Abstract
The formula that fixes how fast an electron gas oscillates (the plasma frequency) and the formula for the force that stops a fast particle in matter (Bethe’s formula) both contain 4π divided by a mass m. Putting in the choice of mass of this corpus, m = 4π, removes π from both formulas. This paper checks whether the two 4π are the same thing, by counting where each comes from. No new theorem or law is claimed.
First, the 4π of the plasma frequency is the solid angle of the field coming out of a sheet of charge. Displacing a slab of electrons slightly makes sheets of charge appear at its ends. The difference between the electric fields on the two sides of a sheet of charge, obtained by numerical integration, is $\mathbf{12.566370614}$ (4π) in three dimensions and $\mathbf{19.739208802}$ (2π²) in four, the surface area of the unit sphere in every dimension. The squared frequency obtained by solving the motion of the slab numerically agreed with 4π to a relative difference of 10⁻⁸.
Second, the 4π of Bethe’s formula is the circumference 2π of a ring multiplied by 2. When a fast particle passes beside an electron, the momentum it gives to the electron is inversely proportional to the distance b, with a coefficient of 2 (the force summed over time). The energy the electron receives is the square of that divided by 2m: 4 divided by 2 leaves 2. The electrons at distance b lie on a ring of radius b, so the circumference 2πb multiplies. 2π × 2 = 4π. In three dimensions the momentum obtained by solving the actual orbit agreed with the exact formula with a ratio of $\mathbf{0.999999500}$.
Third, and this is the core of the paper: when the dimension is changed, the two coefficients become different numbers. The coefficient of the plasma frequency is 4π, 2π², 8π²/3, π³ in three to six dimensions. The coefficient of Bethe’s formula is 4π, π³/2, 16π²/9, 3π⁴/16, numerically $12.566370614$, $15.503138340$, $17.545963380$, $18.264204569$. Their ratio is 1, $\mathbf{0.785398163}$, $\mathbf{0.666666667}$, $\mathbf{0.589048623}$: they are the same number only in three dimensions.
Fourth, that ratio is the fraction of a circumscribed cylinder filled by a ball. The sequence of ratios π/4, 2/3, 3π/16 is the same as the fraction “ball ÷ circumscribed cylinder” counted in Paper 1080 (in two, three and four dimensions). This is because the coefficient of the transferred momentum is the ratio of the surface areas of the unit spheres of neighbouring dimensions. In the three-dimensional Bethe formula this fraction is 1 because the one-dimensional ball (a segment) exactly fills its circumscribed cylinder (the same segment).
Fifth, the collision logarithm appears only in three dimensions. Bethe’s formula carries the logarithm of the ratio of the distances of far and near collisions. Multiplying the upper end of the integral over distance by 10 adds $\mathbf{2.302585}$ (the natural logarithm of 10) in three dimensions, but only $0.009000$ in four and $0.000049$ in five. Only in three dimensions do far collisions count as much as near ones.
To be candid, one check failed once. In solving the orbit, the particle was started at a finite distance with its speed left at 1, and the momentum was off by 0.02 percent. The speed and position at the start were corrected to match the values far away. The mistake was in the measuring, not in the formula.
Keywords: plasma frequency, stopping power, Bethe formula, solid angle, impact parameter, choice of unit, dimension.
Scope (what is not claimed)
No new theorem or law is claimed. Plasma oscillation (Tonks and Langmuir 1929), the estimate of how a particle is stopped by the accumulation of far collisions (Bohr 1913) and its quantum-mechanical formula (Bethe 1930), and the angle of Coulomb scattering (Rutherford 1911) are all known. The inside of the logarithm is not treated. How near and far collisions are cut off inside the logarithm of Bethe’s formula (the quantum-mechanical treatment, the frequencies of bound electrons) is not entered into. Only the coefficient in front is counted. Relativity and temperature are left out. The relativistic corrections for a fast particle and the effect of the temperature of the plasma on the frequency are not treated. No comparison with measurement is made. Measured stopping powers of materials are not used. Setting the mass to 4π is not called a fact about nature. It is a choice of unit. No claim is made that electricity exists in higher dimensions. The force law is only listed against the dimension. Relation to earlier papers: Paper 22 treated the plasma-frequency formula with choices for the mass and the permittivity put in, and in its second edition withdrew the claim about confinement it had attached to that formula. Here we go back to the formula before the choices are put in and count where the 4π comes from. Paper 35 showed that the 4π of the Poisson equation of gravity is a solid angle forced by Gauss’s law. Here the same 4π is counted in electricity as the difference of the fields on the two sides of a sheet of charge. Paper 1082 showed that the 4π in the scattering-length formula is a solid angle that becomes invisible when the mass is set to 4π, and that a formula without the mass keeps its 4π. Here one case is added in which the 4π facing the mass is not a solid angle. Paper 1085 took a census of the 2 in 8π and counted that 8π has readings other than 2×4π. Here it is counted that 4π has the reading 2×2π. Paper 912 treated the family in which the mass is taken as the surface area of the unit sphere in d dimensions. Here it is seen that with that choice π disappears from the plasma frequency in every dimension, while a factor that does not disappear remains in Bethe’s formula. Paper 1080 counted the fractions π/4, 2/3, 3π/16 of a circumscribed cylinder filled by a ball and showed that they are Wallis integrals. Here those fractions appear as the ratio of the two coefficients. What is added: separating the 4π of the two 4π/m into a solid angle and “the 2π of a ring times 2”, changing the dimension to count that the two coefficients are the same number only in three dimensions, finding that their ratio is the fraction of a cylinder filled by a ball, and counting that the collision logarithm appears only in three dimensions.
