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測定の流れを試す
核図表で ¹³³Xe の基底状態と核異性体状態を調べ、励起エネルギーと不確かさを確認してください。核異性体は比較的長寿命の励起核状態です。MR-TOF ではイオンが静電ミラー間を往復します。周回数、ピーク幅、存在比を変えてみましょう。TOF-ICR では高周波励起を共鳴に沿って走査します。混合信号、単一イオン種のフィット、残差（観測値からフィット値を引いた値）を比べます。PI-ICR では運動の位相を検出器に画像化します。蓄積時間を変えて、角度が一周後に戻る様子を観察してください。ピーク分離、統計精度、モデルの偏りは別々の制限です。
ビームが止まっている間に遊んでみてください。最高得点はブラウザ内にのみ保存されます。
⏳ どちらが長生き？
NUBASE2020 から選んだ2つの原子核——半減期が長いほうを選んでください。連続正解を目指そう！
🎯 サイクロトロン共鳴を探せ（TOF-ICR）
励起周波数 ν_rf を設定してイオンを打ち、飛行時間の最小値を探します。7 T トラップ中の ¹³³Cs⁺：ν_c = qB/(2πm) ≈ 808 795.01 Hz。打つ回数が少なく、推定が正確なほど高得点。
💡 この設定の意味と使い方
イオン・電荷数 z・B・励起時間を選択。矩形は RF パルス一つ、Ramsey は分離した二つです。
νrf を変えて測定を重ね、曲線を作ります。理論はモデルを重ね、点/ピクセルは表示を変更。長い励起ほど理想共鳴が狭くなります。
💡 物理のヒント
- ν_rf = ν_c では四重極励起がゆっくりしたマグネトロン運動を速い換算サイクロトロン運動へ完全に変換します。検出器へ向かう途中、動径エネルギー E_r ∝ ω₊²r² が磁場勾配で軸方向エネルギーに変わるため、共鳴イオンは速く飛びます：共鳴は飛行時間の最小値です。
- 線幅：矩形パルスでは FWHM ≈ 0.8/T_rf。励起が長いほど谷は狭く、ν_c は精密になります（統計精度 ∝ 1/(T_rf·√N)）。
- 矩形励起には副極小（sinc² 型）があります。ラムゼー励起（短い2パルス＋待ち時間）は中心縞が約40%狭く、副縞が強くなります——副縞を中心と間違えないように。
- 周波数から質量へ：m = qB/(2πν_c)（B は参照イオンで校正）。808 kHz で 0.1 Hz の誤差は δm/m ≈ 1.2×10⁻⁷、A = 133 で約 15 keV。
TOF-ICR：重なる共鳴とフィットの偏り
Compare individual resonances with a mixed sample. A single-species fit can give a precise-looking centre that differs from the target ion’s resonance.
一部の科学的な説明は英語で表示されます。
💡 この設定の意味と使い方
質量・電荷数・B・トラップ電圧と寸法から設定。ν₊・ν₋・νz は修正サイクロトロン・マグネトロン・軸方向周波数。
プリセットから始め、一度に一項目を変えて読数を比較。半径と位相は初期運動、励起はパルス、速度は再生を指定。Fancier は校正パネルで有効にするとトラップ 1/2 の独立校正を使います。
解釈
詳細パラメータと仮定
Masses: AME2020 ground state + NUBASE2020 excitation energy. # marks extrapolated properties. Unknown uncertainty is not zero; unavailable covariance uses an explicit diagonal approximation. Ionization and molecular binding energies are omitted.
教材の応答：t(F) = t_offset + t_scale / √(1 + κF)。F は無次元のモード変換割合で、範囲は 0–1 です。既定値は t_offset = 62 µs、t_scale = 193 µs、κ = 1.876。未変換背景は 8%、イベント幅 σ は 7 と 9 µs です。混合平均と表示する各成分のイベント密度は存在比で重み付けします。ノイズは例示用です。局所的な Δχ² = 1 区間には混合による偏りを含みません。
TOF-ICR：重なる共鳴とフィットの偏り
Compare individual resonances with a mixed sample. A single-species fit can give a precise-looking centre that differs from the target ion’s resonance.
一部の科学的な説明は英語で表示されます。
💡 この設定の意味と使い方
質量・電荷数・B・トラップ電圧と寸法から設定。ν₊・ν₋・νz は修正サイクロトロン・マグネトロン・軸方向周波数。
プリセットから始め、一度に一項目を変えて読数を比較。半径と位相は初期運動、励起はパルス、速度は再生を指定。Fancier は校正パネルで有効にするとトラップ 1/2 の独立校正を使います。
解釈
詳細パラメータと仮定
Masses: AME2020 ground state + NUBASE2020 excitation energy. # marks extrapolated properties. Unknown uncertainty is not zero; unavailable covariance uses an explicit diagonal approximation. Ionization and molecular binding energies are omitted.
教材の応答：t(F) = t_offset + t_scale / √(1 + κF)。F は無次元のモード変換割合で、範囲は 0–1 です。既定値は t_offset = 62 µs、t_scale = 193 µs、κ = 1.876。未変換背景は 8%、イベント幅 σ は 7 と 9 µs です。混合平均と表示する各成分のイベント密度は存在比で重み付けします。ノイズは例示用です。局所的な Δχ² = 1 区間には混合による偏りを含みません。
🌀 アイソマーを分離せよ（PI-ICR）
Phase-imaging: after the accumulation time t_acc the cyclotron phase of each ion is projected onto a position-sensitive detector. Ground state and isomer differ in mass by Eₓ/c², so their spots move apart by Δφ = 2π Δν_c t_acc. Find a t_acc that separates them.
💡 この設定の意味と使い方
イオン検索または複数イオンを選択。他の状態は追加のカタログ状態を含めます。質量表は出典、B は磁場。蓄積時間で位相が増えますが、角度差は 360° で折り返します。
半径は検出リング、スポット幅の単位は 0.1 mm。比率と汚染率は集団数を指定。σ はスポット幅による分離目標。位相オフセットは始点、クロックは実現可能時間への丸めを指定。検出器練習は角度下限と重心カウント、シードは雑音再現。U₀ と d は理想トラップ周波数を設定します。
基準位相とタイミング
磁気回転基準の座標系で、検出器へ投影したパターンの位相をずらします。反時計回りが正です。入力した最初のイオン、または単一対の基底状態が基準周波数を決めます。重ね合わせを有効にすると、νc t + (δ₊ − δ₋)/360° が整数になる次の解へ指定時間を進めます。標準クロックは PyMassScanner と同様に理想時間を 4 ns 刻みに丸め、残る位相差を表示します。理想タイミングでは正確に重なります。図、周回数、不確かさ、時間比較、出力には実際の時間を使います。共通の原点はすべてのスポットを回転させ、相対的な間隔を保ちます。
α₋ = α₀ + δ₋; α₊ = α₀ + δ₊ + 360°νc t; νc = ν₊ + ν₋.
