a r X i v :n u c l -t h /0701008v 1 3 J a n 2007
Quadrupole collective variables in the natural Cartan-Weyl basis
S.De Baerdemacker,K.Heyde and V.Hellemans
Universiteit Gent,Vakgroep Subatomaire en Stralingsfysica,Proeftuinstraat 86,B-9000Gent,Belgium
E-mail:stijn.debaerdemacker@ugent.be
Abstract.The matrix elements of the quadrupole collective variables,emerging from collective nuclear models,are calculated in the natural Cartan-Weyl basis of O (5)which is a subgroup of a covering SU (1,1)×O (5)structure.Making use of an intermediate set method,explicit expressions of the matrix elements are obtained in a pure algebraic way,?xing the γ-rotational structure of collective quadrupole models.
PACS numbers:02.20Qs,21.60Ev
Submitted to:J.Phys.A:Math.Gen.
1.Introduction
Collective modes of motion play a signi?cant role in the low-energy structure of atomic nuclei.To account for large quadrupole moments,the spherical shell-model picture needed to be extended to spheroidal deformations[1],due to collective polarization e?ects induced by single particles moving in the nuclear medium.As a consequence the nucleus can no longer be regarded as a rigid body,rather a soft object with a surface that can undergo oscillations and rotations in the laboratory framework.The quantized treatment of these excitations in the intrinsic framework[2,3]led to the development of the Bohr Hamiltonian in terms of the intrinsic collective variablesβandγ[4,5], corresponding respectively to the degree of axial and triaxial deformation.The dynamics is determined by the potential contained in the Bohr Hamiltonian,which can either be constructed from a microscopic theory or through phenomenological considerations. When the latter strategy is followed,one can either choose an analytically solvable potential,a topic which has recently gained a considerable amount of interest because of its application to critical points in phase shape transitions[6–8],or a more general expression[4]in terms of the collective variables,determining the surface(1).For an overview on(approximative)analytically solvable potentials,we would like to refer the reader to[9].
Analytically solvable potentials are intended as benchmarks in order to study more general and complex potentials.To handle these potentials,one needs to perform a diagonalization,for which a suitable basis is needed.Within the literature, several methods have been proposed and profoundly discussed.Pioneering work has been carried out by B`e s[10],who determined the explicitγ-soft wavefunctions through a coupled di?erential equation method.Unfortunately,this technique becomes tremendously complicated for spin states,higher than L= 6.Therefore,other techniques have been developed,fully exploiting the SU(1,1)×O(5)structure of the ?ve-dimensional harmonic oscillator.Nevertheless,complications arise.Since the Hamiltonian is an angular momentum scalar,the eigenstates automatically possess good quantum numbers L and M of the physical O(3)?O(2)chain which does not evolve naturally from the Cartan-Weyl group reduction.As a consequence one is forced to construct explicit wavefunctions,starting either from basic building blocks[11–13] or from a projective coherent state formalism[14],constituting an orthonormal basis [15,16].Nevertheless,the alternative Cartan-Weyl[17,18]reduction path may also be followed as it leads to a reduction scheme which is more natural in a mathematical sense,though the physical meaning of the quantum numbers is partially lost.This strategy was followed by Hecht[19]to construct fractional parentage coe?cients for spin-2phonons,that were used by the Frankfurt group[4,20]in the development of the General Collective model.
It is noteworthy that new techniques have been proposed within the last decennium. First,the vector coherent state formalism[21–24]and much more recently the algebraic tractable model[25,26]were developed,enabling the construction of the quadrupole
harmonic oscillator representations,exhibiting good O(3)angular momentum quantum numbers.
In the present paper,the path of the natural Cartan-Weyl reduction is taken.It will be shown that the matrix elements of the quadrupole variable can be extracted within this basis,without making use of the explicit representations in terms of the collective variables.In the end,it will turn out that the basic commutation relations of the collective variables su?ce to?x the complete structure of the algebra,and furthermore the dynamics of the Hamiltonian.
2.The collective model
Within the framework of the geometrical model,the nucleus is regarded as a liquid drop with a surface R(θ,φ)described by a multipole expansion using spherical harmonics in the laboratory system
R(θ,φ)=R0 1+ λ,μα?λ,μYλ,μ(θ,φ) ,(1)
which de?nes the set of collective coordinatesαλ,μof multipolarityλand projection μ.Up to quadrupole deformation,the surface(1)is restricted to spheroidal deforma-tions determined by the variablesα2,μwhich will be abbreviated toαμfrom here on. Although the intrinsic surface is unambiguously described by this set of variables,it is convenient to rotate from the laboratory to the intrinsic framework by means of the Euler angles(θi,i=1,2,3).Doing so,the collective variables(β,γ)are introduced as intrinsic parameters of the ellipsoid,rendering a straightforward interpretation of axial and triaxial deformation[4].
αμ=β cosγD2?μ0(θi)+sinγ2(D2?μ2(θi)+D2?μ?2(θi)) .(2) This set of collective variables is su?cient for the determination of the static properties of nuclear shapes.To build in the essential quantum mechanical dynamics,canonic conjugate momentaπμ′need to be incorporated.These must ful?ll the standard commutation relations[4]
[πμ′,αμ]=?i δμμ′,[πμ′,πμ]=0,[αμ′,αμ]=0.(3) Note that the variables have become operators though we silently omit the operator symbol to avoid notational overload.
To establish the SU(1,1)×O(5)group structure,it is convenient to introduce the following recoupling formula
(α·α)(π?·π?)=(α·π?)(α·π?)+3i (α·π?)
?2([απ?](1)·[απ?](1)+[απ?](3)·[απ?](3)),(4) where the complex conjugateπ?is introduced to ensure for good angular momentum transformation properties and T l·S l=(?)l√
The 3operators (α·α),(π?·π?)and (α·π?)generate the algebra of an SU (1,1)group,which forms a direct product together with the O (5)group,built from the 10operators
[απ?](1)M and [απ?](3)
M .Whereas the SU (1,1)group is strongly linked to the excitations in the radial variable β,the O (5)group encompasses the γvibrations coupled to the rotational structure.In this work,we concentrate on the application of the Cartan-Weyl scheme on the O (5)group,leaving a freedom of choice of a suitable SU (1,1)basis.3.The Cartan-Weyl reduction of O (5)
The commutation relations of the operators L M and O M ,de?ned by
[απ?](1)
M =
i
10L M ,[απ?](3)
M
=
i
10O M
,(5)
span the algebra of the O (5)group.[L m ,L m ′]=?
√
3 1m 3m ′|3m +m ′ O m +m ′,(7)[O m ,O m ′]=?2
√
6 3m 3m ′|3m +m ′ O m +m ′.
(8)
The M =0projections {L 0,O 0}form an intuitive choice of the Cartan subalgebra within the set {L m ,O m ′}.This set has the advantage of incorporating the angular momentum projection operator L 0in the physical group reduction chain O (5)?O (3)?O (2).Nevertheless,it is not explicitly contained in the natural Cartan-Weyl reduction chain O (5)?O (4)～=SU (2)×SU (2)which can be realized through the following rotation [11,21]
X +=?12L +1+
√√5(√3O ?1),Y ?=15O ?3
,X 0=1
10
(3L 0?O 0),
T 12
=
1
10O +2,T ?12=?1
50(
√2O +1),T ?1
2
=?110O ?2
,T 1
2
=1
50(
√2O ?1).
(9)
The group reduction is immediately clear,as the sets {X 0,X ±}and {Y 0,Y ±}both span standard SU (2)algebras.Furthermore all generators of the one SU (2)algebra commute with all generators of the other.The commutation relations are given by
[X 0,X ±]=±X ±,[X +,X ?]=2X 0,
[Y 0,Y ±]=±Y ±,[Y +,Y ?]=2Y 0,[X 0,Y 0]=0,
[X ±,Y ±]=[X ±,Y ?]=0.
