Scattering theory. Chapter 14

Transcription

1 Chapte 14 Scatteing theoy Almost eveything we know about nuclei and elementay paticles has been discoveed in scatteing expeiments, fom Ruthefod s supise at finding that atoms have thei mass and positive chage concentated in almost pointlike nuclei, to the moe ecent discoveies, on a fa smalle length scale, that potons and neutons ae themselves made up of appaently point-like quaks. Moe geneally, the methods that we have to pobe the popeties of condensed matte systems ely fundamentally on the notion of scatteing. In this section, we will povide a bief intoduction to the concepts and methodology of scatteing theoy. As pepaation fo the quantum mechanical scatteing poblem, let us fist conside the classical poblem. This will allow us to develop (hopefully a evision!) some elementay concepts of scatteing theoy, and to intoduce some notation. In a classical scatteing expeiment, one consides paticles of enegy E = 1 2 mv2 0 (mass m and asymptotic speed v 0), incident upon a taget with a cental potential V (). Fo a epulsive potential, paticles ae scatteed though an angle θ (see figue). The scatteing coss-section, σ, can be infeed fom the numbe of paticles dn scatteed into some element of solid angle, dω, at angle (θ, φ), i.e. fo an incident flux j i (numbe of paticles pe unit time pe unit aea), dn = j i σdω. The total coss-section is then obtained as σ T = dω σ(θ, φ) = π 0 sin θdθ 2π 0 dφ σ(θ, φ). The angle of deflection of the beam depends on the impact paamete, b (see figue ight). We theefoe have that dn = j i bdb dφ = j i σ sin θdθdφ and σ(θ, φ) = b db sin θ dθ. Example: Let us conside then the case of classical Coulomb scatteing fom a epulsive potential V () = κ whee κ> 0. Fom classical physics, we know that the paticle will follow a hypebolic tajectoy with = L 2 mκ(e cos ϕ 1), whee =(, ϕ) paameteises the elative coodinates of the paticle and taget, 1 and e = (1 + 2EL2 κ 2 m )1/2 > 1 denotes the eccenticity. Since the potential is cental, the angula momentum L is conseved and can be fixed asymptotically by the condition L = mv 0 b. To obtain the scatteing angle, θ, we can use the elation above to find the limiting angle, cos ϕ 0 =1/e, whee ϕ 0 =(π θ)/2. We theefoe have tan(θ/2) = cot ϕ 0 = 1 Note that the angle ϕ is distinct fom the azimuthal angle φ associated with the axis of scatteing.

2 14.1. BASICS 158 1/ e 2 1=( mκ2 2EL ) 1/2 = κ 2 2Eb. Then, fom this elation, we obtain the coss-section σ = known as the Ruthefod fomula. b db sin θ dθ = κ2 1 16E 2 sin 4 θ/2, 14.1 Basics Let us now tun to the quantum mechanical poblem of a beam of paticles incident upon a taget. The potential of the taget, V (), might epesent that expeienced by a fast electon stiking an atom, o an α paticle colliding with a nucleus. As in the classical poblem, the basic scenaio involves diecting a steam o flux of paticles, all at the same enegy, at a taget and detect how many paticles ae deflected into a battey of detectos which measue angles of deflection. In pinciple, if we assume that all the in-going paticles ae epesented by wavepackets of the same shape and size, ou challenge is to solve the full time-dependent Schödinge equation fo such a wavepacket, ] i t Ψ(,t)= [ 2 2m 2 + V () Ψ(,t), and find the pobability amplitudes fo out-going waves in diffeent diections at some late time afte scatteing has taken place. Howeve, if the incident beam of paticles is switched on fo times vey long as compaed with the time a paticle would take to coss the inteaction egion, steady-state conditions apply. Moeove, if we assume that the wavepacket has a well-defined enegy (and hence momentum), so it is many wavelengths long, and we may conside it a plane wave. Setting Ψ(,t)=ψ()e iet/, we may theefoe look fo solutions ψ() of the time-independent Schödinge equation, ] Eψ() = [ 2 2m 2 + V () ψ(), subject to the bounday condition that the incoming component of the wavefunction is a plane wave, e ik x. Hee E = p 2 /2m = 2 k 2 /2m denotes the enegy of the incoming paticles while thei flux is given by j = i 2m (ψ ψ ψ ψ )= k m. In the one-dimensional geomety, the impact of a plane wave with the localized taget esulted in a potion of the wave being eflected and a potion tansmitted though the potential egion. Fom enegy consevation, we may deduce that both components of the outgoing scatteed wave ae plane waves with wavevecto ±k, while the influence of the potential ae encoded in the amplitude of the eflected and tansmitted beams, and a potential phase shift. Both amplitudes and phase shifts ae then detemined by solving the timeindependent Schödinge equation subject to the bounday conditions which ensue enegy and flux consevation. In the thee-dimensional system, the phenomenology is simila: In this case, the wavefunction well outside the localized taget egion will involve a supeposition of the incident plane wave and the scatteed (spheical wave), 2 ψ() e ik + f(θ, φ) eik, 2 Hee, by localized, we mean a potential which is sufficiently shot-anged. At this stage, it is not altogethe clea what constaint this implies. But it will tun out that it excludes the Coulomb potential!

