see Fig. 1). In this case we
Fig. 1. Three-dimensional superlattice of (10, 10) armchair single-wall nanotubes and Cartesian coordinate system (xyz). The fundamental cell of the structure is presented with the help of the thick lines. The black points are atoms of the nanotubes. ˚ , b = 2.4 A ˚ , radius of circles R = 6.8 A ˚. OA = AB = BC = CO = 17 A
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can write for qj = xji + yjj + zjk (i, j, k are the unit vectors along x, y, z-axes) the following relations: 4pj1 þ u1 ; xj1 ¼ R cos N ð4Þ 4pj2 þ u2 ; xj2 ¼ R cos N 4pj1 þ u1 ; y j1 ¼ R sin N ð5Þ 4pj2 þ u2 ; y j2 ¼ R sin N zj1 ¼ 0;
zj2 ¼ b=2;
ð6Þ
where j1, j2 = 1, 2, . . ., N/2 are the indices corresponding to atoms placed in two parallel planes, R is the radius of the ring, b is the period (i.e. the size of the fundamental cell) in the z-direction, u1, u2 are the angle shifts. The reciprocal lattice vector g with the l, m, n-indices in Cartesian coordinates (see Fig. 1) have the form pffiffiffi 2p 2p ðl mÞi þ ðl þ mÞj= 3 þ nk; ð7Þ g¼ a b where a is the size of the fundamental cell in the xy-plane (a = OA = AB, see Fig. 1). It is well-known [7,16] that there are some numbers (very often they are integer numbers) for the structure factors in monoatomic single crystals. In the case under consideration it is necessary to calculate the structure factor according to Eqs. (3)–(6) for every reciprocal vector. In the general case these factors are complex numbers. The equations presented here describe the three-dimensional potential of the lattice of nanotubes, as shown in Fig. 1. This figure illustrates that the ideal lattice consists of uniformly oriented (10, 10) armchair single-wall nanotubes (the angle shifts are selected in the following way: u1 = 0, u2 = 6). For a real nanotube structure we expect that the angle shifts will be different for every nanotube in the lattice. (Notice that the problem of a nanotube orientation in the superlattice is not simple, see [20].) With the aim to take into account this situation we introduce the continuum three-dimensional potential aver-
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aged over angle shifts. This potential has the form described by Eqs. (1)–(3), where the structure factors read Sðl; m; nÞ ¼ cos2 ðpn=2ÞNJ 0 ðRgðl; m; 0ÞÞ:
ð8Þ
Here J0(x) is the Bessel function of the zeroth order and g(l, m, 0) is the value of the (l, m, n)-reciprocal vector projection on the xy-plane. The physical meaning of this potential is the following: it is the most likely value of the potential at any space point, if the angle shifts of nanotubes are distributed randomly. It should be noted that our consideration of potentials has a model independent form [19], or, in other words, it is valid for any atomic potentials. The last fact finds a reflection in that the dependence on the atomic potential takes place by means of atomic form factors. Besides, it is easy to obtain two-dimensional potentials from Eq. (1) by integration over the z-coordinate, as it was considered in [19]. As a result the two-dimensional potentials are also described by Eq. (1), in which the sum is calculated of only over (l, m, 0)-vectors of the reciprocal lattice. Fig. 2 illustrates the calculations of the twodimensional lattice potentials for (10, 10) armchair single-wall nanotubes in both of the following cases: (a) for a lattice with fixed angle shifts (u1 = 0, u2 = 6), see Fig. 1; (b) for a lattice with averaging over angle shifts. In all calculations in the paper the Molier atomic form factor was employed. The amplitude of the thermal vibrations ˚ (as in a diamond single cryswas equal to 0.039 A tal). The increase of this value by a factor 2 is insignificant for all computed results of the coherent processes in this paper. However, the behavior of potentials noticeably depends on the amplitude of the thermal vibrations, but it takes place in a small volume (A3/2) in the vicinity of the atomic centers. The lattice potentials shown in Fig. 2 are close to similar ones, which were calculated in [4] on the basis of the Doyle–Turner approximation for atomic potentials. Notice that in the cited paper the method of calculation is different from that used here.
