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The electromagnetic field in finite crystals
I. Phys. Chtm. Solids Vol. 40. pp. 339-343 Pergamon Press Ltd., 1979. Printed in Great Britain
THE ELECTROMAGNETIC FIELD IN FINITE CRYSTALS A. REDLACKand J. C&NDLA~ Physics Department,University of Waterloo,Waterloo,Ontario,Canada (Received 1 May 1978;accepted in revisedfom 28 July 1978)
Abstrac-The electromagneticpotentialsof a finite crystalline slab of oscillatingpoiot chargesare separatedinto parts with distinct properties.One part is the Ewaklpotential. 1. lrvlmDucrloN
angular frequency of the motion. This oscillating charge
The determination of the electromagneticfield in a crystal is a basic problem in a number of areas of solid state
physics. Ewald’streatmentrl] provided the basis for all later calculationsof this field. Ewald’smain premise was that while the electromagnetic field in a finite crystal consists of a periodic term plus another, complicated, boundary dependent term, only the former determines the bulk properties of the crystal. To abstract the periodic part he introduced a dampingterm e-@’in the potentials, let the boundaries of the crystal go to inlkrity and then sought the limit /3+O’. The success of this approach, at least in the static case, depends criticallyon the restriction to electrically neutral crystals with no net dipole moment (see, e.g. [2J). In this paper we obtain expressions for the terms neglected by Ewald and put them in forms convenient for discussion.In later papers we shall apply the results to the static and time dependent cases and discuss the circumstancesunder which the neglect is just&d. Our system is a rectangular parallelepipedslab of N identical point charges distributed over the sites of an orthorhombicbravais lattice. We too introduce the factor e-#‘, not to “blur” the effects of the surface as in Ewald’s case, but to permit the application of Fourier transform theory. The main problem discussed in this paper is the separation of the electromagneticpotentials into parts with demonstrably distinct properties; a part with the three dimensionalperiodicity of the lattice (the Ewald potential) a part with two dimensional periodicities, etc. We also discuss an alternative representation for the Ewald potential which is convenient to handle both numericallyand analytically. The analysis and results are presented as follows. In Section 2 we derive the vector and scalar potentials A, 4 for a finite crystal of N point charges,in the limit of long wave lengths and small amplitude vibrations. Section 3 contains a number of useful identities and Section 4 the main results of this paper, the partition of the potentials at points inside the outside-and-abovethe crystal. We derive the alternative representation of the Ewald potential in Section 5.
generates electromagnetic fields E(x, 0, B(x,t). In the long wave length limit, oju’l Q c, and at points far from the charge, in the sense of Iu”I+ 1x1,the Lorentz gauge Li&ard-Wiechert potentials A and I$,generatingE and B, take the form A(x, t) = -
exp [- io(t - r/c)l/r
$(x, t) = [q/4rmol[l/r+ u” . P,(l/r* - i&c) x exp [- io(t - r/c)]]
(1) (2)
where e. is the permittivity of free space (M.K.S.
system), c is the speed of light in the vacuum, 2. is the unit vector in the direction of x and r = 1x1.The first term on the right of (2) is the time independent or electrostatic
potential. The second term is related to the A in (1) through the gauge condition; using this relation we can replace (2) with
2.VRClDRANDSCNkUlFOTRMMS Let a point charge q oscillate about the origin of
coordinates u’exp (-id’);
[i~u”/~oczl
with a displacement vector I@) = II“ is the vector amplitude and o the 339
4(x, t) = (q/4mor) - i(c%)V . A.
(3)
Consider the orthorhombic bravais lattice with points described by the lattice vector X(l)
= 2
i-1
hi&i
(4)
where ui (i = 1,2,3) are the lengths of the sides of the orthorhombicunit cell and 6, the unit vectors parallel to these sides; I represents the triple of positive or negative integers (I,, f2 and la). The volume of the unit cell is 0 = alo2a3. The lattice reciprocal to this bravais lattice is descnid by the lattice vector y(h) = 2 27rhrBlai
(3
where h represents the triple of integers (hl, hz and II,). We construct a slab-shapedsystem of N similar point charges q distributed, one to a site, over the N lattice sites defined by the three ranges of integers l&lI m, i=l, 2, 3 with N=(2mI+1)(2m2t1)(2m3t1). If we associate with each charge an orthorhombiccell centred at the appropriate lattice site, we may think of the N charges as occupying a region in space in the shape of a recmngularparallelepipedwith sides L, = (2mrt l)a, and volume NV.
