re=me2Aof(lnT){1−√1−4Aof(lnT)merec}
dredT=ddT{me2Aof(lnT)}.{1−√1−4Aof(lnT)merec}+me2Aof(lnT).ddT{1−√1−4Aof(lnT)merec}
The gradient just before the kink at re=2rec is,
dredT=ddT{me2Aof(lnT)}
dredT=me2AoddT{1f(lnT)}
dredT=me2Aoddln(T){1f(lnT)}dln(T)dT
dredT=−me2Ao1f(lnT)2.f′(ln(T)).1T
dredT=−me2Ao1T1f(lnT)2.f′(ln(T))
From the post "Band Gap? Just A Kink",
Aof(lnT)=me4rec
f(lnT)2=me216A2or2ec
So,
dredT=−me2Ao1T.16A2or2ecme2f′(ln(T))
dredT=−8Aor2ecme1T.f′(ln(T))
dredT=−8Aor2ecme1T.df(ln(T))dTdTdln(T)
dredT=−8Aor2ecme.df(ln(T))dT
And for the case of a single electron orbiting about a single positive charge, rec the orbiting radius without considering drag is,
rec=q24πεomev2
and v2=2c2
dredT=−8Aome.q464π2ε2om2ec4.df(ln(T))dT
dredT=−Aoq48π2ε2om3ec4.df(ln(T))dT
For a nucleus of atomic number n,
dredT=−Aon2q48π2ε2om3ec4.df(ln(T))dT
and v2=2c2
dredT=−8Aome.q464π2ε2om2ec4.df(ln(T))dT
dredT=−Aoq48π2ε2om3ec4.df(ln(T))dT
For a nucleus of atomic number n,
dredT=−Aon2q48π2ε2om3ec4.df(ln(T))dT
The gradient just after the kink is, theoretically,
dredT→−∞
From the post "Gap, Gap, Gap and Quack",
d(re)dt=d(re)dTdTdx
where dTdx is the temperature profile around the nucleus. Thus, the velocity before the kink at re=2rec,
d(re)dt=−8Aor2ecme.df(ln(T))dTdTdx
the negative sign indicates that the electron is moving towards the nucleus, decreasing re. In general for a nucleus of n positive charge.
d(re)dt=−Aon2q48π2ε2om3ec4.df(ln(T))dTdTdx
And the velocity just after the kink is,
dredt→−∞
theoretically.
