Critical Mass Is Set by Skin Balancing Content, Yet a Cube's Extra Volume Does Not Match Its Surface Coefficient ── in a bare one-group diffusion sphere, neutrons leaking through the skin over those gained in the content equal the square of the critical radius over the radius, and a cube's critical-volume factor 1.240490 is smaller than its equal-volume surface coefficient 1.240701 ── the separator is whether the weight of a shape is set by surface area or by the first eigenvalue ── [Paper 1007]

Abstract

Whether a fission chain reaction keeps going is decided by the balance between neutrons gained in the content and neutrons leaking through the skin. This paper counts that balance in a bare one-group diffusion core (no reflector, no extrapolation distance, lengths measured in the diffusion length L, and the material represented only by the infinite multiplication factor k∞), and compares critical volumes of other shapes with surface-area coefficients. No new theorem or law is claimed, and no values for real fuels are used. Scope of this paper (scope note): No new theorem or law is claimed──one-group diffusion theory, material and geometric buckling, the bare-sphere criticality condition B R = π, the bucklings of cubes and cylinders, zeros of Bessel functions, the inverse-square dependence of critical mass on density, and the isoperimetric inequality are all known. No values for real fuels are given──no one-group constants are used; length is the diffusion length L and the material is only k∞. Reactor physics is not widened──reflectors, extrapolation distances, multigroup or transport treatments, delayed neutrons and control are not included. The meaning of the ratio off criticality is limited──at non-critical radii the ratio of leakage to gain assumes the fundamental shape, and the neutron population then grows or decays in time. Relation to earlier papers: the scope note of Paper 326 said that it does not optimize shapes, leaving aside which shape is favourable and showing only the range of coefficients; this paper sits in that seat, compares shapes by critical volume, and counts that the surface coefficient and the critical coefficient do not match (no earlier occupant). Paper 326 showed that the square-cube relation is an identity and that the equal-volume surface coefficient is 1 for a sphere, 1.240701 for a cube and 1.490004 for a regular tetrahedron──here it is counted that the critical-volume ratio of a cube, 1.240490, departs from that coefficient in the fourth decimal place. Paper 12 read 4πc² as having the same form as the surface area of a sphere of radius c, and set boundary energy beside volume energy──here the skin and content of neutrons are counted as leakage and gain. Paper 465 showed that for cooling, making something smaller beats changing its material by forty times──here it is counted that for criticality, making it smaller raises the leakage-to-gain ratio as the inverse square of the radius. Paper 730 showed that cylindrical insulation has a critical radius equal to the thermal conductivity over the surface heat-transfer coefficient──here it is counted that the neutron critical radius is π L over the square root of k∞ − 1. Paper 52 gave the critical mass of a black hole whose Hawking temperature equals that of the cosmic background──the words are the same, but the balance here is neutron leakage against gain, a