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First principles study of three-component

INTERGRANULAR AND INTERPHASE BOUNDARIES IN MATERIALS

First principles study of three-component SrTiO 3/BaTiO 3/PbTiO 3ferroelectric superlattices

S.H.Shah ?P.D.Bristowe ?A.M.Kolpak ?A.M.Rappe

Received:3August 2007/Accepted:8October 2007/Published online:6March 2008óSpringer Science+Business Media,LLC 2008

Abstract The geometrical,chemical and ferroelectric properties of a new nanoscale short-period three-compo-nent SrTiO 3/BaTiO 3/PbTiO 3perovskite superlattice are investigated using a ?rst principles density functional approach.The study focuses on varying the thickness of each component in the superlattice and determining the resulting lattice distortion and total polarization.Thick-nesses of up to three unit cells in a single component are considered and the in-plane lattice constants normal to the [001]stacking direction are ?xed to the bulk SrTiO 3values to simulate a rigid substrate.It is found that the PbTiO 3layers play a key role in strain and polarization enhance-ment.By increasing the amount of PbTiO 3in the superlattices the strain in the other components increases signi?cantly resulting in an enhanced total polarization of the superlattice relative to bulk BaTiO 3.Increasing the number of BaTiO 3layers also improves the overall polar-ization.All the SrTiO 3layers in each superlattice are found to be highly polarized.Many of the calculated features are similar to those found previously by others for the SrTiO 3/BaTiO 3/CaTiO 3superlattice,although in the present study signi?cantly greater enhancement factors and polarization values are found.The predicted enhancement of the polarization is mostly attributed to lattice strain due to mismatch of the in-plane lattice constant of the three-component materials.

Introduction

There is considerable research activity into the design and characterization of ferroelectric superlattices (SL)since experimental measurements and theoretical investigations have shown that they can display superior functional properties relative to single-component ferroelectric materials in either bulk or thin-?lm form.Ferroelectric SLs are arti?cially structured multilayers composed of two or more thin-?lm perovskite components such as BaTiO 3(BT),PbTiO 3(PT)and CaTiO 3(CT).The components themselves need not all be ferroelectric or chemically uniform.Incipient ferroelectrics such as SrTiO 3(ST),metallic oxides such as SrRuO 3and solid solutions have been used with similar effects obtained.By varying the number of components,their thickness,composition and sequence in the multilayer it may be possible to achieve a signi?cant enhancement in dielectric properties such as spontaneous polarization which will undoubtedly bene?t future device applications.The polarization increase is clearly related to the lattice strain induced in the multilayer by interfacing components with different structural parameters,ferroelectric distortions and ionic charges.The optimal combination of parameters which maximizes the functional properties is not obvious and understanding this has been the focus of many of the studies.

The experimental fabrication of ferroelectric SLs with controlled composition and structure requires the use of epitaxial deposition techniques such as MBE and PLD.With these techniques it is possible to engineer the SLs while minimizing the introduction of lattice defects,dislocations and interfacial steps that may degrade the polarization effects.The structural quality of the multilayers can be assessed using various electron microscopy and diffraction techniques [1].The simplest type of SL involves just two

S.H.Shah áP.D.Bristowe (&)

Department of Materials Science and Metallurgy,University of Cambridge,Cambridge CB23QZ,UK e-mail:pdb1000@https://www.wendangku.net/doc/6817654340.html,

A.M.Kolpak áA.M.Rappe

Department of Chemistry,University of Pennsylvania,Philadelphia,PA 19104-6323,USA

J Mater Sci (2008)43:3750–3760DOI 10.1007/s10853-007-2212-7

components having layers of varying thickness.Recent bilayer systems investigated include BaTiO 3/BaO [2],SrTiO 3/SrO [2],BaTiO 3/SrTiO 3[3–7],PbTiO 3/SrTiO 3[8–11],PbTiO 3/PbZrO 3[12],SrZrO 3/SrTiO 3[13–15],KNbO 3/KTaO 3[16],PbTiO 3/BaTO 3[17–20],PbTiO 3/SrRuO 3[21]and modulated PbZr 1àx Ti x O 3(PZT)[22]and PbSc 1àx Nb x O 3(PSN)[23].In addition to determining strain and polarization effects,the evolution and stability of dif-ferent phases has been studied [24,25].Computationally various atomistic approaches have been taken to calculate the structure and properties of SLs.The level of theory employed determines the size of the system that can be studied and the accuracy of the results.The investigations include density functional (DFT)calculations [2–5,21,26]the ?rst of which was on the Pb(Zr 0.5Ti 0.5)O 3system [27,28].Other studies have used empirical shell-model calcu-lations [29],effective Hamiltonian methods [23,30]and electrostatic methods [10].Emerging from this body of experimental and theoretical work are three key factors which in?uence the enhancement of SL polarization:the mismatch between the lattice parameters of the compo-nents,the difference between the dielectric constants of the components and the structure and chemistry of the interphase boundaries between the components.

Introducing a third perovskite into the superlattice is an obvious extension to the bilayer system and can have the fundamental effect of changing its symmetry.Sai et al.[31]predicted the properties of these superlat-tices,which have general formulae A 1=3A 01=3A 00

1=3

BO 3and A B 1=3B 01=3 B 001=3TO 3,by focusing on the effect of compositionally breaking the inversion symmetry.Fol-lowing the theoretical predictions,experimental studies on the SrTiO 3/BaTiO 3/CaTiO 3(ST/BT/CT)system were per-formed [32,33].Warusawithana et al.[33]utilized the broken inversion symmetry of the ST/BT/CT trilayer SL and obtained a very large dielectric constant (103at room temperature)for the system.Lee et al.[32]also prepared ST/BT/CT superlattices on electrically conducting SrRuO 3substrates using PLD.It was found that coherent super-lattices with large remanent polarization can only be made if the thickness of the BT layer is less than the combined thickness of the ST and CT layers,otherwise partial strain relaxation occurs.The study veri?ed that mis?t strain alone is not enough to explain the observed/predicted enhanced polarization,but two other effects,namely interfaces and inverse symmetry breaking,must also contribute.Nakh-manson et al.[34]performed density functional calculations on ST/BT/CT superlattices and obtained increased ferroelectric polarization in the trilayer super-lattice if the BaTiO 3concentration is larger than 30%.Highly polarized ST and CT layers are found in such su-perlattices.Maximum polarization can be obtained by increasing the concentration of CT or BT layers while

preserving the overall strain.Interestingly it is found that trilayer superlattices have two values of polarization depending on the direction of the ferroelectric displace-ment.Nakhmanson et al.[35]also used DFT calculations to construct a general model that optimizes the combina-tion of CT,BT and ST layers and predicts structures with maximum polarization and minimum lattice mismatch.In this article a ?rst principles study of the properties of another short-period trilayer superlattice,(SrTiO 3)x /(BaTiO 3)y /(PbTiO 3)z ,where x ,y ,z =1,2,3is presented.The possibility of improving the ferroelectric properties of the SL is investigated by varying the layer thicknesses.Some bilayer superlattices are also studied for comparison purposes.The results are compared with other computa-tional and experimental work where possible.Details of the computational method are given in the next section followed by the results,discussion and conclusions.

