arXiv:1803.06198v1 [cond-mat.soft] 16 Mar 2018 safruafrteLro rqec ffe nwihthe which the in via result offset enters main frequency microarchitecture The Larmor magnetic paper. the medium’s this for of formula mat- end a white the is of at models discussed realistic is more ter to way A ceptibility. aey h nlsosaeNRivsbeadhv all have and NMR-invisible same are way.the possible inclusions simplest the the in Namely, treated are properties sions’ nti ok eeo hoyfrtecs ffs wa- in- fast magnetized of case outside the space of for connected clusions theory in a diffusion develop ter I work, this In application. [ biological fraction for volume factor low confounding their recognized serious of been limit has the inclusions as with different shift of [ frequency contribution microarchitecture the tissue of for aban- calculation account been direct has in for construction cylinder, doned cavity and Lorentz sphere of the in- shapes general, for simplest works the this of microscop- While true clusions shift. the mimic frequency specific to calculated selected a ically is with shape construction the shape; cavity Lorentz [ the offset involves frequency Larmor contribute the types to different of inclusions netized construction sphere Lorentz classical to [ complex the by too be ar- described might be magnetic which microscopic the medium, the the NMR- of of to of chitecture contribution linked frequency is the precession spins that the reporting in understood environment is local it Fig. date, water, To of mag- that a from an fol- with different susceptibility what inclusions of netic in microscopic consisting water numerous medium as and a to lows) (referred in phase liquid acquired measurable NMR-reporting the signal of calculate NMR point to physics of is the challenge From the of view, dis- its context. focus from the biomedical the decoupled has original for In problem account this myelin. discussion, the of intensive susceptibility with partic- magnetic matter in tinct me- white properties, in brain magnetic shift ular frequency heterogeneous Larmor with the dia calculating in terest ri thg antcfil [ human field the magnetic in high phase signal at the brain of measurement Precise 4 , 5 ;acrigt h urnl vial hoy mag- theory, available currently the to according ]; position-independent rirr est n geometry and density arbitrary M eie sd rmtecnetvt ftepr,tematr the pore, the of exp connectivity when the from field o Aside magnetic device. case microscopic NMR the aniso heterogeneous location-independent for a consisti distinct matrix induces media a NMR-invisible porous has material an in matrix found within is pore shift connected frequency Larmor The 1 eia hsc,Dp.o ailg,Fclyo eiie U Medicine, of Faculty Radiology, of Dept. 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(darker its contribut ity region to spher- far proportional The a field and image). a the far in a the exaggerated first into greatly The region, divided blue). near be (light can ical fluid sample surrounding The diff the susceptibility equality: as of magnetic shown that a from is with ent spin magnetization dots) NMR-reporting heterogeneous (black an structure The calculating inclusions of position for magnetic arrow). the magnetization microscopic (red at its a field magnetic of with the decomposition sample the A and 1. FIG. medium the representing ( parameter Eq. relevant microarchitecture, single a is case [ the susceptibility for magnetic result isotropic recent of the generalizes formula this low; n h eocpcshr.Tefil rmwti h sphere, the within from field The B sphere. mesoscopic the and meso rpcmgei ucpiiiytno that tensor susceptibility magnetic tropic nE.( Eq. in 9 1 ]. sdt h togmi edo an of field main strong the to osed go M-eotn udfiln a filling fluid NMR-reporting of ng xgoer sarbitrary. is geometry ix atdffso ntefli.The fluid. the in diffusion fast f iest fFebr,Germany Freiburg, of niversity = 6 ,nest ecluae ie h medium the given calculated be to needs ), fmgeie nlsos q ( Eq. inclusions, magnetized of eocpcsphere mesoscopic 18 eo.Ti euti applied is result This below. ) = 11 .I h otgen- most the In ]. teshr ieis size sphere (the B macro 9 , 12 + nE.( Eq. in , , 13 15 ,there ], be- ) 6 er- il- es al ), s 2