1. The question
In a gas of electrons (a plasma), displacing the swarm of electrons slightly makes it oscillate about its original position. The angular frequency $\omega_p$ of that oscillation is given by
$$\omega_p^2=\frac{4\pi n e^2}{m}$$(Gaussian units; $n$ is the number density of electrons, $e$ the charge and $m$ the electron mass). On the other hand, when a fast particle of charge $ze$ moves through matter, the energy it loses per unit length is
$$-\frac{dE}{dx}=\frac{4\pi n z^2e^4}{mv^2}\,\ln\Lambda$$($v$ is the speed of the particle and $\ln\Lambda$ the logarithm of the ratio of the distances of far and near collisions). Both contain $4\pi/m$. Putting in the choice of this corpus, $m=4\pi$, gives $\omega_p^2=ne^2$ and $-dE/dx=nz^2e^4\ln\Lambda/v^2$, and $\pi$ disappears.
Are the two $4\pi$ the same thing? Paper 1082 counted that the $4\pi$ in the scattering-length formula is a solid angle coming from Gauss’s theorem. If the same held here the matter would be quickly settled. On checking, it holds for one of them and not for the other.
2. The $4\pi$ of the plasma frequency is the solid angle of the field from a sheet
Let a uniform slab of electrons lie on top of a slab of fixed positive charge. Displacing the electron slab by $\xi$ along its thickness makes electrons stick out at one end and leaves positive charge at the other. Both are sheets of charge with surface density $ne\xi$. The electric field between the two sheets pulls the electrons back.
The electric field of a sheet of charge of surface density $\sigma$ differs by $4\pi\sigma$ between the two sides of the sheet. This $4\pi$ was checked by summing the forces from all the charge on the sheet. With the force law taken as $e^2/r^{d-1}$, the same calculation can be done in $d$ dimensions.
| $d$ | Field on one side $\div\sigma$ | Difference of the two sides $\div\sigma$ | Surface area of the unit sphere $S_{d-1}$ |
|---|---|---|---|
| 3 | $6.283185307$ | $12.566370614$ | $12.566370614$ |
| 4 | $9.869604401$ | $19.739208802$ | $19.739208802$ |
| 5 | $13.159472535$ | $26.318945070$ | $26.318945070$ |
| 6 | $15.503138340$ | $31.006276680$ | $31.006276680$ |
The difference of the two sides is the surface area of the unit sphere in every dimension. Changing the height above the sheet from $0.37$ to $2.9$ did not change the field. The total field going out in all directions from one point charge is $S_{d-1}$, and the sheet sends half of it to each side. It is the solid angle of Gauss’s law, the same one that Paper 35 showed for the Poisson equation of gravity.
The equation of motion of the electrons is $m\ddot\xi=-4\pi ne^2\xi$. Solving it numerically and measuring the period gave a squared angular frequency of $12.566370614$ in units of $ne^2/m$, with a ratio to $4\pi$ of $1.0000000000$. In $d$ dimensions the coefficient is $S_{d-1}$.
3. The $4\pi$ of Bethe’s formula is twice the $2\pi$ of a ring
A fast particle passes beside an electron at rest, at a distance $b$ (the impact parameter). If the particle is fast, the electron hardly moves while it passes, and the particle goes almost straight. The momentum the electron receives is the transverse force summed over time,
$$\Delta p=\frac{ze^2}{vb}\int_{-\infty}^{\infty}\frac{du}{(1+u^2)^{3/2}}=\frac{2ze^2}{vb} .$$Here one 2 appears. The energy the electron receives is $(\Delta p)^2/(2m)=2z^2e^4/(mv^2b^2)$: 4 divided by 2 leaves 2.