Computational method

The ?rst principles calculations are performed using a density functional,plane-wave pseudopotential approach [36]which is implemented in the CASTEP program [37].The method is used to optimize the geometries of bulk,bilayer and trilayer superlattices involving SrTiO 3,BaTiO 3and PbTiO 3.Optimized norm-conserving pseudopotentials [38]are used for the electron-ion interactions with the following valence states treated explicitly:Ba(5s,5p,6s),Ti(3s,3p,3d,4s),O(2s,2p),Sr(4s,4p,5s)and Pb(5d,6s,6p).The potentials for Ba,Ti and O were constructed using the Opium code [39]and used the designed nonlocal approach of Ramer and Rappe [40–43].A plane-wave kinetic energy cutoff of 700eV is employed which is suf?cient to achieve convergence to less than 0.02eV/atom.The local density approximation (LDA)with the Perdew-Zunger parameter-ization [44]for the exchange-correlation functional is chosen for all the calculations.Monkhorst-Pack k -point meshes with a density of (69696)points are used for bulk and strained crystals of ST,BT and PT.For the bilayers and trilayers,following Nakhmanson et al.[34],meshes with (6969N )points are used where N =6/(x +y +z )for x +y +z B 3,N =2for x +y +z =4and N =1for x +y +z [4.These descriptions of reci-procal space were found suf?cient for convergence purposes in previous calculations [34,45]and in the present study as well.The initial geometries are optimized by the BFGS minimizer [46].The convergence thresholds between geometry optimization cycles for energy change,maximum force,maximum stress and maximum displace-ment are set as 5910-6eV/atom,0.01eV/A

?,0.02GPa and 5910-4A

?,respectively.The optimization terminates when all these criteria are satis?ed.

The trilayers in the computational supercell are con-structed sequentially starting at the lower end with SrTiO 3(cubic),then adding BaTiO 3(tetragonal)and ?nally PbTiO 3(tetragonal)in various proportions.Figure 1illus-trates the equi-component case where x =y =z =1.The supercell is oriented such that the superlattice extends along the [001]direction.For all the bulk,strained,bilayer and trilayer systems studied no particular symmetry is imposed during the calculations.This differs from many previous calculations (e.g.,[34])which impose a restricted symmetry condition that allows relaxation only along [001].In the strained,bilayer and trilayer systems,how-ever,the (001)in-plane lattice parameter is constrained to the optimized value of fully relaxed bulk SrTiO 3so that this material then acts as a substrate.

Results and discussion

As a ?rst step,the structures of all three constituents of the ST/BT/PT superlattice in their bulk and strained phases are optimized.The results are presented in Tables 1–3.It is seen that the optimized geometries are in good agreement with previous DFT calculations and experimental results.

Then for each system,linear response calculations are performed to obtain the Born effective charges of each species in the respective phase.The Born effective charge is a tensor quantity de?ned as

Z ?k ;ab ?X o

o P b

o u k a j E ?0

e1T

where P b is the total polarization per unit cell generated in b direction,u k a is a rigid displacement of the sublattice of atom k in the a direction and X o is the unit cell volume [55].Tables 4–6show the diagonal elements of the calculated Born effective charges with comparisons to values obtained by other DFT calculations.Once again it is seen that the values are in good agreement with previous work.The Born effective charges are used to calculate the spontaneous polarization of the bulk,strained,bilayer and trilayer systems.These charges are sensitive to the atomic displacements,but their values do not change very much for small strains.In the present calculations the use of Born effective charges determined from bulk crystals will not change the basic results derived for the superlattices.To calculate the local polarization in a unit cell the following expression is used [34]:

P k ?

X i

o P

o u k i

eu k i à

u eo Tk i T

?1X k X i Z ?k i

D u k i

e2T

where X k is the volume of a unit cell k ,Z *k i is the Born effective charge of ion i in this cell and D u k i is the displacement of ion i in this cell.The superscript zero refers to the reference structure which is nonpolar and centrosymmetric.The same general expression can be used to calculate the total polarization of a superlattice containing many unit cells by summing over all atoms in the SL structure.Previous studies [34]have shown that the total polarization obtained this way agrees well with equivalent values obtained using the Berry phase approach [57].

Geometrical structure and lattice distortions

One convenient measure of the lattice distortion or strain in a relaxed superlattice is the axial ratio c/a in the component layers and how it changes relative to a reference value that is determined from a bulk crystal or another superlattice.Another way of describing the lattice distortions is to determine the displacements of the ions in the relaxed superlattice with respect to an equivalent centrosymmetric structure,the so-called off-centre displacements.To allow for comparison with previous computational studies both approaches are taken

here.

Fig.1The atomic structure of the SrTiO 3/BaTiO 3/PbTiO 3(ST 1BT 1PT 1)superlattice:(a )unrelaxed and (b )relaxed.The unlabelled black circles are oxygen ions.The circles are not in proportion to the ionic radii.The vertical direction is [001]

To begin with,it is of interest to determine how dis-torted the bulk unit cells of BT and PT become when their in-plane a lattice parameters are constrained to the corre-sponding bulk value of ST.This constraint is performed to simulate the mechanical effect of the rigid ST substrate. The results show that the BT c/a increases by5.05%while the corresponding ratio for PT decreases by5.24%.These values are comparable to results obtained from experi-mental[49,52]and other theoretical investigations[50,51, 53,54].

Consider now the three symmetric bilayer superlattices ST x BT x(x=1,2)and ST1PT1.The c/a ratios for the complete superlattices and for the individual components (averaged when x[1)are given in Table7.For the ST x BT x superlattices,the average c/a in the ST layers increases while that of the BT layers decreases compared to that of the calculated bulk-constrained ST and BT crystals, respectively.For the ST1PT1superlattice,the average c/a increases by a small amount in the ST layer while in the PT layer it decreases when compared to the calculated bulk-constrained ST and PT crystals.

