Abstract
Paper 01 showed that in three dimensions the weight per shell is independent of the radius only for exponent 2, and Paper 496 counted that the sum then grows in proportion to the radius. Paper 1094 counted on a map (two dimensions) that an exponent equal to the dimension is the border where a summation stops converging. This paper counts that border in three dimensions, exponent 3. The field of a dipole weakens exactly as the cube of the distance. No new theorem or law is claimed.
First, a shell at exponent 3 contributes 0 with signs and 1/R in absolute value. Summing dipole fields over spherical shells of a lattice, the signed sum is 0, and the absolute sum times the radius goes from 9.3091 to 9.5850, approaching the spherical integral 16π/(3√3) = 9.673597. Summed in absolute value, the total grows by 22.274279 each time the radius is multiplied by 10 (logarithmically) (Section 2).
Second, and this is the core of the paper: with the same lattice and the same dipoles, the sum changes with the shape of the region summed over. Placing dipoles of the same orientation on a simple cubic lattice and summing the field at one point over a spheroid, the result was 0 for a sphere, 4.1488 for a shape 30 times elongated, and -7.8539 for a shape flattened to 1/30. The standard answer 4π(1/3 - D), with D the depolarization factor, gives 0, 4.1455 and -7.7465 (Section 3).
Third, the limits are 0, 4π/3 and -8π/3, the same numbers as the longitudinal and transverse shares of Paper 1084. The limit of a thin needle is 4π/3 = 4.188790205, that of a thin plate is -8π/3, and their difference is the full solid angle 4π. The longitudinal 4π/3 and transverse 8π/3 that Paper 1084 obtained by integrating the field of one source over a sphere also appear in this discrete sum of many sources at one point (Section 4).
Fourth, the +2 and the 3 of the Clausius-Mossotti relation come from the 1/3 decided by the symmetry of the cubic lattice. Cutting a cubic lattice with a sphere makes the nearby sum 0, and the local field becomes the mean field plus 1/3 of the polarization. Changing the shape of the cut, the cavity field and the nearby sum still add up to the same 1/3 (4.0636 to 4.2137 for the five shapes of Section 3, against 4π/3 = 4.1888). That 1/3 makes the 2 and the 3 of (ε - 1)/(ε + 2). At Nα/ε0 = 1, this relation gives ε = 2.5, while ignoring the local field gives 2.0 (Section 5).
Fifth, the separator is the shape of the region summed over. When the exponent is smaller than the dimension the outside decides the sum, and when it is larger the inside decides. At the border, exactly equal to the dimension, neither can decide, and the shape does.
Keywords: dipole, lattice sum, depolarization factor, Clausius-Mossotti relation, uniqueness theorem, solid angle, separator.
Scope (what is not claimed)
No new theorem or law is claimed. The dipole field, the dependence of lattice sums on the shape of summation, the formula for the depolarization factor of a spheroid, the Lorentz local field and the Clausius-Mossotti relation are all standard. What this paper adds is lining them up with the shell ruler of Paper 01 and counting them as the border where the exponent equals the dimension. It does not fit the dielectric constants of real materials. Non-cubic lattices, quantum corrections and the physics of the polarization catastrophe (ferroelectricity) are not treated. The lattice sums are direct sums over lattice points inside finite spheroids; the Ewald method is not used.
1. The question
With the ruler of Paper 01, when a density (or a field strength) goes as $r^{-n}$, the contribution of one shell of radius $R$ is proportional to $4\pi R^{2}\cdot R^{-n}$. For $n=2$ every shell is the same, and the sum grows in proportion to the radius (Paper 496). For $n$ larger than 3 the sum is decided by the inside and converges. Question: what happens at exactly $n=3$ (exponent equal to the dimension in three dimensions)? Paper 1094 counted on a map (two dimensions) that an exponent equal to the dimension is the border where a summation stops converging. This paper counts that border in three dimensions with the field of dipoles.
2. The shell ruler: exponent 3 is the border
The z component of the field of a z-oriented dipole (strength 1, units with $1/(4\pi\varepsilon_{0})=1$) at displacement $\mathbf{r}$ is $(3z^{2}/r^{2}-1)/r^{3}$. On a simple cubic lattice (spacing 1), only the points in the shell from radius $R$ to $R+1$ were summed.
| $R$ | Signed sum | Absolute sum | Absolute sum $\times R$ |
|---|---|---|---|
| 10 | 0.000000 | 0.930915 | 9.3091 |
| 20 | 0.000000 | 0.469876 | 9.3975 |
| 40 | 0.000000 | 0.237471 | 9.4988 |
| 80 | 0.000000 | 0.119813 | 9.5850 |
Observation 2.1 With signs, every shell gives 0 (summing $3\cos^{2}\theta-1$ over a sphere gives 0). In absolute value, each shell contributes as $1/R$, and multiplying by $R$ approaches the spherical integral $16\pi/(3\sqrt3)=$ 9.673597. Summed in absolute value, the total keeps growing by 22.274279 each time the radius is multiplied by 10 (logarithmically). The sum does not stop in absolute value, and with signs it cancels shell by shell. This is what the border looks like, where the answer depends on the order (the shape) of summation.
