Figure 9: A rectangular particle detector has a non-azimuthally symmetric shape in the θ, φ space of projectilescattering angle. This shape must be defined by entering a number (NFI) of azimuthal angular ranges. Inthe hatched region, NFI =2ranges must be entered specifying the two active regions. The solid linesrepresent θ meshpoints, while the dashed lines indicate θ subdivisions. The rapid variation in active φ rangein the hatched region necessitates input of the active ranges of ∆φ at each subdivision specifying each of thetwo active detection regions shown hatched.θ 1 , θ 2 , ...θ NT Projectile scattering angles (degrees) in the laboratory frame, used as meshpoints. Note,if the target is detected then the projectile scattering angle corresponding to the detected recoilingtarget angle must be input with a negative sign to set the flag specifying the target nucleus detected.The input angles must correspond to the detected particle angular range which exceeds or at leastequals the range of angles subtended by the detector to obtain reliable Lagrange interpolation. Do notinput these angles for the circular detector option.NFIThe number of φ ranges for each θ i meshpoint needed to describe the θ(φ) dependence. OmitNFI input if either the circular detector option or axial symmetry is specified.φ 1 , φ 2 , ... NFI pairs of φ angles describing the φ range for given θ i .Omitφ input if either the circulardetector option or axial symmetry is specified.The above two records must be input for each θ meshpoint specified. NFI should not exceed 4. Inmost cases NFI =1, then the pair of φ angles simply specifies the lower <strong>and</strong> upper f limits for a given θmeshpoint. However, for some geometries, such as for rectangular shaped detectors, it is necessary to includemore than one φ range for some θ values. For example, a rectangular detector placed with its normal at 45 ◦to the incident beam has (θ, φ) contours shown in Figure 9:This ends the input required to calculate the γ-ray yields integrated over azimuthal angle φ at the specifiedset of meshpoints. This part of input must be repeated for all experiments defined in EXPT.The second stage of the input is required for the integration <strong>and</strong> once again has to be entered for allexperiments:NP Number of stopping powers to be input, 3 ≤ NP ≤ 20. If NP =0then the stopping power tableis taken from the previous experiment <strong>and</strong> the following input of energy <strong>and</strong> dE/dx values can beomitted forthiscase. Thisisusefulwhereexperimentsdiffer only with regard to range of scatteringangles or bombarding energies.E 1 , E 2 ,...E NPThe energy meshpoints (in MeV) at which values of the stopping power are to be input.80
¡ dE¢dx1 ..¡ dEdx¢Stopping powers in units of MeV/(mg/cm 2 ). Interpolation between the energy meshpointsof the stopping power table is performed during integration. Consequently it is important toNPensure that the range of energy meshpoints of the stopping power table exceed the range of energiesover which the integration is to be performed.NI1, ±NI2 The number of equal subdivisions of energy (NI1) <strong>and</strong> projectile scattering angle (NI2)used for integration. Lagrange interpolation is performed to interpolate between the (E i ,θ i ) meshpointsat which the full Coulomb <strong>excitation</strong> calculations of the de<strong>excitation</strong> γ-ray yields were performed (SeeSection 4.4). The rapid angle dependence of the Rutherford scattering cross section at forward anglescan cause problems with Lagrange interpolation for the elastic channel if insufficient angle meshpointsare used. Always use many integration subdivisions which has little impact on the speed of theprogram since the full Coulomb <strong>excitation</strong> calculations are performed only at the meshpoints, notat the intergration points, <strong>and</strong> interpolation is fast. NI1 should be even <strong>and</strong> not exceed 100 whileNI2 must not exceed 50. If odd values are given the program increases them to the next larger evennumber. However, the ∆φ input will be confused if NI2 is negative <strong>and</strong> odd. Important: NI2 can benegative <strong>and</strong> must be negative if NT is specified to be negative. Conversely, NI2 must be positive ifNT is positive. If NI2 is negative then the following input must be provided:∆φ 1 , ∆φ 2 , ....∆φ |NI2|+1 where ∆φ i is the total range of φ (in degrees) for each equal subdivision ofprojectile scattering angle as illustrated in Figure 9. That is, ∆φ i equals the sum of all φ ranges for agiven subdivision θ value if the azimuthal angular range also is subdivided into non-contiguous regions.The ∆φ values correspond to equal divisions of projectile scattering angle from θ min to θ max ratherthan equal subdivisions of the meshpoints as in the normal input for NI2 > 0. Note that there is animportant difference in how the interpolation over scattering angle is performed depending on whetherNT <strong>and</strong> NI2 are both positive or both negative. For the positive sign the program interpolates betweenthe calculated yields at each θ meshpoint. This produces excellent results if the φ dependence of theparticle detector is a smooth function of projectile scattering angle, θ. The negative sign option shouldbe used if the φ dependence is more complicated or if φ changes rapidly with θ. When the negative signis used the program stores for each meshpoint the calculated yields per unit angle of azimuthal range,i.e., the calculated yields divided by the total φ range specified at that angle for the exact calculation.The program then interpolates these yields per unit of φ between the meshpoints. These interpolatedvalues then are multiplied by the appropriate ∆φ for each subdivision meshpoint prior to integration.Note that the code uses NI2 equal subdivisions of projectile scattering angles. This is not the same asequal division of geometric angle of the detector if the recoiling target nucleus is detected.The following sample input segment goes with Figure 9. It shows the input of the absolute azimuthalangles under NFI, which are entered if the azimuthal symmetry flag was turned off in sub-option EXPT.It also shows the entry of total azimuthal angular ranges when NT <strong>and</strong> NI2 are set to negative values.OP,INTG5,-10,634,650,24.0,60.0 NE,NT,E min ,E max ,θ min ,θ max634,638,642,646,650 E 1 ,E 2 ,E 3 ,E 4 ,E 524.,28.,32.,36.,40.,44.,48.,52.,56.,60.1 NFI-2.0,2.0 φ 1 φ 21 NFI-12.5,12.5 φ 1 φ 21-18.75,18.75 etc.1-18.0,18.01-17.5,17.51-17.0,17.081
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COULOMB EXCITATION DATA ANALYSIS CO
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10 MINIMIZATION BY SIMULATED ANNEAL
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1 INTRODUCTION1.1 Gosia suite of Co
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104 Ru, 110 Pd, 165 Ho, 166 Er, 186
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Figure 1: Coordinate system used to
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Cλ E =1.116547 · (13.889122) λ (
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Figure 2: The orbital integrals R 2
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2.2 Gamma Decay Following Electroma
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where :d 2 σ= σ R (θ p ) X R kχ
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Formula 2.49 is valid only for t mu
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à XK(α) =exp−iτ i (E γ )x i (
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important to have an accurate knowl
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- Page 31 and 32: with the reduced matrix element M c
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- Page 41 and 42: Zd 2 σ(I → I f )Y (I → I f )=s
- Page 43 and 44: 4.5 MinimizationThe minimization, i
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- Page 47 and 48: However, estimation of the stepsize
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- 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
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- 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
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- Page 85 and 86: 5.13 LEVE (LEVELS)Mandatory subopti
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- Page 89 and 90: Figure 10: Model system having 4 st
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- Page 107 and 108: 2,5,1,-2,23,5,1,-2,23,6,1,-2,2Matri
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- Page 115 and 116: OP,STAR or OP,POIN under OP,GOSI. N
- Page 117 and 118: 5.31 INPUT OF EXPERIMENTAL γ-RAY Y
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- Page 121 and 122: *½P 5 (J) = s(E2 × E2) J ׯh¾
- Page 123 and 124: The expectation value of cos3δ can
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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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*** 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