A general formalism of two-dimensional lattice potential on beam transverse plane for studying channeling radiation
aa r X i v : . [ c ond - m a t . o t h e r] S e p A general formalism of two-dimensional lattice potential on beamtransverse plane for studying channeling radiation
Jack J. Shi and Wade Rush
Department of Physics & Astronomy,The University of Kansas, Lawrence, KS 66045
Abstract
To study channeling radiation produced by an ultra-relativistic electron beam channeling througha single crystal, a lattice potential of the crystal is required for solving the transverse motion of beamelectrons under the influence of the crystal lattice. In this paper, we present a general formalism forthis two-dimensional lattice potential of a crystal with a Lorentz contraction in the beam channelingdirection. With this formalism, the lattice potential can be calculated without approximation fromany given model of electron-ion interaction for an ultra-relativistic beam channeling in any crystaldirection. The formalism presented should be the standard recipe of the lattice potential forstudying the channeling radiation. . INTRODUCTION There has been a renewed interest recently in channeling radiation [1] produced froman electron beam interacting with a crystal lattice for its potential application in hard X-ray production [2–12]. When an ultra-relativistic electron enters a single crystal, ratherthan having random scattering, the electron will channel through the crystal lattice if itsincident angle relative to a specific lattice orientation of the crystal is sufficiently small [13].During the channeling, the non-relativistic motion of the electron in the transverse planethat is perpendicular to the channeling direction could be strongly perturbed by the crystallattice while the ultra-relativistic motion of the electron along the channeling direction isunperturbed. A high-intensity ultra-relativistic electron beam could produce high-brightnesshard X-rays due to the perturbation of the transverse motion of beam electrons in a crystal.The channeling radiation is considered to be from the transitions between bounded Blocheigenstates for the transverse motion of beam electrons during the channeling. To studythe radiation theoretically and numerically, the interaction between the crystal lattice andbeam electrons has been modeled in two different approaches, the one-dimensional (1D)planar [14, 15] and the two-dimensional (2D) axial [15, 16] channeling model. In the planarchanneling model, the transverse motion of the electrons is assumed to be aligned in asingle crystal direction during the channeling and the Bloch eigenstates for the transversemotion are solved from a 1D Schr¨odinger equation. This 1D approximation is valid onlyif the coupling between the original 2D motion of the electrons in the transverse planeis negligible. Note that the lattice potential for the transverse motion of the channelingelectrons is the result of a Lorentz contraction of the three-dimensional (3D) crystal latticein the beam rest frame where the crystal travels with near the speed of light along (oppositeto) the beam channeling direction and the 3D crystal lattice is pancaked into a 2D latticeon the beam transverse plane. The resulting lattice potential is strongly coupled in the2D transverse plane and the transverse motion of a channeling electron in a crystal cannotbe decoupled into 1D motion if the lattice potential dominates the Hamiltonian. At thebounded Bloch eigenstates for the channeling radiation, the transverse energy is negative(inside lattice potential wells) and the lattice potential dominates the transverse kineticenergy. Therefore, the 1D approximation of the planar channeling model cannot be justifiedrigorously and the study of the channeling radiation should be based on the Bloch eigenstates2or the 2D transverse motion of beam electrons in a crystal. In order to solve the Blocheigenstates in the 2D transverse plane, a lattice potential in the transverse plane is required.To obtain this potential of the 2D lattice, in the axial channeling model, the ions in theoriginal crystal are first grouped into strings of the ions along the channeling direction andthe lattice potential is calculated by summing up the contributions of the strings of theions [15, 16, 18]. In this direct calculation of the lattice potential, the identification of thestrings of the ions and the summation over the strings are cumbersome and, in some cases,difficult due to