340
A. REDLACKand J. GRINDLAY
Let each charge in the slab oscillate about its lattice site in such a fashion that the position vector of the Zth charge is x(l) + u’exp i[K. x(l)- ot], where K is some propagation vector and u” a constant vector amplitude. In general the three components of K and the frequency o may be complex. Within the long wavelength, small oscillation approximation (described above) the LiCnardWiechert potentials for this system are, see (1) and (3),
and I=7l-
eca-iK’Xsinh [(a - iK)L/21 x ~ _(L _ a),2 CY sinh [(a - iK)a/2] ’
(*4)
In each equation, (12)-(14), the convention Re a > 0 is used. From (13) we deduce that, for all x,
A(x, t) = - [iqou0/4moc2] Rea > IZmKI. x(~)+o(x-x(I)~/c] exp(_iot)(6) provided Consider the sum I~-~(ol
expi[K. xx I and
4(x, t) = [q/4mol[7
Ix- x(l)l-’ - (ic%)V . Al
(7) which occurs in (13) and (15). It is a periodic function of x with period a. The range
where the 1 below the summation sign indicates a sum over all the N lattice sites within the slab. It is to be understood that if x coincides with a lattice site the corresponding term in the sum is omitted. Let us introduce the sum S(x K ,),zm t 7
I
Ix-x(Ol
(L-a)/25x5(L+a)/2
(16)
is common to (12) and (13). Hence in this range we may equate the right sides of these equations and solve for J to get J(x, K, a) = $ exp (- izK)
* (8)
’
Then (6) and (7) can be written in the form
exp (- az) ev (ad sinh [(a - iK)a/2]‘sinh [(a t iK)a/Z]
1
(17)
A(x, t) = - [iqou”/4neoc2]S(x, K, o) exp i(K *x - ot)
(9)
4(x, r) = (q/4neoKS(x, 0,O) - (u” . V) x [S(x, K, CO) exp i(K . x - or)]}.
(10)
Thus given S we may.determine A and $. The remainder of the paper is concerned with the problem of determining S.
where ,z = x -L/2 = x - ma - a/2. While the derivation of (17) is based on the restricted range (16) the periodic property of .Z permits the use of (17) for any x with the understanding that z t a/2 = (x mod a), with ]zI5 a/2. Thus the existence of the overlap region (16) allows us to sum the infinite series J. 4. PARTITIONING
Consider the sum 3. IDmm
Consider the integral Z(x, L, K, a) = _lexp ;$;;p*] I
~~~~~;~
dy (11)
where x, a are real a, K are complex; L = (2m t 1)a with m a positive integer. A contour integration using the real axis and the semi-circle at infinity (either above or below the real axis) gives Z,ve
-(a+X)x sinb [(a ’ iK)L’2] x 2 (L - a)/2 a sinh [(a t iK)a/2] ’
(12)
exp (i2?rhx/a) ?rexp (- aL/2 - iKx) z=?rh~..[(Kt2&Za)z+a’ 2a Im
1
x exp -(ax t XL/2) + exp (ax + iK42) sinh [(a t iK)a/21 shh [(a - iK)a/21 ’ 1x15 (L t a)/2
(13)
x exp - {iK . [x-x(l)] t (p - io/c)]x - x(Z)]) (18) (x-x(Z)1 where /3 is a complex number. The right side of (18) is a sum over all the I values in the slab. From the definitions (8) and (18) S(x, K, o) = H(x, K, -b/c)
(19)
that is to say H takes the value of S at the origin in the complex J-plane. Since the sum in (18) is finite, H and S are entire functions of K,, Kz and K3 and o and in addition H is an entire function of /3. We shall make use of these points later. From (18) we define the infinite lattice function I&,(X, K, p - b/c) = lim lim lim H(x, K, /3 - h/c). m,-.x?I¶- m,-~ (20)
The electromagneticfield in finite crystals This is the potential discussed by Ewald[l]. The limiting process in (20) is not valid for all values of K, w and j3. We obtain a sufficient condition for the existence of (20) as follows. Let K = K’t iK’, w = or t iw’, p = /3’ + Qi where the subscripts refer to real and imaginary. Then the summand in (18) can be written as exp{i[K’ . (x(l)-x)+@'-
341
H, = - lim lim lim
The remaining HSiand H, canbe generated by an
o'/c)[x(~)-x~]-[~'~o'/C~IK'ICOS f#llX(~)-Xb
(21)
lx(l) -xl
where 4 is the angle between K’ and x(l) - x. When p’to’/c>IK’(
(22)
the square bracket in (21) is positive for all 1 and hence the elements of the sum die off exponentially with increasing lx(i)-xl and the limit exists. Equation (22) is a sufficient condition for the existence of the limit in (20). While the expression (18) is valid for all x (inside and outside the slab), it is not in a form convenient for discussing the properties of the potentials A and I$. We now introduce two rearrangements or partitionings of the sum in (18), one suitable for points inside the slab and the other for points outside the above the slab. “Inside” is defined as jx,llLJ;!
i = 1,2,3
(23)
appropriate permutation of subscripts. We shall use the representation in (26) for points inside the crystal. For the region outside and above we shall use
H=Hb3tH,,+H&H=itHc I=, Hb3= lim lim
2 3 2 g(1). m;- mid Il’-mi I2=-rnj Ij--mj
i= 1,2 X3ZLJ2.
(31)
Again the limiting process invoked here does not exist for all K, o and /3; however since each term in (26) and (30) consists of all or part of the infinite sum H,,clearly (22) is a sufficient condition for the validity of this partitioning process. While (26) is more suitable for points inside and (30) more suitable for points outside and above, both equations are valid for all points; hence
and “outside and above” as lXilILJ2
(30)
H,,,=H,tHa.