different thing. What is added is counting leakage and gain separately from the surface slope and the volume integral and confirming that their ratio is the squared radius ratio, checking the first eigenvalues of the cube and cylinder by finite differences and giving critical-volume ratios, setting critical-volume ratios beside equal-volume surface coefficients and showing they do not match, and counting how critical mass falls as density changes. First, a sphere is critical where the product of the first eigenvalue B and the radius R is π──with 1600 grid points the finite-difference B R was 3.141592150, and the critical radius is π L divided by the square root of k∞ − 1: 3.141593 L for k∞ = 2 and 4.442883 L for 1.5 (Section 2). Second, and this is the core. Neutrons leaking through the skin over those gained in the content equal the square of the critical radius over the radius──counting leakage from the surface slope and gain from the volume integral separately, the ratio was 4, 1 and 0.25 at 0.5, 1 and 2 times the critical radius, agreeing with the squared radius ratio within a relative difference of 1.735×10⁻¹⁶ (Section 3). Third, a cube’s critical volume is 1.240490 times the sphere’s and does not match its surface coefficient 1.240701──the cube’s first eigenvalue is the same from a 3-dimensional sparse matrix and from three times the 1-dimensional one, and the critical-volume ratio is 9√3/(4π). The equal-volume surface coefficient (the cube root of 6/π) is 0.000211 larger, a relative difference of 1.701×10⁻⁴ (Section 4). Fourth, for a cylinder the shape best for criticality differs from the shape with least surface──the cylinder with the smallest critical volume has height over diameter 0.923743 and is 1.141775 times the sphere, while the cylinder with least surface has height equal to diameter. At the critical optimum, the surface coefficient is 1.145522, 0.003747 above the critical ratio (Section 4). Fifth, doubling the density cuts the critical mass to a quarter──with cross sections proportional to density and the diffusion coefficient inversely proportional, the diffusion length and critical radius halve, and the critical mass became 0.250000 times (Section 5). Sixth, the separator is whether the weight of a shape is set by surface area or by the first eigenvalue──the skin-content balance can be written in terms of surface and volume, but the critical volume of another shape is set by the first eigenvalue and departs from the surface coefficient in the fourth decimal place (Section 6). Critical mass is set by skin balancing content, yet a cube’s extra volume does not match its surface coefficient. In a bare one-group diffusion sphere, criticality is where the first eigenvalue times the radius is π, and the ratio of skin leakage to content gain, counted separately, agreed with the squared ratio of critical radius to radius within a relative difference of 1.735×10⁻¹⁶──at half the critical radius leakage is 4 times the gain, and at twice it a quarter. A cube’s critical volume is 1.240490 times the sphere’s, 0.000211 below its equal-volume surface coefficient 1.240701, and for a cylinder the height/diameter best for criticality, 0.923743, differs from the least-surface height = diameter──doubling the density makes the critical mass 0.250000 times. The separator is whether the weight of a shape is set by surface area or by the