The structures of the bilayer SLs can be compared to the trilayer SLs to determine the effect of inserting another component.For example,it is found that adding a PT layer on top of the ST1BT1superlattice to create a trilayer ST1BT1PT1superlattice increases the c/a ratio in the ST layer by about2.60%and decreases the corresponding ratio in the BT layer by1.5%.If the thickness of the PT layer is

Table1The optimized structural parameters of SrTiO3 obtained in the present study compared to experimental work and two other computational investigations Parameters Experimental[47]Present study Reference[48]Reference[2]

a(A?) 3.905 3.856 3.85 3.874

b(A?) 3.905 3.856 3.85 3.874

c(A?) 3.905 3.856 3.85 3.874

c/a1111

V(A?3)59.54757.33957.06758.141

Sr(z-component)0000

Ti(z-component)0.50.50.50.5

O1=O2(z-component)0.50.50.50.5

O3(z-component)0000

Table2The optimized structural parameters of BaTiO3 obtained in the present study compared to experimental work and two other computational investigations

a Calculations performed at

?xed experimental volume Parameters Experimental[49]Present study Reference[50]a Reference[51]a

a(A?) 3.995 3.939 3.994 3.994

b(A?) 3.995 3.939 3.994 3.994

c(A?) 4.034 3.993 4.036 4.036

c/a 1.010 1.0140.9900.990

V(A?3)64.35961.94164.38264.382

Ba(z component)0000

Ti(z component)0.5140.5150.5140.522

O1=O2(z component)0.4880.4860.4810.490

O3(z component)-0.025-0.022-0.031-0.023

Table3The optimized structural parameters of PbTiO3 obtained in the present study compared to experimental work and two other computational investigations

a Optimized c/a at constant volume,LDA,all-electron basis set

b Constant volume and c/a, LDA,ultrasoft pseudopotentials with plane-wave basis Parameters Experimental[52]Present study Reference[53]a Reference[54]b

a(A?) 3.904 3.793 3.834 3.904

b(A?) 3.904 3.793 3.834 3.904

c(A?) 4.152 4.231 4.302 4.151

c/a 1.064 1.116 1.122 1.063

V(A?3)63.28260.88263.23863.266

Pb(z component)0-0.031400

Ti(z component)0.5400.5270.5420.549

O1=O2(z component)0.6120.6330.6340.630

O3(z component)0.1120.1150.1340.125

increased in ST1BT1PT z superlattices it further increases the c/a ratio in the ST layers and also induces an increase in c/a ratio in the BT layers.Similarly,comparing the ST1PT1 superlattice with the ST1BT1PT1superlattice it is found that introducing a single BT layer in between the ST and PT layers also increases the c/a ratio in the ST layer,this time by4.23%,while it slightly decreases c/a in the PT layer by0.42%.On the other hand,if the number of BT layers is increased to2or3,the c/a ratio in the ST and PT layers increases compared to ST1BT1PT1.However, ST1BT2PT1is an exception since the c/a ratio in the ST layer decreases slightly in this case.

The lattice distortion of different trilayer SLs can be compared when the relative composition of the compo-nents is changed.In trilayer ST x BT1PT1superlattices,the addition of ST layers is seen to decrease the average c/a ratio of the ST layers.However,the average c/a ratio of the BT layers in ST1BT y PT1superlattices is found to increase as the thickness of the BT layers is increased.Similarly,in ST1BT1PT z superlattices an increase in the number of PT layers increases the average c/a ratio in the PT layers. Furthermore an increase in the number of BT and PT layers in ST1BT y PT z superlattices also causes an increase in the average c/a ratio of the BT and PT layers when compared to ST1BT1PT1.

By increasing the thickness of the ST layer in ST x BT1PT1superlattices a slight increase in the c/a ratio of the BT layer is observed whereas the ratio in the PT layer remains constant.On the other hand,an increase in the BT layer thickness in ST1BT y PT1superlattices is seen to increase the c/a ratio in the ST layer and slightly in the PT layer as well.Similarly,an increase of the PT layer thickness in ST1BT1PT z superlattices causes a large increase in the c/a ratio in the ST layer with increases in the BT layer also.These results show that the lattice distortion in ST x BT y PT z superlattices can be manipulated by chang-ing the layer thicknesses and the goal is to do so in such as way as to enhance its functional properties.

Figure2a–c illustrates the off-centre displacements of the A cations(Sr,Ba and Pb)for the ST x BT1PT1, ST1BT1PT z and ST1BT y PT1superlattices with respect to the amount of ST,PT and BT present.When there is more than one unit cell of a component in the SL then the dis-placement plotted is an average value for the cation concerned.It is seen that as the thickness of the PT layer increases in the ST1BT1PT z system,the average ferro-electric displacement of the Pb atoms from their centrosymmetric positions also increases quite signi?-cantly.It also increases the off-centre displacement of the Sr atom whereas it does not appreciably move the Ba atom

Table4The Born effective charge tensor for tetragonal SrTiO3 calculated in the present study and compared to two other computa-tional investigations

Elements Present study Reference[55]Reference[56]

Z Sr* 2.56 2.56 2.55

Z Ti*7.407.267.56

Z?O

-2.08-2.15-2.12

-5.81-5.73-5.92

Table5The Born effective charge tensor for tetragonal BaTiO3 calculated in the present study and compared to another computa-tional investigation

Elements Present study Reference[55]

Z Ba*[2.742.742.80][2.722.722.83]

Z Ti*[7.197.196.17][6.946.945.81]

[-5.72-2.15-2.01][-5.53-2.14-1.95]

[-2.15-5.72-2.01][-2.14-5.53-1.95]

[-2.06-2.06-4.95][-1.99-1.99-4.73] The diagonal elements Z xx*,Z yy*and Z zz*of the tensor are presented here as[Z xx*Z yy*Z zz*]

Table6The Born effective charge tensor for tetragonal PbTiO3 calculated in the present study and compared to another computa-tional investigation

Elements Present Study Reference[53]

Z Pb*[3.703.703.08][3.743.743.52]

Z Ti*[5.745.745.21][6.206.205.18]

[-4.80-2.69-2.01][-5.18-2.16-2.16]

[-2.69-4.80-2.01][-2.61-5.18-2.16]

