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tungsten ( w ) is one of the most important constituents of tokamak reactor walls @xcite .
additionally , it radiates strongly over almost all ionisation stages . for example , the most intense emission lines of w ions @xcite are from w xxii to w l in the vuv to the soft x - ray region , covering an electron temperature range from about 0.5 to 5.0 kev .
similarly , ptterich et al .
@xcite have predicted emission features from w lxi to w lxix in the 0.10.15 nm , 1.84.0 nm and around 8 nm ranges .
however , to assess radiation loss and for modelling plasmas , atomic data ( including energy levels and oscillator strengths or radiative decay rates ) are required for many of the w ions .
their need for atomic data for several ions , including those of w , has increased significantly due to the developing iter project .
therefore , several groups of people are actively engaged in producing atomic data .
early calculations for a number of w ions ( w xxxviii to w xlviii ) were performed by fournier @xcite .
he adopted a relativistic atomic structure code , but reported only limited results for energy levels and oscillator strengths ( @xmath0-values ) .
a thorough critical compilation of experimental , theoretical and analytical energy levels of w ions ( w iii through w lxxiv ) has been undertaken by kramida and shirai @xcite and has been further reviewed by kramida @xcite .
these energy levels , along with some spectral lines , are also available on the nist ( national institute of standards and technology ) website at http://www.nist.gov/pml/data/asd.cfm .
recently , spectra in the euv wavelength range ( 420 nm ) have been measured by ralchenko et al .
@xcite , for a number of w ions , namely w lv to w lxiv .
similarly , clementson et al .
@xcite have discussed spectroscopy of many w ions ( w xlvii to w lxxii ) . on the other side , calculations have been performed for several w ions , such as by quinet @xcite for w xlviii to w lxii .
although he adopted the grasp code for the calculations , his reported results for energy levels and radiative rates ( @xmath1-values ) are confined to forbidden lines within the 3p@xmath2 and 3d@xmath2 configurations . however , for the modelling of plasmas , atomic data among a wider range of levels / transitions are preferred .
therefore , we have already reported such data for two w ions , namely w xl @xcite and w lviii @xcite . in this paper , we extend our work to eight other w ions , s - like ( w lix ) to f - like ( w lxvi ) . as in our earlier research @xcite and those of others @xcite ,
we have adopted the fully relativistic multi - configuration dirac - fock ( mcdf ) atomic structure code @xcite , better known as the general - purpose relativistic atomic structure package ( grasp ) @xcite .
this code is based on the @xmath3 coupling scheme , includes higher - order relativistic corrections arising from the breit interaction and qed ( quantum electrodynamics ) effects , and is suitable for the heavy ions considered here .
however , this original version @xcite has undergone several revisions , such as by @xcite , and the one employed here ( and by many other workers ) has been revised by dr .
p. h. norrington , and is freely available at http://web.am.qub.ac.uk / darc/.
extensive configuration interaction ( ci ) has been incorporated in grasp , as described below for each ion , and for the optimisation of the orbitals the option of ` extended average level ' ( eal ) , in which a weighted ( proportional to 2@xmath4 + 1 ) trace of the hamiltonian matrix is minimised , has been adopted .
the grasp code has a few other choices for optimisation , such as average level ( al ) and extended optimal level ( eol ) .
however , in general , the results obtained with the al option are comparable with those of eal as already discussed and demonstrated by us for several other ions , such as those of kr @xcite and xe @xcite .
similarly , the eol option may provide slightly more accurate data for a few predefined levels , but is only useful if the experimental energies are known , which is not the case for a majority of the levels of the ions studied here .
clementson and beiersdorfer @xcite have measured wavelengths for 3 lines of w lix .
they also calculated these with two different codes , i.e. grasp and fac ( flexible atomic code ) , and there is no ( major ) discrepancy among the results . for modelling purposes , feldman et al .
@xcite calculated atomic data for many w ions , including w lix , but did not report the data .
furthermore , they used a simple model consisting of the 3s@xmath53p@xmath6 , 3s3p@xmath7 , 3s@xmath53p@xmath83d , and 3p@xmath9 configurations , generating 48 levels in total . for our work ,
we have performed two sets of calculations using the grasp code . in the first ( grasp1 ) we have included 2762 levels of the all possible combinations of the @xmath10 = 3 orbitals , i.e. 18 configurations in number .
the second ( grasp2 ) involves an additional 28 configurations , which are [ 3s@xmath53p@xmath8 , 3s@xmath53p@xmath53d , 3s3p@xmath6 , 3s3p@xmath83d , 3s@xmath53p3d@xmath5 , 3s3p@xmath53d@xmath5 , and 3p@xmath7]4@xmath12 .
these 46 configurations generate 12 652 levels in total . in table
a we compare the energies obtained from both models , but for only the lowest 20 levels .
differences between the two sets of energies are less than 0.025 ryd and the inclusion of larger ci in the grasp2 calculations has lowered the energies for most of the levels .
therefore , it is necessary to assess the effect of further ci on the energy levels . for this
we have adopted the fac code of gu @xcite , which is also fully relativistic and is available from the website https://www - amdis.iaea.org / fac/. this code is comparatively more efficient to run and generally yields results similar to those obtained with other atomic structure codes , as has already been demonstrated in several of our earlier papers
see for example aggarwal et al . @xcite .
with fac we have also performed two sets of calculations , i.e. fac1 : includes the same 2762 levels as in grasp1 , and fac2 : also includes levels of the 3@xmath134@xmath12 configurations , generating 38 694 levels in total .
energies obtained from both these models are also listed in table a for comparison .
discrepancies between the grasp1 and fac1 energies are up to 0.15 ryd ( see level 13 ) , in spite of including the _ same _ ci .
this is because of the differences in the algorithms of the codes and also in calculating the central potentials .
additionally , the energies obtained from fac are generally lower for most levels .
however , inclusion of additional ci in the fac2 calculations further lowers the energies , but only up to 0.02 ryd for some of the levels .
therefore , it may be reasonable to say that the inclusion of ci in our grasp2 calculations is sufficient to calculate accurate results , but differences with fac2 remain of up to 0.15 ryd .
the nist compilation is only for a few levels of w lix , which are mostly based on the experimental and theoretical work of clementson et al .
however , these energies are not very accurate as indicated on their website , and many levels are also missing from the compilation .
nevertheless , in table a we have included their energies for comparison .
unfortunately , differences between their compiled energies and our ( any of the ) calculations are up to 0.4 ryd for some of the levels , such as 1820 .
therefore , there may be scope to improve upon our calculated energies but the ( in)accuracy can not definitely be determined by the limited comparison shown in table a. our calculated energies from grasp2 are listed in table 1 along with those from fac2 for the lowest 220 levels , which belong to the @xmath14 3 configurations . beyond these , the levels of the @xmath10 = 4 configurations start mixing . discrepancies between the two sets of energies are smaller than 0.4 ryd ( @xmath15 0.5% ) for a majority of levels and the orderings are different only in a few instances , such as 70/71 and 151/152 .
we also note that some differences may be because of a mismatch between the two sets of energies , as it is not always possible to perfectly match these due to their different notations .
also note that the @xmath16 designations of the levels listed in table 1 are not always unambiguous , and a few of these can be ( inter)changed with varying amounts of ci , codes , and authors preferences .
this is inevitable in any calculation because of the strong mixing among some of the levels . as examples
, we list the lowest 20 levels in table b. for some , such as 1 , 2 , 10 , and 12 , there is a clear dominance of one vector ( level ) and hence there is no scope for ambiguity .
however , for others , such as 39 , several vectors ( levels ) dominate and therefore it is not straightforward to designate such levels . for example
, the eigenvector for level 19 is dominant in 19 but is also significant in 4 .
however , the eigenvector for level 105 is dominant in both levels 4 and 105 ( not listed in table b ) .
finally , it may be noted that the degeneracy among the levels of w lix is very large
see for example levels 3 , 5 , 9 , 19 , and 32 of 3s@xmath53p@xmath83d @xmath7d@xmath17 , which are separated by up to @xmath1830 ryd . for the ground state energy
the breit and qed contributions are 28.7 and 21.7 ryd , respectively , although they amount to only @xmath180.1% . for this ion
we have also performed two calculations with grasp using different levels of ci , i.e. grasp1 : includes 1313 levels of the 15 @xmath10 = 3 configurations , which are 3s@xmath53p@xmath8 , 3s@xmath53p@xmath53d , 3s3p@xmath6 , 3s@xmath53p3d@xmath5 , 3s3p@xmath83d , 3s3p@xmath53d@xmath5 , 3p@xmath7 , 3p@xmath63d , 3s@xmath53d@xmath8 , 3p@xmath83d@xmath5 , 3s3p3d@xmath8 , 3p@xmath53d@xmath8 , 3s3d@xmath6 , 3p3d@xmath6 , and 3d@xmath7 . in the other calculation ( grasp2 ) , a further 20 configurations of [ 3s@xmath53p@xmath5 , 3s3p@xmath8 , 3s@xmath53p3d , 3s3p@xmath53d , and 3s@xmath53d@xmath5]4@xmath12 are included , generating in total 3533 levels .
similarly , two calculations with fac are performed , i.e. fac1 with the same ci as in grasp2 , and fac2 , which also includes all possible combinations of 3@xmath204@xmath12 , generating 14 608 levels in total .
energies for the lowest 220 levels from both grasp2 and fac2 are listed in table 2 .
these levels belong to the first 8 configurations listed above . for the higher - lying levels , those of @xmath10 = 4 intermix with @xmath10 = 3 . in table
c we compare our energies for the lowest 25 levels of w lx from grasp1 , grasp2 , fac1 , and fac2 with the nist compilation .
ci for w lx is not as important as for w lix , because differences between our grasp1 and grasp2 energies are smaller than 0.02 ryd .
similarly , discrepancies between the fac1 and fac2 energies are less than 0.03 ryd .
however , differences between the grasp2 and fac2 energies are up to 0.3 ryd for some levels , for reasons already explained in section 2.1 .
the nist compilation is only for the lowest 25 levels , listed in table c , and our grasp2 energies are ( generally ) lower by up to 0.3 ryd
see for example , levels 13 , 17 and 22 .
similar differences remain between the nist and fac2 energies , and therefore are not due to a lack of ci .
however , it is worth emphasising that the compiled energies of nist are mostly based on interpolation / extrapolation and hence are likely not very accurate .
more importantly , there are differences in the designations of a few levels , particularly the ground state , which is ( 3s@xmath53p@xmath8 ) @xmath5d@xmath21 in our work , but @xmath5p@xmath21 in nist .
this is a highly mixed level and the eigenvector for @xmath5p@xmath21 dominates in both levels 1 and 25 see table d in which eigenvectors for the lowest 25 are listed .
however , we have preferred to designate the lower ( ground ) level as @xmath5d@xmath21 , because the placings of @xmath5d@xmath22 and @xmath5p@xmath23 ( levels 5 and 6 ) are unambiguous . there may be similar differences in designations with other calculations because of the very high mixing among some of the levels of w lx .
as for other w ions , we have performed two calculations each with the grasp and fac codes to assess the effect of ci .
these are grasp1 : 518 levels of 12 configurations [ 3s@xmath53p@xmath5 , 3s3p@xmath8 , 3s@xmath53p3d , 3s3p@xmath53d , 3p@xmath6 , 3s@xmath53d@xmath5 , 3p@xmath83d , 3s3p3d@xmath5 , 3p@xmath53d@xmath5 , 3s3d@xmath8 , 3p3d@xmath8 , and 3d@xmath6 ] ; grasp2 : 4364 levels of 48 configurations , the additional 36 are [ 3s@xmath53p , 3s3p@xmath5 , 3s@xmath53d , 3s3p3d , 3p@xmath8 , 3p@xmath53d , 3s3d@xmath5 , 3p3d@xmath5 , and 3d@xmath8]4@xmath12 ; fac1 : 9798 levels of 3 * 4 , 3 * 3 4 * 1 and 3 * 4 5 * 1 ; and finally fac2 : which includes 27 122 levels in total , the additional ones arising from 3 * 3 6 * 1 and 3 * 2 4 * 2 configurations .
energies obtained from these calculations are compared in table e with the nist compilation for the lowest 21 levels of w lxi , which are the only ones in common . as for other ions ,
the ci is not very important for this ion , because the grasp1 and grasp2 energies agree within to 0.02 ryd , and the fac1 and fac2 energies show no appreciable differences .
similarly , the agreement between our grasp2 and fac2 energies is better than 0.2 ryd
see levels 1215 . however , as for other ions , the differences with the nist compilation are larger , up to 0.4 ryd see level 9 for example .
again , the nist energies are not very accurate and therefore such differences are not surprising .
an important difference between our calculations and the nist compilation is the designation for level 4 , i.e. ( 3s3p@xmath8 ) @xmath7s@xmath26 which is @xmath8p@xmath26 ( 64 ) in the latter .
both these levels are highly mixed , as may be seen from the eigenvectors listed in table f for the lowest 21 levels _ plus _ the remaining two of the 3s3p@xmath8 configuration , i.e. @xmath8p@xmath26 and @xmath27p@xmath28 . our recommended energies for the lowest 215 levels of w lxi are listed in table 3 from the grasp2 and fac2 calculations .
these levels belong to the @xmath10 = 3 configurations and beyond these those of @xmath10 = 4 intermix .
finally , there are no major differences in the orderings of the two sets of level energies . for w lxii the experimental energies are also as sparse as for other w ions . however , two sets of theoretical energy levels @xcite are available in the literature .
safronova and safronova @xcite adopted a relativistic many - body perturbation theory ( rmbpt ) and reported energies for the lowest 40 levels belonging to the 3s@xmath53p , 3s3p@xmath5 , 3s@xmath53d , 3s3p3d , 3p@xmath8 , and 3p@xmath53d configurations .
in addition , s. aggarwal et al .
@xcite have calculated energies for the lowest 148 levels of the 3s@xmath53p , 3s3p@xmath5 , 3s@xmath53d , 3s3p3d , 3p@xmath8 , 3p@xmath53d , 3s3d@xmath5 , 3p3d@xmath5 , and 3d@xmath8 ( nine ) configurations , adopting the same version of the grasp code as in the present work .
the rmbpt energies @xcite are closer to the nist compilation and in general are lower than those of s. aggarwal et al .
by up to 0.4 ryd
see table 2 of @xcite .
we have performed several sets of calculations with the grasp code but mention only three here , namely : grasp1 , which includes the basic 148 levels of the 9 configurations listed above ; grasp2 , which considers an additional 776 ( total 924 ) levels of the [ 3s3p , 3s3d , 3p3d , 3s@xmath5 , 3p@xmath5 , and 3d@xmath5]4@xmath12 ( 24 ) configurations ; and finally grasp3 which includes a further 1079 levels ( total 2003 ) of the 30 additional configurations , i.e. [ 3s3p , 3s3d , 3p3d , 3s@xmath5 , 3p@xmath5 , and 3d@xmath5]5@xmath12
. s. aggarwal et al .
@xcite included ci among 35 configurations , which are the basic 9 of grasp1 _ plus _ another 26 , i.e. 3s3p4@xmath12 , 3s3d4@xmath12 , 3p3d4@xmath12 , 3s@xmath54@xmath12 , 3p@xmath54@xmath12 ( except 3p@xmath54d ) , 3p4@xmath29 ( except 3p4p@xmath5 ) , and 3d4@xmath29 .
it is not clear why they overlooked configurations such as : 3p@xmath54d , 3p4p@xmath5 , 3s4@xmath29 , and 3@xmath124@xmath30 .
in addition , their 35 configurations generate 1007 levels in total ( see table 1 of @xcite ) whereas they mention only 894 , and therefore there is an anomaly of 113 levels .
however , we stress that ( particularly ) the omission of the 3p@xmath54d and 3p4p@xmath5 configurations does not affect the energies or the corresponding lifetimes , as already discussed by one of us @xcite .
more importantly , levels of the 3@xmath124@xmath29 configurations lie at energies well above those of our grasp3 calculations , and hence are omitted from our work .
this has been confirmed by our larger calculation with 75 configurations and 2393 levels .
for the same reason we preferred not to include the 4@xmath29 configurations for the calculations of energy levels for other w ions . a complete set of energies for all 148 levels ( of the grasp1 calculations )
are listed in table 4 from grasp3 and fac2 ( see below ) .
we note that levels from all other configurations clearly lie _ above _ these 148 and hence there is no intermixing . as with grasp
, we have also performed several calculations with fac , but focus on only two , i.e. fac1 : includes the same 2003 levels as in grasp3 , and fac2 : contains 12 139 levels in total , the additional ones arising from the 3 * 2 6 * 1 , 3 * 1 4 * 2 , 3 * 1 5 * 2 and 3 * 1 6 * 2 configurations . in table
g we compare our energies from grasp2 , grasp3 , fac1 , and fac2 with those of nist for the lowest 21 levels , which are in common . also included in this table
are the results of safronova and safronova @xcite from rmbpt .
the corresponding data of s. aggarwal et al .
@xcite are not considered because they are similar to our grasp2 calculations and have already been discussed previously @xcite . although a considerably large ci has been included in our calculations , it does not appear to be too important for w lxii , because the grasp2 and grasp3 ( and fac1 and fac2 ) energies are practically identical .
therefore , the discrepancies between the grasp and fac energies ( up to 0.4 ryd , particularly for level 21 ) are not due to different levels of ci but because of the computational and theoretical dissimilarities in the codes . nevertheless ,
although the nist energies are not claimed to be very accurate , their agreements with those from fac and rmbpt are better ( within 0.1 ryd ) than with grasp . regarding all the 148 levels in table 4 , the differences between the grasp and fac energies are up to 0.4 ryd for some ( see levels 77 upwards in the table ) .
finally , as for other w ions , configuration mixing is strong for w lxii also and therefore there is always a possibility of ( inter)change of level designations listed in table 4 .
for the 21 levels listed in table g , their designations and orderings are the same between nist and our calculations , but differ with those of s. aggarwal et al .
@xcite for some , such as levels 10 and 68 , i.e. ( 3p@xmath8 ) @xmath5d@xmath21 and @xmath5p@xmath21 , which are reversed by them .
these two levels ( and many more ) have strong mixing , as may be seen from table h in which we list the eigenvectors for the lowest 21 levels plus 68 , i.e. 3p@xmath8 @xmath5p@xmath21 . similarly , there is a _ disagreement _ for most level designations between our work and nist with those of safronova and safronova @xcite . for this ion , earlier calculations for energy levels are by safronova and safronova @xcite using the rmbpt method for the lowest 35 levels of the 3s@xmath5 , 3s3p , 3p@xmath5 , 3s3d , 3p3d , and 3d@xmath5 configurations , whereas the nist compilation is only for 9 levels see table i. as for other ions we have performed several sets of calculations with grasp and fac and here we only state our final results . for the grasp calculations
we have considered 58 configurations , which are 3@xmath29 , 3s3p , 3s3d , 3p3d , 3@xmath124@xmath12 , 4@xmath29 , 4@xmath30 , 3@xmath125@xmath12 , and 3@xmath126@xmath12 ( except 6h ) , while for fac we include 991 levels , the additional ones arising from 3@xmath127@xmath12 and 4@xmath125@xmath12 .
however , levels of the 4@xmath29 , 4@xmath30 and 4@xmath125@xmath12 configurations mostly lie above those of 3@xmath127@xmath12 and can therefore be neglected .
energy levels from both calculations are listed in table 5 for the lowest 210 levels . in table
i a comparison is shown for the lowest 35 levels with the nist compilation and the rmbpt calculations @xcite .
as for w lxii , the fac and rmbpt energies agree closely with each other as well as with nist , but our grasp energies are higher by up to 0.3 ryd for many levels .
similarly , mixing for the levels is strong for a few as shown in table j for the lowest 35 see in particular levels 22 , 25 and 34 . for this ion
we have gradually increased the number of orbitals to perform grasp calculations for up to 1235 levels .
the configurations included are 2p@xmath9@xmath31 with @xmath14 7 and @xmath32 4 , 2p@xmath73@xmath30 , 2p@xmath73@xmath29 , 2p@xmath74@xmath30 , 2p@xmath74@xmath29 , and 2p@xmath73@xmath124@xmath12 .
however , we note that the levels of 2p@xmath9@xmath31 lie _ below _ those of the other configurations . for this reason we only list the lowest 30 levels in table k , all belonging to 2p@xmath9@xmath31 .
however , with fac we have performed comparatively larger calculations for up to @xmath10 = 20 and all possible values of @xmath12 , i.e. 1592 levels in total .
these results are also listed in table k along with those of nist , which are confined to the @xmath14 5 levels .
the nist energies differ with fac by up to 0.26 ryd for some levels ( see 20 ) , but discrepancies are smaller than 0.15 ryd with those with grasp .
again , the differences between the grasp and fac energies are not because of different levels of ci , but due to methodological variations .
it has not been possible to include higher 2p@xmath9@xmath31 configurations in our grasp calculations , but since the fac energies have been obtained ( as stated above ) in table 6 we list these for the lowest 396 levels , all belonging to 2p@xmath9@xmath31 with @xmath14 20
. this will be helpful for future comparisons .
finally , unlike the other w ions discussed above , there is no ( strong ) mixing and/or ambiguity for the designation of the 2p@xmath9@xmath31 levels listed in tables k and 6 .
safronova et al .
