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as experimental facilities supply fruitful data on nuclei far off the @xmath2-stability , it has been recognized that theoretical approaches to nuclear structure should be renewed in some respects . one of the key ingredients is wave - function asymptotics at large @xmath0 , which sometimes produces neutron ( proton ) halos in vicinity of the neutron ( proton ) drip line . it is also important to reinvestigate effective interactions , in connection to the magic numbers that are different from those near the @xmath2-stability line . we developed a method for the spherical mean - field calculations  @xcite in which the gaussian expansion method ( gem )  @xcite was applied . this method seems suitable to studying the shell structure from stable to drip - line nuclei , owing to its several advantages . a large number of nuclei have quadrupole deformation . the deformation plays a significant role in unstable nuclei as well . for instance , the deformation can be relevant to the disappearance of the @xmath3 magic number on the so - called ` island of inversion '  @xcite , whereas there remain arguments for individual nuclei ; _ e.g. _ spherical description has also been proposed for @xmath4 mg  @xcite . it is desired to implement calculations with deformation , reproducing the wave - function asymptotics and handling a wide variety of effective interactions simultaneously .    in this paper we propose a new method for deformed mean - field calculations by applying the gem . taking advantage of the flexibility in describing the radial degrees of freedom , we adopt a set of the spherical gem bases to represent deformed single - particle ( s.p . ) wave functions , with truncation for the orbital angular momentum @xmath5 . the new method is tested in the hartree - fock ( hf ) and the hartree - fock - bogolyubov ( hfb ) calculations for mg nuclei with the gogny interaction , and the results are compared with those in literatures . from the present calculation , an interesting feature is suggested for the drip - line nucleus @xmath1 mg . in this paper we assume the nuclear mean fields to be axially symmetric and to conserve the parity . the @xmath6 axis is taken to be the symmetry axis . the method can immediately be extended to general cases with no symmetry assumptions on the one - body fields , apart from an additional constraint on the center - of - mass ( c.m . ) position . we represent the s.p . wave functions by superposing the spherical gaussian bases , which have the following form : @xmath7^{(j)}_m\ , ; \nonumber\\ r_{\nu\ell j}(r ) & = & \mathcal{n}_{\nu\ell j}\,r^\ell\exp(-\nu r^2)\ , . \label{eq : basis } \end{aligned}\ ] ] here @xmath8 expresses the spherical harmonics and @xmath9 the spin wave function . we drop the isospin index without confusion . the range parameter of the gaussian basis @xmath10 is a complex number in general  @xcite ; @xmath11 ( @xmath12 ) . via the imaginary part oscillating behavior of the s.p . wave functions can be expressed efficiently  @xcite . formulae for calculating the one- and two - body matrix elements that are required in the hf and the hfb calculations , as well as the constant @xmath13 , are given in refs . wave functions under the axially deformed mean field are represented as @xmath14 where the subscript @xmath15 on the lhs stands for the parity . the sum of @xmath5 and @xmath16 on the rhs runs over all possible values satisfying @xmath17 , @xmath18 and @xmath19 , in principle .    