coulomb excitation data analysis codes; gosia 2007 - Physics and ...

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Figure 6: Model Coulomb excitation case for 200 MeV 56 Ni beam on Z=46, A=110 nucleus.Figure 7: Reorientation effect for the two-level system32
q (20)s (0 + → 2 + ) · M 1 ζ (20)0 + 2 + = qq (20)a (0 + → 2 + ) · M 1 ζ (20)0 + 2 + = q 1 (3.11)q (20)s (2 + → 2 + ) · M 2 ζ (20)2 + 2 + = QNote that there is no antisymmetric component of the interaction for the quadrupole moment, sinceζ =0.The A matrices can then be written as:µ 0 −q1A 1 =(3.12)q 1 0µ 0 qA 2 =q Qyielding:µ cos q1 ¯+sinqexp(±A 1 )=1(3.13)± sin q 1 cos q 1Ãcos p −1exp(−iA 2 )=e iQ/2 2 i Q p sin p − iq p sin p!− iq p sin p cos p + 1 2 i Q p sin pwhereUsing 3.5 gives:p = 1 2 (Q2 +4q 2 ) 1/2a 2 + = 1 sin pieiQ/22 p (Q sin 2q 1 − 2q cos 2q 1 ) (3.14)which results in an excitation probability:P 2+ = 1 sin 2 p4 p 2 (Q sin 2q 1 − 2q cos 2q 1 ) 2 (3.15)The formula 3.15 is a generalization of the second-order perturbation theory result. Taking into accountonly the lowest order terms in q, q 1 ,q 1 and Q gives:P 2+ ≈ q 2 (1 − Q · q1q )2 (3.16)as predicted by second-order perturbation theory. It should be observed, that the influence of the quadrupolemoment is significant only if the ratio of antisymmetric to symmetric q parameters is high, which physicallycorresponds to a large value of ξ. As can be seen from 3.14, theprimaryeffect of the static moment is rotationof the complex excitation amplitude, due to the exp(iQ/2) factor. This allows rather accurate measurementsof static moments in cases of interfering paths of excitation, i.e., when a state is excited in a comparable wayvia two or more sequences of couplings. In this situation the phases that the partial excitation amplitudesare summed are of primary importance.In general, 3.10 is only a semianalytical formula, since for complex cases, exponents of the A matricesmust be evaluated numerically. Nevertheless, numerical determination of exp(A) operators is much fasterthan the integration of a system of differential equations. Application of the A matrix approximation forfitting of matrix elements to reproduce experimental data will be discussed in some more detail in Chapter4. It is worth observing that the truncation of Taylor series approximating the exp(A) operators providesa way to generate perturbation theory of a given order, thus being useful for investigating weak excitationprocesses.33
Page 1: COULOMB EXCITATION DATA ANALYSIS CO
Page 4 and 5: 10 MINIMIZATION BY SIMULATED ANNEAL
Page 6 and 7: 1 INTRODUCTION1.1 Gosia suite of Co
Page 8 and 9: 104 Ru, 110 Pd, 165 Ho, 166 Er, 186
Page 13 and 14: Figure 1: Coordinate system used to
Page 15 and 16: Cλ E =1.116547 · (13.889122) λ (
Page 17 and 18: Figure 2: The orbital integrals R 2
Page 19 and 20: 2.2 Gamma Decay Following Electroma
Page 21 and 22: where :d 2 σ= σ R (θ p ) X R kχ
Page 23 and 24: Formula 2.49 is valid only for t mu
Page 25 and 26: Ã XK(α) =exp−iτ i (E γ )x i (
Page 27 and 28: important to have an accurate knowl
Page 29 and 30: 3 APPROXIMATE EVALUATION OF EXCITAT
Page 31: with the reduced matrix element M c
Page 35 and 36: esults of minimization and error ru
Page 37 and 38: adjustment of the stepsize accordin
Page 39 and 40: approximation reliability improves
Page 41 and 42: Zd 2 σ(I → I f )Y (I → I f )=s
Page 43 and 44: 4.5 MinimizationThe minimization, i
Page 45 and 46: X(CC k Yk c − Yk e ) 2 /σ 2 k =m
Page 47 and 48: However, estimation of the stepsize
Page 49 and 50: It can be shown that as long as the
Page 51 and 52: een exceeded; third, the user-given
Page 53 and 54: where f k stands for the functional
Page 55 and 56: x i + δx i Rx iexp ¡ − 1 2 χ2
Page 57 and 58: method used for the minimization, i
Page 59 and 60: OP,ERRO (ERRORS) (5.6):Activates th
Page 61 and 62: -----OP,SIXJ (SIX-j SYMBOL) (5.25):
Page 63 and 64: 5.3 CONT (CONTROL)This suboption of
Page 65 and 66: I,I1 Ranges of matrix elements to b
Page 67 and 68: CODE DEFAULT OTHER CONSEQUENCES OF
Page 69 and 70: 5.4 OP,CORR (CORRECT )This executio
Page 71 and 72: 5.6 OP,ERRO (ERRORS)ThemoduleofGOSI
Page 73 and 74: 5.7 OP,EXIT (EXIT)This option cause
Page 75 and 76: M AControls the number of magnetic
Page 77 and 78: 5.10 OP,GDET (GE DETECTORS)This opt
Page 79 and 80: 5.12 OP,INTG (INTEGRATE)This comman
Page 81 and 82: ¡ dE¢dx1 ..¡ dEdx¢Stopping powe
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NI1, NI2 Number of subdivisions of
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5.13 LEVE (LEVELS)Mandatory subopti
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5.15 ME (OP,COUL)Mandatory suboptio
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Figure 10: Model system having 4 st
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ME =< INDEX2||E(M)λ||INDEX1 > The
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When entering matrix elements in th
