Analytical results on Casimir forces for conductors with
edges and tips
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Maghrebi, M. F. et al. “Analytical results on Casimir forces for
conductors with edges and tips.” Proceedings of the National
Academy of Sciences 108 (2011): 6867-6871. Web. 9 Nov.
2011. © 2011 National Academy of Sciences
As Published
http://dx.doi.org/10.1073/pnas.1018079108
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National Academy of Sciences
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Thu May 26 04:49:00 EDT 2016
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Detailed Terms
Analytical results on Casimir forces for
conductors with edges and tips
Mohammad F. Maghrebia,1, Sahand Jamal Rahia,b, Thorsten Emigc, Noah Grahamd, Robert L. Jaffea, and Mehran Kardara
a
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139; bCenter for Studies in Physics and Biology, The Rockefeller
University, 1230 York Street, New York, NY 10065; cLaboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, 91405 Orsay, France;
and dDepartment of Physics, Middlebury College, Middlebury, VT 05753
Casimir forces between conductors at the submicron scale are paramount to the design and operation of microelectromechanical
devices. However, these forces depend nontrivially on geometry,
and existing analytical formulae and approximations cannot deal
with realistic micromachinery components with sharp edges and
tips. Here, we employ a novel approach to electromagnetic scattering, appropriate to perfect conductors with sharp edges and tips,
specifically wedges and cones. The Casimir interaction of these
objects with a metal plate (and among themselves) is then computed systematically by a multiple-scattering series. For the wedge,
we obtain analytical expressions for the interaction with a plate,
as functions of opening angle and tilt, which should provide a
particularly useful tool for the design of microelectromechanical
devices. Our result for the Casimir interactions between conducting
cones and plates applies directly to the force on the tip of a scanning tunneling probe. We find an unexpectedly large temperature
dependence of the force in the cone tip which is of immediate
relevance to experiments.
fluctuations ∣ quantum electrodynamics
T
he inherent appeal of the Casimir force as a macroscopic
manifestation of quantum “zero-point” fluctuations has inspired many studies over the decades that followed its discovery
(1). Casimir’s original result (2) for the force between perfectly
reflecting mirrors separated by vacuum was quickly extended to
include slabs of material with specified (frequency-dependent)
dielectric response (3). Quantitative experimental confirmation,
however, had to await the advent of high-precision scanning
probes in the 1990s (4–7). Recent studies have aimed to reduce
or reverse the attractive Casimir force for practical applications
in micron-sized mechanical machines, where Casimir forces
may cause components to stick and the machine to fail. In the
presence of an intervening fluid, experiments have indeed
observed repulsion due to quantum (8) or critical thermal (9)
fluctuations. Metamaterials, fabricated designs of microcircuitry,
have also been proposed as candidates for Casimir repulsion
across vacuum (10).
Although there have been many studies of the role of materials
(dielectrics, conductors, etc.), the treatment of shapes and geometry has remained comparatively less investigated. Interactions
between nonplanar shapes are typically calculated via the proximity force approximation (PFA), which sums over infinitesimal
segments treated as locally parallel plates (11). This approximation represents a serious limitation because the majority of
experiments measure the force between a sphere and a plate
with precision that is now sufficient to probe deviations from
PFA in this and other geometries (12, 13). Practical applications
are likely to explore geometries further removed from parallel
plates. Several numerical schemes (14, –16), and even an analog
computer (17), have recently been developed for computing
Casimir forces in general geometries. However, analytical formulae for quick and reliable estimates remain highly desirable.
The formalism recently implemented in refs. 18 and 19 enables
systematic computations of electromagnetic (EM) Casimir forces
www.pnas.org/cgi/doi/10.1073/pnas.1018079108
in terms of a multipole expansion. Using these methods, we have
been able to compute forces between various combinations of
planes, spheres, and circular and parabolic cylinders (19–24)
(see also refs. 25–27). Both perfect conductors and dielectrics
have been studied. However, with the notable exception of the
knife edge (23–28), which is a limit of the parabolic cylinder
geometry, systems with sharp edges have not yet been studied
analytically.†
In all the above cases, the object corresponds to a surface of
constant radial coordinate. In this paper, we present results on
the quantum and thermal EM Casimir forces between generically
sharp shapes, such as a wedge and a cone, and a conducting plate.
