6.S Overcritical Continuum States
1SS
never exceeds 1. In other words, the highest possible charge for a point nucleus
allowed by QED is
1
(Zmax)point nucleus =  = 137 .
(6.147)
a
Here follows a preliminary consideration of the behaviour of the phase shi
6.2 One Bound State Diving Into One Continuum
v
123
Fig. 6.1. Bound and resonance states in
quantum mechanics
outside the potential well. In many cases it is useful to split up the potential V to
V = Yer+ V', so that Yer stabilizes the state at E 1 Its in
6.5 Overcritical Continuum States
157
and
_ A r lxi,
(x< 0)
,.0
u2(i)(r )
_ A moc2(E Yo) r Ixl+1 .
r+O
21xl +1
(6.157)
Without loss of generality we can restrict ourselves to x < 0 (e.g. the
solutions), where obviously
moc2(E
uf)(r)
Yo)
21xl +1
(6.15
152
6. Resonant States in Supercritical Fields
e 7112y
N=2V1lPlcos 11 e ia + r(2y+1) +sin11eia r(2y+1)
r(y+1+iy)
r(y+1+iy)
and
L1
r(2y+1)
.
= y In2pr+ arg [ eta+
T(y+1+iy)
.
+ tan 11 eta
(6.130)
I
T(2 Y +1)] .
r(y+1+iy)
(6.131)
As mentioned above,
6.5 Overcritical Continuum States
151
Writing now
a + = Neia+cosf/
a_
(6.123)
= Neiasinf/
conditions (6.122) can be rewritten in terms of the phases a+, a_ as
e
2ia
+
(y+iy)e irry
=  '_;:_
. m c2 '
X+lY_O_
E
(y iy) e +irry
(6.124)
. moc 2
X+lYE
He
5.2 Klein's Paradox and Hole Theory
117
Here Pis the penetration factor. Obviously a+ P= 1. For p = moc (Le. for electrons with a velocity amounting to about 80070 of the velocity of light) we obtain
with (5.5)
P=
2
112+1
:; 0.83.
This means that 83% of t
150
6. Resonant States in Supercritical Fields
(6.118)
as x + 0, but I cpl) Iremains limited and therefore integrable.
If CPl is determined, CP2 can be obtained either through (6.115) (as in Sect. 3.5)
or from the coupled system of linear differential eq
6.6 Some Useful Mathematical Relations
173
It is now convenient to introduce the phases d(E) and L1 (E) through
1
Vl
(6.219)
= IVE I e  L1 (E)
(6.220)
Let IJ'E and 'PE be defined by (6.211), but with the amplitudes replaced by the
choice (6.216,217), so
6.2 One Bound State Diving Into One Continuum
133
Fig. 6.5. The charge distribution r2 lIlt ",cfw_r) of K electrons for
heavy. and giant nuclei. The maximum of this distribution
(Bohr radius) shifts towards smaller radii as Z increases while
the tail of t
7. Quantum Electrodynamics of Weak Fields
Two ways of presenting ordinary quantum electrodynamics of weak fields exist.
The first, more formal in nature, starts with the quantization of wave fields; the
second, more easily dealt with, stems from Stuckelbe
178
7. Quantum Electrodynamics of Weak Fields
This means that the interaction is turned on adiabatically. The adiabatic
switching on prevents initial disturbances due to the switching. In the limit
t +  00 the exact wave W(x, t) approaches the impacting
142
6. Resonant States in Supercritical Fields
after utilizing (6.57,59) and the reality of fI(E) in (6.76) follows
L an*(E)an(E') + JdE" cfw_[
n
x cfw_[
P
E'E"
P
+ fI(E) o(E  E")] L V;"nan*(E)
EE"
n
+ fI(E') O(E'E")] L VE"mam(E') = o(EE').
m
(6.78)
156
6. Resonant States in Supercritical Fields
Inserting this expression into the phase shift formula (6.146), dividing numerator
and denominator by the common factor
cos(yln2pR) [U)(R) Imcfw_
.i y i Y2
X+ly(moc )IE
(6.150)
and applying the addition the
164
6. Resonant States in Supercritical Fields
Fig. 6.14. Density distribution of several
electronic shells imbedded in the negative
energy continuum (and the precritical K
shell for reference) obtained with (6.181)
[Mu 72c, 73b]
10
5
200
300 r [Iml
JdE '
6.5 Overcritical Continuum States
15 = LI 15'Og
IEI>moc 2
I
0;
159
(6.168)
and it is immediately evident that it indeed vanishes for p + O. So far the procedure for Coulomb scattering was illustrated for 11r potentials.
For extended nuclei, the phase s
6.3 Two and More Bound States Imbedded in One Continuum
139
Taking matrix elements we arrive at the system of linear equations
Enan+
L L1Enmam+ IdE' v"E,hE,(E) = Ean
(6.56a)
m
L VE'nan(E)+E' hE,(E) = EhE,(E) ,
(6.56b)
n
L an*(E)an(E') + JdE"
n
= (5(EE')
168
6. Resonant States in Supercriticai Fields
1 r e i z d _ 1 2 ' l'
Z    7rl + 1m
2ni cc Z
2ni
R+oo
 J 
J
+rr
qJ 
.
Re 1qJ 2ni
,
.
a
and
1 r e iz d 1' 2texp(icosqJR+RsinqJ)
  J   Z  1m J
.
2ni cc Z
R+O rr
Re 1qJ 2ni
(1)
Hence
I
signc; =
7.2 The S Matrix
177
This is the integral equation for the total Green's function a(x'lx) in terms of
the free Green's function ao(x'lx), which is supposed to be known. It can be
solved by iteration, starting with ao(x' Ix). One then immediately deduces f
172
6. Resonant States in Supercritical Fields
6.6.1 A Different Choice of Phases
The supercritical states are expanded in (sub)critical basis as
IPE(x)
= a(E)
+
m