3. The core: the shape decides the sum
Placing the same z-oriented dipole at every point of the lattice, the z component of the field felt by the dipole at the origin was summed over lattice points inside a spheroid with semi-axes $a,a,c$ (excluding the origin). The polarization is $P=1$ (one dipole per unit volume). The standard answer is $4\pi(1/3-D)$, with $D$ the depolarization factor along z (the same formula Paper 665 used for the shape anisotropy of iron).
| Shape | $a$ | $c$ | Lattice sum | $4\pi(1/3-D)$ | $D$ |
|---|---|---|---|---|---|
| Sphere | 40 | 40 | 0.0000 | 0.0000 | 0.333333 |
| Elongated (10 times) | 18 | 180 | 3.9588 | 3.9339 | 0.020286 |
| Elongated (30 times) | 10 | 300 | 4.1488 | 4.1455 | 0.003444 |
| Flattened (1/10) | 180 | 18 | $-6.7536$ | $-6.6284$ | 0.860804 |
| Flattened (1/30) | 300 | 10 | $-7.8539$ | $-7.7465$ | 0.949778 |
Observation 3.1 With the same lattice, the same dipoles and the same point, summing over a sphere gives 0, an elongated region gives a positive sum, and a flattened region a negative one. For elongated shapes the difference between the lattice sum and the standard answer shrank to 0.0033 at 30 times. For flattened shapes a difference of about 0.1 remained (the plates have only 21 and 37 lattice layers through their thickness; the cause of the difference was not checked).
4. The limits, and the 4π/3 and 8π/3 of Paper 1084
The depolarization factor is $1/3$ for a sphere, 0 in the limit of a thin needle and 1 in the limit of a thin plate (the sum over three directions is always 1). So the limits of the lattice sum are
$$\text{sphere}\ 0,\qquad \text{needle}\ +\frac{4\pi}{3},\qquad \text{plate}\ -\frac{8\pi}{3},$$with $4\pi/3=$ 4.188790205, $8\pi/3=$ 8.377580410, and the difference between needle and plate equal to the full solid angle $4\pi=$ 12.566370614.
Observation 4.1 Paper 1084 counted that integrating the field of one dipole over a sphere gives the longitudinal $4\pi/3$, that the field of a current loop gives the transverse $8\pi/3$, and that the two add up to $4\pi$. The sum in this paper is a discrete sum of many sources at one point, a different quantity from the integral, yet its limits show the same $4\pi/3$ and $8\pi/3$. Both are the shares 1/3 and 2/3 into which the full solid angle $4\pi$ splits, and this paper regards them as the same root.
5. The +2 and the 3 of Clausius-Mossotti
Cutting the cubic lattice with a sphere around the point of interest makes the sum from nearby points inside the sphere 0 (the sphere of Section 3). The local field felt by one molecule then becomes the mean field $E$ plus $1/3$ of the polarization, $E+P/(3\varepsilon_{0})$ (the Lorentz local field). With polarizability $\alpha$ per molecule and number density $N$, solving $P=N\alpha(E+P/(3\varepsilon_{0}))$ gives
$$\frac{\varepsilon-1}{\varepsilon+2}=\frac{N\alpha}{3\varepsilon_{0}}.$$The 2 in the denominator and the 3 on the right both come from the $1/3$ of the local field.
| $N\alpha/\varepsilon_{0}$ | Clausius-Mossotti $\varepsilon$ | Ignoring the local field, $1+N\alpha/\varepsilon_{0}$ | Ratio |
|---|---|---|---|
| 0.1 | 1.103448 | 1.100000 | 1.003135 |
| 0.5 | 1.600000 | 1.500000 | 1.066667 |
| 1.0 | 2.500000 | 2.000000 | 1.250000 |
| 2.0 | 7.000000 | 3.000000 | 2.333333 |
Observation 5.1 In a dilute gas (small $N\alpha/\varepsilon_{0}$) the two are almost equal, but they separate as the medium gets denser, and at $N\alpha/\varepsilon_{0}=3$ the denominator of the relation becomes 0. This $1/3$ is not a number that appeared because a sphere was chosen as the cut.
Observation 5.2 Cut with a cavity of another shape instead of a sphere. The field of the polarization charge on the cavity wall is $4\pi D$ ($P=1$, $D$ the depolarization factor of the cavity shape), and the sum over nearby points inside the cavity is exactly the lattice sum of Section 3. Adding the two approaches $4\pi/3$ for every shape.
| Shape | Lattice sum | Cavity field $4\pi D$ | Total | Difference from $4\pi/3$ |
|---|---|---|---|---|
| Sphere | 0.0000 | 4.1888 | 4.1888 | 0.0000 |
| Elongated (10 times) | 3.9588 | 0.2549 | 4.2137 | 0.0249 |
| Elongated (30 times) | 4.1488 | 0.0433 | 4.1921 | 0.0033 |
| Flattened (1/10) | $-6.7536$ | 10.8172 | 4.0636 | $-0.1252$ |
| Flattened (1/30) | $-7.8539$ | 11.9353 | 4.0813 | $-0.1075$ |
Observation 5.3 The shape of the cut is a step of the calculation and does not change the local field. What decides the $1/3$ is the symmetry of the cubic lattice (all directions are equal, so the nearby sum inside a sphere is 0). In a non-cubic lattice the nearby sum inside a sphere is not 0, and the $1/3$ changes too (not counted in this paper). The difference of about 0.1 left for flattened shapes comes from the same lattice sums as in Section 3.