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dr 2 2 a a x ¼ r0 ð1þ ð1 xÞ Þðw1 þ w1 Þ ð1 xÞðw2 þ w2 Þ ; dx 3 ð9Þ dr 2 ¼ r0 ðx2 þ ð1 xÞ2 Þðwa1 þ w1 Þ þ xð1 xÞðwa2 þ w2 Þ ; dx 3 ð10Þ
where r0 ¼ aQED Z 2 r2e , aQED = 1/137.04, re is the E classical electron radius, x ¼ Ec is the ratio of emitted photon energy Ec to the initial energy E of the electron (or the ratio of the electron energy to the energy of the c-quantum for pair production), wa1 , wa2 are well-known functions in the theory of coherent processes [7,16] (for high energies of the interacting particles they are almost constants), w1, w2-functions are w1 ¼ 4
2 ð2pÞ X dg2 jU ðgÞj2 2? ; NV gk g
w2 ¼ 24
2 2 ð2pÞ2 X 2 d g? ðg k dÞ jU ðgÞj ; NV g4k g
ð11Þ
ð12Þ
where gk is the projection of the g-vector on the direction of the particle motion, g2? ¼ g2 g2k , dvalue is equal to d¼ Fig. 2. Two-dimensional periodic potential of the superlattice (see Fig. 1). The upper and lower half-figures are the potentials described by Eqs. (4)–(6) and (8), respectively. The numbers near equipotential curves are the values of the potential in eV.
3. Cross sections of coherent bremsstrahlung and pair production From Eq. (1) one can see that the potential of the lattice consisting of nanotubes represents a three-dimensional periodic structure. A charged particle or c-quantum moving in this medium will interact periodically with the lattice. The results of this interaction are the following well-known processes: coherent bremsstrahlung (for a charged particle) and e±-pair production (for a c-quantum). If the expansion of the potential in Fourier series is known, the differential cross sections per atom of these processes are, correspondingly
mc2 x 2E 1 x
ð13Þ
for bremsstrahlung and mc2 x ð14Þ d¼ 2E xð1 xÞ for e±-pair production m is the electron mass). The summation in Eqs. (11) and (12) takes place under the following condition: gk P d:
ð15Þ
The d-value has an important physical meaning, i.e. it is the minimal transferred momentum to the medium. The inverse value (1/d) determines the formation length of the radiation process. Eqs. (9) and (10) for cross sections are valid at conditions that the thickness of medium where the particle propagates is more than the formation length and approximately 10 periods (or more) must be to exist in the medium along the particle path [7,18].
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It is significant that in this section we employ the system units in which the value of the reciprocal (direct) lattice vector is measured in units of k1 c ðkc Þ, where kc is the electron Compton length. This system is adopted in the theory of coherent radiation [7,16] and is convenient for performing specific calculations numerically. The equations presented here for cross sections have the same form as for single crystals. The basic difference between single crystals and nanostructures is determined by the difference in the size of the fundamental cells and disposition of atoms, or, in other words, by the values of the structure factors. From Eqs. (11) and (12) one can see that every nonzero term in the sum is proportional to the following factor: 2
M¼
2
jSðgÞj ð1 F ðgÞÞ : NV g2
ð16Þ
We compare the intensities of coherent processes in the nanotube lattice with the intensities of these processes in a single crystal of diamond, for the case when the electron beam moves under a small angle relative to the h0 0 1i axis of the single crystal. Also, from Eqs. (9)–(12) one can see that the intensities of both coherent processes considered
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here are proportional to the inverse volume of the elementary cell. So the volume of such a cell for a (10, 10) armchair single-wall nanotube is ˚ 3. For a diamond crystal the same equal to 600 A ˚ 3. At first sight, from here volume is equal to 46 A it follows that the intensity of the coherent radiation in the nanotube lattice should be approximately 13 times less than in the diamond crystal (at equivalent conditions). However, this conclusion is not true. From Eqs. (9) and (10) follow that the intensity is the sum of various terms. It turns out that in the case of a nanotube lattice the number of effective terms grows with increasing of time. This fact is illustrated in Fig. 3. Here the solid curves present calculations of the MV value the jSðgÞj2 for diamond ¼ 8 at amplitudes of thermal N ˚ as continuous functions vibrations 0 and 0.039 A of the g-value. However, in both cases of lattices the set of g-values is discrete. The values of the MV-factor for discrete g-values are shown in Fig. 3 (the solid intercepts are for a nanotube lattice and dashed ones are for diamond). One can see that only some values for diamond and a lot of values for the lattice of (10, 10) armchair singlewall nanotubes belong to the most effective range. In the latter case small values of the M-factors are partially compensated by the large number of them.
