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pulsars are very stable natural clocks with observed steady pulses . however , many pulsars exhibit significant timing irregularities , i.e. , unpredicted arrival times of pulses . hobbs et al . ( 2010 , hereafter h2010 ) carried out so far the most extensive study of the long - term timing irregularities of 366 pulsars , and ruled out some timing noise models in terms of observational imperfections , random walks ( boynton 1972 ; alpar et al . 1986 ; cheng 1987a , 1987b ) , and planetary companions ( cordes & shannon 2008 ) . lyne et al . ( 2010 , hereafter l2010 ) found that timing behaviors often result from two different spin - down rates , and pulsars switch abruptly between these states , often quasi - periodically , leading to the observed spin - down patterns . the deviations of pulsars spin frequency @xmath0 from its long - term trend are correlated with changes in the pulse shapes ( see fig . 3 and 4 of l2010 ) , and therefore are magnetospheric in origin . by modeling the observed precise timing data of pulsars , we will show that the long - term linear change of @xmath1 is consistent with the timescale of hall drift , and the oscillatory structure of @xmath1 and the changes in pulse shapes can be produced by the hall waves in the crust of a neutron star ( ns ) . consequently the mechanism of magnetic field evolution and the origin of the timing noise of pulsars are better understood , which may improve the sensitivity of gravitational wave detections with pulsars . assuming the pure magnetic dipole radiation in vacuum as the braking mechanism for a pulsar s spin - down ( e.g. lorimer & kramer 2004 ) , we have @xmath2 in which @xmath3 is a constant , @xmath4 is the strength of the dipole magnetic field at the surface of the ns , @xmath5 , @xmath6 , and @xmath7 are the radius , moment of inertia , and angle of magnetic inclination , respectively . previously , we constructed a phenomenological model for @xmath4 with a long - term evolution modulated by short - term oscillations ( zhang & xie 2012a and 2012b , hereafter paper  i and paper  ii , respectively ) : @xmath8 where @xmath9 is the pulsar s age , and @xmath10 , @xmath11 , @xmath12 are the amplitude , phase and period of the @xmath13-th oscillating component , respectively . @xmath14 , in which @xmath15 is the field strength at age @xmath16 , and @xmath17 is the power law index . then from equation ( [ b evolution ] ) and taking only the dominating oscillating component , we obtained the analytic approximation for @xmath18 ( paper  i ) : @xmath19 where @xmath20 is the real age of the pulsar and @xmath21 for the dominating oscillating component . it has been found that equation  ( [ vddot ] ) describes adequately the statistical properties of the observed timing noises of radio pulsars ( paper  i ) . therefore for relatively young pulsars with @xmath22  yr , the first term in equation  ( [ vddot ] ) dominates and we should have @xmath23 if @xmath24 . considering that the characteristic ages ( @xmath25 ) of young pulsars are normally several times larger than @xmath26 , equation  ( [ vddot ] ) thus explains naturally the observed @xmath27 for most young pulsars with @xmath28 yr . similarly for much older pulsars , the second term in equation  ( [ vddot ] ) dominates , in agreement with the observational fact that the numbers of negative and positive @xmath18 are almost equal for the old pulsars in the sample of h2010 . the confirmation of equation  ( [ vddot ] ) with observations naturally suggests that the observed pulsars indeed have evolutionary links , and the equation therefore provides details about the dipole magnetic field evolution of a young pulsar into an old pulsar . however , equation  ( [ vddot ] ) has so far not been tested with the observed evolutions of individual pulsars . following the same approach in paper  i , we obtain @xmath29 where @xmath30 , @xmath31 describes the long - term monotonic variation of @xmath0 . therefore equation  ( [ vdot ] ) can be tested with the long - term monitoring observations of individual pulsars . we can obtain an estimate of @xmath32 by linearly fitting the long - term monotonic variation of reported @xmath0 . consequently we can also determine the time scale of the long - term magnetic field evolution of each pulsar ( see equation  ( 6 ) in paper  i ) @xmath33 for comparisons with theoretical models of neutron star magnetic field evolution . @xmath34 indicates magnetic field decrease and vice versa . the sample of l2010 provides the precise histories of @xmath1 for seventeen pulsars and thus may be applied to test equation  ( [ vdot ] ) . lyne et al . ( 2010 ) found that psr b1828@xmath3511 clearly shows a long - term evolution trend ( noticed in l2010 as a linear increase of its @xmath1 ) , i.e. @xmath36 . we show the comparison between the reported ( taken from l2010 ) and analytically calculated @xmath0 for the pulsar in figure [ fig:1 ] ; all the parameters are listed in table 1 . the one major difference is caused by the decrease of the oscillation periods of the reported data after @xmath37  days , which indicates that the oscillations are unlikely caused by precession of the neutron star . nevertheless , our model describes the general trend of the reported data of b1828 quite well .    