(32)
(24)
We shall now put the partition quantities H, to H, in more convenient forms for later discussion. The end Let m; be an integer greater than m3; then by adding result of the process is a set of representations for these quantities-representations which are analytic for all /3 and subtracting terms we may replace (18) with values to the right of a line /3’= PO’ in the complex &plane and which have singularities to the left of this line; The quantity &’ is a positive function of K and o satisfying (22). This result at first sight seems at odds with the problem to hand, namely the determination of H with g(l) representing the summand in (18). Thus we at fl= 0. However H is an entire function of /3 and have written H for the smaller slab (m ,, m2, m3) in terms hence we can expect that when the sums (26) and (30) of H for the larger slab (m,, m2,m;) and the sum of g(l) are formed the singularities in the separate partition over the lattice sites between the boundaries of the two quantities will cancel; this is indeed what happens and slabs. Similarly by introducing mi >m,, mi>m2 we we are left with an analytic representation for H in may apply this addition and subtraction process to the 11 which we may set B = 0; a knowledge of the precise and l2 summations. In this fashion we can express H as a form of /30’ is not necessary-all we require is that it sum of nine types of terms. If we let the three ml’s go to exists and that as a result there is a region in the complex p-plane in which all our manipulations leading to the infinity, then final expressions for H, to Hs are valid. The Fourier transform of g(l) gives
H=HvtA(H,i+HdtHc i==l
(26)
with Ho defined in (20) and m:I ml 2 2 IIli- ttti- mj- lj=-lVXil*=-mi _-nr*-1 m: .
with y = lyl. Consider first H..Inserting (33) in (20) and car&g out the triple summation results in the equation
H,,= - lim lim lim
(27) H.,=
IVli- #Vii_
mj-0
1 x sin [(Yj- &)L;N} sin [(yj - &had21 FGFG8 with Lj= (2mj t l)a,.
342
A. Rxn~cx and J. GMNJXAY
Each y&tegral can be identifiedin turn with I of eqn (11).The associated m;limit is then given by (15);hence H,(x, K, @- i&) = 5 F la
Hence
(35) x {exp- i[Klvl* t K2vz*lK&~6*)2+
&*)7}
where the sum is over all the reciprocal lattice vectors y(h). Equation(35)holds for all x but is subject to certain where inequalities involving K, fl and o, see the inequality v,*= xl r llai, (41) following eqn (15). However we can always choose a s:= (K,-+ 27&/aj)2 t(B - iold2 j = 1,2,3, (42) sufficientlylarge value of 8’ that the inequalitiescan be satisfied without restrictions on K and o. Thus (35) is and valid for all /3’z/30’. By inspection, H, is a periodic function of x with the period of the direct lattice. Re& >O. (43) The 1, and lz sums in Hss, (27)can be carried out using (33)and the identity (15);hence we tind that The curly brackets denote the sum of four terms obtained from the four permutations (ui-, Ye-), (v,-, ~~7, (Y,+,~~7, (Y,+,~27. Again the choice of a H,,=+g exp [ix - y(h)] lim large value of /3’ is suilicient to satisfy the conditions Jt;Z 3 4+,3-$$+,I imposed by the use of (15) and the demand that the m exp [i&L; 1:$3 - &)I dy, (36) double I-sums converge in (40). Equation (40) holds for 2 2 I -ca all xi, j = 1,2,3; however we note that if lx,1s (& t a)/2, = 1,2, the argumentsof the modifiedBessel functions in with (40)are monotonicincreasingfunctions of 1, and 12.Thus ps2= (K, + 2?rhlla1)2t (K2t 2?rh2/a3)2t @ - i&)2 for points satisfyingthis condition (namelypoints inside, (37) outside-and-aboveor outside-and-belowthe crystal) H., displays no direct lattice periodicity in planes parallel to and no restrictions on x1 or x2. The summationsign with the x3= 0 co-ordinateplane. tt3 does possess the direct the symbol H3 denotes a sum over all reciprocal lattice lattice periodicity in the x3-directionthrough the H3-sum. vectors with hs = 0. To evaluate the &sums we adopt the &,, He2are easily generated from (40). restriction For & we choose to use the summandin (18) rather Ix31 5 (Ls + a3)/2. (38) than the Fourier transform version in (33);then The integralin (36)is a specialcase of I in (11).The sum contains two parts and with the restriction (38) in force eqn (12) is appropriate for Ia> m3 and eqn (14) for I, < - m3. The resultingsums are easily evaluated. Hence Hs3(x,K, B - iolc) =
~~exp[ix.y(~)-~t~~~~~)l x
exp (- pti3 - iUJ2) + exp b 3x3 + iK3kl2) sinh [(p3 t iK3)aJ21 sinh f&3 - %3b3/4
1
(39j
with the convention that Re p3 > 0; this condition stems from the similarconvention imposedon a following(14). The use of (15)to obtain (36)and the demand that ‘the13 sums are convergent impose restrictions on fl, K and CU. Again these are satisfiedby a su5ciently large value of fi’. H,, is periodic in planes parallel to the x3= 0 plane with the equivalentperiodicity of the direct lattice. There is periodicity and decay in the x3-direction,the former determined by Im ~3 and the latter by Re p3. The decay occurs away from the two surface planes x3= *h/2 inwards.The correspondingexpressions for HII, H,2 are easily generated from (39). For the case of Iz3, (28),we can transform the lrsum using (33)and (15); .&e resultingdouble integralcan then be expressed in terms of the mod&d Bessel function of order zero, Ko, (131.3.754.2).