first eigenvalue. Placed among the earlier papers──Paper 326 listed equal-volume surface coefficients and left shape optimization aside; Paper 12 set boundary beside volume energy, Paper 465 showed the power of making things smaller, Paper 730 the critical radius of insulation, and Paper 52 the critical mass of a black hole. Here criticality was counted as the balance of neutron skin and content, and the weight of a shape was shown to be set not by the surface coefficient but by the first eigenvalue. To be honest──values for real fuels, reflectors, extrapolation distances and multigroup or transport treatments are not treated, and the ratio at non-critical radii assumes the fundamental shape. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: critical mass, one-group diffusion, buckling, surface and volume, Bessel function zero, shape optimization. ----- 核分裂の連鎖反応が続くかどうかは、中性子が中身で増える分と、皮から漏れる分の釣り合いで決まる。本稿は、一群拡散の裸の炉心(反射体も外挿距離も置かず、長さを拡散距離 L で測り、材料は無限増倍率 k∞ だけで表す)で、この釣り合いを数え、形を替えたときの臨界体積を表面積の係数と比べる。新しい定理も法則も主張せず、実在の燃料の値は使わない。 本稿の射程(射程注記):新しい定理も法則も主張しない──一群拡散理論、材料と形のバックリング、裸の球の臨界条件 B R=π、立方体と円柱のバックリング、ベッセル関数の零点、臨界質量が密度の二乗に反比例すること、等周不等式は、いずれも既知である。実在の燃料の値を出さない──一群定数を使わず、長さは拡散距離 L、材料は k∞ だけで表した。炉の物理を広げない──反射体、外挿距離、多群や輸送の扱い、遅発中性子、制御は入れない。臨界でない大きさの比の意味を限る──臨界でない半径での漏れる分と増える分の比は、基本の形を仮定した比で、そのとき中性子の数は時間とともに増えるか減る。既刊との関係:論文326 の射程注記は、形の最適化をせず、どの形が有利かは扱わず係数の幅だけを示すと書いた──本稿はその席に座り、臨界体積で形を比べ、表面積の係数と臨界の係数が一致しないことを数える(先客なし)。論文326 は、二乗三乗の関係が恒等式であり、同じ体積での表面積の係数が球 1・立方体 1.240701・正四面体 1.490004 であることを示した──ここでは、立方体の臨界体積の比 1.240490 が、その係数と小数第四位でずれることを数える。論文12 は、4πc² が半径 c の球の表面積と同じ形をしていることを読み、境界のエネルギーと体積のエネルギーを並べた──ここでは、中性子の皮と中身を、漏れる分と増える分として数える。論文465 は、冷え方では材質を替えるより小さくするほうが四十倍効くことを示した──ここでは、臨界では小さくすると漏れる分の比が半径の二乗に反比例して増えることを数える。論文730 は、円筒の断熱では熱伝導率を表面熱伝達率で割った臨界半径があることを示した──ここでは、中性子の臨界半径が π L を k∞−1 の平方根で割った値であることを数える。論文52 は、ホーキング温度が宇宙背景放射の温度に等しくなるブラックホールの臨界質量を示した──同じ臨界質量の語だが、ここでの釣り合いは中性子の漏れと増えで、別のものである。加えたのは、漏れる分と増える分を表面の傾きと体積の積分から別々に数え、比が臨界半径との比の二乗になることを確かめたこと、立方体と円柱の最初の固有値を差分で確かめ、臨界体積の比を出したこと、臨界体積の比と同じ体積での表面積の係数を並べ、一致しないことを示したこと、密度を替えたときの臨界質量の減り方を数えたことである。 第一に、球の臨界は、最初の固有値 B と半径 R の積が π になるところである──差分 1600 点で B R は 3.141592150 で、臨界半径は π L を k∞−1 の平方根で割った値、k∞=2 で 3.141593 L、1.5 で 4.442883 L だった(第2節)。 第二に、これが本稿の芯である。皮から漏れる分と中身で増える分の比は、臨界半径との比の二乗になる──漏れる分を表面の傾きから、増える分を体積の積分から別々に数えると、半径が臨界の 0.5・1・2 倍で比は 4・1・0.25 になり、臨界半径との比の二乗と相対差 1.735×10⁻¹⁶ 以内で一致した(第3節)。 第三に、立方体の臨界体積は球の 1.240490 倍で、表面積の係数 1.240701 と一致しない──立方体の最初の固有値は三次元の疎行列でも一次元の三倍でも同じで、臨界体積の比は 9√3/(4π)。同じ体積での表面積の係数(6/π の立方根)はそれより 0.000211 大きく、相対差は 1.701×10⁻⁴ だった(第4節)。 第四に、円柱では、臨界に最も得な形と表面積が最も小さい形が違う──臨界体積が最小になる円柱は高さ/直径が 0.923743 で球の 1.141775 倍、表面積が最小の円柱は高さ=直径である。臨界が最小の形での表面積の係数は 1.145522 で、臨界の比より 0.003747 大きい(第4節)。 第五に、密度を 2 倍にすると臨界質量は 4 分の 1 になる──断面積が密度に比例し拡散係数が反比例すると、拡散距離と臨界半径は半分になり、臨界質量は 0.250000 倍だった(第5節)。 第六に、分離子は、形の重さを決めるのが表面積か、最初の固有値か──皮と中身の釣り合いは表面積と体積の言葉で書けるが、形を替えたときの臨界体積を決めるのは最初の固有値で、表面積の係数とは小数第四位でずれる(第6節)。 臨界質量は皮と中身の釣り合いで決まり、立方体の割り増しは表面積の係数と一致しない。一群拡散の裸の球では最初の固有値と半径の積が π のときが臨界で、皮から漏れる分と中身で増える分を別々に数えた比は、臨界半径との比の二乗と相対差 1.735×10⁻¹⁶ 以内で一致した──半径が臨界の半分なら漏れは増えの 4 倍、2 倍なら 4 分の 1 である。立方体の臨界体積は球の 1.240490 倍で、同じ体積での表面積の係数 1.240701 より 0.000211 小さく、円柱では臨界に最も得な高さ/直径 0.923743 が表面積の最も小さい高さ=直径と違った──密度を 2 倍にすると臨界質量は 0.250000 倍になる。分離子は、形の重さを決めるのが表面積か、最初の固有値かである。既刊との位置──論文326 は同じ体積での表面積の係数を並べて形の最適化をしないとし、論文12 は境界と体積のエネルギーを、論文465 は小さくする効きを、論文730 は断熱の臨界半径を、論文52 はブラックホールの臨界質量を示した。ここでは、中性子の皮と中身の釣り合いで臨界を数え、形の重さが表面積の係数ではなく最