検出器と練習
カンマ区切り：84Rb · Rb-84 · 133Cs+2 または 133Cs2+（価数）· 45mSc、45Scm または 45Sc*（核異性体、Eₓ は NUBASE2020；m2 = 第2異性体）· 12C16O2 または 12C11（分子・クラスター）· 87Rb:0.3（相対強度）。イオン質量 = 原子質量 − q·mₑ（電子の結合エネルギーは無視）。
第1イオン種が基準です。総周回数 = νc t、差 = (νc − νc,基準)t。残余角は [−180°,180°) に折り返します。最初に反対位置となる時間は 1/(2|Δνc|) で、同じ周波数では分離しません。これは一対の時間で、適用時は 0.01 ms 刻みに丸めます。周回数は既知のシミュレーション周波数から計算します。
蓄積時間の比較
既知のシミュレーション周波数で周回数が決まります。複数の時間で位相の折り返しを示しますが、実験の未知の周回数は推定しません。
角度は反時計回りです。共通の原点は全スポットを回転させ、相対分離は保ちます。検出器の描画半径は 21.00 mm で、スポット半径が 20.00 mm を超えると 32.00 mm に拡大します。検出器の受け入れ損失はモデル化しません。
σ は1事象の角度標準偏差です。全イオン対を 0.01 ms 刻みで 5000 ms まで調べ、位相の一周回り込みを考慮します。フィットの信頼水準ではありません。
外側スポットσは事象幅と位相分解能を設定します。中心スポットσは中央の雲だけを変更します。
σ は個々のイベントの角度分布の標準偏差です。2.355σ は一つのガウスピークの半値全幅（FWHM）の近似で、信頼水準や二つの重心の合成不確かさではありません。プリセットと入力値は同じ最短時間探索と比較線の目標を設定します。
💡 物理のヒント
共通原点から測ると νc = (φ + 2πn)/(2πt)。φ はラジアン、n は完了周回数です。仮想実験は n を知っていますが、実測では別途決定します。同じ電荷で静止エネルギーが E 増加すると |Δνc|/νc = E/(mイオンc² + E)。小角度では σφ ≈ σx,y/r。調整できる r の初期値は 10.00 mm。R = 2πνc t/(2.355 σφ) は半値全幅の分解能指標で、質量不確かさではありません。時間を延ばすと折り返すスポットが再び重なる場合があります。
理想的な運動は速い半径方向運動 ν₊、遅いマグネトロン運動 ν₋、軸方向振動 νz です。Φ = U₀(z² − r²/2)/(2d²)、νc = qB/(2πmイオン)、νz = √(qU₀/(mイオンd²))/(2π)、ν± = [νc ± √(νc² − 2νz²)]/2。閉じ込めには νc² > 2νz² が必要。理想的な関係は ν₊ + ν₋ = νc、ν₊² + ν₋² + νz² = νc²。一階近似は ν₋ ≈ U₀/(4πBd²)。
1 keV はどの角度に対応しますか？
これは静止エネルギー（質量）が 1 keV 増加した場合で、運動エネルギーではありません。δm = 1 keV/c² = 0.00000107354410 u。同じ電荷と B では δνc = −νc δm/(mイオン + δm)、δφ = 360° δνc t です。反時計回りの角度規約で負の値は基準より遅れることを示します。結果は第1イオンと現在の蓄積時間に依存します。仮定した増分であり、測定した異性体エネルギーではありません。
🌀 アイソマーを分離せよ（PI-ICR）
Phase-imaging: after the accumulation time t_acc the cyclotron phase of each ion is projected onto a position-sensitive detector. Ground state and isomer differ in mass by Eₓ/c², so their spots move apart by Δφ = 2π Δν_c t_acc. Find a t_acc that separates them.
💡 この設定の意味と使い方
イオン検索または複数イオンを選択。他の状態は追加のカタログ状態を含めます。質量表は出典、B は磁場。蓄積時間で位相が増えますが、角度差は 360° で折り返します。
半径は検出リング、スポット幅の単位は 0.1 mm。比率と汚染率は集団数を指定。σ はスポット幅による分離目標。位相オフセットは始点、クロックは実現可能時間への丸めを指定。検出器練習は角度下限と重心カウント、シードは雑音再現。U₀ と d は理想トラップ周波数を設定します。
基準位相とタイミング
磁気回転基準の座標系で、検出器へ投影したパターンの位相をずらします。反時計回りが正です。入力した最初のイオン、または単一対の基底状態が基準周波数を決めます。重ね合わせを有効にすると、νc t + (δ₊ − δ₋)/360° が整数になる次の解へ指定時間を進めます。標準クロックは PyMassScanner と同様に理想時間を 4 ns 刻みに丸め、残る位相差を表示します。理想タイミングでは正確に重なります。図、周回数、不確かさ、時間比較、出力には実際の時間を使います。共通の原点はすべてのスポットを回転させ、相対的な間隔を保ちます。
α₋ = α₀ + δ₋; α₊ = α₀ + δ₊ + 360°νc t; νc = ν₊ + ν₋.
