Neutron Production Rates by Inverse-Beta Decay in Fully Ionized Plasmas

0.13 MB·

by  L. Maiani

Neutron Production Rates by Inverse-Beta
4 Decay in Fully Ionized Plasmas
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L. Maiani, A.D. Polosa and V. Riquer
r
a
M Dipartimento di Fisica and INFN, Sapienza Universita` di Roma,
Piazzale Aldo Moro 2, I-00185 Roma, Italy
6
2
March 27, 2014

]
h
t
-
l
c Abstract
u
n Recentlyweshowedthatthenucleartransmutationratesarelargely
[
overestimated in the Widom-Larsen theory of the so called ‘Low En-

2 ergyNuclearReactions’. Hereweshowthatunboundplasmaelectrons
v
are even less likely to initiate nuclear transmutations.
8
8
2
Introduction
5
.
1
Claims ofelectron-protonconversion into aneutron andaneutrino by inverse
0
4 beta decay inmetallic hydrides have recently beenraised [1, 2], inthecontext
1
of the so called Low Energy Nuclear Reactions (LENR). The condition for
:
v
the reaction to occur is a considerable mass renormalization of the electrons,
i
X to overcome the negative Q-value that, otherwise, would forbid the reaction
r to occur. Defining a dimensionless parameter, β, in terms of the electron
a
effective mass, m⋆ 1, one needs:
m⋆ m m
n p
β = − 2.8 (1)
m ≥ m ≈
a reference value, β = 20 was estimated in [1].
1Toavoidconfusion,weunderscorethatthemassrenormalizationin(1)hasnothingto
do with the velocity dependent relativistic mass. We consider extremely non-relativistic
electrons. The situation is closely analogous to muon capture in muonic atoms, in that
⋆
case m being replaced by the muon mass.
1
It is not clear at all if such spectacularly large values of β can be obtained
in metallic hydrides and under which conditions. Nonetheless, assuming a
given value of β, a calculation of the neutron rate can be obtained in a
straightforward fashion from known electroweak physics. A calculation along
these lines has been presented in Ref. [3] for the case of an electron bound
to a proton, superseding the order-of-magnitude estimate presented in [1].
Morerecently, theauthorsofRef. [2] have arguedthatnuclear transmuta-
tions should most likely be started by unbound plasma electrons. Assuming
a fully ionized plasma and completely unscreened electrons, they find a rate
which is enhanced, with respect to the value obtained for bound electrons,
by the so-called Sommerfeld factor, S (c = 1):
0
2πα
S = (2)
0
v
where α is the fine structure constant and v is the average thermal velocity
of the electrons defined by2:
1/2
3kT 3kT T 20
v = = β−1/2 = 3.6 10−4 (3)
th
m⋆ m · 5 103 0K β
r r (cid:20)(cid:18) · (cid:19)(cid:18) (cid:19)(cid:21)
withthenumerical valueincorrespondence toβ = 20andto thetemperature
T 5 103 0K, estimated in [2] as the temperature that can be reached
≈ ·
by hydride cathodes. However, the assumption of completely unscreened
electrons may be unrealistic. We consider here the situation in presence of
Debye screening, which, in a different context, has been recently analysed in
Ref. [4]. We find that at large densities, the plasma enhancement saturates
to a value determined by the Debye length, a :
D
a
D
S S = (4)
0 → a⋆
B
with:
1
a⋆ = = β−1a (5)
B αm⋆ B
and a the Bohr radius.
B
2We shallusethe numericalvalues: k =8.61710−5eV/0K, e2/~c=α=1/137.043and
set c=~c=1.
2
Debye Length
Static charges are screened in a plasma. The potential of the electric field of
a test charge at rest in a plasma is (in Gaussian units)
e
φ = e−r/aD (6)
r
where a is the Debye length defined by:
D
1 1 1
= + (7)
a2 a2 a2
D e i
The two lengths a are associated to electrons and ions respectively and are
e,i
given by [5]
1/2
kT
e
a = (8)
e 4πn e2
(cid:18) e (cid:19)
and:
1/2
kT
i
a = (9)
i 4πn (Ze)2
(cid:18) i (cid:19)
Thedifferenceintemperaturebetween electronsandionsisexpected tooccur
naturally because of the large difference of mass which impedes the exchange
of energy in electron-ion collisions. Here we will make the approximation
a = a , which leads to the numerical value:
D e
T 1020cm−3 1/2
a = 4.87˚A (10)
D × 50000K n
(cid:20)(cid:18) (cid:19)(cid:18) e (cid:19)(cid:21)
or a Debye mass m :
D
ℏ
m = = 404 eV (11)
D
a
D
We therefore get a Debye length of about nine atoms (compared to a =
B
0.5 ˚A) in correspondence to the reference temperature T 5 103 0K and a