in the gem we usually take @xmath10 s belonging to a geometric progression . in ref . @xcite , we found that a certain combination of the real- and complex - range gaussian bases is suitable for nuclear mean - field calculations . in all the following calculations , we take the basis - set of @xmath20 with @xmath21 and @xmath22 , irrespective of @xmath23 . namely , 12 bases are employed for each @xmath23 ; 6 bases have real @xmath10 and the other 6 have complex @xmath10 . this set is quite similar to set  c in ref . @xcite . an appropriate set of the gem bases is capable of describing wave functions with various size . this feature is desirable for self - consistent mean - field calculations with deformation , because in deformed nuclei the density distribution depends on the direction , and degree of the deformation is not known in advance . we here show adaptability of the gem with respect to size of nuclei , by presenting results of the spherical hartree - fock ( hf ) calculations . size of spherical nuclei , which is typically represented by the rms radii , depends on the mass number @xmath24 , apart from exotic structure such as neutron halos near the drip line . many mean - field calculations have been implemented by using the harmonic oscillator ( ho ) bases , particularly when the effective interaction has finite ranges . in the mean - field calculations with the ho bases , the length parameter of the bases @xmath25 depends on @xmath24 . for stable nuclei , @xmath26 is almost proportional to @xmath27 , as @xmath28 , although @xmath26 is often adjusted for individual nuclides so as to minimize their energy @xmath29 . in contrast , since the gem basis - set contains gaussians of various ranges , even a single set can describe many nuclei to good precision . binding energies and rms matter radii calculated with the gogny d1s interaction  @xcite are tabulated in table  [ tab : sph - hf ] , for the doubly - magic nuclei @xmath30o , @xmath31o , @xmath1ca , @xmath32ca , @xmath33zr and @xmath34pb . the values obtained from the gem basis - set of eq . ( [ eq : basis - param ] ) are compared with those from the @xmath24-dependent ho basis - set . the coulomb interaction between protons is handled exactly  @xcite , and the c.m . motion is fully removed from the effective hamiltonian before variation . the influence of the c.m . motion on the rms matter radii is treated in a similar manner  @xcite . in the calculations using the ho bases , the @xmath26 parameter of the bases is determined from @xmath35 , and all the bases up to @xmath36 are included , where @xmath37 is number of the oscillator quanta . because of the variational nature of the hf theory , the lower energy indicates the more reliable result for individual nuclei . in this regard , the @xmath24-independent set of the gem bases gives no worse , even slightly better results than the @xmath24-dependent ho basis - set except for @xmath30o . to show further the adaptability of the gem basis - set to nuclear size , the calculated density distributions of the six nuclei are illustrated in fig . [ fig : rho - sph ] .    .binding energies @xmath38 ( @xmath39 ) and rms matter radii @xmath40 ( @xmath41 ) of @xmath30o , @xmath31o , @xmath1ca , @xmath32ca , @xmath33zr and @xmath34pb in the hf calculations with the d1s interaction . results of the gem bases and the ho bases are compared . [ tab : sph - hf ] [ cols="^,^,>,>",options="header " , ]     we find that the present method gives lower energies in all of @xmath42 mg than the ho calculation in ref .  @xcite . if we take differences in computational procedure irrelevant to the basis - sets ( _ e.g. _ the treatment of the coulomb exchange term ) into consideration , it is fair to say that the present results are no worse than the ho ones in ref . @xcite from the variational viewpoint . we obtain lower energy in the present calculation than in ref . @xcite by greater than @xmath43 for the unstable nucleus @xmath44 mg , while by less than @xmath45 for @xmath46 mg . this suggests that the spatial extention is more important than the mixing of the @xmath47 components particularly in @xmath44 mg . notice that the ho basis - set in ref . @xcite contains up to @xmath48 . as we have pointed out , the gem can describe the wave - function asymptotics at large @xmath0 to reasonable precision . taking this advantage , we investigate density distribution of axially deformed drip - line nuclei , within the mean - field approximation . we here discuss in the hf framework . wave function @xmath49 in eq . ( [ eq : spwf ] ) can be decomposed by a sum of spherical wave functions , @xmath50 here @xmath51 with @xmath52 , which is defined by the @xmath53-projection on @xmath49 