We accomplish this task by considering surfaces of constant angular coordinate. Although the conceptual step—radial to angular—is simple, the practical computation of scattering properties
is nontrivial, necessitating complex mathematical steps. Furthermore, the inclusion of wedges and cones practically exhausts
shapes for which the EM scattering amplitude can be treated
analytically.‡
In what follows, we provide a summary of the approach and
highlights of our results. The first section introduces the scattering formalism, whose main ingredients are scattering amplitudes
from individual objects. A multiple-scattering series then enables
systematic exploration of Casimir interactions between any arrangements of such shapes. Edges are explored in the section
devoted to wedge geometries, where for perfect conductors the
EM field can be parameterized in terms of two scalar fields. The
corresponding scattering amplitudes in this case were previously
known; their application to compute Casimir forces is developed
here. In particular, we obtain simple analytical expressions for the
force between a knife edge and a plate from the first two terms of
the multiple-scattering series, and verify (by numerical computation of higher order terms) that these expressions provide highly
accurate estimates of the force. The limitations of common
approximation schemes can be illustrated by simple examples
in this geometry. Analogous computations relevant to a sharp
Author contributions: M.F.M., S.J.R., T.E., N.G., R.L.J., and M.K. performed research and
wrote the paper.
The authors declare no conflict of interest.
*This Direct Submission article had a prearranged editor.
†
Knife-edge geometries have been studied for scalar fields obeying Dirichlet boundary
conditions in refs. 29 and 30. Wedges and related shapes have been studied in isolation
(31), but computing interactions among such objects requires full scattering and
conversion matrices, which are synthesized in this paper.
‡
Morse and Feshbach (32) enumerate six coordinate systems in which the vector Helmholtz
equation for EM waves is generically solvable in this way: planar, cylindrical (comprising
circular, elliptic, and parabolic), spherical, and conical. Surfaces on which one such
coordinate is constant are candidates for exact analysis. In addition to the plane and
sphere discussed above, the circular (33) (bottom right of Fig. 1) and parabolic (23)
cylinder have already been studied. These shapes are generically smooth, with the
extreme limit of the parabolic cylinder (a “knife edge”) a notable exception. A survey
of the remaining coordinate systems (32) only leads to shapes such as cylinders and cones
with elliptic cross-section, which are generically similar to their circular counterparts.
1
To whom correspondence should be addressed. E-mail: magrebi@mit.edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/
doi:10.1073/pnas.1018079108/-/DCSupplemental.
PNAS ∣ April 26, 2011 ∣ vol. 108 ∣ no. 17 ∣ 6867–6871
PHYSICS
Edited* by John D. Joannopoulos, Massachusetts Institute of Technology, Cambridge, MA, and approved March 5, 2011 (received for review December 3, 2010)
tip are carried out in the section on the cone. Scattering amplitudes for the cone are obtained via a representation of the EM
Green’s function and sketched briefly in Methods. The expressions for the interaction of a cone and a plate (both perfectly
reflecting) are rather complex in general, but reduce to simple
forms in the limit of small opening angle (approaching a needle).
For the interaction between parallel metal plates, thermal corrections at room temperature are known to be practically negligible
(at micron separations). Although these corrections are small for
the wedge, we find that the room temperature result for the cone
is considerably larger than at zero temperature (by roughly a factor of 2 at micron separations). This difference should prove quite
important to experiments and designs involving sharp tips.
In particular, the prospects for experimental detection of the
force on the tip of an atomic force microscope are discussed in
Conclusion and Outlook. Detailed derivations, supporting formulae, and graphs for each of these sections are provided in
the SI Appendix.
Casimir Forces via the Scattering Formalism
The conceptual foundations of the scattering approach can be
traced back to earlier multiple-scattering formalisms (7, 34–36),
but these were not sufficiently efficient to enable more complicated calculations. The ingredients in our method are depicted in
Fig. 1. The Casimir energy associated with a specific geometry
depends on the way that the objects constrain the EM waves that
can bounce back and forth between them. The dependence on
the properties of the objects is completely encoded in the scattering amplitude or T matrix for EM waves. The T matrices are
indexed in a coordinate basis suitable to each object. Fig. 1 summarizes the T matrices for perfectly reflecting planar, wedge,
and conical geometries. Scattering from a plane mirror gives
T plate ¼ 1 for the two polarizations, irrespective of the wavevec~ Cylindrical ðm;kz Þ and spherical ðℓ;mÞ quantum numbers,
tor k.