J hE' (E) lifE' (x) dE'
(6.211)
.
00
Hence
= < IPE )
hE,(E) = <lifE' IIPE) .
a (E)
(6.21
7.3 Propagator for Electrons and Positrons
I
185
Fig. 7.6. Integration path r! leading to the StiickelbergFeynman
propagator. The singularities at
Po =  E and Po = + E are indicated
I
=+lrl Hrcfw_ =E
J
t'>t
dpo exp [  ipo(t'  f)] (
)
2
2
2
p+mo
2n
PO
154
6. Resonant States in Supercritical Fields
,m. (
'PI X
N[ if/
)
e
X+CO
X
r(2iy+1)
iyiy
+e (if/IlY)
r(iy+1+iy)
x+iy(moc 2 )/E
n2iy+1) ]
.
.
r(ly+1+1Y)
.
exp [ tny+ l(pr+ y In2pr)] .
(6.142)
Normalizing the total spinors (6.127) to a functions (6
6.3 Two and More Bound States Imbedded in One Continuum
143
d) Direct evaluation yields
l(E) <5(E'  E)
E an*(E) VJn VEmam(E') .
n,m
(6.84)
The total integral can now be added and inserted into (6.78) and regrouped:
Ean*(E)an(E')+ Ean*(E)cfw_P, [Fnm(E)+1(
140
6. Resonant States in Supercritical Fields
When offdiagonal terms (.1 Enm + F'nm) are small compared to diagonal ones, Em
are approximately given in firstorder perturbation theory by
(6.65)
This approximation certainly holds for the 1 s and higher n
116
5. The Klein Paradox
that (5.13, 17) are again solutions of the problem. However, then the kinetic
energy E  Yo becomes negative, i.e. this is a classically forbidden situation. The
group velocity, given according to (5.16, 1) by
Vgr
=
c2
EYo
_
(5.2
6.4 One Bound State Imbedded in Several Continua
147
in order to focus on the interaction of the continua with the discrete state. From
and x'ffi of (6.98) is dropped, i.e. we start
now on the superscript "critical" on
with the expansion
'PhE = ah(E) rpcr
138
6. Resonant States in Supercritical Fields
applied also to the case where only the bound 1 s state dives, but the influence of
a finite number of bound ns level is considered.
We start with a set of N discrete states
(6.47)
interacting with the contin
5.2 Klein's Paradox and Hole Theory
121
Since the holes created in region II are interpreted as positrons, one can
simply explain the Klein paradox in the following way: the incoming electrons
induce creation of electronpositron pairs at the potential ba
110
4. The Hole Theory
reversed state lfIn' (x', ('). We have stated earlier that the rules to interpret the
wave function
x'=x,
(4.60)
(' =  (
are unchanged, meaning that an observable formed bilinearly from lfIn'(X',(')
and lfIZ, (x', t') has to be int
E res 

6.5 Overcritical Continuum States
V(
mOc
2)2
2
+ const
2
R
t

const
R
165
(6.187)
.
Thus, for a given Z all bound states with Za> Ixl obtain an infinite binding
energy when R  t O. This behaviour demonstrates again that the physical situatio
128
6. Resonant States in Supercritical Fields
one concludes
Imcfw_k(E') =:0
if
Imcfw_E';:O
since Recfw_E' < 0, and therefore
pI VE'sin[k(E')r+o] dE'
EE'
_1 I VE,expcfw_i[k(E')r+OE'] dE'
4i
EE' +ie
=
_
1 J VE,expcfw_i[k(E')r+OE'] dE'
4i
EE' +ie
+_1 I
178
Physiology of Woody Plants
FIGURE 6.4. Simplified scheme of the
carbon biochemistry of growth and respiration.
Arrows indicate fluxes. G, growth; M, maintenance; W, wastage respiration. From Cannell,
M. G. R. and Thornley, J. H. M. Modelling the
compo
134
Physiology of Woody Plants
FIGURE 5.26. Response of photosynthesis to photosynthetic photon flux density (PPFD) in leaves of rain
forest tree seedlings grown in full sunlight (!) and 6% of full sunlight ("): (A) Hymenaea parvifolia, (B)
Hymenaea courb
152
Physiology of Woody Plants
FIGURE 5.45. Effects of high (!) and low (") relative humidity
on leaf water potential (), rate of transpiration (TR), stomatal
diffusive resistance (r1), wateruse efficiency (WUE), and rate of
photosynthesis (Pn) of cacao
172
Physiology of Woody Plants
Isozymes
Enzymes in woody plants exist in several molecular
forms that act on the same substrate. These are known
as isozymes or allozymes. Isozymes may form by
various mechanisms. They may arise through the
binding of a sin
130
Physiology of Woody Plants
system capable of coping with high evaporative
demand. Low rates of photosynthesis during the
winter were associated with low minimum temperatures and a root system unable to cope with any significant evaporative demand.
Win