[-1.94-1.94-4.26][-2.15-2.15-4.38] The diagonal elements Z xx*,Z yy*and Z zz*of the tensor are presented here as[Z xx*Z yy*Z zz*]Table7Calculated c/a ratios for complete superlattices ST x BT y PT z and for individual components ST,BT and PT averaged over the number of layers in the component

Superlattice c/a\c/a[ST\c/a[BT\c/a[PT

ST1BT1 2.056(2.07E,2.042T) 1.023 1.033

ST2BT2 4.121(4.088T) 1.014 1.046

ST1PT1 2.037 1.008 1.030 ST1BT1PT1 3.093 1.050 1.018 1.025 ST2BT1PT1 4.086 1.027 1.015 1.018 ST3BT1PT1 5.093 1.020 1.016 1.018 ST1BT2PT1 4.156 1.050 1.041 1.026 ST1BT3PT1 5.227 1.053 1.049 1.026 ST1BT1PT2 4.138 1.063 1.019 1.028 ST1BT1PT3 5.194 1.076 1.025 1.031 ST1BT2PT2 5.203 1.060 1.045 1.026

Values in parentheses are listed from ref.[34],E=experimental. T=theoretical

from its initial position.Similarly as the BT layer thickness is increased in the ST 1BT y PT 1system,the Pb and Ba atoms gradually increase their off-centre displacement.The Sr atom also moves away from its centrosymmetric position in a similar fashion.On the other hand,increasing the ST layer thickness does not systematically increase the dis-placements of the Sr,Ba or Pb cations in the ST x BT 1PT https://www.wendangku.net/doc/6817654340.html,rge ferroelectric displacements of cations in thick PT and BT superlattices make their local and,con-sequently total,polarization appreciably large.It is also interesting to note the displacement pattern of different ions.In bulk and constrained BT,the ferroelectric dis-placements of the A (Ba),B (Ti)and O ions are similar,that is cations and anions move in opposite directions.Similarly in bulk and constrained PT,Pb ions move in the opposite direction to Ti and O ions.In all the superlattices all the cations move in the same direction as the thickness of a particular layer is increased.

Local and total polarization

The displacement of atoms in ferroelectric perovskites and the distortion of their unit cells induce polarization.Table 8lists the calculated total polarization in the various bulk crystals and ST/BT/PT superlattices studied in this work.Equation 2is used to compute the polarization using the Born effective charges determined from linear response calculations.Also listed is a commonly used measure of the change in polarization,the polarization enhancement factor [3,34]which is calculated relative to the polariza-tion of bulk tetragonal BT.Considering ?rst the bulk structures,it is seen that when constrained to the ST in-plane lattice parameter,bulk BT has a signi?cantly greater polarization compared to its unstrained state (59.6%).However,the polarization in constrained PT decreases by 15.1%compared to its unstrained state.This shows that the ST constraint favours polarization enhancement in BT

but

Fig.2The off-centre displacements of the A cations (Sr,Ba and Pb)in the superlattice with respect to the amount of majority component (a )%ST in ST x BT 1PT 1;(b )%PT in ST 1BT 1PT z ;and (c )%BT in ST 1BT y PT 1

Table 8The total polarization and enhancement factor for bulk and constrained crystals of ST,BT and PT and also ST x BT y PT z superlattices determined using Eq.2

Values in parentheses are listed from other studies,E [ref]=experimental.T [ref]=theoretical

System Total polarization |P|(C/m 2)|P|/|P|Bulk BT

ST 0

0BT 0.265(0.27E [58],0.243T [34],0.30T [59])1PT

1.094(0.75E [59],0.88T [53]) 4.128ST(constrained)0

0BT (constrained)0.423(0.368T [34]) 1.596PT(constrained)0.928(0.73T [10])

3.502

ST 1BT 10.305(0.059E [34],0.231T [34]) 1.151(0.54E [34],0.95T [34])ST 2BT 20.335(0.306T [34]) 1.264(1.26T [34])ST 1PT 10.527 1.989ST 1BT 1PT 10.398 1.502ST 2BT 1PT 10.357 1.347ST 3BT 1PT 10.369 1.392ST 1BT 2PT 10.418 1.577ST 1BT 3PT 1

0.421 1.589ST 1BT 1PT 20.475 1.792ST 1BT 1PT 30.632 2.385ST 1BT 2PT 2

0.503

1.898

Fig.3The calculated total polarization of the superlattice as a function of the number of component layers (a )ST x BT 1PT 1x =1,2,3;(b )ST 1BT 1PT z z =1,2,3;and (c )ST 1BT y PT 1y =1,2,3

not PT.The computed values of the bulk polarization,both strained and unstrained,are encouragingly close to exper-imental data and previous calculations.

The total polarization in the ST x BT x bilayer superlat-tices is enhanced compared to bulk BT and increases as x increases.On the other hand,in the ST1PT1superlattice the total polarization reduces when compared with that of bulk PT.Again the computed values are similar to previous theoretical work but larger than one experimental measurement for ST1BT1.

For the trilayer ST x BT y PT z superlattices,the total polarization is always seen to increase relative to bulk BT irrespective of the layer thicknesses.This is evident from the enhancement factor.The increase in polarization is small in ST x BT1PT1superlattices and even decreases slightly as x increases(see Fig.3a).However,for ST1BT1PT z superlattices the enhancement is strong and increases with increasing PT layer thickness as shown in Fig.3b.The addition of PT layers in the superlattice clearly induces ferroelectric displacements in the ST and BT layers causing a large overall polarization enhancement compared to bulk BT.The structure of the ST1BT1PT3 superlattice which has the largest enhancement factor is shown in Fig.4.By increasing the number of BT layers in ST1BT y PT1superlattices the total polarization also increases but does so at a much slower rate than ST1BT1PT z superlattices(see Fig.3c).

In all the superlattices the ST layers are highly polarized.With the application of periodic boundary conditions the PT layers introduce large ferroelectric displacements in the ST layers making them highly polar. In addition the local polarization in each component of the trilayer superlattices is found to be almost uniform across each superlattice as shown in Fig.5a–h.This is similar to that found previously for the ST/BT/CT superlattice system[34]and has been attributed to the minimization of electrostatic energy across the interfaces [3,15,34].The lattice distortion,as measured through the local c/a ratio in each component of each superlattice, is also found to vary uniformly across the superlattices, which is consistent with the uniformity in local polarization.