Fig. 3. The factor MV as a function of the g-value for a nanotube lattice and a diamond single crystal. The numbers near the arrows are indices of some points of the diamond reciprocal lattice. Incident electrons move under a small angle relative to the h0 0 1i diamond axis (see the explanation in the text).
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It should be noted that coherent radiation has a linear polarization and coherent pair production has a dependence of the cross sections on the linear polarization of the c-quanta. One can also write the relations describing these phenomena. Besides, at very small angles of motion for electrons, positrons or c-quanta with respect to the nanotube axis, the coherent mechanism is violated. Partially this takes place due to channeling in the nanotubes. On the other hand, at angles of motion < Umax/mc2 (where Umax is the potential barrier) the deflection angle of the electron is larger than the inverse Lorentz factor. In this case, the radiation processes have a magneto bremsstrahlung character [18]. For a (10, 10) singlewall nanotube this angle is about 0.3 mrad. Note that at high energies of the particles a similar effect takes place for some strong planar orientations (the corresponding planar angle is less than 0.05 mrad). For calculations of the coherent bremsstrahlung and pair production processes we have found the periodic averaged potential for the nanotube superlattice, which, generally speaking is not periodic in a strict sense. Some explanations and discussion of the coherent radiation in imperfect periodic structures can be found in Appendix A.
4. Calculation of coherent processes The direction of the particle motion relative to the Cartesian coordinate system shown in Fig. 1 can be determined with the help of the a and h angles. We denote by h the angle with the z-axis and a is the azimuthal angle around this axis (a = 0, when the momentum of the incoming particle lies in the yz-plane). Sometimes it is convenient to use an alternative pair of angles, i.e. hx = hcos a and hy = hsin a. Taking into account these angles, we can write (at h 1) 2 2p ðl mÞ2 þ ðl þ mÞ2 =3 ; ð17Þ g2? ¼ a gk ¼
pffiffiffi 2p 2p h ðl mÞ sin a þ ðl þ mÞ cos a= 3 þ n: a b ð18Þ
In the case when g? d (this condition is valid when the particle energy is high enough) the terms corresponding to vectors of the reciprocal lattice with n = 0 yield the main contribution to the cross sections. Further, we carry out a calculation of the radiation coherent processes in this approximation. It means that at high energies the coherent processes depend predominantly on the transversal electric fields in nanostructures. It is well-known [7], that the maximum of the coherent bremsstrahlung spectrum corresponds to the condition gk = d or, in an expanded form, pffiffiffi 2p h ðl mÞ sin a þ ðl þ mÞ cos a= 3 ¼ d: ð19Þ a Let us assume that the particle move in the xzplane. It means that a = p/2. Then Eq. (10) can be rewritten as follows: 2p hðl mÞ ¼ d: a
ð20Þ
The choice of points of the reciprocal lattice under the condition l m = 1 corresponds to selecting a row as shown in Fig. 4. The dashed region corresponds to points of the reciprocal lattice, whose contribution in intensity is large enough. If the initial energy E and the relative energy x of the emitted photon are fixed, then from Eq. (20) we can find the corresponding angle h. The result of a calculation of the intensity of the coherent radiation is presented in Fig. 5 for energy E = 10 GeV and x = 0.5. In this case the angle takes the value h 0.018 radian. Before discussing this result we can draw the following important conclusion: if one knows the intensity (cross section) for some energy E of the particle, then one can find the corresponding intensity for another energy with the help of the simple relation drcnew drc ðEn ; x; hx =k; hy =jÞ ¼ j ðE; x; hx ; hy Þ; dx dx
ð21Þ
where En is the new energy, j = En/E, rc is the coherent part of the cross section. As already noted, the amorphous part of the cross section is practically constant at high energy of the particles. Thus, it is very easy to get the energy behavior of the intensity of the coherent processes.
