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附錄 英文翻譯資料
中文翻譯
正電原子在電離過程中碰撞的理論
摘要
我們回顧過去和現(xiàn)在正子原子在電離過程中碰撞理論的發(fā)展。 從最終狀態(tài)下合并所有相互作用，在一個(gè)同等立足處和保留少量碰撞動(dòng)力學(xué)的一個(gè)確切的物體分析開始, 我們進(jìn)行或重或輕不同的比較, 并且從它們影響電離橫剖面的角度進(jìn)行分析。 終于, 我們發(fā)現(xiàn)了理論碰撞過程中的連續(xù)統(tǒng)一體, 中心點(diǎn)和其它運(yùn)動(dòng)學(xué)機(jī)制 。
主題詞: 電離; 碰撞動(dòng)力學(xué); 驅(qū)散; 電子光譜; 反物質(zhì); 正電子沖擊; 中心點(diǎn)電子; 導(dǎo)軌式電子
1. 介紹
正電原子的簡(jiǎn)單電離碰撞由一個(gè)細(xì)小的結(jié)構(gòu)微粒沖擊, “三體問題”是很多年未解決的一個(gè)物理問題。 1609 年到1687 年“二體”問題由約翰尼.開普勒和由艾薩克?牛頓共同解決了。三體問題比二體問題更加復(fù)雜難懂, 除了一些特殊的現(xiàn)象，它不能被簡(jiǎn)單的分析解決。 1765年, 勒翰得. 依魯爾發(fā)現(xiàn)了原始在線的三大量和依然排列的一種"幾何" 解答。不少年后, 拉格朗日發(fā)現(xiàn)了五平衡點(diǎn)的存在, 今后大家都稱為拉格朗日點(diǎn)。
對(duì)三體驅(qū)散問題的解答，最早的是三百年前天文學(xué)家和數(shù)學(xué)家用數(shù)學(xué)工具和相似比的原理解答出來的。例如, 在大量的中心參考系統(tǒng)下, 我們?cè)?836 年描述三體問題由任何空間座標(biāo)都可能的原因已經(jīng)由杰庫(kù)比介紹。所有這些對(duì)由線形點(diǎn)標(biāo)準(zhǔn)變革關(guān)系, 如所描述[ 1 ] 。在動(dòng)量空間, 系統(tǒng)由伴生的描述(千噸), (千焦) 和(千牛) 。 交換對(duì)實(shí)驗(yàn)室參考框架, 大量電子最后的動(dòng)量m, 許多MT (反沖) 目標(biāo)片段和大量MP 子彈頭可能被寫根據(jù)杰克比沖動(dòng)Kj 通過伽利略變換[ 1 ]得出
數(shù)十年, 電離過程的理論描述承擔(dān)了三體動(dòng)力學(xué)在最終狀態(tài)下的簡(jiǎn)單表示, 根據(jù)事實(shí)表明
（1）對(duì)于離子和原子碰撞, 一個(gè)微粒(電子) 比其它二兩個(gè)原子要輕。
（2）對(duì)于電子和正子原子碰撞, 一個(gè)微粒(目標(biāo)中堅(jiān)力量) 比其它兩個(gè)原子要重的多。
例如, 根據(jù)眾所周知的中心論據(jù), 離子和原子電離碰撞的理論描述的決大多數(shù)使用沖擊參數(shù)來設(shè)置, 那里子彈頭跟隨一條未受干擾的直線彈道在碰撞過程過程中, 并且目標(biāo)中堅(jiān)力量依然是休息[ 2 ] 。 它是確切, 假設(shè), 子彈頭隨后而來一條直線彈道沒有道理在電子或正子原子碰撞的理論描述。 但是, 它通常假設(shè), 目標(biāo)中堅(jiān)力量依然是不動(dòng)。
