Bohrren eredu atomikoa - Wikipedia, entziklopedia askea.

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class="vector-toc-list"> </ul> </li> <li id="toc-Rydberg-en_formula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rydberg-en_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rydberg-en formula</span> </div> </a> <ul id="toc-Rydberg-en_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Franck-Hertzen_esperimentua" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Franck-Hertzen_esperimentua"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Franck-Hertzen esperimentua</span> </div> </a> <ul id="toc-Franck-Hertzen_esperimentua-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bohr-en_ereduaren_akatsak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bohr-en_ereduaren_akatsak"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bohr-en ereduaren akatsak</span> </div> </a> <ul id="toc-Bohr-en_ereduaren_akatsak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Erreferentziak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Erreferentziak"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Erreferentziak</span> </div> </a> <ul id="toc-Erreferentziak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliografia</span> </div> </a> <button aria-controls="toc-Bibliografia-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Erakutsi/ezkutatu Bibliografia azpiatal</span> </button> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> <li id="toc-Gehiago_sakontzeko" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gehiago_sakontzeko"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Gehiago sakontzeko</span> </div> </a> <ul id="toc-Gehiago_sakontzeko-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kanpo_estekak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kanpo_estekak"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Kanpo estekak</span> </div> </a> <ul id="toc-Kanpo_estekak-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Edukiak" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Eduki-taula" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Eduki taularen ikusgarritasuna aldatu" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Eduki taularen ikusgarritasuna aldatu</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Bohrren eredu atomikoa</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Joan beste hizkuntza batean idatzitako artikulu batera. 70 hizkuntzatan eskuragarri." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-70" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">70 hizkuntza</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Bohrsches_Atommodell" title="Bohrsches Atommodell – Suitzako alemana" lang="gsw" hreflang="gsw" data-title="Bohrsches Atommodell" data-language-autonym="Alemannisch" data-language-local-name="Suitzako alemana" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Modelo_atomico_de_Bohr" title="Modelo atomico de Bohr – aragoiera" lang="an" hreflang="an" data-title="Modelo atomico de Bohr" data-language-autonym="Aragonés" data-language-local-name="aragoiera" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D9%85%D9%88%D8%B0%D8%AC_%D8%A8%D9%88%D8%B1" title="نموذج بور – arabiera" lang="ar" hreflang="ar" data-title="نموذج بور" data-language-autonym="العربية" data-language-local-name="arabiera" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Modelu_at%C3%B3micu_de_Bohr" title="Modelu atómicu de Bohr – asturiera" lang="ast" hreflang="ast" data-title="Modelu atómicu de Bohr" data-language-autonym="Asturianu" data-language-local-name="asturiera" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bor_modeli" title="Bor modeli – azerbaijanera" lang="az" hreflang="az" data-title="Bor modeli" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijanera" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%B0%D1%9E%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B0%D0%B4%D1%8D%D0%BB%D1%8C_%D0%B0%D1%82%D0%B0%D0%BC%D0%B0" title="Бораўская мадэль атама – bielorrusiera" lang="be" hreflang="be" data-title="Бораўская мадэль атама" data-language-autonym="Беларуская" data-language-local-name="bielorrusiera" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D0%B5%D0%BB_%D0%BD%D0%B0_%D0%91%D0%BE%D1%80" title="Модел на Бор – bulgariera" lang="bg" hreflang="bg" data-title="Модел на Бор" data-language-autonym="Български" data-language-local-name="bulgariera" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%8B%E0%A6%B0_%E0%A6%AE%E0%A6%A1%E0%A7%87%E0%A6%B2" title="বোর মডেল – bengalera" lang="bn" hreflang="bn" data-title="বোর মডেল" data-language-autonym="বাংলা" data-language-local-name="bengalera" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma – bosniera" lang="bs" hreflang="bs" data-title="Bohrov model atoma" data-language-autonym="Bosanski" data-language-local-name="bosniera" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Model_at%C3%B2mic_de_Bohr" title="Model atòmic de Bohr – katalana" lang="ca" hreflang="ca" data-title="Model atòmic de Bohr" data-language-autonym="Català" data-language-local-name="katalana" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bohr%C5%AFv_model_atomu" title="Bohrův model atomu – txekiera" lang="cs" hreflang="cs" data-title="Bohrův model atomu" data-language-autonym="Čeština" data-language-local-name="txekiera" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bohrs_atommodel" title="Bohrs atommodel – daniera" lang="da" hreflang="da" data-title="Bohrs atommodel" data-language-autonym="Dansk" data-language-local-name="daniera" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bohrsches_Atommodell" title="Bohrsches Atommodell – alemana" lang="de" hreflang="de" data-title="Bohrsches Atommodell" data-language-autonym="Deutsch" data-language-local-name="alemana" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%84%CE%BF%CE%BC%CE%B9%CE%BA%CF%8C_%CF%80%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF_%CF%84%CE%BF%CF%85_%CE%9C%CF%80%CE%BF%CF%81" title="Ατομικό πρότυπο του Μπορ – greziera" lang="el" hreflang="el" data-title="Ατομικό πρότυπο του Μπορ" data-language-autonym="Ελληνικά" data-language-local-name="greziera" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Bohr_model" title="Bohr model – ingelesa" lang="en" hreflang="en" data-title="Bohr model" data-language-autonym="English" data-language-local-name="ingelesa" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Modelo_at%C3%B3mico_de_Bohr" title="Modelo atómico de Bohr – gaztelania" lang="es" hreflang="es" data-title="Modelo atómico de Bohr" data-language-autonym="Español" data-language-local-name="gaztelania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bohri_aatomiteooria" title="Bohri aatomiteooria – estoniera" lang="et" hreflang="et" data-title="Bohri aatomiteooria" data-language-autonym="Eesti" data-language-local-name="estoniera" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D9%84_%D8%A8%D9%88%D8%B1" title="مدل بور – persiera" lang="fa" hreflang="fa" data-title="مدل بور" data-language-autonym="فارسی" data-language-local-name="persiera" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bohrin_atomimalli" title="Bohrin atomimalli – finlandiera" lang="fi" hreflang="fi" data-title="Bohrin atomimalli" data-language-autonym="Suomi" data-language-local-name="finlandiera" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mod%C3%A8le_de_Bohr" title="Modèle de Bohr – frantsesa" lang="fr" hreflang="fr" data-title="Modèle de Bohr" data-language-autonym="Français" data-language-local-name="frantsesa" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Samhail_Bohr" title="Samhail Bohr – irlandera" lang="ga" hreflang="ga" data-title="Samhail Bohr" data-language-autonym="Gaeilge" data-language-local-name="irlandera" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Modelo_at%C3%B3mico_de_Bohr" title="Modelo atómico de Bohr – galiziera" lang="gl" hreflang="gl" data-title="Modelo atómico de Bohr" data-language-autonym="Galego" data-language-local-name="galiziera" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AC%E0%AB%89%E0%AA%B9%E0%AA%B0%E0%AA%A8%E0%AB%8B_%E0%AA%B8%E0%AA%BF%E0%AA%A6%E0%AB%8D%E0%AA%A7%E0%AA%BE%E0%AA%82%E0%AA%A4" title="બૉહરનો સિદ્ધાંત – gujaratera" lang="gu" hreflang="gu" data-title="બૉહરનો સિદ્ધાંત" data-language-autonym="ગુજરાતી" data-language-local-name="gujaratera" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%95%D7%93%D7%9C_%D7%94%D7%90%D7%98%D7%95%D7%9D_%D7%A9%D7%9C_%D7%91%D7%95%D7%94%D7%A8" title="מודל האטום של בוהר – hebreera" lang="he" hreflang="he" data-title="מודל האטום של בוהר" data-language-autonym="עברית" data-language-local-name="hebreera" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%8B%E0%A4%B0_%E0%A4%95%E0%A4%BE_%E0%A4%AA%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A3%E0%A5%81_%E0%A4%AE%E0%A5%89%E0%A4%A1%E0%A4%B2" title="बोर का परमाणु मॉडल – hindia" lang="hi" hreflang="hi" data-title="बोर का परमाणु मॉडल" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma – kroaziera" lang="hr" hreflang="hr" data-title="Bohrov model atoma" data-language-autonym="Hrvatski" data-language-local-name="kroaziera" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bohr-f%C3%A9le_atommodell" title="Bohr-féle atommodell – hungariera" lang="hu" hreflang="hu" data-title="Bohr-féle atommodell" data-language-autonym="Magyar" data-language-local-name="hungariera" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B8%D6%80%D5%AB_%D5%B4%D5%B8%D5%A4%D5%A5%D5%AC" title="Բորի մոդել – armeniera" lang="hy" hreflang="hy" data-title="Բորի մոդել" data-language-autonym="Հայերեն" data-language-local-name="armeniera" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Model_Bohr" title="Model Bohr – indonesiera" lang="id" hreflang="id" data-title="Model Bohr" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiera" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Modello_atomico_di_Bohr" title="Modello atomico di Bohr – italiera" lang="it" hreflang="it" data-title="Modello atomico di Bohr" data-language-autonym="Italiano" data-language-local-name="italiera" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%BC%E3%82%A2%E3%81%AE%E5%8E%9F%E5%AD%90%E6%A8%A1%E5%9E%8B" title="ボーアの原子模型 – japoniera" lang="ja" hreflang="ja" data-title="ボーアの原子模型" data-language-autonym="日本語" data-language-local-name="japoniera" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%91%E1%83%9D%E1%83%A0%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%9D%E1%83%93%E1%83%94%E1%83%9A%E1%83%98" title="ბორის მოდელი – georgiera" lang="ka" hreflang="ka" data-title="ბორის მოდელი" data-language-autonym="ქართული" data-language-local-name="georgiera" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%BE%D1%80_%D2%9B%D0%B0%D2%93%D0%B8%D0%B4%D0%B0%D0%BB%D0%B0%D1%80%D1%8B" title="Бор қағидалары – kazakhera" lang="kk" hreflang="kk" data-title="Бор қағидалары" data-language-autonym="Қазақша" data-language-local-name="kazakhera" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B4%EC%96%B4_%EB%AA%A8%ED%98%95" title="보어 모형 – koreera" lang="ko" hreflang="ko" data-title="보어 모형" data-language-autonym="한국어" data-language-local-name="koreera" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Boro_teorija" title="Boro teorija – lituaniera" lang="lt" hreflang="lt" data-title="Boro teorija" data-language-autonym="Lietuvių" data-language-local-name="lituaniera" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bora_atoma_uzb%C5%ABves_modelis" title="Bora atoma uzbūves modelis – letoniera" lang="lv" hreflang="lv" data-title="Bora atoma uzbūves modelis" data-language-autonym="Latviešu" data-language-local-name="letoniera" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%BE%D0%B2_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB" title="Боров модел – mazedoniera" lang="mk" hreflang="mk" data-title="Боров модел" data-language-autonym="Македонски" data-language-local-name="mazedoniera" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%8B%E0%B5%BC_%E0%B4%AE%E0%B4%BE%E0%B4%A4%E0%B5%83%E0%B4%95" title="ബോർ മാതൃക – malabarera" lang="ml" hreflang="ml" data-title="ബോർ മാതൃക" data-language-autonym="മലയാളം" data-language-local-name="malabarera" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AC%E0%A5%8B%E0%A4%B0%E0%A4%9A%E0%A5%80_%E0%A4%85%E0%A4%A3%E0%A5%82%E0%A4%9A%E0%A5%80_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A5%80" title="बोरची अणूची प्रतिकृती – marathera" lang="mr" hreflang="mr" data-title="बोरची अणूची प्रतिकृती" data-language-autonym="मराठी" data-language-local-name="marathera" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Model_Bohr" title="Model Bohr – malaysiera" lang="ms" hreflang="ms" data-title="Model Bohr" data-language-autonym="Bahasa Melayu" data-language-local-name="malaysiera" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%98%E1%80%AD%E1%80%AF%E1%80%B8%E1%80%A1%E1%80%AC%E1%81%8F_%E1%80%A1%E1%80%80%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%95%E1%80%AF%E1%80%B6%E1%80%85%E1%80%B6" title="ဘိုးအာ၏ အက်တမ်ပုံစံ – birmaniera" lang="my" hreflang="my" data-title="ဘိုးအာ၏ အက်တမ်ပုံစံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmaniera" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Atoommodel_van_Bohr" title="Atoommodel van Bohr – nederlandera" lang="nl" hreflang="nl" data-title="Atoommodel van Bohr" data-language-autonym="Nederlands" data-language-local-name="nederlandera" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Atommodell" title="Atommodell – nynorsk (norvegiera)" lang="nn" hreflang="nn" data-title="Atommodell" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk (norvegiera)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bohrs_atommodell" title="Bohrs atommodell – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Bohrs atommodell" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål (norvegiera)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AC%E0%A9%8B%E0%A8%B9%E0%A8%B0_%E0%A8%AE%E0%A8%BE%E0%A8%A1%E0%A8%B2" title="ਬੋਹਰ ਮਾਡਲ – punjabera" lang="pa" hreflang="pa" data-title="ਬੋਹਰ ਮਾਡਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabera" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Model_atomu_Bohra" title="Model atomu Bohra – poloniera" lang="pl" hreflang="pl" data-title="Model atomu Bohra" data-language-autonym="Polski" data-language-local-name="poloniera" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A8%D9%88%DB%81%D8%B1_%D9%85%D8%A7%DA%88%D9%84" title="بوہر