by linearly fitting @xmath1 of all other pulsars , we find that most of them also exhibit long - term monotonic evolutions modulated by short - term quasi - periodical oscillations . however , three pulsars in the sample , namely , b1822 - 09 , b2035 + 36 and j2043 + 2740 , exhibit much more erratic short - term behaviors , which may significantly bias the fitting results ; here we exclude these three pulsars from further study . we conclude that our phenomenological model can describe adequately the spin - down evolutions of fourteen pulsars in the sample of seventeen pulsars of l2010 ; all the observed and derived parameters for the fourteen pulsars are listed in table 1 . we first investigate the alternative possibility that the long - term linear change of @xmath0 is caused by the change of the inclination angle of a ns , rather than the change of its magnetic field strength . observationally , the inclination angle change would also produce a long - term change on the width of pulses . for a circular beam , the change of its width can be given by the geometrical relation @xmath38^{1/2},\ ] ] where @xmath39 is the magnitude of the inclination angle change that can be obtained from @xmath40 for psr b1828@xmath3511 ( @xmath41 is the time span of the observations ) , @xmath42 is the impact angle , @xmath43 is the half angular width of the radiating cone ( gil et al . 1984 ) , @xmath44^{\frac{1}{2}},\ ] ] @xmath45 is the initial pulse profile width of b1828@xmath3511 ( l2010 ) . thus , @xmath46 only depends on @xmath47 and @xmath42 . there are two observational constraints for @xmath47 and @xmath42 for b1828@xmath3511 : ( a ) @xmath48 ms ( l2010 ) ; and ( b ) its position angle ( p.a . ) gives @xmath49 ( gould & lyne 1998 ) . we show the comparison of the two constraints on @xmath47 and @xmath42 in figure [ fig:3 ] . one can see that condition ( a ) requires @xmath50 , which is much smaller than that from condition ( b ) ; thus the possibility of inclination angle change is ruled out for b1828@xmath3511 . it is therefore reasonable to assume that the long - term linear changes of @xmath0 for other pulsars in the sample are also not caused by the changes of their inclination angles , because the patterns of their spin - down evolutions are very similar to psr b1828@xmath3511 . goldreich & reisenegger ( 1992 ) studied several avenues for magnetic field decay in isolated nss : ohmic decay , ambipolar diffusion , and hall drift . depending on the strength of the magnetic fields , each of these processes may dominate the evolution , and the ambipolar diffusion is only important for magnetars . ohmic decay occurs in both the fluid core and solid crust of a ns ( sang & chanmugam 1987 ; urpin et al . 1994 ; page et al . it is inversely proportional to the electric conductivity and independent of the strength of magnetic fields . if an electric current flows vertically in magnetic fields , the fields exerts a transverse force on the moving charge carriers . the resulting hall drift of the carriers can transport magnetic fields from the inner crust to the outer crust ( jones 1988 ) . the hall effect is non - dissipative and thus can not be a direct cause of magnetic field decay . however , it can enhance the rate of ohmic dissipation , since only electrons are mobile in the solid crust , and their hall angle is large . consequently the evolution of magnetic fields resembles that of vorticity , and then the fields undergo a turbulent cascade terminated by ohmic dissipation at small scales ( goldreich & reisenegger 1992 ; muslimov 1994 ; biskamp & m@xmath51ller 1999 ; urpin & shalybkov 1999 ; rheinhardt & geppert 2002 ; geppert & rheinhardt 2002 ; pons & geppert 2010 ; geppert et al . 2013 ) . cumming et al . ( 2004 , hereafter c2004 ) found that , in isolated nss with relatively pure crusts , the hall effect dominates over ohmic decay after a time @xmath52 , where @xmath53 is in units of @xmath54 g. this is consistent with the fact that @xmath55 is much shorter than the ohmic timescale @xmath56 yr . at lower densities of the crust , the degenerated electrons contribute to the pressure , and the hall timescale ( @xmath57 , where @xmath58 is the decay rate induced by hall effect ) is ( c2004 ) @xmath59 in which @xmath60 , @xmath61 is the number fraction of electrons , and @xmath62 is the local gravity , assumed constant . at densities greater than neutron drip , the neutrons dominate the pressure ( c2004 ) , @xmath63 where @xmath64 is the number fraction of neutrons , and @xmath65 is the factor that accounts for the interactions between neutrons .    