a complicated geometric relationship among lattice ions along the channelingdirection. Moreover, the calculation with the strings of the ions has to be specifically tailoredfor each specific crystal structure and channeling direction, which can hardly be applied toother cases because of the difference in the strings of the ions with different crystal structureand different channeling direction.An accurate lattice potential on the transverse plane is necessary for calculating the Blocheigenstates of the 2D transverse motion of beam electrons in a crystal. With a given interac-tion between a beam electron and individual ion in a crystal, such as the Born approximationfor electron scattering from an ion with Doyle-Turner fitting parameters [19], the questionis what is the best approach, in terms of accuracy, mathematical simplicity, and generalapplicability to any crystal structure and channeling direction, for the construction of thelattice potential on the 2D transverse plane. In this paper, we present a new and betterapproach for this 2D lattice potential calculation. In this new approach, a lattice potentialof the original three-dimensional crystal is first calculated from a given electron-ion interac-tion potential in the original unit cell coordinate of the crystal by taking the advantage ofthe native periodicity of the crystal lattice. This potential of the three-dimensional latticein the unit-cell coordinate is then transferred into the beam coordinate that is aligned withthe beam longitudinal (channeling) and transverse directions using rotational coordinatetransformations. Lastly, the Lorentz contraction of the lattice can be easily accomplishedmathematically by averaging the three-dimensional lattice potential in the beam coordinatealong the channeling direction. In this approach, the calculation of the lattice potential ismathematically clean, systematic, and can be easily applied to any crystal with any chan-neling direction. This generic formulation of the 2D lattice potential is developed in SectionII, Section III presents several examples of the channeling in different crystal direction, andSection IV contains a final remarks. 3
I. FORMULATION OF LATTICE POTENTIAL IN BEAM TRANSVERSE PLANE
When an ultra-relativistic electron beam channels through a crystal lattice with a specificorientation aligned along the beam longitudinal direction, the non-relativistic transversemotion of the electrons is perturbed by the lattice of ions in the crystal. To formulatethis lattice potential, we consider an orthorhombic crystal lattice with each unit cell ofthe lattice containing N ions. Let ~r j with j = 1 , · · · , N be the local coordinates of theions in a unit cell, where the origin of the local coordinate is at a corner of a unit cell, and ~r m = ( m a , m a , m a ) be the global coordinate of a unit cell, where ( a , a , a ) are latticeconstants of the crystal and ~m = ( m , m , m ) are an integer vector. Let ~X = ( X , X , X ) bea global coordinate, referred as lattice coordinate, with the axes of X , X , and X alignedwith the primary crystal axes along [100], [010], and [001] crystal direction, respectively.When an electron travels through the crystal, the interaction between the electron and thecrystal lattice can be calculated by a superposition of the interactions between the electronand each individual ion as V D ( ~X ) = ∞ X ~m = −∞ N X j =1 V ion ( ~X − ~r m − ~r j ) (1)where V ion ( ~r ) is the interaction potential between an electron and a single ion in thecrystal. Note that V D ( ~X ) is periodic in the crystal, i.e. V D ( ~X + ~r l ) = V D ( ~X ), where ~r l = ( l a , l a , l a ) and ( l , l , l ) is any combination of integers. With the lattice nativeperiodicity ( a , a , a ), V D ( ~X ) can be rewritten into a Fourier expansion of V D ( ~X ) = ∞ X ~k = −∞ V ~k e i ~G · ~X (2)where ~G = 2 π ( k /a , k /a , k /a ) is the reciprocal lattice vector of the crystal. The Fourierexpansion coefficient V ~k can be calculated as V ~k = 1 a a a ∞ X ~m = −∞ N X j =1 a Z a Z a Z V ion ( ~X − ~r m − ~r j ) e − i ~G · ~X d ~X = 1 a a a N X j =1 ∞ Z −∞ ∞ Z −∞ ∞ Z −∞ V ion ( ~ξ − ~r j ) e − i ~G · ~ξ d~ξ = − π ¯ h m e v f ion ( ~G ) N X j =1 e − i ~G · ~r j (3)4here v = a a a is the unit-cell volume of the crystal, m e is the electron rest mass, and f ion ( ~G ) = − m e π ¯ h ∞ Z −∞ ∞ Z −∞ ∞ Z −∞ V ion ( ~ξ ) e − i ~G · ~ξ d~ξ (4)is the atomic form factor of a single ion in the crystal [20]. Note that the lattice