[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-512706-105":59,"doc-detail-512706-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","homework-6-thermonuclear-reaction-rates","Homework 6 - Thermonuclear Reaction Rates","","Homework 6 focuses on thermonuclear reaction rates in astrophysical and experimental contexts, starting from the basic definition of a reaction rate for monoenergetic particles and extending it to realistic thermal environments with velocity distributions. It derives the thermally averaged reaction rate \u003Cσv> using Maxwellian particle distributions, then applies transformations to express the integral in center-of-mass variables and further reduce it to a one-dimensional form over relative speed and then energy. Finally, it connects \u003Cσv> to reaction-rate equations for nuclide abundances and introduces the reduced rate λ = NA\u003Cσv>.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/homework-6-thermonuclear-reaction-rates/512706/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/homework-6-thermonuclear-reaction-rates/512706.png","ImageObject",300,407,{"name":92,"@type":93},"Muffet","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-01","2026-10-06",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the thermally averaged reaction rate and why is it needed?","Question",{"text":112,"@type":113},"The thermally averaged reaction rate replaces the monoenergetic definition with averages over particle velocity distributions. It is needed because in stars and in experimental setups particles are not all at one fixed velocity.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do you compute the number density ni for a Maxwellian-distributed particle?",{"text":117,"@type":113},"The homework asks to determine ni for mass mi at temperature T using the Maxwellian distribution. This uses the relationship ni = ∫d^3vi fi(~vi) with fi derived from the Maxwellian form shown in the text.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the problem relate \u003Cσv> to changes in nuclide abundances?",{"text":121,"@type":113},"It states that the reaction rate equals the total time rate of change of the involved nuclides, leading to equations in terms of number densities. It then rewrites them with molar fractions Yi = ni/nbaryon and finally introduces the reduced reaction rate λ = NA\u003Cσv> so the dependence is primarily on temperature T.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},512706,1791506778,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":81,"preview_page_count":4},962090768927,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Homework 6  \n1. Thermonuclear Reaction Rates  \nWe have generalized our discussion somewhat, however the reaction rate is still defined in terms of monoenergetic particles (i.e. the target and projectile particles have fixed velocities, ~vt and ~vp respectively) . In general, an experimental setup will create beams and targets with a distribution of particle velocities. One then speaks in terms of averaged quantities. The same is true in astrophysical environments such as stars. Usually these environments are in thermal equilibrium, which makes calculating the averages more simple. This brings us to the thermally-averaged reaction rate.  \n\u003C reaction rate > = np nt \u003C σv > (1)  \nThe averaged reaction rate is given by:  \n\u003C reaction rate > = Z Z d3 ~vp d3 ~vt fp (~vp) ft (~vt) σ (E)v (2)  \nremembering that ni = R d3 ~vi fi (~vi ), and fi (~vi ) = gi ¡ ~~m~~2π¢3 exp ³ µ~~c~~−kTE~~i~~ ´ , where Ei = mic2 + 12miv2i .  \n(a) What is ni for a Maxwellian distributed particle with mass mi at a temperature T?  \n(b) With the definitions of the center-of-mass total and relative velocities (~V = mp~v~~ ~~pm~~ ~~pm~~ ~~tm~~ ~~t~vt and ~v = ~vp −~vt ), and assuming that the particle distributions are Maxwellian (with the same temperature, T ), transform the reaction rate integral such that it only depends on center-of-mass velocity variables.  \n(c) Show that if the cross-section depends only on the magnitude of the relative velocity (thus E) that you can reduce this integral down to a 1-dimensional integral over v.  \n(d) Transform this 1-dimensional integral from being over v to being over E.  \n(e) Solve for the quantity \u003C σv >= \u003C reaction rate > /np nt .  \nIn chemistry, one speaks in terms of a rate of reaction. The reaction rate is equal to the total rate of change of the nuclides involved:  \n 1 dnA   1 dnB   1 dnC   1 dnD − = − = = = \u003C reaction rate > (3)  \nNA dt NB dt NC dt ND dt  \nwhere, Ni are the number of nuclides of type i produced or destroyed in the reaction A + B → C + D.  \nSince nuclear reactions conserve the number of nucleons, only shuffling them around into different nuclides, it is useful to rewrite things in terms of a molar fraction. Yi ≡ ni /nbaryon is the molar fraction, defined asthe number of nuclides of type i per baryon, or the ratio of their number densities. Since the total baryon number is conserved in nuclear reactions, dnbaryon /dt = 0 . Therefore, we can rewrite our equations:  \n 1 dYA   1 dYB   1 dYC   1 dYD − = − = = = YAYBnbaryon \u003C σv > (4)  \nNA dt NB dt NC dt ND dt  \nSince the typical mass of a baryon is ∼ mu , we can substitute the baryon number density with the baryon mass density (ρbaryon ≈ nbaryon mu ) and Avogadro’s number (NA ) .  \n 1 dYA   1 dYB   1 dYC   1 dYD − = − = = = YA YB ρbaryon NA \u003C σv > (5)  \nNA dt NB dt NC dt ND dt  \nWhen we talk about reaction rates, we usually are referring to the reduced reaction rate λ = NA \u003C σv > . This way, we do not have to specify the concentrations of nuclides or the baryon density. We are left with a quantity (λ) that usually only depends on the temperature, T.","cbCaicBTGUadLKMD","https://ap.wps.com/l/cbCaicBTGUadLKMD","pdf",36276,"English","# Thermonuclear Reaction Rates\n## Average reaction rate and thermal averaging\n## Maxwellian number density\n## Transforming the reaction rate integral\n## Reduction to energy-dependent form\n## Relating reaction rate to nuclide abundances and reduced rate","[{\"question\":\"What is the thermally averaged reaction rate and why is it needed?\",\"answer\":\"The thermally averaged reaction rate replaces the monoenergetic definition with averages over particle velocity distributions. It is needed because in stars and in experimental setups particles are not all at one fixed velocity.\"},{\"question\":\"How do you compute the number density ni for a Maxwellian-distributed particle?\",\"answer\":\"The homework asks to determine ni for mass mi at temperature T using the Maxwellian distribution. This uses the relationship ni = ∫d^3vi fi(~vi) with fi derived from the Maxwellian form shown in the text.\"},{\"question\":\"How does the problem relate \\u003cσv\\u003e to changes in nuclide abundances?\",\"answer\":\"It states that the reaction rate equals the total time rate of change of the involved nuclides, leading to equations in terms of number densities. It then rewrites them with molar fractions Yi = ni/nbaryon and finally introduces the reduced reaction rate λ = NA\\u003cσv\\u003e so the dependence is primarily on temperature T.\"}]","Homework 6 - Thermonuclear Reaction Rates | PDF",1791274686]