2. SETTING THE SCENE because the condition of averaging over the molecular scale does not fulfill for the closest environment of the spin. Following the original idea of Lorentz [18], the effect The above described medium belongs to the class of of this environment should be considered with account for porous media and the corresponding terminology is used its microscopic structure and dynamics. In isotropic liq- hereafter. Consider a macroscopic sample that consists uids such as water, this leads to the zero on average field of a water-filled connected pore and an NMR-invisible from the nearest environment, which is taken effectively matrix occupying the fraction ζ of the sample volume. into account by using the macroscopic Eq. (2) with an r The matrix is described by an indicator function v( ) infinitesimal spherical cavity, the Lorentz sphere [16, 17]. that is unity inside the matrix and zero in the pore. The matrix material is characterised by anisotropic magnetic In media with microscopic structure, many orders susceptibility described by a tensor χab, where a, b . . . of magnitude coarser than the molecular dimensions, label the three spatial components of vectors and ten- Eq.(2) is valid on the microscopic scale. However, the sors. To simplify equations, the magnetic susceptibility effect of heterogeneous field should be specifically aver- is considered relative to water, which means that any aged to yield the overall Larmor frequency from the whole measured Larmor frequency implies subtraction of the macroscopic sample (or an MRI voxel). This averaging frequency measured in a sample of the same shape, but occurs on the scale much coarser than the microscopic without the matrix. The matrix material is considered to one, but much finer than the macroscopic sample size. mesoscopic be non-ferromagnetic, which means that χab 1. This Such a scale is called in physics. Performing condition enable finding the macroscopic| matrix| ≪ magne- the necessary averaging on this scale is the overarching tization as goal of the present study. This is achieved by replicating the original idea of Lorentz Ma = χabB0,b , (1) on the mesoscopic scale [19, 20]: The whole sample is subdivided in a near and a far regions relative to any where B is the b-component of the main field, B . The 0,b 0 considered NMR-reporting spin. While the far region additional macroscopic magnetic field induced by the ma- contributes the field according to the macroscopically trix takes the form averaged medium parameters, the field in the near re- 3 gion should be calculated with account for the medium ∆Ba(r)= d r0 ab(r r0) Mb(r0) , (2) Z Y − microscopic structure and the diffusive motion of water molecules. It is convenient to select the near region in the where the Einstein convention about the summation over form of a sphere, Fig. 1. This sphere is called the meso- repeated indices is used. ab is the elementary dipole scopic sphere in what follows. Its size is much smaller Y field, than the sample dimensions, but large enough to enable a smooth transition from the local environment of a given 3ˆrarˆb δab ab(r)= − , (3) proton to the macroscopically averaged medium prop- Y r3 erties. In other words, the sphere size is large enough to include a statistically representative portion of the where the hat denotes unit vectors,r ˆa = ra/r and δab is the Kronecker delta, which is unity for a = b and zero medium. In this study, I consider the case of fast dif- fusion, which means that the diffusion length in each otherwise. Since Eq. (3) has the form of a convolution, the field can be efficiently calculated in the Fourier do- direction is very large on the microscopic scale for the typical time of the signal acquisition, still it should be main, well below the size of the mesoscopic sphere. Deviations from this condition for isotropic media are discussed else- ∆Ba(k)= ab(k)Mb(k) . (4) Y where [11]. Note that the same root letters are used throughout this Accordingly to the sample decomposition, the averaged paper for the original and Fourier-transformed quantities; deviation from the main field, ∆B¯ , experienced by a wa- the argument is always given explicitly to avoid confu- ter proton consists of two terms, sion. The Fourier transform of the dipole field, Eq. (3), takes the following form in the cgs system [14, 15]: ∆B¯ = Bmacro + Bmeso , (6) δab kakb ab(k)=4π . (5) Y  3 − k2  where B is the field created by a macroscopically ho- This expression takes into account the field of the Lorentz macro mogeneous sample of the given shape with a small spher- sphere for water molecules, 8πδ /3 [16, 17], which is ab ical cavity in the place of the mesoscopic sphere around discussed in more detail below. the reporting spin and Bmeso is the field of the meso- The time-averaged magnetic field experienced by a water scopic sphere, Fig. 1. Finding the latter is the goal of the proton does not coincide with the value given by Eq. (2) subsequent calculations. 3

3. FREQUENCY OFFSET INSIDE motional narrowing, the reported Larmor frequency, Ω, MESOSCOPIC SPHERE is defined by the spatial averaging of the local frequency, Ω(r), over the pore space, The local Larmor frequency offset within the mesoscopic d3r sphere is given by the longitudinal projection of the sus- Ω= Ω(r) [1 v(r)] . (8) ceptibility induced magnetic field, Z (1 ζ)V − − r meso r Ω( )= γnaBa ( ) , (7) While the integration here is performed over the whole volume of mesoscopic sphere, V , the matrix volume is where n = B /B is the unit vector in the direction of the 0 0 excluded by the indicator function 1 v(r) with the de- main magnetic field. This field varies over the character- nominator, (1 ζ)V , written for the normalization− on the istic length defined by the medium structure. The case pore volume. − of fast diffusion considered here implies that during the measurement the typical spin samples a significant por- The field within the sphere, Ω(r), is found according to tion of the medium. In this case, called the diffusion or Eq.(2) with the magnetization from Eq. (1). This gives