検出器と練習
カンマ区切り：84Rb · Rb-84 · 133Cs+2 または 133Cs2+（価数）· 45mSc、45Scm または 45Sc*（核異性体、Eₓ は NUBASE2020；m2 = 第2異性体）· 12C16O2 または 12C11（分子・クラスター）· 87Rb:0.3（相対強度）。イオン質量 = 原子質量 − q·mₑ（電子の結合エネルギーは無視）。
第1イオン種が基準です。総周回数 = νc t、差 = (νc − νc,基準)t。残余角は [−180°,180°) に折り返します。最初に反対位置となる時間は 1/(2|Δνc|) で、同じ周波数では分離しません。これは一対の時間で、適用時は 0.01 ms 刻みに丸めます。周回数は既知のシミュレーション周波数から計算します。
蓄積時間の比較
既知のシミュレーション周波数で周回数が決まります。複数の時間で位相の折り返しを示しますが、実験の未知の周回数は推定しません。
角度は反時計回りです。共通の原点は全スポットを回転させ、相対分離は保ちます。検出器の描画半径は 21.00 mm で、スポット半径が 20.00 mm を超えると 32.00 mm に拡大します。検出器の受け入れ損失はモデル化しません。
σ は1事象の角度標準偏差です。全イオン対を 0.01 ms 刻みで 5000 ms まで調べ、位相の一周回り込みを考慮します。フィットの信頼水準ではありません。
外側スポットσは事象幅と位相分解能を設定します。中心スポットσは中央の雲だけを変更します。
σ は個々のイベントの角度分布の標準偏差です。2.355σ は一つのガウスピークの半値全幅（FWHM）の近似で、信頼水準や二つの重心の合成不確かさではありません。プリセットと入力値は同じ最短時間探索と比較線の目標を設定します。
💡 物理のヒント
共通原点から測ると νc = (φ + 2πn)/(2πt)。φ はラジアン、n は完了周回数です。仮想実験は n を知っていますが、実測では別途決定します。同じ電荷で静止エネルギーが E 増加すると |Δνc|/νc = E/(mイオンc² + E)。小角度では σφ ≈ σx,y/r。調整できる r の初期値は 10.00 mm。R = 2πνc t/(2.355 σφ) は半値全幅の分解能指標で、質量不確かさではありません。時間を延ばすと折り返すスポットが再び重なる場合があります。
理想的な運動は速い半径方向運動 ν₊、遅いマグネトロン運動 ν₋、軸方向振動 νz です。Φ = U₀(z² − r²/2)/(2d²)、νc = qB/(2πmイオン)、νz = √(qU₀/(mイオンd²))/(2π)、ν± = [νc ± √(νc² − 2νz²)]/2。閉じ込めには νc² > 2νz² が必要。理想的な関係は ν₊ + ν₋ = νc、ν₊² + ν₋² + νz² = νc²。一階近似は ν₋ ≈ U₀/(4πBd²)。
1 keV はどの角度に対応しますか？
これは静止エネルギー（質量）が 1 keV 増加した場合で、運動エネルギーではありません。δm = 1 keV/c² = 0.00000107354410 u。同じ電荷と B では δνc = −νc δm/(mイオン + δm)、δφ = 360° δνc t です。反時計回りの角度規約で負の値は基準より遅れることを示します。結果は第1イオンと現在の蓄積時間に依存します。仮定した増分であり、測定した異性体エネルギーではありません。
🪞 同重体を分離せよ（MR-TOF）
イオンは2枚の静電ミラー間を往復し、周回ごとに飛行時間が延びます。同重体の時間差は増え、ピーク幅はゆっくりしか広がりません。まずクーラー・バンチャーの初期バンチ幅を決め、ピークが分かれるまで周回を増やそう。
💡 この設定の意味と使い方
イオン対または手動入力を選択。比率は存在比、カウントは模擬イベント数、周回数はミラー間の往復回数です。
Δt₀ は初期時間幅、Δt/周は追加広がり、t/周は周回時間、t₀ は初期飛行時間。表示単位を使ってください。周回増加は分離と広がりの両方を生みます。合計はピークを加算し、log R は目盛を変えます。
💡 物理のヒント
t(N) = t₀ + N·T_lap ∝ √m、Δt = t·Δm/(2m)、FWHM² = Δt₀² + (N·δ)²、R = t/(2·FWHM)。Δt ≥ FWHM（R ≥ m/Δm）で分離。
MR-TOF：イオン種と核異性体の分離
Choose up to three ion species. Compare individual peaks with their combined spectrum, then see how separation changes with lap count.
一部の科学的な説明は英語で表示されます。
💡 この設定の意味と使い方
質量・電荷数・B・トラップ電圧と寸法から設定。ν₊・ν₋・νz は修正サイクロトロン・マグネトロン・軸方向周波数。
プリセットから始め、一度に一項目を変えて読数を比較。半径と位相は初期運動、励起はパルス、速度は再生を指定。Fancier は校正パネルで有効にするとトラップ 1/2 の独立校正を使います。
解釈
詳細パラメータと仮定
Masses: AME2020 ground state + NUBASE2020 excitation energy. # marks extrapolated properties. Unknown uncertainty is not zero; unavailable covariance uses an explicit diagonal approximation. Ionization and molecular binding energies are omitted.
Flight time scales exactly as √(mion/q); fixed electronic delay is added afterwards. Width is phenomenological: FWHM² = initial width² + (laps × broadening)². Populations are detected fractions; displayed survival does not simulate decay products.
🧲 イオンを留めよ（線形パウルトラップ / RFQ 質量フィルター）
イオンは4本のロッドが高速に切り替わる四重極場の中を運動します。動作点 (q, a) が安定領域内のイオンだけが r₀ 内に留まり、他はロッドに衝突します（火花）。U と V を調整して目標質量だけを通そう。
💡 この設定の意味と使い方
イオンと対象を選択。カスタム例：100Sn, 99In, 133Cs+2。A は質量数、V は RF 電圧、U は直流電圧、r₀ は電極半径、Ω/2π は RF 周波数。
エミッタンスは初期広がり、周期数は RF 周期単位の滞在時間、速度は再生速度。同じシードは同じ初期集団を再現。安定境界は理想 Mathieu モデル、透過率はその集団の軌道計算結果です。
V はゼロからピークまでのRF電圧振幅、U は直流電圧です。無衝突の横方向モデルで有限時間の透過を示します。軸方向輸送、緩衝ガス、空間電荷、端部電場は含みません。 マシュー方程式
Trap design, inverse calculations and derivations
RFQ rod potentials are ±(U−V cos Ωt). V is the zero-to-peak amplitude of one rod pair relative to ground; the differential amplitude is 2V. The cooling panel uses differential V_RF, so equal numerical voltages are not interchangeable.
Φ=(U−V cos Ωt)(x²−y²)/r₀² → Eₓ=−2(U−V cos Ωt)x/r₀², Eᵧ=+2(U−V cos Ωt)y/r₀².
τ=Ωt/2 → x″+(a−2q cos 2τ)x=0, y″+(−a+2q cos 2τ)y=0.
a=8QU/(mr₀²Ω²), q=4QV/(mr₀²Ω²), a/q=2U/V.
νx,y=βx,y f_RF/2; ν≈f_RF√(a+q²/2)/2 in the adiabatic approximation only.