ماڈل – Western Punjabi" lang="pnb" hreflang="pnb" data-title="بوہر ماڈل" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81tomo_de_Bohr" title="Átomo de Bohr – portugesa" lang="pt" hreflang="pt" data-title="Átomo de Bohr" data-language-autonym="Português" data-language-local-name="portugesa" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Modelul_atomic_Bohr" title="Modelul atomic Bohr – errumaniera" lang="ro" hreflang="ro" data-title="Modelul atomic Bohr" data-language-autonym="Română" data-language-local-name="errumaniera" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_%D0%B0%D1%82%D0%BE%D0%BC%D0%B0" title="Боровская модель атома – errusiera" lang="ru" hreflang="ru" data-title="Боровская модель атома" data-language-autonym="Русский" data-language-local-name="errusiera" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Mud%C3%A8llu_at%C3%B2micu_di_Bohr" title="Mudèllu atòmicu di Bohr – siziliera" lang="scn" hreflang="scn" data-title="Mudèllu atòmicu di Bohr" data-language-autonym="Sicilianu" data-language-local-name="siziliera" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Bohr_model" title="Bohr model – eskoziera" lang="sco" hreflang="sco" data-title="Bohr model" data-language-autonym="Scots" data-language-local-name="eskoziera" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma – serbokroaziera" lang="sh" hreflang="sh" data-title="Bohrov model atoma" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaziera" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bohr_model" title="Bohr model – Simple English" lang="en-simple" hreflang="en-simple" data-title="Bohr model" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Bohrov_model_at%C3%B3mu" title="Bohrov model atómu – eslovakiera" lang="sk" hreflang="sk" data-title="Bohrov model atómu" data-language-autonym="Slovenčina" data-language-local-name="eslovakiera" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma – esloveniera" lang="sl" hreflang="sl" data-title="Bohrov model atoma" data-language-autonym="Slovenščina" data-language-local-name="esloveniera" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%BE%D0%B2_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB_%D0%B0%D1%82%D0%BE%D0%BC%D0%B0" title="Боров модел атома – serbiera" lang="sr" hreflang="sr" data-title="Боров модел атома" data-language-autonym="Српски / srpski" data-language-local-name="serbiera" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bohrs_atommodell" title="Bohrs atommodell – suediera" lang="sv" hreflang="sv" data-title="Bohrs atommodell" data-language-autonym="Svenska" data-language-local-name="suediera" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%8B%E0%AE%B0%E0%AF%8D_%E0%AE%85%E0%AE%A3%E0%AF%81_%E0%AE%AE%E0%AE%BE%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AE%BF" title="போர் அணு மாதிரி – tamilera" lang="ta" hreflang="ta" data-title="போர் அணு மாதிரி" data-language-autonym="தமிழ்" data-language-local-name="tamilera" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AC%E0%B1%8B%E0%B0%B0%E0%B1%8D_%E0%B0%A8%E0%B0%AE%E0%B1%82%E0%B0%A8%E0%B0%BE" title="బోర్ నమూనా – telugua" lang="te" hreflang="te" data-title="బోర్ నమూనా" data-language-autonym="తెలుగు" data-language-local-name="telugua" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%A5%E0%B8%AD%E0%B8%87%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%82%E0%B8%9B%E0%B8%A3%E0%B9%8C" title="แบบจำลองของโปร์ – thailandiera" lang="th" hreflang="th" data-title="แบบจำลองของโปร์" data-language-autonym="ไทย" data-language-local-name="thailandiera" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Modelong_Bohr" title="Modelong Bohr – tagaloa" lang="tl" hreflang="tl" data-title="Modelong Bohr" data-language-autonym="Tagalog" data-language-local-name="tagaloa" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bohr_modeli" title="Bohr modeli – turkiera" lang="tr" hreflang="tr" data-title="Bohr modeli" data-language-autonym="Türkçe" data-language-local-name="turkiera" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D1%82%D0%BE%D0%BC%D0%BD%D0%B0_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_%D0%91%D0%BE%D1%80%D0%B0" title="Атомна модель Бора – ukrainera" lang="uk" hreflang="uk" data-title="Атомна модель Бора" data-language-autonym="Українська" data-language-local-name="ukrainera" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Bor_postulatlari" title="Bor postulatlari – uzbekera" lang="uz" hreflang="uz" data-title="Bor postulatlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekera" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%C3%B4_h%C3%ACnh_Bohr" title="Mô hình Bohr – vietnamera" lang="vi" hreflang="vi" data-title="Mô hình Bohr" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamera" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8E%BB%E5%B0%94%E6%A8%A1%E5%9E%8B" title="玻尔模型 – wu txinera" lang="wuu" hreflang="wuu" data-title="玻尔模型" data-language-autonym="吴语" data-language-local-name="wu txinera" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%8E%BB%E5%B0%94%E6%A8%A1%E5%9E%8B" title="玻尔模型 – txinera" lang="zh" hreflang="zh" data-title="玻尔模型" data-language-autonym="中文" data-language-local-name="txinera" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%8E%BB%E7%88%BE%E6%A8%A1%E5%9E%8B" title="玻爾模型 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="玻爾模型" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%8E%BB%E7%88%BE%E6%A8%A1%E5%9E%8B" title="玻爾模型 – kantonera" lang="yue" hreflang="yue" data-title="玻爾模型" data-language-autonym="粵語" data-language-local-name="kantonera" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q172948#sitelinks-wikipedia" title="Aldatu hizkuntzen arteko loturak" class="wbc-editpage">Aldatu loturak</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Bohr_atom_model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/304px-Bohr_atom_model.svg.png" decoding="async" width="304" height="265" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/456px-Bohr_atom_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/608px-Bohr_atom_model.svg.png 2x" data-file-width="310" data-file-height="270" /></a><figcaption><a href="/wiki/Hidrogeno" title="Hidrogeno">Hidrogeno</a> atomoaren <b>Bohrren eredu atomikoa</b>, non negatiboki kargaturiko elektroiak positiboki kargatutako nukleoari biraka dabiltzan. Elektroiak energia maila batetik bestera pasatzeko, (irudian marratxo beltzez irudikatuta) <i>hν</i> energia elektromagnetiko kantitate bat igorri edo jaso behar dute. Orbiten erradioa <i>n</i><sup>2</sup>gaz haunditzen da, <i>n</i> <a href="/wiki/Zenbaki_kuantiko_nagusi" title="Zenbaki kuantiko nagusi">zenbaki kuantiko nagusia</a> delarik. Elektroia hirugarren mailatik bigarrenera pasatzen denean (irudian bezala) <a href="/w/index.php?title=Balmer_serie&action=edit&redlink=1" class="new" title="Balmer serie (sortu gabe)">Balmer seriearen</a> lehenengo marra sortzen du, eta hidrogenoarentzat (Z = 1) igorritako fotoiaren <a href="/wiki/Uhin-luzera" title="Uhin-luzera">uhin-luzera</a> 656 nm da, <a href="/wiki/Claudio_Etxeberria" title="Claudio Etxeberria">gorria</a></figcaption></figure> <p><b>Bohr-en eredu