potentialcalculated from Eqs. (3) and (4) includes exactly all the contributions of ions in a crystal andonly approximation involved is in the modeling of the electron-ion interaction V ion . With theDoyle-Turner approximation of the form factor [19], the electron-ion interaction potentialcan be expressed as [15] V ion ( ~ξ ) = − π ¯ h m e M X i =1 α i ( β i /π ) / e − π | ~ξ | /β i (5)where α i and β i with i = 1 , · · · , M are the parameters for fitting f ion ( ~G ) to the relativisticHartree-Fock calculation of the form factor at values of | ~G | [19]. In the original Doyle-Turner’s calculation, four Gaussian functions ( M = 4) were used in Eq. (5). To improve theaccuracy at relatively large value of | ~G | , Chouffani and ¨Uberall used six Gaussian functions( M = 6) for the fitting. The obtained values of α i and β i for diamond, silicon and germaniumwere given in Ref. [18]. With the electron-ion interaction potential in Eq. (5), the Fouriertransformation of the lattice potential of a three-dimensional crystal can then be calculatedfrom Eqs. (3) and (4) as V ~k = − π ¯ h m e v M X i =1 α i e − λ i | ~k | / (2 a ) N X j =1 e − i π ( ~k · ~r j ) /a (6)where λ i = β i for V ion given in Eq. (5) and λ i = β i + 8 π h µ i for including the effect ofthermal vibrations of the lattice with p h µ i being the root-mean-square displacement ofthe thermal vibration of the lattice [15].The lattice potential V D ( ~X ) in Eq. (2) is expressed in the lattice coordinate that isaligned with the primary crystal axes but not aligned with the beam channeling direc-tion. In order to study channeling radiation that is generated from the perturbation of thetransverse motion of beam electrons under the influence of the lattice potential, the latticepotential in the beam transverse plane is needed. Since the beam is ultra-relativistic, theinteraction between a beam electron channeling through the crystal and the ions in thecrystal is too weak to have a significant effect on the longitudinal motion of the electron.5he dependence of the lattice potential on the longitudinal coordinate of the electron cantherefore be neglected by averaging the potential over the longitudinal direction. It is thusnecessary to transform the lattice potential into a beam coordinate that is aligned with thebeam channeling direction. Physically, this is just rotating the crystal lattice so that thecrystal axis for the channeling is aligned with the beam direction. Let ~r = ( x, y, z ) be thebeam coordinate where z is along the beam channeling (longitudinal) direction of the beamand ( x, y ) are two orthogonal coordinates in the beam transverse plane. The transformationfrom the lattice coordinate ~X to the beam coordinate ~r can be accomplished by a 3 × R , ~r = R ~X . For a beam channeling along the [ hkl ] crystal direction, wherethree integers h , k , and l are the Miller indices of crystal lattices [20], R can be in generalconstructed with three consecutive transformations (see Fig. 1 for an illustration). The firsttransformation rotates the lattice coordinate ( X , X , X ) in the X − X plane by an angleof θ = arctan( k/h ) and the transformed coordinate is labeled as ( X ′ , X ′ , X ). The axesof X ′ and X ′ are aligned with the [ hk
0] and [¯ kh
0] crystal direction, respectively (see Fig.1b). The second transformation is rotating ( X ′ , X ′ , X ) in the X ′ − X plane by an angleof θ = arctan( l/ √ h + k ) and the transformed coordinate is labeled as ( X ′′ , X ′ , X ′ ). Theaxes of X ′′ , X ′ , and X ′ are aligned with [ hkl ], [¯ kh h y k y l y ] direction, respectively,(see Fig. 1c) where( h y , k y , l y ) = ( h , k , l ) × ( − k , h ,
0) = ( − hl , − kl , h + k ) (7)The third transformation is to switch the coordinate axes such that x = X ′ , y = X ′ , and z = X ′′ , where z along the [ hkl ] direction is the beam longitudinal coordinate (channel-ing direction) and x along the [¯ kh
0] direction and y along the [ h y k y l y ] direction are twoorthogonal coordinates in the beam transverse plane, respectively. These three coordinatetransformations can be expressed as ~X = R − ~r = cos θ − sin θ θ cos θ