reference density n = 1020 cm−3. When considering the≈n de·pendence, we
e
shall restrict to the range:
1014 cm−3 n 6 1023 cm−3 (12)
≤ ≤ ×
Values between 108 and 1014 cm−3 are typical of glow discharges and arcs
whereas a value of about 1022 cm−3 is the free electron density in Copper [6].
Around 2.5 1021 cm−3 the Debye length equals the Bohr radius3.
×
3Electroncaptureoccursspontaneouslyduringtheformationofneutronstars,whenthe
3
Critical Velocity
The Sommerfeld factors in a plasma, Eqs. (40) and (43), can be obtained
from an intuitive argument as follows (see the Appendix for a derivation
from the Schro¨dinger equation following [4]).
We consider a critical value of the velocity, defined as:
2πα
= αm⋆a (13)
D
v
crit
In this condition, the de Broglie wavelength of the particle, is equal to the
Debye length4:
2π
λ = = a (14)
m⋆v D
crit
For larger velocities, the wavelength is smaller and the particle probes a
region of space smaller than a , where it sees an essentially unscreened
D
Coulomb potential. In these conditions, we have to use S , Eq. (2).
0
For smaller velocities, as v 0, the wavelength gets larger than a .
D
→
The Sommerfeld factor saturates to the value on the r.h.s. of (13) since
the particle explores increasingly large portions of neutral plasma, and the
screened Sommerfeld factor in Eq. (4) has to be considered.
The critical velocity defined by (13) is:
20 n 50000K
v = 2.48 10−4 (15)
crit · β 1020cm−3 T
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)
We consider our electrons to be at v , Eq. (3). At the reference point, this
th
is larger than v , hence we should apply the unscreened result, S . With
crit 0
increasing density, however, v goes above v (at n 2 1020 cm−3) and
crit th
∼ ·
one should apply the screened result, S.
Transmutation Rates
Totranslatethepreviousdiscussion intotheexpectedratesfortransmutation
from electrons in a plasma, we first recall the rate for the transmutation from
Fermienergyofthe electronsincreasesabovethe thresholdvalue,due tothe gravitational
pressure. This occurs at electron densities &1031 cm−3.
4we use ~=1, so that h=2π.
4
bound electrons [3]:
2
1 g
Γ(e˜p nν ) = ψ(0) 2 (G m )2 1+3 A (β β )2;
e bound F e 0
→ | | × 2π g × −
" (cid:18) V (cid:19) #
β3
ψ(0) 2 =
| | πa3
B
Γ [β = 20] = 1.8 10−3 Hz (16)
bound
·
The total rate is obtained by multiplying the result Γ by the volume and
bound
by the ion density, which we take equal to the electron density, n, because of
global neutrality:
Rate = n V Γ (17)
bound bound
· ·
In the case of plasma electrons, screened and unscreened rates are ob-
tained by the substitution:
ψ(0) 2 n (S or S ) (18)
0
| | → ·
and the rate is proportional to n2:
Γ
bound
Rate = n V n (S or S ) (19)
plasma · · ψ(0) 2 · · 0
| |
S and S corresponding respectively to the screened Debye plasma and to
0
the unscreened Coulomb case.
For convenience, we normalize the rates in plasma to the rate in Eq. (17),
computed for β = 20, already a considerably large rate, although a factor of
300 smaller than claimed in [1], and see if we can get anywhere close to
∼
unity or higher.
The formulae are
Rate πa3 (β β )2
η (n,β) = Debye = n B − 0 S =
Debye
Rate [β = 20] β3 (20 β )2
bound 0
−
π(na3 )a (β β )2
= B D − 0 (20)
β2 a (20 β )2
B 0
−
and
Rate 2παπa3 (β β )2
η (n,β) = Coul = n B − 0 (21)
Coul Rate [β = 20] v β3 (20 β )2
bound 0
−
for the two cases.
5
0.01
Β=20 U nscreened
10-4
L
Β
, 10-6
n
H
R Screened
10-8
10-10
16 18 20 22 24
Log @nD
10
Figure 1: Ratios correspondingtothescreenedplasma(Sommerfeldfactor S)and
to the unscreened one (Sommerfeld factor S ), for the case β = 20. The previous
0
discussion indicates that we must use S for v v and S for v v . The
0 crit th crit th
≤ ≥
result is represented by the thick line.
In Fig. 1 we display the ratios corresponding to the screened plasma
(Sommerfeld factor S) and to the unscreened one (Sommerfeld factor S ),
0
for the case β = 20. The previous discussion indicates that we must use S
0
for v v and S for v v . The result is represented by the thick
crit th crit th
≤ ≥
line.
The rate for electron capture from plasma never goes anywhere close to
the capture rate for bound electrons derived in [3] for the same value of β,