atomikoa</b><sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr-ek</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> proposatutako eredu atomikoa da, <a href="/wiki/Geiger-Marsden_saiakuntza" title="Geiger-Marsden saiakuntza">Rutherforden urre xafla saiakuntzan</a> lortutako emaitzetan oinarrituta dago. Bohr-en eredua, <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Rutherfordenaren</a> antzera, planetarioa da, hau da, erdian positiboki <a href="/wiki/Karga_elektriko" title="Karga elektriko">kargaturiko</a> nukleo bat dauka eta <a href="/wiki/Elektroi" title="Elektroi">elektroiak</a>, beraien karga negatiboarekin, nukleo honi biraka dabiltza. <a href="/wiki/Grabitazio" title="Grabitazio">Grabitazio</a> indarraren ordez, <a href="/w/index.php?title=Energia_elektrostatiko&action=edit&redlink=1" class="new" title="Energia elektrostatiko (sortu gabe)">energia elektrostatikoak</a> tokian mantentzen ditu. <a href="/wiki/Max_Planck" title="Max Planck">Plancken</a> energiaren izaera kuantikoan oinarrituta, Bohrrek elektroien orbitak diskretuak direla proposatu zuen, hau da, elektroien orbitak energia zehatz batzuk bakarrik izan ditzakete. <a href="/wiki/Thomsonen_eredu_atomikoa" title="Thomsonen eredu atomikoa">Thomsonen eredu atomikoaren</a> jarraitzailea izan zen eta <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Rutherfordek</a> lehenago proposatutako eredu atomikoaren hobekuntza bat zenez, askotan Rutherford-Bohr eredu atomiko bezala ere ezagutzen da. </p><p><a href="/wiki/1913" title="1913">1913an</a> proposatu zuen Bohr-ek bere eredua eta arrakasta <a href="/w/index.php?title=Rydbergen_formula&action=edit&redlink=1" class="new" title="Rydbergen formula (sortu gabe)">Rydberg formulak</a> hidrogenorako aurresaten zituen espektroen lerroak azaltzearekin batera etorri zen. Bohrren ereduaren aurretik Rydberg formula esperimentalki frogatuta zegoen baina teorikiko ezin zen azaldu. Bohrren azalpenak Rydbergen formularen emaitzei oinarri teorikoa emateaz gain, formulan <a href="/wiki/Konstante_(matematika)" title="Konstante (matematika)">funtsezko konstante</a> fisiko hainbat zeudela ere frogatu zuen. </p><p>Bohrren eredua hidrogenozko atomoaren eredu sinple bat da baina hainbat sistemetarako balio du. Horregatik gaur egun <a href="/wiki/Mekanika_kuantiko" title="Mekanika kuantiko">mekanika kuantikoaren</a> sarrera bezala erakusten da ikastokietan, nahiz eta berez Bohrren eredua teoria zientifiko zaharkitua izan. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=1" title="Aldatu atal hau: «Historia»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=1" title="Edit section's source code: Historia"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/XX._mendea" title="XX. mendea">XX. mendearen</a> hasieran <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a> eta beste hainbat fisikariek egindako hainbat saiakuntzetan <a href="/wiki/Atomo" title="Atomo">atomoak</a> erdialdean positiboki <a href="/wiki/Karga_(argipena)" class="mw-disambig" title="Karga (argipena)">kargatutako</a> <a href="/wiki/Atomo_nukleo" title="Atomo nukleo">nukleo</a> txiki eta dentso bat zeukala egiaztatu zuten, negatiboki kargatutako <a href="/wiki/Elektroi" title="Elektroi">elektroi</a> laino lauso bat ingurunean dagoela. Informazio esperimental honekin Rutherfordek atomoaren <a href="/w/index.php?title=Eredu_planetarioa&action=edit&redlink=1" class="new" title="Eredu planetarioa (sortu gabe)">eredu planetarioa</a> aurkeztu zuen, <a href="/wiki/1911" title="1911">1911ko</a> <a href="/wiki/Rutherforden_eredu_atomikoa" title="Rutherforden eredu atomikoa">Rutherforden eredua</a> izenarekin ezagutzen dena. </p><p>Eredu honen arabera <a href="/wiki/Atomo" title="Atomo">atomoaren nukleoa</a> zentroan dago eta elektroiak horren inguruan <a href="/wiki/Orbita" title="Orbita">orbitak</a> deskribatzen dituzte, baina eredu honek akatz handi bat erakusten zuen: <a href="/wiki/Mekanika_klasiko" title="Mekanika klasiko">Mekanika klasikoaren</a> arabera (<a href="/w/index.php?title=Larmoren_formula&action=edit&redlink=1" class="new" title="Larmoren formula (sortu gabe)">Larmoren formula</a>) elektroiak azeleratzean nukleoek erreadiazio elektromagnetikoa igorriko lukete; honen ondorioz, nukleo atomiko bat orbitatzean, azelerazio zentripetuagatik, elektroiak energia galtzen joango lirateke espiralean nukleora jauziz. Hau ezinezkoa da, atomoak estruktura egonkorrak baitira. Beste aldetik,  <a href="/wiki/XIX._mendea" title="XIX. mendea">XIX. mendearen</a> bukaeran hidrogenoaren atomoaren <a href="/wiki/Espektro_(argipena)" class="mw-disambig" title="Espektro (argipena)">espektroaren</a> hutsuneak demostratzen zuten bakarrik uhin luzera diskretu zehatzetan erradiazioa igortzen zela, <a href="/wiki/Johann_Jakob_Balmer" title="Johann Jakob Balmer">Balmer</a>, hain zuzen, izan zen hidrogenoaren atomoan hutsune hauen arteko erlazio matematikoa aurkitu zuen lehenengoa </p><p>Honako arazo teorikoak konpontzeko Bohr-ek 1913 honako lau postulatuetan oinarritu zuen bere modelo atomikoa proposatzeko: </p> <ol><li>Elektroiek ez dute <a href="/wiki/Erradiazio_elektromagnetikoa" title="Erradiazio elektromagnetikoa">erradiazio elektromagnetikorik</a> igortzen orbitatzean, nukleoa orbitatzen dute distantzia zehatzeko orbitetan, elektroiak orbita horien zehar soilik mugi daitezke.</li> <li>Elektroia eta nukleoaren arteko elkarrakzioa elektrostatikoa da (<a href="/wiki/Coulomben_legea" title="Coulomben legea">Coulomb-en indarra</a>). Elektroien orbitak nukleoarekiko zirkularrak dira.</li> <li>Elektroi orbita bakoitzak momentu angeluar bakarra dauka finkatuta, momentu angeluar hori kuantizatuta dago, hau da, balio diskretuak hartzen ditu soilik. Balio horiek <a href="/wiki/Plancken_konstantea" title="Plancken konstantea">Plancken konstantearekiko</a> proportzionaltasuna mantentzen dute <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=n\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=n\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac7d2abfd338c95f2ba503eb0be903cc74a3a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.382ex; height:2.176ex;" alt="{\displaystyle L=n\hbar }"></span> non <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> zenbaki osoa den eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar =h/2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =h/2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f836f3a2de168369eb9f80f4057518222ecde88b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.401ex; height:2.843ex;" alt="{\displaystyle \hbar =h/2\pi }"></span> Planck-en konstante laburtua.</li> <li>Planken ereduaren arabera elektroiek energia gal edo irabaz dezakete bakarrik orbita batetik bestera pasatzeko. Energia minimo hori gainditzen ez bada elektroiak ez du energia xurgatuko; era berean, elektroiak fotoi bat askatzen badu fotoiak lehen hurbilketan bi orbiten arteko energia diferentzia izango du.