@xcite have reported energies for 242 levels of w lxiv from three independent codes , namely rmbpt , hullac ( hebrew university lawrence livermore atomic code @xcite ) and the atomic structure code of r.d .
cowan available at http://das101.isan.troitsk.ru/cowan.htm .
although nist energies for this ion are only available for a few levels , as already seen in table k , their rmbpt results are closest to the measurements .
additionally , based on the comparisons made for other w ions , their rmbpt energies should be the most accurate .
nevertheless , the rmbpt energy for level 2 ( 2p@xmath73s @xmath8p@xmath26 ) differs by 1.3% and 6.4% with those from hullac and cowan , respectively .
corresponding differences for the remaining levels are up to 0.3% and 1% , respectively .
only the lowest 5 levels of table k are common with their work , as the remaining 237 belong to the 2p@xmath73@xmath30 configurations .
therefore , our listed energies in table 6 supplement their data . the nist compilation of energies for this ion is limited to only 10 levels of the 2p@xmath73@xmath12 configurations
. however , vilkas et al .
@xcite have reported energies for 141 levels of the 2p@xmath9 , ( 2s2p@xmath9)3@xmath12 , 4@xmath12 , 5@xmath12 ( except 5 g ) , and ( 2p@xmath7 ) 3@xmath12 , 4@xmath12 , 5@xmath12 ( except 5 g ) configurations . for their calculations they adopted the relativistic multi - reference many - body mller - plesset ( mrmp ) perturbation theory , and included ci up to the @xmath10 = 5 orbitals .
we have included the same configurations for our calculations with grasp , which generate 157 levels in total because we have also considered the 5 g orbital .
however , in table 7 we list energies for only the lowest 121 , because beyond this the levels of the 2s2p@xmath96@xmath12 configurations start mixing in the same way as of 2s2p@xmath95 g with those of 2s2p@xmath94@xmath12 see levels 9299 in the table .
additionally , we have performed larger calculations with fac with up to 1147 levels , belonging to the 2 * 8 , ( 2 * 7 ) 3 * 1 , 4 * 1 , 5 * 1 , 6 * 1 , 7 * 1 , and 2 * 6 3 * 2 configurations .
these results are also listed in table 7 for comparison .
differences between the grasp and fac energies are up to 0.5 ryd ( 0.07% ) for some levels , but the level orderings are almost identical . similarly , there is no difference in level orderings with the mrmp calculations @xcite and the energies differ only by less than 0.6 ryd ( 0.06% ) with grasp
see levels 63 and 7783 .
therefore , overall there is no ( significant ) discrepancy between the three independent calculations .
however , in general the fac energies are lower than those from grasp for a majority of levels , whereas those of mrmp are higher . in table
l , we compare energies with the nist compilation for only the _ common _ levels .
there is no uniform pattern for ( dis)agreement between the theoretical and experimental energies . in general , the mrmp energies are closer to those of nist whereas those from fac differ the most .
unfortunately , these comparisons are not sufficient for accuracy determination , particularly when the nist energies are not based on direct measurements .
finally , as for most w ions , for w lxv also there is a strong mixing for some levels and therefore the level designations listed in table 7 can vary , although the mrmp calculations @xcite have the same labels as in our work .
nevertheless , in table m we list the eigenvectors for the lowest 33 levels , which include all of the nist compilation .
note particularly the mixing for levels 24 , 25 and 31 . for this ion
we have performed a series of calculations with grasp with gradually increasing ci and our final set includes 501 levels of 38 configurations , which are : 2s@xmath52p@xmath7 , 2s2p@xmath9 , ( 2s@xmath52p@xmath6 , 2s2p@xmath7 , 2p@xmath9)3@xmath12 , 4@xmath12 , 5@xmath12 .
similarly , calculations with fac have been performed for up to 1113 levels from the 2 * 7 and ( 2 * 6 ) 3 * 1 , 4 * 1 , 5 * 1 , 6 * 1 , 7 * 1 configurations .
these levels span an energy range of up to 1360 ryd .
opening the 1s shell gives rise to levels above 5000 ryd and therefore has not been included in the calculations .
energies from both of these calculations are listed in table 8 for the lowest 150 levels , because beyond this the levels of the @xmath10 = 5 configurations start mixing .
however , the listed levels include all of the @xmath10 = 3 configurations .
differences between the two sets of energies are up to 0.5 ryd for some levels , except three ( 145147 ) for which the discrepancies are slightly larger , up to 0.7 ryd . the level orderings are also the same for a majority of levels , but slightly differ in a few instances , such as for 93112 .
nist listings are available for only two levels , namely 2s@xmath52p@xmath7 @xmath5p@xmath23 and 2s2p@xmath9 @xmath5s@xmath33 , and the energy for the latter is lower by 0.5 ryd than the theoretical results .
no other similar theoretical energies are available for this ion for comparison purposes .
finally , this ion is no exception for level mixing and examples of this are listed in table n for the lowest 48 levels see in particular 13 , 15 , 40 , and 42 .
apart from energy levels , calculations have been made for absorption oscillator strengths ( @xmath0-values , dimensionless ) , radiative rates ( @xmath1-values , s@xmath34 ) and line strengths ( @xmath35-values , in atomic units , 1 a.u .
= 6.460@xmath3610@xmath37 cm@xmath5 esu@xmath5 ) .
however , @xmath0- and @xmath1-values for all types of transition ( @xmath39 ) are connected by the following expression : @xmath40 where @xmath41 and @xmath42 are the electron mass and charge , respectively , @xmath43 the velocity of light , @xmath44 the transition wavelength in @xmath45 , and @xmath46 and @xmath47 the statistical weights of the lower @xmath48 and upper @xmath4 levels , respectively . similarly , @xmath0- and
@xmath1-values are related to @xmath35 by the standard equations given in @xcite . in tables
916 we present results for energies ( wavelengths , @xmath44 in @xmath49 ) , @xmath1- , @xmath0- and @xmath35- values for electric dipole ( e1 ) transitions in w ions , which have been obtained with the grasp code . for other types of transitions , namely magnetic dipole ( m1 ) , electric quadrupole ( e2 ) , and magnetic quadrupole ( m2 ) ,
only the @xmath1-values are listed , because the corresponding results for @xmath0- or @xmath35-values can be obtained using eqs .
( 1 - 5 ) given in @xcite . additionally
, we have also listed the ratio ( r ) of the velocity ( coulomb gauge ) and length ( babushkin gauge ) forms which often ( but not necessarily ) give an indication of the accuracy .
the _ indices _ used to represent the lower and upper levels of a transition are defined in tables 18 .
furthermore , only a limited range of transitions are listed in tables 916 , but full tables are available online in the electronic version . for the w ions considered here ,
existing @xmath1- ( or @xmath0- ) values are available mostly for three ions , i.e. al - like w lxii @xcite , mg - like w lxiii @xcite and na - like w lxiv @xcite .
therefore , we confine our comparisons to these three ions . in table o we compare the @xmath0-values for common e1 transitions with the results of safronova and safronova @xcite .
both sets of data agree very well for all transitions .
similarly , for a few weak transitions ( @xmath0 @xmath18 10@xmath50 ) , such as 122 , 23 and 1419 , the ratio r is up to 1.7 and is closer to unity for the comparatively strong transitions .
similar comparison with their results for transitions in w lxiii is shown in table p. for the common transitions listed here , r is unity for all , and @xmath0-values agree closely for most with only a few exceptions , such as 2032 , 2130 and 2634 for which discrepancies are a factor of two . however , we note that the @xmath0- ( or @xmath1- ) values of @xcite are only for a small number of transitions whereas our results listed in tables 12 and 13 cover a much wider range .
vilkas et al .
@xcite have listed @xmath1-values for some ( not all ) transitions of w lxv and in table
q we compare their results with our calculations with grasp , but only from the lowest three to higher excited levels .
additionally we have listed the @xmath0-values to indicate the strength of transitions . as for other w ions
, r is also listed for these transitions and is within a few percent of unity , irrespective of the @xmath0-value .
there are no appreciable differences between the two sets of @xmath1-values and discrepancies , if any , are ( generally ) within @xmath1820% .
the comparisons of @xmath1- ( @xmath0- ) values discussed above are only for a subset of transitions . considering a wider range , for a majority of strong transitions ( @xmath0 @xmath51 0.01 )
r is often within 20% of unity , as already seen in tables o , p and q. however , there are ( as always ) some exceptions .
for example , there are only six transitions of w lxiii with @xmath0 @xmath52 0.01 for which r is up to 1.6 , namely 148166 ( @xmath0 = 0.011 , r = 1.3 ) , 158173 ( @xmath0 = 0.021 , r = 1.3 ) , 160174 ( @xmath0 = 0.028 , r = 1.6 ) , 161175 ( @xmath0 = 0.025 , r = 1.4 ) , 162176 ( @xmath0 = 0.027 , r = 1.4 ) , and 163177 ( @xmath0 = 0.029 , r = 1.6 ) . therefore , based on this and other comparisons
already discussed , our assessment of accuracy for the @xmath0-values for a majority of strong transitions is @xmath1820% . finally ,
for much weaker transitions ( often with @xmath0 @xmath53 10@xmath50 ) , r can be several orders of magnitude and it is very difficult to assess the accuracy of the @xmath0-values because results are often much more variable with ci and/or codes .
generally , such transitions do not make an appreciable contribution to plasma modelling and their results are mostly required for completeness .
the lifetime @xmath54 of a level @xmath4 is given by 1.0/@xmath55@xmath56 and the summation includes @xmath1-values from all types of transitions , i.e. e1 , e2 , m1 , and m2 . since this is a measurable quantity it helps to assess the accuracy of @xmath1-values , particularly when a single ( type of ) transition dominates .
unfortunately , to our knowledge no measurements of @xmath54 are available for the levels of the w ions considered here , but in tables 18 we list our calculated results .
previous theoretical results are available for two ions , i.e. w lxii @xcite and w lxv @xcite .
unfortunately , the @xmath54 of s. aggarwal et al . @xcite
contain large errors , by up to 14 orders of magnitude , for over 90% of the levels of w lxii and bear no relationship to the @xmath1-values , as already discussed @xcite . for w lxv , the reported @xmath54 of vilkas et al .
@xcite are included in table 7 , and there is no significant discrepancy for any level .
energy levels and radiative rates for e1 , e2 , m1 , and m2 transitions are reported for eight w ions ( w lix to w lxvi ) .
a large number of levels are considered for each ion and the data sets reported here are significantly larger than available in the literature . for our calculations
the grasp code has been adopted , although fac has also been utilised for the determination of energy levels to assess the importance of ci , larger than that considered in grasp .
it is concluded that ci beyond a certain level does not appreciably improve the level energies .
differences between the grasp and fac energies , and the available experimental and theoretical values , are often smaller than 0.5 ryd , or equivalently the listed energy levels for all w ions are assessed to be accurate to better than 1% , but scope remains for improvement .
a similar assessment of accuracy for the corresponding @xmath1-values is not feasible , mainly because of the paucity of other comparable results .
however , for strong transitions ( with large @xmath0-values ) , the accuracy for @xmath1-values and lifetimes may be @xmath1820%
. lifetimes for these levels are also listed although no measurements are currently available in the literature .
however , previous theoretical values are available for most levels of w lxv and there is no discrepancy with our work .
kma is thankful to awe aldermaston for financial support .
owing to space limitations , only parts of tables 916 are presented here , the full tables being made available as supplemental material in conjunction with the electronic publication of this work .
supplementary data associated with this article can be found , in the online version , at doi : nn.nnnn / j.adt.2016.nn.nnn . 999 t. ptterich , r. neu , r. dux , a.d .
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data tables 94 ( 2008 ) 50 . @rlllrrrr@ index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + + index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + 1 & 3s@xmath53p@xmath6 & @xmath8p@xmath57 & 00.0000 & 0.0000 & 0.0000 & 0.00000 & 0.0000 + 2 & 3s@xmath53p@xmath6 & @xmath27s@xmath58 & 01.394 & 1.4629 & 1.4691 & 1.46006 & 1.4675 + 3 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath59 & 17.2585 & 17.3063 & 17.2920 & 17.23239 & 17.2200 + 4 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8p@xmath60 & 17.852 & 17.9536 & 17.9413 & 17.85940 & 17.8689 + 5 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath61 & 17.852 & 17.9342 & 17.9213 & 17.87821 & 17.8489 + 6 & 3s@xmath53p@xmath8(@xmath5p)3d & @xmath8f@xmath62 & 17.9173 & 17.9812 & 17.9637 & 17.90613 & 17.8913 + 7 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath27g@xmath63 & 23.4589 & 23.4945 & 23.4758 & 23.42838 & 23.4095 + 8 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8d@xmath59 & 23.94 & 23.9785 & 23.9637 & 23.91315 & 23.8975 + 9 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath62 & 25.29 & 25.4172 & 25.3923 & 25.35176 & 25.3264 + 10 & 3s@xmath53p@xmath6 & @xmath8p@xmath64 & & 25.4932 & 25.4899 & 25.52002 & 25.5134 + 11 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8p@xmath61 & 25.96 & 26.2537 & 26.2281 & 26.18601 & 26.1616 + 12 & 3s@xmath53p@xmath6 & @xmath27d@xmath57 & & 26.2096 & 26.2037 & 26.23460 & 26.2258 + 13 & 3s@xmath53p@xmath5(@xmath27s)3d@xmath5(@xmath8f ) & @xmath8f@xmath57 & & 35.3490 & 35.3341 & 35.19957 & 35.1818 + 14 & 3s@xmath53p@xmath5(@xmath8p)3d@xmath5(@xmath8p ) & @xmath7d@xmath58 & & 37.1186 & 37.1256 & 36.96614 & 36.9719 + 15 & 3s3p@xmath7 & @xmath8p@xmath59 & 39.0268 & 39.1209 & 39.1426 & 39.03447 & 39.0438 + 16 & 3s3p@xmath7 & @xmath27p@xmath61 & 40.62 & 40.6371 & 40.6465 & 40.56564 & 40.5641 + 17 & 3s@xmath53p@xmath5(@xmath27s)3d@xmath5(@xmath8f ) & @xmath8f@xmath65 & & 41.8457 & 41.8247 & 41.70542 & 41.6775 + 18 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8f@xmath59 & 42.43 & 42.0799 & 42.0634 & 42.03196 & 42.0150 + 19 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath60 & 42.53 & 42.1816 & 42.1679 & 42.13380 & 42.1196 + 20 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8d@xmath61 & 42.60 & 42.2933 & 42.2833 & 42.23549 & 42.2231 + + nist : http://www.nist.gov/pml/data/asd.cfm + grasp1 : present results with the grasp code from 18 configurations and 2762 levels + grasp2 : present results with the grasp code from 46 configurations