The electrons at distance $b$ lie on a ring of radius $b$ about the path of the particle. With $db$ the width of the ring, the particle meets $n\times2\pi b db dx$ electrons while it advances a length $dx$. Multiplying by the energy and summing over $b$ gives
$$-\frac{dE}{dx}=n\int\frac{2z^2e^4}{mv^2b^2}\,2\pi b\,db=\frac{4\pi nz^2e^4}{mv^2}\int\frac{db}{b} .$$The $4\pi$ is the $2\pi$ of the circumference of the ring multiplied by the 2 of the energy. A field spreading in all directions appears nowhere. Summing numerically from $b=0.01$ to 100 gave $115.740550599$, in agreement with $4\pi\ln(b_{\max}/b_{\min})$.
The view that the particle “passes straight by” was checked with actual orbits ($ze^2=1$, $m=1$, $v=1$, a repulsive force).
| $b$ | $\Delta p$ from the orbit | $2ze^2/(vb)$ | Ratio | Exact formula $mv\sin\theta$ |
|---|---|---|---|---|
| 5 | $0.384615192$ | $0.400000000$ | $0.961538$ | $0.384615385$ |
| 20 | $0.099750574$ | $0.100000000$ | $0.997506$ | $0.099750623$ |
| 100 | $0.019997990$ | $0.020000000$ | $0.999900$ | $0.019998000$ |
The values from the orbit agreed with the exact formula in terms of the scattering angle $\theta$ with a ratio of $0.999999500$ in all three cases. $2ze^2/(vb)$ agrees to a ratio of $0.999900$ for a far collision ($b=100$) and only to $0.961538$ for a near one ($b=5$). The coefficient 2 is the value that is right for far collisions.
4. When the dimension is changed, the two coefficients become different numbers
The same calculation is done in $d$ dimensions. The transferred momentum is $(ze^2/(vb^{d-2}))\times I_d$ with $I_d=\int du/(1+u^2)^{d/2}$. The ring becomes a sphere of dimension $d-2$ and radius $b$, of size $S_{d-2}b^{d-2}$. The coefficient of the stopping force is $K_d=S_{d-2}\times I_d^2/2$.
| $d$ | $I_d$ | Ring coefficient $S_{d-2}$ | $K_d$ (stopping) | $S_{d-1}$ (plasma) | Ratio $K_d/S_{d-1}$ |
|---|---|---|---|---|---|
| 3 | $2.000000000$ | $6.283185307$ | $12.566370614$ | $12.566370614$ | $1.000000000$ |
| 4 | $1.570796327$ | $12.566370614$ | $15.503138340$ | $19.739208802$ | $0.785398163$ |
| 5 | $1.333333333$ | $19.739208802$ | $17.545963380$ | $26.318945070$ | $0.666666667$ |
| 6 | $1.178097245$ | $26.318945070$ | $18.264204569$ | $31.006276680$ | $0.589048623$ |
In terms of $\pi$, $I_d$ is 2, $\pi/2$, $4/3$, $3\pi/8$ and $K_d$ is $4\pi$, $\pi^3/2$, $16\pi^2/9$, $3\pi^4/16$. The plasma coefficient $S_{d-1}$ is $4\pi$, $2\pi^2$, $8\pi^2/3$, $\pi^3$. The two are the same number only in three dimensions; in four they separate into $\pi^3/2$ and $2\pi^2$. Since one can be moved without the other, the two $4\pi$ come from different places.
The difference also shows in the choice of mass. Setting the mass to $4\pi$ removes $\pi$ from both formulas in three dimensions. Taking the mass as the surface area $S_{d-1}$ of the unit sphere, following Paper 912, makes the coefficient of the plasma frequency $1.000000$ in every dimension, but leaves $0.785398$ in four dimensions, $0.666667$ in five and $0.589049$ in six in the coefficient of the stopping force. The $4\pi$ of the plasma always disappears when the mass is matched to the number of the solid angle. The $4\pi$ of Bethe’s formula disappeared because in three dimensions it happened to be the same number.
5. That ratio is the fraction of a cylinder filled by a ball
The sequence of ratios 1, $\pi/4$, $2/3$, $3\pi/16$ looks familiar. Paper 1080 counted, dimension by dimension, the fraction of a circumscribed cylinder that a ball fills, and showed it to be $\pi/4$ in two dimensions, $2/3$ in three (the ratio of Archimedes) and $3\pi/16$ in four. It is the same sequence.
The reason lies in the coefficient $I_d$ of the transferred momentum. $I_d$ equals the ratio $S_{d-1}/S_{d-2}$ of the surface areas of the unit spheres of neighbouring dimensions (in three to six dimensions the ratio of the two is $1.000000000$). Hence $K_d=S_{d-2}I_d^2/2=S_{d-1}\times I_d/2$, and the ratio $K_d/S_{d-1}$ is $I_d/2$. $I_d/2$ agreed, in the four dimensions, to within $10^{-9}$ with the fraction of a circumscribed cylinder filled by a ball of dimension $d-2$.