in the top panel of figure [ fig:4 ] , it is shown that there is no significant correlation between @xmath66 and @xmath67 . in the second and third panels , we show that @xmath68 is consistent with @xmath67 for eight pulsars and @xmath69 is for the other six pulsars . here , the outer and inner crusts are defined by @xmath70 and @xmath71 , respectively , yielding the upper and lower limits for @xmath72 plotted in the second and third panels of figure [ fig:4 ] . this implies that the hall drift is responsible for their observed long - term evolutions in @xmath73 . in other words , the evolution of their crustal magnetic fields , rather than that in their cores , dominates the observed long term spin - down evolution of these young pulsars . assuming @xmath74 , we obtain an effective density @xmath75 , which may be useful to indicate the location of the majority of the magnetic field lines in the ns crusts . the values of @xmath75 are shown in the bottom panel of figure [ fig:4 ] . the hall drift can also pump energy from an internal strong toroidal field to the dipolar poloidal component on a timescale of @xmath76 ( pons et al . 2012 ; gourgouliatos & cumming 2014 ) , resulting in increased @xmath73 for a pulsar . therefore the hall drift can cause both long term decrease and increase of @xmath73 in a pulsar . figure 3 shows that all the fourteen pulsars are quite young ( with @xmath77  yr ) and only three of the four pulsars are observed with long term increase of @xmath73 , i.e. , @xmath78 . this is consistent with our previous result that the long - term decrease of magnetic fields are more frequently observed for young pulsars ( paper  i ) , suggesting that the hall drift tends to generate more magnetic field decay than increase . very recently , geppert et al . ( 2013 ) found that the hall drift can produce small - scale strong surface magnetic field anomalies ( spots ) due to the interaction of the  initially " dipolar field with the strong toroidal crustal component , which is fundamental for generating observable radio emission . besides , there is a large scale dipolar poloidal component which has impact on the braking behavior , and its evolution due to hall drift may be responsible for the long - term monotonic evolutions of @xmath1 .      the above discussed theoretical expectation that the hall drift can produce both increase and decrease of @xmath73 suggests that an oscillatory mode of @xmath73 in a ns crust may also be produced ; however , the hall timescales ( either @xmath79 or @xmath80 ) are too long for the observed periodicities in pulsar timing noises ( with @xmath81  yr as listed in table 1 ) . on the other hand , the diffusive motion of the magnetic fields can perturb the background magnetic fields ( here the dipole magnetic fields ) at the base of the ns crust . such perturbations propagate as circularly polarized hall waves " along the background field lines upward into the lower density regions in the crusts ( thompson & duncan 1996 ) . the hall waves can strain the crust with a wave period ( c2004 ) , @xmath82 where @xmath83 is the hall timescale at the base of the crust , and @xmath84 is the number of nodes over the crust . the elastic response of the crust to the hall wave can induce angular displacement @xmath85 , where @xmath86 is the fluid displacement , @xmath87 is the depth of the crust , and @xmath88 is the shear modulus . the maximum strain occurs at or above the turning point . using the wkb scaling , it is found that at the turning point ( c2004 ) , @xmath89 in which @xmath90 is the amplitude of the mode at the base of the crust . the density at the turning point can then be calculated from ( c2004 ) @xmath91    since most of the dipole field lines are collected at the polar regions of nss , the strains at the regions are especially strong . the elastic responses of the crusts to the waves induce changes of the areas of polar regions , producing both pulse shape changes and oscillations in @xmath1 . the quasi - periodical oscillations in @xmath1 and time residuals can be caused by the hall waves , i.e. , @xmath92 . the calculated parameters from the above equations are also listed in table 1 . the values of @xmath93 are moderate for the theoretically predicted density at the turning point . above the turning point , the perturbation on magnetic fields @xmath94 decreases to match the boundary condition @xmath95 . the variation in @xmath96 roughly correlates with @xmath97 as @xmath98 , where @xmath99 is the amplitude of the oscillations in @xmath100 .    