Electron Screening effect on

the Big Bang Nucleosynthesis

by

Biao Wang

A Thesis

Presented for the

Master of Science Degree

CWID: 50047332

Department of Physics and Astronomy

TAMU-Commerce

March 2011

Commerce Texas

Abstract

We study the effects of electron screening on nuclear reaction rates occurring during the

Big Bang nucleosynthesis epoch. The sensitivity of the predicted elemental abundances on

electron screening is studied in details. It is shown that electron screening does not pro-

duce noticeable results in the abundances unless the traditional Debye-Huckel model for

the treatment of electron screening in stellar environments is enhanced by several orders of

magnitude. The present work rules out electron screening as a relevant ingredient to Big

Bang nucleosynthesis confirming a previous study by Itoh et al. [1] and ruling out exotic

possibilities for the treatment of screening, beyond the mean-field theoretical approach.

Keywords Electron screening effect, Big Bang Nucleosynthesis, Debye model, Early

Universe

ii

Acknowledgements

First of all, I will state that it is difficult to overstate my gratitude to my supervisor,

Prof. Carlos Bertulani who giving me countless insightful guidance and advices with great

supports patiently on all aspects of this work. Carlos is one of best advisors around world

I met so far, he treats his students as his best friends and share his brilliant knowledge

and scientific experience to us without any hesitate. Also, beside the research, he gave me

an example about how to educate and inspire the beginners in physics and how to form a

virtuous behavior and find the true meaning of their lives. His sincere encourage always

make us feel confident for the future career. It is really my great honor to work under his

instruction.

Also I would like to thank Dr. Li, Bao-An, Dr. William Newton and Dr. Kent Mont-

gomery, who have gave us excellent lectures and invited so many famous scientists and

engineers to make a presentation in commerce which really broaden our outlook of research

fields and industrial technology. I really appreciate all the efforts they made.

In addition, we thank Professor A. Baha Balantekin at University of Wisconsin-Madison

who gave us some amendments and ideas in our paper. Thanks are also given to Profes-

sor Michael Smith at Oak Ridge National laboratory, who held the EBSS 2010 and gave us

an detailed explanation on the numerical computational method during the synthesis process.

Last but not least, I would like to thank the classmates and friends in Commerce like

Taylor, John, Weikang Lin and so on. From physics to getting me settled in Commerce, they

iii

helped me a lot.

This work was partially supported by the U.S. Department of Energy grants DE-FG02-

08ER41533, DE-FC02-07ER41457 (UNEDF, SciDAC-2) and the U.S. National Science Foun-

dation Grant No. PHY-0855082.

iv

Contents

Abstract ii

Acknowledgements iii

List of Tables vii

List of Figures viii

1 A brief introduction to the early universe 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thermal history in the early universe . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Nuclear reaction rate 18

2.1 Nuclear reaction rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Electron screening effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Electron screening effects on Big Bang Nucleosynthesis 27

3.1 About the neutron decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Big Bang NucleoSynthesis with electron screening . . . . . . . . . . . . . . . 29

v

Bibliography 40

vi

List of Tables

1.1 Mass excess of light nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

vii

List of Figures

1.1 The thermal history of universe . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 degrees of freedom for the particles in early universe . . . . . . . . . . . . . . 10

1.3 The evolution of the effective degrees of freedom . . . . . . . . . . . . . . . . 11

1.4 Diagram of nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Relation between η and final abundance . . . . . . . . . . . . . . . . . . . . 16

2.1 Network of Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Differential cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Nuclear reaction patten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 All the possible nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Electron screening effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Number density of electrons (solid curve) and of baryons (dashed curve) during

the early universe as a function of the temperature in units of billion degrees

Kelvin, T9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Baryon density (solid curve) during the early universe as a function of the

temperature in units of billion degrees Kelvin, T9. The dashed curves are

obtained from Eq. (3.11) with h ∼ 2.1× 10−5 and h ∼ 5.7× 10−5, respectively. 34

viii

3.3 Debye radius during the BBN as a function of the temperature in units of

billion degrees Kelvin (solid line). The dotted lines are the approximation

given by Eq. (3.14) with R(0)D ∼ 6.1 × 10−5 cm and R

(0)D ∼ 3.7 × 10−5 cm,

respectively. The inter-ion distance is shown by the isolated dashed-line. . . 36

3.4 Variation of final abundances . . . . . . . . . . . . . . . . . . . . . . . . . . 38

ix

Chapter 1

A brief introduction to the early

universe

1.1 Introduction

Cosmologists are committed to explain the origin point and evolution of the entire matter

in the universe. By using physical processes, we may get a more accurate description of the

universe, and thereby a deeper understanding of the laws of physics due to the observational

evidence of those processes. Some people may be curious in tracing the beginning of the

universe and its history, then let’s break down in details.