3 14.1. BASICS 159 whee the function f(θ, φ) ecods the elative amplitude and phase of the scatteed components along the diection (θ, φ) elative to the incident beam. To place these ideas on a moe fomal footing, conside the following: If we define the diection of the incoming wave k to lie along the z-axis, a plane wave can be ecast in the fom of an incoming and an outgoing spheical wave, e ik = i 2k i l (2l + 1) [ e i(k lπ/2) ei(k lπ/2) ] P l (cos θ), whee P l (cos θ) = ( 4π 2l+1 )1/2 Y l0 (θ) denote the Legende polynomials. If we assume that the potential petubation, V () depends only on the adial coodinate (i.e. that it is spheically symmetic) and that the numbe of paticles ae conseved by the potential (the flux of incoming paticles is matched by the flux of outgoing), 3 when the potential is sufficient shot-anged (deceasing faste than 1/), the scatteing wavefunction takes the asymptotic fom ψ() i 2k i l (2l + 1) [ e i(k lπ/2) S l (k) ei(k lπ/2) ] P l (cos θ), subject to the constaint S l (k) = 1 following fom the consevation of paticle flux (i.e. S l (k) =e 2iδ l(k) ). Physically, the incoming component of the spheical wave is undistubed by the potential while the sepaate components of the outgoing spheical wave ae subject to a set of phase shifts, δ l (k). Recast in the fom of a petubation, the asymptotic fom of the wavefunction can be staightfowadly ewitten as ψ() e ik + f(θ) eik, whee the second component of the wavefunction denotes the change in the outgoing spheical wave due to the potential, and f(θ) = (2l + 1)f l (k)p l (cos θ), (14.1) with the coefficents f l (k) = 1 2ik (S l(k) 1) defining the patial wave scatteing amplitudes. The coesponding asymptotic flux is then given by j = i m Re { [ e ik + f(θ) eik ] ] [e } ik + f(θ) eik. In geneal, an expansion then leads to a fomidible collection of contibuting tems. Howeve, fo most of these contibutions, thee emains an exponential facto, e ±ik(1 cos θ) whee θ denotes the angle between k and. Fo, the small angula integation implied by any physical measuement leads to a fast oscillation of this facto. As a esult, such tems ae stongly suppessed and can be neglected. Retaining only those tems whee the phase cancellation is complete, one obtains, j = k m + k f(θ) 2 m ê 2 + O(1/ 3 ). 3 Note that this assumption is not innocent. In a typical high enegy physics expeiment, the collision enegies ae high enough to lead to paticle poduction.