問題的這些簡(jiǎn)單化被介紹了在18 世紀(jì)。 unsolvable t三體問題被簡(jiǎn)化了, 對(duì)所謂的有限的三體問題, 那里一個(gè)微粒被承擔(dān)有一許多足夠小不影響其它二個(gè)微粒的行動(dòng)。 雖則介紹作為手段提供近似解答對(duì)系統(tǒng)譬如太陽(yáng)行星彗星在古典技工范圍內(nèi), 它廣泛被應(yīng)用在原子物理在所謂的沖擊參量略計(jì)對(duì)離子原子電離碰撞。
三體問題的其它簡(jiǎn)單化廣泛被使用在19 世紀(jì)假設(shè), 一個(gè)微粒比其它二巨型的并且依然是在大量的中心鎮(zhèn)定自若由其它二。 這略計(jì)廣泛被應(yīng)用在電子或正子原子電離碰撞。
2. 多個(gè)有差別的橫剖面
一個(gè)三體連續(xù)流最后狀態(tài)的一個(gè)運(yùn)動(dòng)學(xué)上完全描述在任一原子碰撞會(huì)要求, 原則上, 九可變物知識(shí), 譬如動(dòng)量的組分聯(lián)系了對(duì)每個(gè)三個(gè)微粒在最終狀態(tài)。 但是, 動(dòng)量和能源節(jié)約的情況減少這個(gè)數(shù)字
到五。 此外, 每當(dāng)最初的目標(biāo)不準(zhǔn)備在任何優(yōu)先方向, 多個(gè)有差別的橫剖面必須是相稱由三體系統(tǒng)的自轉(zhuǎn)在子彈頭的行動(dòng)的最初的方向附近。 因而, 擱置一邊三個(gè)片段的內(nèi)部結(jié)構(gòu)在最終狀態(tài), 只四
喪失九可變物是必要完全地描述驅(qū)散過程。 所以, 電離過程的一個(gè)完全描述特性也許被獲得以一個(gè)四倍有差別的橫剖面:
有許多可能的套四可變物使用。 為,事例, 我們能選擇了電子的方位角角度和其它二個(gè)微粒的當(dāng)中一個(gè), 相對(duì)角度在行動(dòng)之間飛機(jī), 并且一個(gè)微粒能量。
這樣選擇是任意的, 但完成在感覺, 其他套可變物可能與這一個(gè)有關(guān)。 獨(dú)立可變物一個(gè)相似的選擇是標(biāo)準(zhǔn)的為原子電離的描述由電子沖擊, 理論上和實(shí)驗(yàn)性地[ 3,4 ] 。
非常一般四倍有差別的橫剖面的圖片不是可行的。 因而, 它通常是必要減少可變物的數(shù)量在橫剖面。 這可能由修理達(dá)到一兩他們?cè)谀承┨厥鈨r(jià)值或情況。 例如, 我們也許任意地制約自己描述coplanar (i.e. ?=?0) 或a collinear motion (i.e. ?=?0 and θ1?=?θ2), 以便使問題的依賴性降低到三或二獨(dú)立可變物, 各自地。
另一選擇將集成四倍有差別的橫剖面在一個(gè)或更多可變物。
前廣泛被應(yīng)用學(xué)習(xí)電子碰撞, 當(dāng)后者是主要工具描繪離子原子和正子原子電離碰撞。 特別重要對(duì)唯一微粒分光學(xué)的用途, 那里動(dòng)量的微粒的當(dāng)中一個(gè)被測(cè)量。
3. 單個(gè)微粒的動(dòng)量分布