I&(x, K, /3- i&) = x exp - [iK *vf t
(B- i&)lv*ll I
where v* is a vector with components(Y,*,u2*,Ye*)and the curly brackets represent the sum of eight terms obtained from the eight permutations (Y,+,v2+,~~7, (v*+, y2+, v3-). . . , @I-, v2-, v3-1. Finally we discuss Hb3, (31). In this case we use (15) for the II, h sums, adopt the restriction x3 2 (L3 -
a3)/2
(45)
and use (12)to get Hb3(x,K, /I - i&z) = T
8 exp [ix +y(h)1 3
X
eXP [-X3&3 t CL3
iK3)] &lb
[(pr
sinh [(p3
+ x3&3/21 +
iK3M21 (4a)
with the convention that Re p3 > 0. As before the choice of a large B’ guarantees the vaM.ity of this expression. Ha,, lb2 are easily generated from (46).Iis is periodic Y .planesparallel to the co-ordinate plane x3 = 0 with the perk&city of the direction lattice. As in the case of HS3, this partition quantity exhibits periodicty and decay
The electromagnetic field in finite crystals
along the x4irection determined by the imaginary and real parts of p3 respectively. Equations (35), (39), (@), (44) and (46) contain representations for H,, Hsj, H+ H, and HL,,respectively. The ranges of validity of these representations may be summarized as follows: (i) H,, H,, Hej: the representations hold for all xk, (ii) Hlj: the representation holds for l&l s(L) + a3)/2 and all xk, (k # j), (iii) Hbi: the representation holds for xi 2 (Lj - a,)/2 and all xkt (k# j). It follows immediately that partition (26) with representations (35), (39), (40) and (44) holds for all points inside the crystal, case (a), and partition (30) with representation (39), @IO),(44) and (46) holds for all points outside and above the crystal, case (b). We note in passing that both partitions and the associated representations hold in the common region (L3 - a3)/2 5x3 5 (Ls + a3)/2 and all xk, k # 3. 5. ANALTERNATIVE II,, REPRIISENTATION The expression for H, in eqn (35) is a slowly convergent triple sum through reciprocal lattice space. The convergence is sufficiently slow that (a) (35) is not convenient to use to obtain numerical values of H, and 0) space derivatives do not commute with the summation operation. This latter restriction is particularly inconvenient for a quantity which is to be identified with a potential. In this section we derive another representation for H, which does not suffer from the disadvantages of (35). In the region (L3 - a,)/2 I x3 I (La t a3)/2, eqn (39) for H,, and eqn (46) for Hb3 are both valid (see Section 4). Also H, = Hh3 - H,, for all x, (32); hence in this restricted region we express H, in terms of the representation of Hb3 and Hs3 to get Hdx, K, fi - idc) = $+ exp (- iz3K3) 8
[
exp [ix - y(h)] 3
CL3
1
exp (- ~323) p 613231 sinh [(F3 t iK)a3/21t sinh iL3 - iKa)a3/21
(47)
with z3 = x3 - m3a3 - a3/2. However, H. is periodic in x and therefore (47) may be applied at any x provided 2, t aa/ = (x3 mod a3) and jz3i= a3/2, see discussion in Section 3. In eqn (47) H, is expressed as a double sum which represents a saving in labour in comparitin with the triple sum of (35); moreover the real exponentials in
343
(47) result in a rapid convergence of the sum (except on the planes z3 = ?a3/2) and permit the commutation of space derivatives and the summation operation. The method of planewise summation[4] applied to (35) also yields (47). Ewald[l] developed another widely used, rapidly convergent representation for Ho. However this involves two triple sums and the incomplete gamma function and as a result is somewhat more unwieldy in applications than (47). 6. SUMMARY ANDCOMMENT The vector and scalar potentials for the oscillating charges in the slab are generated from the quantity H (9), (IO) and (19). We have partitioned H (26) and (30), into quantities with distinct properties. The term H, (35) and (47), has the full three dimensional periodicity of the direct lattice, H,,, H,,, (34) and (46) have a two dimensional lattice periodicity, He, (40), a one dimensional lattice periodicity and H, no lattice periodicity. We conclude by remarking on the choice of subscripts on the partition quantities. The expressions Hsj, Hbi have the additional property that each consists of a sum of two terms, one numerically significant only near the slab planes x1= +L,/2 and the other significant throughout the crystal. He, has a similar property except that it is significant only near the slab edges parallel to the xi-axis, see, e.g. eqns (17H20) in the following paper. Thus the partition quantities provide us with a measure of microscopic fringing effects at the surface and edges. It was these considerations that led to our choice of subscripts for volume, surface, boundary and edge. At an early stage in our analysis we believed that H, had a term which was significant only near the comer-hence our choice of c for comer-but this proved not to be the c&e. We describe the application of our results to the static case in the following paper. The dynamic case will be discussed at a later date. Acknowledgement-The authors wish to thank the National Research Council of Canada for financial support.
REFERENCES 1. Ewald P. P., Ann. d. Phys. 54,519 (1917).
2. Redlack A. and Grindlay J., J. Phys. Chem. Solids 36,73,953 (1975). 3. Gradshteyn I. S. and Ryzhik I. M., Table of Integra/s, Series and Products. Academic Press, New York (1%5). 4. Nijboer B. R. A. and de Wette F. W., Physica 24,422 (1958).