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Elektroien_energia_mailak">Elektroien energia mailak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=2" title="Aldatu atal hau: «Elektroien energia mailak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=2" title="Edit section's source code: Elektroien energia mailak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bohrren ereduak zehaztasun handiko emaitzak ematen ditu atomo hidrogenoideetan, hau da, nukleo bat eta elektroi bakar bat duten atomoak. Atomo hidrogenoide batzuk dira: <a href="/wiki/Hidrogeno" title="Hidrogeno">hidrogenoa</a> (H), <a href="/wiki/Helio" title="Helio">helio</a> ionizatua (He+), bi aldiz ionizaturiko <a href="/wiki/Litio" title="Litio">litioa</a> (Li++), <a href="/wiki/Positroi" title="Positroi">positroia</a> (Ps) eta Rydberg-en formula<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> jarraitzen duten atomoak. </p><p><a href="/w/index.php?title=Indar_elektroestatiko&action=edit&redlink=1" class="new" title="Indar elektroestatiko (sortu gabe)">Indar elektroestatikoen</a> eraginez, <a href="/wiki/Elektroi" title="Elektroi">elektroiak</a> orbita zirkularrak egingo ditu nukleoaren inguruan. Nukleoaren <a href="/wiki/Masa" title="Masa">masa</a> elektroiarena baino askoz ere handiagoa dela onartzen da. Guzti hau kontutan hartuz, mekanika klasikoaren arabera, sistema honetan <a href="/wiki/Indar_zentripetu" title="Indar zentripetu">indar zentripetua</a> Coulomb-en indarraren berdina izango da, hau da: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {m_{\mathrm {e} }v^{2} \over r}={Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {m_{\mathrm {e} }v^{2} \over r}={Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c76c02f478184a00f66c44c036746f15d66fbdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.551ex; height:6.343ex;" alt="{\displaystyle {m_{\mathrm {e} }v^{2} \over r}={Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r^{2}}}"></span> </p><p>non <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8303b668e94e02d8f3db8c5b3ebd069ca5da9ba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.039ex; height:2.009ex;" alt="{\displaystyle m_{e}}"></span> elektroiaren masa den, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> elektroiaren <a href="/wiki/Karga_elektriko" title="Karga elektriko">karga elektrikoa</a>,  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> <a href="/wiki/Atomo-zenbaki" title="Atomo-zenbaki">zenbaki atomikoa</a> eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> elektroiaren nukleoarekiko distantzia (elektroiaren orbitaren erradioa). Ekuazio honetatik, elektroiaren <a href="/wiki/Abiadura" title="Abiadura">abiadura</a> lor dezakegu edozein erradioko orbitetarako: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\sqrt {Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }m_{\mathrm {e} }r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }m_{\mathrm {e} }r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c78b4f3c438415e11dbdf1f8b0775e973ee3efd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.93ex; height:7.676ex;" alt="{\displaystyle v={\sqrt {Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }m_{\mathrm {e} }r}}}"></span> </p><p>Gainera, Bohr-en 3. postulatuaren arabera, orbitatzen ari den elektroiaren <a href="/wiki/Momentu_angeluar" title="Momentu angeluar">momentu angeluarra</a> hurrengoa da: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=m_{e}vr=n\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>v</mi> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=m_{e}vr=n\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d96cbe69732cdd3555b0eaf2a6d422e031c442b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.696ex; height:2.509ex;" alt="{\displaystyle L=m_{e}vr=n\hbar }"></span> </p><p>non n <a href="/wiki/Zenbaki_oso" title="Zenbaki oso">zenbaki osoa</a> den. </p><p>Momentu angeluarraren adierazpena eta abiadurarena berdinduz eta r askatuz, eredu honek baimenduriko erradioen hurrengo adierazpena dugu: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}={n^{2} \over Z}{4\pi \epsilon _{\mathrm {0} }\hbar ^{2} \over m_{e}e^{2}}={n^{2} \over Z}{a_{\mathrm {0} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Z</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Z</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}={n^{2} \over Z}{4\pi \epsilon _{\mathrm {0} }\hbar ^{2} \over m_{e}e^{2}}={n^{2} \over Z}{a_{\mathrm {0} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa515e8751c92a1a04c81f8a4eaeabffd495b4fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.023ex; height:6.343ex;" alt="{\displaystyle r_{n}={n^{2} \over Z}{4\pi \epsilon _{\mathrm {0} }\hbar ^{2} \over m_{e}e^{2}}={n^{2} \over Z}{a_{\mathrm {0} }}}"></span> </p><p>non <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\mathrm {0} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {0} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029c795c7e8a74f45316403c069fde1a93a2b515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{\mathrm {0} }}"></span> Bohr-en erradioa den. Ikus daitekenez, erradioak kuantizatuta daude, eta Z = 1 kasurako oinarrizko orbitaren erradioa (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\mathrm {0} }\simeq 0.53{\overset {o}{\mathrm {A} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mo>≃</mo> <mn>0.53</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mi>o</mi> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {0} }\simeq 0.53{\overset {o}{\mathrm {A} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46ea9b2394ab4a873ea0f42242c1a51e95fe28da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.26ex; height:4.009ex;" alt="{\displaystyle a_{\mathrm {0} }\simeq 0.53{\overset {o}{\mathrm {A} }}}"></span> da. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\mathrm {0} }={4\pi \epsilon _{\mathrm {0} }\hbar ^{2} \over m_{e}e^{2}}\simeq 0.53{\overset {o}{\mathrm {A} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≃</mo> <mn>0.53</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mi>o</mi> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {0} }={4\pi \epsilon _{\mathrm {0} }\hbar ^{2} \over m_{e}e^{2}}\simeq 0.53{\overset {o}{\mathrm {A} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30319225d67adbade45be13583d1410eac74c23d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.063ex; height:6.343ex;" alt="{\displaystyle a_{\mathrm {0} }={4\pi \epsilon _{\mathrm {0} }\hbar ^{2} \over m_{e}e^{2}}\simeq 0.53{\overset {o}{\mathrm {A} }}}"></span> </p><p>Abiadura ere kuantizatuta dago. Erradioaren adierazpena Bohr-en momentu angeluarraren adierazpenan sartuz eta abiadura bakanduz hurrengoa lor dezakegu: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={n\hbar \over mr}={Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }\hbar }{1 \over n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={n\hbar \over mr}={Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }\hbar }{1 \over n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d053209701ff0941ebd504ab236266b044bcf1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.116ex; height:6.176ex;" alt="{\displaystyle v={n\hbar \over mr}={Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }\hbar }{1 \over n}}"></span> </p><p>Behin abiadura jakinda, elektroiaren <a href="/wiki/Energia" title="Energia">energia</a> totala ere jakin dezakegu edozien orbitetarako: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=T+V={1 \over 