An Accurate Image Simulation Method for High-Order Laue Zone Effects

CHIN.PHYS.LETT.Vol.25,No.5(2008)1772 An Accurate Image Simulation Method for High-Order Laue Zone Effects∗CAI Can-Ying(蔡灿英)1,2,ZENG Song-Jun(曾松军)1,2,LIU Hong-Rong(刘红荣)1,2,YANG Qi-Bin(杨奇斌)1,2∗∗1Institute of Modern Physics,Xiangtan University,Xiangtan411105 2Key Laboratory of Low-Dimensional Materials and Application Technology of Ministry of Education,XiangtanUniversity,Xiangtan411105(Received16October2007)A completely different formulation for simulation of the high order Laue zone(HOLZ)diffractions is derived.It refers to the new method,i.e.the Taylor series(TS)method.To check the validity and accuracy of the TS method,we take polyvinglidenefluoride(PVDF)crystal as an example to calculate the exit wavefunction by the conventional multi-slice(CMS)method and the TS method.The calculated results show that the TS method is much more accurate than the CMS method and is independent of the slice thicknesses.Moreover,the purefirst order Laue zone wavefunction by the TS method can reflect the major potential distribution of thefirst reciprocal plane.PACS:61.05.Jd,61.50.AhIt is well known that the most crystal structure information can be extracted from the exit wavefunc-tion contributed by the zero order Laue zone(ZOLZ) diffraction.The ZOLZ diffraction is caused by the projection of the crystal potential along the incident beam direction.Thus,the potential variation along the incident beam direction is lost if only the ZOLZ diffraction is considered.In order to obtain the poten-tial variation along the incident beam direction or the distribution of plane defects,for example,GP zone and stacking fault etc,high-order Laue zone(HOLZ) diffractions must be considered.[1−4]Nowadays,the most widely used image simulation method is the multi-slice(MS)theory[5]which was originally proposed by Cowley and Moodie[6]based on the physical–optical principles.The most important reason for the MS method gaining extensive applica-tion is its relatively fast calculation speed especially after Ishizuka and Uyeda introduced the fast Fourier transformation(FFT)[7]in this method.Twentyfive years later,Van Dyck[8]successfully tackled a new method,i.e.the real space(RS)method,which is directly based on the solution of the Schr¨o dinger equa-tion.In order to carry out the simulation by both the MS and the RS methods,a crystal has to be divided into a series of thin slices perpendicular to the incident beam direction.For the HOLZ diffraction calculation, the slices should be thin enough to be able to dis-play the potential variation along the incident beam direction.[9,10]The crystal potential was described as a series of steps in the RS scheme as shown in Fig.1(a), which is the approximation of the real potential as shown in Fig.1(b).Mathematically,any function can be expanded as Taylor series,which includes many high-order differential terms.Unexceptionally,the exit wavefunction can also be expanded as Taylor se-ries,implying that the exit wavefunction is influenced by not only the magnitude of the potential itself,but also every order differential of the potential.To ig-nore the potential differential effects on the wave func-tion in the RS scheme,the slice thickness should be infinitely thin,which is impossible for the practical application.To solve this problem,we adapt a com-pletely different way to derive the exit wavefunction from the Schr¨o dinger equation.In this study,we as-sume the exit wavefunction to be a Taylor series,then substitute it into the Schr¨o dinger equation and derive the coefficients of the Taylor series,which will be de-scribed in the following in detail.As usual,we start with the Schr¨o dinger equation[11]∆Ψ(r)+4π2k2Ψ(r)+V(r)Ψ(r)=0,(1) where∆is the Laplacian operator in the x−y plane, and V(r)=iσU(r)(λ,σand U(r)are the electron wavelength,the interaction constant and the crystal potential,respectively).For high-energy electrons(≥100keV),the in-fluence of the specimen potential can be considered as perturbation.Thus,the exit wavefunctionΨ(r) can be written as a modulated plane waveΨ(r)=φ(r)exp(2πikr).Substituting the above equation into Eq.