Inverse design uses the selected target ion, radius and RF frequency. Preview before applying; any edit invalidates the previous preview. Nominal depth QqV/4 is for U=0 at r=r₀ in the ideal adiabatic model, rather than a measured rod-surface barrier. Mass cutoffs describe the infinite-time first stability region; finite-residence transmission may differ. NIST DLMF
動作直線と質量カットオフ
理想 Mathieu 境界は測定された透過効率ではありません。V はゼロからピークまでの RF 振幅で、ピーク間電圧は 2V です。カットオフ質量は電荷に比例し、ここでは m/z を u/e で示します。
💡 物理のヒント
RF 周期数は滞在時間であり、電極の長さではありません。周波数 f MHz で N 周期なら t = N/f マイクロ秒です。初期値の 1.00 MHz、60 周期は 60.00 µs です。実際のフィルター長には軸方向速度も必要で、L = v_z t となります。ここでは軸方向運動を計算しません。r₀ は軸から電極表面までの距離で、初期値は 10.00 mm です。V は RF の片振幅、U は直流電圧、Ω = 2πf です。z は正の素電荷の個数、m/z は電荷あたりの質量です。無次元の a、q が Mathieu 安定図上の位置を決め、動作直線は a/q = 2U/V です。LMCO と HMCO は理想的な質量電荷比の下限と上限で、透過効率ではありません。初期広がりは発射位置円盤の半径を r₀ に対する百分率で指定します。発射速度にもこの教材用広がりを使い、実測ビームのエミッタンスではありません。アニメーション速度は再生だけを変え、a、q やカットオフは変えません。乱数シードは発射系列を再現します。緑の安定領域は理想的な無限時間方程式での有界運動を示し、有限時間だけ生き残ることは安定性の証明ではありません。
❄️ レーザー冷却 — 光モラセス — 原子雲を冷やす
赤方離調したレーザー光は原子やイオンを減速させます。粒子が向かってくる光をドップラー効果で共鳴に近く見るため、その光から多くの光子を散乱します。吸収ごとに運動と逆向きの反跳 ħk を受け、ランダムな自然放出は加熱します。その釣り合いがドップラー温度です。
x 方向に対向する 2 本のビーム。離調と強度を調整してレーザーをオンにし、できるだけ早くドップラー限界 ħΓ/2k_B に到達しましょう。
目標：T < 1.2 × ドップラー限界
📐 原理（物理の詳細）
散乱率：R = (Γ/2)·s₀/(1 + s₀ + (2Δ/Γ)²)。Δ = δ − k·v は速度 v の原子が感じる離調（ドップラーシフト k·v、k = 2π/λ）。
放射圧：吸収した光子ごとに光の方向へ運動量 ħk を受け、自然放出の方向はランダム。1本のビームの力は F = ħk·R。
光モラセス：対向する2本の赤方離調ビームで F(v) = ħk[R(δ − kv) − R(δ + kv)] ≈ −β·v、β = −8ħk²δs₀/(Γ(1 + s₀ + (2δ/Γ)²)²)。捕獲速度 ≈ (Γ/2k)√(1+s₀) 以内でのみ有効。
加熱：ランダムな反跳が運動量拡散 D₀ = ħ²k²R_tot(1+η)/2 を生む（1次元教科書モデルで η = 1、等方放出で 1/3）。
ドップラー温度：k_BT = D₀/β = ħΓ(1+η)(1 + s₀ + (2δ/Γ)²)/(16|δ|/Γ)。δ = −(Γ/2)√(1+s₀)、s₀ → 0 で最小 T_D = ħΓ/2k_B。反跳温度 T_r = ħ²k²/(m k_B) が下限。
シミュレーションは各原子・各ビームでポアソン分布の光子数を引くため、温度曲線に実際のショットノイズが現れる。青方離調（δ > 0）は加熱する。
💡 ヒント
最良のドップラー温度：低強度で δ = −Γ/2。s₀ を大きくすると速く冷えますがパワー広がりがあります：T ∝ (1 + s₀ + (2δ/Γ)²)/(2|δ|/Γ)。
原子が捕獲速度 ≈ Γ/2k より速いとほとんど相互作用しません：低温から始めるか、まず大きな離調を使いましょう。
2 準位モデル：超微細構造、暗状態、マイクロモーションは含みません。Ca⁺ などにはリポンプ光が必要です。値は教育用の見積もりで、実験結果ではありません。
📚 出典
💡 Parameter guide, physical basics and experiments
The 3D motion and cooling curves use the same clock. Spatial trajectories are projected onto the screen; rotating the camera does not change the ions. Dynamics use ideal fields, classical ions and independent two-level beams.
Each beam scatters at R = (Γ/2)s/[1+s+(2(δ−kv)/Γ)²]. Absorption gives ±ℏk momentum; spontaneous emission gives random 3D recoil. Axis temperatures use velocity variance after removing centre-of-mass drift, Tᵢ=m Var(vᵢ)/kB; mean temperature averages the axes. Micromotion and collective motion can enter this diagnostic; it is not automatically an equilibrium temperature.
| Parameter / quantity | Meaning, effect and use |
|---|---|
| Species / transition | Mass sets recoil and trap frequencies. Wavelength sets photon momentum; linewidth sets scattering and Doppler scales. The simplified beams omit repumper dynamics. |
| Detuning δ/Γ | Negative detuning usually damps velocity; positive detuning heats. Start at −0.5, then compare +0.5. Γ is the angular decay rate: Γ = 2π × linewidth in MHz. |
| Saturation s | Larger s increases scattering but also recoil and power broadening; it does not guarantee a colder cloud. Original/analytic panels use per-beam s; the 3D panel divides its intensity scale by six per active beam. |
| Initial temperature | Sets the initial Gaussian velocity variance, not the final equilibrium. A small sample need not equal the requested temperature exactly. Use a fixed seed when comparing conditions. |
| Particle count | More particles smooth statistical noise, but Coulomb pair computation grows as N². Begin with 8–12; use a small count for the actual RF model. |
| Emission recoil / photon model | Random photon emission heats while red-detuned absorption cools. Isotropic emission spreads recoil over three axes. The desktop mean-force option removes this diffusion and cannot predict a Doppler equilibrium. |
| Magnetic field B (T) | Penning only. Larger B raises the cyclotron frequency and changes the magnetron frequency and cooling window. Require ωc² > 2ωz². Fine decimal entry is a numerical setting, not field accuracy. |
| U₀ and characteristic length d | Analytic Penning model: ωz² = Q U₀/(m d²), so larger U₀ or smaller d tightens axial confinement but can destroy radial stability. The 3D model takes axial frequency directly. |
| Axial frequency νz (kHz) | Confines z. In a Paul trap its DC field defocuses x/y; in a Penning trap it changes both radial modes. Recheck stability after increasing it. |
| RF frequency and Mathieu q | Paul only. The time-averaged radial frequency obeys ωr² ≈ Ω²q²/8 − ωz²/2. Increasing q tightens confinement but increases micromotion. The displayed stability check is a secular/first-band guide, not a complete Floquet map. |
| Time-averaged versus actual RF | The averaged potential removes fast micromotion. Actual RF integrates the oscillating quadrupole and includes micromotion in kinetic temperature. Compare at the same seed and duration. Penning has no RF quadrupole in this model. |
| Radial beam offset yb (µm) | Penning: offset creates an intensity-gradient torque. Zero offset cools cyclotron motion but can heat magnetron motion. Suggest offset searches the analytic small-amplitude window; numerical 3D results need not have the same optimum. |
| Beam waist w (µm) | Gaussian intensity is proportional to exp[−2(y−yb)²/w²]. A narrower beam strengthens gradients but covers less of the orbit. Analytic rates are local approximations; use amplitude much smaller than w. |
| Beam tilt θ | Analytic Penning: increasing tilt supplies an axial cooling component while reducing the radial component. The fancier 3D Penning model instead has one radial +x beam and two axial ±z beams; it is a different beam geometry. |
| Initial r₊, r₋ and z amplitudes | Analytic Penning starts the three envelopes at this amplitude. Large values can leave the linear cooling regime. Start with 20 µm and a 100 µm waist. Inspect time shows later amplitudes; it does not change the initial value. |
| Initial cloud σ / relaxation | 3D: σ is the Gaussian width on each spatial axis. A random interacting cloud can release electrostatic energy and warm initially. Paul Relax initial positions prepares an approximate local configuration before the clock starts; it is not simulated cooling or a proven global minimum. |
| Coulomb repulsion | Repulsion redistributes energy and may create ordered structures. The force is softened at 0.3 µm to avoid a singular close encounter. Shape alone is not proof of a crystal phase. Penning begins with repulsion off for a clear single-particle comparison. |
| Observation duration / inspect time | Duration sets the physical end time; playback speed changes presentation. Analytic Show this time jumps to the chosen envelope time. Reset restarts at t=0. All 3D projections and temperatures share one clock. |
| Radial/axial ranges, zoom and camera | Ranges are half-widths in µm. Fixed limits reveal shrinkage; fit reveals detail. Zoom and camera never alter the physics. A clipped orbit is marked; clipping is distinct from an ion crossing the model aperture. |
| Random seed | Same seed plus identical settings and numerical steps repeats the stochastic run. Different seeds change photon histories. Compare trends over several seeds before interpreting small differences. |
| βrω₋ < F′ < βrω₊ | βr is the radial friction coefficient (N·s/m); F′ is the transverse force gradient (N/m). For βr>0, F′/(2πβr) must lie between ν₋ and ν₊. Negative amplitude rates cool; positive rates heat. The logarithmic frequency axis makes both boundaries visible. |
| PNG / CSV / settings | PNG saves the views; CSV saves physical traces and a parameter header. Browsers normally save to their configured download folder; choose your preferred local folder. Desktop exports include particle coordinates, inputs, structure diagnostics and PNG/PDF/SVG. No data are uploaded. |
| Centre view on cloud | Subtracts the cloud mean position from the drawing only. Combine with Fit whole view to resolve a cold cloud that has drifted off the trap centre. Coordinates are then relative to the cloud centre; exported physics and temperature remain unchanged. |
Suggested sequence: (1) compare red and blue detuning in optical molasses; (2) compare zero and suggested offset in analytic Penning, show 50 ms and fit the orbit; (3) choose Paul or Penning in the fancier panel and compare all three axes; (4) compare averaged and actual RF for Paul; (5) save CSV after fixing parameters and seed. Penning confinement has no positive-definite static 3D electric-potential minimum, so Paul static relaxation must not be applied to Penning.