4 14.2. METHOD OF PARTIAL WAVES 160 The fist tem epesents the incident flux, while the emainde descibes the adial flux of scatteed paticles. In paticula, the numbe of paticles cossing the aea that subtends a solid angle dω at the oigin (the taget) is given by j ê da = k f(θ) 2 m 2 2 dω+ O(1/). Dopping tems of ode 1/, negligible in the asymptotic limit, one thus obtains the diffeential coss-section, the atio of the scatteed flux to the incident flux, dσ = m k j ê da = f(θ) 2 dω, i.e. dσ dω = f(θ) 2. The total coss-section is then given by σ tot = dσ = f(θ) 2 dω. Then, making use of the identity dωp l (cos θ)p l (cos θ) = 4π 2l+1 δ ll, and Eq. (14.1) one obtains (execise) σ tot = 4π k 2 (2l + 1) sin 2 δ l (k). In paticula, noting that P l (1) = 1, fom Eq. (14.1) it follows that Im f(0) = k 4π σ tot, a elation known as the optical theoem Method of patial waves Having established the basic concepts fo the scatteing poblem, we tun now to conside opeationally how the scatteing chaacteistics can be computed. Hee, fo simplicity, we will focus on the popeties of a centally symmetic potential, V (), whee the scatteing wavefunction, ψ() (and indeed that scatteing amplitudes, f(θ)) must be symmetical about the axis of incidence, and hence independent of the azimuthal angle, φ. In this case, the wavefunction can be expanded in a seies of Legende polynomials, ψ(, θ) = R l ()P l (cos θ). Each tem in the seies is known as a patial wave, and is a simultaneous eigenfunction of the angula momentum opeatos ˆL 2 and ˆL z having eigenvalue 2 l(l + 1) and 0 espectively. Following standad spectoscopic notation, l = 0, 1, 2, ae efeed to as s, p, d, waves. The patial wave amplitudes, f l ae detemined by the adial functions, R l (), defined by [ ] l(l + 1) 2 U()+k 2 R l () =0, whee U() =2mV ()/ 2 epesents the effective potential. Example: To develop the patial wave scatteing method, we will conside the poblem of quantum scatteing fom an attactive squae well potential, U() = U 0 θ(r ). In this case, the adial wave equation takes the fom [ ] l(l + 1) 2 + U 0 θ(r )+k 2 R l () =0.

5 14.2. METHOD OF PARTIAL WAVES 161 At high enegies, many channels contibute to the total scatteing amplitude. Howeve, at low enegies, the scatteing is dominated by the s-wave (l = 0) channel. In this case, setting u() = R 0 (), the adial equation takes the simple fom, ( 2 + U 0 θ(r ) +k 2 )u() = 0, with the bounday condition that u(0) = 0. We theefoe obtain the solution { C sin K < R u() = sin(k + δ 0 ) > R, whee K 2 = k 2 + U 0 >k 2. The continuity condition of the wavefunction and its deivative at = R tanslates to the elation K cot(kr)=kcot(kr + δ 0 ). Fom this expession, we obtain the l = 0 phase shift, δ 0 = tan 1 ( k K tan(kr)) kr,4 i.e. tan δ 0 (k) = k tan(kr) K tan(kr) K + k tan(kr) tan(kr), Then, unless tan(kr) = (see below), an expansion at low enegy (small k) obtains δ 0 kr( tan(kr) KR 1), and the l = 0 patial coss-section, σ 0 = 4π k 2 sin2 δ 0 (k) = 4π 1 k cot 2 δ 0 (k) 4π ( ) 2 tan(kr) k 2 δ2 0 =4πR 2 KR 1. Fom this esult, we find that when tan(kr) KR = 1, the scatteing coss-section vanishes. An expansion in small k obtains k cot δ 0 = 1 a k 2 +, whee a 0 = (1 tan γ 1/2 γ )R, with γ = U0 R, defines the scatteing length, and 0 is the effective ange of the inteaction. At low enegies, k 0, the scatteing cosssection, σ 0 =4πa 2 0 is fixed by the scatteing length alone. If γ 1, a 0 is negative. As γ is inceased, when γ = π/2, both a 0 and σ 0 divege thee is said to be a zeo enegy esonance. This condition coesponds to a potential well that is just able to suppot an s-wave bound state. If γ is futhe inceased, a 0 tuns positive as it would be fo an effective epulsive inteaction until γ = π when σ 0 = 0 and the pocess epeats with the appeaance of a second bound state at γ =3π/2, and so on. Moe geneally, the l-th patial coss-section σ l = 4π k 2 (2l + 1) cot 2 δ l (k), takes its maximum value is thee is an enegy at which cot δ l vanishes. If this occus as a esult of δ l (k) inceasing apidly though an odd multiple of π/2, the coss-section exhibits a naow peak as a function of enegy and thee is said to be a esonance. Nea the esonance, cot δ l (k) = E R E Γ(E)/2, whee E R is the esonance enegy. If Γ(E) vaies slowly in enegy, the patial cosssection in the vicinity of the esonance is given by the Beit-Wigne fomula, σ l (E) = 4π k 2 (2l + 1) Γ 2 (E R )/4 (E E R ) 2 +Γ 2 (E R )/4. (14.2) Scatteing wavefunction, u(), fo thee-dimensional squae well potential fo kr =0.1 and γ =1 (top), π/2 (middle) and 2 (bottom). Note that the scatteing length, a 0 changes fom negative to positive as system passes though bound state. Scatteing phase shift fo kr = 0.1 as a function of γ. 4 Moe geneally, choosing the solution to be finite at the oigin, we find that R l () =N l (K)j l (K), < R, whee N l (K) is a nomalization constant. In the exteio egion, the geneal solution can be witten as R l () =B l (k)[j l (k) tan δ l (k)η l (k)]. Continuity of R l and the deivative R l at the bounday, = R, lead to the following expession fo the phase shifts Hee j l(x) = xj l (x) and similaly η l. tan δ l (k) = kj l(kr)j l (KR) Kj l (kr)j l(kr) kη l (kr)j l(kr) Kη l (kr)j l (KR).