(1),one hasdφdz=iλ4π(∆+V)φ,(2)∗Supported by the National Natural Science Foundation of China under Grant No10374077,and Key Foundation of Education Department of Hunan Province under Grant No05C097.∗∗To whom correspondence should be addressed.Email:yqb@ or Yangqibin02002@c 2008Chinese Physical Society and IOP Publishing LtdNo.5CAI Can-Ying et al.1773 where V can be divided into two parts,i.e.V=V P+V1.Fig.1.(a)Step potential used in the RS method(b)realpotential,versus distance along the a-axis in a unit cellwith unit a being the lattice constant.LetV P=kkF(hk0)exp[−2πi(hx+ky)],(3)V1=kkl=0F(hkl)exp[−2πi(hx+ky+lz)],(4)where V P is the projection of the crystal potential and is used to calculate the ZOLZ diffraction,V P+V1are used to calculate the ZOLZ and the HOLZ diffpared with the ZOLZ diffraction,the HOLZ diffractions are so weak that they can be con-sidered as perturbation.Among all the HOLZ diffrac-tions,thefirst one is the strongest;and the others can be neglected in the most practical cases.Thus one just needs to consider l=1in Eq.(4),V1=kkF(hk1)exp[−2πi(hx+ky+z)]=V2exp(−2πiz),(5)whereV2=kkF(hk1)exp[−2πi(hx+ky)].(6)Then Eq.(2)becomesdφdz =iλ4π[∆+V P+V2exp(−2πiz)]φ.(7)To solve Eq.(7),we assume thatφ(R,z)=φ(R,0)+dφdzz+12d2φdz2z2+13!d3φdz3z3+···+1n!d nφdz nz n+···=∞n=0z nn!d nφdz n,(8)orφ(R,(z+∆z))=φ(R,z)+dφdz∆z+12d2φdz∆z2+13!d3φdz3∆z3+···+1n!d nφdz n∆z n+···=∞n=0∆z nn!d nφdz,(9)where R is a plane vector.From Eq.(7),one mayfindthatdφdz=[∆+V P+V2exp(−2πiz/c)]φ,d2φdz2=[∆+V P+V2exp(−2πiz/c)]dφdz+−2πicV2exp(−2πiz/c)φ,d3φdz3=[∆+V P+V2exp(−2πiz/c)]d2φdz2+2−2πicV2exp(−2πiz/c)dφdz+−2πic2V2exp(−2πiz/c)φ,d4φdz=[∆+V P+V2exp(−2πiz/c)]d3φdz+3−2πicV2exp(−2πiz/c)d2φdz2+3−2πic2V2exp(−2πiz/c)dφdz+−2πic3V2exp(−2πiz/c)φ,······d nφdz n=[∆+V P+V2exp(−2πiz/c)]d n−1φdz n−1+(n−1)−2πicV2exp(−2πiz/c)d n−2φdz+(n−1)(n−2)2−2πic2V2·exp(−2πiz/c)d n−3φdz+(n−1)(n−2)(n−3)2×3−2πic3V2·exp(−2πiz/c)d n−4φdz n−4+···+(−2πic)n−1V2exp(−2πiz/c)φ.1774CAI Can-Ying et al.Vol.25Letηk =(∆z )k k !d k φdz k,(10)then we haveφ(R ,z +∆z )=∞ n =0ηn ,(11)whereηn =(∆+V p )∆z n ηn −1+V 2exp(−2πiz/c )∆zn· ηn −1+ −2πi ∆z cηn −2+12! −2πi ∆z c2ηn −3+···+1k ! −2πi ∆z ckηn −k −1+···+ .(12)Let∆zc=γ,the above equation becomes a recur-sive formula as follows:ηn =∆z n(∆+V p )ηn −1+V 2exp(−2πiz/c )n −1 k =0αk ηn −k −1,(13)whereαk =(−2πiγ)kk !.(14)To check the validity and the accuracy of expres-sions (11)–(14),we calculate the exit wavefunction of the ZOLZ and the HOLZ diffractions by the CMS method and the new expressions which we referred as the TS method using polyvinglidene fluoride (PVDF)crystal ((CH 2CF 2)2)as an example;then we compare the results calculated by those two different methods to find out which one is more accurate.PVDF belongs to an orthogonal crystal and the space group is P m 2m with the lattice constants a =0.858nm,b =0.491nmFig.2.(a)Amplitude and (b)phase of exit wavefunction calculated by the TS method,versus distance along the b -axis in a unit cell with unit b being the lattice constant,for z =0.5c .Fig.3.Phase map of two-dimensional exit wavefunctions:(a)calculated by the TS method with slice thickness a/32;(b)calculated by the TS method with slice thickness a/128;(c)calculated by the CMS method with slice thickness a/32;(d)calculated by the CMS method with slice thickness a/128.No.5CAI Can-Ying et al.1775 Fig.4.Maps of the potential and the purefirst order Laue zone wavefunction:(a)map of the potential of the zeroth reciprocal plane;(b)map the potential of thefirst reciprocal plane;(c)map of the purefirst order Laue zone wavefunction (amplitude)calculated by the TS method.and c=0.256nm.Thefitting constants for calcu-lating the structure factors of(PVDF)are taken from Ref.