THE ELECTROMAGNETIC FIELD IN FINITE CRYSTALS A. REDLACKand J. C&NDLA~ Physics Department,University of Waterloo,Waterloo,Ontario,Canada (Received 1 May 1978;accepted in revisedfom 28 July 1978)
Abstrac-The electromagneticpotentialsof a finite crystalline slab of oscillatingpoiot chargesare separatedinto parts with distinct properties.One part is the Ewaklpotential. 1. lrvlmDucrloN
angular frequency of the motion. This oscillating charge
The determination of the electromagneticfield in a crystal is a basic problem in a number of areas of solid state
physics. Ewald’streatmentrl] provided the basis for all later calculationsof this field. Ewald’smain premise was that while the electromagnetic field in a finite crystal consists of a periodic term plus another, complicated, boundary dependent term, only the former determines the bulk properties of the crystal. To abstract the periodic part he introduced a dampingterm e-@’in the potentials, let the boundaries of the crystal go to inlkrity and then sought the limit /3+O’. The success of this approach, at least in the static case, depends criticallyon the restriction to electrically neutral crystals with no net dipole moment (see, e.g. [2J). In this paper we obtain expressions for the terms neglected by Ewald and put them in forms convenient for discussion.In later papers we shall apply the results to the static and time dependent cases and discuss the circumstancesunder which the neglect is just&d. Our system is a rectangular parallelepipedslab of N identical point charges distributed over the sites of an orthorhombicbravais lattice. We too introduce the factor e-#‘, not to “blur” the effects of the surface as in Ewald’s case, but to permit the application of Fourier transform theory. The main problem discussed in this paper is the separation of the electromagneticpotentials into parts with demonstrably distinct properties; a part with the three dimensionalperiodicity of the lattice (the Ewald potential) a part with two dimensional periodicities, etc. We also discuss an alternative representation for the Ewald potential which is convenient to handle both numericallyand analytically. The analysis and results are presented as follows. In Section 2 we derive the vector and scalar potentials A, 4 for a finite crystal of N point charges,in the limit of long wave lengths and small amplitude vibrations. Section 3 contains a number of useful identities and Section 4 the main results of this paper, the partition of the potentials at points inside the outside-and-abovethe crystal. We derive the alternative representation of the Ewald potential in Section 5.
generates electromagnetic fields E(x, 0, B(x,t). In the long wave length limit, oju’l Q c, and at points far from the charge, in the sense of Iu”I+ 1x1,the Lorentz gauge Li&ard-Wiechert potentials A and I$,generatingE and B, take the form A(x, t) = -
exp [- io(t - r/c)l/r
$(x, t) = [q/4rmol[l/r+ u” . P,(l/r* - i&c) x exp [- io(t - r/c)]]
(1) (2)
where e. is the permittivity of free space (M.K.S.
system), c is the speed of light in the vacuum, 2. is the unit vector in the direction of x and r = 1x1.The first term on the right of (2) is the time independent or electrostatic
potential. The second term is related to the A in (1) through the gauge condition; using this relation we can replace (2) with
2.VRClDRANDSCNkUlFOTRMMS Let a point charge q oscillate about the origin of
coordinates u’exp (-id’);
[i~u”/~oczl
with a displacement vector I@) = II“ is the vector amplitude and o the 339
4(x, t) = (q/4mor) - i(c%)V . A.
(3)
Consider the orthorhombic bravais lattice with points described by the lattice vector X(l)
= 2
i-1
hi&i
(4)
where ui (i = 1,2,3) are the lengths of the sides of the orthorhombicunit cell and 6, the unit vectors parallel to these sides; I represents the triple of positive or negative integers (I,, f2 and la). The volume of the unit cell is 0 = alo2a3. The lattice reciprocal to this bravais lattice is descnid by the lattice vector y(h) = 2 27rhrBlai
(3
where h represents the triple of integers (hl, hz and II,). We construct a slab-shapedsystem of N similar point charges q distributed, one to a site, over the N lattice sites defined by the three ranges of integers l&lI m, i=l, 2, 3 with N=(2mI+1)(2m2t1)(2m3t1). If we associate with each charge an orthorhombiccell centred at the appropriate lattice site, we may think of the N charges as occupying a region in space in the shape of a recmngularparallelepipedwith sides L, = (2mrt l)a, and volume NV.