(10)
So the reduction is O (5)?O (4)～=SU (2)X ×SU (2)Y .The non-O (4)operators T μνcan be identi?ed as the 4components of a bitensor of character {12}within the SU (2)×SU (2)scheme,according to Racah’s de?nition [27].The index μdenotes
the bitensor component relative to the SU(2)X group,whileνis the component with respect to SU(2)Y
[X0,Tμν]=μTμν,
[X±,Tμν]= 2?μ)(1
(1
2±ν+1)Tμν±1.(11)
The internal commutation relations of the T bitensor completes the Cartan-Weyl structure,which can be found in table(1)
?2?12?12121
T?1
22
Y?1
2
(X0+Y0)
T1
22
Y?01
2
X+
T?1
22
X??12Y+
T1
22
(X0+Y0)1
2
Y+0
Table1.Multiplication table for the internal commutation relations of the T bitensor. The multiplication?symbolizes the standard commutation.
Once the commutation relations have been determined within the Cartan-Weyl basis, it is instructive to construct the root diagram.Figure1shows2di?erent realizations of the same root diagram,depending on the choice of the Cartan subalgebra.On the left side(Fig.1a)a standard root diagram with respect to the{X0,Y0}Cartan subalgebra is depicted,while on the right side(Fig.1b),a more physical subalgebra{L0,O0}is chosen
as a reference frame.The latter framework has a visual advantage,since the projection of the generators on the L0-axis is readily established.This enhances the insight in the problem of constructing wavefunctions with good angular momentum from the weight diagrams in the Cartan-Weyl basis(see section(6)).
4.Representations of O(5)
Every subgroup in the group reduction chain provides an associated Casimir operator. The quadratic Casimir operator of O(5)can be constructed from the Killing form[18] C2[O(5)]=1
2
(X+X?+X?X+),(14) Y2=Y20+1
Y 0
X Y 0
Figure 1.The root diagrams of the O (5)algebra in the Cartan-Weyl basis for either the (a)natural {X 0,Y 0}or (b)physical {L 0,O 0}Cartan subalgebra
Starting from the explicit expressions of the generators (see Appendix A)in terms of the collective variables and the canonic conjugate momenta,the following operator identity can be proven
X 2?Y 2≡0,
(16)
which is true in general for symmetric representations [11].The consequence of this identity is that we are left with 4operators that commute among each others,i.e.the quadratic Casimir operator of O (5),the quadratic Casimir operator of SU (2)X and SU (2)Y (X 2≡Y 2)and the Cartan subalgebra {X 0,Y 0}which are the respective linear Casimir operators of the O (2)X and O (2)Y subgroups.As a result,we obtain a representation which is determined by 4independent quantum numbers
|vXM X M Y
(17)with
C 2[O (5)]|vXM X M Y =v (v +3)|vXM X M Y ,(18)
X 2|vXM X M Y =Y 2|vXM X M Y =X (X +1)|vXM X M Y ,
(19)
X 0|vXM X M Y =M X |vXM X M Y ,(20)
Y 0|vXM X M Y =M Y |vXM X M Y .
(21)
Now that the basis to work in is ?xed,we can study the action of the generators as they hop through the representations with ?xed quantum number v .Acting with the O (4)～=SU (2)X ×SU (2)Y generators on |vXM X M Y is trivial because of the well-known angular momentum theory
X ±|vXM X M Y =
(X ?M Y )(X ±M Y +1)|vXM X ,M Y ±1 ,
(24)
Y 0|vXM X M Y =M Y |vXM X M Y .
(25)
The action of T μνon |vX (M X ,M Y ) is less trivial,though the bitensorial character of T can be well exploited.Since T is a {12}bitensor,it can only connect representations that di?er 1
2
,M X +μ,M Y +ν
+b |v,X ?
1
2,M X +μ,M Y +ν|T μν|vXM X M Y
=(?)k
X +12X ?M X ?μμM X X +1
2X ?M Y ?ννM Y
vX +
12
,M X +μ,M Y +ν|T μν|vXM X M Y
=(?)k
X ?1
2X ?M X ?μμM X X ?12X ?M Y ?ννM Y
vX ?12
?1
2
1
2
(X 0+Y 0),and sandwich it with the state |vXM X M Y
vXM X M Y |T ?1
2
T 12
|vXM X M Y ? vXM X M Y |T 12
T ?12
|vXM X M Y
=
1
2
?1
2
1
21
2
?1
2
(M X +M Y ).
(30)
?We formally use the single reduced matrix notation in order to express the double reduced matrix,
as any confusion between normal and double reduced matrix element is excluded within this
work.
Due to symmetry considerations,a large amount of the matrix elements in the summation are identically zero.First of all the SU(2)X×SU(2)Y bitensor character of T dictates strict selection rules with respect to X,M X and M Y.As a result the summation over X′,M′X and M′X is restricted to speci?c values which are completely governed by the Wigner-3j symbol in(27,28).Secondly,the components Tμνof T are O(5)generators,which cannot alter the seniority quantum number v.So,the summation over v′is reduced to one state v′=v.
Once the restriction in the summation is carried out,it is convenient to apply the Wigner-Eckart theorem(27,28)and after some tedious algebra we obtain a relationship for the double reduced matrix elements
vX||T||vX+12||T||vX 2 vX?1
2X =
(2X+1)2
21
2?12?1212?121212?1 2 vX+1
2 ?=? v,X+1
2 |2=1
2 |2=1
2 and vX||T||vX?1
2
1
(X+M X+1)(X+M Y+1)(v?2X)(v+2X+3)
(2X+1)(2X+2)
|vX+12,M Y+1 (X?M X)(X?M Y)(v?2X+1)(v+2X+2)
(2X)(2X+1)
|vX?12,M Y+1
T 1
2
|vXM X M Y ==
2
2
,M X +
1
2 +
2
2
,M X +
1
2
,
(39)
T ?12
|vXM X M Y
= 2
2
,M X ?
1
2 +
2
2
,M X ?1
2
,
(40)
T ?1
2
|vXM X M Y
=
2
2,M X ?1
2 ?
2
2
,M X ?1
2
.
(41)
From these expressions it is clearly seen that no representations can be constructed
with X >v
Figure v =2.Every sphere denotes in planes with distinct X states.
respect to SU (2)X ×SU (2)Y .Calculating the commutation relations of αμwith the SU (2)generators (which is done most conveniently using the explicit expressions given in Appendix A),we can summerize them as
[X 0,αλλμν]=μαλλ
μν,
(44)
[X ±,αλλ
μν]=
(λ?ν)(λ±ν+1)αλλ
μν±1,
(47)
where the 5collective variables have been relabelled as follows
α2=α1212,α1=α12?12,α?1=α12
12,α?2=α1
2?1
2
,(48) α0=α00
00
.
(49)
This clearly states that the 5projections of αcan be divided into the 4components
of a {12}bispinor and a single biscalar,according to Racah [27].We can again de?ne double reduced matrix elements
vXM X M Y |αλλμν
|v ′X ′M ′X M ′Y =(?)k
X λX
′
?M X μM ′
X
X λX
′
?M Y νM ′
Y
vX ||αλ
||v ′X ′
,
(50)
with k =2X ?M X ?M Y .It is noteworthy that,contrary to the matrix elements of the generators T μν,v ′is not necessarily equal to v .To obtain explicit expressions for the double reduced matrix elements,we start from the commutation relations
[T μν,α
1
2μ′ν′]=(?)(μ+ν)2
δ?μμ′δ?νν′α0000
,(51)
[Tμν,α0000]=1
2
α12μν,(52)
[αλλμν,αλ′λ′
μ′ν′
]=0.(53) First a relationship between vX||α0||v′X′ and vX||α1
21
2
1
2
1
2
,M X+1
2|[T12,α
1
2
1
2
]|v′X?1
2
,M Y?1
2
1
2
1
2
1
2
1
2
1
2
1 2||α1
(v′?2X)(v′+2X+3)
2X
? vX||α12 √
2||α1
(v′?2X+1)(v′+2X+2)
2X+2
? vX||α12 √
2||α1
2X?(v+1)(v+2)
2||v′X+1
2X?(v′+1)(v′+2)
2
1
2
1
2
1
2||α1
2
||v′X+1
2||α12||v′X+1
Finally,we repeat the procedure for [T 12
,α1
2?1
2
]=?