動(dòng)量發(fā)行為散發(fā)的電子和正子禮物幾個(gè)結(jié)構(gòu)。 首先, 我們能觀察門限在高電子或正子速度因?yàn)橛幸粋€(gè)極限在任一個(gè)微?？赡芪諒南到y(tǒng)的動(dòng)能。 第二個(gè)結(jié)構(gòu)是土坎被設(shè)置沿圈子。 它對(duì)應(yīng)于正子的二進(jìn)制碰撞與散發(fā)的電子, 用目標(biāo)中堅(jiān)力量充當(dāng)實(shí)際角色。 終于, 有尖頂和anticusp 在零速度在電子和正子動(dòng)量分布, 各自地。 第一個(gè)對(duì)應(yīng)于電子的勵(lì)磁于目標(biāo)的一個(gè)低能源連續(xù)流狀態(tài)。秒鐘是取盡由于正子的捕獲的不可能的事由目標(biāo)中堅(jiān)力量。 這些動(dòng)量發(fā)行允許我們學(xué)習(xí)電離碰撞的主要特征。 但是, 我們必須記住, 分析只微粒的當(dāng)中一個(gè)在最后狀態(tài)的任一個(gè)實(shí)驗(yàn)性技術(shù)可能只提供部份洞察入電離過程。 四倍有差別的橫剖面也許顯示由綜合化洗滌在這實(shí)驗(yàn)的碰撞物產(chǎn)。
4. 理論模型
我們想要討論在這通信的主要問題是如果有一些重要碰撞物產(chǎn)在正子原子碰撞, 那不是可測(cè)的，總共, 單或雙有差別的電離橫剖面, 并且那因?yàn)槲幢话l(fā)現(xiàn)。 為了了解這些結(jié)構(gòu)的起源, 我們對(duì)應(yīng)的橫剖面與那些比較被獲得在離子原子碰撞。 履行這個(gè)宗旨它是必要的有一種充分的量子機(jī)械治療能同時(shí)應(yīng)付電離碰撞由重和輕的子彈頭的沖擊是因此相等地可適用的- 例如- 對(duì)離子原子或正子原子碰撞。 一種理論與這特征將允許我們學(xué)習(xí)倍數(shù)任一個(gè)指定的特點(diǎn)的變動(dòng)
有差別的橫斷面當(dāng)許多聯(lián)系在片段之中變化。 特別是, 它會(huì)允許我們學(xué)習(xí)變異當(dāng)改變?cè)诙g制約了運(yùn)動(dòng)學(xué)情況。
第二重要點(diǎn)將對(duì)待所有互作用在最終狀態(tài)在一個(gè)同等立足處。 如同我們解釋了, 在離子原子碰撞, internuclear 互作用不充當(dāng)實(shí)際在散發(fā)的電子的動(dòng)量發(fā)行的角色和因此未被考慮在對(duì)應(yīng)的演算。 在這工作, 這假定被避免了。 橫剖面利益在這范圍內(nèi)是
轉(zhuǎn)折矩陣可能供選擇地被寫在崗位或預(yù)先的形式
那里擾動(dòng)潛力被定義
為出生類型初始狀態(tài)
哪些包括子彈頭的自由行動(dòng)和最初的一定的狀態(tài)Ui 目標(biāo), 并且擾動(dòng)潛力vi 簡(jiǎn)單地是正子電子和正子中堅(jiān)力量互作用的總和。 轉(zhuǎn)折矩陣也許然后被分解入二個(gè)期限依靠是否正子首先與目標(biāo)中堅(jiān)力量或電子相處融洽。
為了是一致的與動(dòng)力學(xué)的我們充分的治療, 它是必要描述最終狀態(tài)Wf 通過考慮所有互作用在同樣立足處的wavefunction 。 因而, 我們采取一個(gè)被關(guān)聯(lián)的C3 波浪作