340
A. REDLACKand J. GRINDLAY
Let each charge in the slab oscillate about its lattice site in such a fashion that the position vector of the Zth charge is x(l) + u’exp i[K. x(l)- ot], where K is some propagation vector and u” a constant vector amplitude. In general the three components of K and the frequency o may be complex. Within the long wavelength, small oscillation approximation (described above) the LiCnardWiechert potentials for this system are, see (1) and (3),
and I=7l-
eca-iK’Xsinh [(a - iK)L/21 x ~ _(L _ a),2 CY sinh [(a - iK)a/2] ’
(*4)
In each equation, (12)-(14), the convention Re a > 0 is used. From (13) we deduce that, for all x,
A(x, t) = - [iqou0/4moc2] Rea > IZmKI. x(~)+o(x-x(I)~/c] exp(_iot)(6) provided Consider the sum I~-~(ol
expi[K. xx I and
4(x, t) = [q/4mol[7
Ix- x(l)l-’ - (ic%)V . Al
(7) which occurs in (13) and (15). It is a periodic function of x with period a. The range
where the 1 below the summation sign indicates a sum over all the N lattice sites within the slab. It is to be understood that if x coincides with a lattice site the corresponding term in the sum is omitted. Let us introduce the sum S(x K ,),zm t 7
I
Ix-x(Ol
(L-a)/25x5(L+a)/2
(16)
is common to (12) and (13). Hence in this range we may equate the right sides of these equations and solve for J to get J(x, K, a) = $ exp (- izK)
* (8)
’
Then (6) and (7) can be written in the form
exp (- az) ev (ad sinh [(a - iK)a/2]‘sinh [(a t iK)a/Z]
1
(17)
A(x, t) = - [iqou”/4neoc2]S(x, K, o) exp i(K *x - ot)
(9)
4(x, r) = (q/4neoKS(x, 0,O) - (u” . V) x [S(x, K, CO) exp i(K . x - or)]}.
(10)
Thus given S we may.determine A and $. The remainder of the paper is concerned with the problem of determining S.
where ,z = x -L/2 = x - ma - a/2. While the derivation of (17) is based on the restricted range (16) the periodic property of .Z permits the use of (17) for any x with the understanding that z t a/2 = (x mod a), with ]zI5 a/2. Thus the existence of the overlap region (16) allows us to sum the infinite series J. 4. PARTITIONING
Consider the sum 3. IDmm
Consider the integral Z(x, L, K, a) = _lexp ;$;;p*] I
~~~~~;~
dy (11)
where x, a are real a, K are complex; L = (2m t 1)a with m a positive integer. A contour integration using the real axis and the semi-circle at infinity (either above or below the real axis) gives Z,ve
-(a+X)x sinb [(a ’ iK)L’2] x 2 (L - a)/2 a sinh [(a t iK)a/2] ’
(12)
exp (i2?rhx/a) ?rexp (- aL/2 - iKx) z=?rh~..[(Kt2&Za)z+a’ 2a Im
1
x exp -(ax t XL/2) + exp (ax + iK42) sinh [(a t iK)a/21 shh [(a - iK)a/21 ’ 1x15 (L t a)/2
(13)
x exp - {iK . [x-x(l)] t (p - io/c)]x - x(Z)]) (18) (x-x(Z)1 where /3 is a complex number. The right side of (18) is a sum over all the I values in the slab. From the definitions (8) and (18) S(x, K, o) = H(x, K, -b/c)
(19)
that is to say H takes the value of S at the origin in the complex J-plane. Since the sum in (18) is finite, H and S are entire functions of K,, Kz and K3 and o and in addition H is an entire function of /3. We shall make use of these points later. From (18) we define the infinite lattice function I&,(X, K, p - b/c) = lim lim lim H(x, K, /3 - h/c). m,-.x?I¶- m,-~ (20)
The electromagneticfield in finite crystals This is the potential discussed by Ewald[l]. The limiting process in (20) is not valid for all values of K, w and j3. We obtain a sufficient condition for the existence of (20) as follows. Let K = K’t iK’, w = or t iw’, p = /3’ + Qi where the subscripts refer to real and imaginary. Then the summand in (18) can be written as exp{i[K’ . (x(l)-x)+@'-
341
H, = - lim lim lim
The remaining HSiand H, canbe generated by an
o'/c)[x(~)-x~]-[~'~o'/C~IK'ICOS f#llX(~)-Xb
(21)
lx(l) -xl
where 4 is the angle between K’ and x(l) - x. When p’to’/c>IK’(
(22)
the square bracket in (21) is positive for all 1 and hence the elements of the sum die off exponentially with increasing lx(i)-xl and the limit exists. Equation (22) is a sufficient condition for the existence of the limit in (20). While the expression (18) is valid for all x (inside and outside the slab), it is not in a form convenient for discussing the properties of the potentials A and I$. We now introduce two rearrangements or partitionings of the sum in (18), one suitable for points inside the slab and the other for points outside the above the slab. “Inside” is defined as jx,llLJ;!
i = 1,2,3
(23)
appropriate permutation of subscripts. We shall use the representation in (26) for points inside the crystal. For the region outside and above we shall use
H=Hb3tH,,+H&H=itHc I=, Hb3= lim lim
2 3 2 g(1). m;- mid Il’-mi I2=-rnj Ij--mj
i= 1,2 X3ZLJ2.
(31)
Again the limiting process invoked here does not exist for all K, o and /3; however since each term in (26) and (30) consists of all or part of the infinite sum H,,clearly (22) is a sufficient condition for the validity of this partitioning process. While (26) is more suitable for points inside and (30) more suitable for points outside and above, both equations are valid for all points; hence
and “outside and above” as lXilILJ2
(30)
H,,,=H,tHa.