2}m_{e}v^{2}-{Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r}=-{1 \over 2}{Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r}=-{m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}{Z^{2} \over n^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>T</mi> <mo>+</mo> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=T+V={1 \over 2}m_{e}v^{2}-{Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r}=-{1 \over 2}{Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r}=-{m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}{Z^{2} \over n^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6acf55e3fda32b7f338f91ec1023a109d730b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:64.193ex; height:6.509ex;" alt="{\displaystyle E=T+V={1 \over 2}m_{e}v^{2}-{Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r}=-{1 \over 2}{Ze^{2} \over 4\pi \epsilon _{\mathrm {0} }r}=-{m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}{Z^{2} \over n^{2}}}"></span> </p><p>Beraz, oinarrizko energia (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span>) hau da, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}=-{m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}{Z^{2} \over n^{2}}=-13.6Z^{2}eV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>13.6</mn> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>e</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}=-{m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}{Z^{2} \over n^{2}}=-13.6Z^{2}eV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c5d35a08ddea87244307486100de572ac698f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.68ex; height:6.509ex;" alt="{\displaystyle E_{0}=-{m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}{Z^{2} \over n^{2}}=-13.6Z^{2}eV}"></span> </p><p>Ohartu gaitezke energia kuantizatuta dagoela ere eta negatiboa izateaz gain erradioarekiko alderantzizko proportzionala dela. Hau hurrengo moduan interpreta daiteke: elektroia nukleotik urruntzeko energia behar da. Beraz, elektroia nukleotik izugarri urrun degoenean (distantzia infinitutzat hartzeko beste), energia hau zero izango da. </p> <div class="mw-heading mw-heading2"><h2 id="Rydberg-en_formula">Rydberg-en formula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=3" title="Aldatu atal hau: «Rydberg-en formula»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=3" title="Edit section's source code: Rydberg-en formula"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Atomic_emission_spectrum_of_helium.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Atomic_emission_spectrum_of_helium.svg/293px-Atomic_emission_spectrum_of_helium.svg.png" decoding="async" width="293" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Atomic_emission_spectrum_of_helium.svg/440px-Atomic_emission_spectrum_of_helium.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Atomic_emission_spectrum_of_helium.svg/586px-Atomic_emission_spectrum_of_helium.svg.png 2x" data-file-width="1342" data-file-height="619" /></a><figcaption>Hidrogeno eta helio atomoek xurgatutako uhin luzerak espektro elektromagnetikoan (tarte ikuskorrean)</figcaption></figure> <p>Rydberg-en formula<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>, Bohr-en formula baino lehen enpirikoki ezaguna zena, energia maila orbital desberdinen arteko trantsizio energiak deskribatzen ditu Bohr-en teorian. Bohr-en formulak jadanik ezaguna zen <a href="/w/index.php?title=Rydbergen_konstantea&action=edit&redlink=1" class="new" title="Rydbergen konstantea (sortu gabe)">Rydberg-en konstantearen</a> balioa ematen du, baina naturako oinarriagoak diren konstanteen funtzioan, elektroiaren karga eta Planck-en konstantea batik bat. </p><p>Elektroi bat hasierako energia maila batetik maila handiago batera pasatu ondoren hasierako posiziora saltoka bueltatu egiten da tarteko energia mailetatik pasatzen eta salto bakoitzean fotoi bat emititzen du. Atomo hidrogenoideen energia mailetarako deribatutako formula erabiliz atomo hauek emititu ditzaketen argi izpien uhin-luzerak ondorioztatu daitezke. </p><p>Atomo hidrogenoide batek emititutako fotoi baten energia atomo honen bi energia mailen arteko energien aldeagatik emanda dago. Aurretik lortutako energiaren adierazpenetik: </p><p><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E_{i}-E_{f}=E_{\mathrm {0} }\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right)\ =-E_{\mathrm {0} }\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right)\ ={m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}Z^{2}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>=</mo> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{i}-E_{f}=E_{\mathrm {0} }\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right)\ =-E_{\mathrm {0} }\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right)\ ={m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}Z^{2}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2216720bf0310f45d027c5ac36bca26ab0e866c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:87.223ex; height:7.676ex;" alt="{\displaystyle E=E_{i}-E_{f}=E_{\mathrm {0} }\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right)\ =-E_{\mathrm {0} }\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right)\ ={m_{\mathrm {e} }e^{4} \over (4\pi \epsilon _{\mathrm {0} })^{2}2\hbar ^{2}}Z^{2}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right)\,}"></span> </p><p><br /> non <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0172eb646140c291d0afb8fcd0d550f828d1affd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.531ex; height:2.343ex;" alt="{\displaystyle n_{f}}"></span> amaierako energia maila den, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}"></span> hasierako energia maila eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"></span> atomoaren energia maila txikiena den. </p><p><a href="/w/index.php?title=Planck-en_formularen&action=edit&redlink=1" class="new" title="Planck-en formularen (sortu gabe)">Planck-en formularen</a> arabera fotoi baten energia hurrengoa da: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {hc}{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>c</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {hc}{\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a8c83479c76652ec6c8bc1a7cb6b4a7e89c3d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.056ex; height:5.509ex;" alt="{\displaystyle E={\frac {hc}{\lambda }}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {hc}{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>c</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {hc}{\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a8c83479c76652ec6c8bc1a7cb6b4a7e89c3d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.056ex; height:5.509ex;" alt="{\displaystyle E={\frac {hc}{\lambda }}}"></span> </p><p>Aurreko bi formulak erabiliz, atomo hidrogenoideak emititutako fotoi baten uhin luzera hau da: </p><p><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\lambda }}={\frac {E_{i}-E_{f}}{hc}}=R_{\infty }Z^{2}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mi>h</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}={\frac {E_{i}-E_{f}}{hc}}=R_{\infty }Z^{2}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f692366facef775769ab41fae5198cbc5285c21b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.815ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{\lambda }}={\frac {E_{i}-E_{f}}{hc}}=R_{\infty }Z^{2}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right).