Leibfried et al. (2003) · Hendricks et al., Penning cooling · Ca⁺ transition measurement
🌀 レーザー冷却 — ペニングトラップ中の 1 個のイオン
径方向のビームはサイクロトロン運動を冷やしますが、マグネトロン運動（エネルギーが負）を加熱します。ビームを中心からずらし、強度勾配でマグネトロン軌道を内側へ押します。傾ければ軸方向運動も冷やせます。
目標：100 ms 以内に 3 つの振幅すべてを 10 µm 未満に
Compare beam offset 0 with Suggest offset: the indicator enters the green window and the magnetron rate changes from positive to negative. Keep a fixed range to compare cooling; Fit current orbit magnifies the current motion. Inspect time changes the observation point without refitting.
📐 原理（物理の詳細）
ペニングトラップの運動：修正サイクロトロン ω₊、マグネトロン ω₋、軸方向 ω_z。ω₊ + ω₋ = ω_c = qB/m、ω₊ω₋ = ω_z²/2。
マグネトロン運動はエネルギーが負なので、摩擦で半径が増える：中心のビームは ω₊ を冷却するが ω₋ を加熱する。
Itano–Wineland 法：ビームをずらすと力の勾配 F′ が生じる。γ₊ = (F′ − β_rω₊)/(2mΔω)、γ₋ = (β_rω₋ − F′)/(2mΔω)。窓 β_rω₋ < F′ < β_rω₊ の内側でのみ両方が冷却される。和は γ₊ + γ₋ = −β_r/(2m)。
傾き θ により軸方向減衰 γ_z = −β sin²θ/(2m)、β_r = β cos²θ。
平衡振幅 A_eq = √(D/(−2γ))；A(t)² = A₀²e^{2γt} + D(e^{2γt} − 1)/(2γ)。出典：Itano & Wineland PRA 25, 35 (1982)；Hendricks ら arXiv:0709.3817。
💡 ヒント
ペニングトラップ：径方向の減衰は γ₊ + γ₋ = −β_r/2m と分配され、ビームのずれが減衰をサイクロトロンからマグネトロンへ移します。
2 準位モデル：超微細構造、暗状態、マイクロモーションは含みません。Ca⁺ などにはリポンプ光が必要です。値は教育用の見積もりで、実験結果ではありません。
💡 Parameter guide, physical basics and experiments
The 3D motion and cooling curves use the same clock. Spatial trajectories are projected onto the screen; rotating the camera does not change the ions. Dynamics use ideal fields, classical ions and independent two-level beams.
Each beam scatters at R = (Γ/2)s/[1+s+(2(δ−kv)/Γ)²]. Absorption gives ±ℏk momentum; spontaneous emission gives random 3D recoil. Axis temperatures use velocity variance after removing centre-of-mass drift, Tᵢ=m Var(vᵢ)/kB; mean temperature averages the axes. Micromotion and collective motion can enter this diagnostic; it is not automatically an equilibrium temperature.