(32)
(24)
We shall now put the partition quantities H, to H, in more convenient forms for later discussion. The end Let m; be an integer greater than m3; then by adding result of the process is a set of representations for these quantities-representations which are analytic for all /3 and subtracting terms we may replace (18) with values to the right of a line /3’= PO’ in the complex &plane and which have singularities to the left of this line; The quantity &’ is a positive function of K and o satisfying (22). This result at first sight seems at odds with the problem to hand, namely the determination of H with g(l) representing the summand in (18). Thus we at fl= 0. However H is an entire function of /3 and have written H for the smaller slab (m ,, m2, m3) in terms hence we can expect that when the sums (26) and (30) of H for the larger slab (m,, m2,m;) and the sum of g(l) are formed the singularities in the separate partition over the lattice sites between the boundaries of the two quantities will cancel; this is indeed what happens and slabs. Similarly by introducing mi >m,, mi>m2 we we are left with an analytic representation for H in may apply this addition and subtraction process to the 11 which we may set B = 0; a knowledge of the precise and l2 summations. In this fashion we can express H as a form of /30’ is not necessary-all we require is that it sum of nine types of terms. If we let the three ml’s go to exists and that as a result there is a region in the complex p-plane in which all our manipulations leading to the infinity, then final expressions for H, to Hs are valid. The Fourier transform of g(l) gives
H=HvtA(H,i+HdtHc i==l
(26)
with Ho defined in (20) and m:I ml 2 2 IIli- ttti- mj- lj=-lVXil*=-mi _-nr*-1 m: .
with y = lyl. Consider first H..Inserting (33) in (20) and car&g out the triple summation results in the equation
H,,= - lim lim lim
(27) H.,=
IVli- #Vii_
mj-0
1 x sin [(Yj- &)L;N} sin [(yj - &had21 FGFG8 with Lj= (2mj t l)a,.
342
A. Rxn~cx and J. GMNJXAY
Each y&tegral can be identifiedin turn with I of eqn (11).The associated m;limit is then given by (15);hence H,(x, K, @- i&) = 5 F la
Hence
(35) x {exp- i[Klvl* t K2vz*lK&~6*)2+
&*)7}
where the sum is over all the reciprocal lattice vectors y(h). Equation(35)holds for all x but is subject to certain where inequalities involving K, fl and o, see the inequality v,*= xl r llai, (41) following eqn (15). However we can always choose a s:= (K,-+ 27&/aj)2 t(B - iold2 j = 1,2,3, (42) sufficientlylarge value of 8’ that the inequalitiescan be satisfied without restrictions on K and o. Thus (35) is and valid for all /3’z/30’. By inspection, H, is a periodic function of x with the period of the direct lattice. Re& >O. (43) The 1, and lz sums in Hss, (27)can be carried out using (33)and the identity (15);hence we tind that The curly brackets denote the sum of four terms obtained from the four permutations (ui-, Ye-), (v,-, ~~7, (Y,+,~~7, (Y,+,~27. Again the choice of a H,,=+g exp [ix - y(h)] lim large value of /3’ is suilicient to satisfy the conditions Jt;Z 3 4+,3-$$+,I imposed by the use of (15) and the demand that the m exp [i&L; 1:$3 - &)I dy, (36) double I-sums converge in (40). Equation (40) holds for 2 2 I -ca all xi, j = 1,2,3; however we note that if lx,1s (& t a)/2, = 1,2, the argumentsof the modifiedBessel functions in with (40)are monotonicincreasingfunctions of 1, and 12.Thus ps2= (K, + 2?rhlla1)2t (K2t 2?rh2/a3)2t @ - i&)2 for points satisfyingthis condition (namelypoints inside, (37) outside-and-aboveor outside-and-belowthe crystal) H., displays no direct lattice periodicity in planes parallel to and no restrictions on x1 or x2. The summationsign with the x3= 0 co-ordinateplane. tt3 does possess the direct the symbol H3 denotes a sum over all reciprocal lattice lattice periodicity in the x3-directionthrough the H3-sum. vectors with hs = 0. To evaluate the &sums we adopt the &,, He2are easily generated from (40). restriction For & we choose to use the summandin (18) rather Ix31 5 (Ls + a3)/2. (38) than the Fourier transform version in (33);then The integralin (36)is a specialcase of I in (11).The sum contains two parts and with the restriction (38) in force eqn (12) is appropriate for Ia> m3 and eqn (14) for I, < - m3. The resultingsums are easily evaluated. Hence Hs3(x,K, B - iolc) =
~~exp[ix.y(~)-~t~~~~~)l x
exp (- pti3 - iUJ2) + exp b 3x3 + iK3kl2) sinh [(p3 t iK3)aJ21 sinh f&3 - %3b3/4
1
(39j
with the convention that Re p3 > 0; this condition stems from the similarconvention imposedon a following(14). The use of (15)to obtain (36)and the demand that ‘the13 sums are convergent impose restrictions on fl, K and CU. Again these are satisfiedby a su5ciently large value of fi’. H,, is periodic in planes parallel to the x3= 0 plane with the equivalentperiodicity of the direct lattice. There is periodicity and decay in the x3-direction,the former determined by Im ~3 and the latter by Re p3. The decay occurs away from the two surface planes x3= *h/2 inwards.The correspondingexpressions for HII, H,2 are easily generated from (39). For the case of Iz3, (28),we can transform the lrsum using (33)and (15); .&e resultingdouble integralcan then be expressed in terms of the mod&d Bessel function of order zero, Ko, (131.3.754.2).