6. The separator: exponent versus dimension
| Exponent and dimension | Contribution per shell | What decides the sum |
|---|---|---|
| Exponent = dimension - 1 (2 in 3D; Papers 01, 496) | the same for every shell | the outside (sum grows with radius) |
| Exponent = dimension (3 in 3D, this paper; 2 in 2D, Paper 1094) | 0 with signs, $1/R$ in absolute value | the shape of the region |
| Exponent > dimension (4 or more in 3D) | smaller outward | the inside (sum converges) |
Observation 6.1 At the border where the exponent equals the dimension, neither the outside nor the inside can decide the sum, and the shape of the region decides it. The separator is the shape of the region summed over (the depolarization factor $D$). If the $n=2$ of Paper 01 is the flatness of "every shell the same", the next step $n=3$ is the border where "every shell cancels and only the shape remains".
7. What the checks showed
When first checking the absolute contribution of a shell with a trapezoidal integral, the numerical tool at hand lacked the function name used, and the check stopped. The spherical integral closes by hand as $2\pi\cdot 8/(3\sqrt3)$ after substituting $u=\cos\theta$, so both the closed form and a trapezoidal integral under the older function name were printed, and they agreed at 9.673597. The lattice sums were taken layer by layer in z to save memory. For the 2nd edition, Section 5 was re-derived by hand after publication, and the sums with cavities of other shapes (Observation 5.2) were added to the checks.
Closing
Closing: one step beyond the uniqueness theorem, a field with exponent 3 in three dimensions contributes 0 with signs and 1/R in absolute value per shell, so its sum stands on a logarithmic border. Summing the field of dipoles on a simple cubic lattice at one point, the answer is decided by the shape of the region: 0 for a sphere, 4.1488 for a shape 30 times elongated, -7.8539 for a shape flattened to 1/30 (standard answers 0, 4.1455, -7.7465). The limits are 0, 4π/3 and -8π/3, the same numbers as the longitudinal and transverse shares of Paper 1084. The +2 and the 3 of the Clausius-Mossotti relation come from the 1/3 decided by the symmetry of the cubic lattice, and cutting the cavity in any shape returns the same 1/3 (4.0636 to 4.2137 for five shapes, against 4π/3 = 4.1888). The separator is the shape of the region summed over.
Revision Record
2nd Edition (October 2026): Section 5, the fourth item of the abstract, the subtitle and the closing were corrected. The 1st edition treated the $1/3$ of the Clausius-Mossotti relation as "a number that appears because one cuts with a sphere" and wrote that "if the shape were not a sphere, the 1/3 of the local field would change, and so would the 2 and the 3 of the relation"; this was wrong. The shape of the cavity cut out to find the local field is a step of the calculation: adding the cavity field and the nearby sum returns the same $1/3$ for every shape. The 2nd edition adds a table checking this with the five shapes of Section 3 (Observation 5.2) and states that the $1/3$ is decided by the symmetry of the cubic lattice (Observation 5.3). The numbers up to Section 4, and the core that the lattice sum is decided by the shape of the region summed over, are unchanged.
References
- Yamagishi, Y. (2026). The Uniqueness Theorem of n = 2 for Spherically Symmetric Density Distributions [Paper 01]. YOI: https://www.yamagishiyuuki.com/yoi/1; DOI: 10.5281/zenodo.21403186
- Yamagishi, Y. (2026). The 4π Appears Twice, Cancels, and Then the Sum Diverges [Paper 496]. YOI: https://www.yamagishiyuuki.com/yoi/496; DOI: 10.5281/zenodo.22760174
- Yamagishi, Y. (2026). The Same Iron Has No Direction as a Sphere and Thirty-Eight Times the Crystal’s as a Thin Film [Paper 665]. YOI: https://www.yamagishiyuuki.com/yoi/665; DOI: 10.5281/zenodo.23014711
- Yamagishi, Y. (2026). 4π Splits into a Longitudinal 4π/3 and a Transverse 8π/3, and the 21 cm Line of Hydrogen Measures the Transverse Part [Paper 1084]. YOI: https://www.yamagishiyuuki.com/yoi/1084
- Yamagishi, Y. (2026). The “Squared Distance” in Retail Gravity Models Is, on a Map, the Exponent at the Border Where the Total Number of Customers Stops Converging [Paper 1094]. YOI: https://www.yamagishiyuuki.com/yoi/1094
Reference
Yamagishi, Y. (2026). A Lattice Sum of Dipole Fields Is Decided by the Shape of the Summation: 0 for a Sphere, 4π/3 for a Thin Needle, −8π/3 for a Thin Plate [Paper 1097]. YOI: https://www.yamagishiyuuki.com/yoi/1097
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1097. This paper is a preprint and has not been peer reviewed.