00 0 1 cos θ − sin θ θ θ xyz (8)With the transformation matrix R in Eq. (8), 3D lattice potential V D ( ~X ) in Eq. (2) canbe expressed in the beam coordinate as V D ( R − ~r ) = ∞ X ~k = −∞ V ~k exp h i ( ~G T R − ) · ~r i (9)6here ~G T = 2 π ( k /a , k /a , k /a ) denotes the row vector of ~G for the matrix multipli-cation. Since the crystal lattice has also periodicities along the orientations of the beamcoordinate, V D is periodic in ~r . Let ( b , b , b ) be the periodicities of the lattice along the( x, y, z ) direction, respectively. The periodicity of V D ( ~r ) requires e ib ( ~G T R − ) = e ib ( ~G T R − ) = e ib ( ~G T R − ) = 1which yields a transformation of the reciprocal lattice vector (cid:18) k a , k a , k a (cid:19) R − = (cid:18) n b , n b , n b (cid:19) (10)where ( n , n , n ) are integers. The relationships between ( b , b , b ) and ( a , a , a ) andbetween ( n , n , n ) and ( k , k , k ) can be obtained from Eq. (10) with the condition thatboth ( n , n , n ) and ( k , k , k ) are integers for a set of smallest values of ( b , b , b ). Sincethe transformation in Eq. (9) is from the expansion in ~X with ( k , k , k ) as the indicesto an expansion in ~r with ( n , n , n ) as the indices, ( k , k , k ) as functions of ( n , n , n )are needed here. For each given set of ( n , n , n ) in integers, therefore, ( k , k , k ) have tobe solved from Eq. (10) as integers. With the coordinate of the 3D lattice potential of acrystal correctly aligned with the beam directions, the average of the lattice potential alongthe beam channeling ( z ) direction can easily be calculated as V D ( x, y ) = 1 b Z b V D ( R − ~r ) dz = ∞ X ~k = −∞ V ~k e i π ( n x/b + n y/b ) δ n (11)where δ n is the Kronecker delta for n = 0. The Fourier expansion of this projected latticepotential on the transverse plane can then be written as V D ( x, y ) = ∞ X n = −∞ ∞ X n = −∞ V n n e i π ( n x/b + n y/b ) (12)and the expansion coefficient can be obtained from V ~k of the 3D lattice potential that iscalculated using Eqs. (3) and (4) as V n n = ∞ X ~k = −∞ V ~k δ (cid:16) ~k − ~k ( n , n ) (cid:17) (13)where ~k ( n , n ) is the solution of ~k = ( k , k , k ) as functions of ( n , n , n ) solved from Eq.(10) with n = 0 and the delta functions are the Kronecker delta for ~k = ~k ( n , n ).7t should be noted that the minima (potential wells) of V D ( x, y ) in the transverse planeform periodically a 2D lattice that results from the projection of the original crystal latticeonto the transverse plane. The Lorentz contraction along the channeling direction of athree-dimensional lattice can create a more condense structure of ions in the transverseplane. The periodicity of V D ( x, y ) could be smaller than b and b due to the averaging of3D lattice potential V D along the beam channeling direction. Mathematically, the changeof the periodicity occurs when the summation over the ions in a unit cell in Eq. (6) zerosperiodically for certain combinations of ( k , k , k ) with n ( ~k ) = 0. If that occurs, thesummation for the Fourier expansion of V D ( x, y ) in Eq. (12) can be rearranged with thecorrect values of the periods. To solve the Bloch eigenstates for the transverse motion ofbeam electrons, moreover, one needs to identify the primitive unit cell of the 2D lattice of V D ( x, y ) and the x and y coordinate should be aligned with the axes of the primitive unitcell [20]. The crystal axes of the primitive unit cell and the lattice constants that are theperiods of V D ( x, y ) can be easily identified by examining the contour plot of V D ( x, y ). Ifthe coordinate ( x, y ) in Eq. (12) is not aligned with the primitive unit cell, an additionalrotational transformation in the transverse plane is needed for V D ( x, y ). III. EXAMPLES
To illustrate this general method for the lattice potential in the beam transverse plane,the potential is calculated for the channeling along [001], [110], [111], and [210] direction ofa cubic crystal with a diamond-like atomic structure that has eight ions in a unit cell (e.g.diamond, silicon, germanium). The coordinate of the ions in a unit cell is listed in Table Iand the lattice constant of the crystal is denoted by a = a = a = a . For the electron-ion interaction V ion ( ~r ), Eq. (6) of the Doyle-Turner model is used in the calculation. Allfigures for the calculated V D ( x, y ) are plotted with germanium (Ge) lattice and the fittingparameters in the Doyle-Turner model obtained by Chouffani and ¨Uberall [18]. For theconvenience of reading, those fitting parameters are re-listed in Table II. The value usedfor the root-mean-square amplitude of the lattice thermal vibration of Ge in Eq. (6) is p h µ i = 0 . a = 5 . . Channeling along [001] Crystal Axis For the channeling along the [001] crystal axis, the beam longitudinal coordinate z isaligned with the lattice coordinate