那包括畸變Dj 為三活躍互作用。 在連續(xù)流波浪作用這個(gè)選擇的最后渠道擾動(dòng)潛力是[ 5 ]
在純凈的庫(kù)侖潛力情況下, 畸變被給
關(guān)于這個(gè)模型由佳瑞波帝和馬瑞吉拉[ 6 ] 提議為離子原子碰撞, 并且由Brauner 和布里格斯六年后為正子原子和電子碰撞[ 7 ] 。 但是, 在所有這些箱子問題的動(dòng)力學(xué)被簡(jiǎn)化了, 依照被談?wù)撛谠缦炔糠? 根據(jù)大非對(duì)稱在介入的片段的大量之間。 另外, Garibotti 和Miraglia 忽略了互作用潛力的矩陣元素在接踵而來的子彈頭和目標(biāo)離子之間, 并且做銳化的略計(jì)評(píng)估轉(zhuǎn)折矩陣元素。 這進(jìn)一步略計(jì)被取消了在紙由Berakdar 等。 (1992), 雖然他們保留許多制約在他們的離子沖擊電離分析。
5. 電子捕獲對(duì)連續(xù)流尖頂
讓我們回顧一些結(jié)果在立體幾何。 我們選擇作為二個(gè)獨(dú)立參量散發(fā)的電子動(dòng)量組分, 平行和垂線對(duì)正子子彈頭的行動(dòng)的最初的方向。 子彈頭的能量是1 keV 。圖2, 我們觀察三個(gè)不同結(jié)構(gòu): 二個(gè)極小值和土坎。
圖2
土坎的起源很好被了解。 它對(duì)應(yīng)于電子捕獲于連續(xù)流(ECC) 尖頂被發(fā)現(xiàn)在離子原子碰撞三十年前由Crooks 和Rudd [ 8 ] 。 他們測(cè)量了電子能量光譜在向前方向和確切地觀察了尖頂形狀峰頂在子彈頭的速度。 第一理論解釋[ 9 ] 表示, 它分流以與1 相似的方式k 。 這個(gè)尖頂結(jié)構(gòu)是很多實(shí)驗(yàn)性和理論研究焦點(diǎn)。
因?yàn)镋CC 尖頂是一個(gè)推測(cè)橫跨捕獲電離極限入高度激動(dòng)的一定的狀態(tài), 這個(gè)同樣作用必須是存在在正子原子碰撞。 實(shí)際上, 這樣作用的觀察聯(lián)系了假定物體的形成, 當(dāng)被預(yù)言的二十年前由布朗勒和布里格斯, 依然是一個(gè)有爭(zhēng)議的問題。
這爭(zhēng)執(zhí)的原因是那, 與離子對(duì)比盒, 正子外出的速度與那不是相似沖擊, 但主要傳播在角度和巨大。 因而沒有特殊速度在哪里尋找尖頂。 并且這一定是如此。 如果我們?cè)u(píng)估雙重有差別的橫剖面, 我們看見, 尖頂清楚地是可看見的在離子原子碰撞, 但非常溫和和被傳播的肩膀在正子原子碰撞。 因而, 觀察這結(jié)構(gòu)它是必要增加橫剖面的維度。 例如由考慮四倍有差別的橫剖面的零的程度裁減在collinear 幾何。
Kover 和Laricchia 測(cè)量了在1998 dr/dEedXkdXK 橫剖面在一個(gè)collinear 情況在零的程度, 為H2 的電離分子由100 keV 正子沖擊[ 10 ] 。 結(jié)構(gòu)依照為沖擊對(duì)重的離子被觀察那么尖銳不被定義由于占實(shí)驗(yàn)性窗口在正子的卷積
