| Summary: | Young pulsars are thought to be highly magnetized neutron stars (NSs). The crustal magnetic field of a NS usually decays at different timescales in the forms of Hall drift and Ohmic dissipation. The magnetization parameter <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>ω</mi> <mi>B</mi> </msub> <mi>τ</mi> </mrow> </semantics> </math> </inline-formula> is defined as the ratio of the Ohmic timescale <inline-formula> <math display="inline"> <semantics> <msub> <mi>τ</mi> <mrow> <mi>O</mi> <mi>h</mi> <mi>m</mi> </mrow> </msub> </semantics> </math> </inline-formula> to the Hall drift timescale <inline-formula> <math display="inline"> <semantics> <msub> <mi>τ</mi> <mrow> <mi>H</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> </mrow> </msub> </semantics> </math> </inline-formula>. During the first several million years, the inner temperature of the newly born neutron star cools from <inline-formula> <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>9</mn> </msup> <mspace width="0.166667em"></mspace> </mrow> </semantics> </math> </inline-formula>K to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>8</mn> </msup> <mspace width="0.166667em"></mspace> </mrow> </semantics> </math> </inline-formula>K, and the crustal conductivity increases by three orders of magnitude. In this work, we adopt a unified equations of state for cold non-accreting neutron stars with the Hartree–Fock–Bogoliubov method, developed by Pearson et al. (2018), and choose two fiducial dipole magnetic fields of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>13</mn> </msup> <mspace width="0.166667em"></mspace> </mrow> </semantics> </math> </inline-formula>G and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>14</mn> </msup> <mspace width="0.166667em"></mspace> </mrow> </semantics> </math> </inline-formula>G, four different temperatures, T, and two different impurity concentration parameters, Q, and then calculate the conductivity of the inner crust of NSs and give a general expression of magnetization parameter for young pulsars: <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>ω</mi> <mi>B</mi> </msub> <mi>τ</mi> <mo>≃</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mn>50</mn> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mn>0</mn> </msub> <mrow> <mo>/</mo> <mo>(</mo> </mrow> <msup> <mn>10</mn> <mn>13</mn> </msup> <mspace width="0.166667em"></mspace> </mrow> </semantics> </math> </inline-formula>G) by using numerical simulations. It was found when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>≤</mo> <msup> <mn>10</mn> <mn>15</mn> </msup> </mrow> </semantics> </math> </inline-formula> G, due to the quantum effects, the conductivity increases slightly with the increase in the magnetic field, the enhanced magnetic field has a small effect on the matter in the low-density regions of the crust, and almost has no influence the matter in the high-density regions. Then, we apply the general expression of the magnetization parameter to the high braking-index pulsar PSR J1640-4631. By combining the observed arrival time parameters of PSR J1640-4631 with the magnetic induction equation, we estimated the initial rotation period <inline-formula> <math display="inline"> <semantics> <msub> <mi>P</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, the initial dipole magnetic field <inline-formula> <math display="inline"> <semantics> <msub> <mi>B</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, the Ohm dissipation timescale <inline-formula> <math display="inline"> <semantics> <msub> <mi>τ</mi> <mrow> <mi>O</mi> <mi>h</mi> <mi>m</mi> </mrow> </msub> </semantics> </math> </inline-formula> and Hall drift timescale <inline-formula> <math display="inline"> <semantics> <msub> <mi>τ</mi> <mrow> <mi>H</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> </mrow> </msub> </semantics> </math> </inline-formula>. We model the magnetic field evolution and the braking-index evolution of the pulsar and compare the results with its observations. It is expected that the results of this paper can be applied to more young pulsars.
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