Resonance Theor y

Basi cs

• Dea ls with th e de scr ip tio n of n ucle us - nu cleus int era ction an d aim s at t he pre dictio n of the expe rime nta l structure of cross - se ction s

• Int era ction mo de l whi ch trea ts the nu cleus as a bl ack bo x

– Pote nti al is un kn own so mod el s ca nn ot pr ed ic t acc ur ate ly

– Only car e a t w hat ca n b e o bs erv ed bef ore an d afte r a c ol lis io n

R - matrix theory

• Intr odu ce d by W ign er and Eisen bud (1 947 )

• Requ ire s no in for mati on ab ou t i nte rn al s tru ct ur e of the nu cl eu s

• It is mathe matic all y rig oro us

– Usu all y app rox imated

– Most physi cal and appropriate of resona nce fr amework

• Cros s - se cti ons are par ametr ize d in ter ms of

– Int eraction radii & boun dary cond itio n

– Re sona nce ene rgy & wi dths

– Quan tum nu mber (angu la r momentum, s pi n, …)

Why bother?

• Cou ld n’ t we j ust use th e me asure d da ta ?

– Too muc h i nfo rma tio n, too lit tle un de rs tan di ng

• x .s. vs ene rgy wo uld requi res 100 ,000’ s of ex pe rimental po in ts

• An gul ar distributio ns wo uld requi re even mor e

– Need fo r ext ra po la tio n

• Di ff erent en ergi es

• Temperature cha nge s

• Geometr y consi derations (self - shi eld ing, …)

• Un stabl e or rare nucl id es

R - matrix theory As sumpti ons

• Ap pl ica bi lity of no n - rela tivi stic qu an tum mecha ni cs

• Un importance of p rocesses wh ere mor e than t wo product nucl ei are f orm ed

• Un importance of a ll processes of creation or d estr uction

• Ex istence of a fin ite radia l distance beyo nd wh ich no nucl ear interact io n occurs

• Ba sed on the no tion that w e can de scrib e accu rately wh at’s f ar eno ugh fr om the compound nucl eus but not wh at’s insi de

Definiti on

• R - matri x i s call ed a c han nel - ch ann el matri x

• Chan ne l

– De sig nates a possi ble pai r of n ucle us and part icl e an d the spi n of the pa ir

– Incoming chann el (c)

– Outgoing channe l (c’)

– De fine d by pa ir of pa rt icl es, mass, ch arge, spi n

• Many pos sib le ch anne ls exist

• In comi ng chan ne l (c)

– We can co ntr ol th e i nc omin g ch an ne l b y the way we se t up t he ex pe rime nt

• Ne utron energy

• Target

• Outg oi ng chan ne l (c’)

– We can ob se rv e t he ou tgo in g c ha nn el with pr ec is e meas ur eme nt

Total spi n of the c hannel

Cross - sec tion

• In 22 .1 01 , you used th e ph ase shift th eo ry to de term ine an expre ss ion for the sc att erin g cross - section

– This ex pr es si on c an be de fin ed in te rms of the co lli si on matr ix U

– Diffe re nt re la tio ns be twee n x. s an d U ex is t for oth er in ter ac tio n t yp e

Goal of R - matrix

• Phase sh ift theo ry req uir es kn owled ge of t he po ten tia l V (r )

– Ap prox imated by squa re we ll

• R - matri x t heo ry bu ild s a rel ati ons hip bet ween a matr ix R tha t d ep en ds on ly on ob se rv ab le , mea su ra bl e qu an tit ie s and th e c ol lis io n ma tri x

– Byp asses the n eed for the p otential

– Re qui res ex perimental data

• We will d er iv e a s impl is tic c as e of a n eu tro n in ter ac tio n with n o s pi n d ep en de nc e

R - Matrix Derivati on

• Sta rt with th e stea dy - state Schr ö di ng er eq ua tio n with a com ple x po ten tia l

– Eige nv al ue pr ob le m

• The wavefu nctio n is expressed in the fo rm of pa rtia l wa ves

• In rad ia l g eo me try, th e mo me nt is a solut ion of the fol lowin g equ ati on

(1)

• Add iti on al ly, th e mo me nt can be rep resent ed by an expa nsion in term s of the eig en vectors of the solu tio n

– Eigen ve cto rs are al so so lut ion s of the ab ov e eq ua tio n

• Eig en vectors are also a sol uti on of : (2)

• Bou nd ary cond iti on s

– Both eq ua tio ns mus t b e f in ite at r = 0

– Log ari thmic de riv ati ve at nuc lea r s urf ac e i s tak en to be c on st an t (wher e B l is re al )

• Th e ei ge nvecto rs fo rm a ba sis se t, if no rmal ized pro pe rly, the y have the fol lowin g pro pe rty:

– They form an ort hon orma l bas is se t

• From thi s con dit ion , th e e xpan sion coef ficie nt s can be d ef in ed a s:

• Our go al is to e lim in at e the p ot en tia l V (r)

– Mult ip ly eq (1 ) b y the ei ge nv ec tor an d mult ip ly eq (2 ) b y the mome nt