6 14.3. THE BORN APPROXIMATION 162 Execise. Fo a had-coe inteaction, U() =U 0 θ(r ), with U 0, show that at low enegy δ 0 = kr. As k 0, show that the diffeential coss-section is isotopic and given by dσ dω = R2 k 0 and σ tot 4πR 2. Info. Ultacold atomic gases povide a topical aena in which esonant scatteing phenomena ae exploited. In paticula, expeimentalists make use of Feshbach esonance phenomena to tune the effective inteaction between atoms. This tunability aises fom the coupling of fee unbound atoms to a molecula state in which the atoms ae tightly bound. The close this molecula level lies with espect to the enegy of two fee atoms, the stonge the inteaction between them. In the example on the left, the two fee atoms ae both spin up, wheeas the molecula state is a singlet, in which the atoms have opposite spin, adding up to zeo total magnetic moment. Thus, a magnetic field shifts the enegies of two fee atoms elative to the molecula state and theeby contols the inteatomic inteaction stength. The inteaction between two atoms can be descibed by the scatteing length, shown ight as a function of magnetic field close to a Feshbach esonance. On the side whee the scatteing length is positive, the molecula enegy level is lowe in enegy than the enegy of two unbound atoms. The molecula state is thus eal and stable, and atoms tend to fom molecules. If those atoms ae femions, the esulting molecule is a boson. A gas of these molecules can thus undego Bose-Einstein condensation (BEC). This side of the esonance is theefoe called BEC-side. On the side of the esonance whee the scatteing length is negative, isolated molecules ae unstable. Nevetheless, when suounded by the medium of othes, two femions can still fom a loosely bound pai, whose size can become compaable to o even lage than the aveage distance between paticles. A Bose-Einstein condensate of these fagile pais is called a BCS-state, afte Badeen, Coope and Schieffe. This is what occus in supeconductos, in which cuent flows without esistance thanks to a condensate of electon pais ( Coope pais ) The Bon appoximation The patial wave expansion is tailoed to the consideation of low-enegy scatteing pocesses. At highe enegies, when many patial waves contibute, the expansion is not vey convenient and it is helpful to develop a diffeent methodology. By developing a geneal expansion of the scatteing wavefunction, ψ k (), in tems of the Geen function of the scatteing potential one may show that, ψ k () =e ik 1 4π d 3 eik U( )ψ k ( ). (14.3) Hee the subscipt k eminds us that the solution is fo a paticula incoming plane wave. This integal epesentation of the scatteing wavefunction, known as the Lippmann-Schwinge equation, povides a moe useful basis to addess situations whee the enegy of the incoming paticles is lage and the scatteing potential is weak. The elements of the deivation of this equation ae summaised in the info box below: Info. Lippmann-Schwinge equation: Fo the time-independent Schödinge

