最密な、最密充填の

WordNet

packed especially tightly
used of hair or haircuts; "a close military haircut"
finish a game in baseball by protecting a lead; "The relief pitcher closed with two runs in the second inning"
inclined to secrecy or reticence about divulging information; "although they knew her whereabouts her friends kept close about it" (同)closelipped, closemouthed, secretive, tightlipped
in an attentive manner; "he remained close on his guard" (同)closely, tight
come together, as if in an embrace; "Her arms closed around her long lost relative" (同)come_together
crowded; "close quarters" (同)confining
marked by fidelity to an original; "a close translation"; "a faithful copy of the portrait"; "a faithful rendering of the observed facts" (同)faithful
fill or stop up; "Can you close the cracks with caulking?" (同)fill_up
become closed; "The windows closed with a loud bang" (同)shut
move so that an opening or passage is obstructed; make shut; "Close the door"; "shut the window" (同)shut
fitting closely but comfortably; "a close fit" (同)snug, close-fitting
(of a contest or contestants) evenly matched; "a close contest"; "a close election"; "a tight game" (同)tight
of textiles; "a close weave"; "smooth percale with a very tight weave" (同)tight
at or within a short distance in space or time or having elements near each other; "close to noon"; "how close are we to town?"; "a close formation of ships"
bar access to; "Due to the accident, the road had to be closed for several hours"
be priced or listed when trading stops; "The stock market closed high this Friday"; "My new stocks closed at $59 last night"
bring together all the elements or parts of; "Management closed ranks"
cause a window or an application to disappear on a computer desktop
change ones body stance so that the forward shoulder and foot are closer to the intended point of impact
close in relevance or relationship; "a close family"; "we are all...in close sympathy with..."; "close kin"; "a close resemblance"
complete a business deal, negotiation, or an agreement; "We closed on the house on Friday"; "They closed the deal on the building"
confined to specific persons; "a close secret"
draw near; "The probe closed with the space station"
engage at close quarters; "close with the enemy"
finish or terminate (meetings, speeches, etc.); "The meeting was closed with a charge by the chairman of the board"
rigorously attentive; strict and thorough; "close supervision"; "paid close attention"; "a close study"; "kept a close watch on expenditures"
strictly confined or guarded; "kept under close custody"
set up a committee or legislative body with ones own supporters so as to influence the outcome; "pack a jury"
treat the body or any part of it by wrapping it, as with blankets or sheets, and applying compresses to it, or stuffing it to provide cover, containment, or therapy, or to absorb blood; "The nurse packed gauze in the wound"; "You had better pack your swollen ankle with ice"
a cream that cleanses and tones the skin (同)face_pack
a bundle (especially one carried on the back)
a convenient package or parcel (as of cigarettes or film)
a sheet or blanket (either dry or wet) to wrap around the body for its therapeutic effect
a complete collection of similar things
a group of hunting animals
compress into a wad; "wad paper into the box" (同)bundle, wad, compact
load with a pack (同)load down
arrange in a container; "pack the books into the boxes"
carry, as on ones back; "Pack your tents to the top of the mountain"
fill to capacity; "This singer always packs the concert halls"; "The murder trial packed the court house"
seal with packing; "pack the faucet"
with shutters closed
blocked against entry; "a closed porch" (同)closed in
used especially of mouth or eyes; "he sat quietly with closed eyes"; "his eyes were shut against the sunlight" (同)shut
not having an open mind; "a closed mind unreceptive to new ideas" (同)unsympathetic
(set theory) of an interval that contains both its endpoints
not open or affording passage or access; "the many closed streets made travel difficult"; "our neighbors peeped from behind closed curtains"
not open to the general public; "a closed meeting"
requiring union membership; "a closed shop"
final or ending; "the closing stages of the election"; "the closing weeks of the year"; "the closing scene of the film"; "closing remarks"
approaching a particular destination; a coming closer; a narrowing of a gap; "the ships rapid rate of closing gave them little time to avoid a collision" (同)closure
pressed together or compressed; "packed snow"