along with polarization, label appropriate bases for the cylinder
and sphere. As detailed in the following sections on the wedge
and cone, imaginary angular momenta, labeled by μ for the wedge
and λ for the cone, enumerate the possible scattering waves. The
cone’s scattering waves are also labeled by the real integer m,
corresponding to the z component of angular momentum. The
corresponding T matrices depend on the opening angle θ0 . The
T matrix for the wedge (top right in Fig. 1) is independent of
the axial wavevector kz . In addition to the usual polarizations,
for the cone we needed to introduce an extra “ghost” field
(labeled Gh) and the corresponding T matrix (top left in Fig. 1).
We will also need the matrix U that captures the appropriate
translations and rotations between the scattering bases for each
object. This matrix encodes the objects’ relative positions and
orientations. (The U matrices needed for our calculations are
presented in SI Appendix.) The expression for the Casimir interaction energy,
ℏc
E¼
2π
Z
0
∞
ℏc
dκ tr ln½1 − N ¼ −
2π
Z
0
∞
1
2
dκ tr N þ tr N þ ⋯ ;
2
[1]
involves integration over the imaginary wave number κ, an implicit argument of the above matrices. For an object in front of an
infinite plate, N ¼ UT object U† T plate . An expansion of tr ln½1 − N
in powers of N corresponds to multiple scatterings of quantum
fluctuations of the EM field between the two objects; the trace
operation sums over the full spectrum of scattering channels
(plane waves for example). The presence of both the speed of
light c and Planck’s constant ℏ indicate that this energy is a result
of relativistic quantum electrodynamics. This procedure can be
generalized to multiple objects, with the material properties
and shape of each body encoded in its T matrix. Analytical results
are restricted to objects for which EM scattering can be solved
exactly in a multipole expansion, a familiar problem of mathematical physics with classic applications to radar and optics.
Edges via the Wedge
For a perfectly reflecting wedge, translation symmetry makes it
possible to decompose the EM field into two scalar components:
an E-polarization field that vanishes on its surface (Dirichlet
boundary condition) and an M-polarization field that has vanishing normal derivative (Neumann). In the cylindrical coordinate
system ðr;ϕ;zÞ, a wedge has surfaces of constant ϕ ¼ θ0 .
Whereas for describing scattering from cylinders, a natural
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1Þ
basis is eimϕ H m ði κ 2 þ k2z rÞeikz z with Bessel-H ð1Þ functions
indexed by m ¼ 0;1;2;…, for a wedge we must choose
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1Þ
eμϕ H iμ ði κ 2 þ k2z rÞeikz z with real μ ≥ 0, corresponding to imaginary angular momenta that are no longer quantized (see SI
Appendix, Section I). The T matrices, diagonal in μ, take the
simple forms indicated in Fig. 1. Dimensional analysis indicates
that the interaction energy of a wedge of edge length L at a
separation d from a plate is E ¼ −ðℏcL∕d2 Þf ðθ0 ;ϕ0 Þ, where
f ðθ0 ;ϕ0 Þ is a dimensionless function of the opening angle θ0
and inclination ϕ0 to the plate. This geometry, and the corresponding function f ðθ0 ;ϕ0 Þ, are plotted in Fig. 2, Middle.
The limit θ0 → 0 corresponds to a knife edge which was
previously studied as a limiting form of a parabolic cylinder
(23, 28). The matrix N for the wedge simplifies in this limit,
enabling exact calculation of the first few terms in the expansion
of tr ln½1 − N in Eq. 1,
Fig. 1. Ingredients in the scattering theory approach to EM Casimir forces. See the text for further discussion.
6868 ∣
www.pnas.org/cgi/doi/10.1073/pnas.1018079108
Maghrebi et al.
Fig. 2. The Casimir interaction energy of a wedge at a distance d above
a plane, as a function of its semiopening angle θ0 and tilt ϕ0 . The rescaled
energy as a function of θ0 and ϕ0 is shown in the middle panel. The symmetric
case, ϕ0 ¼ 0, is displayed in the top panel and the interesting case where
the back side of the wedge is hidden from the plane is shown in the bottom
panel. See the text for further discussion.