7 14.3. THE BORN APPROXIMATION 163 equation ( 2 + k 2 )ψ() =U()ψ(), the geneal solution can be witten fomally as ψ() =φ()+ d 3 G 0 (, )U( )ψ( ), whee φ() is a solution of the homogeneous (fee paticle) Schödinge equation, ( 2 + k 2 )φ() = 0, and G 0 (, ) is a Geen function of the Laplace opeato, ( 2 + k 2 )G 0 (, )=δ 3 ( ). Fom the asymptotic behaviou of the bounday condition, it is evident that φ() =e ik. In the Fouie basis, the Geen function is diagonal and given by G 0 (k, k ) = (2π) 3 δ 3 (k k ) 1 k. Tansfomed back into eal space, we 2 have G 0 (, )= 1 e ik 4π. Substituted back into the expession fo the scatteing wavefunction, we obtain the Lippmann-Schwinge equation (14.3). In the fa-field egion, ˆ +, i.e. e ik eik e ik, whee the vecto k = kê is oiented along the diection of the scatteed paticle. We theefoe find that the scatteing wavefunction ψ k () =e ik + f(θ, φ) eik can be expessed in integal fom, with the scatteing amplitude given by f(θ, φ) = 1 4π φ k U ψ k 1 4π d 3 e ik U( )ψ k ( ). (14.4) The coesponding diffeential coss-section can then be expessed as dσ dω = f 2 = m2 (2π) 2 4 T k,k 2, whee, cast in tems of the oiginal scatteing potential, V () = 2 U()/2m, T k,k = φ k V ψ k denotes the tansition matix element. Eq. (14.3) povides a natual means to expand the scatteing wavefunction in powes of the inteaction potential. At zeoth ode in V, the scatteing wavefunction is specified by the unpetubed incident plane wave, φ (0) k () = φ k (). Using this appoximation, Eq. (14.3) leads to the fist ode coection, ψ (1) k () =φ k()+ d 3 G 0 (, )U( )ψ (0) k ( ). Fom this equation, we can use (14.3) to obtain the next tem in the seies, ψ (2) k () =φ k()+ d 3 G 0 (, )U( )ψ (1) k ( ), and so on, i.e. f = 1 4π φ k U + UG 0U + UG 0 UG 0 U + φ k. Physically, an incoming paticle undegoes a sequence of multiple scatteing events fom the potential (see schematic on the ight). This seies expansion is

8 14.4. INFO: SCATTERING OF IDENTICAL PARTICLES 164 known as the Bon seies, and the leading tem in known as the fist Bon appoximation to the scatteing amplitude, f Bon = 1 4π φ k U φ k. (14.5) Setting = k k, whee denotes the momentum tansfe, the Bon scatteing amplitude fo a cental potential is given by (execise) f Bon ( ) = 1 d 3 e i U() = d sin( ) 4π U(), whee, noting that k = k, = 2k sin(θ/2). Coulomb scatteing: Due to the long ange natue of the Coulomb scatteing potential, the bounday condition on the scatteing wavefunction does not apply. We can, howeve, addess the poblem by woking with the e sceened (Yukawa) potential, U() =U /α 0, and taking α. Fo this potential, one may show that (execise) f Bon = U 0 /(α ). Theefoe, fo α, we obtain σ(θ) = f(θ) 2 = which is just the Ruthefod fomula. 0 U k 4 sin 4 θ/2, Info. Peviously, we have used time-dependent petubation theoy to develop an expession fo the tansition ate between states. In the leading ode of petubation theoy, we found that the tansition ate between states and i and f induced by a potential V is given by Femi s Golden ule, Γ i f = 2π f V i 2 δ(e (E f E i )). In a thee-dimensional scatteing poblem, we should conside the initial state as a plane wave state of wavevecto k and the final state as the continuum of states with wavevectos k. In this case, the total tansition (o scatteing) ate into a fixed solid angle, dω, is given by Γ k k = k dω 2π k V k 2 δ(e (E k E k )) = 2π k V k 2 g(e), whee g(e) = dn de denotes the density of states and both states k and k have enegy E = 2 k 2 /2m = 2 k 2 /2m they ae said to be on-shell. As a esult, we obtain the density of states g(e) = dn dk dk de = k2 dω m (2π/L) 3 2 k while the incident flux pe unit volume is given by k/ml 3. As a esult, we obtain the scatteing coss-section, dσ dω = Γ k k k/ml 3 dσ dω = 1 (4π) 2 k 2mV 2 k 2. We can theefoe ecognize that Femi s Golden ule is equivalent to the fist ode Bon appoximation Info: Scatteing of identical paticles Until now, we have assumed that the paticles involved in the scatteing pocess, the incoming paticle and the taget, ae distinguishable. Howeve, vey often we ae inteested in the scatteing of identical quantum paticles. In such cases, we