besides a constant factor . the nuclear force becomes negligible at sufficiently large @xmath0 . there the coordinate - represented hf hamiltonian for neutrons becomes approximately spherically symmetric and the components having different @xmath5 decouple to one another  @xcite . the spin - orbit coupling , which is an effect of the nuclear force , also becomes negligible , and thereby the @xmath54 ( and @xmath55 ) value is frozen . after each @xmath53 component propagates over a certain region of @xmath0 , @xmath49 will be dominated by the @xmath56 component of the lowest possible @xmath5 ( _ i.e. _ @xmath57 , with the sign fixed by the parity ) due to the centrifugal barrier  @xcite , although the degree of the dominance depends on characters of the deformed orbit . the asymptotic form of @xmath58 at large @xmath0 is @xmath59 with @xmath60 ( @xmath61 is the s.p . hence the neutron density distribution asymptotically behaves as @xmath62 where the subscript @xmath63 represents the highest occupied level , @xmath64 is determined by @xmath65 and @xmath66 , and @xmath67 stands for the occupation number on the level @xmath63 . we hereafter assume @xmath68 without loss of generality , owing to the reflection symmetry .    in connection to halos , the @xmath69 and @xmath70 cases are particularly interesting . for @xmath69 , which implicates @xmath71 and @xmath72 , eq . ( [ eq : asymp ] ) is reduced to @xmath73 giving isotropic density in the asymptotic region . hence halos formed by @xmath74 last nucleon are spherically symmetric , as long as the @xmath75 components can be neglected . for @xmath76 which presumes @xmath77 , there are two possibilities @xmath72 and @xmath78 . in the @xmath79 case , ( [ eq : asymp ] ) becomes @xmath80 this asymptotic component is obviously deformed , having vanishing contribution in the @xmath6 direction . in the @xmath72 case , we obtain @xmath81 \ , . \nonumber\\   \label{eq : asymp1b}\end{aligned}\ ] ] the asymptotic component of eq . ( [ eq : asymp1b ] ) can also be deformed , and its degree depends on @xmath82 and @xmath83 . thus , within the hf approximation , a deformed halo is expected in axially deformed drip - line nuclei with the last nucleon occupying a @xmath84 or @xmath85 orbit . connected to the nilsson model , the asymptotic quantum numbers have widely been used in description of well - deformed nuclei , comprising @xmath55 ( denoted by @xmath86 in the nilsson model ) . therefore @xmath54 becomes an approximate quantum number . if this is the case , one of @xmath87 is much greater than the other in eq . ( [ eq : asymp1b ] ) . then the asymptotic behavior of @xmath88 depends largely on direction in the @xmath89 case . in the extreme case where one of @xmath90 vanishes , the damping factor @xmath91 is missing in a certain direction ; if @xmath92 , @xmath93 damps according to the @xmath91 factor in the @xmath6 direction , but damps faster in the @xmath94 and @xmath95 directions the absence of the @xmath91 asymptotic component is significant only in the vicinity of the @xmath96 plane . hence a neck structure of a halo is expected for drip - line nuclei with @xmath89 that is dominated by the @xmath97 component . in contrast , the density in nuclei with @xmath98 or @xmath99 dominated by @xmath100 may have dips on the @xmath6 axis , forming a halo with an apple - like shape . in the mg isotopes , the neutron - drip line has been predicted to lie at @xmath101 in several mean - field calculations so far  @xcite . in a recent experiment @xmath1 mg has been confirmed to be bound  @xcite . we obtain the drip line at @xmath101 , as reported in ref . @xcite ; @xmath1 mg is bound while @xmath102 mg has higher energy than @xmath1 mg . in the present hf result @xmath1 mg has a prolate shape at its energy minimum , with @xmath103 , @xmath104 and @xmath105 . it is noted that this energy is appreciably lower than in the @xmath106 ho calculation  @xcite . this is probably connected to the spatial extension of the wave function shown below . the last two neutrons , which have @xmath107 , occupy a @xmath84 orbit that has large portion of the @xmath108 component by @xmath109 . moreover , we have @xmath110 for this orbit . applying the above argument , it is likely that @xmath88 of @xmath1 mg has a halo oriented to the @xmath6 direction , but this halo component is almost missing on the @xmath96 plane , leading to neck structure of the neutron halo .    