12
α0000.Besides (56)and (57),we
obtain the expression vX +
1
2
||v ′X
(v ?2X )(v +2X +3)
X (v ′+1)(v ′+2)?(X +1)(v +1)(v +2)+2(2X +1)2
+
vX ||α
1
2
(v ′?2X )(v ′+2X +3)
X (v +1)(v +2)?(X +1)(v ′+1)(v ′+2)+2(2X +1)2
=?2
√
(2X +1)(2X +2) vX ||α0||v ′X .(60)
Solving the set of equations (59)and (60)(or equivalently (58)and (60))results in
expressions of all possible double reduced matrix elements of α
1
2
||α1
√
2X +2v ?2X 2
||v +1,X +
1
√
2X +2
v +2X +4
2
||α1
√
2X +2v +2X +32
||v ?1,X +1
√
2X +2
v ?2X ?1
2
and λ′=
1
2
||v ′X +1
2
||α1
2X +2
?
v
′ vX ||α1
2
v ′X ?
1
2
||vX
(v ?2X +1)(v +2X +3)
vX ||α0||v +1X v +1X ||α0||vX
=
(2v +1)
connections,though the seniority connection(v to v′=v±1)was?xed.Now,(66) relates matrix elements with di?erent seniority connection,leaving the X quantum number unaltered.
At last,in order to obtain explicit expressions,we return to the geometry of the problem. It has been mentioned earlier that the operatorα·αcommutes with all the generators of O(5),making it an O(5)scalar.Therefore,this operator can be treated as a constant with respect to the O(5)scheme.We call this constantβ2,referring to the radial deformation parameter in(2).As a consequence,we can write
vXM X M Y|α·α|vXM X M Y =β2.(67) The procedure used culminates into closed expressions of the matrix elements.By inserting a complete set of basis states between the variables of(67),we can rewrite this expression in terms of double reduced matrix elements.
1
β2=
2||v′X+12||α1
2||v′X?12||α1
(2X+1)2β2,(69)
(2v+3)(2v+5)
v,X||α0||v?1,X v?1,X||α0||v,X =(v?2X)(v+2X+2)
(2X+1)2β2,(72)
(2v+3)(2v+5)
| v,X||α0||v?1,X |2=(v?2X)(v+2X+2)
(v?2X+1)(v+2X+3)
(v?2X)(v+2X+2)
the total matrix elements of theαvariable can easily be derived.As an example we evaluate the matrix element
vXM X M Y|α0000|v′XM X M Y =(?)k X0X
?M X0M X X0X
?M Y0M Y
vX||α0||v′,X ,(76)
(with k=2X?M X?M Y).This leads to the closed expression
vXM X M Y|α0000|v+1,XM X M Y =β (2v+3)(2v+5),(77)
vXM X M Y|α0000|v?1,XM X M Y =β (2v+1)(2v+3).(78) There is a subtlety involved with equation(67).β2can be either be regarded as the radial variable in the5-dimensional Euclidean space,which is a constant by de?nition under rotations of the O(5)orthogonal group,or it can be recognized as a generator of the aforementioned SU(1,1)algebra.In the latter scheme,the O(5)Hilbert space needs to be extended to incorporate this SU(1,1)basis.Then,it is more convenient to move over to a boson creation and annihilation realization
b?μ=1
2(
√
√√kαμ?i k π?μ),(79)
with[bμ,b?ν]=δμνand?bμ=(?1)μb?μ,as it gives immediately rise to the SU(1,1) algebra spanned by
B+=1
2?b·?b,B
=1
a rotation from the natural group chain O (5)?O (4)～=SU (2)×SU (2)to the physical chain O (5)?O (3)?O (2)is needed.Fortunately,the Casimir operator L 0associated with the physical O (2)group is diagonal in the Cartan-Weyl basis,leaving L ·L the only operator to diagonalize.
L ·L =L 20+
1
12T ?1
2
,(82)L 0=X 0+3Y 0,
(83)
so that L ·L can be written as
L ·L =4X 2?3[(X 0?3Y 0+
1
2
)?1
3[T ?12
X ?+T 1
2
X +]+12T 1
2
T ?12
.
(84)
The action of all generators are known in the natural basis (see section 4),so the matrix
elements of a matrix representation can easily be calculated.The dimension of the matrix is governed by the seniority quantum number since the generators involved in the expression L ·L cannot alter the O (5)quantum number,which means that there is an associated matrix with every v .This matrix is even further reducible if one takes the L 0operator into account.Since L 0can be written as L 0=X 0+3Y 0,it is immediately diagonal in the Cartan-Weyl basis,making M =M X +3M Y a good quantum number.As a consequence,the total matrix representation of L ·L can be divided in separate sub-matrices with distinct M quantum number.The possible basis states |vXM X M Y spanning the sub-matrices with M =M X +3M Y can easily be recognized in the tilted weight diagrams (Figure 3).The diagram is tilted with respect to the angular momentum operator L 0,so that the vertical projection of every basis state immediately gives the L 0component.As a result,all basis states,lying on the same vertical projection line form a subspace of states for which M =M X +3M Y holds.Although the rotation from the natural towards the physical basis corresponds to a standard diagonalization problem,it is useful to study some speci?c cases.It is readily seen from ?gure 3that the projection M =2v can only be constructed from one single
basis state |v,X =v 2
,M Y =v
2
(v 2)|L ·L |v,v
2
v 2,M X =v
2
,giving the same eigenvalue 2v (2v +1).For M =2v ?2,there are two di?erent states:|v,X =v 2
?2,M Y =v
2?12?12
?14(v ?1)
3
34(v ?1)
L O 0
Figure 3.Projections of the v =4irreps on the L 0axis,connecting all basis states for which M =M X +3M Y is equal.The line most on the right only projects the M X =v/2,M Y =v/2state on the L 0axis,resulting in an M =2v state.For v =4the maximal L 0projection corresponds to M =8
So it is clear that the associated eigenvector of λ+belongs to the L =2v multiplet while the eigenvector of λ?will be the heighest M state of the L =2v ?2multiplet.Basically,this procedure can be repeated up to M =0by means of a symbolic mathematical computer program or by means of numerical procedures.
Finally we discuss the dimension of the matrix representations,as they are important in actual calculations.The total number of basis states within a representation v can be determined as
v/2 X =0
(2X +1)2=
12
)|3?1][(X m +
3
2
)|3][(X m +
32
)|3]2+[(X m ?
1
2
)|3)+1].