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Fig. 4. Reciprocal lattice for a structure of nanotubes (see Fig. 1). The numbers near the points are (l, m)-indices of the reciprocal lattice vectors. The arrow Ph is the projection of the particle momentum on the xy-plane.
Fig. 5. The bremsstrahlung spectrum x dr of 10-GeV electrons dx moving at angles h = 0.018 and a = p/2 radian (curves 1 and 2). Curves 1 and 2 correspond to potentials, which are described by Eqs. (4)–(6) and (8). Curve 3 is calculated for hx = 0.006, hy = 0.002 radian and for a potential, which is described by Eq. (8). Curve 4 is the amorphous part of the cross section (see the explanation in the text).
The calculations presented in Fig. 5 were carried out for both potentials (see Eqs. (4)–(8)). Fig. 5 shows that the intensity of the first potential Eqs. (4)–(7) is 1.5 times higher than for the other. However, we think that the second potential is more realistic. We have also compared these calculations with similar ones for the (1 0 0) plane of the reciprocal lattice of a diamond single crystal. For a diamond single crystal the maximum intensity takes place at h = 0.00133 radian (for x = 0.5). In this case the electron momentum lies in the (0 1 1) plane of the single crystal. The absolute value of the coherent cross section is 4.3 and 6.5 times larger in comparison with the nanotube lattice for the first and second potentials. Thus, the decrease in intensity is less than it may be expected. The purpose and mechanism of this effect were discussed already (see also Fig. 3). We call attention to the fact that the coherent processes in nanostructures have a tens of times wider angle sensitivity. This result
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can be seen from Eqs. (19) and (20): the angle h is proportional to the characteristic minimal period of a structure. The curves 1 and 2 (see Fig. 5) present a special orientation of the electron beam when the momentum of electron is perpendicular to rows of reciprocal lattice points (see also Fig. 4). In the general case the direction of the electron beam is arbitrary relative to the reciprocal lattice and the quasimonochromaticity of the emitted photons is lost. Curve 3 presents a sample of a similar behavior. It should be noted that curves 1 and 2 (see Fig. 5) are shown for the purposes of comparison of the obtained data with the analogous ones for the diamond single crystal. Besides, similar orientations are widely covered in the literature. The authors of the paper [16,21] proposed to name such a kind of spectra as ideal ones. Due to the specific motion of electrons, the form of the real spectra differ from ideal ones as it is illustrated by the dashed curve, which corresponds to curve 1.
Fig. 6. The behavior of the bremsstrahlung cross section for a 10 GeV electron beam, as a function of the hx and hy orientation angles. The numbers near the curves are the values of the cross section in comparison with the amorphous one. Some insignificant details in the figure were deleted for the sake of clarity of the picture. Calculations are made for the potential described by Eq. (8). The black points are narrow maxima of the distribution, with the peak value being approximately equal to 4.5.
Fig. 7. The behavior of the pair production cross section for 10 GeV unpolarized c-quanta, as a function of the hx and hy orientation angles. The numbers near the curves are the values of the cross section in comparison with the amorphous one. Some insignificant details in the figure were deleted for the sake of clarity of the picture. Calculations are made for the potential described by Eq. (8).