from equation  ( [ vdot ] ) , we have @xmath101 . for b1828@xmath3511 , @xmath102 , which is roughly the same as @xmath103 ( listed in table 1 ) ; here @xmath104 is defined as the peak - to - peak amplitude of the oscillations . the small difference between @xmath105 and @xmath106 is due to the partial cancellation of the two harmonics with the same amplitude , but different phases . for the other thirteen pulsars , @xmath107 is not available and we thus assume @xmath108 , which is reasonable because only one significant periodicity has been observed for each of them . assuming @xmath109 , @xmath110 can be calculated , as listed in table 1 . finally we can constrain @xmath111 ( the last column in the two tables ) , which is consistent with @xmath112 usually assumed in theoretical calculations ( e.g. thompson & duncan 1996 ) . in this work , we applied the phenomenological model of pulsar timing noise we developed recently to the observed precise timing data of young radio pulsars , and analyzed the influence of the hall effects in ns crusts on the magnetic field evolutions . this is the first time that the two aspects of the hall effect , i.e. , hall drift and waves , are correlated with observational data for individual pulsars . we obtained the following conclusions :    1 . the @xmath1 evolutions for most of the pulsars in l2010 sample can be described adequately with our phenomenological model consisting a long - term monotonic change and short - term oscillations in @xmath0 ; 2 . the observed long - term monotonic changes in @xmath0 can be interpreted with the hall drifts in ns crusts ; 3 . the observed short - term oscillations in @xmath0 and in pulse shapes can be produced by the hall waves in ns crusts .    in a unified view , the long - term evolutions in @xmath1 can be understood as that they are caused by the hall waves with very low harmonics ( @xmath113 ) . actually , goldreich & reisenegger ( 1992 ) conjectured a turbulent hall cascade , transferring energy from large to small scales , with an energy spectrum @xmath114 ( where @xmath115 is the wavenumber ) . however , it is still unclear that why the high harmonics ( e.g. @xmath116 ) dominate in all pulsars we studied here . it is probably due to some selection biases that the observational time spans for all pulsars are only a few decades . indeed , hobbs et al . ( 2010 ) found that the structures seen in the timing noise vary with data span ; as more data are collected , more quasi - periodic features are observed . to ultimately address these questions , the power spectrum and the physical origin of the hall waves need to be investigated in detail in future works . as we have discussed above and shown previously , the long - term evolution of @xmath73 of young pulsars are quite similar to that of the ( young ) pulsars studied here . in addition , the huge spans of the reported braking indices of all pulsars can be naturally explained by short - term oscillations of their @xmath73 with characteristics also similar to that of these pulsars . therefore it is reasonable to assume that the hall effects we studied here are also responsible for the observed spin evolutions of essentially all pulsars , including millisecond radio pulsars . physically both a long - lived core based dipolar field and a crustal field should coexist and be superimposed for a young pulsar . however , it is likely that the  observed " features , with e.g. @xmath117  yr , are caused by the crustal field , whose evolution proceeds faster than that of the core field , due to hall effect and finite conductivity in its crust . as the pulsar ages , its crustal field is gradually dissipated and the core field begins to dominate its spin - down evolution . in this scenario we would expect much longer @xmath118 for millisecond pulsars . unfortunately , currently the data on the evolutions of @xmath1 of millisecond pulsars are not available with comparable details as for those normal radio pulsars studied here ( l2010 ) . this is because that millisecond pulsars have much weaker dipole magnetic fields , implying much smaller @xmath110 and @xmath119 , which is actually the basis for using them to detect gravitational waves . availability of higher quality timing data of millisecond pulsars from the current on - going pulsar timing array observations ( haasteren et al . 2011 ; demorest et al . 2013 ; manchester et al . 2013 ) may allow both the long - term monotonic evolutions and short oscillations to be identified for individual millisecond pulsars , which will improve the current gravitational - wave limits ( shannon et al . 