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There are no restrictions concernin
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5.18 OP,POIN (POINT CALCULATION)Thi
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5.20 OP,RAW (RAW UNCORRECTED γ YIE
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5.21 OP,RE,A (RELEASE,A)This option
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5.25 OP,SIXJ (SIXJ SYMBOL)This stan
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5.27 OP,THEO (COLLECTIVE MODEL ME)C
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2,5,1,-2,23,5,1,-2,23,6,1,-2,2Matri
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5.29 OP,TROU (TROUBLE)This troubles
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to that of the previous experiment,
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To reduce the unnecessary input, on
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OP,STAR or OP,POIN under OP,GOSI. N
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5.31 INPUT OF EXPERIMENTAL γ-RAY Y
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6 QUADRUPOLE ROTATION INVARIANTS -
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*½P 5 (J) = s(E2 × E2) J ×¯h¾
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The expectation value of cos3δ can
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where ē is an arbitratry vector. D
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achieved using “mixed“ calculat
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TAPE9 Contains the parameters neede
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TAPE18 Input file, containing the i
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7.4.4 CALCULATION OF THE INTEGRATED
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OP,EXITInput: TAPE4,TAPE7,TAPE9Outp
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OP,ERRO0,MS,MEND,1,0,RMAXand the fi
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8 SIMULTANEOUS COULOMB EXCITATION:
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4, 3, 1kr88.corKr corrected yields
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0 Correction for in-flight decay ch
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OP, ERRO Estimation of errors of fi
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9 COULOMB EXCITATION OF ISOMERIC ST
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configurations with a probability e
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The average range covered by each m
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SFX,NTOTI1(1),I2(1),RSIGN(1)I1(2),I
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11.2 LearningtoWriteGosiaInputsThe
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(1.6 MeV)1.1 MeV0.75 MeV0.4 MeV0.08
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Define the germaniumdetector geomet
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Figure 15: Flow diagram for Gosia m
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gosia < 2-make-correction-factors.i
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Issue the commandgosia < 9-diag-err
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At this point, it is suggested to c
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calculation.) In this case, a copy
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4,-4, -3.705, 3,44,5, 4.626, 3.,7.5
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90145901459014590145901459014590145
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.10.028921.10.026031.10.023431.10.0
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5,5,634,650,82.000,84.000634,638,64
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***********************************
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*** CHISQ= 0.134003E+01 ***MATRIX E
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CALCULATED AND EXPERIMENTAL YIELDS
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11.7 Annotated excerpt from a Coulo
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11.8 Accuracy and speed of calculat
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18,10.056,0.068,0.082,0.1,0.12,0.15
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line 152 Eu 182 Tanumber (keV) (keV
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1.6 Normalization between data sets
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13 GOSIA 2007 RELEASE NOTESThese no
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Matrix elements 500(April 1990, T.
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14 GOSIA Manual UpdatesDATE UPDATE2
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[KIB08]T.Kibédi,T.W.Burrows,M.B.Tr
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