| Parameter / quantity | Meaning, effect and use |
|---|---|
| Species / transition | Mass sets recoil and trap frequencies. Wavelength sets photon momentum; linewidth sets scattering and Doppler scales. The simplified beams omit repumper dynamics. |
| Detuning δ/Γ | Negative detuning usually damps velocity; positive detuning heats. Start at −0.5, then compare +0.5. Γ is the angular decay rate: Γ = 2π × linewidth in MHz. |
| Saturation s | Larger s increases scattering but also recoil and power broadening; it does not guarantee a colder cloud. Original/analytic panels use per-beam s; the 3D panel divides its intensity scale by six per active beam. |
| Initial temperature | Sets the initial Gaussian velocity variance, not the final equilibrium. A small sample need not equal the requested temperature exactly. Use a fixed seed when comparing conditions. |
| Particle count | More particles smooth statistical noise, but Coulomb pair computation grows as N². Begin with 8–12; use a small count for the actual RF model. |
| Emission recoil / photon model | Random photon emission heats while red-detuned absorption cools. Isotropic emission spreads recoil over three axes. The desktop mean-force option removes this diffusion and cannot predict a Doppler equilibrium. |
| Magnetic field B (T) | Penning only. Larger B raises the cyclotron frequency and changes the magnetron frequency and cooling window. Require ωc² > 2ωz². Fine decimal entry is a numerical setting, not field accuracy. |
| U₀ and characteristic length d | Analytic Penning model: ωz² = Q U₀/(m d²), so larger U₀ or smaller d tightens axial confinement but can destroy radial stability. The 3D model takes axial frequency directly. |
| Axial frequency νz (kHz) | Confines z. In a Paul trap its DC field defocuses x/y; in a Penning trap it changes both radial modes. Recheck stability after increasing it. |
| RF frequency and Mathieu q | Paul only. The time-averaged radial frequency obeys ωr² ≈ Ω²q²/8 − ωz²/2. Increasing q tightens confinement but increases micromotion. The displayed stability check is a secular/first-band guide, not a complete Floquet map. |
| Time-averaged versus actual RF | The averaged potential removes fast micromotion. Actual RF integrates the oscillating quadrupole and includes micromotion in kinetic temperature. Compare at the same seed and duration. Penning has no RF quadrupole in this model. |
| Radial beam offset yb (µm) | Penning: offset creates an intensity-gradient torque. Zero offset cools cyclotron motion but can heat magnetron motion. Suggest offset searches the analytic small-amplitude window; numerical 3D results need not have the same optimum. |
| Beam waist w (µm) | Gaussian intensity is proportional to exp[−2(y−yb)²/w²]. A narrower beam strengthens gradients but covers less of the orbit. Analytic rates are local approximations; use amplitude much smaller than w. |
| Beam tilt θ | Analytic Penning: increasing tilt supplies an axial cooling component while reducing the radial component. The fancier 3D Penning model instead has one radial +x beam and two axial ±z beams; it is a different beam geometry. |
| Initial r₊, r₋ and z amplitudes | Analytic Penning starts the three envelopes at this amplitude. Large values can leave the linear cooling regime. Start with 20 µm and a 100 µm waist. Inspect time shows later amplitudes; it does not change the initial value. |
| Initial cloud σ / relaxation | 3D: σ is the Gaussian width on each spatial axis. A random interacting cloud can release electrostatic energy and warm initially. Paul Relax initial positions prepares an approximate local configuration before the clock starts; it is not simulated cooling or a proven global minimum. |
| Coulomb repulsion | Repulsion redistributes energy and may create ordered structures. The force is softened at 0.3 µm to avoid a singular close encounter. Shape alone is not proof of a crystal phase. Penning begins with repulsion off for a clear single-particle comparison. |
| Observation duration / inspect time | Duration sets the physical end time; playback speed changes presentation. Analytic Show this time jumps to the chosen envelope time. Reset restarts at t=0. All 3D projections and temperatures share one clock. |
| Radial/axial ranges, zoom and camera | Ranges are half-widths in µm. Fixed limits reveal shrinkage; fit reveals detail. Zoom and camera never alter the physics. A clipped orbit is marked; clipping is distinct from an ion crossing the model aperture. |
| Random seed | Same seed plus identical settings and numerical steps repeats the stochastic run. Different seeds change photon histories. Compare trends over several seeds before interpreting small differences. |
| βrω₋ < F′ < βrω₊ | βr is the radial friction coefficient (N·s/m); F′ is the transverse force gradient (N/m). For βr>0, F′/(2πβr) must lie between ν₋ and ν₊. Negative amplitude rates cool; positive rates heat. The logarithmic frequency axis makes both boundaries visible. |
| PNG / CSV / settings | PNG saves the views; CSV saves physical traces and a parameter header. Browsers normally save to their configured download folder; choose your preferred local folder. Desktop exports include particle coordinates, inputs, structure diagnostics and PNG/PDF/SVG. No data are uploaded. |
| Centre view on cloud | Subtracts the cloud mean position from the drawing only. Combine with Fit whole view to resolve a cold cloud that has drifted off the trap centre. Coordinates are then relative to the cloud centre; exported physics and temperature remain unchanged. |
Suggested sequence: (1) compare red and blue detuning in optical molasses; (2) compare zero and suggested offset in analytic Penning, show 50 ms and fit the orbit; (3) choose Paul or Penning in the fancier panel and compare all three axes; (4) compare averaged and actual RF for Paul; (5) save CSV after fixing parameters and seed. Penning confinement has no positive-definite static 3D electric-potential minimum, so Paul static relaxation must not be applied to Penning.
Leibfried et al. (2003) · Hendricks et al., Penning cooling · Ca⁺ transition measurement
Orbits use the physical ν₊, ν₋ and νz phases at the displayed time; the trail covers the last six cyclotron periods. Playback uses an accelerated logarithmic clock and fast motion can visually alias. Drag the 3D view to rotate; wheel to zoom.
❄ Fancier simulation: 3D laser cooling in Paul / Penning traps
Compare the actual RF drive with the time-averaged potential. Watch micromotion, ion repulsion, red-detuned cooling and blue-detuned heating. All views share one simulation clock.
Trap design and inputs
Axial frequency is calculated from endcap voltage, geometry and ion mass. Enter physical voltages normally; advanced mode sets Mathieu q directly. Species changes preserve voltages: check stability again. V_RF is the differential zero-to-peak amplitude between rod pairs.
Equations, variables and stability
Φ = (U_dc − V_RF cos Ωt)(x² − y²)/(2r₀²) + κ U_end (z² − (x²+y²)/2)/z₀².
q = 2 Q V_RF/(m r₀² Ω²); aₓ,ᵧ = 4Q(±U_dc/r₀² − κ U_end/z₀²)/(mΩ²); ωz² = 2QκU_end/(mz₀²); νx,y = βx,y Ω/(4π).
Penning: Φ = U₀(z² − ρ²/2)/(2d²), ωz² = QU₀/(md²), ωc = QB/m, ω± = (ωc ± √(ωc²−2ωz²))/2. d² = (z₀²+r₀²/2)/2 for ideal hyperbolic electrodes. Real plate electrodes require a calibrated field coefficient; plate spacing alone does not establish this ideal potential. Geometry and display range are different quantities. NIST Mathieu stability · Brown & Gabrielse
3D view: drag to rotate, wheel to zoom (use View zoom on touch screens); Fit restores the scale. Video records a 2–30 second clip under the current conditions. Short runs slow their display while recording; the physical clock remains explicit. Figures and video include a condition summary; save full conditions as JSON and numbers as CSV. Choose your preferred folder in the browser’s save dialog.
Initial phase-space area and ion-ball size
εₓ = √(Var(x)Var(vₓ) − Cov(x,vₓ)²), in µm·m/s. An uncorrelated Gaussian start has ε ≈ σₓ√(kBT/m). Set T from ε uses this relation and disables initial-position relaxation. Finite samples differ from the requested target; the status shows actual initial and current ε. Temperature uses the velocity sample variances after centre-of-mass subtraction; mean is the average of x/y/z. Initial T is a distribution target, so a small sample differs. Potential-energy release and micromotion can cause transient heating; curves are not forced to decrease. The Doppler line is a reference scale rather than a prediction for every condition. This is a phase-space area, rather than beam emittance in mm·mrad, which requires a longitudinal velocity. Ball radius changes the drawn marker only; it is not the ion’s physical radius.
Model, controls and sources
Defaults: ⁴⁰Ca⁺, 397 nm, Γ/2π=23.05 MHz. Run cooling, then compare blue heating or the RF preset. Changing physical parameters resets the trajectory; zoom changes only the drawing. Temperature uses velocity variance after subtracting centre-of-mass drift. Actual-RF temperature includes micromotion and is not a pure secular temperature.