PrepTutorEJDIC

…'を'『閉じる』,閉める(shut) / …'を'ふさぐ(fill) / 〈事務・仕事・話など〉'を'『終える』,済ませる / 〈通路・施設など〉'を'一時的に閉じる,‘の'使用を一時中止する / (一つにまとまるように)…'を'つめる / 〈ドア・目・花などが〉『閉じる』,閉まる;ふさがる / 〈話・相談・契約などが〉『終る』,終了する,〈店などが〉終業(休業)する / くっつく,接近する(come together) / 終結,結末,終り(end) / 締め切り / (音楽の)終止
(距離・時間・関係が)『近い』,接近している・《名詞の前にのみ用いて》『親密な』,親しい・ぎっしり詰まった,密集した;目のつんだ,きめの細かい・綿密な,厳重な(strict) ・息がつまる,重苦しい(stuffy) / 厳重に隠された,秘密の / ほとんど同等の,匹敵する・《補語にのみ用いて》(金銭が)乏しい;(…に)けちな(stingy)《+『with』+『名』》・(音声が)閉鎖音の(舌を口蓋(こうがい)に近づけて発音する) / (…に)『近接して』,(…の)すぐそば(近く)に;(…に)ぴったりくっついて《+『to(behind, by)』+『名』》 / きっちりと;ぎっしりと・(大寺院の)囲い地;(学校の)構内,校庭・《英》袋小路
『包み』,『束』;荷物 / (…の一つの)『包み』,組,そろい《+『of』+『名』》 / (人・動物などの一つの)『隊』,群れ《+『of』+『塁』》 / 《a~》たくさん(の…)《+『of』+『名』》 / (傷口などに当てる)湿布 / (美容のための)パック / 〈物など〉‘を'『包む』,『束ねる』,荷作りする / (物)〈容器など〉‘に'『詰める』,入れる《+『名』+『with』+『名』》 / (容器に)〈物など〉‘を'『詰め込む』,いっぱいに入れる《+『名』+『into』(『in』)+『名』》 / 《しばしば受動態で》(人・物が)〈城所・乗り物など〉‘を'『いっぱいにする』 / 《米話》…‘を'身につけている / (運搬・保管などのために)『荷作りする』,『包装する』《+『up』》 / 〈物が〉荷作りできる,包装し得る / (…に)〈が〉ぎっしりはいる《+『into』+『名』》
〈人材・カードなど〉‘を'えり好みする(stack)
閉じる,終りの / 〈U〉閉じること,閉鎖 / 〈C〉〈U〉終了,締め切り;決算
満員の,すし詰めの

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/06/10 00:41:40」(JST)

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Regular arrangement of equal spheres in a plane changing to an irregular arrangement of unequal spheres (bubbles).

hcp and fcc close-packing of spheres

In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is

The same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture is now widely considered proven by T. C. Hales.^[1]^[2]

Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles.

fcc and hcp lattices

There are two simple regular lattices that achieve this highest average density. They are called face-centered cubic (fcc) (also called cubic close packed) and hexagonal close-packed (hcp), based on their symmetry. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; they differ in how the sheets are stacked upon one another. The fcc lattice is also known to mathematicians as that generated by the A₃ root system.^[3]

Cannonball problem

Cannonballs piled on a triangular (front) and rectangular (back) base, both fcc lattices.

The problem of close-packing of spheres was first mathematically analyzed by Thomas Harriot around 1587, after a question on piling cannonballs on ships was posed to him by Sir Walter Raleigh on their expedition to America.^[4] Cannonballs were usually piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground.

Positioning and spacing

In both the fcc and hcp arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres (octahedral) and two smaller gaps surrounded by four spheres (tetrahedral). The distances to the centers of these gaps from the centers of the surrounding spheres is for the tetrahedral, and for the octahedral, when the sphere radius is 1.

Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.

The most regular ones are:

fcc = ABCABCA (every third layer is the same)
hcp = ABABABA (every other layer is the same).

In close-packing, the center-to-center spacing of spheres in the x–y plane is a simple honeycomb-like tessellation with a pitch (distance between sphere centers) of one sphere diameter. The distance between sphere centers, projected on the z (vertical) axis, is:

where d is the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres.

The coordination number of hcp and fcc is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74.

Comparison between hcp and fcc

Figure 1 – The hcp lattice (left) and the fcc lattice (right). The outline of each respective Bravais lattice is shown in red. The letters indicate which layers are the same. There are two "A" layers in the hcp matrix, where all the spheres are in the same position. All three layers in the fcc stack are different. Note the fcc stacking may be converted to the hcp stacking by translation of the upper-most sphere, as shown by the dashed outline.


Figure 2 – Shown here is a stack of eleven spheres of the hcp lattice illustrated in Figure 1. The hcp stack differs from the top 3 tiers of the fcc stack shown in Figure 3 only in the lowest tier; it can be modified to fcc by an appropriate rotation or translation.	Figure 3 – Thomas Harriot, circa 1585, first pondered the mathematics of the cannonball arrangement or cannonball stack, which has an fcc lattice. Note how adjacent balls along each edge of the regular tetrahedron enclosing the stack are all in direct contact with one another. This does not occur in an hcp lattice, as shown in Figure 2.

Lattice generation

When forming any sphere-packing lattice, the first fact to notice is that whenever two spheres touch a straight line may be drawn from the center of one sphere to the center of the other intersecting the point of contact. The distance between the centers along the shortest path namely that straight line will therefore be r₁ + r₂ where r₁ is the radius of the first sphere and r₂ is the radius of the second. In close packing all of the spheres share a common radius, r. Therefore two centers would simply have a distance 2r.

Simple hcp lattice

An animation of close-packing lattice generation. Note: If a third layer (not shown) is directly over the first layer then the HCP lattice is built, if the third layer is placed over holes in the first layer then the FCC lattice is created

To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to hcp. The box would be placed on the x-y-z coordinate space.