Conclusions

To summarize the present study,a series of?rst principles calculations have been performed on a new nanoscale tri-layer ST x BT y PT z superlattice to investigate the effect of different stacking thicknesses and periods on its structure and total polarization.It is found that the PT layers play a key role in strain and polarization enhancement.By increasing the amount of PT in ST1BT y PT z superlattices the strain in the other layers i.e.,ST and BT,increases signi?cantly resulting in an enhanced total polarization of the superlattice relative to bulk BT.Similarly increasing the number of BT layers also improves the overall polari-zation of the trilayer superlattices.An increment in the number of ST layers,although increasing the total polari-zation,is not as signi?cant an effect as increasing the Fig.5The calculated local polarization and c/a ratio in individual layers across the superlattice(a)ST1BT1PT1;(b)ST2BT1PT1;(c) ST3BT1PT1;(d)ST1BT2PT1;(e)ST1BT3PT1;(f)ST1BT1PT2;(g) ST1BT1PT3;and(h)ST1BT2PT2

c Fig.4The atomic structure of the ST1BT1PT3superlattice:(a)

unrelaxed and(b)relaxed.The unlabelled black circles are oxygen

ions.The circles are not in proportion to the ionic radii.The vertical

direction is[001]

amount of BT and particularly PT in the trilayer superlat-tices.All the ST layers in each superlattice are found to be highly polarized.In addition to that the results show approximately uniform local polarization and distortion in all layers of each superlattice studied.Many of these cal-culated features are similar to those found previously for the ST x BT y CT z superlattice[34],although for ST x BT y PT z we?nd signi?cantly greater enhancement factors and absolute magnitudes of the local and total polarization. This predicted enhancement of the polarization is mostly attributed to lattice strain due to mismatch of the in-plane lattice constant of the three constituent perovskite materi-als.Finally,we note that the enhancement factor has been calculated with respect to bulk BT which is usually chosen as the standard reference material.A more stringent com-parison would be to use bulk PT since it has a larger spontaneous polarization.The results show that although the ST x BT y PT z superlattices studied so far do not exhibit enhancement with respect to bulk PT there is a clear trend towards increasing polarization as the number of PT and BT layers are increased and therefore the system merits further investigation,both computational and experimental, to determine the effect of longer periods and different compositional sequences.

Acknowledgements Support for this work was provided by the Higher Education Commission of Pakistan,the US Of?ce of Naval Research under grant N00014-00-1-0372and by the US Department of Energy,Of?ce of Basic Energy Sciences under grant DE-FG02-07ER15920.The calculations were performed using the high-per-formance computing facility at the University of Cambridge. References

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如何写先进个人事迹

如何写先进个人事迹篇一：如何写先进事迹材料如何写先进事迹材料一般有两种情况：一是先进个人，如先进工作者、优秀党员、劳动模范等；一是先进集体或先进单位，如先进党支部、先进车间或科室，抗洪抢险先进集体等。无论是先进个人还是先进集体，他们的先进事迹，内容各不相同，因此要整理材料，不可能固定一个模式。一般来说，可大体从以下方面进行整理。 (1)要拟定恰当的标题。先进事迹材料的标题，有两部分内容必不可少，一是要写明先进个人姓名和先进集体的名称，使人一眼便看出是哪个人或哪个集体、哪个单位的先进事迹。二是要概括标明先进事迹的主要内容或材料的用途。例如《王鬃同志端正党风的先进事迹》、《关于评选张鬃同志为全国新长征突击手的材料》、《关于评选鬃处党支部为省直机关先进党支部的材料》等。 (2)正文。正文的开头，要写明先进个人的简要情况，包括：姓名、性别、年龄、工作单位、职务、是否党团员等。此外，还要写明有关单位准备授予他(她)什么荣誉称号，或给予哪种形式的奖励。对先进集体、先进单位，要根据其先进事迹的主要内容，寥寥数语即应写明，不须用更多的文字。然后，要写先进人物或先进集体的主要事迹。这部分内容是全篇材料

的主体，要下功夫写好，关键是要写得既具体，又不繁琐；既概括，又不抽象；既生动形象，又很实在。总之，就是要写得很有说服力，让人一看便可得出够得上先进的结论。比如，写一位端正党风先进人物的事迹材料，就应当着重写这位同志在发扬党的优良传统和作风方面都有哪些突出的先进事迹，在同不正之风作斗争中有哪些突出的表现。又如，写一位搞改革的先进人物的事迹材料，就应当着力写这位同志是从哪些方面进行改革的，已经取得了哪些突出的成果，特别是改革前后的．经济效益或社会效益都有了哪些明显的变化。在写这些先进事迹时，无论是先进个人还是先进集体的，都应选取那些具有代表性的具体事实来说明。必要时还可运用一些数字，以增强先进事迹材料的说服力。为了使先进事迹的内容眉目清晰、更加条理化，在文字表述上还可分成若干自然段来写，特别是对那些涉及较多方面的先进事迹材料，采取这种写法尤为必要。如果将各方面内容材料都混在一起，是不易写明的。在分段写时，最好在每段之前根据内容标出小标题，或以明确的观点加以概括，使标题或观点与内容浑然一体。最后，是先进事迹材料的署名。一般说，整理先进个人和先进集体的材料，都是以本级组织或上级组织的名义；是代表组织意见的。因此，材料整理完后，应经有关领导同志审定，以相应一级组织正式署名上报。这类材料不宜以个人名义署名。写作典型经验材料-般包括以下几部分： (1)标题。有多种写法，通常是把典型经验高度集中地概括出来，一

first集和follow集生成算法模拟

课程设计(论文)任务书软件学院学院软件测试专业 1 班一、课程设计(论文)题目 first集和follow集生成算法模拟二、课程设计(论文)工作自2015 年 6 月16 日起至2013 年6 月 19 日止。三、课程设计(论文) 地点: 软件学院实训中心四、课程设计(论文)内容要求： 1．本课程设计的目的进一步培养学生编译器设计的思想，加深对编译原理和应用程序的理解，针对编译过程的重点和难点内容进行编程，独立完成有一定工作量的程序设计任务，同时，强调好的程序设计风格，并综合使用程序设计语言、数据结构和编译原理的知识, 熟悉使用开发工具VC /JA V A/C#/.NET 。 2．课程设计的任务及要求 1）课程设计任务：设计一个由正规文法生成First集和Follow集并进行简化的算法动态模拟。 2）创新要求：动态模拟算法的基本功能是：（１）输入一个文法Ｇ（２）输出由文法G构造的FIRST集算法（３）输出FIRST算法（４）输出由文法G构造的FOLLOW集算法（５）输出FOLLOW集 3）课程设计论文编写要求（1）课程设计任务及要求（2）设计思路--工作原理、功能规划（3）详细设计---数据分析、算法思路、功能实现（含程序流程图、主要代码及注释）、界面等。（4）运行调试与分析讨论---给出运行屏幕截图，分析运行结果，有何改进想法等。