let alone to the larger rate quoted in [1]. Our results are in line with the lack
of observation of neutrons in plasma discharge experiments recently reported
in [8].
Acknowledgements
We thank Giancarlo Ruocco and Massimo Testa for interesting discussions.
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APPENDIX: Sommerfeld factor for electrons in screened
and unscreened plasma
Let us consider an attractive screened potential in the plasma in the form:
α
V(r) = emDr (22)
−r
−
The radial Schro¨dinger equation for the two body (e –ion) wave-function,
χ(r), reads:
d2χ(r) v2
+2m⋆ m⋆ V(r) χ(r) = 0 (23)
d2r 2 −
(cid:18) (cid:19)
Changing r into the adimensional variable x:
1
r = a⋆ x = x (24)
B αm∗
we get:
v2 2
χ′′(x)+ + e−ǫx χ(x) = 0 (25)
α2 x
(cid:18) (cid:19)
In the limit of small or vanishing v we write the equation as:
χ′′(x)+k2(x)χ(x) = 0 (26)
in terms of an effective momentum:
2
k2(x) = e−ǫx (27)
x
and solve it by the WKB method, which gives:
χ(x) = A 1 e±iRxk(x′)dx′ (28)
k(x)
We can use the WKB approximatpion as long as
′
k (x)
1 (29)
k2(x) ≪
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
that is:
(cid:12) (cid:12)
eǫx/2
(1+ǫx) 1 (30)
2√2x ≪
7
At the value where the exponential bends, namely ǫx = 1, we have:
′
k (x) ǫ ǫ ǫ
= e1/2 = (31)
k2(x) 2 k(x = 1/ǫ) ≡ k
(cid:12) (cid:12)x=1/ǫ r eff
(cid:12) (cid:12)
(cid:12) (cid:12)
and the condition(cid:12) that(cid:12)this region is within the range of validity of WKB is
then:
v a⋆ a
eff = k ǫ = B = B (32)
eff
α ≫ a βa
D
with β defined as in Eq. (1).
For β = 20 and a from Eq. (10), we find:
D
αa
v > B v 3.9 10−5 (33)
eff WKB
βa ≡ ≈ ·
D
On the other hand, the smallest velocity we consider is the thermal velocity,
Eq. (3), which is safely within the region of validity of the WKB approxi-
mation. Note that v is simply proportional to the critical velocity v
WKB crit
defined in (13):
v
crit
v = (34)
WKB
2π
We are interested in the square modulus of the wavefunction at the origin
relativetoitsunperturbedvalue(transmutationistakingplaceattheorigin),
the ratio being the Sommerfeld enhancement:
2 2
R (x = 0) χ (0)
S ψ (0) 2 = k,ℓ=0 = k (35)
k k
∼ | | Ak Axk
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where we have used the fact th(cid:12)at Rkℓ(x) x(cid:12)ℓ as(cid:12)x (cid:12)0. The constant
∼ →
A depends on the normalization of the radial function at large distances 5.
Since R goes to a constant as x 0, we need that χ (x) 0 as x 0
k,ℓ=0 k
→ → →
or
′
χ (x) xχ (0) as x 0 (36)
k → k →
thus giving
′ 2
χ (0)
S k (37)
k
∼ Ak
(cid:12) (cid:12)
(cid:12) (cid:12)
5In the conventions of [7], A=2 (cid:12) (cid:12)
(cid:12) (cid:12)
8
Within the region of validity of the WKB approximation, k & ε, we have
χ(x) = A 1 e±iRxdx′k(x′) (38)
k(x)
where A is chosen to be the samepconstant which appears in (35). Therefore
2
′
S 1 e±iRxdx′k(x′) i 1 k (x) (39)
k
∼ (cid:12) k(x) ± − 2k2(x) (cid:12)
(cid:12) (cid:18) (cid:19)(cid:12)x=0
(cid:12) (cid:12)
the last term in pare(cid:12)npthesis being much smaller than on(cid:12)e. The maximum
(cid:12) (cid:12)
value attainable by S is at the border of the WKB approximation limit, i.e.
k
for k ǫ, Eq. (32)
∼
1 a a
S = = β (40)
∼ ǫ a⋆ a
B B
In the limit ǫ 0, the Schro¨dinger equation (23) is solved analytically.
′ ′ →
The in wavefunction in the continuous spectrum of the attractive Coulomb
field is given by:
ψ(+) = eπk/2Γ(1 i/k)eik·rF(i/k,1,ikr ikr) (41)
k − −
where F = F is the Kummer function (hypergeometric confluent). Here
1 1
k r corresponds to mv r, measured in units 1/m. Thus it is the adimen-
· ×
sional quantity v/α. The same would hold writing kr = (k/αm)(αmr).
In these respects k/αm k is dimensionless, k = v/α, and we under-
→
stand the factor eπk/2, or the term Γ = (1 i/k). The k = v/α appears in
−
the Schro¨dinger equation (23).
The action of the attractive Coulomb field on the motion of the particle
near the origin can be characterized by the ratio of the square modulus of
ψ(+)(0) to the square modulus of the wave function for free motion ψ (r) =
k
eik·r. Using that Γ∗(z) =Γ(z∗), F(i/k,1,0) = 1 and:
π
Γ(1+i/k)Γ(1 i/k) = (42)
− ksinh(π/k)
we get the result:
2 2π 2πα
S = S = ψ(+)(0) 2 = = (43)
0 | k | k(1 e−2π/k) ≈ k v
−
for small velocities [4, 2].
9
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