并且電子偵查。 從目標(biāo)反沖不充當(dāng)在這個(gè)實(shí)驗(yàn)性情況的重大角色, 當(dāng)前一般理論給結(jié)果相似與那些由Berakdar [ 11 ] 獲得, 并且兩個(gè)跟隨嚴(yán)密實(shí)驗(yàn)性價(jià)值。
這同樣實(shí)驗(yàn)由Sarkadi 和工友執(zhí)行了在氬電離由75 keV 氫核沖擊。 他們第一次測(cè)量了四倍有差別的電離橫剖面在collinear 幾何為離子原子碰撞, 并且發(fā)現(xiàn)ECC 尖頂和在正子沖擊在大角度。 在這種情況下, 我們必須保留動(dòng)力學(xué)的一個(gè)完全帳戶為了再生產(chǎn)實(shí)驗(yàn)性結(jié)果[ 12 ] 。
6. 托馬斯機(jī)制
現(xiàn)在讓我們走回到H2 的電離由1 keV 正子沖擊。 一個(gè)結(jié)構(gòu)在45 可能被觀察, 1993 年哪些象由于被預(yù)言了和被解釋了由Brauner 和布里格斯二個(gè)等效雙重碰撞機(jī)制干涉。 每個(gè)這些過程包括正子電子二進(jìn)制碰撞, 被偏折跟隨被90 輕的微粒的當(dāng)中一個(gè)被重的中堅(jiān)力量。 這個(gè)機(jī)制由托馬斯[ 13 ] 提議作為扼要負(fù)責(zé)任電子捕獲由快速的重的離子。 在這種情況下, 從電子和正子大量是相等的, 這兩個(gè)過程干涉在45 。 如果我們降低能量從1000 年eV 到100 eV, 這個(gè)結(jié)構(gòu)在45 消失, 與想法是一致的結(jié)果托馬斯機(jī)制是一個(gè)高能作用。 但有其它結(jié)構(gòu), 在大約22.5。我們?cè)谙聜€(gè)部分將考慮這個(gè)結(jié)構(gòu)。
7. 備鞍點(diǎn)機(jī)制
結(jié)構(gòu)的起源在大約22.5 一定更難辨認(rèn)。 對(duì)我們的最佳的知識(shí), 它以前未被預(yù)言在正子原子碰撞, 即使機(jī)制負(fù)責(zé)任它的起源幾乎已經(jīng)提議在離子原子碰撞二十年之內(nèi)以前。 想法是, 電子能從離子原子碰撞涌現(xiàn)由在在子彈頭和殘余的目標(biāo)離子潛力的備鞍點(diǎn)。 1772 年這個(gè)機(jī)制清楚地與平衡點(diǎn)的當(dāng)中一個(gè)有關(guān)由拉格朗日發(fā)現(xiàn), 或?qū)C(jī)制由Wannier 提議為低能源電子放射。 在 離子原子碰撞案件, 查尋這個(gè)機(jī)制的理論和實(shí)驗(yàn)性證據(jù)是陰暗由生動(dòng)的爭(zhēng)論[ 14-18 ] 。
在正子原子碰撞情況下, 為電子被困住在正子和殘余離子潛力的馬鞍, 電子和正子必須首先執(zhí)行二進(jìn)制碰撞以便最終獲得正確的速度
那里ei 是目標(biāo)的結(jié)合能在初始狀態(tài)。
能量和動(dòng)量保護(hù)原則的應(yīng)用表示, 正子偏離在角度
終于, 為電子涌現(xiàn)在方向和正子一樣, 它必須遭受隨后碰撞以殘余中堅(jiān)力量在a 托馬斯象過程。 在這第二碰撞, 電子由90 和殘余目標(biāo)離子反沖偏轉(zhuǎn)在形成大約135 角度與電子和正子的方向。 這個(gè)機(jī)制被描述在圖4.