3r Ω d 1 3 = d r0 [1 v(r1)] na ab(r1 r0)χbcnc v(r0) , (9) Ω Z (1 ζ)V − Y − 0 −

where Ω0 = γB0 is the nominal Larmor frequency. The wheret Γ(k) is discontinuous at k = 0; its value at this unity in the brackets gives rise to the integral over r1, point is zero, which is easy to show by integrating Eq. (11) which is proportional to the field induced by a homoge- over r. The final result in the Fourier domain takes the neously magnetized sphere. Since the elementary dipole form field, Eq. (5), takes into account the field of the Lorentz Ω 1 d3k sphere, the result is identically zero. In the remaining k k = 3 na ab( )χbcnc Γ( ) , (14) r r Ω0 −1 ζ Z (2π) Y integral, which is bilinear in v( ), the variable 1 is sub- − stituted with r0 + r, which gives where the elementary dipole field, ab(k), is defined 3r 3r in Eq. (5). Note that the tensor n Yχ n is constant. Ω d d 0 r r r r a bc c = v( 0 + )v( 0)na ab( )χbcnc , (10) Another observation is that the correlation function in Ω0 − Z (1 ζ)V Y − Eq.(15) is the only quantity that depends on the length r r For a fixed , the integration over 0 is performed over the of vector k. The radial integration can be performed r r overlap of two mesoscopic spheres, r0 < R and 0 + < resulting in the expression R, Fig. 2. The overlap volume is large for r of| the order| of magnitude of the medium correlation length, therefore Ω 1 2ˆ ˆ ˆ r r r r = d kna ab(k)χbcnc Γ(k) , (15) the 0-integrated product v( 0 + )v( 0) gives rise to the Ω0 −1 ζ Z Y density-density correlation function of the matrix, − where the hats are written to underscore the exclusive r d 0 2 dependence of all quantities on the orientation of kˆ, the Γ(r)= v(r0 + r)v(r0) ζ , (11) Z V − integration is performed over the full solid angle in k- where ζ appears as the sample mean of v(r). The integral space and from this expression is substituted in Eq. (10), where the ∞ 1 2 ζ2 integrated with (r) gives zero by the same reason Γ(kˆ)= dkk Γ(k) . (16) ab (2π)3 Z as above taking intoY account that the r integration is 0 performed over the sphere of the radius 2R. The final It is important to notice that the medium’s magnetic expression thus takes the form microarchitechture is represented in the observable fre- Ω 1 3 quency shift by only five relevant coefficients. This fol- = d r na ab(r)χbcnc Γ(r) . (12) Ω0 −1 ζ Z Y lows from the observation that the elementary dipole − field, Eq. (5) is a trace-free tensor of the second rank build For further analysis, it is convenient to formulate this of the unit vector kˆ. There is one-to-one correspondence result in terms of the Fourier-transformed quantities. between the components of such tensors and the spherical m ˆ Straightforward transformation of Eq. (11) gives harmonics, Yl (k), of the order ℓ = 2 with explicit for- mulas given in Appendix A. In other words, the expres- k 1 k k Γ( )= v( )v( ) , (13) sion na ab(kˆ)χbcnc in Eq. (15) is a linear combination of V − Y 4