(90)
These two dimension formulas are plotted in ?gure 4.From this ?gure,it is clear that the
total dim M=0 dim
1
10 100 1000 10000
100000 1e+06 1 10
100
dim
v
Figure 4.Total dimension of a representations with a given seniority v ,compared to the dimension of the subspace of states with M =0.It is clear that the dimension of the M =0subspace increases quadratically with increasing seniority,however it remains feasible for realistic calculations.For v =100,the dimension of M =0space is 1717,while the total dimension is 348551.
dimension of the M =0subspace stays reasonable with respect to modern computation standards,as long as relatively low-order seniorities are considered.Anyhow,when performing realistic calculations,the transformation from natural to physical basis does not need to be repeated for every calculation,as the rotation is independent of the speci?c physical system (Hamiltonian)under study.In practical calculations,the rotation only has to be carried out once and stored for later use.7.Conclusions and outlook
Collective modes of motions have proven to be very important in atomic nuclei,away from the shell-closures.Therefore it is of major interest to construct schematic Hamiltonians in the signi?cant degrees of freedom.Throughout the last decades much attention has been given to solving the Schr¨o dinger equation for general potential energy surfaces in the collective variables.This resulted in a number of techniques,based on combinations of analytical and algebraic considerations.The present manuscript adds a method which is completely algebraic in the sense that no normalized highest weight states need to be constructed.As a matter of fact,although the γ-rotational structure of the collective model is completely contained in the O (5)subgroup of SU (1,1)×O (5),no explicit representations in terms of γhad to be constructed to obtain matrix elements of the collective variables,relevant for general potential energy surfaces.This suggests that the proposed technique can be extended to higher rank algebras,such as the O (7)
orthogonal group,emerging from octupole degrees of freedom in atomic nuclei.
Now the theoretical framework is set,it is interesting to study to what extend the geometrical model can be applied to the collective behaviour of atomic nuclei with respect to the recent developments in exotic nuclei.This will be the subject of further investigations.
Acknowledgments
The authors wish to thank Piet Van Isacker and John Wood for interesting discussions and suggestions.Financial support from the University of Ghent and the”FWO-Vlaanderen”that made this research possible is acknowledged.Also the Interuniversity Attraction Pool(IUAP)under project P5/07is acknowledged for?nancial support.
Appendix A.Explicit expressions of the O(5)generators in the
Cartan-Weyl basis
The generators L M and O M can explicitly be expressed in terms of the collective variables and their canonic conjugate momenta according to the de?nition(5)
L M=?i
√
[απ?]1M=?
i
√
μμ′ 2μ2μ′|1M αμπ?μ′,
O M=?i
√
[απ?]3M=?
i
√
μμ′ 2μ2μ′|3M αμπ?μ′,
(A.1)
where j1m1j2m2|j3m3 are the commonly known Clebsch Gordan coe?cients.Taking the rotation to the Cartan representation into account(9),explicit and relatively simple expressions for the generators can be obtained
X+=i
(α2π?1?α1π?2),
X?=?i
(α?1π??2?α?2π??1),
X0=?i
2
(α2π??2?α1π??1+α?1π?1?α?2π?2),
T1
2
=?i2(α2π?0?α0π?2),T?12=?i2(α1π?0?α0π?1),
T1
2=?i2(α?1π?0?α0π??1),T?12=?i2(α?2π?0?α0π??2).
(A.2)
References
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JOIN IN 学生用书1 Word List Starter Unit 1.Good afternoon 下午好 2.Good evening 晚上好 3.Good morning 早上好 4.Good night 晚安 5.Stand 站立 Unit 1 6.count [kaunt] （依次）点数 7.javascript:;eight [eit] 八 8.eleven [i'levn] 十一 9.four [f?:] 四 10.five [faiv] 五 11.flag [fl?g] 旗 12.guess [ges] 猜 13.jump [d??mp] 跳 14.nine [nain] 九 15.number ['n?mb?] 数字 16.one [w?n] 一 17.seven ['sevn] 七 18.six [siks] 六 19.ten [ten] 十 20.three [θri:] 三 21.twelve [twelv] 十二 22.two [tu:] 二 23.your [ju?] 你的 24.zero ['zi?r?u] 零、你们的 Unit 2 25.black [bl?k] 黑色26.blue [blu:] 蓝色 27.car [kɑ:] 小汽车 28.colour ['k?l?] 颜色 29.door [d?:] 门 30.favourite [feiv?rit]javascript:; 特别喜爱的 31.green [gri:n] 绿色 32.jeep [d?i:p] 吉普车 33.orange ['?:rind?] 橙黄色 34.pin k [pi?k] 粉红色 35.please [pli:z] 请 36.purple ['p?:pl] 紫色 37.red [red] 红色 38.white [wait] 白色 39.yellow ['jel?u] 黄色 Unit 3 40.blackboard ['bl?kb?:d] 黑板 41.book [buk] 书 42.chair [t???] 椅子 43.desk [desk] 桌子 44.pen [pen] 钢笔 45.pencil ['pensl] 铅笔 46.pencil case [keis] 笔盒 47.ruler ['ru:l?] 尺、直尺 48.schoolbag [sku:l] 书包 49.tree [tri:] 树 50.window ['wind?u] 窗、窗口 Unit 4 51.brown [braun] 棕色 52.cat [k?t] 猫