Classical point ions, ideal fields, independent two-level beams and stochastic recoil; Coulomb softening 0.3 µm, adjustable rod radius/aperture. At least 40 integration steps per RF period. Multilevel repumping dynamics are omitted; this teaching model is not experimental or crystal-phase evidence. Leibfried et al., RMP 75, 281 (2003) · Ca⁺ transition source
Ion masses: AME2020 atomic mass minus electron mass; ionization binding corrections are omitted. Voltage mode keeps voltages and geometry when changing species; q and motional frequencies change with mass. The JSON conditions record the mass. AME2020
💡 Parameter guide, physical basics and experiments
The 3D motion and cooling curves use the same clock. Spatial trajectories are projected onto the screen; rotating the camera does not change the ions. Dynamics use ideal fields, classical ions and independent two-level beams.
Each beam scatters at R = (Γ/2)s/[1+s+(2(δ−kv)/Γ)²]. Absorption gives ±ℏk momentum; spontaneous emission gives random 3D recoil. Axis temperatures use velocity variance after removing centre-of-mass drift, Tᵢ=m Var(vᵢ)/kB; mean temperature averages the axes. Micromotion and collective motion can enter this diagnostic; it is not automatically an equilibrium temperature.
| Parameter / quantity | Meaning, effect and use |
|---|---|
| Species / transition | Mass sets recoil and trap frequencies. Wavelength sets photon momentum; linewidth sets scattering and Doppler scales. The simplified beams omit repumper dynamics. |
| Detuning δ/Γ | Negative detuning usually damps velocity; positive detuning heats. Start at −0.5, then compare +0.5. Γ is the angular decay rate: Γ = 2π × linewidth in MHz. |
| Saturation s | Larger s increases scattering but also recoil and power broadening; it does not guarantee a colder cloud. Original/analytic panels use per-beam s; the 3D panel divides its intensity scale by six per active beam. |
| Initial temperature | Sets the initial Gaussian velocity variance, not the final equilibrium. A small sample need not equal the requested temperature exactly. Use a fixed seed when comparing conditions. |
| Particle count | More particles smooth statistical noise, but Coulomb pair computation grows as N². Begin with 8–12; use a small count for the actual RF model. |
| Emission recoil / photon model | Random photon emission heats while red-detuned absorption cools. Isotropic emission spreads recoil over three axes. The desktop mean-force option removes this diffusion and cannot predict a Doppler equilibrium. |
| Magnetic field B (T) | Penning only. Larger B raises the cyclotron frequency and changes the magnetron frequency and cooling window. Require ωc² > 2ωz². Fine decimal entry is a numerical setting, not field accuracy. |
| U₀ and characteristic length d | Analytic Penning model: ωz² = Q U₀/(m d²), so larger U₀ or smaller d tightens axial confinement but can destroy radial stability. The 3D model takes axial frequency directly. |
| Axial frequency νz (kHz) | Confines z. In a Paul trap its DC field defocuses x/y; in a Penning trap it changes both radial modes. Recheck stability after increasing it. |
| RF frequency and Mathieu q | Paul only. The time-averaged radial frequency obeys ωr² ≈ Ω²q²/8 − ωz²/2. Increasing q tightens confinement but increases micromotion. The displayed stability check is a secular/first-band guide, not a complete Floquet map. |
| Time-averaged versus actual RF | The averaged potential removes fast micromotion. Actual RF integrates the oscillating quadrupole and includes micromotion in kinetic temperature. Compare at the same seed and duration. Penning has no RF quadrupole in this model. |
| Radial beam offset yb (µm) | Penning: offset creates an intensity-gradient torque. Zero offset cools cyclotron motion but can heat magnetron motion. Suggest offset searches the analytic small-amplitude window; numerical 3D results need not have the same optimum. |
| Beam waist w (µm) | Gaussian intensity is proportional to exp[−2(y−yb)²/w²]. A narrower beam strengthens gradients but covers less of the orbit. Analytic rates are local approximations; use amplitude much smaller than w. |
| Beam tilt θ | Analytic Penning: increasing tilt supplies an axial cooling component while reducing the radial component. The fancier 3D Penning model instead has one radial +x beam and two axial ±z beams; it is a different beam geometry. |
| Initial r₊, r₋ and z amplitudes | Analytic Penning starts the three envelopes at this amplitude. Large values can leave the linear cooling regime. Start with 20 µm and a 100 µm waist. Inspect time shows later amplitudes; it does not change the initial value. |
| Initial cloud σ / relaxation | 3D: σ is the Gaussian width on each spatial axis. A random interacting cloud can release electrostatic energy and warm initially. Paul Relax initial positions prepares an approximate local configuration before the clock starts; it is not simulated cooling or a proven global minimum. |
| Coulomb repulsion | Repulsion redistributes energy and may create ordered structures. The force is softened at 0.3 µm to avoid a singular close encounter. Shape alone is not proof of a crystal phase. Penning begins with repulsion off for a clear single-particle comparison. |
| Observation duration / inspect time | Duration sets the physical end time; playback speed changes presentation. Analytic Show this time jumps to the chosen envelope time. Reset restarts at t=0. All 3D projections and temperatures share one clock. |
| Radial/axial ranges, zoom and camera | Ranges are half-widths in µm. Fixed limits reveal shrinkage; fit reveals detail. Zoom and camera never alter the physics. A clipped orbit is marked; clipping is distinct from an ion crossing the model aperture. |
| Random seed | Same seed plus identical settings and numerical steps repeats the stochastic run. Different seeds change photon histories. Compare trends over several seeds before interpreting small differences. |
| βrω₋ < F′ < βrω₊ | βr is the radial friction coefficient (N·s/m); F′ is the transverse force gradient (N/m). For βr>0, F′/(2πβr) must lie between ν₋ and ν₊. Negative amplitude rates cool; positive rates heat. The logarithmic frequency axis makes both boundaries visible. |
| PNG / CSV / settings | PNG saves the views; CSV saves physical traces and a parameter header. Browsers normally save to their configured download folder; choose your preferred local folder. Desktop exports include particle coordinates, inputs, structure diagnostics and PNG/PDF/SVG. No data are uploaded. |
| Centre view on cloud | Subtracts the cloud mean position from the drawing only. Combine with Fit whole view to resolve a cold cloud that has drifted off the trap centre. Coordinates are then relative to the cloud centre; exported physics and temperature remain unchanged. |
Suggested sequence: (1) compare red and blue detuning in optical molasses; (2) compare zero and suggested offset in analytic Penning, show 50 ms and fit the orbit; (3) choose Paul or Penning in the fancier panel and compare all three axes; (4) compare averaged and actual RF for Paul; (5) save CSV after fixing parameters and seed. Penning confinement has no positive-definite static 3D electric-potential minimum, so Paul static relaxation must not be applied to Penning.
Leibfried et al. (2003) · Hendricks et al., Penning cooling · Ca⁺ transition measurement
❄ 3D optical molasses: cooling and cloud expansion
Six counterpropagating Gaussian beams act on independent neutral atoms. Molasses damps velocity without a restoring force; a MOT combines a quadrupole magnetic field and circular polarization to restore displaced atoms. Compare temperature, bulk drift and cloud size separately.