and 12 652 levels + fac1 : present results with the fac code from 2762 levels + fac2 : present results with the fac code from 38 694 levels + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath53p@xmath6 & @xmath8p@xmath57 & 0.67 ( 1)+0.31 ( 12 ) + 2 & 3s@xmath53p@xmath6 & @xmath27s@xmath58 & 0.36 ( 46)+0.64 ( 2 ) + 3 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath59 & 0.16 ( 3)+0.08 ( 41)+0.19 ( 18)+0.06 ( 45)+0.10 ( 26)+0.23(108)+0.12(132 ) + 4 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8p@xmath60 & 0.26 ( 19)+0.14 ( 4)+0.12 ( 37)+0.48(105 ) + 5 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath61 & 0.24 ( 5)+0.14 ( 20)+0.10 ( 43)+0.18 ( 30)+0.27(107 ) + 6 & 3s@xmath53p@xmath8(@xmath5p)3d & @xmath8f@xmath62 & 0.13 ( 9)+0.12(106)+0.21 ( 22)+0.05 ( 44)+0.27 ( 6)+0.06(136)+0.14 ( 40 ) + 7 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath27g@xmath63 & 0.26 ( 32)+0.14 ( 25)+0.10 ( 7)+0.48(130 ) + 8 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8d@xmath59 & 0.10 ( 3)+0.14 ( 41)+0.16 ( 8)+0.07 ( 28)+0.04(108)+0.22 ( 39)+0.19(132 ) + 9 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath62 & 0.14 ( 9)+0.09(106)+0.06 ( 22)+0.17 ( 27)+0.30(136)+0.17 ( 40 ) + 10 & 3s@xmath53p@xmath6 & @xmath8p@xmath64 & 0.98 ( 10 ) + 11 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8p@xmath61 & 0.22 ( 35)+0.14 ( 11)+0.08 ( 29)+0.09 ( 30)+0.14(107)+0.24(147 ) + 12 & 3s@xmath53p@xmath6 & @xmath27d@xmath57 & 0.31 ( 1)+0.67 ( 12 ) + 13 & 3s@xmath53p@xmath5(@xmath27s)3d@xmath5(@xmath8f ) & @xmath8f@xmath57 & 0.23 ( 59)+0.06 ( 88)+0.10 ( 96)+0.05(285)+0.05(350)+0.08 ( 38)+0.06(277)+0.22 ( 13)+0.07(325 ) + 14 & 3s@xmath53p@xmath5(@xmath8p)3d@xmath5(@xmath8p ) & @xmath7d@xmath58 & 0.28 ( 14)+0.16 ( 85)+0.04(114)+0.21 ( 42)+0.21(371)+0.10(342 ) + 15 & 3s3p@xmath7 & @xmath8p@xmath59 & 0.07 ( 28)+0.85 ( 15 ) + 16 & 3s3p@xmath7 & @xmath27p@xmath61 & 0.12 ( 5)+0.05 ( 11)+0.11 ( 43)+0.24 ( 86)+0.36 ( 16 ) + 17 & 3s@xmath53p@xmath5(@xmath27s)3d@xmath5(@xmath8f ) & @xmath8f@xmath65 & 0.23 ( 57)+0.16(221)+0.05(262)+0.08 ( 87)+0.10(139)+0.05(282)+0.30 ( 17 ) + 18 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8f@xmath59 & 0.29 ( 3)+0.10 ( 41)+0.40 ( 18)+0.13 ( 8)+0.04 ( 26 ) + 19 & 3s@xmath53p@xmath8(@xmath6s)3d & @xmath7d@xmath60 & 0.53 ( 19)+0.21 ( 4)+0.24 ( 37 ) + 20 & 3s@xmath53p@xmath8(@xmath5d)3d & @xmath8d@xmath61 & 0.32 ( 5)+0.05 ( 35)+0.40 ( 20)+0.07 ( 11)+0.06 ( 86)+0.05 ( 16 ) + + @rlllrrrr@ index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + + index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + 1 & 3s@xmath53p@xmath8 & @xmath5d@xmath66 & 00.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 + 2 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath67 & 16.8821 & 16.9403 & 16.9357 & 16.8671 & 16.8529 + 3 & 3s@xmath53p@xmath5(@xmath27s)3d & @xmath5d@xmath68 & 23.903 & 23.8688 & 23.8613 & 23.8040 & 23.7839 + 4 & 3s@xmath53p@xmath8 & @xmath6s@xmath66 & 25.060 & 25.0648 & 25.0619 & 25.0905 & 25.0865 + 5 & 3s@xmath53p@xmath8 & @xmath5d@xmath69 & 25.9556 & 25.9490 & 25.9411 & 25.9727 & 25.9639 + 6 & 3s@xmath53p@xmath8 & @xmath5p@xmath70 & 27.019 & 27.0845 & 27.0831 & 27.1065 & 27.1049 + 7 & 3s3p@xmath6 & @xmath6p@xmath68 & 37.9315 & 37.9970 & 38.0129 & 37.9087 & 37.9134 + 8 & 3s3p@xmath6 & @xmath5p@xmath67 & 40.242 & 40.1880 & 40.1973 & 40.1132 & 40.1082 + 9 & 3s3p@xmath6 & @xmath5s@xmath71 & 40.205 & 40.3454 & 40.3636 & 40.2582 & 40.2632 + 10 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath68 & 42.01 & 41.8139 & 41.8076 & 41.7643 & 41.7479 + 11 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6d@xmath71 & 42.16 & 42.0470 & 42.0473 & 41.9973 & 41.9888 + 12 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6d@xmath67 & 42.24 & 42.0827 & 42.0820 & 42.0263 & 42.0159 + 13 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5g@xmath72 & 42.97 & 42.7366 & 42.7255 & 42.6856 & 42.6644 + 14 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5d@xmath68 & 44.848 & 44.9721 & 44.9564 & 44.9235 & 44.8978 + 15 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5p@xmath71 & 45.51 & 45.7510 & 45.7334 & 45.6952 & 45.6683 + 16 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5d@xmath67 & 45.6572 & 45.8196 & 45.8091 & 45.7563 & 45.7341 + 17 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath72 & 47.96 & 47.7759 & 47.7717 & 47.7352 & 47.7188 + 18 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5p@xmath67 & 48.90 & 48.7073 & 48.7034 & 48.6656 & 48.6501 + 19 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5g@xmath73 & 48.98 & 48.7536 & 48.7404 & 48.7119 & 48.6861 + 20 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5f@xmath68 & 49.19 & 48.9732 & 48.9665 & 48.9329 & 48.9141 + 21 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6p@xmath68 & 50.74 & 50.5135 & 50.4973 & 50.4698 & 50.4412 + 22 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5f@xmath72 & 50.87 & 50.5992 & 50.5800 & 50.5583 & 50.5262 + 23 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5d@xmath67 & 51.38 & 51.2398 & 51.2202 & 51.1970 & 51.1682 + 24 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5s@xmath71 & 51.67 & 51.5331 & 51.5132 & 51.4871 & 51.4582 + 25 & 3s@xmath53p@xmath8 & @xmath5p@xmath66 & 52.18 & 52.2859 & 52.2799 & 52.3345 & 52.3257
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp1 : present results with the grasp code from 15 configurations and 1313 levels + grasp2 : present results with the grasp code from 35 configurations and 3533 levels + fac1 : present results with the fac code from 1313 levels + fac2 : present results with the fac code from 14 608 levels + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath53p@xmath8 & @xmath5d@xmath66 & 0.25 ( 4)+0.27 ( 1)+0.48 ( 25 ) + 2 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath67 & 0.34 ( 2)+0.12 ( 12)+0.10 ( 23)+0.11 ( 18)+0.31 ( 60 ) + 3 & 3s@xmath53p@xmath5(@xmath27s)3d & @xmath5d@xmath68 & 0.17 ( 10)+0.20 ( 57)+0.14 ( 21)+0.15 ( 20)+0.30 ( 3 ) + 4 & 3s@xmath53p@xmath8 & @xmath6s@xmath66 & 0.55 ( 4)+0.45 ( 1 ) + 5 & 3s@xmath53p@xmath8 & @xmath5d@xmath69 & 1.00 ( 5 ) + 6 & 3s@xmath53p@xmath8 & @xmath5p@xmath70 & 0.98 ( 6 ) + 7 & 3s3p@xmath6 & @xmath6p@xmath68 & 0.66 ( 7)+0.27 ( 45 ) + 8 & 3s3p@xmath6 & @xmath5p@xmath67 & 0.08 ( 2)+0.06 ( 58)+0.11 ( 18)+0.06 ( 16)+0.11 ( 41)+0.32 ( 8)+0.24 ( 51 ) + 9 & 3s3p@xmath6 & @xmath5s@xmath71 & 0.05 ( 59)+0.07 ( 24)+0.24(137)+0.07 ( 48)+0.53 ( 9 ) + 10 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath68 & 0.46 ( 10)+0.16 ( 20)+0.29 ( 74 ) + 11 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6d@xmath71 & 0.79 ( 11)+0.04 ( 59)+0.14 ( 70 ) + 12 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6d@xmath67 & 0.32 ( 2)+0.28 ( 12)+0.12 ( 18)+0.12 ( 16)+0.05 ( 73)+0.04 ( 51 ) + 13 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5g@xmath72 & 0.18 ( 17)+0.14 ( 56)+0.53 ( 13)+0.12 ( 22 ) + 14 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5d@xmath68 & 0.10 ( 10)+0.10 ( 57)+0.12 ( 21)+0.30 ( 14)+0.24 ( 74)+0.06 ( 72 ) + 15 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5p@xmath71 & 0.30 ( 59)+0.35 ( 15)+0.18 ( 24)+0.08 ( 9 ) + 16 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5d@xmath67 & 0.22 ( 58)+0.08 ( 18)+0.26 ( 16)+0.14 ( 73)+0.13 ( 8)+0.07 ( 51 ) + 17 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6f@xmath72 & 0.37 ( 17)+0.48 ( 67)+0.07 ( 56)+0.07 ( 22 ) + 18 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5p@xmath67 & 0.16 ( 12)+0.23 ( 58)+0.05 ( 23)+0.25 ( 18)+0.06 ( 16)+0.20 ( 73 ) + 19 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5g@xmath73 & 0.37 ( 69)+0.62 ( 19 ) + 20 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5f@xmath68 & 0.04 ( 10)+0.22 ( 57)+0.07 ( 21)+0.32 ( 20)+0.04 ( 74)+0.25 ( 72 ) + 21 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath6p@xmath68 & 0.35 ( 21)+0.10 ( 20)+0.14 ( 14)+0.08 ( 74)+0.27 ( 72 ) + 22 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5f@xmath72 & 0.17 ( 17)+0.05 ( 67)+0.18 ( 56)+0.15 ( 13)+0.44 ( 22 ) + 23 & 3s@xmath53p@xmath5(@xmath8p)3d & @xmath5d@xmath67 & 0.05 ( 2)+0.11 ( 12)+0.46 ( 23)+0.14 ( 16)+0.22 ( 73 ) + 24 & 3s@xmath53p@xmath5(@xmath27d)3d & @xmath5s@xmath71 & 0.31 ( 70)+0.23 ( 15)+0.36 ( 24 ) + 25 & 3s@xmath53p@xmath8 & @xmath5p@xmath66 & 0.19 ( 4)+0.28 ( 1)+0.50 ( 25 ) + + @rlllrrrr@ index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + + index & configuration & level & nist & grasp1 & grasp2 & fac1 & fac2 + + + 1 & 3s@xmath53p@xmath5 & @xmath8p@xmath58 & 00.0000 & 00.0000 & 0.0000 & 0.0000 & 0.0000 + 2 & 3s@xmath53p@xmath5 & @xmath8p@xmath64 & 25.5337 & 25.5392 & 25.5395 & 25.5650 & 25.5675 + 3 & 3s@xmath53p@xmath5 & @xmath27d@xmath57 & 26.2946 & 26.2402 & 26.2351 & 26.2574 & 26.2587 + 4 & 3s3p@xmath8 & @xmath7s@xmath59 & 38.2094 & 38.0581 & 38.0606 & 37.9699 & 37.9709 + 5 & 3s3p@xmath8 & @xmath8d@xmath61 & 39.9800 & 40.1129 & 40.1071 & 40.0211 & 40.0210 + 6 & 3s@xmath53p3d & @xmath8f@xmath59 & & 41.7903 & 41.7744 & 41.7215 & 41.7222 + 7 & 3s@xmath53p3d & @xmath8d@xmath61 & & 45.1086 & 45.0826 & 45.0158 & 45.0129 + 8 & 3s@xmath53p3d & @xmath8p@xmath59 & 49.57 & 49.1925 & 49.1738 & 49.1246 & 49.1247 + 9 & 3s@xmath53p3d & @xmath8f@xmath62 & 49.65 & 49.2768 & 49.2540 & 49.2059 & 49.2052 + 10 & 3s@xmath53p@xmath5 & @xmath8p@xmath57 & 52.27 & 52.3039 & 52.2993 & 52.3463 & 52.3487 + 11 & 3s@xmath53p@xmath5 & @xmath27s@xmath58 & 53.71 & 53.8224 & 53.8202 & 53.8658 & 53.8642 + 12 & 3s3p@xmath5(@xmath6p)3d & @xmath7f@xmath64 & & 54.6247 & 54.6092 & 54.4384 & 54.4378 + 13 & 3s3p@xmath5(@xmath6p)3d & @xmath8p@xmath57 & & 55.1766 & 55.1588 & 54.9863 & 54.9850 + 14 & 3s3p@xmath5(@xmath6p)3d & @xmath7p@xmath65 & & 61.6560 & 61.6353 & 61.4690 & 61.4679 + 15 & 3s3p@xmath5(@xmath6p)3d & @xmath8f@xmath57 & & 62.7077 & 62.6817 & 62.5113 & 62.5082 + 16 & 3s3p@xmath8 & @xmath8d@xmath59 & 63.01 & 62.9640 & 62.9654 & 62.9020 & 62.9054 + 17 & 3s3p@xmath8 & @xmath8d@xmath62 & 64.46 & 64.3393 & 64.3357 & 64.2689 & 64.2712 + 18 & 3s3p@xmath8 & @xmath8p@xmath60 & 65.29 & 65.1798 & 65.1832 & 65.1238 & 65.1237 + 19 & 3s3p@xmath8 & @xmath8p@xmath61 & 66.52 & 66.2880 & 66.2870 & 66.2268 & 66.2275 + 20 & 3s3p@xmath8 & @xmath27d@xmath59 & 66.51 & 66.2776 & 66.2699 & 66.2141 & 66.2160 + 21 & 3s3p@xmath8 & @xmath8s@xmath61 & 67.24 & 67.1867 & 67.1794 & 67.1047 & 67.1036
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp1 : present results with the grasp code from 12 configurations and 518 levels + grasp2 : present results with the grasp code from 48 configurations and 4364 levels + fac1 : present results with the fac code from 9798 levels + fac2 : present results with the fac code from 27 122 levels + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath53p@xmath5 & @xmath8p@xmath58 & 0.69 ( 1)+0.31 ( 11 ) + 2 & 3s@xmath53p@xmath5 & @xmath8p@xmath64 & 1.00 ( 2 ) + 3 & 3s@xmath53p@xmath5 & @xmath27d@xmath57 & 0.35 ( 10)+0.64 ( 3 ) + 4 & 3s3p@xmath8 & @xmath7s@xmath59 & 0.27 ( 4)+0.16 ( 16)+0.08 ( 20)+0.46 ( 64 ) + 5 & 3s3p@xmath8 & @xmath8d@xmath61 & 0.14 ( 21)+0.28 ( 5)+0.16 ( 19)+0.24 ( 69)+0.09 ( 7)+0.05 ( 29 ) + 6 & 3s@xmath53p3d & @xmath8f@xmath59 & 0.74 ( 6)+0.19 ( 27 ) + 7 & 3s@xmath53p3d & @xmath8d@xmath61 & 0.04 ( 21)+0.04 ( 5)+0.05 ( 69)+0.44 ( 7)+0.14 ( 24)+0.25 ( 29 ) + 8 & 3s@xmath53p3d & @xmath8p@xmath59 & 0.34 ( 23)+0.45 ( 8)+0.14 ( 27 ) + 9 & 3s@xmath53p3d & @xmath8f@xmath62 & 0.50 ( 9)+0.22 ( 28)+0.26 ( 22 ) + 10 & 3s@xmath53p@xmath5 & @xmath8p@xmath57 & 0.64 ( 10)+0.35 ( 3 ) + 11 & 3s@xmath53p@xmath5 & @xmath27s@xmath58 & 0.30 ( 1)+0.67 ( 11 ) + 12 & 3s3p@xmath5(@xmath6p)3d & @xmath7f@xmath64 & 0.38 ( 12)+0.07 ( 32)+0.05 ( 57)+0.08 ( 42)+0.05(137)+0.30(110 ) + 13 & 3s3p@xmath5(@xmath6p)3d & @xmath8p@xmath57 & 0.18 ( 30)+0.11(105)+0.08(134)+0.12 ( 13)+0.12 ( 45)+0.21(113)+0.09(135 ) + 14 & 3s3p@xmath5(@xmath6p)3d & @xmath7p@xmath65 & 0.13 ( 33)+0.19(106)+0.15 ( 14)+0.04 ( 49)+0.08 ( 60)+0.05(116)+0.30(132 ) + 15 & 3s3p@xmath5(@xmath6p)3d & @xmath8f@xmath57 & 0.09 ( 30)+0.05(105)+0.25 ( 15)+0.07(134)+0.08(119)+0.07 ( 63)+0.09(113)+0.21(135 ) + 16 & 3s3p@xmath8 & @xmath8d@xmath59 & 0.45 ( 4)+0.49 ( 16 ) + 17 & 3s3p@xmath8 & @xmath8d@xmath62 & 0.94 ( 17 ) + 18 & 3s3p@xmath8 & @xmath8p@xmath60 & 0.86 ( 18)+0.14 ( 25 ) + 19 & 3s3p@xmath8 & @xmath8p@xmath61 & 0.18 ( 21)+0.20 ( 5)+0.34 ( 19)+0.12 ( 69)+0.04 ( 7)+0.12 ( 24 ) + 20 & 3s3p@xmath8 & @xmath27d@xmath59 & 0.08 ( 4)+0.08 ( 16)+0.55 ( 20)+0.07 ( 6)+0.12 ( 23)+0.08 ( 27 ) + 21 & 3s3p@xmath8 & @xmath8s@xmath61 & 0.32 ( 21)+0.20 ( 5)+0.26 ( 19)+0.18 ( 69 ) + ... + 64 & 3s3p@xmath8 & @xmath8p@xmath59 & 0.17 ( 4)+0.16 ( 16)+0.14 ( 20)+0.50 ( 64 ) + 69 & 3s3p@xmath8 & @xmath27p@xmath61 & 0.23 ( 21)+0.23 ( 5)+0.14 ( 19)+0.35 ( 69 ) + + @rlllrrrrr@ index & configuration & level & nist & grasp2 & grasp3 & fac1 & fac2 & rmbpt + + + + index & configuration & level & nist & grasp2 & grasp3 & fac1 & fac2 & rmbpt + + + 1 & 3s@xmath53p & @xmath5p@xmath70 & 00.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 + 2 & 3s3p@xmath5 & @xmath6p@xmath71 & 12.3076 & 12.4425 & 12.4422 & 12.3220 & 12.3204 & 12.3018 + 3 & 3s@xmath53p & @xmath5p@xmath66 & 26.7311 & 26.7061 & 26.7060 & 26.7306 & 26.7314 & 26.7056 + 4 & 3s3p@xmath5 & @xmath6p@xmath67 & 36.7742 & 36.8823 & 36.8835 & 36.7935 & 36.7959 & 36.7790 + 5 & 3s3p@xmath5 & @xmath5d@xmath68 & 38.109 & 38.2380 & 38.2375 & 38.1447 & 38.1457 & 38.1110 + 6 & 3s3p@xmath5 & @xmath5d@xmath67 & 39.6875 & 39.7869 & 39.7857 & 39.7028 & 39.7033 & 39.6637 + 7 & 3s3p@xmath5 & @xmath5p@xmath71 & 40.4238 & 40.5824 & 40.5806 & 40.4826 & 40.4819 & 40.4024 + 8 & 3s@xmath53d & @xmath5d@xmath67 & 43.9039 & 44.0171 & 44.0103 & 43.9482 & 43.9448 & 43.8775 + 9 & 3s@xmath53d & @xmath5d@xmath68 & 49.263 & 49.3268 & 49.3215 & 49.2778 & 49.2759 & 49.2626 + 10 & 3p@xmath8 & @xmath5d@xmath66 & & 51.9171 & 51.9167 & 51.7504 & 51.7496 & 51.7270 + 11 & 3s3p3d & @xmath6f@xmath66 & & 52.9289 & 52.9278 & 52.7221 & 52.7218 & 52.6746 + 12 & 3s3p3d & @xmath6f@xmath69 & & 53.7715 & 53.7680 & 53.5988 & 53.5989 & 53.5817 + 13 & 3s3p3d & @xmath6d@xmath70 & & 55.7589 & 55.7524 & 55.5868 & 55.5849 & 55.5323 + 14 & 3s3p3d & @xmath6d@xmath66 & & 56.2688 & 56.2632 & 56.0908 & 56.0883 & 56.0220 + 15 & 3s3p3d & @xmath6p@xmath69 & & 59.7857 & 59.7835 & 59.6243 & 59.6251 & 59.6240 + 16 & 3s3p3d & @xmath6f@xmath74 & & 60.8471 & 60.8418 & 60.6804 & 60.6797 & 60.6611 + 17 & 3s3p(@xmath8p)3d & @xmath5f@xmath69 & & 61.6177 & 61.6112 & 61.4480 & 61.4457 & 61.4177 + 18 & 3s3p(@xmath8p)3d & @xmath5d@xmath66 & & 62.0001 & 61.9931 & 61.8237 & 61.8206 & 61.7773 + 19 & 3s3p@xmath5 & @xmath6p@xmath68 & 64.372 & 64.4986 & 64.4980 & 64.4306 & 64.4321 & 64.3709 + 20 & 3s3p@xmath5 & @xmath5s@xmath71 & 67.115 & 67.2861 & 67.2849 & 67.2135 & 67.2094 & 67.1156 + 21 & 3p@xmath5(@xmath8p)3d & @xmath6f@xmath67 & 67.479 & 67.8062 & 67.8002 & 67.3757 & 67.3714 & 67.4809