m ˆ spherical harmonics Y2 (k). Therefore, it performs a pro- jection of all terms in the spherical harmonic expansion of Γ(kˆ) onto the subspace with ℓ = 2. By the orthogonality of spherical harmonics, only the projections of the terms with ℓ = 2 are non-zero. Since Γ(kˆ) is real according to Eq. (13), the remaining subspace is five-dimensional 2 spanning the complex-valued coefficients in front of Y2 , 1 0 Y2 and a real-valued one in front of Y2 . Note that for the most conventional case of isotropic media, Γ(kˆ) is m a constant and all coefficients in front of Y2 are zero. The mesoscopic sphere thus does not contribute to the FIG. 2. Illustration of the integration variable change on the average field, in other words, it can be treated as an ex- transition from Eq. (9) to Eq.(10). The original integration tension of the classical molecular Lorentz sphere on the over r0 and r1 spans the sphere centered around the origin mesoscopic scale. (the cross). The substitution r1 = r0 + r shifts this sphere by the vector r, but the original limits on the r1 integration reduces the r0 integration volume to the intersection of the two spheres. This volume is large enough to provide for the 4. THE CASE OF AXIAL SYMMETRY averaging that gives the correlation function, Eq. (11). This condition is violated for large r for which the intersection collapses, but such values are not relevant because Γ(r) is r The axial symmetry of the medium implies that the essentially nonzero only for much smaller than the sphere size. susceptibility tensor χab has the eigenvalues χ⊥, χ⊥, χk when the third direction is selected along the symmetry axis. When conducting experiments with such media, it makes sense to selects samples of cylindrical shape co- The observed frequency shift is found by the addition of axial with the microscopic symmetry axis, which is as- the macroscopic contribution defined by Bmacro for the sumed throughout this section. Such a setup serves as overall cylindrical shape with the macroscopically aver- a (strongly simplified) model of experiments performed aged susceptibility tensor, ζχab (the first term on the with excised nerve segments [9, 12]. The aim of this right-hand side of Eq. (6) and its graphical representa- section is to find the frequency shift for an arbitrary ori- tion in Fig. 1). A straightforward calculation yields entation of the sample relative to the main field. macro The axial symmetry also implies that Γ(kˆ) does not de- Ω 2π 2 = ζ (2χk + χ⊥)cos θ χ⊥ . (19) pend on the azimuthal angle, ϕ, in the selected reference Ω0 3 − frame. Therefore, its only relevant component is propor-   0 tional to Y2 , It is instructive to see how the present general approach Γ(kˆ)= C0Y 0(kˆ)+ ..., (17) results in the identical zero frequency shift for a sample 2 2 made of a bunch of parallel magnetized cylinders with the overall macroscopic cylindrical shape [4]. For such media, where the dotted terms do not contribute to the fre- 0 quency shift. the coefficient C2 can be found for arbitrary structure in the transverse plane (Appendix B), This fact significantly simplifies the calculation of the linear combination n (kˆ)χ n in Eq. (15). In the se- a ab bc c √ lected reference frame,Y the direction of the main field 0 5π C2 = ζ(1 ζ) , (20) can be chosen as n = (0, sin θ, cos θ)† and the product − 4π − na ab(kˆ) can be calculated explicitly using Eq. (5) and Y 0 ˆ Equation (18) with this coefficient identically cancels the omitting all terms that are not contributed by Y2 (k). There are only two terms that should be kept accord- macroscopic contribution, Eq. (19). The same zero field is obvious when the cylinder-shaped mesoscopic cavity is ing to Eqs. (A7) and (A8), which gives na ab(kˆ) = Y used instead of the sphere in this special case [4, 5]. (0, 22 sin θ, 33 cos θ). The remaining factor is simply Y Y † 0 χbcnc = (0,χ⊥ sin θ,χk cos θ) . Taking the product, us- Note that the coefficient C2 is negative in media with ing the explicit form of the dipole field, Eq. (5) and per- elongated microstructure as in the extreme example given 0 forming the integration using the normalization of Y2 on by Eq. (20). In such media, Γ(r) is elongated towards the unity, transform Eq. (15) in the following expression for cigar shape due to stronger correlation along the main the mean field inside the mesoscopic sphere axis. The Fourier transform, Γ(k), is correspondingly closer to a disk-shaped form, which, along with the ex- 2 0 0 Ω 8π C2 2 plicit form of Y2 , Eq.(A3), leads to negative values of = (2χk + χ⊥)cos θ χ⊥ . (18) 0 Ω0 3√5π 1 ζ − C2 . −   5