\,}"></span> </p><p><br /> Formula hau Rydberg-en formula eta aplikagarria izateko <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}>n_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}>n_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf3163d8fa82bb099684be9070c2b939d65f2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.824ex; height:2.509ex;" alt="{\displaystyle n_{i}>n_{f}}"></span> izan behar da, bestela uhin luzera negatiboa dugu, zentzu fisikorik gabeko emaitza. Izan ere, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}<n_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}<n_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e3c2acdf614878f4a4d1019e75711dce0e3e273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.824ex; height:2.509ex;" alt="{\displaystyle n_{i}<n_{f}}"></span> kasuan elektroia orbitaz aldatzeko energia behar du eta ondorioz fotoi bat emititu beharrean xurgatu egingo du. </p><p>Formulan agertzen den <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639146705058e9d335aed47b55a9f802bc290b5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.639ex; height:2.509ex;" alt="{\displaystyle R_{\infty }}"></span> balioari Rydberg-en konstantea deritzo. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639146705058e9d335aed47b55a9f802bc290b5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.639ex; height:2.509ex;" alt="{\displaystyle R_{\infty }}"></span> konstante fundamentalen mende idazten badugu Rydberg-en konstanterako hurrengo adierazpena lortzen dugu: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\infty }=-{\frac {E_{0}}{hc}}={\frac {m_{e}\ e^{4}}{8\ (\epsilon _{0})^{2}\ h^{3}\ c}}=10973731.6m^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mi>h</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mtext> </mtext> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>10973731.6</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\infty }=-{\frac {E_{0}}{hc}}={\frac {m_{e}\ e^{4}}{8\ (\epsilon _{0})^{2}\ h^{3}\ c}}=10973731.6m^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0656b3ed06d62a8880e5c868e3a2ef6d6eca52b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.833ex; height:6.509ex;" alt="{\displaystyle R_{\infty }=-{\frac {E_{0}}{hc}}={\frac {m_{e}\ e^{4}}{8\ (\epsilon _{0})^{2}\ h^{3}\ c}}=10973731.6m^{-1}}"></span> </p><p>Formula hau jadanik XIX. mendean ezaguna zen <a href="/wiki/Espektroskopia" title="Espektroskopia">espektroskopian</a> aritutako zientizalarientzat, baina azalpen teorikorik ez zegoen Bohr-en teoriara arte. Horregatik, Bohr-en eredutik Rydberg-en konstantea ondorioztatu ahal izatea eta Bohr-en formulak esperimentalki behatutako  <a href="/wiki/Lyman" title="Lyman">Lyman</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0172eb646140c291d0afb8fcd0d550f828d1affd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.531ex; height:2.343ex;" alt="{\displaystyle n_{f}}"></span> =1), <a href="/w/index.php?title=Balmer&action=edit&redlink=1" class="new" title="Balmer (sortu gabe)">Balmer</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0172eb646140c291d0afb8fcd0d550f828d1affd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.531ex; height:2.343ex;" alt="{\displaystyle n_{f}}"></span>=2), eta <a href="/w/index.php?title=Paschen&action=edit&redlink=1" class="new" title="Paschen (sortu gabe)">Paschen</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0172eb646140c291d0afb8fcd0d550f828d1affd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.531ex; height:2.343ex;" alt="{\displaystyle n_{f}}"></span>=3) serieetako lerro espektralekin erakutsitako erabateko adostasunak, eta garaian ezezagunak ziren beste serie batzuen aurreikuspen arrakastatzuak garrantzi handikoak izan ziren ereduak izan zuen berehalako onarpenean. </p> <div class="mw-heading mw-heading2"><h2 id="Franck-Hertzen_esperimentua">Franck-Hertzen esperimentua</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=4" title="Aldatu atal hau: «Franck-Hertzen esperimentua»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=4" title="Edit section's source code: Franck-Hertzen esperimentua"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Franck-Hertzen_esperimentua" class="mw-redirect" title="Franck-Hertzen esperimentua">Franck-Hertzen esperimentua</a>, <a href="/wiki/1914" title="1914">1914. urtean</a>, <a href="/wiki/James_Franck" title="James Franck">James Franck</a> eta <a href="/wiki/Gustav_Hertz" title="Gustav Hertz">Gustav Hertz</a>-ek Bohren ereduak proposatutako energia mailen kuantizazioa demostratzeko egindako esperimentua da<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>. Esperimentu horretan elektroiak azeleratzen dira <a href="/wiki/Tentsio_(elektrizitatea)" title="Tentsio (elektrizitatea)">potentzial diferentzia</a> batean zehar; azeleratutako elektroi askeek <a href="/wiki/Zilarbizi" class="mw-redirect" title="Zilarbizi">merkurio</a> atomoetakoko elektroiak kitzikatzeko energia minimoa lortzean energia zinetikoa galtzen dute, ondorioz zirkuituko tentsioa txikitzen da. <a href="/wiki/Tentsio_(elektrizitatea)" title="Tentsio (elektrizitatea)">Tentsio</a> honen galera bakarrik lortzen da balio konkretu batzuetan, honen bidez demostratu zen atomoen energia mailak Bohrek deskribatu zituen bezalakoak direla. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Experiencia_Franck_Hertz.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Experiencia_Franck_Hertz.svg/387px-Experiencia_Franck_Hertz.svg.png" decoding="async" width="387" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Experiencia_Franck_Hertz.svg/581px-Experiencia_Franck_Hertz.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Experiencia_Franck_Hertz.svg/774px-Experiencia_Franck_Hertz.svg.png 2x" data-file-width="474" data-file-height="283" /></a><figcaption>1914 urtean egindako Franck-Hertz esperimentuaren diagrama. </figcaption></figure> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Franck-Hertz_en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Franck-Hertz_en.svg/310px-Franck-Hertz_en.svg.png" decoding="async" width="310" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Franck-Hertz_en.svg/465px-Franck-Hertz_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Franck-Hertz_en.svg/620px-Franck-Hertz_en.svg.png 2x" data-file-width="521" data-file-height="390" /></a><figcaption>Franck-Hertz-en esperimentuko emaitzak korrontea tentsioarekiko jartzean. Klasikoki tentsioa handitzean korrontea handitu beharko litzateke baina ikusten denez periodikoki korrontea behera doa, Bohr-en teorioak aurreikusten zuen bezala.