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp2 : present results with the grasp code from 33 configurations and 928 levels + grasp3 : present results with the grasp code from 63 configurations and 2003 levels + fac1 : present results with the fac code from 2003 levels + fac2 : present results with the fac code from + rmbpt : earlier results of safronova and safronova @xcite + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath53p & @xmath5p@xmath70 & 1.00 ( 1 ) + 2 & 3s3p@xmath5 & @xmath6p@xmath71 & 0.52 ( 2)+0.17 ( 7)+0.30 ( 20 ) + 3 & 3s@xmath53p & @xmath5p@xmath66 & 0.98 ( 3 ) + 4 & 3s3p@xmath5 & @xmath6p@xmath67 & 0.88 ( 4)+0.08 ( 6 ) + 5 & 3s3p@xmath5 & @xmath5d@xmath68 & 0.37 ( 19)+0.61 ( 5 ) + 6 & 3s3p@xmath5 & @xmath5d@xmath67 & 0.28 ( 22)+0.49 ( 6)+0.18 ( 8) + 7 & 3s3p@xmath5 & @xmath5p@xmath71 & 0.29 ( 2)+0.69 ( 7 ) + 8 & 3s@xmath53d & @xmath5d@xmath67 & 0.08 ( 22)+0.12 ( 6)+0.79 ( 8) + 9 & 3s@xmath53d & @xmath5d@xmath68 & 0.96 ( 9 ) + 10 & 3p@xmath8 & @xmath5d@xmath66 & 0.23 ( 11)+0.11 ( 18)+0.06 ( 37)+0.12 ( 25)+0.16 ( 10)+0.26 ( 68 ) + 11 & 3s3p3d & @xmath6f@xmath66 & 0.55 ( 11)+0.10 ( 31)+0.07 ( 25)+0.08 ( 10)+0.14 ( 68 ) + 12 & 3s3p3d & @xmath6f@xmath69 & 0.42 ( 12)+0.07 ( 17)+0.18 ( 35)+0.24 ( 32 ) + 13 & 3s3p3d & @xmath6d@xmath70 & 0.58 ( 13)+0.06 ( 28)+0.10 ( 39)+0.26 ( 33 ) + 14 & 3s3p3d & @xmath6d@xmath66 & 0.32 ( 14)+0.18 ( 27)+0.20 ( 37)+0.13 ( 31)+0.05 ( 40)+0.04 ( 68 ) + 15 & 3s3p3d & @xmath6p@xmath69 & 0.11 ( 12)+0.31 ( 29)+0.38 ( 15)+0.07 ( 17)+0.08 ( 35 ) + 16 & 3s3p3d & @xmath6f@xmath74 & 0.42 ( 16)+0.23 ( 30)+0.08 ( 36)+0.26 ( 38 ) + 17 & 3s3p(@xmath8p)3d & @xmath5f@xmath69 & 0.13 ( 12)+0.12 ( 15)+0.46 ( 17)+0.05 ( 32)+0.21 ( 41 ) + 18 & 3s3p(@xmath8p)3d & @xmath5d@xmath66 & 0.11 ( 14)+0.05 ( 27)+0.35 ( 18)+0.16 ( 37)+0.20 ( 40 ) + 19 & 3s3p@xmath5 & @xmath6p@xmath68 & 0.61 ( 19)+0.38 ( 5 ) + 20 & 3s3p@xmath5 & @xmath5s@xmath71 & 0.18 ( 2)+0.14 ( 7)+0.67 ( 20 ) + 21 & 3p@xmath5(@xmath8p)3d & @xmath6f@xmath67 & 0.06 ( 22)+0.04 ( 6)+0.30 ( 21)+0.10 ( 43)+0.08 ( 59)+0.10 ( 99)+0.27 ( 93 ) + ... + 68 & 3p@xmath8 & @xmath5p@xmath66 & 0.19 ( 25)+0.28 ( 10)+0.50 ( 68 ) + + @rlllrrr@ index & configuration & level & nist & grasp & fac & rmbpt + + + + index & configuration & level & nist & grasp & fac & rmbpt + + + 1 & 3s@xmath5 & @xmath27s@xmath58 & 00.0000 & 0.0000 & 0.0000 & 0.0000 + 2 & 3s3p & @xmath8p@xmath60 & 10.261 & 10.3595 & 10.2414 & 10.2650 + 3 & 3s3p & @xmath8p@xmath61 & 11.4036 & 11.5247 & 11.4028 & 11.4104 + 4 & 3p@xmath5 & @xmath8p@xmath58 & & 24.7520 & 24.5032 & 24.4911 + 5 & 3s3p & @xmath8p@xmath59 & 37.398 & 37.4521 & 37.3609 & 37.3992 + 6 & 3s3p & @xmath27p@xmath61 & 40.0821 & 40.2273 & 40.1296 & 40.1225 + 7 & 3p@xmath5 & @xmath27d@xmath57 & & 50.6187 & 50.4140 & 50.4418 + 8 & 3p@xmath5 & @xmath8p@xmath64 & & 50.7934 & 50.5757 & 50.5885 + 9 & 3s3d & @xmath8d@xmath64 & 53.100 & 53.2554 & 53.0898 & 53.0968 + 10 & 3s3d & @xmath8d@xmath57 & 54.0418 & 54.2279 & 54.0506 & 54.0421 + 11 & 3s3d & @xmath8d@xmath65 & 59.214 & 59.3590 & 59.1988 & 59.2129 + 12 & 3s3d & @xmath27d@xmath57 & 60.490 & 60.6497 & 60.4812 & 60.4926 + 13 & 3p3d & @xmath8f@xmath59 & & 64.4267 & 64.1328 & 64.1616 + 14 & 3p3d & @xmath8d@xmath61 & & 67.0032 & 66.7097 & 66.6958 + 15 & 3p3d & @xmath8p@xmath59 & & 71.9267 & 71.6393 & 71.6688 + 16 & 3p3d & @xmath8f@xmath62 & & 72.1483 & 71.8606 & 71.8875 + 17 & 3p@xmath5 & @xmath8p@xmath57 & & 78.4319 & 78.2413 & 78.2584 + 18 & 3p@xmath5 & @xmath27s@xmath58 & & 79.8512 & 79.6567 & 79.6637 + 19 & 3p3d & @xmath8d@xmath59 & & 92.4948 & 92.2324 & 92.2537 + 20 & 3p3d & @xmath8p@xmath60 & & 93.2392 & 92.9761 & 92.9927 + 21 & 3p3d & @xmath8p@xmath61 & & 93.2818 & 93.0183 & 93.0311 + 22 & 3p3d & @xmath27f@xmath62 & & 93.2541 & 92.9898 & 92.9946 + 23 & 3p3d & @xmath8f@xmath63 & & 97.7310 & 97.4746 & 97.5357 + 24 & 3p3d & @xmath27d@xmath59 & & 98.6141 & 98.3571 & 98.4030 + 25 & 3p3d & @xmath8d@xmath62 & & 100.0066 & 99.7469 & 99.7687 + 26 & 3p3d & @xmath27p@xmath61 & & 100.9429 & 100.6812 & 100.6989 + 27 & 3d@xmath5 & @xmath8f@xmath57 & & 107.5330 & 107.1956 & 107.2079 + 28 & 3d@xmath5 & @xmath8p@xmath58 & & 109.5116 & 109.1710 & 109.1670 + 29 & 3d@xmath5 & @xmath8f@xmath65 & & 113.3167 & 112.9856 & 113.0263 + 30 & 3d@xmath5 & @xmath8p@xmath57 & & 114.2228 & 113.8902 & 113.9201 + 31 & 3d@xmath5 & @xmath27g@xmath75 & & 114.3842 & 114.0519 & 114.0755 + 32 & 3d@xmath5 & @xmath8p@xmath64 & & 114.6216 & 114.2887 & 114.3189 + 33 & 3d@xmath5 & @xmath8f@xmath75 & & 119.7747 & 119.4490 & 119.5071 + 34 & 3d@xmath5 & @xmath27d@xmath57 & & 120.5499 & 120.2230 & 120.2754 + 35 & 3d@xmath5 & @xmath27s@xmath58 & & 122.6951 & 122.3631 & 122.3938
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp : present results with the grasp code from 58 configurations and 509 levels + fac : present results with the fac code from 991 levels + rmbpt : earlier results of safronova and safronova @xcite + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 3s@xmath5 & @xmath27s@xmath58 & 1.00 ( 1 ) + 2 & 3s3p & @xmath8p@xmath60 & 1.00 ( 2 ) + 3 & 3s3p & @xmath8p@xmath61 & 0.72 ( 3)+0.27 ( 6 ) + 4 & 3p@xmath5 & @xmath8p@xmath58 & 0.69 ( 4)+0.30 ( 18 ) + 5 & 3s3p & @xmath8p@xmath59 & 1.00 ( 5 ) + 6 & 3s3p & @xmath27p@xmath61 & 0.27 ( 3)+0.72 ( 6 ) + 7 & 3p@xmath5 & @xmath27d@xmath57 & 0.28 ( 17)+0.56 ( 7)+0.05 ( 10)+0.12 ( 12 ) + 8 & 3p@xmath5 & @xmath8p@xmath64 & 1.00 ( 8) + 9 & 3s3d & @xmath8d@xmath64 & 1.00 ( 9 ) + 10 & 3s3d & @xmath8d@xmath57 & 0.06 ( 17)+0.07 ( 7)+0.69 ( 10)+0.18 ( 12 ) + 11 & 3s3d & @xmath8d@xmath65 & 1.00 ( 11 ) + 12 & 3s3d & @xmath27d@xmath57 & 0.26 ( 10)+0.69 ( 12 ) + 13 & 3p3d & @xmath8f@xmath59 & 0.76 ( 13)+0.20 ( 24 ) + 14 & 3p3d & @xmath8d@xmath61 & 0.53 ( 14)+0.18 ( 21)+0.29 ( 26 ) + 15 & 3p3d & @xmath8p@xmath59 & 0.35 ( 19)+0.48 ( 15)+0.14 ( 24 ) + 16 & 3p3d & @xmath8f@xmath62 & 0.52 ( 16)+0.22 ( 25)+0.26 ( 22 ) + 17 & 3p@xmath5 & @xmath8p@xmath57 & 0.64 ( 17)+0.36 ( 7 ) + 18 & 3p@xmath5 & @xmath27s@xmath58 & 0.31 ( 4)+0.69 ( 18 ) + 19 & 3p3d & @xmath8d@xmath59 & 0.19 ( 13)+0.52 ( 19)+0.04 ( 15)+0.25 ( 24 ) + 20 & 3p3d & @xmath8p@xmath60 & 1.00 ( 20 ) + 21 & 3p3d & @xmath8p@xmath61 & 0.36 ( 14)+0.59 ( 21)+0.04 ( 26 ) + 22 & 3p3d & @xmath27f@xmath62 & 0.48 ( 16)+0.18 ( 25)+0.35 ( 22 ) + 23 & 3p3d & @xmath8f@xmath63 & 1.00 ( 23 ) + 24 & 3p3d & @xmath27d@xmath59 & 0.11 ( 19)+0.46 ( 15)+0.40 ( 24 ) + 25 & 3p3d & @xmath8d@xmath62 & 0.61 ( 25)+0.38 ( 22 ) + 26 & 3p3d & @xmath27p@xmath61 & 0.11 ( 14)+0.23 ( 21)+0.66 ( 26 ) + 27 & 3d@xmath5 & @xmath8f@xmath57 & 0.74 ( 27)+0.23 ( 34 ) + 28 & 3d@xmath5 & @xmath8p@xmath58 & 0.71 ( 28)+0.30 ( 35 ) + 29 & 3d@xmath5 & @xmath8f@xmath65 & 1.00 ( 29 ) + 30 & 3d@xmath5 & @xmath8p@xmath57 & 0.21 ( 27)+0.49 ( 30)+0.30 ( 34 ) + 31 & 3d@xmath5 & @xmath27g@xmath75 & 0.29 ( 33)+0.71 ( 31 ) + 32 & 3d@xmath5 & @xmath8p@xmath64 & 1.00 ( 32 ) + 33 & 3d@xmath5 & @xmath8f@xmath75 & 0.71 ( 33)+0.29 ( 31 ) + 34 & 3d@xmath5 & @xmath27d@xmath57 & 0.06 ( 27)+0.48 ( 30)+0.46 ( 34 ) + 35 & 3d@xmath5 & @xmath27s@xmath58 & 0.30 ( 28)+0.69 ( 35 ) + + @rlllrrr@ index & configuration & level & nist & grasp & fac & @xmath54 ( s ) + + + + index & configuration & level & nist & grasp & fac & @xmath54 ( s ) + + + 1 & 2p@xmath93s & @xmath5s@xmath76 & 000.0 & 0.0000 & 0.0000 & ........ + 2 & 2p@xmath93p & @xmath5p@xmath23 & 011.7280 & 11.8989 & 11.7457 & 2.218@xmath7711 + 3 & 2p@xmath93p & @xmath5p@xmath21 & 039.1890 & 39.3365 & 39.2218 & 5.664@xmath7713 + 4 & 2p@xmath93d & @xmath5d@xmath78 & 052.9692 & 53.1127 & 52.9352 & 6.986@xmath7713 + 5 & 2p@xmath93d & @xmath5d@xmath79 & 059.2105 & 59.3372 & 59.1730 & 4.987@xmath7712 + 6 & 2p@xmath94s & @xmath5s@xmath76 & 239.12 & 239.0661 & 238.9973 & 1.501@xmath7714 + 7 & 2p@xmath94p & @xmath5p@xmath23 & 243.92 & 243.9788 & 243.8505 & 1.267@xmath7714 + 8 & 2p@xmath94p & @xmath5p@xmath21 & 255.18 & 255.2154 & 255.0981 & 2.010@xmath7714 + 9 & 2p@xmath94d & @xmath5d@xmath78 & 260.37 & 260.4510 & 260.3002 & 8.821@xmath7715 + 10 & 2p@xmath94d & @xmath5d@xmath79 & 263.09 & 263.1426 & 262.9954 & 8.466@xmath7715 + 11 & 2p@xmath94f & @xmath5f@xmath22 & 265.94 & 265.8618 & 265.7361 & 4.087@xmath7715 + 12 & 2p@xmath94f & @xmath5f@xmath80 & 267.12 & 267.0446 & 266.9176 & 4.198@xmath7715 + 13 & 2p@xmath95s & @xmath5s@xmath76 & & 345.5593 & 345.3305 & 1.888@xmath7714 + 14 & 2p@xmath95p & @xmath5p@xmath23 & & 348.0234 & 347.7664 & 1.600@xmath7714 + 15 & 2p@xmath95p & @xmath5p@xmath21 & & 353.6728 & 353.4209 & 2.398@xmath7714 + 16 & 2p@xmath95d & @xmath5d@xmath78 & & 356.2383 & 355.9695 & 1.189@xmath7714 + 17 & 2p@xmath95d & @xmath5d@xmath79 & 357.54 & 357.6240 & 357.3573 & 1.168@xmath7714 + 18 & 2p@xmath95f & @xmath5f@xmath22 & 358.84 & 358.9640 & 358.7180 & 7.736@xmath7715 + 19 & 2p@xmath95f & @xmath5f@xmath80 & 359.46 & 359.5722 & 359.3256 & 7.962@xmath7715 + 20 & 2p@xmath95 g & @xmath5g@xmath81 & 359.77 & 359.7585 & 359.5057 & 1.361@xmath7714 + 21 & 2p@xmath95 g & @xmath5g@xmath82 & 360.11 & 360.1191 & 359.8662 & 1.378@xmath7714 + 22 & 2p@xmath96s & @xmath5s@xmath76 & & 401.9007 & 401.5572 & 2.653@xmath7714 + 23 & 2p@xmath96p & @xmath5p@xmath23 & & 403.3052 & 402.9603 & 2.262@xmath7714 + 24 & 2p@xmath96p & @xmath5p@xmath21 & & 406.5339 & 406.2203 & 3.266@xmath7714 + 25 & 2p@xmath96d & @xmath5d@xmath78 & & 407.9823 & 407.6849 & 1.742@xmath7714 + 26 & 2p@xmath96d & @xmath5d@xmath79 & & 408.7855 & 408.5064 & 1.735@xmath7714 + 27 & 2p@xmath96f & @xmath5f@xmath22 & & 409.5470 & 409.2872 & 1.310@xmath7714 + 28 & 2p@xmath96f & @xmath5f@xmath80 & & 409.8998 & 409.6444 & 1.351@xmath7714 + 29 & 2p@xmath96 g & @xmath5g@xmath81 & & 410.0204 & 409.7698 & 2.328@xmath7714 + 30 & 2p@xmath96 g & @xmath5g@xmath82 & & 410.2293 & 409.9788 & 2.357@xmath7714
+ + nist : http://www.nist.gov/pml/data/asd.cfm + grasp : present results with the grasp code from 50 configurations and 1235 levels + fac : present results with the fac code from 1592 levels + @rllrrrr@ index@xmath83 & configuration & level & nist & grasp & fac & mrmp + + + + index@xmath83 & configuration & level & nist & grasp & fac & mrmp + + + 1 & 2s@xmath52p@xmath9 & @xmath27s@xmath58 & 0.000 & 0.0000 & 0.0000 & 0.0000 + 3 & 2s@xmath52p@xmath73s & @xmath27p@xmath61 & 610.640 & 610.2292 & 610.1423 & 610.5354 + 9 & 2s@xmath52p@xmath73p & @xmath27s@xmath58 & 653.859 & 653.7288 & 653.5037 & 653.7409 + 11 & 2s@xmath52p@xmath73d & @xmath8p@xmath61 & 661.507 & 660.9754 & 660.7169 & 661.1325 + 17 & 2s@xmath52p@xmath73d & @xmath27p@xmath61 & 670.246 & 670.5722 & 670.2893 & 670.6958 + 19 & 2s@xmath52p@xmath73s & @xmath8p@xmath61 & 711.936 & 711.7088 & 711.6628 & 712.0517 + 21 & 2s@xmath52p@xmath73p & @xmath8p@xmath58 & 726.088 & 725.8494 & 725.6751 & 725.9370 + 27 & 2s2p@xmath93p & @xmath8p@xmath61 & 758.302 & 758.6381 & 758.2086 & 758.3025 + 29 & 2s@xmath52p@xmath73d & @xmath8d@xmath61 & 765.027 & 764.8414 & 764.5743 & 764.9308 + 33 & 2s2p@xmath93p & @xmath27p@xmath61 & 786.651 & 787.2457 & 786.8073 & 786.8504 + + @xmath84 : see table 7 for definition of all levels + nist : http://www.nist.gov/pml/data/asd.cfm + grasp : present results with the grasp code from 25 configurations and 157 levels + fac : present results with the fac code from 1147 levels + mrmp : earlier calculations of vilkas et al
. @xcite @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 2s@xmath52p@xmath9 & @xmath27s@xmath58 & 1.00 ( 1 ) + 2 & 2s@xmath52p@xmath73s & @xmath8p@xmath59 & 1.00 ( 2 ) + 3 & 2s@xmath52p@xmath73s & @xmath27p@xmath61 & 0.34 ( 19)+0.66 ( 3 ) + 4 & 2s@xmath52p@xmath73p & @xmath8p@xmath64 & 0.09 ( 20)+0.49 ( 4)+0.31 ( 25)+0.10 ( 6 ) + 5 & 2s@xmath52p@xmath73p & @xmath8d@xmath57 & 0.50 ( 5)+0.17 ( 8)+0.34 ( 24 ) + 6 & 2s@xmath52p@xmath73p & @xmath27p@xmath64 & 0.08 ( 20)+0.36 ( 25)+0.56 ( 6 ) + 7 & 2s@xmath52p@xmath73p & @xmath8d@xmath65 & 1.00 ( 7 ) + 8 & 2s@xmath52p@xmath73p & @xmath8p@xmath57 & 0.67 ( 8)+0.34 ( 24 ) + 9 & 2s@xmath52p@xmath73p & @xmath27s@xmath58 & 0.37 ( 21)+0.62 ( 9 ) + 10 & 2s@xmath52p@xmath73d & @xmath8p@xmath60 & 1.00 ( 10 ) + 11 & 2s@xmath52p@xmath73d & @xmath8p@xmath61 & 0.32 ( 29)+0.66 ( 11 ) + 12 & 2s@xmath52p@xmath73d & @xmath8f@xmath62 & 0.53 ( 12)+0.07 ( 16)+0.40 ( 31 ) + 13 & 2s@xmath52p@xmath73d & @xmath8d@xmath59 & 0.18 ( 28)+0.55 ( 13)+0.10 ( 30)+0.18 ( 15 ) + 14 & 2s@xmath52p@xmath73d & @xmath8f@xmath63 & 1.00 ( 14 ) + 15 & 2s@xmath52p@xmath73d & @xmath27d@xmath59 & 0.04 ( 28)+0.06 ( 13)+0.41 ( 30)+0.49 ( 15 ) + 16 & 2s@xmath52p@xmath73d & @xmath8d@xmath62 & 0.71 ( 16)+0.27 ( 31 ) + 17 & 2s@xmath52p@xmath73d & @xmath27p@xmath61 & 0.18 ( 29)+0.18 ( 11)+0.62 ( 17 ) + 18 & 2s@xmath52p@xmath73s & @xmath8p@xmath60 & 1.00 ( 18 ) + 19 & 2s@xmath52p@xmath73s & @xmath8p@xmath61 & 0.66 ( 19)+0.34 ( 3 ) + 20 & 2s@xmath52p@xmath73p & @xmath8d@xmath64 & 0.74 ( 20)+0.23 ( 6 ) + 21 & 2s@xmath52p@xmath73p & @xmath8p@xmath58 & 0.62 ( 21)+0.37 ( 9 ) + 22 & 2s2p@xmath93s & @xmath8s@xmath64 & 0.08 ( 4)+0.86 ( 22 ) + 23 & 2s2p@xmath93s & @xmath27s@xmath58 & 1.00 ( 23 ) + 24 & 2s@xmath52p@xmath73p & @xmath27d@xmath57 & 0.49 ( 5)+0.17 ( 8)+0.34 ( 24 ) + 25 & 2s@xmath52p@xmath73p & @xmath8s@xmath64 & 0.07 ( 20)+0.42 ( 4)+0.26 ( 25)+0.10 ( 6)+0.14 ( 22 ) + 26 & 2s2p@xmath93p & @xmath8p@xmath60 & 1.00 ( 26 ) + 27 & 2s2p@xmath93p & @xmath8p@xmath61 & 0.66 ( 27)+0.31 ( 33 ) + 28 & 2s@xmath52p@xmath73d & @xmath8f@xmath59 & 0.74 ( 28)+0.20 ( 15 ) + 29 & 2s@xmath52p@xmath73d & @xmath8d@xmath61 & 0.48 ( 29)+0.15 ( 11)+0.35 ( 17 ) + 30 & 2s@xmath52p@xmath73d & @xmath8p@xmath59 & 0.36 ( 13)+0.48 ( 30)+0.14 ( 15 ) + 31 & 2s@xmath52p@xmath73d & @xmath27f@xmath62 & 0.44 ( 12)+0.22 ( 16)+0.34 ( 31 ) + 32 & 2s2p@xmath93p & @xmath8p@xmath59 & 1.00 ( 32 ) + 33 & 2s2p@xmath93p & @xmath27p@xmath61 & 0.32 ( 27)+0.67 ( 33 ) + + @rllr@ index & configuration & level & eigenvectors + + + + index & configuration & level & eigenvectors + + + 1 & 2s@xmath52p@xmath7 & @xmath5p@xmath85 & 1.00 ( 1 ) + 2 & 2s@xmath52p@xmath7 & @xmath5p@xmath86 & 1.00 ( 2 ) + 3 & 2s2p@xmath9 & @xmath5s@xmath87 & 1.00 ( 3 ) + 4 & 2s@xmath52p@xmath63s & @xmath6p@xmath88 & 0.69 ( 4)+0.31 ( 28 ) + 5 & 2s@xmath52p@xmath63s & @xmath5p@xmath89 & 0.12 ( 26)+0.56 ( 5)+0.32 ( 29 ) + 6 & 2s@xmath52p@xmath63s & @xmath5s@xmath87 & 0.23 ( 86)+0.12 ( 27)+0.66 ( 6 ) + 7 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6p@xmath85 & 0.07 ( 31)+0.31 ( 7)+0.18 ( 92)+0.09 ( 13)+0.16 ( 43)+0.17 ( 33 ) + 8 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5d@xmath90 & 0.27 ( 38)+0.12 ( 10)+0.29 ( 8)+0.25 ( 32)+0.07 ( 45 ) + 9 & 2s@xmath52p@xmath6(@xmath27s)3p & @xmath5p@xmath86 & 0.19 ( 87)+0.04 ( 30)+0.08 ( 40)+0.66 ( 9 ) + 10 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6p@xmath90 & 0.40 ( 10)+0.27 ( 8)+0.07 ( 32)+0.24 ( 45 ) + 11 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5s@xmath86 & 0.09 ( 30)+0.18 ( 40)+0.37 ( 11)+0.32 ( 46 ) + 12 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6d@xmath91 & 0.67 ( 12)+0.32 ( 41 ) + 13 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5p@xmath85 & 0.24 ( 92)+0.15 ( 39)+0.21 ( 13)+0.13 ( 43)+0.12 ( 33)+0.15 ( 14 ) + 14 & 2s@xmath52p@xmath6(@xmath27s)3p & @xmath5p@xmath85 & 0.13 ( 31)+0.12 ( 7)+0.16 ( 13)+0.04 ( 43)+0.50 ( 14 ) + 15 & 2s@xmath52p@xmath6(@xmath27d)3d & @xmath5p@xmath89 & 0.32 ( 50)+0.23 ( 58)+0.07 ( 62)+0.22 ( 55)+0.08 ( 15 ) + 16 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath6d@xmath88 & 0.13 ( 51)+0.35 ( 16)+0.06(106)+0.10 ( 23)+0.22 ( 54)+0.10 ( 60 ) + 17 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath6p@xmath87 & 0.12 ( 49)+0.50 ( 17)+0.05 ( 22)+0.22 ( 63)+0.10 ( 53 ) + 18 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath5f@xmath92 & 0.28 ( 56)+0.06 ( 20)+0.34 ( 18)+0.28 ( 52 ) + 19 & 2s@xmath52p@xmath6(@xmath27s)3d & @xmath5d@xmath89 & 0.17(101)+0.07 ( 