X . A simple choice of the transverse coordinate is x and y aligned with [100] and [010] crystal direction, respectively, and the beam coordinate ( x, y, z )is aligned with the lattice coordinate ( X , X , X ). In Eq. (8), on the other hand, θ = 0 and θ = π/ hkl ] = [001], which yields a transformation matrix R that switches only the x and y coordinate and has no any physical consequence on the Fourier transformation of thelattice potential. No coordinate transformation is therefore needed, i.e. R can be chosen asan identity matrix and the lattice potential V D ( x, y ) can easily be calculated from Eqs. (6)and (13). In the contour plot of the calculated V D ( x, y ), however, the axes of the primitiveunit cell of the two-dimensional lattice on the transverse plane were observed to be alignedwith the [110] and [¯110] crystal direction. A rotational transformation in the transverseplane is therefore needed to align x and the y axes with the [110] and [¯110] crystal direction,respectively, i.e. X X X = cos( π/ − sin( π/
4) 0sin( π/
4) cos( π/
4) 00 0 1 xyz = R − xyz (14)where π/ (cid:18) k + k √ a , k − k √ a , k a (cid:19) = (cid:18) n b , n b , n b (cid:19) (15)Since for each given set of integers ( n , n , n ), ( k , k , k ) have to be solved from Eq. (15)as integers, the solution with the minimal values of ( b , b , b ) is ( b , b , b ) = (cid:0) √ a/ , √ a/ , a (cid:1) ( k , k , k ) = ( n − n , n + n , n ) (16)The lattice potential with x and y coordinate aligned with the [110] and [¯110] crystal axesis then calculated from Eqs. (12), (13), and (6) as V D ( x, y ) = ∞ X n = −∞ ∞ X n = −∞ V n n exp (cid:20) i π (cid:18) n xa/ √ n ya/ √ (cid:19)(cid:21) (17)9here V n n = − π ¯ h m e v X ~k M X i =1 α i e − λ i | ~k | / (2 a ) N X j =1 e − i π ( ~k · ~r j ) /a ! δ k , ( n − n ) δ k , ( n + n ) δ k , = − π ¯ h m e a M X i =1 α i a e − λ i ( n + n ) / a N X j =1 e − i π [ n ( x j + y j ) − n ( x j − y j )] /a (18)With the coordinate ~r j of the ions in a unit cell given in Table I, the second summation inEq. (18) can be evaluated as N X j =1 e − i π [ n ( x j + y j ) − n ( x j − y j )] /a = , for ( n , n ) = (even, even)0 , otherwise (19)To purge the terms of zero in the summation of Eq. (17), let n = 2 k and n = 2 k , andthe lattice potential in the transverse plane can be rewritten as V D ( x, y ) = ∞ X k = −∞ ∞ X k = −∞ v k k exp (cid:20) i πa/ (2 √
2) ( k x + k y ) (cid:21) (20)where v k k = V (2 k )(2 k ) = − π ¯ h m e a M X i =1 α i a e − λ i ( k + k ) /a (21)The periods of V D ( x, y ) is therefore a x = a y = a/ (2 √
2) in both x and y direction when x and y axes are aligned with the [110] and [¯110] crystal axis that are of the primitive cellin the transverse plane of a beam channeling in the [001] direction. The lattice potentialin Eqs. (20) and (21) has also been obtained previously using the method of strings ofions in the axial channeling model [18], as this is the easiest case for the axial channelingcalculation. As shown in Fig. 2a, V D ( x, y ) has a single potential well in each square unitcell on the beam transverse plane. Near the bottom of the potential wells, where the stringsof ions is located, the interaction is almost rotationally symmetric on the transverse plane.The rotational symmetry of the lattice potential has been previously used to simplifyingthe study of the Bloch eigenstates and transitions between the eigenstates for the transversemotion of beam electrons [12, 16–18]. b. Channeling along [110] Crystal Axis For a beam channeling in [ hkl ] = [110] direction, θ = arctan( k/h ) = π/ θ = 0 fortransformation matrix R in Eq. (8) and h y = k y = 0 and l y = 2 for the Miller indices of the10rystal directions on the transverse plane. The x and y coordinate on the transverse planeare therefore aligned with [¯ khl ] = [¯110] and [ h y k y l y ] = [001] crystal axes, respectively. Thetransformation of the reciprocal lattice vector is calculated from Eq. (10) as (cid:18) k − k √ a , k a , k + k √ a (cid:19) = (cid:18) n b , n b , n b (cid:19) (22)For integer ( k , k , k ) and ( n , n , n ), the solution of Eq. (22) with the minimal values of( b , b , b ) is ( b , b , b ) = (cid:0) √ a/ , a , √ a/ (cid:1) ( k , k , k ) = ( n − n , n + n , n ) (23)The lattice potential with x and y coordinate aligned with the [¯110] and [001] crystal axesis then calculated from Eqs. (12), (13), and (6) as V D ( x, y ) = ∞ X n = −∞ ∞ X n = −∞ V n n exp (cid:20) i π (cid:18) n xa/ √ n ya (cid:19)(cid:21) (24)and V n n = − π ¯ h m e a M X i =1 α i a e − λ i (2 n + n ) / (2 a ) N X j =1 e − i π [ n ( y j − x j )+ n z j ] /a (25)Since there is no additional periodic zero in the summation over ( x j , y j , z j ) in Eq. (25), theexpansion in Eq. (24) is the correct Fourier expansion of V D ( x, y ) for solving the Blocheigenstates. As shown in Fig. 2b, there are four potential wells in each unit cell and thelattice potential is highly anisotropic in the transverse plane. The period of V D ( x, y ) are a x = a/ √ a y = a along the [¯110] and [001] crystal axes that are the axes of the primitiveunit cell in the transverse plane for a beam channeling through a diamond-like crystal alongthe [110] crystal axis. c. Channeling along [111] Crystal Axis For a beam channeling in [ hkl ] = [111] direction, θ = arctan( k/h ) = π/ θ =arctan( l/ √ h + k ) = arctan(1 / √
2) for transformation matrix R in Eq. (8) and h y = − hl = − k y = − kl = −
1, and l y = h + k = 2 for the Miller indices of one crystaldirection for the transverse coordinates. The x and y coordinate are therefore aligned with[¯ kh
0] = [¯110] and [ h y k y l y ] = [¯1¯12] crystal axes, respectively. The transformation of thereciprocal lattice vector is calculated from Eq. (10) as (cid:18) k − k √ a , k − k − k √ a , k + k + k √ a (cid:19) = (cid:18) n b , n b , n b (cid:19) (26)11or integer ( k , k , k ) and ( n , n , n ), the solution of Eq. (26) with the minimal values of( b , b , b ) at n = 0 is ( b , b , b ) = (cid:0) a/ √ , a/ √ , √ a (cid:1) ( k , k , k ) = ( − n − n , n − n , n ) (27)The 2D lattice potential can then be calculated from Eqs. (12), (13), and (6) as V D ( x, y ) = ∞ X n = −∞ ∞ X n = −∞ V n n exp (cid:20) i π (cid:18) n xa/ √ n ya/ √ (cid:19)(cid:21) (28)where V n n = − π ¯ h m e a M X i =1 α i a e − λ i ( n +3 n ) / a N X j =1 e i π [ n ( x j − y j )+ n ( x j + y j − z j )] /a (29)With the ion coordinates ( x j , y j , z j ) given in Table I, the second summation in Eq. (29) canbe evaluated as N X j =1 e i π [ n ( x j − y j )+ n ( x j + y j − z j )] /a = , for n + n = even0 , otherwise (30)and V n n can thus be rewritten as V n n = − π ¯ h m e a M X i =1 α i a e − λ i ( n +3 n ) / a ! δ n + n , even (31)Because of different periodicity of V D ( x, y ) in Eq. (28) along the x and y direction, theperiodic occurrence of zero in V n n cannot be removed from the summations in Eq. (28) byrelabeling the summation indices and a rotation of ( x, y ) coordinate while keeping the formof a Fourier expansion. This can also be seen in the contour plot of V D ( x, y ) in Fig. 2c,where there are two identical potential wells at the corners and center of a unit cell and itcannot be simplified into a single square lattice because of different lattice constants alongthe x and y direction. The [¯110] and [¯1¯12] crystal axes are therefore the orthogonal axes ofthe primitive unit cell in the transverse plane and a x = a/ √ a y = a/ √ V D ( x, y ) for a beam channeling in the [111] direction. d. Channeling along [210] Crystal Axis For a beam channeling along the [ hkl ] = [210] crystal axis, θ = arctan( k/h ) =arctan(1 /
2) and θ = arctan( l/ √ h + k ) = 0 and the solution of Eq. (10) is ( b , b , b ) = (cid:0) a/ √ , a , a/ √ (cid:1) ( k , k , k ) = ( − n + 2 n , n + n , n ) (32)12he lattice potential with x and y coordinate aligned with [¯ khl ] = [¯120] and [ h y k y l y ] = [001]crystal axes can be calculated from Eqs. (12), (13), and (6) as V D ( x, y ) = ∞ X n = −∞ ∞ X n = −∞ V n n exp (cid:20) i π (cid:18) n xa/ √ n ya (cid:19)(cid:21) (33)where V n n = − π ¯ h m e a M X i =1 α i a e − λ i (5 n + n ) / (2 a ) N X j =1 e − i π [ n (2 y j − x j )+ n z j ] /a (34)With ion coordinates ~r j in a unit cell given in Table I, the summation over the ions in Eq.(34) is N X j =1 e − i π [ n (2 y j − x j )+ n z j ] /a = (cid:2) − ( n + n ) / (cid:3) , for ( n , n ) = (even, even)0 , otherwise (35)Similar to the case of the channeling in the [001] direction, this periodic occurrence of zeroin the Fourier expansion can be removed by relabelling the summation indices as n = 2 k and n = 2 k in Eq. (33) without altering the form of the Fourier expansion. The periodsof V D ( x, y ) in this case is thus a x = a/ (2 √
5) and a y = a/ V D ( x, y ) = ∞ X k = −∞ ∞ X k = −∞ v k k exp (cid:20) i π (cid:18) k xa/ (2 √