（5）设计体会与小结---设计遇到的问题及解决办法，通过设计学到了哪些新知识，巩固了哪些知识，有哪些提高。（6）报告按规定排版打印，要求装订平整，否则要求返工；（7）课设报告的装订顺序如下：封面---任务书---中文摘要---目录----正文---附录(代码及相关图片) （8）严禁抄袭，如有发现，按不及格处理。 4）课程设计评分标准：（1）学习态度：20分；（2）系统设计：20分；（3）编程调试：20分；（4）回答问题：20分；（5）论文撰写：20分。 5）参考文献：（1）张素琴，吕映芝. 编译原理[M]., 清华大学出版社（2）蒋立源、康慕宁等，编译原理（第2版）[M]，西安：西北工业大学出版社 6）课程设计进度安排 1．准备阶段（4学时）：选择设计题目、了解设计目的要求、查阅相关资料 2．程序模块设计分析阶段（4学时）：程序总体设计、详细设计 3．代码编写调试阶段（8学时）：程序模块代码编写、调试、测试 4．撰写论文阶段（4学时）：总结课程设计任务和设计内容，撰写课程设计论文学生签名： 2015 年 6 月19 日课程设计(论文)评审意见（1）学习态度（20分）：优（）、良（）、中（）、一般（）、差（）；（2）系统设计（20分）：优（）、良（）、中（）、一般（）、差（）；（3）编程调试（20分）：优（）、良（）、中（）、一般（）、差（）；（4）回答问题（20分）：优（）、良（）、中（）、一般（）、差（）；（5）论文撰写（20分）：优（）、良（）、中（）、一般（）、差（）；评阅人：职称：讲师 2015 年 6 月19 日

《女人的资本》读后感,-女性必读的三十部经典-最新范文

《女人的资本》读后感,:女性必读的三十部经典篇一：女人一生必读的本书女人一生必读的本书世界有十分美丽,但如果没有女人,将失掉七分色彩;女人有十分美丽,但如果远离书籍,将失掉七分内蕴。有人说,书是女人最好的化妆品。做时尚的女性,不需要掌握男人,只需要善待自己。读书的女人是美丽的。正如著名女作家毕淑敏所说：“日子一天一天地走,书要一页一页地读。清风朗月水滴石穿,一年几年一辈子地读下去。书就像微波,从内到外震荡着我们的心,徐徐地加热,精神分子的结构就改变了、成熟了,书的效力就凸现出来了。”女人不可不读书,世界因女人的存在而美丽,女人因书籍的滋养而变得更加聪慧. 1《第二性》被《时代》周刊评为20世纪改变人类思想和生活的10本书之一当代西方女权主义运动的“圣经” 迄今为止对女性问题研究得最为透彻的一本书 2《写给女人》畅销全球的女性缔造成熟之爱、获取人生幸福的经典之作当代女性的必读文本为广大女性朋友们提供了切实可行的人生指导和精神启迪 3《红楼梦》中国古今第一奇书中国古典小说的巅峰之作

一本不读就是人生极大遗憾的书 4《谁是最美的女人》一本女人塑造个性形象的建议书打开这本书你会发现原来美丽并不远做一个标准女孩的必读书 5《简·爱》一个平凡女人不平凡的生活经历一段曲折离奇而又缠绵动人的爱情故事一部历久不衰的经典名著 6《居里夫人传》一部内容详实的科学家传记一位影响过世界进程的伟大女性不平凡的一生一垃为子女树立光辉榜样的伟大母亲的精神世界 7《圣经》一部包罗万象的古代文化百科全书打开西方精神世界的钥匙要了解西方文学首先应该了解《圣经》 8《世界美术名作二十讲》艺术史著作中一部难得的佳作有利于女性读者增强美术修养的艺术杰作一本为广大女性读者指点艺术迷津的经典之作 9《金锁记》

【8A版】编译原理实验报告FIRST集和FOLLOW集

编译原理实验报告实验名称计算first集合和follow集合实验时间院系计算机科学与技术班级软件工程1班学号姓名

输入：任意的上下文无关文法。输出：所输入的上下文无关文法一切非终结符的first 集合和follow 集合。 2. 实验原理设文法G[S]＝（V N ，V T ，P ，S ），则首字符集为： FIRST （α）＝{a|α?* a β，a ∈V T ，α,β∈V G }。若α?* ε，ε∈FIRST （α）。由定义可以看出，FIRST （α）是指符号串α能够推导出的所有符号串中处于串首的终结符号组成的集合。所以FIRST 集也称为首符号集。设α＝G 1G 2…G n ，FIRST （α）可按下列方法求得：令FIRST （α）＝Φ，i ＝1；（1）若G i ∈V T ，则G i ∈FIRST （α）；（2）若G i ∈V N ； ①若ε?FIRST （G i ），则FIRST （G i ）∈FIRST （α）； ②若ε∈FIRST （G i ），则FIRST （G i ）－{ε}∈FIRST （α）；（3） i ＝i+1，重复（1）、（2），直到G i ∈V T ，（i ＝2，3，…，n ）或G i ∈V N 且若ε?FIRST （G i ）或i>n 为止。当一个文法中存在ε产生式时，例如，存在A →ε，只有知道哪些符号可以合法地出现在非终结符A 之后，才能知道是否选择A →ε产生式。这些合法地出现在非终结符A 之后的符号组成的集合被称为FOLLOW 集合。下面我们给出文法的FOLLOW 集的定义。设文法G[S]＝（V N ，V T ，P ，S ），则 FOLLOW （A ）＝{a|S ?…Aa …，a ∈V T }。若S ?* …A ，#∈FOLLOW （A ）。由定义可以看出，FOLLOW （A ）是指在文法G[S]的所有句型中，紧跟在非终结符A 后的终结符号的集合。 FOLLOW 集可按下列方法求得：（1）对于文法G[S]的开始符号S ，有#∈FOLLOW （S ）；（2）若文法G[S]中有形如B →GAy 的规则，其中G ，y ∈V G ，则FIRST （y ）－{ε}∈FOLLOW （A ）；（3）若文法G[S]中有形如B →GA 的规则，或形如B →GAy 的规则且ε ∈FIRST （y ），其中G ，y ∈V G ，则FOLLOW （B ）∈FOLLOW （A ）；