58)+0.07 ( 62)+0.62 ( 19 ) + 20 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath6d@xmath92 & 0.06 ( 56)+0.41 ( 20)+0.22 ( 18)+0.28 ( 61 ) + 21 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath6f@xmath93 & 0.67 ( 21)+0.32 ( 59 ) + 22 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath5p@xmath87 & 0.05 ( 49)+0.12 ( 17)+0.50 ( 22)+0.10 ( 63)+0.22 ( 53 ) + 23 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath5d@xmath88 & 0.05 ( 16)+0.22(106)+0.14 ( 57)+0.26 ( 23)+0.09 ( 54)+0.21 ( 60 ) + 24 & 2s@xmath52p@xmath6(@xmath8p)3d & @xmath5p@xmath89 & 0.18 ( 58)+0.19 ( 62)+0.26 ( 24)+0.10 ( 55)+0.21 ( 15 ) + 25 & 2s@xmath52p@xmath6(@xmath27s)3d & @xmath5d@xmath88 & 0.10 ( 51)+0.11 ( 16)+0.04 ( 57)+0.09 ( 23)+0.61 ( 25 ) + 26 & 2s@xmath52p@xmath63s & @xmath6p@xmath89 & 0.86 ( 26)+0.12 ( 5 ) + 27 & 2s@xmath52p@xmath63s & @xmath5p@xmath87 & 0.34 ( 86)+0.67 ( 27 ) + 28 & 2s@xmath52p@xmath63s & @xmath5d@xmath88 & 0.31 ( 4)+0.67 ( 28 ) + 29 & 2s@xmath52p@xmath63s & @xmath5d@xmath89 & 0.32 ( 5)+0.66 ( 29 ) + 30 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6p@xmath86 & 0.27 ( 87)+0.52 ( 30)+0.21 ( 11 ) + 31 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6d@xmath85 & 0.56 ( 31)+0.14 ( 39)+0.13 ( 13)+0.07 ( 43)+0.07 ( 33 ) + 32 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5f@xmath90 & 0.12 ( 38)+0.05 ( 10)+0.14 ( 8)+0.53 ( 32)+0.15 ( 45 ) + 33 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5p@xmath85 & 0.12 ( 7)+0.13 ( 92)+0.18 ( 13)+0.25 ( 43)+0.29 ( 33 ) + 34 & 2s2p@xmath7(@xmath8p)3s & @xmath6p@xmath90 & 0.96 ( 34 ) + 35 & 2s2p@xmath7(@xmath8p)3s & @xmath5p@xmath85 & 0.27 ( 89)+0.67 ( 35 ) + 36 & 2s2p@xmath7(@xmath27p)3s & @xmath5p@xmath86 & 0.05 ( 30)+0.06 ( 11)+0.06 ( 46)+0.10 ( 88)+0.20 ( 90)+0.52 ( 36 ) + 37 & 2s2p@xmath7(@xmath27p)3s & @xmath5p@xmath85 & 0.04 ( 92)+0.16 ( 89)+0.22 ( 35)+0.52 ( 37 ) + 38 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath6d@xmath90 & 0.55 ( 38)+0.28 ( 10)+0.14 ( 8) + 39 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5d@xmath85 & 0.24 ( 7)+0.16 ( 92)+0.46 ( 39 ) + 40 & 2s@xmath52p@xmath6(@xmath8p)3p & @xmath5p@xmath86 & 0.09 ( 87)+0.23 ( 30)+0.23 ( 40)+0.19 ( 11)+0.10 ( 46)+0.10 ( 36 ) + 41 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5f@xmath91 & 0.32 ( 12)+0.67 ( 41 ) + 42 & 2s2p@xmath7(@xmath8p)3p & @xmath6s@xmath89 & 0.11 ( 95)+0.44(130)+0.29 ( 42)+0.13 ( 65 ) + 43 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5d@xmath85 & 0.16 ( 92)+0.15 ( 13)+0.29 ( 43)+0.27 ( 33)+0.04 ( 89)+0.05 ( 37 ) + 44 & 2s2p@xmath7(@xmath8p)3p & @xmath5d@xmath88 & 0.40(133)+0.17 ( 66)+0.44 ( 44 ) + 45 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5d@xmath90 & 0.14 ( 10)+0.15 ( 8)+0.14 ( 32)+0.52 ( 45 ) + 46 & 2s@xmath52p@xmath6(@xmath27d)3p & @xmath5p@xmath86 & 0.07 ( 87)+0.34 ( 40)+0.05 ( 11)+0.52 ( 46 ) + 47 & 2s2p@xmath7(@xmath27p)3p & @xmath5p@xmath87 & 0.10 ( 91)+0.18 ( 99)+0.08 ( 70)+0.40 ( 47)+0.22(135 ) + 48 & 2s2p@xmath7(@xmath27p)3p & @xmath5d@xmath89 & 0.21 ( 95)+0.05(134)+0.08 ( 65)+0.52 ( 48)+0.10 ( 69 ) + + @rrlll@ i & j & rmbpt & grasp & r + + + + i & j & rmbpt & grasp & r + + + 1 & 2 & 3.17@xmath772 & 3.17@xmath772 & 9.8@xmath771 + 1 & 4 & 2.07@xmath773 & 2.07@xmath773 & 1.0@xmath940 + 1 & 6 & 1.05@xmath771 & 1.05@xmath771 & 1.0@xmath940 + 1 & 8 & 4.99@xmath771 & 4.99@xmath771 & 1.0@xmath940 + 1 & 20 & 2.07@xmath774 & 2.07@xmath774 & 1.3@xmath940 + 1 & 22 & 1.22@xmath774 & 1.22@xmath774 & 7.0@xmath771 + 2 & 3 & 3.17@xmath774 & 3.23@xmath774 & 1.7@xmath940 + 3 & 4 & 2.48@xmath773 & 2.48@xmath773 & 1.0@xmath940 + 3 & 5 & 2.01@xmath772 & 2.01@xmath772 & 1.1@xmath940 + 3 & 9 & 1.07@xmath772 & 1.07@xmath771 & 1.1@xmath940 + 3 & 19 & 9.34@xmath772 & 9.34@xmath772 & 1.0@xmath940 + 3 & 20 & 7.73@xmath772 & 7.73@xmath772 & 9.9@xmath771 + 3 & 22 & 2.69@xmath771 & 2.69@xmath771 & 1.0@xmath940 + 5 & 17 & 1.27@xmath772 & 1.65@xmath772 & 1.0@xmath940 + 12 & 22 & 4.23@xmath773 & 4.23@xmath773 & 9.0@xmath771 + 14 & 19 & 1.06@xmath774 & 1.06@xmath774 & 1.4@xmath940 + 19 & 29 & 1.38@xmath774 & 1.38@xmath774 & 7.7@xmath771 + 19 & 31 & 2.43@xmath774 & 2.43@xmath774 & 1.2@xmath940 + 19 & 35 & 5.52@xmath772 & 5.52@xmath772 & 1.0@xmath940 + 19 & 36 & 9.69@xmath772 & 9.69@xmath772 & 1.1@xmath940 + 19 & 37 & 1.16@xmath772 & 1.16@xmath772 & 1.0@xmath940 + 19 & 38 & 6.93@xmath772 & 6.93@xmath772 & 1.1@xmath940 + 19 & 40 & 3.91@xmath773 & 3.91@xmath773 & 9.1@xmath771 + 22 & 27 & 1.14@xmath774 & 1.14@xmath774 & 9.1@xmath771 + 22 & 28 & 1.03@xmath773 & 1.02@xmath773 & 1.0@xmath940 + 22 & 33 & 1.01@xmath773 & 1.01@xmath773 & 1.3@xmath940 + 22 & 35 & 1.87@xmath773 & 1.87@xmath773 & 1.2@xmath940 + 22 & 37 & 1.97@xmath772 & 1.97@xmath772 & 1.1@xmath940 + 22 & 39 & 3.71@xmath772 & 3.71@xmath773 & 9.9@xmath771 + 22 & 40 & 9.91@xmath773 & 9.91@xmath773 & 1.0@xmath940 + + rmbpt : earlier results of safronova and safronova @xcite + grasp : present results with the grasp code from 63 configurations and 2003 levels + r : ratio of velocity and lrength forms of @xmath0-values + @rrlll@ i & j & rmbpt & grasp & r + + + + i & j & rmbpt & grasp & r + + + 1 & 6 & 5.97@xmath771 & 6.08@xmath771 & 1.0@xmath940 + 2 & 8 & 2.76@xmath771 & 2.81@xmath771 & 1.0@xmath940 + 2 & 9 & 2.23@xmath771 & 2.27@xmath771 & 1.0@xmath940 + 3 & 4 & 3.11@xmath772 & 3.18@xmath772 & 1.0@xmath940 + 3 & 7 & 6.43@xmath772 & 6.45@xmath772 & 1.0@xmath940 + 3 & 8 & 5.27@xmath772 & 5.36@xmath772 & 1.0@xmath940 + 3 & 9 & 3.61@xmath772 & 3.67@xmath772 & 1.0@xmath940 + 3 & 10 & 3.12@xmath771 & 3.19@xmath771 & 1.0@xmath940 + 3 & 12 & 1.85@xmath772 & 1.85@xmath772 & 1.0@xmath940 + 4 & 14 & 4.36@xmath771 & 4.43@xmath771 & 1.0@xmath940 + 5 & 11 & 9.10@xmath772 & 9.27@xmath772 & 1.1@xmath940 + 5 & 17 & 1.36@xmath771 & 1.39@xmath771 & 1.0@xmath940 + 6 & 17 & 2.63@xmath771 & 2.67@xmath771 & 1.0@xmath940 + 6 & 18 & 9.49@xmath772 & 9.66@xmath772 & 1.0@xmath940 + 7 & 16 & 5.51@xmath772 & 5.63@xmath772 & 1.1@xmath940 + 7 & 19 & 1.03@xmath771 & 1.06@xmath771 & 1.0@xmath940 + 7 & 22 & 3.45@xmath772 & 3.44@xmath772 & 1.1@xmath940 + 8 & 15 & 9.74@xmath772 & 9.90@xmath772 & 1.1@xmath940 + 8 & 19 & 8.62@xmath772 & 8.76@xmath772 & 1.0@xmath940 + 8 & 20 & 3.39@xmath772 & 3.45@xmath772 & 1.0@xmath940 + 8 & 21 & 9.14@xmath772 & 9.31@xmath772 & 1.0@xmath940 + 9 & 19 & 1.19@xmath771 & 1.21@xmath771 & 1.0@xmath940 + 9 & 20 & 5.15@xmath772 & 5.24@xmath772 & 1.0@xmath940 + 9 & 21 & 1.16@xmath771 & 1.18@xmath771 & 1.0@xmath940 + 10 & 14 & 2.57@xmath772 & 2.19@xmath772 & 1.1@xmath940 + 10 & 19 & 2.20@xmath772 & 2.20@xmath772 & 1.0@xmath940 + 10 & 22 & 2.97@xmath771 & 3.04@xmath771 & 1.0@xmath940 + 11 & 23 & 1.77@xmath771 & 1.79@xmath771 & 1.0@xmath940 + 11 & 24 & 2.50@xmath772 & 2.53@xmath772 & 1.0@xmath940 + 11 & 25 & 7.50@xmath772 & 7.65@xmath772 & 1.0@xmath940 + 12 & 24 & 1.17@xmath771 & 1.19@xmath771 & 1.0@xmath940 + 12 & 25 & 7.35@xmath772 & 7.47@xmath772 & 1.0@xmath940 + 12 & 26 & 8.57@xmath772 & 8.71@xmath772 & 1.0@xmath940 + 13 & 27 & 1.09@xmath771 & 1.11@xmath771 & 1.0@xmath940 + 14 & 27 & 1.05@xmath771 & 1.78@xmath771 & 1.0@xmath940 + 14 & 28 & 7.59@xmath772 & 7.72@xmath772 & 1.0@xmath940 + 15 & 29 & 6.22@xmath772 & 6.33@xmath772 & 1.0@xmath940 + 15 & 30 & 8.59@xmath772 & 8.73@xmath772 & 1.0@xmath940 + 15 & 32 & 6.76@xmath772 & 6.87@xmath772 & 1.0@xmath940 + 16 & 29 & 6.51@xmath772 & 6.61@xmath772 & 1.0@xmath940 + 16 & 30 & 1.87@xmath772 & 1.90@xmath772 & 1.0@xmath940 + 16 & 31 & 1.09@xmath771 & 1.12@xmath771 & 1.0@xmath940 + 17 & 25 & 1.66@xmath771 & 1.69@xmath771 & 1.1@xmath940 + 18 & 26 & 1.98@xmath771 & 2.02@xmath771 & 1.1@xmath940 + 20 & 32 & 1.95@xmath771 & 1.06@xmath771 & 1.1@xmath940 + 21 & 30 & 3.50@xmath772 & 6.40@xmath772 & 1.1@xmath940 + 21 & 32 & 3.31@xmath772 & 3.35@xmath772 & 1.1@xmath940 + 22 & 31 & 1.22@xmath771 & 9.22@xmath772 & 1.1@xmath940 + 23 & 33 & 3.03@xmath772 & 3.11@xmath772 & 1.1@xmath940 + 24 & 34 & 8.49@xmath772 & 8.64@xmath772 & 1.1@xmath940 + 25 & 33 & 9.07@xmath772 & 1.24@xmath771 & 1.1@xmath940 + 26 & 34 & 3.78@xmath772 & 5.90@xmath772 & 1.1@xmath940 + 26 & 35 & 5.39@xmath772 & 5.39@xmath772 & 1.1@xmath940 + + rmbpt : earlier results of safronova and safronova @xcite + grasp : present results with the grasp code from 58 configurations and 509 levels + r : ratio of velocity and length forms of @xmath0-values + @rrrrrr@ i & j & mrmp & grasp & f ( grasp ) & r + + + + i & j & mrmp & grasp & f ( grasp ) & r + + + 1 & 3 & 1.206@xmath9414 & 1.5309@xmath9414 & 1.5354@xmath771 & 1.0@xmath770 + 1 & 11 & 6.551@xmath9413 & 8.2270@xmath9413 & 7.0330@xmath772 & 9.8@xmath771 + 1 & 17 & 2.613@xmath9415 & 2.8077@xmath9415 & 2.3320@xmath940 & 9.8@xmath771 + 1 & 19 & 2.694@xmath9413 & 3.8180@xmath9413 & 2.8152@xmath772 & 9.9@xmath771 + 1 & 27 & 6.243@xmath9414 & 7.7623@xmath9414 & 5.0372@xmath771 & 1.0@xmath770 + 1 & 29 & 1.227@xmath9415 & 1.3590@xmath9415 & 8.6763@xmath771 & 9.8@xmath771 + 1 & 33 & 3.350@xmath9414 & 4.4021@xmath9414 & 2.6529@xmath771 & 1.0@xmath770 + 1 & 39 & 4.193@xmath9413 & 5.3006@xmath9413 & 2.7134@xmath772 & 9.6@xmath771 + 1 & 53 & 1.021@xmath9415 & 1.0126@xmath9415 & 4.8992@xmath771 & 9.7@xmath771 + 1 & 83 & 2.365@xmath9414 & 2.3392@xmath9414 & 9.1872@xmath772 & 9.7@xmath771 + 1 & 101 & 8.690@xmath9414 & 8.9169@xmath9414 & 3.4857@xmath771 & 9.7@xmath771 + 1 & 111 & 1.118@xmath9414 & 1.3923@xmath9414 & 5.2373@xmath772 & 9.9@xmath771 + 1 & 113 & 1.850@xmath9414 & 2.2983@xmath9414 & 8.4470@xmath772 & 9.9@xmath771 + 1 & 129 & 2.953@xmath9414 & 3.0675@xmath9414 & 9.9018@xmath772 & 9.7@xmath771 + 1 & 143 & 6.038@xmath9413 & 7.5371@xmath9413 & 2.3136@xmath772 & 9.9@xmath771 + 1 & 145 & 9.992@xmath9413 & 1.2558@xmath9414 & 3.8142@xmath772 & 9.8@xmath771 + 2 & 4 & 2.917@xmath9410 & 3.0224@xmath9410 & 1.8434@xmath772 & 9.2@xmath771 + 2 & 6 & 2.395@xmath9411 & 2.4114@xmath9411 & 1.1881@xmath772 & 9.8@xmath771 + 2 & 7 & 1.669@xmath9412 & 1.6814@xmath9412 & 1.9361@xmath771 & 1.1@xmath770 + 2 & 8 & 9.092@xmath9411 & 9.1611@xmath9411 & 7.1771@xmath772 & 9.7@xmath771 + 2 & 22 & 1.580@xmath9413 & 1.5536@xmath9413 & 6.1622@xmath772 & 9.5@xmath771 + 2 & 40 & 4.805@xmath9413 & 5.4832@xmath9413 & 6.6094@xmath772 & 1.0@xmath770 + 2 & 41 & 2.845@xmath9413 & 3.2448@xmath9413 & 6.5169@xmath772 & 1.0@xmath770 + 2 & 42 & 3.484@xmath9413 & 4.0782@xmath9413 & 1.0485@xmath771 & 9.9@xmath771 + 2 & 68 & 2.616@xmath9413 & 2.9775@xmath9413 & 1.7587@xmath772 & 1.0@xmath770 + 2 & 71 & 2.101@xmath9413 & 2.4627@xmath9413 & 3.2866@xmath772 & 9.8@xmath771 + 3 & 6 & 1.342@xmath9412 & 1.3518@xmath9412 & 1.1467@xmath771 & 1.0@xmath770 + 3 & 8 & 8.614@xmath9411 & 8.6757@xmath9411 & 1.1693@xmath771 & 1.0@xmath770 + 3 & 9 & 2.411@xmath9412 & 2.4572@xmath9412 & 5.3889@xmath772 & 9.5@xmath771 + 3 & 23 & 2.171@xmath9413 & 2.1941@xmath9413 & 4.6553@xmath772 & 9.5@xmath771 + 3 & 41 & 2.894@xmath9413 & 3.2979@xmath9413 & 1.1095@xmath771 & 9.9@xmath771 + 3 & 43 & 2.956@xmath9413 & 3.4575@xmath9413 & 6.3790@xmath772 & 9.9@xmath771 + 3 & 45 & 3.309@xmath9413 & 3.8940@xmath9413 & 2.3629@xmath772 & 1.0@xmath770 + 3 & 73 & 2.062@xmath9413 & 2.4239@xmath9413 & 7.6916@xmath773 & 1.0@xmath770 + + mrmp : earlier results of vilkas et al . @xcite + grasp : present results with the grasp code from 25 configurations and 157 levels + r : ratio of velocity and lrength forms of @xmath0-values +
[ cols= " < , < " , ]
|
calculations of energy levels , radiative rates and lifetimes are reported for eight ions of tungsten , i.e. s - like ( w lix ) to f - like ( w lxvi ) .
a large number of levels has been considered for each ion and extensive configuration interaction has been included among a range of configurations . for the calculations , the general - purpose relativistic atomic structure package ( grasp ) has been adopted , and radiative rates ( as well as oscillator strengths and line strengths ) are listed for all e1 , e2 , m1 , and m2 transitions of the ions .
comparisons have been made with earlier available experimental and theoretical energies , although these are limited to only a few levels for most ions .
therefore for additional accuracy assessments , particularly for energy levels , analogous calculations have been performed with the flexible atomic code ( fac ) .
+ + _ received _ : 28 september 2015 , _ accepted _ : 12 february 2016 s - like to f - like tungsten ions , energy levels , radiative rates , oscillator strengths , line strengths , lifetimes
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the understanding of the implications of non - markovianity and the reasons for its occurrence are still largely elusive . yet
, they are stimulating a growing interest in light of their potential impact on many disciplines , from quantum information and nano - technology up to quantum biology @xcite . an important contribution to
this quest came from the formulation of quantitative measures of the degree of non - markovianity of a process @xcite .
in general , these tools address different _ features _ of non - markovianity , from the lack of divisibility of a map @xcite to the ability of the environment to reciprocate the information transfer from the system .
this process occurs unidirectionally in a markovian dynamics @xcite , while the re - focusing of information on the system is the signature of memory effects , as verified in all - optical set - ups @xcite .
the handiness of such instruments has recently triggered the analysis of non - markovianity in quantum many - body systems such as quantum spin chains @xcite or impurity - embedded ultra - cold atomic systems @xcite and in excitation - transfer processes in photosynthetic complexes @xcite .
while these studies relate non - markovian features to the critical behavior of a quantum many - body system @xcite , they also provide a promising arena where the roots for non - markovianity can be researched in physically motivated contexts . in this paper
we explore the competition between two profoundly different mechanisms in a simple open quantum model that is relevant for the physics of nitrogen - vacancy centers in diamonds @xcite and molecular nanomagnets @xcite .
specifically , we address the interplay between the dynamics induced on a two - level system by its coherent interaction with other ( environmental ) spins , and the markovian process describing the relaxation of the latter .
one would expect that , when such memoryless dissipative coupling determines the shortest dynamical timescale of the system , markovianity should emerge preponderantly , especially as the number of environmental spins increases .
indeed , one could imagine that a sort of `` markovianity - mixing '' property would hold as a result of the increasing difficulty to re - build the coherence of the system when many decoherence channels are open .
quite strikingly , we show that this is not generally true . in order to do this using a physically relevant model , general enough to encompass
the unexpected features that we would like to highlight , we consider a spin - star configuration whose peripheral sites are coupled to _ rigid _ boson environments , assumed to induce a memoryless dissipative dynamics . while certainly not exhausting the possible scenarios that can be tackled , our choice is illustrative since the degree of non - markovianity ( as defined in ref .