– Subt ra ct re su lti ng eq ua tio ns

– Int eg ra te be twee n 0 a nd a

– Resu lt: Gives an ex pre ss ion for

• Which c an be used t o find the ex pan sio n coefficie nts

• We can no w fin d an e xpression fo r th e mo me nt at r = a

• Whe re we can extract a d efi nit ion of th e R - ma trix

• Or mo re commo nly

– γ λ l is the red uc ed width ampl itu de for lev el λ

an d ang ul ar mome ntu m l

– λ is the re so na nc e

– E λ is the ene rgy at the res ona nc e p eak

– γ λ l ’s an d E λ ’s ar e u nk no wn pa ra mete rs an d ca n be ev al ua ted by ob se rv in g me as ur ed cr os s - se ct io ns

• E λ is the ener gy val ue at the peak

• γ λ l is a m eas ure of the width of the re sonanc e at a cer ta in am plit ude f or the n uclei at rest

– Re la te d to the m or e com m on Γ thro ug h a m at rix t ran sfo rm

– No t easy to m easur e becau se of te m per at ur e eff ect s (D op pler)

– Usu al ly infer red fr om the reso nan ce integral

General Form

Advantages /Disadv atages of R - matrix theory

• Disad va nta ges

– Mat rix i nversion is al wa ys requi red

– Ch ann el radii and bou nda ry cond itio n app ear arbitrar y

– Di ff icu lt to acco mmodate di rect reactio ns (i.e.

potential scat tering)

• Adva nta ge s

– Ch an ne l radi i an d bo un da ry con di tion ha ve na tural

defin itio ns wh ich makes t hem app eal ing

– Re duce d wi dth conce pt has an appe ali n g relatio n to nu cle ar spectroscop y

Boundary conditon

• In the ea rly da ys, the re was much confusi on in the

choi ce of cha nne l radii and bou nda ry cond itio n

– This topic has been deba ted heavily ov er the las t 40 y ear s!

– Early pape rs des cr ibed their ch oic e as ar bitrar y

– Opt ical model has facilit ated the choic e of these parameters

• “Natural” ch oice s ex ist

– De sc ribe d in more details in pdf R - matrix theory (2 )

– B l mus t be ke pt re al to pr es er ve the nature of the eigen value

problem

– Ch oic e of boun dar y co ndition is to se t it equa l to the sh ift function at so me point in t he ener gy interval of meas ur ement.

• Ke ep on ly rea l pa rt of t he log ari thmic de riva tive of the ou tgo in g w ave

– Mat ching radii usua lly selected base d on squa re - well int erac ti on

Relation with colli sio n matrix

• We found an ex pre ss ion for the so lut ion of t he wav efu nc tio n tha t doe sn ’t de pe nd on th e po ten tia l

– De pe nd s on R - matrix

– R - mat rix d epe nds on ex perimental ly measured data

• Tota l w av e fun ct io n in r eg io n ou ts id e nuc le ar po ten tia l inte ra ct io n c an be ex pr es se d a s a lin ea r comb in ati on of the in co ming an d o utg oi ng wav es

• From R - ma trix an al ysis , we f ou nd

• We can the n f ind th at

• Defi ni ng

• We ge t

General form

• No ap pro xima tio n ha s be en ma de

– Exac t re pr es en tat io n b etwe en U an d R

Level matrix

• Th e R - ma trix is fa irly sma ll bu t f ai rly comp lex to bu ilt

• Wign er int rod uced a cle are r rep resent ati on calle d the A - ma trix whose ele me nts correspon d to en erg y l evels

– A is much lar ger

– But its pa ra mete rs ar e cl ea rly de fin ed

– Summati on is ov er in co ming ch an ne ls

A - matrix

• Very la rge

– Corr es po nd s to the tot al nu mbe r of re so na nc es

– Symmetr ic mat rix

– Diag on al ter ms d ep en d o n e ac h l ev el ind epe nde ntl y

– Off - di ag on al te rms ar e mi xe d t er ms t ha t in tro du ce the in flu en ce of di ffe re nt lev el s on eac h othe r

Mult i - lev el Breit Wigner

• Neg le cting of f - di ag on al te rms yield s the Breit Wign er a pp roxima tio n

– Anal yz in g a s in gl e l ev el at a t ime yi el ds th e Single le ve l B rei t Wig ner (SLBW) ap pr ox imat io n

• Works we ll if resonance s are wel l space d

• Origi na ll y de vel op ed by Wig ne r ba sed on an

ana lo gy to the disp ersion of l ig ht

– In so me ca se s, off - di ag on al ter ms ma tte r

Reic h Moore For mali sm

• Curre nt me th od o f cho ice

– Keep s most of f - di ag on al te rms

– Negl ec ts impa ct of ga mma ch an ne ls

• Measurements have show n that fluctuatio ns betwee n gamm a chann els at d ifferent leve ls must be small

• ML BW is mo re restricti ve th an Rei ch Mo ore

– Poor tre atme nt of mult i - ch an ne l eff ec ts

• SLBW is mo re restrictive tha n ML BW

– Can gi ve ne ga tiv e c ro ss - se ct io n val ue s

Reich Moore vs SLBW (U235 fis sion)

• Solid line : S LBW

• Dotte d line : RM

Fe - 56: RM, MLBW, SLBW

• So li d li ne : RM

• Da shed li ne: MLBW

• Do tt ed li ne: SL BW

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22.106 Neutron Interactions and Applications

Spring 20 10

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