编译原理一点就通first follow LL()

1 编译原理 2013年11月28日 LL 的含义－自左向右扫描分析输入符号串－从识别符号开始生成句子的最左推导 LL(1):向前看一个输入符号，便能唯一确定当前应选择的规则LL(k):向前看k 个输入符号，才能唯一确定当前应选择的规则 4.2.3 LL(1)文法的判别要构造确定的自顶向下分析程序要求描述文法必须是LL(1)文法 2 编译原理 2013年11月28日同一非终结符有多个候选式时引起回溯的原因【例4.1】α=acb G[S]:S →aAb A →cd|c （1）候选式的终结首符号相同（2）候选式的终结首符号相同【例4.8】S →Aa A →a|

3 编译原理 2013年11月28日 1. FIRST 集 FIRST(α)：从α可能推导出的所有开头终结符号或ε对于文法G 的所有非终结符的每个候选式α，其终结首符号集称为FIRST 集，定义如下： ε，则规定ε∈FIRST(α) 若α 【例】S →aAb A →cd|c a …，a ∈V T FIRST(α)={a|α FIRST(aAb )={a}FIRST(cd )={c}FIRST(c )={c} 【例】S →Aa A →a|ε FIRST(a )={a}FIRST(ε)= {ε}FIRST(Aa)={a} FIRST(S )={a}FIRST(A )={c} FIRST(S )={a}FIRST(A )={a, ε} 4 编译原理 2013年11月28日（1）若α=a α′，且a ∈V T ，则a ∈FIRST(α)；例：FIRST(i)={i} FIRST(+TE')={+} E →TE'E'→+TE'|ε T →FT'T'→*FT'|ε F →(E)|i 构造FIRST 集的算法（2）若α=X α′，X ∈V N ，且有产生式X →b …，则把b 加入到FIRST(α)中；例：FIRST(FT')={(,i} ??

编译原理实验报告记录FIRST集和FOLLOW集

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编译原理实验报告实验名称计算first集合和follow集合实验时间院系计算机科学与技术班级软件工程1班学号姓名

输入：任意的上下文无关文法。输出：所输入的上下文无关文法一切非终结符的first 集合和follow 集合。 2. 实验原理设文法G[S]＝（V N ，V T ，P ，S ），则首字符集为： FIRST （α）＝{a | α?* a β，a ∈V T ，α,β∈V *}。若α?* ε，ε∈FIRST （α）。由定义可以看出，FIRST （α）是指符号串α能够推导出的所有符号串中处于串首的终结符号组成的集合。所以FIRST 集也称为首符号集。设α＝x 1x 2…x n ，FIRST （α）可按下列方法求得：令FIRST （α）＝Φ，i ＝1；（1）若x i ∈V T ，则x i ∈FIRST （α）；（2）若x i ∈V N ； ① 若ε?FIRST （x i ），则FIRST （x i ）∈FIRST （α）； ② 若ε∈FIRST （x i ），则FIRST （x i ）－{ε}∈FIRST （α）；（3） i ＝i+1，重复（1）、（2），直到x i ∈V T ，（i ＝2，3，…，n ）或x i ∈V N 且若ε?FIRST （x i ）或i>n 为止。当一个文法中存在ε产生式时，例如，存在A →ε，只有知道哪些符号可以合法地出现在非终结符A 之后，才能知道是否选择A →ε产生式。这些合法地出现在非终结符A 之后的符号组成的集合被称为FOLLOW 集合。下面我们给出文法的FOLLOW 集的定义。设文法G[S]＝（V N ，V T ，P ，S ），则 FOLLOW （A ）＝{a | S ?… Aa …，a ∈V T }。若S ?* …A ，#∈FOLLOW （A ）。由定义可以看出，FOLLOW （A ）是指在文法G[S]的所有句型中，紧跟在非终结符A 后的终结符号的集合。 FOLLOW 集可按下列方法求得：（1）对于文法G[S]的开始符号S ，有#∈FOLLOW （S ）；（2）若文法G[S]中有形如B →xAy 的规则，其中x ，y ∈V *，则FIRST （y ）－{ε}∈FOLLOW （A ）；（3）若文法G[S]中有形如B →xA 的规则，或形如B →xAy 的规则且ε ∈FIRST （y ），其中x ，y ∈V *，则FOLLOW （B ）∈FOLLOW （A ）；

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求first集和follow集

编译原理实验实验名称：求first集和follow集姓名：学号：教师签字：成绩：

一．实验目的： .掌握和了解first集和follow集的求解过程。二．实验原理： 1.first集的求解：（1）若X∈Vt,则FIRST(X)={X}; (2)若X∈Vn,且有产生式X->a……,a∈Vt，则a∈FIRST(X); (3)若X∈Vn,X->@,则@∈FIRST(X) (4)若X,Y1,Y2,Y3,Y4…………Yn都∈Vn,而产生式X->Y1,Y2……Yn.当 Y1,Y2,Y3,Y4…………Yn都能=>@那么FIRST(X)=并集的 FIRST(Yi)-{@}(0<=i<=n) (5)若Yi=>@(i=1,2,3……)，则FIRST(X)=并集的FIRST(Yi)-{@}并上{@} 2.follow集的求解：（1）若为文法开始符号S，则FOLLOW(S)={#} (2)若为文法A->aBb是一个产生式，则把FIRST（b）的非空元素加入 FOLLOW(B)中。如果b->@则把FOLLOW(A)也加入FOLLOW(B)中。三．实验代码 #include #include #include #include #include using namespace std; //********************* struct define //产生式 { char left; string right; }; //*************** int N,K1=0,K2=0; char B; struct define *p=new define[10]; //************************ bool find(char b) //查找是否有产生空的产生式 { int i; for(i=0;i