−
E
sec ϕ0
1
csc3 ϕ0 sec ϕ0 ð2ϕ0 − sin 2ϕ0 Þ
þ
¼
þ
ℏcL∕d2
192π 3
256π 3
16π 2
Tips via the Cone
Computations for a cone—the surface of constant θ in spherical
coordinates ðr;θ;ϕÞ—require a similar passage from spherical
waves labeled by ðℓ;mÞ to complex angular momentum, ℓ → iλ−
1∕2. In this case λ ≥ 0 is real, whereas m remains quantized to
integer values. However, unlike the wedge case, the EM field
can no longer be separated into two scalar parts; the more complicated representation that we report in the SI Appendix,
Section II involves an additional field, similar to the ghost fields
that appear elsewhere in quantum field theory. This representation of EM scattering can also be of use for describing reflection
of ordinary EM waves from cones. Dimensional analysis indicates
that, for a cone poised vertically at a distance d from a plate, the
interaction energy scales as (ℏc∕d) times a function of its opening
angle θ0 . This arrangement and the resulting interaction energy
are depicted in Fig. 3, Left, with the energy scaled by cos2 θ0 to
remove the divergence as the cone opens up to a plate for
θ0 → π∕2. The PFA (11) (depicted by the dashed line) becomes
exact in this limit, but it is progressively worse as θ0 decreases
from π∕2. In particular, this approximation predicts that the energy vanishes linearly as θ0 → 0, although in fact it vanishes as
E∼−
þ ⋯:
[2]
The first square brackets corresponds to tr N (depicted by an orange line in Fig. 2) and the second to tr N2 ∕2; their sum (depicted
by a red line) is in remarkable agreement with the full result (blue
surface). As ϕ0 → π∕2, the knife edge becomes parallel to the
plate and the interaction energy diverges as it becomes proportional to the area rather than L. For parallel plates, we know that
the terms in the multiple-scattering series tr½Nn ∕n are proportional to 1∕n4 (1). Numerically, we find that the convergence
is more rapid for ϕ0 < π∕2, and that the first two terms in
Eq. 2 are accurate to within 1%. Including more than three terms
in the series will not modify the curve at the level of accuracy for
this figure. Casimir’s calculation for parallel plates gives an exact
result at ϕ0 ¼ π∕2, marked with an × in Fig. 2, Middle.
The panels in Fig. 2 display some of the more interesting
aspects of the wedge-plate geometry. In the middle panel, the
energy, rescaled by an overall factor of ℏcL∕d2 and multiplied by
cosðθ0 þ ϕ0 Þ, is plotted versus θ0 and ϕ0 . The factor of cosðθ0 þ
ϕ0 Þ is introduced to remove the divergence as one face of the
wedge becomes parallel to the plate and the energy becomes
proportional to the area rather than just the length of the wedge.
Maghrebi et al.
ℏc ln 4 − 1 1
;
d 16π j ln θ20 j
[3]
where the logarithmic divergence is characteristic of the remnant
line in this limit (33). The EM results shown as the bold blue
curve in Fig. 3 are obtained by including two terms in the series
of Eq. 1, with N constructed from the plate and cone T matrices
in Fig. 1, including the additional ghost field. We have also included the corresponding curves for scalar fields subject to
Dirichlet and Neumann boundary conditions which are depicted
as fine red curves where the top (bottom) one corresponds to the
Dirichlet (Neumann) boundary condition. The limit of θ0 → 0 is
shown in the left panel of Fig. 3 as an orange dashed line. As the
sharp tip is tilted by an angle β, the prefactor ðln 4 − 1Þ∕16π is
replaced by gðβÞ∕ cos β, where gðβÞ can be computed from integrals of trigonometric functions (see SI Appendix, Section II). We
plot this quantity in the right panel of Fig. 3.