因而, 檢查備鞍點(diǎn)的提案是正確的, 我們看是否我們的演算顯示與備鞍點(diǎn)電子生產(chǎn)的這個(gè)描述是一致的結(jié)構(gòu)。
圖 3
圖 4
極小值被觀察在無效性QDCS 。 圖3 和圖4 精確地設(shè)置早先條件在任何能量和角度三個(gè)微粒符合的那些點(diǎn)。 我們做了其它測(cè)試在備鞍點(diǎn)機(jī)制的有效性和無效性。 圖5 表示, 結(jié)構(gòu)完全出現(xiàn)從tp 期限。 這個(gè)結(jié)果與提出的機(jī)制是一致的, 那里備鞍點(diǎn)結(jié)構(gòu)出現(xiàn)從第一正子電子碰撞之后, 正子和電子被中堅(jiān)力量驅(qū)散。
圖 5
8. 結(jié)論
總結(jié)結(jié)果提出了在這通信, 我們由正子的沖擊調(diào)查了分子氫的電離。 被獲得的四倍有差別的橫斷面為電子和正子涌現(xiàn)在同樣方向顯示三個(gè)統(tǒng)治結(jié)構(gòu)。 你是知名的電子捕獲對(duì)連續(xù)流峰頂。 另外一個(gè)是托馬斯機(jī)制。 終于, 有被解釋對(duì)象由于所謂的"備鞍點(diǎn)" 電離機(jī)制的極小值。
雖然主要結(jié)論研究的非常充分但也有一些不足。橫剖面也許會(huì)被很多巨大的困難所阻礙, 但值得高興的是, 我們一直沒有錯(cuò)過對(duì)問題許多不同的全方位的觀察, 唯一的遺憾就是對(duì)總橫剖面的研究。
英文原文
Theory of ionization processes in positron–atom collisions
Abstract
We review past and present theoretical developments in the description of ionization processes in positron–atom collisions. Starting from an analysis that incorporates all the interactions in the final state on an equal footing and keeps an exact account of the few-body kinematics, we perform a critical comparison of different approximations, and how they affect the evaluation of the ionization cross section. Finally, we describe the appearance of fingerprints of capture to the continuum, saddle-point and other kinematical mechanisms.
Keywords: Ionization; Collision dynamics; Scattering; Electron spectra; Antimatter; Positron impact; Saddle-point electrons; Wannier; CDW
PACS classification codes: 34.10.+x; 34.50.Fa
1. Introduction
The simple ionization collision of a hydrogenic atom by the impact of a structureless particle, the “three-body problem”, is one of the oldest unsolved problems in physics. The two-body problem was analyzed by Johannes Kepler in 1609 and solved by Isaac Newton in 1687. The three-body problem, on the other hand, is much more complicated and cannot be solved analytically, except in some particular cases. In 1765, for instance, Leonhard Euler discovered a “collinear” solution in which three masses start in a line and remain lined-up. Some years later, Lagrange discovered the existence of five equilibrium points, known as the Lagrange points. Even the most recent quests for solutions of the three-body scattering problem use similar mathematical tools and follow similar paths than those travelled by astronomers and mathematicians in the past three centuries. For instance, in the center-of-mass reference system, we describe the three-body problem by any of the three possible sets of the spatial coordinates already introduced by Jacobi in 1836. All these pairs are related by lineal point canonical transformations, as described in [1]. In momentum space, the system is described by the associated pairs (kT,KT), (kP,KP) and (kN,KN). Switching to the Laboratory reference frame, the final momenta of the electron of mass m, the (recoil) target fragment of mass MT and the projectile of mass MP can be written in terms of the Jacobi impulses Kj by means of Galilean transformations [1]
For decades, the theoretical description of ionization processes has assumed simplifications of the three-body kinematics in the final state, based on the fact that
? in an ion–atom collision, one particle (the electron) is much lighter than the other two,
? in an electron–atom or positron–atom collision, one particle (the target nucleus) is much heavier than the other two.
For instance, based on what is known as Wick’s argument, the overwhelming majority of the theoretical descriptions of ion–atom ionization collisions uses an impact-parameter approximation, where the projectile follows an undisturbed straight line trajectory throughout the collision process, and the target nucleus remains at rest [2]. It is clear that to assume that the projectile follows a straight line trajectory makes no sense in the theoretical description of electron or positron–atom collisions. However, it is usually assumed that the target nucleus remains motionless.
These simplifications of the problem were introduced in the eighteenth century. The unsolvable three-body problem was simplified, to the so-called restricted three-body problem, where one particle is assumed to have a mass small enough not to influence the motion of the other two particles. Though introduced as a means to provide approximate solutions to systems such as Sun–planet–comet within a Classical Mechanics framework, it has been widely used in atomic physics in the so-called impact-parameter approximation to ion–atom ionization collisions.