常用整流二极管 型号VRM/Io IFSM/ VF /Ir 封装用途说明1A5 600V/1.0A 25A/1.1V/5uA[T25] D2.6X3.2d0.65 1A6 800V/1.0A 25A/1.1V/5uA[T25] D2.6X3.2d0.65 6A8 800V/6.0A 400A/1.1V/10uA[T60] D9.1X9.1d1.3 1N4002 100V/1.0A 30A/1.1V/5uA[T75] D2.7X5.2d0.9 1N4004 400V/1.0A 30A/1.1V/5uA[T75] D2.7X5.2d0.9 1N4006 800V/1.0A 30A/1.1V/5uA[T75] D2.7X5.2d0.9 1N4007 1000V/1.0A 30A/1.1V/5uA[T75] D2.7X5.2d0.9 1N5398 800V/1.5A 50A/1.4V/5uA[T70] D3.6X7.6d0.9 1N5399 1000V/1.5A 50A/1.4V/5uA[T70] D3.6X7.6d0.9 1N5402 200V/3.0A 200A/1.1V/5uA[T105] D5.6X9.5d1.3 1N5406 600V/3.0A 200A/1.1V/5uA[T105] D5.6X9.5d1.3 1N5407 800V/3.0A 200A/1.1V/5uA[T105] D5.6X9.5d1.3 1N5408 1000V/3.0A 200A/1.1V/5uA[T105] D5.6X9.5d1.3 RL153 200V/1.5A 60A/1.1V/5uA[T75] D3.6X7.6d0.9 RL155 600V/1.5A 60A/1.1V/5uA[T75] D3.6X7.6d0.9 RL156 800V/1.5A 60A/1.1V/5uA[T75] D3.6X7.6d0.9 RL203 200V/2.0A 70A/1.1V/5uA[T75] D3.6X7.6d0.9 RL205 600V/2.0A 70A/1.1V/5uA[T75] D3.6X7.6d0.9 RL206 800V/2.0A 70A/1.1V/5uA[T75] D3.6X7.6d0.9 RL207 1000V/2.0A 70A/1.1V/5uA[T75] D3.6X7.6d0.9 RM11C 1000V/1.2A 100A/0.92V/10uA D4.0X7.2d0.78 MR750 50V/6.0A 400A/1.25V/25uA D8.7x6.3d1.35 MR751 100V/6.0A 400A/1.25V/25uA D8.7x6.3d1.35 MR752 200V/6.0A 400A/1.25V/25uA D8.7x6.3d1.35 MR754 400V/6.0A 400A/1.25V/25uA D8.7x6.3d1.35 MR756 600V/6.0A 400A/1.25V/25uA D8.7x6.3d1.35 MR760 1000V/6.0A 400A/1.25V/25uA D8.7x6.3d1.35 常用整流二极管(全桥） 型号VRM/Io IFSM/ VF /Ir 封装用途说明RBV-406 600V/*4A 80A/1.10V/10uA 25X15X3.6 RBV-606 600V/*6A 150A/1.05V/10uA 30X20X3.6 RBV-1306 600V/*13A 80A/1.20V/10uA 30X20X3.6 RBV-1506 600V/*15A 200A/1.05V/50uA 30X20X3.6 RBV-2506 600V/*25A 350A/1.05V/50uA 30X20X3.6 常用肖特基整流二极管SBD 型号VRM/Io IFSM/ VF Trr1/Trr2 封装用途说明EK06 60V/0.7A 10A/0.62V 100nS D2.7X5.0d0.6 SK/高速 EK14 40V/1.5A 40A/0.55V 200nS D4.0X7.2d0.78 SK/低速 D3S6M 60V/3.0A 80A/0.58V 130p SB340 40V/3.0A 80A/0.74V 180p SB360 60V/3.0A 80A/0.74V 180p SR260 60V/2.0A 50A/0.70V 170p MBR1645 45V/16A 150A/0.65V <10nS TO220 超高速
1.塑封整流二极管 序号型号IF VRRM VF Trr 外形 A V V μs 1 1A1-1A7 1A 50-1000V 1.1 R-1 2 1N4001-1N4007 1A 50-1000V 1.1 DO-41 3 1N5391-1N5399 1.5A 50-1000V 1.1 DO-15 4 2A01-2A07 2A 50-1000V 1.0 DO-15 5 1N5400-1N5408 3A 50-1000V 0.95 DO-201AD 6 6A05-6A10 6A 50-1000V 0.95 R-6 7 TS750-TS758 6A 50-800V 1.25 R-6 8 RL10-RL60 1A-6A 50-1000V 1.0 9 2CZ81-2CZ87 0.05A-3A 50-1000V 1.0 DO-41 10 2CP21-2CP29 0.3A 100-1000V 1.0 DO-41 11 2DZ14-2DZ15 0.5A-1A 200-1000V 1.0 DO-41 12 2DP3-2DP5 0.3A-1A 200-1000V 1.0 DO-41 13 BYW27 1A 200-1300V 1.0 DO-41 14 DR202-DR210 2A 200-1000V 1.0 DO-15 15 BY251-BY254 3A 200-800V 1.1 DO-201AD 16 BY550-200～1000 5A 200-1000V 1.1 R-5 17 PX10A02-PX10A13 10A 200-1300V 1.1 PX 18 PX12A02-PX12A13 12A 200-1300V 1.1 PX 19 PX15A02-PX15A13 15A 200-1300V 1.1 PX 20 ERA15-02～13 1A 200-1300V 1.0 R-1 21 ERB12-02～13 1A 200-1300V 1.0 DO-15 22 ERC05-02～13 1.2A 200-1300V 1.0 DO-15 23 ERC04-02～13 1.5A 200-1300V 1.0 DO-15 24 ERD03-02～13 3A 200-1300V 1.0 DO-201AD 25 EM1-EM2 1A-1.2A 200-1000V 0.97 DO-15 26 RM1Z-RM1C 1A 200-1000V 0.95 DO-15 27 RM2Z-RM2C 1.2A 200-1000V 0.95 DO-15 28 RM11Z-RM11C 1.5A 200-1000V 0.95 DO-15 29 RM3Z-RM3C 2.5A 200-1000V 0.97 DO-201AD 30 RM4Z-RM4C 3A 200-1000V 0.97 DO-201AD 2.快恢复塑封整流二极管 序号型号IF VRRM VF Trr 外形 A V V μs （1）快恢复塑封整流二极管 1 1F1-1F7 1A 50-1000V 1.3 0.15-0.5 R-1 2 FR10-FR60 1A-6A 50-1000V 1. 3 0.15-0.5 3 1N4933-1N4937 1A 50-600V 1.2 0.2 DO-41 4 1N4942-1N4948 1A 200-1000V 1.3 0.15-0. 5 DO-41 5 BA157-BA159 1A 400-1000V 1.3 0.15-0.25 DO-41 6 MR850-MR858 3A 100-800V 1.3 0.2 DO-201AD
Join in 小学五年级英语教案 介休市宋古二中上站小学庞汝君 Unit 6 Friends 单元目标： 1、单词：sport music kite cap car book dog cat rat frog spider butterfly 2、句型：Who is your best friend ? （重点） How old is he ? When is his birthday ? What’s his favourite food ? 3、段落：介绍自己的好朋友 My best friend’s name is Toby . He is ten years old . His favourite colour is red . His birthday is in May . He has got a dog . 4、语法：第三人称的人称代词和物主代词（难点） 他he 他的his 她she 她的her 一般现在时第三人称单数动词+s
5、故事：Emma , Jackie and Diana . Are you all right ? What’s the matter ? Don’t be silly . We can play together . The first class 一、教学目标： 两个句型1、Who is your best friend ? 2、How old is he ? 二、教学过程： 1、Talk about your best friend . （1）教师说句子Who is your best friend ? My best friend is Anna. 让学生先领会句子的意思，然后模仿， 小组练习对话并上台表演。 （2）第二个句子How old is he ? He is nine years old . 象上面一样练习，等到学生掌握后把两个问句连起来做 问答操练。
二极管封装大全 篇一：贴片二极管型号、参数 贴片二极管型号.参数查询 1、肖特基二极管SMA（DO214AC） 2010-2-2 16:39:35 标准封装： SMA 2010 SMB 2114 SMC 3220 SOD123 1206 SOD323 0805 SOD523 0603 IN4001的封装是1812 IN4148的封装是1206 篇二：常见贴片二极管三极管的封装 常见贴片二极管/三极管的封装 常见贴片二极管/三极管的封装 二极管： 名称尺寸及焊盘间距其他尺寸相近的封装名称 SMC SMB SMA SOD-106 SC-77A SC-76/SC-90A SC-79 三极管： LDPAK