5. DISCUSSION 5.2. Implications for interpretation of previous experiments

The most general result of this study is the mean fre- quency offset inside the mesoscopic sphere, Eq.(15). The result expressed in Eq.(18) can be considered as a When applied to an experiment, this expression has to simplified model of experiments in which the signal phase be combined with frequency shift for the specific macro- was measured in excised segments of animal optic nerves scopic sample shape calculated according to Eqs. (4) and as a function of the segment orientation relative to the (5). In addition to the magnetic susceptibility tensor, the main field [9, 12]. In both experiments, the frequency medium microarchitecture is represented with only five shift created by the sample in the embedding fluid was parameters due to the isomorphism between the elemen- used to find the overall sample-averaged magnetic sus- tary dipole field and the spherical harmonics of the order ceptibility that was compared with the frequency shift ℓ = 2. inside the nerve. Luo et al. [12] interpreted the discov- ered anisotropy from the microstructural point of view; the sample was considered as consisting of isotropic and cylindrical magnetic susceptibility inclusions, either with 5.1. Relevant medium parameters and the fate of a scalar (isotropic) magnetic susceptibility. As discussed the Lorentz cavity above this is a correct quantitative description for inclu- sions with low volume fraction. Beyond this assumption the decomposition in isotropic and cylindrical inclusion The present results add to the polemic about the usage of should be considered as an effective representation of the the Lorentz cavity in media with nontrivial microarchi- more general result, Eq.(18). tecture [5, 10]. The Lorentz cavity is a simplified way to take into account the near field in Eq. (6) by subtracting Wharton and Bowtell [9] circumvented the lack of the- the field of a cavity of a predefined shape [5], the tradi- oretical description for dense media by finding the mi- tional Lorentz sphere for isotropic media [19, 20] or the crostructural contribution as the difference between the cylinder for media composed with parallel cylindrical ob- measured frequency shift and the traditional approach for the sample with anisotropic magnetic susceptibility. jects [4]. One can hypothesize about an ellipsoid as the 2 interpolating shape. Indeed, the aspect ratios and ori- They described the difference as fR = A sin θ + b, where entation of an ellipsoid are described by five parameters A represented the microstructure contribution and b was that might be mapped on the five relevant parameters due to both the microstructure and the possible chemi- of the medium magnetic microarchitechture. This is not cal exchange [21]. Analysing their framework from the an easy task though and, moreover, it makes little sense: present point of view, the traditional approach coincides When the near field is found according to Eq. (15), the with Eq. (19) because it takes into account the sphere problem is already solved in the non-simplified manner. of Lorentz. Identifying the sample-averaged components The Lorentz cavity construction is thus abandoned for of the susceptibility tensor, ζχk and ζχ⊥, with χI + χA and χI χA/2 from Ref. [9], respectively, results in the the direct calculation of the magnetic field in agreement − with the recent literature [6–9]. following expression for the empirical coefficient A: To my opinion, the most advanced, but still traditional 0 usage of the Lorentz cavity appears in the calculation of 4π C2 χA the frequency shift in mixtures of isotropic and long par- A = χI + γB0 (21) −√5 ζ(1 ζ)  2  allel magnetized inclusions with the overall low volume − fraction, ζ 1, [6–8]. In these calculations, either inclu- sion type is≪ assigned the own Lorentz cavity, a spherical in the cgs system; in SI, the factor 4π is absorbed in and a cylindrical one, respectively. This is justified by the the correspondingly larger numerical values of magnetic property of the correlation function to be proportional to susceptibilities. the sum of correlation functions of individual inclusions The numerical value of this coefficient as found by Whar- when ζ 1. Otherwise, the cross-correlations should be ton and Bowtell results in an essential contribution to the taken into≪ account, which breaks the additivity. frequency shift. As they showed by simulations in a re- In axially symmetric media, the microarchitecture con- alistic phantom obtained by translating properties of the tributes the single parameter in the frequency shift, the optic nerve to the whole human brain, both the quantita- 0 coefficient C2 in Eq. (18). Further parameters of inter- tive susceptibility mapping (QSM, see Ref. [22] and refer- est are the axial and transverse magnetic susceptibili- ences therein) and the susceptibility tensor imaging (STI, ties, χk and χ⊥. According to Eq. (18), all three param- see Ref. [23] and references therein) cannot quantify the eters cannot be found from measuring the orientation white matter microstructure until the microstructural ef- dependence of the frequency shift. The only available fects are taken into account. This account, however, re- are the products of the magnetic susceptibilities with the quires performing at least diffusion tensor imaging for the 0 microarchitecture-defined coefficient C2 . determination of local fiber configuration [9]. 6