@xcite ) _ can actually increase _ with the number of peripheral spins , while stronger interactions with the boson baths only affect its rate of growth .
the features of the system at hand are quite complex and a rich non - markovianity phase diagram emerges , spanning degrees of memory - keeping effects all the way down to zero values .
this can be exploited to qualitatively modify the character of the dynamics by engineering its features via accessible control parameters such as the detuning between the central and the outer spins . in turn , this opens up the possibility to implement qubit - state preparation protocols in an open - system scenario that exploits non - markovinity , along the lines of refs .
@xcite and beyond the well - established markovian dissipative framework @xcite . in the following ,
we first present the model and its solution in the simplest terms in sec . [ model ] , while the microscopic description and more sophisticated solution method are presented in the appendices .
we then proceed to the analysis of the non - markovianity of the dynamics in sec .
[ nonmasec ] and sec .
[ timenonma ] .
some concluding remarks are given in sec .
[ conclu ] .
the physical set - up that we describe is sketched in fig .
[ bloch_graph ] a , which shows a central spin ( labelled @xmath0 ) coupled to @xmath1 outer spins , with bonds along the branch of a star .
each environmental spin is further coupled to a local boson reservoir .
the evolution of the central spin is ruled by the master equation @xmath2+\sum_{j=1}^n\hat{\mathcal{l}}_j[\rho(t)]\}\ ] ] with @xmath3 the density matrix of the whole system .
each lindblad superoperator @xmath4 describes local dissipation at temperature @xmath5 ( the same for all the baths ) as @xcite @xmath6 where @xmath7 describes the effective coupling of each external spin to its thermal reservoir , populated by @xmath8 excitations ( @xmath9 , where @xmath10 is the boltzmann constant ) . in what follows
, we will consider the peripheral spins to be initially prepared in @xmath11 . *
( a ) * interacts with @xmath1 peripheral spins , each affected by its own local environment .
( b ) evolution of states @xmath12 ( red trajectory ) and @xmath13 ( blue one ) for a star with @xmath14 peripheral sites .
the bloch spheres in the left ( right ) column correspond to the isotropic ( anisotropic ) spin - spin coupling .
the top ( bottom ) row is for the resonant ( off - resonant at @xmath15 ) case with @xmath16 and @xmath17 . in the isotropic cases ,
the final state of spin @xmath0 is pure , while for @xmath18 it is mixed .
@xmath19 prevents the intersections of the trajectories , which are the dynamical points at which the trace distance is strictly null.,width=321 ] to solve the master equation , we use the damping basis @xcite made out of tensor products of eigenoperators of @xmath4 . in this basis , the density matrix of the system reads @xmath20 where @xmath21 , @xmath22 , @xmath23 and @xmath24 are right eigenoperators of @xmath4 with eigenvalues @xmath25 .
the set of operators @xmath26 is composed of the tensor product of @xmath1 damping - basis elements , one for each peripheral spin . due to the symmetry of the hamiltonian ,
if @xmath27 and @xmath28 consist of the same elements of the damping bases ( although differing for their order ) , the respective coefficients must satisfy @xmath29 .
this simple observation allows us to reduce the number of relevant operators from @xmath30 to @xmath31 . with the help of the single - spin dual damping basis @xmath32 , made of left eigenoperators of @xmath33 s , and using the orthogonality condition @xmath34=\delta_{kk'}\delta_{jl}$ ] , we find @xmath35 with @xmath36\rbrace{+}\lambda_m\delta_{rn}\delta_{ms}$ ] and @xmath37 . by calling @xmath38 , the state of the spin star at time
@xmath39 is @xmath40 tracing over the degrees of freedom of the peripheral spins , we find @xmath41 this gives the exact solution for the dynamics of the central spin , valid for any @xmath1 once the expressions for @xmath42 are taken . with this at hand , in the next section we evaluate the amount non - markovianity of the time evolution .
to quantify the degree of non - markovianity of the dynamical evolution of the central spin described in eq .
( [ exactsol ] ) , we employ the measure put forward in ref .
@xcite , which is based on the idea that memory effects can be characterized by the information flowing out of the open system @xmath0 and quantified in terms of the trace distance @xmath43=\text{tr } |\rho_{0,1 } ( t ) -\rho_{0,2 } ( t)|/2 $ ] between any two of its states @xmath44 .
the trace distance quantifies the distinguishability of two states and leads to measure non - markovianity as @xmath45 , \label{nonmarkovianity}\]]where @xmath46 is the union of the intervals where @xmath47 . to provide a general assessment of the dynamics of spin @xmath0
, we consider the coupling with the external spins to be described by the anisotropic xy model @xmath48,\ ] ] where @xmath49 is an anisotropy parameter and @xmath50 is the spin - spin coupling strength . for isotropic coupling ( @xmath51 ) and zero temperature ,
we obtain a simple scaling law @xcite : for any @xmath52 @xmath53 is obtained from the expression valid for @xmath54 with the re - definition @xmath55 .
this enables the analytic optimization over the input states entering @xmath56 . by calling @xmath57
, we have @xmath58{=}\sqrt{\delta \rho^{00}(t ) @xmath59 and we have introduced @xmath60[{(g+i\delta)\sinh(zt)+z\cosh(zt)}]/{2z}$ ] , @xmath61 and the energy mismatch @xmath62 between the central and outer spins .
the maximum in eq . is achieved for the pure states @xmath63 with @xmath64 . here , @xmath65 are the angles that identify the respective bloch vector .
@xmath56 is optimized by equatorial antipodal states ( _ i.e. _ states with @xmath66 and @xmath67 ) . in [ appb ] , we provide an alternative analytic approach to the evolution of spin @xmath0 and the dependence of the trace distance on such angles .
the trajectories described on the bloch sphere by the evolved states are shown in fig .
[ bloch_graph ] ( b ) [ top row , left - most sphere ] where we see that the states tend to intersect , giving @xmath68 . for @xmath19 ,
the states that optimize the measure of non - markovianity are those with @xmath69 ( the phases being immaterial ) as shown in fig .
[ trd6 ] ( b ) .
interestingly , non - zero values of @xmath70 hinder the intersections of the state trajectories [ cf .
[ bloch_graph ] ( b ) ] .
however , this does not prevent the dynamics to become markovian at proper working points , as we show later on .
the evolution of spin @xmath0 can be characterized using @xmath56 .
when the peripheral spins are detached from their respective baths , any information seeded in the central site undergoes coherent oscillations from the center to the periphery of the star and back . for @xmath71 and peripheral spins prepared in @xmath72 , the dynamics induced by @xmath73 with @xmath74 is ( strongly ) non - markovian at all times @xcite . in our case
, the interaction of the outer spins with their environments radically modifies this picture . as an example , in fig .
[ trd6 ] ( a ) we plot the trace distance for the optimal states at @xmath75 .
we ramp up the spin - bath interaction strength @xmath7 , at set values of the intra - star coupling @xmath50 , looking for the influences that an explicitly markovian mechanism has on the degree of non - markovianity that arises from the dynamical environment to which particle @xmath0 is exposed .
we find a non - monotonic behavior of the trace distance that results in non - markovianity .
the quantitative features of @xmath76 depend on the actual strength of the markovian process : as @xmath7 increases , the revivals of the trace distance become less pronounced . as @xmath56 depends on the number of temporal regions where @xmath77 , fig .
[ trd6 ] ( a ) tells us that @xmath56 decreases as @xmath7 increases , thus showing that , at resonance , a strong influence from the rigid environmental baths over the peripheral spins is sufficient to make the whole process markovian .
for @xmath78 peripheral spins with @xmath79 and @xmath80 ( dot - dashed line ) , @xmath16 ( dashed line ) and @xmath81 ( solid line ) . as the relaxation time becomes shorter , the revivals of @xmath82 are suppressed as a result of a reduction of information back - flow from the baths .
( b ) @xmath56 against @xmath70 for @xmath83 .
the two lines correspond to @xmath84 ( solid blue curve ) and @xmath85 ( dashed red curve ) , which are the optimal states in different detuning regions : @xmath56 is the topmost curve in each region .
there is a finite window of detunings ( light - shadowed region marked as m ) where @xmath86 [ nm marks regions where @xmath87 .
inset : @xmath56 against @xmath70 for @xmath88 and 1.5 ( from top to bottom curve ) . , width=264 ] this
is expected as the excitations distributed to the peripheral spins by spin @xmath0 find the _ sink _ embodied by the baths . the reduced ability to feed back information sets @xmath86 . however , the general picture is more involved : it is sufficient to move to the off - resonant case to face a rather rich _ phase diagram _ of non - markovianity . fig .
[ trd6 ] ( b ) considers the case of coupling mechanisms such that @xmath83 and explores the effect that an energy mismatch between spin @xmath0 and the peripheral sites has on @xmath56 .
we find two ranges of values of @xmath70 for which @xmath86 , symmetrically with respect to @xmath89 . in between and beyond such regions
, @xmath90 behaves quite distinctively : at resonance , the measure of non - markovianity achieves a global maximum ( equatorial states realize the maximum upon which @xmath56 depends ) . for larger detunings ,
@xmath56 changes slowly with @xmath70 ( @xmath91 being the optimal states ) .
clearly , the trend followed by @xmath90 also depends on @xmath92 : small values of @xmath92 push the dynamics towards strong non - markovianity , regardless of @xmath70 , as many coherent oscillations occur between site @xmath0 and the periphery before the initial excitation is lost into the environments . at the same time , the range of detunings for which @xmath86 increases with @xmath7 [ cf .
inset of fig .
[ trd6 ] ( b ) ] .
however non - markovianity persists , both on and off resonance , even when @xmath7 becomes the largest parameter .
this demonstrates an effective control of the degree of non - markovianity of the dynamics undergone by spin @xmath0 , which can be tuned by both the energy mismatch between the outer spins and the central one , @xmath70 , and the intra - star coupling strength .
, @xmath50 . against @xmath1 and @xmath70 for @xmath93 and @xmath94 .
differently from @xmath17 , except for a small range of values , the detuning has no effect on the _ character _ of the dynamics of spin @xmath0 .
strikingly , @xmath56 grows with @xmath1 ( almost linearly for @xmath95 ) .
( b ) analytic behavior of @xmath96 versus @xmath1 for @xmath51 , @xmath75 , @xmath97 at @xmath17 ( @xmath98 ) .
inset : we present the case corresponding to @xmath99 [ other parameters as in panel ( b)].,width=264 ] our discussions so far were restricted to the isotropic coupling at zero temperature , @xmath100 . when the peripheral spins interact with baths populated by @xmath101 thermal excitations , the markovianity regions disappear .
this is seen in fig .
[ nm - ani ] ( a ) where we show a typical case of the behavior of @xmath96 against @xmath70 and @xmath1 .
the anisotropy of the intra - star coupling is crucial for the determination of the dynamics : for @xmath102 the pair of states that maximize @xmath56 changes with the number of peripheral spins . a numerical search for the optimal states can be performed , leading to quite surprising results concerning the scaling of @xmath56 with the size of the spin environment .
intuitively , one would conclude that , as @xmath1 grows , the dynamics of spin @xmath0 will be pushed towards markovianity .
this is not the case : as shown in fig .
[ nm - ani ] ( a ) , @xmath56 _ increases _ with @xmath1 if @xmath79 , regardless of @xmath70 .
this shows that the non - markovian character resists such markovianity - enforcing mechanisms and , counter - intuitively , overcomes them .
we have checked this behavior for the exact analytical expression obtained at @xmath103 [ cf .
[ nm - ani ] ( b ) ] .
the picture somehow changes for @xmath104 : @xmath56 decreases with the growing dimension of the star .
however , even for @xmath95 the non - markovian character is preserved and @xmath56 achieves a non - null quasi - asymptotic value .
the non - markovianity measure gives an integral characterization of the dynamics .
more details on the time dependence of the system - environment information - exchange process is obtained by considering the ratio of in - flowing to out - flowing information , up to a given value @xmath105 of the evolution time . to this end
, we define @xmath106 , where the in - flow [ out - flow ] @xmath107 [ @xmath108 is defined as [ minus ] the integral of @xmath109 , over the time intervals in which it is positive ( negative ) , but only up to @xmath105 . to evaluate these quantities explicitly , we chose as input states the same @xmath110 that optimize the non - markovianity measure @xmath111 .
the ratio @xmath112 gives the fraction of the lost information that returns to the system within @xmath105 , and its behavior is quite different in the various dynamical regimes that we have identified so far . in fig .
[ seigrafici ] , @xmath113 is shown for three values of @xmath70 corresponding to the three regions of fig . [ trd6 ] ( b ) .
the diverse evolutions of @xmath112 signal qualitatively different dynamical behaviors of the system , depending on both the detuning and the anisotropy parameter . at short times
, @xmath112 is always zero ( information has to flow out of the system before it can come back ) , while its first peak is determined by the first revival of the trace distance [ see fig . [ trd6 ] ( a ) ] .
then , its features become strongly dependent on @xmath70 . at long times and at resonance , where a maximum of @xmath56 is found for @xmath79 , information oscillates between the star and spin @xmath0 and @xmath114 [ cf . fig .
[ seigrafici ] ( a ) ] .
the overall dynamics is non - markovian also for the case of fig .
[ seigrafici ] ( c ) , where the time behavior of @xmath115 is shown for a large detuning . in this case , however , @xmath113 decays to zero at long times .
thus , the regions of non - markovianity in fig .
[ trd6 ] ( b ) correspond to different behaviors : near resonance , a fraction of information comes back to the system , different input states remain distinguishable even at long times and thus no equilibrium state is found . for large detunings ,
non - markovianity is built up at short times , while different input states converge towards a long - time equilibrium .
on the other hand , for intermediate values of the detuning [ _ i.e. _ for @xmath70 in the markovianity region of fig .
[ trd6 ] ( b ) ] and @xmath79 , there is no back - flow . even for @xmath104 ,
the fraction of information that comes back is quite small .
the picture changes when @xmath50 increases , the evolution becoming increasingly non - markovian and the role of the anisotropy being fully reversed : @xmath79 implies a larger @xmath113 , persisting for longer times at resonance . versus @xmath49 for a star of @xmath116 sites at @xmath17 , with @xmath117 ( left plots ) and @xmath16 ( right plots ) for three different values of the detunig : @xmath118 for the plots ( a ) and ( d ) , @xmath119 for ( b ) and ( e ) , while @xmath120 for ( c ) and ( f ) . ]
we have used a measure of non - markovianity to show the possibility to control the dynamics of an open quantum system coupled to many independent decohering channels .
we have highlighted the key role played by the detuning and the degree of anisotropy of the system - environment coupling : both can be used to explore a rich non - markovianity phase diagram , where qualitatively different scaling laws with the number of decoherence channels are found . the ability to switch from a markovian to a non - markovian regime by means of a local parameter
could be used to prepare a quantum system in a desired state : indeed the markovian character of processes can be employed for state engineering and information manipulation @xcite . on other hand , while the formation of a steady entangled state is supported by non - markovianity , a purely markovian dynamics produces separable steady states @xcite .
we acknowledge financial support from the uk epsrc ( ep / g004759/1 ) .
sl thanks the centre for theoretical atomic , molecular and optical physics for hospitality during the early stages of this work .
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the total hamiltonian of the spin - star system @xmath121 consists of a few contributions .
the first one is the system s free energy [ we take units such that @xmath122 throughout the paper ] @xmath123 , describing the free evolution of @xmath124 spin-@xmath125 particles [ here , @xmath126 is the @xmath127-pauli matrix of spin @xmath128 ( @xmath129 ) ] , each with transition frequency @xmath130 between spin states @xmath131 and @xmath132 . the second term in @xmath133
describes the energy of @xmath1 sets of @xmath134 harmonic modes ( one set per peripheral spin of the star ) with creation ( annihilation ) operators @xmath135 ( @xmath136 ) which satisfy the commutation relations
@xmath137 = \delta_{kk'}\delta_{jl}$ ] .
the central and peripheral spins are coupled by @xmath138 , whose explicit form will be specified later on .
each peripheral spin interacts with its own bath as @xmath139 , where @xmath140 with @xmath141 .
we assume that the their local bath induce a markovian dynamics of the peripheral spins and take uniform couplings , so that the evolution of the central spin is ruled by @xmath142+\sum_{j=1}^n\hat{\mathcal{l}}_j[\rho(t)]\}\ ] ]
here we provide an alternative solution to the dynamics of the system . in the interaction picture with respect to @xmath133 the schrdinger equation reads @xmath143 where the interaction hamiltonian is given by @xmath144 with @xmath145 the operator @xmath146 counts the number of excitations in the system and commutes with the total hamiltonian @xmath147 , so that any initial state of the form @xmath148 evolves after time @xmath39 into the state @xmath149 where the state @xmath150 denotes the product state @xmath151 and @xmath152 for the sites on the star ; @xmath153 is the vacuum state of all the reservoirs , and @xmath154 the state with one particle in mode @xmath155 in the @xmath128th reservoir .
+ the amplitude @xmath156 is constant in time because of @xmath157 .
+ substituting eq . into the schrdinger equation one finds @xmath158 we assume in the following that @xmath159 .
this means that the two level systems on the star are in the @xmath160 state and that each environment is in the vacuum state initially .
+ the total initial state is given by the product state @xmath161 formally integrating eq . and substituting into eq .
one obtains the system for the amplitude @xmath162 , @xmath163 we can define the kernels @xmath164 describing the two - point correlation function of each reservoir , which are the fourier transform of the respective environmental spectral density @xmath165 for the moment , we do not make any restrictive hypothesis on the form of @xmath166 , so that our results will be valid for an environment with a generic spectral density . in order to solve the system above it
is convenient to pass in the laplace domain : @xmath167&=&\;c_1(0)-i j
\sum_{j=1}^n \tilde{c_j}[s+i(\epsilon_0-\epsilon_j)]\\ & s \tilde{c_j}[s]&=&-i j \tilde{c_1}[s - i(\epsilon_0-\epsilon_j)]- \tilde{c_j}[s]\tilde{f}_j[s ] \end{aligned}\end{cases } \label{eq-22}\ ] ] solving the second of eq . [ eq-22 ] respect to @xmath168$],assuming that all the reservoirs are the same ( @xmath169 ) , and substituting in the first we get @xmath170=c_1(0)\dfrac{s - i\delta - f[s - i(\epsilon_0-\epsilon)]}{s^2-is(\epsilon_0-\epsilon)-i s f[s - i(\epsilon_0-\epsilon)]+j^2 n}\label{c1s}\nonumber\ ] ] where @xmath171 ( @xmath172 ) . to specify the model , but still retaining a general enough description , we consider a lorentzian spectral density for each bath ( which gives rise to an exponentially decaying correlation function ) : @xmath173 here @xmath174 is the detuning of the center frequency of the bath @xmath175 and the frequency of the two - level system @xmath130 , the parameter @xmath49 defines the spectral width of the environment , which is associated with the reservoir correlation time by the relation @xmath176 and the parameter @xmath7 is related to the relaxation time scale @xmath177 by the relation @xmath178 .
+ we will consider @xmath179 , and in this case we may distinguish between the markovian and the non - markovian regimes ( for the dynamics of the environmental spins themselves ) using the ratio of @xmath7 and @xmath49 : @xmath180 gives a markovian regime and @xmath181 corresponds to non - markovian regime .
+ substituting in eq .
[ c1s ] and anti - transforming we have @xmath182 with @xmath183}{(\alpha_1-\alpha_2)(\alpha_2-\alpha_3)(\alpha_1-\alpha_3 ) } , \end{aligned}\label{gt}\ ] ] where @xmath184 , @xmath185 and for @xmath186 . here ,
@xmath187 s are the roots of the equation @xmath188 already at this point , it is evident how the only effects of increasing @xmath1 is to redefine the coupling constant @xmath50 .
the solution of the schrdinger equation of the total system with initial states of the form lyes in the sector of the hilbert space corresponding to zero or one excitations .
+ we can construct the exact dynamical map describing the time - evolution of the reduced density matrix of the central spin which is given by @xmath189 where @xmath190 for @xmath191 . using eq . and eq . we find @xmath192 the optimization of the initial states in eq .
( [ nonmarkovianity ] ) obtains the maximally possible non - markovianity of a particular quantum evolution . + in our case , the maximization is achieved by pure states , thus we choose as initial states for eq .
[ statoiniziale ] @xmath193 where we used the fact that , since @xmath147 is invariant under rotations along z - axis , the maximum is obtained for @xmath194 .