DPAK SC-63 SOT-223 SC-73 TO-243/SC-62/UPAK/MPT3 SC-59A/SOT-346/MPAK/SMT3 SOT-323 SC-70/CMPAK/UMT3 SOT-523 SC-75A/EMT3 SOT-623 SC-89/MFPAK SOT-723 SOT-923 VMT3 篇三：常用二极管的识别及ic封装技术 常用晶体二极管的识别 晶体二极管在电路中常用“D”加数字表示，如： D5表示编号为5的二极管。 1、作用：二极管的主要特性是单向导电性，也就是在正向电压的作用下，导通电阻很小;而在反向电压作用下导通电阻极大或无穷大。正因为二极管具有上述特性，无绳电话机中常把它用在整流、隔离、稳压、极性保护、编码控制、调频调制和静噪等电路中。 电话机里使用的晶体二极管按作用可分为：整流二极管(如1N4004)、隔离二极管(如1N4148)、肖特基二极管(如BAT85)、发光二极管、稳压二极管等。 2、识别方法：二极管的识别很简单，小功率二极管的N极(负极)，在二极管外表大多采用一种色圈标出来，有些二极管也用二极管专用符号来表示P极(正极)或N极(负极)，也有采用符号标志为“P”、“N”来确定二极管极性的。发光二极管的正负极可从引脚长短来识别，长
五年级下英语月考试卷-全能练考-Join in剑桥英语 姓名：日期：总分：100 一、根据意思，写出单词。（10′） Fanny:Jackie, ________[猜] my animal,my ________最喜欢的 animal.OK? Jackie:Ok!Mm…Has it got a nose? Fanny:Yes,it’s got a big long nose,…and two long ________[牙齿]. Jackie:Does it live in ________[美国]? Fanny:No,no.It lives ________[在] Africa and Asia. Jackie:Can it ________[飞]? Fanny:I’m sorry it can’t. Jackie:Is it big? Fanny:Yes,it’s very big and ________[重的]. Jackie:What ________[颜色] is it?Is it white or ________[黑色]? Fanny:It’s white. Jackie:Oh,I see.It’s ________[大象]. 二、找出不同类的单词。（10′） （）1 A.sofa B.table C.check （）2 A.noodle https://www.doczj.com/doc/d1615962.html,k C.way （）3 A.fish B.wolf C.lion （）4 A.bike B.car C.write （）5 A.tree B.farm C.grass （）6 A.big B.small C.other ( ）7 A.eat B.listen C.brown
Join In剑桥小学英语(改编版)入门阶段Unit 1Hello,hello第1单元嗨,嗨 and mime. 1 听,模仿 Stand up Say 'hello' Slap hands Sit down 站起来说"嗨" 拍手坐下来 Good. Let's do up Say 'hello' Slap hands Sit down 好. 我们来再做一遍.站起来说"嗨"拍手坐下来 the pictures. 2 再听一遍给图画编号. up "hello" hands down 1 站起来 2 说"嗨" 3 拍手 4 坐下来说 3. A ,what's yourname 3 一首歌嗨,你叫什么名字 Hello. , what's yourname Hello. Hello. 嗨. 嗨. 嗨, 你叫什么名字嗨,嗨. Hello, what's yourname I'm Toby. I'm Toby. Hello,hello,hello.嗨, 你叫什么名字我叫托比. 我叫托比 . 嗨,嗨,嗨. I'm Toby. I'm Toby. Hello,hello, let's go! 我是托比. 我是托比. 嗨,嗨, 我们一起! Hello. , what's yourname I'm 'm Toby. 嗨.嗨.嗨, 你叫什么名字我叫托比.我叫托比. Hello,hello,hello. I'm 'm Toby. Hello,hello,let's go! 嗨,嗨,嗨.我是托比. 我是托比. 嗨,嗨,我们一起! 4 Listen and stick 4 听和指 What's your name I'm Bob. 你叫什么名字我叫鲍勃. What's your name I'm Rita. What's your name I'm Nick. 你叫什么名字我叫丽塔. 你叫什么名字我叫尼克. What's your name I'm Lisa. 你叫什么名字我叫利萨. 5. A story-Pit'sskateboard. 5 一个故事-彼德的滑板. Pa:Hello,Pit. Pa:好,彼德. Pi:Hello,:What's this Pi:嗨,帕特.Pa:这是什么 Pi:My new :Look!Pi:Goodbye,Pat! Pi:这是我的新滑板.Pi:看!Pi:再见,帕特! Pa:Bye-bye,Pit!Pi:Help!Help!pi:Bye-bye,skateboard! Pa:再见,彼德!Pi:救命!救命!Pi:再见,滑板! Unit 16. Let's learnand act 第1单元6 我们来边学边表演.
1N 系列常用整流二极管的主要参数
反向工作 峰值电压 URM/V 额定正向 整流电流 整流电流 IF/A 正向不重 复浪涌峰 值电流 IFSM/A 正向 压降 UF/V 反向 电流 IR/uA 工作 频率 f/KHZ 外形 封装
型 号
1N4000 1N4001 1N4002 1N4003 1N4004 1N4005 1N4006 1N4007 1N5100 1N5101 1N5102 1N5103 1N5104 1N5105 1N5106 1N5107 1N5108 1N5200 1N5201 1N5202 1N5203 1N5204 1N5205 1N5206 1N5207 1N5208 1N5400 1N5401 1N5402 1N5403 1N5404 1N5405 1N5406 1N5407 1N5408
25 50 100 200 400 600 800 1000 50 100 200 300 400 500 600 800 1000 50 100 200 300 400 500 600 800 1000 50 100 200 300 400 500 600 800 1000
1
30
≤1
＜5
3
DO-41
1．5
75
≤1
＜5
3
DO-15
2
100
≤1
＜10
3
3
150
≤0.8
＜10
3
DO-27
常用二极管参数： 05Z6.2Y 硅稳压二极管 Vz=6~6.35V,Pzm=500mW,
公司登记（备案）申请书 注：请仔细阅读本申请书《填写说明》，按要求填写。 □基本信息 名称 名称预先核准文号/ 注册号/统一 社会信用代码 住所 省（市/自治区）市（地区/盟/自治州）县（自治县/旗/自治旗/市/区）乡（民族乡/镇/街道）村（路/社区）号 生产经营地 省（市/自治区）市（地区/盟/自治州）县（自治县/旗/自治旗/市/区）乡（民族乡/镇/街道）村（路/社区）号 联系电话邮政编码 □设立 法定代表人 姓名 职务□董事长□执行董事□经理注册资本万元公司类型 设立方式 （股份公司填写） □发起设立□募集设立经营范围 经营期限□年□长期申请执照副本数量个
□变更 变更项目原登记内容申请变更登记内容 □备案 分公司 □增设□注销名称 注册号/统一 社会信用代码登记机关登记日期 清算组 成员 负责人联系电话 其他□董事□监事□经理□章程□章程修正案□财务负责人□联络员 □申请人声明 本公司依照《公司法》、《公司登记管理条例》相关规定申请登记、备案，提交材料真实有效。通过联络员登录企业信用信息公示系统向登记机关报送、向社会公示的企业信息为本企业提供、发布的信息，信息真实、有效。 法定代表人签字：公司盖章（清算组负责人）签字：年月日
附表1 法定代表人信息 姓名固定电话 移动电话电子邮箱 身份证件类型身份证件号码 （身份证件复印件粘贴处） 法定代表人签字：年月日
附表2 董事、监事、经理信息 姓名职务身份证件类型身份证件号码_______________ （身份证件复印件粘贴处） 姓名职务身份证件类型身份证件号码_______________ （身份证件复印件粘贴处） 姓名职务身份证件类型身份证件号码_______________ （身份证件复印件粘贴处）
小学三年级英语（上册）重要知识点归纳 一、单词 Unit 1学习文具：pen (钢笔) pencil (铅笔) pencil-case ( 铅笔盒) ruler(尺子) eraser(橡皮) crayon (蜡笔) book (书) bag (书包) sharpener (卷笔刀) school (学校) Unit 2身体部位：head (头) face( 脸) nose (鼻子) mouth (嘴) eye (眼睛)leg (腿) ear (耳朵) arm (胳膊)finger (手指) leg (腿) foot (脚)body (身体) Unit 3颜色：red (红色的) yellow (黄色的)green (绿色的)blue (蓝色的) purple (紫色的) white (白色的) black (黑色的) orange (橙色的) pink (粉色的)brown (棕色的) Unit 4动物：cat (猫)dog (狗)monkey (猴子)panda (熊猫)rabbit( 兔子) duck (鸭子)pig (猪)bird (鸟) bear (熊)elephant (大象)mouse (老鼠) squirrel (松鼠) Unit 5食物：cake (蛋糕)bread (面包) hot dog (热狗) hamburger (汉堡包) chicken (鸡肉)French fries (炸薯条)coke (可乐)juice (果汁)milk (牛奶) water (水)tea (茶) coffee (咖啡) Unit 6数字：one (一) two (二) three (三)four (四) five (五)six( 六)seven (七) eight (八) nine( 九) ten( 十)doll (玩具娃娃) boat (小船)ball (球) kite (风筝) balloon (气球) car (小汽车)plane (飞机) 二、对话 1、向别人问好应该说――A: Hello! (你好!) B: Hi! (你好!) 2、问别人的名字应该说-――A：What's your name? 你的名字是什么? B：My name's Chen Jie. 我的名字是陈洁。 3、跟别人分手应该说――A: Bye.\ Good bye!(再见) B: See you.(再见) \ Goodbye.(再见) 4、A: I have a pencil\ bag\ruler 我有一只铅笔\书包\尺子。 B: Me too . 我也有。 5、早上相见应该说-――A: Good morning. 早上好!