Browser semiclassical teaching model: ideal F=0 → F′=1, instantaneous steady scattering, shared-saturation approximation and isotropic spontaneous-emission diffusion. Alkali entries are effective-line examples; their real hyperfine structure, repumping, polarization-gradient, sub-Doppler and gray-molasses dynamics are omitted.
Transition data and approximations
Γ/2π is given in MHz; the kernel uses Γ (rad/s), λ (m) and atomic mass (kg). Recoil-resolved transitions are rejected rather than treated as valid diffusion cooling.
Beams, magnetic field and initial cloud
Integration, duration and loss criterion
Stochastic diffusion integration with additional damping, restoring-force, beam-transit and Zeeman-change timestep limits. Optical cycles, internal-state transients and individual sparse photon events are not resolved; photon counts are expectations. Playback speed never changes the physical timestep. Work is capped at 20 million particle steps; halve the timestep to check convergence. The loss radius is a user-defined diagnostic boundary, not a MOT wall; survivor temperature uses only atoms still inside it.
View and value inspection
Drag the 3D plot to rotate and use the wheel to zoom. Hover over a point or curve for values; click to pin them in the textbox. View changes keep the physics; physical edits stop and reset the run. Hidden method tabs pause immediately without background simulation.
PNG and PDF use the same selected views; the PDF contains a high-resolution image. JSON saves conditions, view, actual timestep and statistics; restore starts from the same seed rather than resuming the random state midway. CSV saves the sampled history and particle snapshot. Recorded video has a separate save link below its browser preview; choose your folder in the system save dialog.
Physics, equations, limits and sources
Rᵢ = (Γ/2) Σ_q sᵢ |εᵢ,q|² / [1 + Σⱼ sⱼ + 4(Δᵢ,q/Γ)²], Δᵢ,q = δ − kᵢ·v − q μ_eff |B|/ℏ. F = Σᵢ ℏ kᵢ Rᵢ. B = (−b x/2, −b y/2, b z).
Circular polarization is projected onto the local magnetic-field direction; strong and transverse axes have opposite helicities. Absorption shot noise and isotropic spontaneous emission give Var(dvⱼ)/dt = vᵣ²(Rⱼ,+ + Rⱼ,− + R_total/3). Tⱼ = m Σ(vⱼ−v̄ⱼ)²/[(N−1)k_B]; centroid drift is excluded. Recoil off is a deterministic comparison, not a physical cooling limit. The Doppler scale ℏΓ/(2k_B) is a reference; finite beams, saturation, magnetic fields, gravity and escape alter equilibrium and capture.
This program does not call PyLCP; its heuristic equation supplies the formula reference. At high velocity or saturation this approximation differs from full rate equations or optical Bloch equations. Multiple scattering, collisions, dark states, coherence, dipole forces, magnetic-dipole forces and multilevel repumping are omitted.
Eckel et al., PyLCP, Computer Physics Communications 270 (2022) 108166 · Eq. 35 · Author paper (NIST) · PyLCP documentation · Alkali transition data · AME2020
❄ Magneto-optical trap: six-beam 3D capture and cooling
Six counterpropagating Gaussian beams act on independent neutral atoms. Molasses damps velocity without a restoring force; a MOT combines a quadrupole magnetic field and circular polarization to restore displaced atoms. Compare temperature, bulk drift and cloud size separately.
Browser semiclassical teaching model: ideal F=0 → F′=1, instantaneous steady scattering, shared-saturation approximation and isotropic spontaneous-emission diffusion. Alkali entries are effective-line examples; their real hyperfine structure, repumping, polarization-gradient, sub-Doppler and gray-molasses dynamics are omitted.
Transition data and approximations
Γ/2π is given in MHz; the kernel uses Γ (rad/s), λ (m) and atomic mass (kg). Recoil-resolved transitions are rejected rather than treated as valid diffusion cooling.
Beams, magnetic field and initial cloud
Integration, duration and loss criterion
Stochastic diffusion integration with additional damping, restoring-force, beam-transit and Zeeman-change timestep limits. Optical cycles, internal-state transients and individual sparse photon events are not resolved; photon counts are expectations. Playback speed never changes the physical timestep. Work is capped at 20 million particle steps; halve the timestep to check convergence. The loss radius is a user-defined diagnostic boundary, not a MOT wall; survivor temperature uses only atoms still inside it.
View and value inspection
Drag the 3D plot to rotate and use the wheel to zoom. Hover over a point or curve for values; click to pin them in the textbox. View changes keep the physics; physical edits stop and reset the run. Hidden method tabs pause immediately without background simulation.
PNG and PDF use the same selected views; the PDF contains a high-resolution image. JSON saves conditions, view, actual timestep and statistics; restore starts from the same seed rather than resuming the random state midway. CSV saves the sampled history and particle snapshot. Recorded video has a separate save link below its browser preview; choose your folder in the system save dialog.
Physics, equations, limits and sources
Rᵢ = (Γ/2) Σ_q sᵢ |εᵢ,q|² / [1 + Σⱼ sⱼ + 4(Δᵢ,q/Γ)²], Δᵢ,q = δ − kᵢ·v − q μ_eff |B|/ℏ. F = Σᵢ ℏ kᵢ Rᵢ. B = (−b x/2, −b y/2, b z).
Circular polarization is projected onto the local magnetic-field direction; strong and transverse axes have opposite helicities. Absorption shot noise and isotropic spontaneous emission give Var(dvⱼ)/dt = vᵣ²(Rⱼ,+ + Rⱼ,− + R_total/3). Tⱼ = m Σ(vⱼ−v̄ⱼ)²/[(N−1)k_B]; centroid drift is excluded. Recoil off is a deterministic comparison, not a physical cooling limit. The Doppler scale ℏΓ/(2k_B) is a reference; finite beams, saturation, magnetic fields, gravity and escape alter equilibrium and capture.
This program does not call PyLCP; its heuristic equation supplies the formula reference. At high velocity or saturation this approximation differs from full rate equations or optical Bloch equations. Multiple scattering, collisions, dark states, coherence, dipole forces, magnetic-dipole forces and multilevel repumping are omitted.
Eckel et al., PyLCP, Computer Physics Communications 270 (2022) 108166 · Eq. 35 · Author paper (NIST) · PyLCP documentation · Alkali transition data · AME2020
🧪 周期表と元素の起源
化学ブロック、天体物理的な起源（元素ごとに複数）、状態、または物性のヒートマップで色分けし、起源ハイライトゲームで遊べます。
…
🧩 魔法数と元素
核図表についてのクイズ。
🦓 トラップ動物園
さまざまな装置がイオンを閉じ込め・導く様子のアニメーション（縮尺は不正確）。
アニメーションによる模式図（縮尺は不正確）。赤/青 = 電極の極性（RF 装置では RF とともに反転）；色付きの軌跡は1個のイオンまたは原子。