Thermal Corrections
For practical applications, the above results have to be corrected
for finite temperature. These are easily incorporated by replacing
the integral in Eq. 1 with a sum over Matsubara frequencies κ n ¼
ð2πkB T∕ℏcÞn (37). For the knife edge, the first (single-scattering)
term in the force resulting from Eq. 2 is modified to
PNAS ∣
April 26, 2011 ∣
vol. 108 ∣
no. 17 ∣
6869
PHYSICS
The blue surface is obtained by numerical evaluation of the
first three terms in the multiple-scattering series of Eq. 1, with
N constructed in terms of the plate and wedge T matrices given
in Fig. 1. The front curve (θ0 ¼ 0) corresponds to a knife edge
as previously mentioned. The top panel depicts the “butterfly”
configuration where the wedge is aligned symmetrically with
respect to the normal to the plate. As the wings open up to a full
plate at θ0 ¼ π∕2, the energy again approaches the classic parallel
plate result, also marked by an ×. Finally, and perhaps most
interestingly from a qualitative point of view, the bottom panel
depicts the case where one wing is fixed at π∕4, and the other
opens up by ψ ¼ 2θ0 . This case displays the sensitivity of the
Casimir energy to the back side of the wedge, which is “hidden”
from the plate. In the proximity force approximation, the energy
is independent of the orientation of the back side of the wedge,
and thus the PFA result (solid line) is constant until the back
surface becomes visible to the plate. The correct result (dotted
line) varies continuously with the opening angle and differs from
the proximity force estimate by nearly a factor of 2, showing that
the effects responsible for the Casimir energy are more subtle
than can be captured by the PFA.
Fig. 3. Casimir interaction of a cone of semiopening angle θ0 a distance d above a plane, scaled by the dimensional factor of ℏc∕d. The left panel corresponds
to a vertical orientation, with the energy multiplied by cos2 θ0 to remove the divergence as the energy becomes proportional to the area for θ0 ¼ π∕2. The right
panel shows the force, F, suitably scaled, for a tilted, sharp cone (θ0 → 0, evocative of an atomic force microscope tip) as a function of tilt angle β and
temperatures T ¼ 300, 80, and 0 K (top to bottom), at a separation of 1 μm. See the text for further discussion.
knife
edge
F EM
¼−
ℏcL 1 1
1 d 4
1
þ
þ
⋯
;
45 λT
8π 2 cos ϕ0 d3
[4]
where λT ¼ 2πkℏcB T is the thermal wavelength. At room temperature
λT ≈ 7 μm, indicating that thermal corrections are insignificant at
separations d ≈ 1 μm. (A similar correction appears in the first
scattering term for parallel plates.)
By contrast, the thermal corrections for a cone are quite significant, and the force from Eq. 3 on a sharp cone is modified at
low temperatures to
F needle
EM
−ℏc
ln 4 − 1
2
2d 0.810
∼
− 2 ln þ 2 þ ⋯ :
d2
3λT λT
λT
16πj ln θ20 j
[5]
More generally, for a sharp cone titled by an angle β, and at a
temperature T, the result of Eq. 3 is multiplied by a (semi)analytic
function gðβ;d∕λT Þ∕ cos β as reported in SI Appendix, Section III.
The function gðβ;d∕λT Þ is an increasing function of both β and the
temperature T. We plot this function for d ¼ 1 μm in Fig. 3.
Interestingly, the room temperature force is more than 100%
higher than the force for T ¼ 0. In contrast, the corresponding
increase for parallel plates at d ¼ 1 μm is only about 0.1%. This
enhanced role of thermal corrections for specific geometries has
been noted before (38, 39) and appears essential to the design of
microelectromechanical (MEM) devices.
Conclusion and Outlook
Construction of electromechanical devices at the micrometer
scale requires mastery of Casimir forces, which depend nontrivially on geometry and shape. The few analytic solutions available
previously apply generically to smooth objects such as cylinders
and spheres. Mechanical devices, however, can optimally manipulate forces by adjusting the alignment and proximity of sharp
components such as tips and edges. In this paper, we have presented a synthesis of EM scattering results from wedges and
cones to compute Casimir forces for perfect conductors with
sharp edges and tips. The interaction of these objects with a metal
plate can then be computed systematically by a multiple-scattering series, which provides an analytical expressions for the interaction of a plate and a knife edge with arbitrary opening and tilt
angles, and the singular behavior of the force on the tip of a sharp
scanning needle.