1N系列稳压管
快恢复整流二极管
常用整流二极管型号和参数 05Z6.2Y 硅稳压二极管 Vz=6~6.35V,Pzm=500mW, 05Z7.5Y 硅稳压二极管 Vz=7.34~7.70V,Pzm=500mW, 05Z13X硅稳压二极管 Vz=12.4~13.1V,Pzm=500mW, 05Z15Y硅稳压二极管 Vz=14.4~15.15V,Pzm=500mW, 05Z18Y硅稳压二极管 Vz=17.55~18.45V,Pzm=500mW, 1N4001硅整流二极管 50V, 1A,(Ir=5uA,Vf=1V,Ifs=50A) 1N4002硅整流二极管 100V, 1A, 1N4003硅整流二极管 200V, 1A, 1N4004硅整流二极管 400V, 1A, 1N4005硅整流二极管 600V, 1A, 1N4006硅整流二极管 800V, 1A, 1N4007硅整流二极管 1000V, 1A, 1N4148二极管 75V, 4PF,Ir=25nA,Vf=1V, 1N5391硅整流二极管 50V, 1.5A,(Ir=10uA,Vf=1.4V,Ifs=50A) 1N5392硅整流二极管 100V,1.5A, 1N5393硅整流二极管 200V,1.5A, 1N5394硅整流二极管 300V,1.5A, 1N5395硅整流二极管 400V,1.5A, 1N5396硅整流二极管 500V,1.5A, 1N5397硅整流二极管 600V,1.5A, 1N5398硅整流二极管 800V,1.5A, 1N5399硅整流二极管 1000V,1.5A, 1N5400硅整流二极管 50V, 3A,(Ir=5uA,Vf=1V,Ifs=150A) 1N5401硅整流二极管 100V,3A, 1N5402硅整流二极管 200V,3A, 1N5403硅整流二极管 300V,3A, 1N5404硅整流二极管 400V,3A,
有限合伙企业登记注册操作指南 风险控制部 20xx年x月xx日
目录 一、合伙企业的概念 (4) 二、有限合伙企业应具备的条件 (4) 三、有限合伙企业设立具备的条件 (4) 四、注册有限合伙企业程序 (5) 五、申请合伙企业登记注册应提交文件、证件 (6) （一）合伙企业设立登记应提交的文件、证件： (6) （二）合伙企业变更登记应提交的文件、证件： (7) （三）合伙企业注销登记应提交的文件、证件： (8) （四）合伙企业申请备案应提交的文件、证件： (9) （五）其他登记应提交的文件、证件： (9) 六、申请合伙企业分支机构登记注册应提交的文件、证件 (9) （一）合伙企业分支机构设立登记应提交的文件、证件 (10) （二）合伙企业分支机构变更登记应提交的文件、证件： (10) （三）合伙企业分支机构注销登记应提交的文件、证件： (11) （四）其他登记应提交的文件、证件： (12) 七、收费标准 (12) 八、办事流程图 (12) （一）有限合伙企业创办总体流程图（不含专业性前置审批） (12) （二）、工商局注册程序 (15)
（三）、工商局具体办理程序（引入网上预审核、电话预约方式） (16) 九、有限合伙企业与有限责任公司的区别 (16) （一）、设立依据 (16) （二）、出资人数 (16) （三）、出资方式 (17) （四）、注册资本 (17) （五）、组织机构 (18) （六）、出资流转 (18) （七）、对外投资 (19) （八）、税收缴纳 (20) （九）、利润分配 (20) （十）、债务承担 (21) 十、常见问题解答与指导 (21)
JOIN IN英语三年级下册课本重点Start unit 1 Words morning afternoon evening night 2 Sentences Good morning !早上好 Good afternoon !下午好 Good evening!晚上好 Good night!晚安 3 Phrases clap your hands 拍拍手 jump up high 往高跳 shake your arms and your legs 晃动你的胳膊和腿 bend your knees 弯曲你的膝盖 touch your toes 摸摸你的脚指
stand nose to nose 鼻子对鼻子站着 Unit 1 Pets 1 Words cat猫dog狗bird 鸟mouse老鼠fish鱼rabbit 兔子frog青蛙hamster仓鼠 budgie鹦鹉tiger老虎monkey 猴子panda熊猫giraffe 长颈鹿elephant 大象bear 熊run跑sit坐fly飞swim游泳roar吼叫eat吃 2 Grammar ★名词的复数:一般在词尾直接加s,不规则变化要牢记: fish-----fish mouse------mice 3 Sentences 1.Have you got a pet ? 你有宠物吗?Yes ,I have. 是的,我有。/No, I haven’t. 不,我没有。 2.2. What have you got ? 你有什么宠物吗?I’ve got a dog . / A dog. 我有一只狗。 3.What colour is the cat ? 你的猫是什么颜色的?It’s black. 它是黑色的。 What iswizard’s pet? 巫师的宠物是什么? 4.What is it ? 它是什么? It’s a rabbit .它是只兔子。
→客户提供：场所证明租赁协议身份证委托书三张一寸相片 →需准备材料：办理税务登记证时需要会计师资格证与财务人员劳动合同 →提交名称预审通知书→公司法定代表人签署的《公司设立登记申请书》→全体股东签署的《指定代表或者公共委托代理人的证明》（申请人填写股东姓名）→全体股东签署的公司章程（需得到工商局办事人员的认可）→股东身份证复印件→验资报告（需到计师事务所办理：需要材料有名称预审通知书复印件公司章程股东身份证复印件银行开具验资账户进账单原件银行开具询证函租赁合同及场所证明法人身份证原件公司开设临时存款账户的复印件）→任职文件（法人任职文件及股东董事会决议）→住所证明（房屋租赁合同）→工商局（办证大厅）提交所有材料→公司营业执照办理结束 →需带材料→公司营业执照正副本原件及复印件→法人身份证原件→代理人身份证→公章→办理人开具银行收据交款元工本费→填写申请书→组织机构代码证办
理结束 →需带材料→工商营业执照正副本复印件原件→组织机构正副本原件及复印件→公章→公司法定代表人签署的《公司设立登记申请书》→公司章程→股东注册资金情况表→验资报告书复印件→场所证明（租赁合同）→法人身份证复印件原件→会计师资格证（劳动合同）→税务登记证办理结束 →需带材料→工商营业执照正副本复印件原件→组织机构正副本原件及复印件→税务登记证原件及复印件→公章→法人身份证原件及复印件→代理人身份证原件及复印件→法人私章→公司验资账户→注以上复印件需四份→办理时间个工作日→办理结束 →需带材料→工商营业执照正副本复印件原件→组织机构正副本原件及复印件→公章→公司法定代表人签署的《公司设立登记申请书》→公司章程→股东注册资金情况表→验资报告书复印件→场所证明（租赁合同）→法人身份证复印件原件→会计师资格证（劳动合同）→会计制度→银行办理的开户许可证复印件→税务登记证备案办理结束
Starter Unit Good to see you again知识总结 一. 短语 1. dance with me 和我一起跳舞 2. sing with me 和我一起唱歌 3. clap your hands 拍拍你的手 4. jump up high 高高跳起 5.shake your arms and your legs晃晃你的胳膊和腿 6. bend your knees 弯曲你的膝盖 7. touch your toes 触摸你的脚趾8. stand nose to nose鼻子贴鼻子站 二. 句子 1. ---Good morning. 早上好。 ---Good morning, Mr Li. 早上好，李老师。 2. ---Good afternoon. 下午好。 ---Good afternoon, Mr Brown. 下午好，布朗先生。 3. ---Good evening，Lisa. 晚上好,丽莎。 ---Good evening, Bob. 晚上好，鲍勃。 4. ---Good night. 晚安。 ----Good night. 晚安。 5. ---What’s your name? 你叫什么名字？ ---I’m Bob./ My name is Bob. 我叫鲍勃。 6. ---Open the window, please. 请打开窗户。 ---Yes ,Miss. 好的，老师。 7. ---What colour is it? 它是什么颜色？ 它是蓝红白混合的。 ---It’s blue, red and white. 皮特的桌子上是什么？ 8. ---What’s on Pit’s table? ---A schoolbag, an eraser and two books. 一个书包，一个橡皮和两本书。 9. ---What time is it? 几点钟？ 两点钟。 ---It’s two. 10.---What’s this? 这是什么？ ---My guitar. 我的吉他。