Calculations were carried out for thickness equal to 0.01 cm. One can see that at x > 0.1 the dashed curve and curve 1 practically coincide. Figs. 6 and 7 illustrate the behavior of cross sections of both latter processes considered, for 10 GeV incoming particles at x = 0.5 as a function of the hH, hV-angles. From here one can see that the intensities have a complicated enough angle dependence and a wide angle range, where coherent processes exceed the noncoherent ones. By using Eq. (20) it is easy to recalculate the data in Fig. 7 (or in Fig. 6) for c-quanta energies higher than 10 GeV. For example, for an energy of 100 GeV we must reduce both angle ranges by 10 times (from ±0.1 to ±0.01 radian). Besides, there is a need to recalculate the numbers near the curves; so, the quantity 1.8 is replaced by 10(1.8 1) + 1 = 9 and so on. 5. Conclusion We consider the processes of the coherent bremsstrahlung of electrons and coherent pair pro-
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duction of c-quanta moving in a three-dimensional lattice of nanotubes. At present, these structures are created in small amounts, however the development of nanotechnology allows one to assume a significant progress in manufacturing much larger quantities of them in the near future. The main result of our consideration for a lattice based on (10, 10)-armchair single-wall nanotubes is that both coherent processes take place in the large angle range with high intensity. A more detailed analysis of coherent processes in a nanotube lattice, including the polarization characteristics of cross sections and the behavior of processes at low energies of the moving particles, will be published elsewhere. However, based on our consideration of the specific mechanism for the formation of coherent radiation processes, we can claim that these processes exist in different periodic nanostructures and their intensity may be very high at high energies of incident particles in a wide angle range. Besides, we think that the results of our investigations may be also useful for the X-ray analysis of periodic nanostructures: for example, the jU(g)j2-values in Eqs. (11) and (12) are similar to scattering strengths for X-rays. It should be noted that the results obtained here (at the condition of solving the problem of synthesizing nanotube lattices in large volumes) bear important consequences for creating various sources of positrons and c-radiation, polarimeters and so on. Besides, presently one can begin carrying out some experimental investigations of existing samples in electron beams.
Acknowledgements This work is supported in part from the Italian Ministry MIUR under grant COFIN 2002022534.
Appendix A. Coherent bremsstruhlung and pair production in the periodic imperfect structures The process of coherent bremsstrahlung was investigated and utilized mainly in simple single crystals. Such crystals as a diamond and silicon have the negligible small degree of mozaicity and
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admixture. Because of this, the problem of calculations of the bremsstrahlung in imperfect structures did not practically appear with one exception. It is taking into account of the influence of atomic thermal motion on the intensity of radiation. In [7] (where this process was investigated) point out the fundamental analogy between coherent bremsstrahlung and diffraction of X-ray in single crystals. Starting from the famous article [22] the process of the X-ray diffraction gain worldwide recognition as a powerful tool for investigations of various atomic structures, due to real impetus was given in development of the theory of the phenomenon, in particularly, for imperfect periodic structures. Thus, we can use the results of this theory for calculations of the process of coherent bremsstrahlung in the periodic imperfect structures, for example, such as the nanotube superlattices. The main result of the theory [23], which we apply to coherent bremsstrahlung, is the following: the differential cross section of the process is calculated according to Eqs. (9)–(12), in which U(g) is the Fourier component of the averaged potential of a structure. Besides, in the general case the noncoherent contribution in the cross section is possible. The theory gives some methods for obtaining of averaged potentials. However, in the general case they are enough complicated. In this paper we have introduced the averaged potential on the basis of the physical