[12].The accelerate voltage is200kV correspond-ing toλ=0.00251nm.The incident beam is along the[100]direction.The y−z plane is divided into 64×32pixels.The thickness of the crystal along the incident beam direction(the a axis)is3.432nm corre-sponding to4times of the lattice constant a.A series of slice thicknesses a/128,a/64,a/32,a/16,a/8,a/4 and a/2are designed for comparing the effects of slice thicknesses on the results calculated by two different methods.The calculated results are shown in Figs.2–4.The calculated results along the[010]direction with z=0.5are schematically shown in Fig.2.From thesefigures we can see that both the amplitude and the phase almost keep constant for the different slice thicknesses(a/128,a/8,a/4and a/2).Figure3shows the phase of wavefunction in the exit plane.The results shown in Figs.3(a)and3(b) are calculated by the TS method with the different slice thicknesses(a/32and a/128).Figures3(c)and 3(d)show the calculated results by the CMS method with the different slice thicknesses.From those maps it can also be seen that the results calculated by the TS method are independent of the slice thickness.How-ever,those calculated by the CMS method are sensi-tive to the slice thickness especially in the peak area. However,when the slice thickness is very thin(here a/128),the results calculated by the CMS is in agree-ment with those calculated by the TS method.It can be concluded that the TS method is a more accurate method than the CMS method.Figure4(a)is the potential map of the zeroth re-ciprocal plane,from which the atomic positions pro-jected along the[100]direction can be obtained.Fig-ure4(b)is the potential map of thefirst reciprocal plane,i.e.V2in Eq.(6),which carries3D potential in-formation and is different from Fig.4(a).Figure4(c)is the map of the purefirst order Laue zone wavefunction (i.e.whole wave function subtracting ZOLZ wavefunc-tion)calculated by the TS pared with Fig.4(b),one mayfind that purefirst order Laue zone wavefunction can reflect major potential distribution of thefirst reciprocal plane although there are some weak differences between Figs.4(b)and4(c),which are probably caused by the interaction between ZOLZ and HOLZ diffraction.In conclusion,the CMS neglects the effect of the high-order differentials of the crystal potential on the HOLZ diffractions so that the slice thickness should be very thin.The TS method takes all the high-order differentials of the potential into account.The calcu-lated results are independent of the slice thicknesses.The purefirst order Laue zone wavefunction by the TS method can reflect the major potential distri-bution of thefirst reciprocal plane which is the3D potential information of the crystal.The TS method is suitable to the simulation requiring very accurate results.References[1]Brown J F and Clark D1952Acta Cryst.5615[2]Munson D and Wheeler M J1968J.Inst.Metal.96252[3]Gard J A1956Brit.J.Appl.Phys.7361[4]Gard J A and Taylor H F W1958Amer.Min.431[5]Spence J C H and Zuo J M1992Electron Microdiffraction(New York:Plenum)[6]Cowley J M and Moodie A F1957Acta Cryst.10609[7]Ishizuka K and Uyeda N1977Acta Cryst.A33740[8]Dyck D Van1980J.Microsc.119141[9]Chen J H,Beek M Op de,Dyck D Van1996Microsc.Mi-croanal.Microstruct.727[10]Self P G,O’Keefe M A,Buseck P R and Spargo A E C1983Ultramicroscopy1135[11]Dyck D Van1983J.Microsc.13231[12]Peng L.M,Ren G,Dudarev S and Whelan M1996ActaCryst.A52257。