Another simplification of the three-body problem widely employed in the nineteenth century assumes that one of the particles is much more massive than the other two and remains in the center of mass unperturbed by the other two. This approximation has been widely used in electron–atom or positron–atom ionization collisions.
2. The multiple differential cross section
A kinematically complete description of a three-body continuum final-state in any atomic collision would require, in principle, the knowledge of nine variables, such as the components of the momenta associated to each of the three particles in the final state. However, the condition of momentum and energy conservation reduces this number to five. Furthermore, whenever the initial targets are not prepared in any preferential direction, the multiple differential cross section has to be symmetric by a rotation of the three-body system around the initial direction of motion of the projectile. Thus, leaving aside the internal structure of the three fragments in the final state, only four out of nine variables are necessary to completely describe the scattering process. Therefore, a complete characterization of the ionization process may be obtained with a quadruple differential cross section:
There are many possible sets of four variables to use. For, instance, we can chose azimuthal angles of the electron and of one of the other two particles, the relative angle between the planes of motion, and the energy of one particle.
Such a choice is arbitrary, but complete in the sense that any other set of variables can be related to this one. A similar choice of independent variables has been standard for the description of atomic ionization by electron impact, both theoretically and experimentally [3] and [4].
A picture of the very general quadruple differential cross section is not feasible. Thus, it is usually necessary to reduce the number of variables in the cross section. This can be achieved by fixing one or two of them at certain particular values or conditions. For instance, we might arbitrarily restrict ourselves to describe a coplanar (i.e. ?=?0) or a collinear motion (i.e. ?=?0 and θ1?=?θ2), so as to reduce the dependence of the problem to three or two independent variables, respectively.
The other option is to integrate the quadruple differential cross section over one or more variables.
The former has been widely used to study electron–atom collisions, while the latter has been the main tool to characterize ion–atom and positron–atom ionization collisions. Particularly important has been the use of single particle spectroscopy, where the momentum of one of the particles is measured.
3. Single particle momentum distributions
In ionization by positron impact it is feasible to study the momentum distribution of any of the involved fragments. As is shown in Fig. 1, the momentum distributions for the emitted electron and the positron present several structures. First, we can observe a threshold at high electron or positron velocities because there is a limit in the kinetic energy that any particle can absorb from the system. The second structure is a ridge set along a circle. It corresponds to a binary collision of the positron with the emitted electron, with the target nucleus playing practically no role. Finally, there is a cusp and an anticusp at zero velocity in the electron and positron momentum distributions, respectively. The first one corresponds to the excitation of the electron to a low-energy continuum state of the target. The second is a depletion due to the impossibility of capture of the positron by the target nucleus. These momentum distributions allow us to study the main characteristics of ionization collisions. However, we have to keep in mind that any experimental technique that analyzes only one of the particles in the final-state can only provide a partial insight into the ionization processes. The quadruple differential cross sections might display collision properties that are washed out by integration in this kind of experiments.
Fig. 1.?Electron and positron momentum distributions for the ionization of helium by impact of positrons with incident velocity v?=?12?a.u.
4. Theoretical model
The main question that we want to address in this communication is if there are some important collision properties in positron–atom collisions, that are not observable in total, single or double differential ionization cross sections, and that therefore have not yet been discovered. In order to understand the origin of these structures, we compare the corresponding cross sections with those obtained in ion–atom collisions. To fulfill this objective it is necessary to have a full quantum-mechanical treatment able to deal simultaneously with ionization collisions by impact of both heavy and light projectiles that is therefore equally applicable – for instance – to ion–atom or positron–atom collisions. A theory with this characteristics will allow us to study the changes of any given feature of multiple-differential cross-sections when the mass relations among the fragments vary. In particular, it would allow us to study the variation when changing between the two restricted kinematical situations.
The second important point is to treat all the interactions in the final state on an equal footing. As we have just explained, in ion–atom collisions, the internuclear interaction plays practically no role in the momentum distribution of the emitted electron and has therefore not been considered in the corresponding calculation. In this work, this kind of assumption has been avoided.