representations for the case of superlattice when the angle shifts are distributed randomly. One can see that this potential have the clear and transparent physical sense. Further we suggest the method for determination of the averaged potential in more general cases. For simplicity we consider our method for a two-dimensional square supperlattice. Besides, we assume that all Na atoms (see Eqs. (3)–(6) and Fig. 1) are distributed with the equal step around the nanotube circle. Now we can build the square superlattice consisting of N2 nanotubes. Then we can enumerate every nanotube in the superlattice as ij (i, j = 1,2,3, . . ., N), where i and j are the horizontal and vertical numbers of the square cell of side a, correspondingly. And after we enumerate sequentially the atoms in every nanotube as k where k = 1, 2, 3, . . ., Na. Thus, every atom in the
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r = (i 1)aix + (j 1)ajy in the basic coordinate system. In Eq. (A.3)P the value S ij ð~ gÞ is the followN ing structure factor: 1 a exp i~gqijk , where in the local coordinate system vector qijk is
superlattice get own ijk-number. We can also determinate the angle shifts uij1 for every atom with the 1st number in the nanotube. Then the other atoms obtain the angle shifts uijk according to relation: uijk ¼ uij1 þ 2pðk 1Þ=N a :
qijkx ¼ R cos uijk þ a=2;
ðA:1Þ
ðA:4Þ
By this means we determinate all the geometrical parameters of the superlattice. Let be specific, we select the basic Cartesian coordinate system in the left and bottom corner of the superlattice with x and y axes in the horizontal and vertical directions, correspondingly. In a similar manner, we introduce the local Cartesian coordinate systems for every cell with i and j numbers. With this lattice (of Na · Na sizes) as the basic element we can build (by using of parallel translations in x and y directions with the period of Na) the twodimensional infinite periodic system and calculate its potential with the help of Eq. (1). For this pur~ gÞ for every pose, we find the structure factors Sð~ reciprocal vector ~ g: S~ ¼
N X N X i¼1
exp i~ grij S ij ð~ gÞ;
Recall that the averaged over angle shifts potential of the superlattice (in limit N ! 1) are determined by (1)l+mNaJ0(gR) factor, where (1)l+mmultiplay is due to our choice of the coordinate (the shift of coordinate system on (a/2, a/2)-vector corresponds to Eq. (8)). First we consider the superlattice where the orientation of all the nanotubes is the same, i.e. the angle shifts of the different nanotubes are equal in between. It is obvious that in this case our periodic structure is one which based on the square fundamental cell with the period of a. The all structure factors (see Eq. (A.2)) for which l/N and m/N are nonintegral numbers are equal to zero, and the others structure factors are equal to N2S, where S is structure factor for cell with the a-size. One can see that in this case the reciprocal space consists of the following vectors (corresponding to nonzero structure factors): g ¼ ~gN . Taking into account that S~ ¼ N 2 S and the corresponding volumes of both the cells are connected by relation V~ ¼ N 2 V we get the expected result: at our choice of the angles shifts the real funda-
ðA:2Þ
j¼1
where the reciprocal vector ~ g is ~ g ¼ G0 lix þ G0 mjy ;
l; m ¼ 0; 1; 2; . . . ;
qijky ¼ R sin uijk þ a=2:
ðA:3Þ
where G0 = 2p/(Na), ix and jy are the unit vectors in x and y directions and the translation vector
Table 1 Results of simulation of averaged potential. Normalized structure factors in 2–5 columns are calculated for reciprocal vectors e g ¼ 2plix =ðaN Þ l/N
STh
S1
S2
S3
S4
10 20 30 40 50 60 70 80 90 100 200
4.478713 3.175097 2.594649 2.247977 2.011157 1.836234 1.700225 1.590557 1.499696 1.422815 1.006334
4.478714 3.066866 2.604257 2.268821 2.183495 1.771523 1.702413 1.640649 1.460587 1.442049 1.068331
4.478713 3.160334 2.605523 2.246414 2.048165 1.817817 1.662717 1.560811 1.492919 1.438173 1.001786
4.478713 3.174532 2.591840 2.249317 2.018196 1.845523 1.700150 1.584478 1.499754 1.416810 1.014237
4.478713 3.175091 2.594654 2.247972 2.011253 1.836263 1.700272 1.590533 1.499752 1.422829 1.006339
STh is the theoretical value of the factor. S2 is the factor calculated at N = 100 and averaged over number of tests NT = 10. S1, S3, S4 are calculated factors for condition NT = 1 and N = 100, 1000, 10000, correspondingly. We do not illustrate factors less than 10 due to excellent agreement with the theory.