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we study the interplay between forgetful and memory - keeping evolution enforced on a two - level system by a multi - spin environment whose elements are coupled to local bosonic baths .
contrarily to the expectation that any non - markovian effect would be _ buried _ by the forgetful mechanism induced by the spin - bath coupling , one can actually induce a full markovian - to - non - markovian transition of the two - level system s dynamics , controllable by parameters such as the mismatch between the energy of the two - level system and of the spin environment . for a symmetric coupling
, the amount of non - markovianity surprisingly grows with the number of decoherence channels .
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the classical theory of force - free motions of rigid particle systems has a long history in connection with investigation in mathematics and mechanics . from the differential calculus , in the case of translational particle motions in euclidean space ( newton axioms of mechanics ) , to the riemann geometry .
hence , the definition of geodesics has immediately concerned with force - free particle motions on surfaces .
the introduction of riemannian manifolds and the geometry of their geodesics were motivated by the mechanics of constrained particle systems . in the last few years
, the study of force - free motion systems on riemannian manifolds of constant sectional curvature has attracted the interest of several authors @xcite , @xcite , @xcite , @xcite , @xcite and @xcite . in particular ,
in @xcite the authors define the notion of a pendulum on a surface of constant gaussian curvature @xmath0 and they study the motion of a mass at a fixed distance from a pivot .
so , a pendulum problem on a surface of constant curvature is defined as a pivot point and a mass connected to that point by a rigid massless rod of fixed length @xmath1 .
it is assumed that the pivot is constrained to move along some fixed curve with prescribed motion .
the rod provides the only force on the mass in order to keep the mass at the fixed distance from the pivot .
no torque is applied to the rod , ( fig .
1 ) . in @xcite
it is studied the pendulum problem when the pivot moves along a geodesic path , and the space is a surface of constant ( not zero ) curvature .
it is considered the surface immersed in @xmath2 or @xmath3 and it is obtained the differential motion equation by newtonian procedure doing a laborious calculation . moreover , the cases @xmath4 and @xmath5 are necessary to come from different forms . in this work
, we deal with the pendulum problem from an analytic point of view .
this procedure allows approach simultaneously the cases of positive and negative curvature and also of zero curvature ( as a limit case ) , with very simple computations .
the key of our study is a lagragian approach , which has a 1-dimensional configuration space .
as adirect consequence the internal curvature force is conservative and its potential function is easily calculated .
moreover , this method can be used to another related problems , as the elastic pendulum , or the quantum pendulum .
mainly , this work concerns the case in which the pivot moves along a geodesic .
let @xmath6 be the angle between the rod and the motion direction of the pivot .
we will assume the following convention : if the constant curvature of the space is @xmath4 , @xmath7 if the constant curvature of the space is @xmath5 , @xmath8 the main result is the following : using the previous result and geometric arguments , we obtain the motion equation of the system when the pivot accelerates along a geodesic ( see section 4 ) .
further developements are discussed in section 5 .
it is well known that a complete , simply - connected riemannian @xmath9-manifold with constant sectional curvature is isometric to one of model spaces @xmath10 , @xmath11 or @xmath12 ( see @xcite ) .
let @xmath13 be the 2-dimensional sphere of radius @xmath14 , endowed with the metric induced from @xmath15 , where @xmath16 consider the coordinate system @xmath17 in @xmath18 with @xmath19 and @xmath20 , being @xmath21 in this coordinate system the metric is given by @xmath22 and the non - zero christoffel symbols are @xmath23 and @xmath24 .
let s consider also the hyperbolic plane @xmath25 , where @xmath26 and @xmath27 is the induced metric from the lorentz - minkowski spacetime @xmath3 .
let @xmath28 be the coordinate system in @xmath29 with @xmath30 and @xmath31 , where @xmath32 the metric is given by @xmath33 and the non - zero christoffel symbols are @xmath34 and @xmath35 .
firstly , we deal with the rigid pendulum problem in the case that the pivot moves along a geodesic of its space , with constant speed . we will do analogously the study for @xmath4 and @xmath5 . note that in the case @xmath4 we must require that @xmath36 to guarantee that the mass and the rod are on the same side of the geodesic line .
suppose that the pivot moves along the geodesic @xmath37 with constant speed .
let s take a reference associated to the pivot .
denote by @xmath6 the angle between the rigid rod and the direction of pivot motion .
the lagrangian of the system has two components , on the one hand , the kinetic energy of the mass @xmath38 , and on the other hand , the potential energy @xmath39 , which is due to the curvature of space , i.e. , @xmath40 if the space is flat .
consider the isosceles geodesic triangle formed in an infinitesimal temporal interval , by the rod and the arc @xmath41 traced by the mass ( fig .
2 ) .
the sine theorem for spherical geometry allows us to write @xmath42 thus @xmath43 therefore , if the particle has mass @xmath44 , the kinetic energy is given by @xmath45 consider the coordinate system @xmath17 with the metric @xmath46 and suppose that the pivot moves along the geodesic @xmath37 with constant speed @xmath47 . obtain us the mass acceleration due to the curvature , so suppose that the mass moves along the curve @xmath48 its acceleration is given by @xmath49 as consequence , the external force that we must be to apply on the rod will be @xmath50 and the curvature potential energy @xmath39 satisfies @xmath51 , where @xmath52 . hence , @xmath53 where we have taken @xmath54 as origin energy . again , using the sine theorem ( fig .
3 ) we obtain @xmath55 thus @xmath56 now , we can enunciate the following theorem , suppose that the pivot of pendulum moves with constant speed @xmath47 along a geodesic on a surface with constant curvature @xmath4 .
let @xmath57 the angle at the time @xmath58 between the rigid rod and the direction of pivot motion .
then the lagrangian of the system is @xmath59 therefore , the motion differential equation is given by @xmath60 if we do @xmath61 , @xmath62 and @xmath63 obtaining the lagrangian function of the pendulum moving in the euclidean plane , when the pivot moves along a straigh line with constant speed .
consider us now , the pivot moving along the geodesic @xmath64 , with constant speed @xmath47 .
analogously to the spherical case , we take a reference joint with the pivot and we denote @xmath6 the angle between the rigid rod and the direction of pivot motion . if we consider the isosceles hyperbolic differential triangle ( fig .
2 ) , and making use of the corresponding theorems of the hyperbolic geometry , we can to conclude that the kinetic energy of the system is given by @xmath65 reasoning as the in spherical case , we consider that the pivot moves along the geodesic @xmath64 . if the mass moves along the curve @xmath66 we can to calculate the curvature potential function , @xmath67 where we have taken @xmath68 as origin energy . again , using the sine theorem in the hyperbolic case , we obtain @xmath69 thus , @xmath70 suppose that the pivot of the pendulum moves with constant speed @xmath47 along a geodesic on a surface with constant curvature @xmath5 .
let @xmath57 the angle at the time @xmath58 between the rigid rod and the direction of pivot motion .
then the lagrangian of the system is given by @xmath71 therefore , the motion differential equation is @xmath72 with our convention , we can to enunciate ( compare with @xcite , theorem a ) : suppose that the pivot of pendulum moves with constant speed @xmath47 along a geodesic on a surface with constant curvature @xmath0 .
let @xmath57 the angle at the time @xmath58 between the rigid rod and the direction of pivot motion .
then the lagrangian of the system is given by @xmath73 therefore , the motion differential equation is @xmath74 suppose @xmath4 and consider the motion equation ( [ motion1 ] ) . making the change @xmath75 , we obtain the differential equation @xmath76 which is the equation of planar pendulum of length @xmath77 in euclidean space subject to a constant gravitational field of magnitude @xmath78 .
its stable and unstable equilibria at @xmath79 and @xmath80 , correspond to the stable @xmath81 and unstable @xmath82 equilibria of the pendulum on the spherical surface . on the other hand ,
if @xmath5 , making @xmath83 in the motion equation ( [ motion ] ) , we obtain , in similar form , that @xmath82 is stable equilibria and @xmath81 is unstable equilibria .
moreover , it is well known that the equation ( [ planar ] ) can be exactly solved in term of a elliptic integral of first kind ( see @xcite , for instance ) , which can not be evaluated in a closed form . a first approximation to this problem in arbitrary constant curvature is given for small oscillations around the stable equilibria points of this physical system .
that is , for small @xmath84 , @xmath85 , and the equation of motion is approximated by @xmath86 observe that the previous equation represents a simple harmonic oscillator of frequence @xmath87 . since the hamiltonian function @xmath88 is time
independent , it is an integral on phase space and it represents the system energy . using the standard notation @xmath89 in the phase space , we can write @xmath90 the legendre transformations allow us to give @xmath91 moreover , it is easy to see that @xmath88 is constant if and only if @xmath92 is constant ( compare with ( * ? ? ? * prop .
suppose now that the pivot moves along a geodesic path with lineal acceleration @xmath93 .
we come to compute the acceleration induced by the accelerated pivot at the mass making use of a geometric argument .
we follow the following steps : \a ) translate to the mass the pivot acceleration @xmath93 .
\b ) transform this acceleration in angular acceleration between the rod and the geodesic path traced by the pivot .
\c ) project on the orthogonal direction to the rod .
indeed , consider the geodesic triangle of the figure 4 .
the translated acceleration to the mass is given by @xmath94 and the angular acceleration by @xmath95 finally , in order to project , it is enough to multiply by @xmath96 @xmath97 on another hand , taking into account the previous triangle , from the pythagoras and cosine theorem in non - euclidean geometries , we have @xmath98 thus @xmath99 . as a consequence ,
the searched acceleration is @xmath100 so , we prove in a different approach the result ( ( * ? ? ?
* theorem d ) ) assume that the pivot moves with speed @xmath101 along a geodesic on a surface with constant curvature @xmath0 .
let @xmath57 the angle that the rigid rod makes with the direction of the motion of the pivot .
then the pendulum satisfies the differential equation @xmath102 where @xmath103
as a first application , consider us the case of a pendulum whose pivot moves along a geodesic on a surface of constant curvature with constant speed @xmath47 and whose rod is elastic .
let @xmath104 be the elastic constant of the rod , let @xmath1 be the length of the rod and let @xmath105 be the elongation of the rod at the time @xmath58 .
our analytical approach allows us to initiate the study of dynamical system in similar form as the rigid case .
so , it is not difficult to see that its lagrangian function is given by @xmath106 secondly , we will make an approximation to the quantum system .
denote @xmath107 .
then , ( [ hamiltonian ] ) can be written , for @xmath108 , in terms of @xmath6 and @xmath109 as @xmath110 observe that the same expression , up an additive term , is obtained for @xmath5 .
therefore , a suitable time independient schrodinger equation for this system can be described as @xmath111 where @xmath112 is a complex function representing the wave function of the mass particle in this physical system .
observe that it is expected that the energy should take quantized values .
in fact , if it is assumed that @xmath113 is confined in a region where @xmath6 is close to @xmath68 , the previous equation , in this first approximation , corresponds to the quantum harmonic oscillator , whose energy is found to be @xmath114 observe that , within this simplification , in the quantum of energy , @xmath115 , appears the speed of the rod and the curvature of the ambient space .
the authors are partially supported by the spanish mec - feder grant mtm2010 - 18099 .
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the dynamics of force free motion of pendulums on surfaces of constant gaussian curvature is addressed when the pivot moves along a geodesic obtaining the lagragian of the system . as a application it is possible the study of elastic and quantum pendulums .
_ ams classification scheme numbers _ : 70h03 , 53a35 , 53b20 .
_ keywords _ : dynamical systems , lagrangian and hamiltonian mechanics , semi - riemannian geometry , constant curvature , pendulum .
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( hereafter ) is a 33ms radio pulsar discovered in the green bank telescope ( gbt ) 350 mhz drift - scan pulsar survey @xcite . with a dispersion measure of 3.27pc@xmath5
, it appeared to be one of the closest pulsars to the earth .
further observations showed was in a binary system with an orbital period of 2.45days and a minimum companion mass of about 1@xmath6 .
this sort of system straddles the line between potential companion types .
it could be a double - neutron star ( dns ) , of which there are only roughly 12 and whose study is crucial to understanding the formation of sources of khz gravitational waves ( e.g. , * ? ? ?
* ) and testing general relativity ( e.g. , * ? ? ?
, it could be a pulsar with a massive white dwarf companion a so - called `` intermediate - mass binary pulsar '' ( imbp)that descended from a binary with a more massive companion than in traditional systems with pulsars and low - mass white dwarfs @xcite .
imbp systems are rare , with fewer than 20 known , and massive white dwarfs are themselves rare , with fewer than 8% of the white dwarfs ( wds ) from optical surveys having masses above @xmath7 @xcite .
understanding the formation and evolution of imbp systems provides a crucial piece in our understanding of binary evolution and pulsar recycling , and helps delineate evolutionary paths between low - mass nss and high - mass white dwarfs @xcite .
@xcite used very long baseline interferometry astrometry to measure the parallax of with exquisite precision .
they find a distance of @xmath0pc ( it is the second closest binary pulsar system and one of the closest nss of any type ) .
the astrometric data also suggested an edge - on orbit , opening up the possibility of a measurement of the shapiro delay @xcite , which gives two post - keplerian @xcite parameters for the system and hence determines the component masses ( e.g. , * ? ? ? * ) .
here we present the detailed timing analysis of the system , including the measurement of the shapiro delay and the determination of the masses ( ) .
we then present deep optical and near - infrared searches for the companion to ( ) , which we use to constrain models of its formation and evolution ( ) .
we find that the system almost certainly must be an imbp system , but that we do not detect the companion , constraining it to be one of the coolest white dwarfs ever observed .
unlike some sources where temperature inferences are highly dependent on white dwarf model atmospheres ( e.g. , * ? ? ?
* ) , this measurement is robust , given the small uncertainties on the mass and ( especially ) distance .
we conclude in .
radio observations of to measure the shapiro delay occurred in the last week of 2011 may with the 100 m robert c. byrd gbt .
we had a 6hr observation taken around superior conjunction of the binary system augmented by five 2hr observations at each of the other five shapiro extrema , all using the green bank ultimate pulsar processing instrument ( guppi ; @xcite ) .
the 800mhz of bandwidth centered at 1500mhz in two orthogonal polarizations was separated into 512 nyquist - sampled frequency channels of width 1.5625mhz via a polyphase filter bank .
these channels , sampled at 8-bits , provided full polarization information and an effective time resolution of 0.64@xmath8s .
each channel was coherently dedispersed at the nominal dispersion measure ( dm ) of the pulsar ( 3.27761pc@xmath9 at the time , although we later refined this measurement ) .
each observing session was broken into 30-minute observations of separated by 60s calibration scans of the extragalactic radio source 3c 190 .
the calibration scans were taken in the same mode as the pulsar observations , but also included a 25hz noise diode inserted into the receiver . ) .
the position angle of the linear polarization is given in the upper panel . as is the case with most msps
, the polarization position angle variations do not permit a rotating vector model fit , so we can not constrain the emission geometry .
[ fig : profile ] , scaledwidth=40.0% ] data reduction was performed using the ` psrchive ` package @xcite .
flux calibration used the on- and off - source scans of 3c 190 .
this was followed by removal of radio frequency interference by the psrzap utility .
the calibrated pulse profile determined from the long observation covering conjunction is given in .
the data were aligned in time using the best ephemeris ( below ) , divided into 16 frequency channels , and re - fit for dispersion measure and rotation measure using a bootstrap error analysis .
we found that the period - averaged flux density varied by a factor of a few over the course of long observations due to scintillation , with an average of 12mjy at 1500mhz .
individual times - of - arrival ( toas ) were measured from the folded total - intensity profiles using the frequency domain algorithm in ` psrchive ` @xcite .
a template was created by fitting three gaussians to the summed pulse profile . from these gaussian components
, we created a noise - free template with the phase of the fundamental component in the frequency domain rotated to zero .
the observations were divided into 2minute segments , with one toa measured for each segment . note that since interstellar scintillation caused the flux to vary considerably , there was a proportional change in the toa precision that varied over the data set .
these data were combined with previous data taken for the discovery observations of @xcite to produce a timing model .
we used the `` dd '' model @xcite in ` tempo ` , which incorporates the shapiro delay .
the astrometric data for this model were taken from @xcite , and we used the de421 jpl ephemeris @xcite .
timing fits with no shapiro delay were statistically unacceptable , with an rms residual of @xmath10s ( @xmath11 for 931 degrees - of - freedom ) , and a clear shapiro delay signature was obvious in the residuals ( ) . with the shapiro delay included in the fit the rms residual was 4.2@xmath8s ( @xmath12 for 929 degrees - of - freedom ) , with no obvious remaining structure in the residuals ( varying the astrometric parameters within the uncertainties from @xcite changed the timing results by @xmath13 ) .
the shapiro delay determines the inclination of the orbit and the companion mass ; this is then combined with the binary mass function to determine the pulsar s mass .
due to the combination of several different and much less precise observing modes from earlier monitoring with the high - precision shapiro delay campaign , we estimated the timing parameters with a bootstrap error analysis .
we give the full timing results , with 1-@xmath14 error estimates from the bootstrap analysis , in .
timing residuals for , using the new data from this paper ( blue : mjd 55,60055,921 ) and older data ( gray ) , as a function of orbital phase ( true anomaly plus longitude of periastron ) .
top : residuals computed from the best - fit model without shapiro delay ( the rms residual is @xmath10s ) .
middle : residuals computed including shapiro delay .
the red curve is the best - fit shapiro delay profile .
bottom : residuals computed relative to the best - fit model including shapiro delay ( the rms residual is @xmath15s ) .
conjunction is at a phase of 0.25 . in all panels
the left axis shows the residuals in @xmath8s , while the right axis shows the residuals in milliperiods .
note the different @xmath16-axis scales.,scaledwidth=50.0% ] our data consist of high - quality coherently dedispersed data from an intensive 1 week campaign and a few other epochs .
the remainder of the data were both less precise and less uniform , with a wider range of observation frequency and instrumental setup .
this makes it difficult ( if not impossible ) to robustly constrain long - term secular changes like periastron precession ( @xmath17 ; @xcite ) . nonetheless , we tried a fit with @xmath17 fixed to the value predicted by general relativity ( @xmath18 ) .
the resulting fit was good , with the rms decreasing to 3.8@xmath8s .
the pulsar and companion masses each increased by about 1@xmath14 compared to the values in .
given the small eccentricity and inhomogeneous data set with large gaps we do not believe that fitting for @xmath17 is viable at this time , but encourage further long - term monitoring of this system to establish its secular behavior .
l c + spin period ( s ) & 0.032817859053065(3 ) + period derivative ( ss@xmath19 ) & @xmath20 + dispersion measure ( pc@xmath9 ) & 3.2842(6 ) + rotation measure ( radm@xmath21 ) & + 2.6(1 ) + reference epoch ( mjd ) & 55743 + right ascension ( j2000 ) & 22:22:05.969101(1 ) + declination ( j2000 ) & @xmath22:37:15.72441(4 ) + r.a . proper motion ( mas@xmath23 ) & 44.73(4 ) + dec proper motion ( mas@xmath23 ) & @xmath245.68(6 ) + parallax ( mas ) & @xmath25 + position epoch ( mjd ) & 55743 + span of timing data ( mjd ) & 5500555922 + number of toas & 943 + rms residual ( @xmath8s ) & 4.2 + + orbital period ( days ) & 2.4457599929(3 ) + projected semi - major axis ( lt - s ) & 10.8480276(12 ) + epoch of periastron ( mjd ) & 55742.13242(0 ) + orbital eccentricity & @xmath26 + longitude of periastron ( deg ) & 119.778(12 ) + mass function ( @xmath6 ) & 0.22907971(8 ) + @xmath27 & 0.9985(3 ) + companion mass ( @xmath6 ) & 1.05(6 ) + + distance ( pc ) & 267.3@xmath28 + transverse velocity ( km@xmath29 ) & 57.1@xmath30 + orbital inclination @xmath31 ( deg ) & 86.8(4 ) + shklovskii period derivative ( @xmath32 ) & @xmath33 + intrinsic period derivative ( @xmath32 ) & @xmath34 + surface magnetic field ( @xmath35 gauss ) & 0.719 + spin - down luminosity ( @xmath36 ) & 1.72 + characteristic age ( gyr ) & 33.8 + pulsar mass ( @xmath6 ) & 1.20(14 ) + flux density at 1500mhz ( mjy ) & 12 + we observed the position of at optical and near - infrared wavelengths , as listed in .
the deepest keck observations used the red side of the low - resolution imaging spectrometer ( lris ; @xcite ) on the 10 m keck i telescope .
the data were reduced using standard procedures in ` iraf ` , subtracting the bias , dividing by flatfields , and combining the individual exposures .
the seeing was about @xmath37 in the combined @xmath38 image , and @xmath39 in the combined @xmath40 image .
we computed an astrometric solution fitting for a shift and separate scales and rotations along each axis ( i.e. , a six - parameter fit ) using 100 non - saturated stars identified from the sloan digital sky survey ( sdss ) data release 10 ( dr10 ; @xcite ) , giving rms residuals of @xmath41 in each coordinate .
we did photometric calibration relative to sdss photometry , identifying 23 well - detected , well - separated , non - saturated stars , and transforming from the sdss filter set to johnson cousins using the appropriate transformation equations .
the zero - point uncertainty was @xmath42mag , although there are systematic uncertainties coming from our filter transformations .
we see no object at the position of the pulsar ( ) ; the closest object is about @xmath43 from the position of the pulsar ( about @xmath44 away ) and appears extended ( @xmath45 and statistical position uncertainties of @xmath46 in each coordinate ) .