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Bohr-en_ereduaren_akatsak">Bohr-en ereduaren akatsak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=5" title="Aldatu atal hau: «Bohr-en ereduaren akatsak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=5" title="Edit section's source code: Bohr-en ereduaren akatsak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bohren ereduak zailtasunak ditu espektro atomikoan gertatzen den beste fenomeno bat azaltzeko. Hasieran hidrogenoaren espektroaren espekro-lerroak azaltzeko eredu egokia zela zirudien, baina benetan espektro-lerro bakoitzaren ingurune txiki batean begiratzean beste espektro-lerro batzuk agertzen ziren, eta hutsune hauek ezin dira azaldu bohren eredua erabiliz. Beste hutsune berri hauek azaldu ahal izateko Wilson eta <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeldek</a> aldaketa batzuk egin izan behar zuten, aldaketa horiek Wilson eta <a href="/wiki/Sommerfelden_eredu_atomikoa" title="Sommerfelden eredu atomikoa">Sommerfelden ereduan</a><sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> batzen dira. </p> <div class="mw-heading mw-heading2"><h2 id="Erreferentziak">Erreferentziak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=6" title="Aldatu atal hau: «Erreferentziak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=6" title="Edit section's source code: Erreferentziak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-:0-1">↑ <a href="#cite_ref-:0_1-0"><sup><b>a</b></sup></a> <a href="#cite_ref-:0_1-1"><sup><b>b</b></sup></a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Gaztelaniaz dago aipu honen iturburua"><b>(Gaztelaniaz)</b></span></span> <span class="citation" id="CITEREFMartínez2007">Martínez, Jorge Enríquez Figueroa.  (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=v3P8NP6M5P4C"><i>Física Moderna Edición Revisada. </i></a> Pearson Educación <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/9789702607892" title="Berezi:BookSources/9789702607892">9789702607892</a>.</span>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Gaztelaniaz dago aipu honen iturburua"><b>(Gaztelaniaz)</b></span></span> <span class="citation"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mjvKG4BJ0xwC"><i>Química i. </i></a> EUNED <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/9789968316262" title="Berezi:BookSources/9789968316262">9789968316262</a>.</span>.</span> </li> <li id="cite_note-:1-3">↑ <a href="#cite_ref-:1_3-0"><sup><b>a</b></sup></a> <a href="#cite_ref-:1_3-1"><sup><b>b</b></sup></a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Frantsesez dago aipu honen iturburua"><b>(Frantsesez)</b></span></span> <span class="citation" id="CITEREFRydberg1889">Rydberg, J.R..  (1889). <i>Researches sur la constitution des spectres d'émission des éléments chimiques. </i> Kongliga Svenska Vetenskaps-Akademiens Handlingar [Proceedings of the Royal Swedish Academy of Science], 1-177 or.</span>.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> <span class="reference-text"><span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Alemanez dago aipu honen iturburua"><b>(Alemanez)</b></span> Franck, J. & Hertz, G. (1914): <i>Über Zusammenstöße zwischen Elektronen und den Molekülen des Quecksilberdampfes und die Ionisierungsspannung desselben</i>. Verh. Deutsche Phys. Ges. <b>16</b>, 457-467</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=7" title="Aldatu atal hau: «Bibliografia»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=7" title="Edit section's source code: Bibliografia"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation"></span> <span class="citation" id="CITEREFBohr">Bohr, Niels. <a rel="nofollow" class="external text" href="http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf"><i>On the Constitution of Atoms and Molecules, Part I. </i></a> Philosophical Magazine, 1–24 or.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F14786441308634955">10.1080/14786441308634955</a></span>. <a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>: <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1913PMag...26....1B">1913PMag...26....1B</a>.</span>.</li> <li><span class="citation"></span> <span class="citation" id="CITEREFBohr1913">Bohr, Niels.  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(1914). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1429570"><i>The spectra of helium and hydrogen. </i></a> Nature, 231-232 or.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1038%2F092231d0">10.1038/092231d0</a></span>. <a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>: <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1913Natur..92..231B">1913Natur..92..231B</a>.</span>.</li> <li><span class="citation"></span> <span class="citation" id="CITEREFBohr1921">Bohr, Niels.  (1921). <a rel="nofollow" class="external text" href="http://www.chemteam.info/Chem-History/Bohr-Nature-1921.html"><i>Atomic Structure. </i></a> Nature, 104–107 or.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1038%2F107104a0">10.1038/107104a0</a></span>. <a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>: <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1921Natur.107..104B">1921Natur.107..104B</a>.</span>.</li> <li><span class="citation"></span> <span class="citation" id="CITEREFEinstein1917">Einstein, Albert.  (1917). <i>Zum Quantensatz von Sommerfeld und Epstein. </i> Verhandlungen der Deutschen Physikalischen Gesellschaft</span>. Reprinted in <i>The Collected Papers of Albert Einstein</i>, A. Engel translator, (1997) Princeton University Press, Princeton. <b>6</b> p. 434. (provides an elegant reformulation of the Bohr–Sommerfeld quantization conditions, as well as an important insight into the quantization of non-integrable (chaotic) dynamical systems.)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Gehiago_sakontzeko">Gehiago sakontzeko</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=8" title="Aldatu atal hau: «Gehiago sakontzeko»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=8" title="Edit section's source code: Gehiago sakontzeko"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>R. Eisberg & R. Resnick, "Física Cuántica", Editorial Limusa 1978, <a href="/wiki/Berezi:BookSources/978-968-18-0419-0" title="Berezi:BookSources/978-968-18-0419-0">ISBN 978-968-18-0419-0</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Kanpo_estekak">Kanpo estekak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&veaction=edit&section=9" title="Aldatu atal hau: «Kanpo estekak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrren_eredu_atomikoa&action=edit&section=9" title="Edit section's source code: Kanpo estekak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.sc.ehu.es/sbweb/fisika/elecmagnet/movimiento/bohr/bohr.htm">Bohr-en eredu atomikoa (EHU)</a></li> <li><a rel="nofollow" class="external text" href="https://www.hiru.eus/eu/fisica/el-modelo-atomico-de-bohr-sommerfeld">Bohr Sommerfeld-en eredu atomikoa (hiru.eus)</a></li></ul> <div style="clear:both;"></div><style data-mw-deduplicate="TemplateStyles:r7786466">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control,.mw-parser-output .mw-mf-linked-projects{border:1px solid #a2a9b1;font-size:88%}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}</style><div class="mw-authority-control"><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r9236167">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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