First form a row of spheres. The centers will all lie on a straight line. Their x-coordinate will vary by 2r since the distance between each center if the spheres are touching is 2r. The y-coordinate and z-coordinate will be the same. For simplicity, say that the balls are the first row and that their y- and z-coordinates are simply r, so that their surfaces rest on the zero-planes. Coordinates of the centers of the first row will look like (2r, r, r), (4r, r, r), (6r ,r, r), (8r ,r, r), ... .

Now, form the next row of spheres. Again, the centers will all lie on a straight line with x-coordinate differences of 2r, but there will be a shift of distance r in the x-direction so that the center of every sphere in this row aligns with the x-coordinate of where two spheres touch in the first row. This allows the spheres of the new row to slide in closer to the first row until all spheres in the new row are touching two spheres of the first row. Since the new spheres touch two spheres, their centers form an equilateral triangle with those two neighbors' centers. The side lengths are all 2r, so the height or y-coordinate difference between the rows is . Thus, this row will have coordinates like this:

The first sphere of this row only touches one sphere in the original row, but its location follows suit with the rest of the row.

The next row follows this pattern of shifting the x-coordinate by r and the y-coordinate by . Add rows until reaching the x and y maximum borders of the box.

In an A-B-A-B-... stacking pattern, the odd numbered planes of spheres will have exactly the same coordinates save for a pitch difference in the z-coordinates and the even numbered planes of spheres will share the same x- and y-coordinates. Both types of planes are formed using the pattern mentioned above, but the starting place for the first row's first sphere will be different.

Using the plane described precisely above as plane #1, the A plane, place a sphere on top of this plane so that it lies touching three spheres in the A-plane. The three spheres are all already touching each other, forming an equilateral triangle, and since they all touch the new sphere, the four centers form a regular tetrahedron.^[5] All of the sides are equal to 2r because all of the sides are formed by two spheres touching. The height of which or the z-coordinate difference between the two "planes" is . This, combined with the offsets in the x and y-coordinates gives the centers of the first row in the B plane:

The second row's coordinates follow the pattern first described above and are:

The difference to the next plane, the A plane, is again in the z-direction and a shift in the x and y to match those x- and y-coordinates of the first A plane.^[6]

In general, the coordinates of sphere centers can be written as:

$\begin{bmatrix} 2i + ((j\ +\ k)\ \bmod{2})\\ \sqrt{3}\left[j + \frac{1}{3}(k\ \bmod{2})\right]\\ \frac{2\sqrt{6}}{3}k\\ \end{bmatrix}r$

where , and are indices starting at for the , and coordinates.

Miller indices

Miller–Bravais index for hcp lattice

Crystallographic features of hcp systems, such as vectors and atomic plane families can be described using a four-value Miller index notation ( hkil ) in which the third index i denotes a convenient but degenerate component which is equal to −h − k. The h, i and k index directions are separated by 120°, and are thus not orthogonal; the l component is mutually perpendicular to the h, i and k index directions.

Filling the remaining space

The FCC and HCP packings are the densest known packings of equal spheres. Denser sphere packings are known, but they involve unequal sphere packing. A packing density of 1, filling space completely, requires non-spherical shapes, such as honeycombs.

Replacing each contact point between two spheres with an edge connecting the centers of the touching spheres produces tetrahedrons and octahedrons of equal edge lengths. The FCC arrangement produces the tetrahedral-octahedral honeycomb. The HCP arrangement produces the gyrated tetrahedral-octahedral honeycomb. If, instead, every sphere is augmented with the points in space that are closer to it than to any other sphere, the duals of these honeycombs are produced: the rhombic dodecahedral honeycomb for FCC, and the trapezo-rhombic dodecahedral honeycomb for HCP.

Spherical bubbles in soapy water in a FCC or HCP arrangement, when the water in the gaps between the bubbles drains out, also approach the rhombic dodecahedral honeycomb or trapezo-rhombic dodecahedral honeycomb. However, such FCC of HCP foams of very small liquid content are unstable, as they do not satisfy Plateau's laws. The Kelvin foam an the Weaire Phelan foam are more stable, having smaller interfacial energy in the limit of a very small liquid content. ^[7]

Notes

^ Hales, T. C. (1998). "An overview of the Kepler conjecture". v2. arXiv:math/9811071.
^ "Mathematics: Does the proof stack up?".
^ Conway, John Horton; Sloane, Neil James Alexander; & Bannai, Eiichi. Sphere packings, lattices, and groups. Springer, 1999, Section 6.3.
^ David Darling. "Cannonball Problem". The Internet Encyclopedia of Science.
^ "on Sphere Packing". Grunch.net. Retrieved 2014-06-12.
^ Weisstein, Eric W., "Hexagonal Close Packing", MathWorld.
^ Foams, Structure and Dynamics, I. Cantat et al, Oxford University Press 2013

External links

P. Krishna & D. Pandey, "Close-Packed Structures" International Union of Crystallography by University College Cardiff Press. Cardiff, Wales. PDF
"3D Sphere Packing Applet" Close-Packing of Spheres java applet

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