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优秀党务工作者事迹简介范文

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三、生活朴素，乐于助人平时重视与同事间的关系，主动与同事打成一片，善于发现他人的难处，及时妥善地给予帮助。在其它同志遇到困难时，积极主动伸出援助之手，尽自己最大努力帮助有需要的人。养成了批评与自我批评的优良作风，时常反省自己的工作，学习和生活。不但能够真诚的指出同事的缺点，也能够正确的对待他人的批评和意见。面对误解，总是一笑而过，不会因为误解和批评而耿耿于怀，而是诚恳的接受，从而不断的提高自己。在生活上勤俭节朴，不铺张浪费。身为一名老党员，我感到责任重大，应该做出表率，挤出更多的时间来投入到**党总支的工作中，不找借口，不讲条件，不畏困难，将总支建设摆在更重要的位置，解开工作中的思想疙瘩，为攻坚克难铺平道路，以支部为纽带，像战友一样团结，像家庭一样维系，像亲人一样关怀，践行入党誓言。把握机遇，迎接挑战，不负初心。

构造FIRST集和FOLLOW集的方法

构造FIRST集和FOLLOW集的方法 1、构造FIRST集的算法 (1) 对于G中的每个文法符号X,为求FIRST(X),反复应用如下规则,直到集合不再增大: ①若X∈V T，则FIRST(X)是{X} ②若X∈V N ,且X→aα(a∈V T )，则{ a } ? FIRST(X) X→ε，则{ε} ? FIRST(X) ③若X->Y1Y2…Y i-1 Y i…Y K∈P,Y1∈V N ,则 FIRST(Y1)-{ε} ? FIRST(X) 而对所有的j(1≤j ≤i-1), Y j∈V N，且Y j??ε，则令 FIRST(Y j)-{ε} ? FIRST(X) (1≤j ≤i) 特别，当ε∈FIRST(Y j) (1≤j ≤k)时，令ε∈FIRST(X) (2) 对文法G的任何符号串α=X1X2…X n构造集合FIRST(α) ①置FIRST(X1)-{ε} ? FIRST(α) ②若对任何1≤j≤i-1，ε∈FIRST(X j)，则FIRST(X i) -{ε} ? FIRST(α) 特别是，若所有的FIRST(X j)均含有ε，1≤j≤n，则{ε} ? FIRST(α)。显然，若α＝ε则FIRST(α)＝{ε}。 2、构造FOLLOW集的算法对于G中的每一A∈V N，为构造FOLLOW(A)，可反复使用如下的规则，直到每个FOLLOW集不再增大为止: ①对于文法的开始符号S，令# ∈FOLLOW(S)。 ②对于每一A→αBβ∈P，令FIRST(β) - {ε} ? FOLLOW(B) 。 ③对于每一A→αB∈P，或A→αBβ∈P，且ε∈FIRST(β)，则令FOLLOW(A) ? FOLLOW(B) 。

主要事迹简介怎么写(2020年最新)

主要事迹简介怎么写概括?简要地反映?个单位(集体)或个?事迹的材料。简要事迹不?定很短，如果情况多的话，也有?千字的。简要事迹虽然“简要”，但切忌语?空洞，写得像?学?期末鉴定。 ?应当以事实来说话。简要事迹是对某单位或个?情况概括?简要地反映情况，?如有三个??很突出，就写三个??，只是写某???时，要把主要事迹突出出来。简要事迹?般来说，?少要包括两个??的内容。?是基本情况。简要事迹开头，往往要??段?字来表述?些基本情况。如写?个单位的简要事迹，应包括这个单位的?员、承担的任务以及?段时间以来取得的主要成绩。如写个?的简要事迹，应包括该同志的性别、出?年?、参加?作时间、籍贯、民族、?化程度以及何时起任现职和主要成绩。这样上级组织在看了材料的开头，就会对这个单位或个?有?个基本印象。?是主要特点。这是简要事迹的主体部分，最突出的事例有?个??就写成?块，并按照?定的逻辑关系进 ?排列，把同类的事例排在?起，?个??通常由?个?然段或?个?然段组成。写作时，特别要注意以下四点： 1.?第三?称。就是把所要写的对象，是集体的?“他们”来表述，是个?的称之为“他(她)”。 (她)”，单位可直接写名称，个?可写其姓名。为了避免连续出现?个“他们”或“他 2.掌握好时限。?论是单位或个?的简要事迹，都有?个时间跨度，既不要扯得太远，也不要故意混淆时间概念，把过去的事当成现在的事写。这个时间跨度多长，要根据实际情况 ?定。如上级要某个同志担任乡长以来的情况就写他任乡长以来的事迹;上级要该同志两年来的情况，就写两年来的事迹。当然，有时为了需要，也可适当地写?点超过这个时间的背景情况。 3.?点他?的语?。就是在写简要事迹时，可?些群众的语?或有关?员的语?，这样会给??种?动、真切的感觉，衬托出写作对象?较?的思想境界。在?他?语?时，可适当加?，但不能造假。 4.?事实说话。简要事迹的每?个??可分为多个层次，?个层次先??句话作为观点，再???两个突出的事例来说明。?事实说话时，要尽量把?个事例说完整，以给?留下深刻印象。

正规文法的First集合Follow集求解过程动态模拟-实验报告

华东交通大学课程设计（论文）任务书软件学院专业项目管理班级2005－4一、课程设计（论文）题目正规文法的First集合Follow集求解过程动态模拟二、课程设计（论文）工作：自2008年6月23 日起至2008年 6 月27 日止。三、课程设计（论文）的内容要求： 1、基本要求：进一步培养学生编译器设计的思想，加深对编译原理和应用程序的理解，针对编译过程的重点和难点内容进行编程，独立完成有一定工作量的程序设计任务，同时强调好的程序设计风格，并综合使用程序设计语言、数据结构和编译原理的知识，熟悉使用开发工具VC 6.0 或其它软件编程工具。为了使学生从课程设计中尽可能取得比较大的收获，对课程设计题目可根据自己的兴趣选题（须经老师审核），或从老师给定题目中选择完成（具体见编译原理课程设计题目要求）。通过程序实现、总结报告和学习态度综合考评，并结合学生的动手能力，独立分析解决问题的能力和创新精神。成绩分优、良、中、及格和不及格五等。

2、具体要求设计一个由正规文法生成Fisrt集Follow集的动态过程模拟动态模拟算法的基本功能是： ●输入一个正规文法； ●输出由文法构造的First集的算法； ●输出First集； ●输出由文法构造的Follow集的算法； ●输出Follow集；学生签名： 2008 年 6 月 27 日课程设计（论文）评阅意见评阅人职称副教授 2008 年 6 月 27 日

目录一、需求分析 (3) 二、总体设计 (4) 三、详细设计 (9) 四、课设小结 (12) 五、谢辞 (13) 六、参考文献 (14)

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