5) + k ya/ (cid:19)(cid:21) (36)where v k k = − π ¯ h m e a M X i =1 α i a e − λ i (5 k + k ) /a (cid:2) − k + k (cid:3) (37)where v k k = 0 for k + k = odd . As shown in Fig. 2d, there are two identical potential wellsin each unit cell of V D ( x, y ). Similar to the case of channeling in the [111] direction, this2D body-center rectangular unit cell cannot be simplified into a simple rectangular unit cellbecause of different periodicity of V d ( x, y ) along the x and y direction. The [¯120] and [001]crystal axis are the orthogonal axes of the primitive unit cell in the transverse plane and a x = a/ (2 √
5) and a y = a/ V D ( x, y ) for a beam channelingthrough a diamond-like crystal along the [210] crystal axis.13 V. FINAL REMARKS
To solve the Bloch eigenstates for the transverse motion of beam electrons in a crystalnumerically, the Fourier expansion of the lattice potential in Eq. (12) needs to be truncatedas V D ( x, y ) = K max X n = − K max K max X n = − K max V n n e i π ( n x/b + n y/b ) (38)This truncation is possible due to a fast decay of expansion coefficient V n n which is theconsequence of an electron-ion interaction potential that decays faster then 1 /r . The con-vergence of the truncation, however, needs to be checked to ensure the accuracy of thenumerically calculated potential. To examine the convergence, we estimate the truncationerror using Truncation Error = 1 a x a y a y Z a x Z (cid:12)(cid:12)(cid:12)(cid:12) V D ( x, y, K max ) V D ( x, y, K max + ∆) − (cid:12)(cid:12)(cid:12)(cid:12) dxdy (39)where V D ( x, y, K max ) and V D ( x, y, K max + ∆) are the lattice potential V D ( x, y ) calculatedwith the truncation at K max and K max + ∆, respectively. Figure 3 plots this truncationerror as a function of K max for the examples in Fig. 2 and shows that the numericallycalculated potential is sufficiently accurate (with a truncation error smaller than 10 − ) for K max <
50. Note that solving the Bloch eigenstates numerically from a two-dimensionalSchr¨odinger equation requires diagonalizations of (2 K max + 1) × (2 K max + 1) matrices forthe Hamiltonian operator and, with K max <
50, this computational task can be handledwith well-configured pc computers nowadays.In summary, we have developed a general formalism for the lattice potential for studyingthe transverse motion of beam electrons when an ultra-relativistic beam channels througha crystal lattice. As shown by the examples, this two-dimensional lattice potential caneasily be calculated for a beam channeling through a crystal along any crystal direction.With the availability of the 2D lattice potential and increasing power of pc computers, the2D calculation of the Bloch eigenstates should becomes the standard for the study of thechanneling radiation and the 1D approximation in modeling the channeling radiation is no14onger needed. [1] M.A. Kumakhov, Phys. Lett. , 17 (1976).[2] P. Piot et al, FERMILAB-CONF-13-086-AD-APC.[3] T. Sen and C. Lynn, Int. J. Mod. Phys. A , 1450179 (2014).[4] B. Azadegan, W. Wagner, and J. Pawelke, Phys. Rev. B , 045209 (2006).[5] D. Mihalcea, D.R. Edstrom, P. Piot, W.D. Rush, and T. Sen, “Channeling Radiation Experi-ment at Fermilab ASTA”, Proc. 6th Int. Particle Accelerator Conf. , Richmond, VA, 2015, pp.95–98. http://jacow.org/IPAC2015/papers/mopwa009.pdf ,[6] B. Azadegan, L.Sh. Grigoryan, J. Pawelke, and W. Wagner, J. Phys. B: At. Mol. Opt. Phys. , 235101 (2008).[7] B. Azadegan, S.B. Dabagov, Eur. Phys. J. Plus , 58 (2011).[8] B. Azadegan, Comp. phys. Commu. , 1064 (2013).[9] C.K. Gary, A.S. Fisher, R.H. Pantell, J. Harris, M.A. Piestrup, Phys. Rev. B , 7 (1990).[10] C.K. Gary, R.H. Pantell, M. ¨Ozcan, M.A. Piestrup, D.G. Boyers, J. Appl. Phys. , 2995(1991).[11] H. Genz, H.D. Gr¨af, P. Hoffmann, W. Lotz, U. Nething, A. Richter, H. Kohl, A. Weickenmeier,W. Kn¨upfer, J.P.F. Sellschop, Appl. Phys. Lett. , 2956 (1990).[12] H. Genz, L. Groening, P. Hoffmann-Staschek, A. Richter, M. H¨ofer, J. Hormes, U. Nething,J.P.F. Sellschop, C. Toepffer, M. Weber, Phys. Rev. B , 8922 (1996).[13] J. Lindhard, Phys. Lett. , 126 (1964).[14] J.U. Andersen, K.R. Eriksen and E. Laegsgaard, Physica Scripta. , 588 (1981).[15] J.U. Andersen, E. Bonderup, and E. Laegsgaard, in Coherent Radiation Sources , Eds. A.W.Saenz and H. ¨Uberall, (Springer-Verlag, Berlin, 1985).[16] J.U. Andersen, E. Bonderup, E. Laegsgaard, B.B. Marsh, and A.H. Sorensen, Nucl. Instrum.Meth. & Inst. , 209 (1982).[17] R. Klein, J. Kephart, R. Pantell, H. Park, B. Berman, R. Swent, S. Datz, and R. Fearick,Phys. Rev. B , 68 (1985).[18] K. Chouffani and H. ¨Uberall, Phys. Stat. Sol. B , 107 (1999).[19] P.A. Doyle and P.S. Turner, Acta Cryst. A , 390 (1968).