9 14.5. SCATTERING BY AN ATOMIC LATTICE 165 have to conside the influence of quantum statistics on the scatteing pocess. As a peliminay execise, conside the classical pictue of scatteing between two identical positively chaged paticles, e.g. α-paticles viewed in the cente of mass fame. If an outgoing α paticle is detected at an angle θ to the path of the ingoing α-paticle, it could be (a) deflected though an angle θ, o (b) deflected though π θ. Classically, we could tell which one it was by watching the collision as it happened, and keeping tack. Howeve, in a quantum mechanical scatteing pocess, we cannot keep tack of the paticles unless we bombad them with photons having wavelength substantially less than the distance of closest appoach. This is just like detecting an electon at a paticula place when thee ae two electons in a one dimensional box: the pobability amplitude fo finding an α paticle coming out at angle θ to the ingoing diection of one of them is the sum of the amplitudes (not the sum of the pobabilities!) fo scatteing though θ and π θ. Witing the asymptotic scatteing wavefunction in the standad fom fo scatteing fom a fixed taget, ψ() e ikz + f(θ) eik, the two-paticle wavefunction in the cente of mass fame, in tems of the elative coodinate, is given by symmetizing: ψ() e ikz + e ikz +(f(θ)+f(π θ)) eik. How does the paticle symmety affect the actual scatteing ate at an angle θ? If the paticles wee distinguishable, the diffeential coss section would be ( dσ dω ) dist. = f(θ) 2 + f(π θ) 2, but quantum mechanically we must compute, ( ) dσ = f(θ)+f(π θ) 2. dω indist. This makes a big diffeence! Fo example, fo scatteing though 90 o, whee f(θ) = f(π θ), the quantum mechanical scatteing ate is twice the classical (distinguishable) pediction. Futhemoe, if we make the standad expansion of the scatteing amplitude f(θ) in tems of patial waves, f(θ) = (2l + 1)a lp l (cos θ), then f(θ)+f(π θ) = (2l + 1)a l (P l (cos θ)+p l (cos(π θ))). Since P l ( x) = ( 1) l P l (x), the scatteing only takes place in even patial wave states. This is the same thing as saying that the oveall wavefunction of two identical bosons is symmetic. So, if they ae in an eigenstate of total angula momentum, fom P l ( x) =( 1) l P l (x) it has to be a state of even l. Fo femions in an antisymmetic spin state, such as poton-poton scatteing with the two poton spins foming a singlet, the spatial wavefunction is symmetic, and the agument is the same as fo the boson case above. Fo paallel spin potons, howeve, the spatial wavefunction has to be antisymmetic, and the scatteing amplitude will then be f(θ) f(π θ). In this case thee is zeo scatteing at 90 o! Note that fo (non-elativistic) equal mass paticles, the scatteing angle in the cente of mass fame is twice the scatteing angle in the fixed taget (lab) fame Scatteing by an atomic lattice Finally, to close this section, let us say a few wods about scatteing phenomena in solid state systems. If we ignoe spin degees of feedom, so that we do not have to woy whethe an electon does o does not flip its spin duing the scatteing pocess, then at low enegies, the scatteing amplitude of paticles fom a cystal f(θ) becomes independent of angle (s-wave). In this case, the solution of the Schödinge equation by a single atom i located at a point R i has the asymptotic fom, ψ() =e ik ( R i) + f eik R i R i.

10 14.5. SCATTERING BY AN ATOMIC LATTICE 166 Now, since k R i = k ( 2 2 R i + R 2 i ) 1/2 k (1 2 R ) 1/2 i 2 k kê R i, and kê = k, we have ψ() =e ikr i [e ik + fe i(k k) R i e ik ]. As a esult, we can deduce the effective scatteing amplitude, f(θ) =fe i R i, = k k. If we conside scatteing fom a cystal lattice, we must sum ove all atoms. In this case, the total diffeential scatteing coss-section is given by dσ dω = f R i e i R i 2. X-ay diffaction patten of a quasi-cystal. In the case of a peiodic cystal, the sum ove atoms tanslates to the Bagg condition, dσ (2π)3 = f 2 dω L 3 δ(3) (k k 2πn/L), whee L epesents the size of the (cubic) lattice, and n denote a vecto of integes the Mille indices of the Bagg planes. We theefoe expect that the diffeential coss-section is vey small expect when k k =2πn/L. These elations can be genealised staightfowadly to addess moe complicated cystal stuctues.