The cross section of interest within this framework is
The transition matrix can be alternatively written in post or prior forms as
where the perturbation potentials are defined by (H???E)Ψi?=?Vi Ψi and (H???E)Ψf?=?VfΨf.
For the Born-type initial state
which includes the free motion of the projectile and the initial bound state Φi of the target, and the perturbation potential Vi is simply the sum of the positron–electron and positron–nucleus interactions. The transition matrix may then be decomposed into two terms
depending on whether the positron interacts first with the target nucleus or the electron.
In order to be consistent with our full treatment of the kinematics, it is necessary to describe the final state by means of a wavefunction that considers all the interactions on the same footing. Thus, we resort to a correlated C3 wave function
that includes distortions for the three active interactions. The final-channel perturbation potential for this choice of continuum wave function is [5]
(1)
In the case of pure coulomb potentials, the distortions are given by
with νj?=?mjZj/kj. This model was proposed by Garibotti and Miraglia [6] for ion–atom collisions, and by Brauner and Briggs six years later for positron–atom and electron–atom collisions [7]. However, in all these cases the kinematics of the problem was simplified, as discussed in the previous section, on the basis of the large asymmetry between the masses of the fragments involved. In addition, Garibotti and Miraglia neglected the matrix element of the interaction potential between the incoming projectile and the target ion, and made a peaking approximation to evaluate the transition matrix element. This further approximation was removed in a paper by Berakdar et al. (1992), although they kept the mass restrictions in their ion-impact ionization analysis.
5. The electron capture to the continuum cusp
Let us review some results in a collinear geometry. We choose as the two independent parameters the emitted electron momentum components, parallel and perpendicular to the initial direction of motion of the positron projectile. The energy of the projectile is 1?keV. In Fig. 2, we observe three different structures: two minima and a ridge.
Fig. 2.?QDCS for ionization of H2 by impact of 1?keV positrons for emission of electrons in the direction of the projectile deflection.
The origin of the ridge is very well understood. It corresponds to the electron capture to the continuum (ECC) cusp discovered in ion–atom collisions three decades ago by Crooks and Rudd [8]. They measured the electron energy spectra in the forward direction and observed a cusp-shape peak at exactly the projectile’s velocity. The first theoretical explanation [9] showed that it diverges in the same way as 1/k. This cusp structure was the focus of a large amount of experimental and theoretical research.
Since the ECC cusp is an extrapolation across the ionization limit of capture into highly excited bound states, this same effect has to be present in positron–atom collisions. In fact, the observation of such an effect associated with positronium formation, while predicted two decades ago by Brauner and Briggs, remained a controversial issue. The reason for this dispute was that, in contrast to the case of ions, the positron outgoing velocity is not similar to that of impact, but is largely spread in angle and magnitude. Thus there is no particular velocity where to look for the cusp. And this is certainly so. If we evaluate the double differential cross section, we see that the cusp is clearly visible in ion–atom collisions, but just a very mild and spread shoulder in positron–atom collisions. Thus, to observe this structure it is necessary to increase the dimension of the cross section. For instance by considering a zero degree cut of the quadruple differential cross section in collinear geometry.
Kover and Laricchia measured in 1998 the dσ/dEe?dΩk?dΩK cross section in a collinear condition at zero degree, for the ionization of H2 molecules by 100?keV positron impact [10]. The structure is not so sharply defined as for impact observed for heavy ions because of the convolution that accounts for the experimental window in the positron and electron detection. Since the target recoil plays no significant role in this experimental situation, the present general theory gives results similar to those obtained by Berakdar [11], and both closely follow the experimental values.
The same kind of experiment was performed by Sarkadi and coworkers in Argon ionization by 75?keV proton impact. They measured the quadruple differential ionization cross section in a collinear geometry for ion–atom collisions for the first time, and found the ECC cusp as in positron impact at large angles. In this case, we have to keep a compl