5.3. Towards realistic model of white matter APPENDICES

Applications of the obtained results to brain white mat- Appendix A: Spherical harmonics of order ℓ = 2 vs. ter is hindered by essential simplification of the present second-rank symmetric trace-free tensor model. The main one is the location independence of magnetic susceptibility tensor, χ . This allowed the fac- ab The second-rank symmetric trace-free tensor is the struc- torization of χab with the remaining terms giving the r ture appearing in particular in Eq. (5). Consider a pure correlation function, Γ, of the structure, v( ); for n the present discussion the notation can be specified as unit three-dimensional vector, , that can be spec- ified via its Cartesian components, na or the two Γvv. In white matter, the susceptibility tensor follows the orientation of myelinated axons [6–9, 24–26], which angles of the spherical co-ordinates, (nx,ny,nz) = are known to have notable orientation dispersion [27– (sin θ cos ϕ, sin θ sin ϕ, cos θ). Using this relation, it is 30]. Extending the present approach to this case re- straightforward to express the spherical harmonics of the quires working with the structure-susceptibility correla- second order, ℓ = 2, in terms of na: tion function, Γ . While Γ can be found using, e.g. vχ vv 1 15 electron microscopy [31, 32] finding Γvχ is more difficult −2 2 Y2 = (nx iny) (A1) because of the need to assign each point a local magnetic 4r2π − susceptibility, which is invisible in histological images. −1 1 15 Y2 = (nx iny)nz (A2) Another problem is the multicompartment structure of 2r2π − white matter. At least two compartment, the intra- 1 5 Y 0 = (2n2 n2 n2) (A3) axonal and extra-axonal water should be taken into ac- 2 4rπ z − x − y count in any measurement. This is possible in principle, 1 15 but requires further structuring of the correlation func- Y 1 = (n + in )n (A4) 2 r x y z tions in intra- and cross-compartment contributions. It −2 2π is also worth to note the nontrivial effect of the radially 1 15 Y 2 = (n + in )2 (A5) oriented local magnetic susceptibility of myelin sheets 2 4r2π x y [6, 8, 24]. Solving this system gives the inverse transformation, The above problems are not unsolvable, but they require further work to create an adequate theoretical descrip- 1 2 π 2π tion of the phase contrast in brain white matter and n2 = Y 0 + Y 2 + Y −2 (A6) x − 3 −3r 5 2 r 15 2 2 perhaps other anisotropic tissues. Even the oversimpli-
fied example considered here shows that the account for 1 2 π 2π n2 = Y 0 Y 2 + Y −2 (A7) microstructure should be essentially more detailed than y − 3 −3r 5 2 − r 15 2 2
the simple subtraction of the field of the Lorentz sphere. 1 4 π The account of microstructure is feasible in terms of n2 = Y 0 (A8) z − 3 3r 5 2 microstructural correlation functions, which are broadly used in physics to describe the structure of disordered 2π 2 −2 nxny = i Y Y (A9) media. − r 15 2 − 2
2π n n = Y 1 + Y −1 (A10) x z r 15 2 2
2π 1 −1 nynz = i Y Y (A11) − r 15 2 − 2
The quantities on the left-hand sides define all compo- nents of the tensor nanb δab/3. −

0 Appendix B: Coefficient C2 for the case of parallel cylinders

ACKNOWLEDGMENTS The correlation function in this case does not depend on the third co-ordinate, which implies the following form I am grateful to Dmitry S. Novikov for fruitful discus- in the Fourier domain: sions. This work was partially supported by the German (2d) Research Foundation (DFG), grant KI 1089/6-1. Γ(k)=2πδ(k3)Γ (k1, k2) , (B1) 7

(2d) where δ(kz) is the Dirac delta-function and Γ the two- This results in Eq. (20). dimensional correlation function in the transverse cross- Substitution of this coefficient in the general expression, section of the sample. Eq.(18), results in the following contribution of the meso- 0 It is now straightforward to calculate the coefficient C2 scopic sphere to the frequency shift in Eq. (18),

2 2ˆ 0 dkk d k (2d) 0 C = 2πδ(k3)Γ (k1, k2)Y (kˆ) . (B2) 2 Z (2π)3 2

The three-dimensional integration is restored in this ex- Ω 2π 2 = ζ (2χ + χ⊥)cos θ χ⊥ . (B4) pression. It is therefore convenient to substitute Y 0(kˆ) Ω − 3 k − 2 0   with its form in terms of the Cartesian components of kˆ, Eq.(A3), and set k3 = 0 due to the presence of δ(k3). The two-dimensional correlation function is integrated in the transverse plane according to its relation to the variance of the indicator function and the property v(r)2 = v(r): Since this is exactly opposite to the macroscopic contri- d2k bution, Eq. (19), the whole cylindrical sample does not Γ(2d)(k) = Γ(2d)(r =0) = ζ(1 ζ) . (B3) result in any frequency shift. Z (2π)2 −

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