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mental cell is square with the side is equal to a and the potential of this structure is described by Eq. (1). Note the presence of volume V in Eq. (1) for
Fig. 8. Simulation of the potential in nanotube superlattice. The calculated absolute values of the normalized structure factors for lattice with N = 10 are shown as functions of l, mnumbers in the following cases: for one test (top) and averaged over 100 tests (bottom). The area of the circles are proportional to their quantity. The quantity of point with l, m = 0 is equal to 40. The points with the quantities less than 10-4 are invisible. Figure illustrates the different behavior of the structure factors at averaging and gives some insight into the diffusion noise properties.
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a two-dimensional structure one can to explain that this structure are disposed in the three-dimensional space. Let us consider the second case when the angle shifts of the nanotubes are distributed randomly in the interval (0, 2p). This problem we can solve by computer simulation. The angle shifts one can define with the help of the generator of the pseudorandom numbers. Using the above-mentioned ~ gÞ=N 2 -values (normalized relations we calculate Sð~ structure factors) for enough large numbers of the first reciprocal vectors ~g. For calculations we ˚ , the number of select the a = 17 and R = 6.8 A atoms Na = 40. The number N we select in the range from some tens to some thousands. Our calculations show that the normalized structure factors have valuable quantity (as a rule from 0.1 to 5) when l/N and m/N-values are the integer numbers. These quantities are practically independent of N. The values of others normalized structure factors depend on N-number: when N arise the values of the pointed structure factors decrease. Taking the different sets of random numbers one can obtain the corresponding sets of structure factors. Obtained in such a manner, the sets may be averaged over the number of tests. The values of averaged factors of the 1st kind with high accuracy lþm equal to values ð1Þ N a J 0 ðR~ gÞ and approximately to zero for factors of the 2nd kind. Recall previous example it is easy to see that the structure factors of the 1st kind represent the averaged potential of the cell with a-size and the factors of the 2nd kind represent the diffusion noise. Table 1 and Fig. 8 illustrate some results of calculations. One can see that the calculated values of structure factors are really close to theoretical ones and the agreement became better with increase N or number of tests. Besides, this agreement is practically conserved down to very large numbers l and m. Now we can make conclusions that potential (8) introducing with the help of physical reasons is really averaged potential. It should be noted, the experimental investigations of the nanotube superlattice by X-ray diffraction support the random character of angle shifts [1] and the experimental electron density is in agreement with our calculations of the averaged potential.
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It is obvious that diffusion structure factors one can use for determination of the additional noncoherent contribution in the cross section. In the case of random angle shifts we found that diffusion factors are enough small when l, m-numbers less than 10N. From here we have estimated that noise contribution is less than some percents of incoherent one. In a whole we think that the method considered here may be utilized for determination of averaged potentials in more complicated cases. For this purpose, the known dependence of the angle shifts should be simulated. Analyzing the calculated values of structure factors we can find ones independent of N, NT-numbers and depending on these numbers. The first factors are relative to the averaged potential and others correspond to diffusion noise. References [1] A. Thess, R. Lee, P. Nikolaev, et al., Science 273 (1996) 484. [2] V.V. Klimov, V.S. Letokhov, Phys. Lett. A 222 (1996) 424. [3] L.A. Gevorgian, K.A. Ispirian, R.K. Ispirian, Nucl. Instr. and Meth. B 145 (1998) 155. [4] N.K. Zhevago, V.I. Glebov, Phys. Lett. A 250 (1998) 360. [5] V.M. Biryukov, S. Bellucci, Phys. Lett. B 542 (2002) 111. [6] S. Bellucci et al., Phys. Rev. ST Accel. Beams 6 (2003) 033502.
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