in fig . [ fig : rho - mg40 ] , @xmath111 and @xmath112 @xmath113 of @xmath1 mg are depicted . although @xmath114 is elongated in comparison to @xmath115 to certain extent because of the prolate deformation , they damp with nearly equal slope at @xmath116 . we find halo structure in @xmath117 at @xmath118 with the asymptotic behavior that is consistent with @xmath119 to good approximation . it is remarked that the component with the same asymptotics is highly suppressed in the @xmath94 direction . mg in the @xmath94 ( dashed lines ) and the @xmath6 ( solid lines ) directions , calculated with the d1s interaction . [ fig : rho - mg40 ] ]    to observe the shape of @xmath1 mg at low density , equi - density lines on the @xmath120-plane are drawn in fig . [ fig : rhcnt - mg ] . for comparison , similar plots for @xmath31 mg and @xmath44 mg are given as well . all the three mg nuclei have prolate deformation in the present hf calculation . the equi - density lines distribute with almost equal intervals from @xmath121 to @xmath122 , indicating exponential decrease of @xmath123 . with the exception of @xmath93 in @xmath1 mg , the size of the intervals does not depend on direction . although the @xmath124 line in @xmath31 mg is slightly constricted in the @xmath94 direction , it could be influenced by deviation from the exponential asymptotics due to numerical errors . the contour plot of @xmath93 in @xmath1 mg is intriguing . the large intervals of the equi - density lines correspond to the slow damping of @xmath93 due to the small @xmath119 . however , the intervals are comparable to those in @xmath44 mg in the @xmath94 direction . we view the neck structure in the low @xmath93 region . ( @xmath125 ) on the @xmath120-plane for @xmath31 mg , @xmath44 mg and @xmath1 mg , calculated with the d1s interaction . positions of @xmath126 , @xmath121 , @xmath127 , @xmath128 , @xmath129 and @xmath122 are presented . [ fig : rhcnt - mg ] ]    because it mixes the @xmath97 and @xmath70 components , the pair correlation tends to restore the spherical symmetry in the density distribution in the asymptotic region . depending on degree of the mixing , the neck structure may survive or disappear . however , in the present hfb calculation with the d1s interaction , we do not find a superfluid solution having lower energy than the hf minimum in @xmath1 mg . thus the density distribution of @xmath1 mg shown in figs . [ fig : rho - mg40 ] and [ fig : rhcnt - mg ] is not altered . if the @xmath75 components remain sizable in the asymptotic region , deformed halos ( including neck structure ) with other @xmath130 may be present . in ref . @xcite , possibility of deformed halos depending on the direction was pointed out for @xmath131be and @xmath132c , based on the skyrme hfb calculations . those halos are formed by a @xmath133 orbit . there remain problems with respect to deformed halos : ( a ) how correlations beyond hfb ( including restoration of the rotational symmetry ) affect them , and ( b ) whether they are detectable in experiments . although both questions are very important , it is not easy to give satisfactory answers at this moment and we leave these problems to future studies . we here emphasize that such exotic structure of nuclear halos can be investigated only via numerical methods that are capable of describing the wave - function asymptotics appropriately . we extensively develop a new method of implementing the hartree - fock ( hf ) and the hartree - fock - bogolyubov ( hfb ) calculations of nuclei with deformation , applying the gaussian expansion method ( gem ) . owing to the adaptability in describing radial degrees of freedom , which is confirmed by the hf calculations from @xmath30o to @xmath34pb , we adopt the spherical gem bases . we argue how large @xmath5 should be taken