we determined the 3@xmath14 upper limits using ` sextractor ` @xcite to determine the magnitude that gave a 0.3mag uncertainty ( verified with fake - star tests ) , which we give in .
we observed in @xmath47-band with the goodman spectrograph @xcite on the 4.1 m southern astrophysical research ( soar ) telescope over two nights in 2013 july .
all exposures were dithered and binned by a factor of two in both dimensions .
the frames were bias - subtracted and flattened with a dome flat .
we then used a median of the data ( having masked the scattered - light halos of three saturated stars ) from the second night constructed without registration to create a sky flat , which we smoothed with a @xmath48pixel boxcar filter .
this corrects for larger - scale brightness variations .
cosmic rays were interpolated on individual exposures using the ` lacosmic ` routine @xcite .
the seeing varied considerably over the course of the observations , going from @xmath49 to @xmath43 .
we then shifted each exposure by an integer number of pixels for registration and summed them .
the final summed image has an effective seeing of @xmath50 and a total exposure time of 2.6hr .
the photometric zero - point was again computed relative to the sdss dr10 data , using 31 stars .
the astrometric solution was done using six 30s exposures through http://astrometry.net @xcite . as with the keck data , we see no object at the position of the pulsar ( ) and
give a 3@xmath14 upper limit in .
l c c c c c soar / goodman & 2013 jul 2 & @xmath47 & @xmath51 & 26.4 & 19.2 + soar / goodman & 2013 jul 3 & @xmath47 & @xmath52 & & + keck
i / lris(red ) & 2013 aug 4 & @xmath38 & @xmath53 & 26.3 & 19.1 + keck
i / lris(red ) & 2013 aug 4 & @xmath40 & @xmath53 & 26.0 & 18.9 + keck ii / nirc2 & 2013 oct 12 & @xmath54 & @xmath55 & 21.0 & 13.9 + while they were taken through different filters and with very different instruments / resolutions , we tried combining the keck @xmath38-band and soar @xmath47-band images using ` swarp ` @xcite .
we still see no source at the position of the pulsar .
the data are sufficiently different that a limiting flux is difficult to compute , but it could be as much as 0.3mag fainter than the limits in .
the near - infrared observations come from the nirc2 camera square field of view . ] on the 10 m keck ii telescope , and used the laser guide star adaptive optics ( ao ) system @xcite .
the data were taken through thin clouds and the ao corrections were not optimal , resulting in a delivered image quality of @xmath41 fwhm .
the images were reduced using a custom pipeline implemented with ` python ` and ` pyraf ` using dark frames and dome - flats .
a sky fringe frame was created by combining dithered images of multiple targets with the bright stars masked .
we used ` sextractor ` @xcite for the preliminary detection and masking of stars .
the fringe frame was subtracted from the flat - fielded data after being scaled to the appropriate sky background level . before coadding the frames ,
each frame was corrected for optical distortion using a distortion solution measured for nirc2 .
a faint glare has been visible in the lower right ( south - west ) corner of the nirc2 wide camera images starting in 2009 august .
the shape and amplitude of the glare vary with telescope orientation , resisting correction through surface fitting or modeling .
instead we masked the glare using a triangular region .
there was no independent photometric calibration that night , and only a single star is visible on the co - added image .
to determine a photometric zero - point , we used photometry for that star from the sdss dr10 .
we then employed the empirical main - sequence color relations from @xcite , inferring the @xmath56 color from the observed @xmath57 color ( we ignore differences between @xmath58 and @xmath54 filters ) . for this star ( sdss [email protected] )
we infer a spectral type of k2.5 and predict @xmath59 . we expect zero - point uncertainties of @xmath60mag or so based on comparison of the other sdss colors to those predicted using @xcite .
again we see no object at the position of the pulsar , and give 3@xmath14 upper limits in .
since we do not detect the optical counterpart of the companion , the first inference is that the companion could be a low - mass ns .
it would be the lowest mass ns known @xcite , although it is only a roughly 23 @xmath14 excursion from the mean of the companions in dns systems @xcite : rare , given the @xmath61dns systems , but not impossible . in that case , its eccentricity of @xmath62 would be a factor of @xmath63 lower than any other dns system ( has the lowest eccentricity of @xmath64 , although this may be an ns wd system ; @xcite ; van leeuwen et al .
2014 , , submitted ) . in
we show the eccentricity versus component masses for all dns and ns wd systems with well - determined masses .
in fact there are three ns wd systems with higher eccentricities : , which was likely not recycled @xcite ; , which has had its eccentricity increased by dynamical interactions @xcite ; and ( likely an imbp , with the eccentricity the result of unstable mass transfer ; @xcite ) .
the normal formation scenario for a dns involves two core - collapse supernova explosions , with the eccentricity the result of the second explosion and its kick , and no final mass - transfer phase to circularize the orbit ( e.g. , * ? ? ?
in contrast , formation via an electron - capture supernova ( ecs ; @xcite ) could result in a significantly lower ns mass ( @xcite ; @xcite ) along with a lower supernova kick @xcite .
has a low transverse velocity ( @xmath65 ) , although higher than some systems thought to be the products of ecss ( given the age of the system , this velocity may be more related to motion in the galactic potential than birth conditions ) .
this may reflect the velocity dispersion of the progenitor systems .
however , the contrast between and other systems thought to be the results of ecss ( e.g. , psr j1906 + 0746 or ; @xcite ) is extreme , with the ratio of eccentricities above 200 as mentioned previously . in a scenario without a kick we can place an upper limit on the amount of material that could have been ejected by the explosion to @xmath66 ( with @xmath67 the pulsar mass and @xmath68 the current companion mass ; e.g. , * ? ? ?
this is a much tighter bound than in any of the other systems proposed for this mechanism , and difficult to reconcile with the change in binding energy needed to collapse to a ns @xmath69 ( with @xmath70 km for an ns ) , presumably released as neutrinos ( e.g. , * ? ? ?
* ) : this leads to the horizontal line in , above which all confirmed dns systems are found . in order to have a dns system with such a low eccentricity
, we need to invoke increasingly exotic ( and perhaps implausible ) evolutionary scenarios .
for instance , if the system began as a hierarchical triple @xcite , then the inner components could have formed a standard eccentric dns system early on
. later evolution of the outer member could have led to a circum - binary accretion disk that would have worked to circularize the inner system , after which the outer object would have exploded or otherwise been ejected from the system .
the other possible scenario is that the companion could be a massive wd , making the system an imbp .
its orbital eccentricity is somewhat high compared to most low - mass binary pulsars of similar periods ( based on @xcite ) , but not nearly as high as a dns , consistent with an imbp classification @xcite .
it falls in the locus of other co wds in the `` corbet '' ( binary period versus spin period ) diagram in @xcite . the pulsar mass is lower than most pulsar
wd binaries , but is consistent with the short orbital - period imbp discussed by @xcite which may indicate a similar formation mechanism involving a common envelope @xcite .
however , as a wd it would be extremely faint : far fainter than any of the optical companions to imbps currently known @xcite or indeed any wd companion to a millisecond pulsar ( msp ) with a similar mass @xcite ; it is perhaps the faintest wd ever observed . with the apparent magnitude limits from
, we can compute absolute magnitude limits in each band .
we use the distance @xmath71pc @xcite , and we estimate the extinction to be @xmath72mag from @xcite . in terms of bolometric luminosity the most constraining limit ends up coming from the @xmath38-band data , where we limit @xmath73 ( the @xmath47-band limit of @xmath74 is very similar , given slight differences in bolometric correction ) . for comparison , the companion to with a median companion mass of @xmath75 has @xmath76 @xcite . in
we plot the absolute magnitude against mass for pulsar+wd systems as well as select cool wds with parallax distances : even compared to the observed truncation of the cooling sequence in old halo globular clusters like ngc 6397 @xcite or m4 @xcite , the putative companion is far fainter : at the distance of ngc 6397 , our limit of @xmath73 translates to an apparent magnitude of @xmath77 , compared to @xmath78 , or @xmath79 for the coolest wds seen in ngc 6397 .
some of the difference comes from the change in radius : a @xmath80 wd has a radius about 65% of that of a typical @xmath81 wd , leading to a 1mag change in brightness at the same effective temperature .
but the difference in is more like 2.5mag , so the companion to must also be cooler than the known thick disk / halo wds . beyond the absolute magnitude , which is directly computable from observable quantities , we can limit the radius / temperature of a putative wd by using our @xmath38-band absolute magnitude limit to constrain the bolometric luminosity .
this is more complicated , as it involves atmosphere calculations in an uncertain and poorly tested regime , but it should be reasonably reliable .
we use the synthetic photometry and evolutionary models from @xcite and @xcite for h and he atmospheres , respectively . ] . for isolated wds pure he atmospheres
can be largely excluded because of bondi - hoyle accretion from the ism @xcite , and even small amounts of hydrogen mixed into the helium can cause near - infrared flux deficiencies like pure hydrogen ( see below ; @xcite ) . however , the binary orbit and msp wind in this system could have inhibited such accretion and therefore a he atmosphere is possible . in any case
a pure he atmosphere will serve as a limiting case compared to the h models .
these models are used to convert the absolute magnitude limits into temperature limits , so for simplicity we use the @xmath80 models ( differences in bolometric corrections as a function of mass are small , @xmath82mag ) .
the most constraining limit is again from the @xmath38-band data , where @xmath73 implies @xmath83k ( see ) for a h atmosphere .
the he - atmosphere models do not extend to sufficiently cool temperatures but stop at @xmath84k with @xmath85 . at lower temperatures the details of the atmospheric physics are rather uncertain , but a blackbody is likely an acceptable approximation ( p. bergeron , 2014 , private communication ) . with a he atmosphere
an effective temperature @xmath86k would be required ( ) . the h limits are more constraining since more of the flux appears in the optical regime rather than the near - infrared a consequence of collisionally induced absorption by molecular h@xmath87 @xcite .
these limits change slightly with mass given the small but finite mass uncertainties , since the radius would change with mass : going to the @xmath88 h model we can constrain @xmath89k ( at our nominal mass of @xmath90 the radius of a c / o wd is about @xmath91 , and it scales as @xmath92 ) . as inferred from
, the companion to would be far cooler than any known wd from other surveys ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) , where the coolest objects tend to have @xmath93k .
however , we can not exclude such a very cool wd on age grounds .
wd cooling curves , which start out having more massive objects warmer at the same age , eventually cross to have more massive objects cooler at the same age (; this is also visible in ) .
this is because massive wds crystallize earlier , at a higher @xmath94 ( but at a similar internal temperature ) , at which point the faster debye cooling takes over @xcite .
cooling ages for these models may not be reliable , as the impacts of state changes , sedimentation , and chemical processes are not precisely known , and the atmospheres are not trivial to calculate @xcite .
but we believe conservatively that the cooling age is close to 10gyr , almost certainly @xmath95gyr . in we
show example cooling curves , computed for thin and thick da atmospheres and c / o wds ( likely irradiation is a negligible perturbation to the wds surface temperature , given the measured spin - down luminosity of the pulsar ) . for the model closest to the best - fit mass of we would infer that the true age is near 9gyr , with the possible range from 612gyr .
the upper limit provided by the pulsar s characteristic spin - down age ( 34gyr after correction for the shklovskii effect [ @xcite ] ) is not constraining ; the assumption that the pulsar s initial spin period is much shorter than the current spin period is clearly not valid .
instead , we take as our upper limit to the age that of the milky way s halo ( @xmath96gyr ; @xcite ) minus the @xmath97myr required for the main - sequence lifetime of a @xmath98 progenitor @xcite , although this does not really exclude any models .
such an age would , however , imply a lower limit to the ( re-)birth period of about 25ms , assuming spin - down with a braking index @xmath99 ( magnetic dipole radiation ) .
we note that the cooling models in may not be the only solution for this progenitor : changing the wd composition ( likely it is below the transition to o / ne / mg wds based on @xcite , although binary evolution could change that ; also see @xcite ) or atmosphere ( helium , carbon , etc ) could lead to different solutions , and to draw robust conclusions we need to explore a wider range of models with better observational constraints .
there are also considerable complications and uncertainties in models for these temperatures : for instance , the models of ( * ? ? ?
* the basti database ) give rather different ages as @xmath94 never drops below @xmath100k for @xmath80 models , even for ages of @xmath101gyr , while @xcite and @xcite do have @xmath102 models go below 4000k ( note that the models in @xcite are primarily o / ne rather than c / o ) .
however , we believe the @xmath94 upper limits to be more robust , as they do tend to agree between different calculations . while extreme , the companion to may not be especially unique .
similar ultra - cool wds are presumably present in globular clusters and in the field even if they are often too faint to identify on their own .
individual ultra - cool wds can be identified but only if very nearby , like the two objects in @xcite at @xmath103pc . if we correct roughly for the different progenitor masses between the @xcite systems and @xcite and use a @xcite initial mass function , we would estimate @xmath104 massive wds of a similar age within 300pc , which is of the same order as the luminosity function from ( * ? ? ?
* also see @xcite ) extrapolated and @xcite say that at most a few percent of wds are lost off the faint end of the luminosity function . ] to @xmath105 .
instead , binary systems are the best way to identify cold wds ( e.g. , * ? ? ?
* ) , which is effectively the technique used here .
but even in binary systems where we know that a source is present , the systems will often be too distant for good constraints ( i.e. , psr j1454@xmath245846 in ; @xcite ) .
we still require a fortuitously nearby system for useful observations .
the occurrence of a nearby massive wd like the companion to is reasonably consistent with expectations based on the observed binary population : there are five pulsar binaries from the atnf pulsar catalog @xcite within 300pc , and the other four have low - mass he wd companions .
this @xmath106 ratio is similar to that for co wd compared to he wd companions in the whole atnf catalog ( also see @xcite ) , and the pulsars spin - down ages appear to have similar distributions for both companion types .
finally , we can ask whether an ns is the most likely companion to an ultra - cool wd .
most binaries are assumed to have mass ratios near one ( @xcite , but see @xcite ) , but a binary composed of two ultra - cool wds would be just as hard to detect optically as a single object .
if the companion were a lower - mass wd or a main - sequence star the binary could be visible , although it would require spectroscopic follow - up to identify the companion and in the absence of _ gaia _ this has not been done for the majority of stars within a few hundred pc .
so the situation of , with an ns companion , is reasonably plausible as the initial mass ratio would have been close to one and the chances of companion follow - up and identification after discovery of the pulsar are high .
we have determined an accurate mass for the partially recycled pulsar and its companion ; the latter is value consistent with both an ns and a wd . despite not finding the companion in a deep optical / near - infrared search
, we reject a dns explanation as the binary system shows evidence of circularization requiring mass transfer after the last supernova .
instead the companion is likely a high - mass wd . using the extremely precise distance determination from @xcite
, we can set a robust limit of @xmath107 .
this implies an very old and cool wd : fainter than all other pulsar companions by a factor of about 100 , and fainter than the lower - mass `` ultra - cool '' wd in the solar neighborhood by a factor of about four .
converting this limit to a temperature depends somewhat on the assumed mass and composition , but we believe an effective temperature limit of @xmath108k is a robust upper limit .
for such an object to not be older than the milky way requires that it have already entered the faster debye cooling regime , i.e. , that it already crystallized ( also see @xcite ) . future searches , if they can detect the companion to , will be a unique probe of the very late stages of wd evolution , with a well - determined mass and radius that are not usually available for studies of such objects .
we thank an anonymous referee for useful suggestions , and t. tauris , m. van kerkwijk , i. stairs , p. bergeron and r. oshaughnessy for helpful discussions .
is supported by the national science foundation grant ast-1312822 .
m.a.m . and d.r.l .
are supported by wvepscor , the nsf pire program , and the research corporation for scientific advancement .
jrb acknowledges support from wvepscor , the national radio astronomy observatory , the national science foundation ( ast 0907967 ) , and the smithsonian astrophysical observatory ( chandra proposal 12400736 ) .
a.t.d . was supported by an nwo veni fellowship .
some of the data presented herein were obtained at the w. m. keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california and the national aeronautics and space administration .
the observatory was made possible by the generous financial support of the w. m. keck foundation .
the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community .
we are most fortunate to have the opportunity to conduct observations from this mountain . based on observations obtained at the southern astrophysical research ( soar ) telescope , which is a joint project of the ministrio da cincia , tecnologia , e inovao ( mcti ) da repblica federativa do brasil , the u.s
. national optical astronomy observatory ( noao ) , the university of north carolina at chapel hill ( unc ) , and michigan state university ( msu ) . funding for sdss - iii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , and the u.s .
department of energy office of science .
the sdss - iii web site is http://www.sdss3.org/. we made extensive use of simbad , ads , and astropy ( http://www.astropy.org ; @xcite ) . ,
i. h. 2010 , in iau symp .
261 , relativity in fundamental astronomy : dynamics , reference frames , and data analysis , ed .
s. a. klioner , p. k. seidelmann , & m. h. soffel , ( cambridge : cambridge univ . press ) , 218
|
the recycled pulsar is one of the closest known neutron stars , with a parallax distance of @xmath0pc and an edge - on orbit .
we measure the shapiro delay in the system through pulsar timing with the green bank telescope , deriving a low pulsar mass ( @xmath1 ) and a high companion mass ( @xmath2 ) consistent with either a low - mass neutron star or a high - mass white dwarf .
we can largely reject the neutron star hypothesis on the basis of the system s extremely low eccentricity ( @xmath3)too low to have been the product of two supernovae under normal circumstances .
however , despite deep optical and near - infrared searches with soar and the keck telescopes we have not discovered the optical counterpart of the system .
this is consistent with the white dwarf hypothesis only if the effective temperature is @xmath4k , a limit that is robust to distance , mass , and atmosphere uncertainties .
this would make the companion to one of the coolest white dwarfs ever observed .
for the implied age to be consistent with the age of the milky way requires the white dwarf to have already crystallized and entered the faster debye - cooling regime .
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"a convex polyhedron @xmath0 is the intersection of half - spaces of the @xmath1-dimensional euclide(...TRUNCATED)
| "the probability content of a convex polyhedron with a multivariate normal distribution can be regar(...TRUNCATED)
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"a couple of decades ago it was found that radio emission is observed by about three times more ofte(...TRUNCATED)
| "three hundred fifty three radio sources from the nrao vla sky survey ( nvss ) ( condon et al .\n@xc(...TRUNCATED)
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"modern lattice qcd simulations are mostly based on direct evaluation of the path integral of the th(...TRUNCATED)
| "we propose a stochastic method for solving schwinger - dyson equations in large-@xmath0 quantum fie(...TRUNCATED)
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"the question of neutrino mass is one of the most profound in modern particle physics .\nmost plausi(...TRUNCATED)
| "we present a markov chain monte carlo global analysis of neutrino parameters using both cosmologica(...TRUNCATED)
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"recent discoveries of photoinduced metallic phases in several mott insulators@xcite made us conside(...TRUNCATED)
| "we performed optical - pump terahertz - probe measurements of a mott insulator ytio@xmath0 and a ba(...TRUNCATED)
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"traffic flow is one of the most interesting phenomena of many - body systems which may be controlle(...TRUNCATED)
| "in the optimal velocity model proposed as a new version of car following model , it has been found (...TRUNCATED)
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in Data Studio
ArXiv Summarization Dataset - 20K Preprocessed
A preprocessed dataset of 20,000 ArXiv papers with their full articles and abstracts, designed for abstract generation and summarization tasks.
Dataset Description
This dataset contains 20,000 ArXiv papers that have been filtered and preprocessed to ensure quality for training summarization models. Each example contains the full article text and its corresponding abstract.
Dataset Structure
The dataset has the following structure:
- article: The full text of the ArXiv paper
- abstract: The abstract/summary of the paper
Dataset Statistics
- Total Papers: 20,000
- Article Word Count:
- Mean: 5,875.84 words
- Median: 5,217 words
- Range: 2,000 - 14,998 words
- Abstract Word Count:
- Mean: 179.86 words
- Median: 166 words
- Range: 50 - 500 words
- Length Ratio (article/abstract):
- Mean: 36.00
- Median: 32.43
- Range: 5.01 - 99.98
Filtering Criteria
The dataset was filtered using the following criteria:
- Minimum article words: 2,000
- Maximum article words: 15,000
- Minimum abstract words: 50
- Maximum abstract words: 500
- Minimum length ratio (article/abstract): 5
- Maximum length ratio (article/abstract): 100
Usage
from datasets import load_dataset
# Load the dataset
dataset = load_dataset("yilmazzey/arxiv_summarization_20k_preprocessed")
# Access the data
print(dataset['train'][0])
# Output: {'article': '...', 'abstract': '...'}
Use Cases
This dataset is suitable for:
- Training abstract generation models
- Fine-tuning language models for summarization
- Research on long-form text summarization
- Evaluating summarization metrics (ROUGE, BLEU, etc.)
Citation
If you use this dataset, please cite:
@dataset{arxiv_summarization_20k_preprocessed,
title={ArXiv Summarization Dataset - 20K Preprocessed},
author={Yilmaz, Zeynep},
year={2024},
url={https://huggingface.co/datasets/yilmazzey/arxiv_summarization_20k_preprocessed}
}
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