In practice, the ideal Casimir results have to be corrected due
to imperfect conductivity and finite temperatures. We find that
thermal fluctuations at room temperature are quite significant
6870 ∣
www.pnas.org/cgi/doi/10.1073/pnas.1018079108
for a cone and have to be properly accounted for. Fortuitously,
for a (semiinfinite) needle, a shape-induced, drastic reduction of
magnetic against electric polarizability limits the influence of
imperfect conductivity, which for low frequencies can be easily
understood from the polarizability of a needle (40). In particular,
there is ongoing debate in the literature (41–44) regarding the
use of Drude or Plasma models of conductivity—the former does
not shield low-frequency magnetic modes leading to decreased
Casimir force at high temperatures (24). The magnetic modes
actually penetrate the needle at all temperatures and do not contribute to the Casimir force plotted in Fig. 3, Right. Imperfect
conductivity can also modify the electric modes. However,
although we do not compute forces for a general frequencydependent dielectric response ϵðωÞ, we argue in the SI Appendix,
Section IV that for typical metals (e.g., Au or Al), even at zero
temperature and for needles (the cases where these corrections
are largest), the corrections due to imperfect conductivity are at
most around 5% for d ¼ 0.3 μm.
Thus for separations 0.3 μm ≲ d ≲ 10 μm, relevant to MEM
devices and experiments, the perfect conductor results, with
the important finite temperature corrections, should suffice.
For example, let us consider the tip of an atomic force microscope: At separations, d ≈ 0.3 μm, where the tip may be well
approximated as a metal cone, our results predict a force that
is a fraction of a piconewton. Such forces are at the limit of current sensitivities (45) and will likely become accessible with future
improvements. Current experiments are performed on spheres
of relatively large radius R, where the force is greater by a factor
of (R∕d) (a typical R is 100 μm). A rounded wedge with radius of
curvature
pffiffiffiffiffiffiffiffiffi R falls in an intermediate range, with forces larger (23)
by R∕d than the sharp case. The force can also be enhanced by
using arrays of cones or wedges, at the cost of the difficulty of
maintaining their alignment.
Methods
The primary tool in finding the scattering matrix of an object is the (free)
Green’s function given in a coordinate system appropriate to the object’s
shape. The scalar Green’s function for the wedge and cone geometries were
previously known (46, 47). Although the translation symmetry of a perfect
conducting wedge makes it possible to treat the EM field as a combination
of two scalar problems, the vector nature of the field is unavoidable in the
case of the cone.
We obtain the EM Green’s function for the cone by analytic continuation
of the standard representation in terms of spherical partial waves. Electromagnetism, unlike the scalar field, does not allow monopole fluctuations,
and this constraint introduces an additional subtlety. The full Green’s function becomes
Maghrebi et al.
κ ∞
4π ∑
Z
m¼−∞
×
∞
0
dλλ tanhðλπÞ
1
λ2 þ 1∕4
which have no analog in other representations of the Green’s function. The
total angular momentum of the latter modes is zero, which might appear in
contradiction with the absence of the ℓ ¼ 0 channel in electromagnetism.
Here, however, these modes cancel the unphysical contributions of the other
two modes to this channel. For this reason, we designate them as ghost fields.
This Green’s function is the primary tool for the result in Tips via the Cone.
reg
0
ðMout
iλ−1∕2;m ðrÞMiλ−1∕2;m ðr Þ
reg
0
− Nout
iλ−1∕2;m ðrÞNiλ−1∕2;m ðr ÞÞ
κ
4π
∞
where, in addition to the usual magnetic (transverse electric) modes M and
electric (transverse magnetic) modes N, we also find a set of discrete modes R,
ACKNOWLEDGMENTS. We wish to thank Umar Mohideen for discussions on
the atomic force microscope tip. This work was supported by the National
Science Foundation through Grants PHY08-55426 (to N.G.), DMR-08-03315
(to S.J.R. and M.K.), Defense Advanced Research Projects Agency Contract
S-000354 (to M.F.M., S.J.R., M.K., and T.E.), by the Deutsche Forschungsgemeinschaft through Grant EM70/3 (to T.E.), and by the US Department
of Energy under cooperative research agreement DE-FG02-05ER41360
(to M.F.M., R.L.J.).
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−
∑
0
ΓðjmjÞΓðjmj þ 1ÞRout
0;m ðrÞR0;m ðr Þ;
reg
m¼−∞;m≠0
[6]
Maghrebi et al.
PNAS ∣
April 26, 2011 ∣
vol. 108 ∣
no. 17 ∣
6871
PHYSICS
G0 ðr; r0 ;iκÞ ¼ −