20] C. Kittel,
Introduction to Solid State Physics , 8th ed., (John Wiley & Sons, New York, 2005).[21] D.S. Gemmell, Rev. Mod. Phys. , 129 (1974). ~r j in a unit cell of diamond-like crystals where the origin of thecoordinate is at a corner of the unit cell and a is the lattice constant. j x j /a y j /a z j /a α j and β j for atomic form factor of germanium obtainedby Chouffani and ¨Uberall [18], where a = 5 . i α i /a β i /a ✷ ❂ ❬(cid:0)✶(cid:0)❪✁ ✸✂✄☎☎✆✝✞ ✟ ✠ ✡☛✵✵☞ ✌❤✍✎✏✡✑✒❧☞✦✓ ✦✔ ✕✖✗ ✘✰ ❦ ✘ ✭✙✚ ❳✶ ❂ ❬(cid:0)✵✵❪✁ ✷✂✄☎✆☎✝ ✞✦ ✟ ✠ ✡❤☛☞✌✍ ✎✏ ✑ ✒ ø❦✓✔✕ ✖✗ ✘✘✙ ✭✚✮❳ ✦✶ ❂ (cid:0)❤✁✵✂✄ ✸☎❬✆✆✝❪ ✞✟✟ ✠ ✡ ☛☞✌❧✍✎ ✏✑ ✒ ✓✔② ❦② ✕② ✖ ✗✷ ✘✘✙ ✚✛ ✰✜✛ ✭❝✢ Figure 1. (a) The [ hkl ] and [ hk
0] crystal axis in the lattice coordinate ( X , X , X ) ofan orthorhombic crystal lattice, (b) the 1st coordinate rotation on the X – X plane and(c) the 2nd coordinate rotation on the X ′ – X plane for the construction of R for a beamchanneling in [ hkl ] direction, where θ = arctan( k/h ) and θ = arctan( l/ √ h + k ). Thecrystal axis [ h y k y l y ] is along the direction of integer vector ( h y , k y , l y ) = ( h, k, l ) × ( − k, h,
0) =( − hl, − kl, h + k ) 18 / a y x/a x -0.5 -0.25 0.0 0.25 0.5-0.5-0.250.00.250.5 (a) [001]V min = -120 eV, V max = -5 eV y / a y x/a x -0.5 -0.25 0.0 0.25 0.5-0.5-0.250.00.250.5 (b) [110]V min = -170 eVV max = -2 eV y / a y x/a x -0.5 -0.25 0.0 0.25 0.5-0.5-0.250.00.250.5 (c) [111]V min = -138 eVV max = -5 eV y / a y x/a x -0.5 -0.25 0.0 0.25 0.5-0.5-0.250.00.250.5 (d) [210]V min = -57 eVV max = -9 eV Figure 2. Contour plots of lattice potential V D ( x, y ) on the beam transverse plane for abeam channeling through a diamond-like crystal along (a) [001], (b) [110], (c) [111], and(d) [210] direction, where a x and a y are the periods of V D ( x, y ) in the x and y direction,respectively, and V min and V max are the minimum and maximum of V D ( x, y ). The centersof equipotential loops are the bottoms of potential wells.19
10 20 30 40 50 60 T r un ca ti on E rr o r K max -10 -8 -6 -4 -2 [001][110][111][210] Figure 3. Truncation error of the truncated Fourier expansion of V D ( x, y ) in Eq. (38) v.s.K maxmax