In this reading the coefficient of Bethe’s formula is “the solid angle $S_{d-1}$ multiplied by the fraction of ball to cylinder”. In three dimensions $d-2=1$; the one-dimensional ball is a segment and its circumscribed cylinder is the same segment, so the fraction is 1. The $4\pi$ of Bethe’s formula can be written as $2\pi\times2$ or as $4\pi\times1$. In either way of writing, it equals the $4\pi$ of the plasma only in three dimensions. In five dimensions the fraction is the $2/3$ of Archimedes.
6. The collision logarithm appears only in three dimensions
The last factor of Bethe’s formula, $\int db/b$, is a logarithm: far and near collisions count equally, decade of distance by decade. In $d$ dimensions this integral becomes $\int db/b^{d-2}$. Its value from $b=0.01$ to 100, and the increase when the upper end is extended to 1000, were counted.
| $d$ | $\int db/b^{d-2}$ (from $0.01$ to 100) | Increase when the upper end is 1000 |
|---|---|---|
| 3 | $9.210340$ | $2.302585$ |
| 4 | $99.990000$ | $0.009000$ |
| 5 | $4999.999950$ | $0.000049$ |
In three dimensions, multiplying the upper end by 10 adds the natural logarithm of 10, $2.302585$. In four or more dimensions it adds almost nothing. Distant electrons are many, but each receives little energy. In three dimensions the growth of the circumference of the ring in proportion to $b$ and the fall of the energy as the inverse square of $b$ leave exactly $1/b$. In four or more dimensions the fall wins and only near collisions count. The plasma frequency has no such summing over distance. Here too the two formulas are built differently.
7. Relation to earlier papers, and a mistake in measuring
Paper 1082 counted a case in which the $4\pi$ facing the mass is a solid angle, and showed that a formula without the mass keeps its $4\pi$. This paper has added a case in which the $4\pi$ facing the mass is not a solid angle. Seeing the form $4\pi/m$ does not tell what the $4\pi$ is. What Paper 1085 counted for $8\pi$ happens for $4\pi$ as well.
Paper 22 treated the plasma-frequency formula with choices for the mass and the permittivity put in, and in its second edition withdrew the claim about confinement it had attached to that formula. This paper treats the formula before the choices are put in, and has limited what it says to one point: that the $4\pi$ is the solid angle of the field coming out of a sheet of charge.
One check failed once. In solving the orbit, the particle was started at a distance of 4000 with its speed left at 1. At that distance some force still remains, so the speed far away differed from 1 and the momentum disagreed with the exact formula by $0.02$ percent. The start was moved to a distance of two million, and the speed and transverse position at the start were corrected so that the speed far away and the impact parameter take the intended values. The mistake was in the measuring, not in the formula.
Closing
Closing: the plasma frequency and Bethe’s formula contain the same 4π/m, but the two 4π have different origins. The 4π of the plasma frequency is the solid angle of the field coming out of a sheet of charge; counted as the difference of the fields on the two sides of the sheet it is $\mathbf{12.566370614}$ in three dimensions and the surface area of the unit sphere in every dimension. The 4π of Bethe’s formula is the 2π of the circumference of a ring multiplied by the 2 that comes from half the square of the transferred momentum. When the dimension is changed the two coefficients separate, with ratios 1, $\mathbf{0.785398163}$, $\mathbf{0.666666667}$, $\mathbf{0.589048623}$. This sequence is the fraction of a circumscribed cylinder filled by a ball. The logarithm over collision distances also appears only in three dimensions. To be candid, the disappearance of π when the mass is set to 4π does not tell what the 4π is. What has been found is that moving the dimension by one distinguishes a 4π that disappears from one that does not. The separator is whether a field comes out of a sheet or a particle passes through a ring.
References
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- Yamagishi, Y. (2026). Taking $m$ as the Area of the Unit Sphere Makes the Coefficient of Temperature the Volume of the Unit Ball in Every Dimension [Paper 912]. YOI: https://www.yamagishiyuuki.com/yoi/912; DOI: 10.5281/zenodo.22804990
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- Yamagishi, Y. (2026). The $4\pi$ in $g=4\pi\hbar^2a/m$ Is a Solid Angle, and Setting $m=4\pi$ Turns It into $g=\hbar^2a$ [Paper 1082]. YOI: https://www.yamagishiyuuki.com/yoi/1082
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Reference
Yamagishi, Y. (2026). The Same 4π/m, but the 4π of the Plasma Frequency Is a Solid Angle and the 4π of Bethe’s Formula Is Twice the 2π of a Ring [Paper 1088]. YOI: https://www.yamagishiyuuki.com/yoi/1088
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 1088. This paper is a preprint and has not been peer reviewed.