I&(x, K, /3- i&) = x exp - [iK *vf t
(B- i&)lv*ll I
where v* is a vector with components(Y,*,u2*,Ye*)and the curly brackets represent the sum of eight terms obtained from the eight permutations (Y,+,v2+,~~7, (v*+, y2+, v3-). . . , @I-, v2-, v3-1. Finally we discuss Hb3, (31). In this case we use (15) for the II, h sums, adopt the restriction x3 2 (L3 -
a3)/2
(45)
and use (12)to get Hb3(x,K, /I - i&z) = T
8 exp [ix +y(h)1 3
X
eXP [-X3&3 t CL3
iK3)] &lb
[(pr
sinh [(p3
+ x3&3/21 +
iK3M21 (4a)
with the convention that Re p3 > 0. As before the choice of a large B’ guarantees the vaM.ity of this expression. Ha,, lb2 are easily generated from (46).Iis is periodic Y .planesparallel to the co-ordinate plane x3 = 0 with the perk&city of the direction lattice. As in the case of HS3, this partition quantity exhibits periodicty and decay
The electromagnetic field in finite crystals
along the x4irection determined by the imaginary and real parts of p3 respectively. Equations (35), (39), (@), (44) and (46) contain representations for H,, Hsj, H+ H, and HL,,respectively. The ranges of validity of these representations may be summarized as follows: (i) H,, H,, Hej: the representations hold for all xk, (ii) Hlj: the representation holds for l&l s(L) + a3)/2 and all xk, (k # j), (iii) Hbi: the representation holds for xi 2 (Lj - a,)/2 and all xkt (k# j). It follows immediately that partition (26) with representations (35), (39), (40) and (44) holds for all points inside the crystal, case (a), and partition (30) with representation (39), @IO),(44) and (46) holds for all points outside and above the crystal, case (b). We note in passing that both partitions and the associated representations hold in the common region (L3 - a3)/2 5x3 5 (Ls + a3)/2 and all xk, k # 3. 5. ANALTERNATIVE II,, REPRIISENTATION The expression for H, in eqn (35) is a slowly convergent triple sum through reciprocal lattice space. The convergence is sufficiently slow that (a) (35) is not convenient to use to obtain numerical values of H, and 0) space derivatives do not commute with the summation operation. This latter restriction is particularly inconvenient for a quantity which is to be identified with a potential. In this section we derive another representation for H, which does not suffer from the disadvantages of (35). In the region (L3 - a,)/2 I x3 I (La t a3)/2, eqn (39) for H,, and eqn (46) for Hb3 are both valid (see Section 4). Also H, = Hh3 - H,, for all x, (32); hence in this restricted region we express H, in terms of the representation of Hb3 and Hs3 to get Hdx, K, fi - idc) = $+ exp (- iz3K3) 8
[
exp [ix - y(h)] 3
CL3
1
exp (- ~323) p 613231 sinh [(F3 t iK)a3/21t sinh iL3 - iKa)a3/21
(47)
with z3 = x3 - m3a3 - a3/2. However, H. is periodic in x and therefore (47) may be applied at any x provided 2, t aa/ = (x3 mod a3) and jz3i= a3/2, see discussion in Section 3. In eqn (47) H, is expressed as a double sum which represents a saving in labour in comparitin with the triple sum of (35); moreover the real exponentials in
343
(47) result in a rapid convergence of the sum (except on the planes z3 = ?a3/2) and permit the commutation of space derivatives and the summation operation. The method of planewise summation[4] applied to (35) also yields (47). Ewald[l] developed another widely used, rapidly convergent representation for Ho. However this involves two triple sums and the incomplete gamma function and as a result is somewhat more unwieldy in applications than (47). 6. SUMMARY ANDCOMMENT The vector and scalar potentials for the oscillating charges in the slab are generated from the quantity H (9), (IO) and (19). We have partitioned H (26) and (30), into quantities with distinct properties. The term H, (35) and (47), has the full three dimensional periodicity of the direct lattice, H,,, H,,, (34) and (46) have a two dimensional lattice periodicity, He, (40), a one dimensional lattice periodicity and H, no lattice periodicity. We conclude by remarking on the choice of subscripts on the partition quantities. The expressions Hsj, Hbi have the additional property that each consists of a sum of two terms, one numerically significant only near the slab planes x1= +L,/2 and the other significant throughout the crystal. He, has a similar property except that it is significant only near the slab edges parallel to the xi-axis, see, e.g. eqns (17H20) in the following paper. Thus the partition quantities provide us with a measure of microscopic fringing effects at the surface and edges. It was these considerations that led to our choice of subscripts for volume, surface, boundary and edge. At an early stage in our analysis we believed that H, had a term which was significant only near the comer-hence our choice of c for comer-but this proved not to be the c&e. We describe the application of our results to the static case in the following paper. The dynamic case will be discussed at a later date. Acknowledgement-The authors wish to thank the National Research Council of Canada for financial support.
REFERENCES 1. Ewald P. P., Ann. d. Phys. 54,519 (1917).
2. Redlack A. and Grindlay J., J. Phys. Chem. Solids 36,73,953 (1975). 3. Gradshteyn I. S. and Ryzhik I. M., Table of Integra/s, Series and Products. Academic Press, New York (1%5). 4. Nijboer B. R. A. and de Wette F. W., Physica 24,422 (1958).