JOIN IN英语三年级下册 Start unit 1 Words morning afternoon evening night 2 Sentences Good morning !早上好Good afternoon ！下午好Good evening！晚上好 Good night！晚安 3 Phrases clap your hands 拍拍手 jump up high 往高跳 shake your arms and your legs 晃动你的胳膊和腿 bend your knees 弯曲你的膝盖 touch your toes 摸摸你的脚指 stand nose to nose 鼻子对鼻子站着 Unit 1 Pets 1 Words cat猫dog狗bird 鸟mouse老鼠fish鱼rabbit 兔子frog青蛙hamster仓鼠 budgie鹦鹉tiger老虎monkey 猴子panda熊猫giraffe 长颈鹿elephant 大象bear 熊run跑sit坐fly飞swim游泳roar吼叫eat吃 2 Grammar
★名词的复数：一般在词尾直接加s,不规则变化要牢记: fish-----fish mouse------mice 3 Sentences 1.Have you got a pet ? 你有宠物吗？ Yes ,I have. 是的,我有。/No, I haven’t. 不，我没有。 2.What have you got ? 你有什么宠物吗？I’ve got a dog . / A dog. 我有一只狗。 3.What colour is the cat ? 你的猫是什么颜色的？It’s black. 它是黑色的。 What is wizard’s pet? 巫师的宠物是什么？ 4.What is it ? 它是什么? It’s a rabbit .它是只兔子。 5.How many budgies /mice are there? 这里有多少只鹦鹉/老鼠？ There are + 数字budgies/mice. 这里有------只鹦鹉/老鼠。 6.Fly like a budgie. 像鹦鹉一样飞。Run like a rabbit. 像兔子一样跑。 Swim like a fish. 像鱼一样游泳。Eat like a hamster. 像仓鼠一样吃东西。 Sit like a dog. 像狗一样坐。Roar like a tiger. 像老虎一样吼叫。 7.What are in the pictures. 图片里面是什么？Animals. 动物。 8. What animals? 什么动物？ 9.How many pandas (elephants /bears/ giraffes/ monkeys/ budgies) are there?有多少.？ How many + 可数名词的复数形式 Unit 2 The days of the week
G ENERAL PURPOSE RECTIFIERS – P LASTIC P ASSIVATED J UNCTION 1.0 M1 M2 M3 M4 M5 M6 M7 SMA/DO-214AC G ENERAL PURPOSE RECTIFIERS – G LASS P ASSIVATED J UNCTION S M 1.0 GS1A GS1B GS1D GS1G GS1J GS1K GS1M SMA/DO-214AC 1.0 S1A S1B S1D S1G S1J S1K S1M SMB/DO-214AA 2.0 S2A S2B S2D S2G S2J S2K S2M SMB/DO-214AA 3.0 S3A S3B S3D S3G S3J S3K S3M SMC/DO-214AB F AST RECOVERY RECTIFIERS – P LASTIC P ASSIVATED J UNCTION MERITEK ELECTRONICS CORPORATION
U LTRA FAST RECOVERY RECTIFIERS – G LASS P ASSIVATED J UNCTION
S CHOTTKY B ARRIER R ECTIFIERS
S WITCHING D IODES Power Dissipation Max Avg Rectified Current Peak Reverse Voltage Continuous Reverse Current Forward Voltage Reverse Recovery Time Package Part Number P a (mW) I o (mA) V RRM (V) I R @ V R (V) V F @ I F (mA) t rr (ns) Bulk Reel Outline 200mW 1N4148WS 200 150 100 2500 @ 75 1.0 @ 50 4 5000 SOD-323 1N4448WS 200 150 100 2500 @ 7 5 0.72/1.0 @ 5.0/100 4 5000 SOD-323 BAV16WS 200 250 100 1000 @ 7 5 0.8 6 @ 10 6 5000 SOD-323 BAV19WS 200 250 120 100 @ 100 1.0 @ 100 50 5000 SOD-323 BAV20WS 200 250 200 100 @ 150 1.0 @ 100 50 5000 SOD-323 BAV21WS 200 250 250 100 @ 200 1.0 @ 100 50 5000 SOD-323 MMBD4148W 200 150 100 2500 @ 75 1.0 @ 50 4 3000 SOT-323-1 MMBD4448W 200 150 100 2500 @ 7 5 0.72/1.0 @ 5.0/100 4 3000 SOT-323-1 BAS16W 200 250 100 1000 @ 7 5 0.8 6 @ 10 6 3000 SOT-323-1 BAS19W 200 250 120 100 @ 100 1.0 @ 100 50 3000 SOT-323-1 BAS20W 200 250 200 100 @ 150 1.0 @ 100 50 3000 SOT-323-1 BAS21W 200 250 250 100 @ 200 1.0 @ 100 50 3000 SOT-323-1 BAW56W 200 150 100 2500 @ 75 1.0 @ 50 4 3000 SOT-323-2 BAV70W 200 150 100 2500 @ 75 1.0 @ 50 4 3000 SOT-323-3 BAV99W 200 150 100 2500 @ 75 1.0 @ 50 4 3000 SOT-323-4 BAL99W 200 150 100 2500 @ 75 1.0 @ 50 4 3000 SOT-323- 5 350mW MMBD4148 350 200 100 5000 @ 75 1.0 @ 10 4 3000 SOT-23-1 MMBD4448 350 200 100 5000 @ 75 1.0 @ 10 4 3000 SOT-23-1 BAS16 350 200 100 1000 @ 75 1.0 @ 50 6 3000 SOT-23-1 BAS19 350 200 120 100 @ 120 1.0 @ 100 50 3000 SOT-23-1 BAS20 350 200 200 100 @ 150 1.0 @ 100 50 3000 SOT-23-1 BAS21 350 200 250 100 @ 200 1.0 @ 100 50 3000 SOT-23-1 BAW56 350 200 100 2500 @ 70 1.0 @ 50 4 3000 SOT-23-2 BAV70 350 200 100 5000 @ 70 1.0 @ 50 4 3000 SOT-23-3 BAV99 350 200 100 2500 @ 70 1.0 @ 50 4 3000 SOT-23-4 BAL99 350 200 100 2500 @ 70 1.0 @ 50 4 3000 SOT-23-5 BAV16W 350 200 100 1000 @ 75 0.86 @ 10 6 3000 SOD-123 410-500mW BAV19W 410 200 120 100 @ 100 1.0 @ 100 50 3000 SOD-123 BAV20W 410 200 200 100 @ 150 1.0 @ 100 50 3000 SOD-123 BAV21W 410 200 250 100 @ 200 1.0 @ 100 50 3000 SOD-123 1N4148W 410 150 100 2500 @ 75 1.0 @ 50 4 3000 SOD-123 1N4150W 410 200 50 100 @ 50 0.72/1.0 @ 5.0/100 4 3000 SOD-123 1N4448W 500 150 100 2500 @ 7 5 1.0 @ 200 4 3000 SOD-123 1N4151W 500 150 75 50 @ 50 1.0 @ 10 2 3000 SOD-123 1N914 500 200 100 25 @ 20 1.0 @ 10 4 1000 10000 DO-35 1N4148 500 200 100 25 @ 20 1.0 @ 10 4 1000 10000 DO-35 LL4148 500 150 100 25 @ 20 1.0 @ 10 4 2500 Mini-Melf SOT23-1 SOT23-2 SOT23-3 SOT23-4 SOT23-5 SOT323-1 SOT323-2 SOT323-3 SOT323-4 SOT323-5