into account , by comparing the numerical solutions obtained from the spherical gem bases with the analytic ones in the axially deformed harmonic oscillator . the present method maintains three notable advantages of the gem algorithm for the mean - field calculations : ( i ) we can efficiently describe the energy - dependent asymptotics of single - particle ( s.p . ) wave functions at large @xmath0 , ( ii ) we can handle various effective interactions , including those having non - locality , and ( iii ) a single - set of bases is applicable to wide mass range of nuclei and therefore is suitable to systematic calculations . the present method is applied to magnesium nuclei with the gogny force . we show that the present results are no worse than those in the literatures , from the variational viewpoint . compared with the conventional method using the harmonic oscillator bases , the present method is suitable particularly to nuclei far from the @xmath2 stability . for @xmath1 mg , we suggest neck structure of a neutron halo , which arises due to the asymptotic behavior depending on the direction . such a possibility can be argued only via methods describing the asymptotic form of the s.p . wave functions appropriately . + this work is financially supported in part as grant - in - aid for scientific research ( c ) , no .  19540262 , by japan society for the promotion of science . numerical calculations are performed on hitac sr11000 at institute of media and information technology , chiba university , and at information technology center , university of tokyo . 99 h. nakada and m. sato , nucl . a699 * ( 2002 ) 511 ; _ ibid . _ * a714 * ( 2003 ) 696 . h. nakada , nucl . phys . * a764 * ( 2006 ) 117 ; _ ibid . _ * a801 * ( 2008 ) 169 . m. kamimura , phys . rev . a * 38 * ( 1988 ) 621 . e. k. warburton , j. a. becker and b. a. brown , phys . c * 41 * ( 1990 ) 1147 . m. yamagami and n. van giai , phys . c * 69 * ( 2004 ) 034301 . e. hiyama , y. kino and m. kamimura , prog . * 51 * ( 2003 ) 223 . j. f. berger , m. girod and d. gogny , comp . * 63 * ( 1991 ) 365 . h. nakada , phys . c * 68 * ( 2003 ) 014316 . a. bohr and b. r. mottelson , _ nuclear structure _ vol .  1 ( benjamin , new york , 1969 ) , p.  220 . j. decharg and d. gogny , phys . c * 21 * ( 1980 ) 1568 . r. blmel and k. dietrich , nucl . * a471 * ( 1987 ) 453 . y. sugawa , m. kimura and h. horiuchi , prog . * 106 * ( 2001 ) 1129 . r. rodrguez - guzmn , j. l. egido and l. m. robledo , phys . b * 474 * ( 2000 ) 15 . t. misu , w. nazarewicz and s. berg , nucl . phys . * a614 * ( 1997 ) 44 . i. hamamoto , phys . c * 69 * ( 2004 ) 041306 . j. terasaki , h. flocard , p .- h . heenen and p. bonche , nucl . phys . * a621 * ( 1997 ) 706 . r. rodrguez - guzmn , j. l. egido and l. m. robledo , nucl . phys . * a709 * ( 2002 ) 201 . t. baumann _ et al . _ , nature * 449 * ( 2007 ) 1022 . j. c. pei , f. r. xu and p. d. stevenson , nucl . * a765 * ( 2006 ) 29 .
we extensively develop a method of implementing mean - field calculations for deformed nuclei , using the gaussian expansion method ( gem ) . 
 this gem algorithm has the following advantages : ( i ) it can efficiently describe the energy - dependent asymptotics of the wave functions at large @xmath0 , ( ii ) it is applicable to various effective interactions including those with finite ranges , and ( iii ) the basis parameters are insensitive to nuclide , thereby many nuclei in wide mass range can be handled by a single set of bases . 
 superposing the spherical gem bases with feasible truncation for the orbital angular momentum of the single - particle bases , we obtain deformed single - particle wave - functions to reasonable precision . 
 we apply the new algorithm to the hartree - fock and the hartree - fock - bogolyubov calculations of mg nuclei with the gogny interaction , by which neck structure of a deformed neutron halo is suggested for @xmath1 mg . 
 pacs numbers : 21.60.jz , 21.10.gv , 21.10.dr , 27.30.+t    keywords : mean - field calculation ; gaussian expansion method ; axially - symmetric deformation ; finite - range interaction ; deformed neutron halo