2013 ) . we thank j. y. liao , j. xu and p. f. wang for useful discussions . snz acknowledges partial funding support by 973 program of china under grant 2009cb824800 , by the national natural science foundation of china under grant nos . 11133002 , 11373036 and 10725313 , and by the qianren start - up grant 292012312d1117210 . hobbs , g. , lyne , a. g. , & kramer , m. 2010 , , 402 , 1027 ( h2010 ) boynton , p. e. et al . 1972 , , 175 , 217 alpar , m. a. , nandkumar , r. & pines , d. 1986 , , 311 , 197 cheng , k. s. 1987 , , 321 , 799 ( 1987a ) . cheng , k. s. 1987 , , 321 , 805 ( 1987b ) . cordes , j. m. , & shannon , r. m. 2008 , , 682 , 1152 lyne , a. et al . 2010 , science , 329 , 408 ( l2010 ) lorimer , d. r. , & kramer , m. 2004 , handbook of pulsar astronomy , ( cambridge university press , cambridge ) zhang , s. n. , & xie , y. 2012 , , 757 , 153 ( paper  i ) zhang , s. n. , & xie , y. 2012 , , 761 , 102 ( paper  ii ) gil , j. , gronkowski , p. , & rudnicki , w. 1984 , a&a , 132 , 312 goldreich , p. , & reisenegger , a. 1992 , , 395 , 250 gould , d. m. & lyne , a. g. 1998 , , 301 , 235 gourgouliatos , k. n. & cumming , a. 2014 , mnras , 438 , 1618 sang , y. m. , & chanmugam , g. 1987 , , 323 , l61 urpin , v. a. , chanmugam , g. , & sang , y. m. 1994 , , 433 , 780 page , d. , prakash , n. , lattimer , j. m. , & steiner , a. w. 2000 , phys . lett . , 85 , 2048 jones , p.b . 1988 , , 233 , 875 muslimov , a. g. 1994 , , 267 , 523 biskamp , d. , & m@xmath51ller , w. c. 1999 , phys . , 82 , 2195 urpin , v. , & shalybkov , d. 1999 , , 304 , 451 rheinhardt , m. , & geppert , u. 2002 , phys . lett . , 88 , 101103 geppert , u. , & rheinhardt , m. 2002 , a & a , 392 , 1015 pons , j. a. & geppert , u. 2010 , a&a , 513 , l12 geppert , u. , gil , j. , & melikidze , g. 2013 , , 435 , 3262 cumming , a. , arras , p. , & zweibel , e. 2004 , , 609 , 999 ( c2004 ) pons , j. a. , vigano , d. , & geppert , u. 2012 , a&a , 547 , a9 thompson , c. , & duncan , r. c. 1996 , , 473 , 322 van haasteren , r. , et al . 2011 , , 414 , 3117 demorest , p. b. , et al . 2013 , , 762 , 94 manchester , r. n. , et al . 2013 , pasa , 30 , e017 shannon , r. m. , et al . 2013 , science , 342 , 334    .observed and derived parameters for the pulsars with long - term monotonic changes and short - term quasi - periodical oscillations in @xmath1 ; the former are taken from l2010 . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^ " , ]      _ upper panel _ : @xmath120 for psr b1828@xmath3511 during the past 20 years . the reported data ( taken from l2010 ) are represented by red stars ; and the solid black line is calculated from equation  ( [ vdot ] ) with the following parameters : @xmath121  days , @xmath122 , @xmath123 , @xmath124 . the dotted , solid , dot - dashed and dashed lines are linear fits for the major peak values of reported data and analytical calculation , the minor peak values of reported data and analytical calculation , respectively . _ lower panel _ : time differences between the peak positions of reported data and analytical calculation . the differences for major peak positions and minor peak positions are represented by triangles and diamonds , respectively.,width=12 ]       @xmath66 , @xmath126 , @xmath80 and @xmath75 versus @xmath67 . in all panels : the crosses and circles represent @xmath127 and @xmath128 , respectively . in the second and third panels : the crosses , circles , solid line and dotted line represent @xmath129 , @xmath130 , @xmath131 and @xmath132 , respectively . in bottom panel : the outer crust and inner crust mean the density values that are derived from the equations @xmath131 , and @xmath132 , respectively.,width=18 ]
the observed long - term spin - down evolution of isolated radio pulsars can not be explained by the standard magnetic dipole radiation with a constant braking torque . 
 however , how and why the torque varies still remains controversial , which is a major issue in understanding neutron stars . 
 many pulsars have been observed with significant long - term changes of their spin - down rates modulated by quasi - periodic oscillations . applying the phenomenological model of pulsar timing noise we developed recently to the observed precise pulsar timing data of isolated neutron stars , here we show that the observed long - term evolutions of their spin - down rates and quasi - periodic modulations can be explained by hall effects in their crusts . 
 therefore the evolution of their crustal magnetic fields , rather than that in their cores , dominates the observed long term spin - down evolution of these young pulsars . 
 understanding of the nature of pulsar timing noise not only reveals the interior physics of neutron stars , but also allows physical modeling of pulsar spin - down and thus improves the sensitivity of gravitational wave detections with pulsars .