According to general relativity, the scale of a clock or meter is related to its gravitational

potential and velocity. So if we want to make an accurate measurement in terms of time and

distance, we must exactly know the trajectories and velocities of the clock and meter. But

in quantum mechanics, we can not measure any trajectories exactly correct in principle. So,

then we can conclude that the time and distance can not be measured exactly correct, the

1

definitions of classic time and space can only be applied within a certain range.

The uncertainty principle tell us,

Et ∼ ~ (1.1)

then we get the energy accuracy of a clock by

E ∼ ~t

. (1.2)

Also, from this we know the mass accuracy of the clock

m =E

c2, (1.3)

then we get the uncertainty of the gravitational potential is

ϕ =Gm

x=

GE

c2x, (1.4)

where x is the geometric scale of clock.

So, the degree of accuracy in time measurement caused by red shift effect is

t

t=

ϕ

c2=

GE

c4x. (1.5)

This value should be no larger than 1, otherwise there is meaningless in the measurement.

So we get

t

t6 1 =⇒ x > GE

c4(1.6)

then fulfill x ∼ ct and E ∼ ~t

, we find

t >√

~Gc5

, (1.7)

2

x >√

~Gc3

. (1.8)

These two values are defined as Plank time and Plank length

tP =

√~Gc5

= 5.39124(27)× 10−44s, (1.9)

lP =

√~Gc3

= 1.616252(81)× 10−35m. (1.10)

The Plank time is the earliest time we have seen in the classic cosmology, the grand unifica-

tion epoch, electroweak epoch, inflationary epoch and so on all comes after that.

In the epoch from 0 to plank time, it is called quantum cosmology, which means the time

weren’t continuous any more and the gravitational fields were quantized. Some of the ap-

proach derived from super string theory has developed several models in which the quantum

gravitational effects became significant important and thus may be possible to address the

question of initial conditions in Big Bang Cosmology. But unfortunately, there is no suffi-

cient evidence to support any of those theoretical assumptions, and therefore I will skip this

discussion in this thesis.

For a homogeneous and isotropic evolving space and time assumed by the standard big bang

cosmological model, we can choose a system of spherical coordinates in which the proper

interval between two events is given by the Robertson-Walker metric

ds2 = gµνdxµdxν = dt2 −R2(t)[

dr2

1− kr2+ r2(dθ2 + sin2 θdϕ2)] (1.11)

where R(t) is the cosmic scale factor describing the expansion of the universe and k is the

three dimensional space curvature signature which is conventionally scaled to be -1, 0, +1

corresponding to an elliptic, Euclidean or hyperbolic space.

3

Assume a particle of mass m at radial distance r receding from the center of a mass distri-

bution of density ρ. The gravitational potential of the particle is

V = −GMm

r= −4

3πGmρr2 (1.12)

where M is the total mass of the volume with density ρ inside the radius r.

Since the universe expands linearly, the kinetic energy T of the particle with velocity v is

T =1

2mv2 =

1

2mH2

0r2. (1.13)

Thus the total energy is given by

E = T + V =1

2mH2

0r2 − 4

3πGmρr2 = mr2(

1

2H2

0 −4

3πGρ). (1.14)

If the mass density ρ is too large, the expansion will be stopped by the gravitational force,

and it will happen when

E = 0, (1.15)

then we call this density is critical density

ρc =3H2

0

8πG= 1.0539× 1010h2eV m−3. (1.16)

Also we know that ρ is time dependent, and we use ρc to express the present value of the

density. Then we define the dimensionless density parameter by

Ω0 =ρ0ρc

=8πGρ03H2

0

, (1.17)

the value of density parameter determines the type of the universe’s evolution.

From the Einstein field equations, we know that

R

R= −4

3πG(ρ+ 3P ) +

Λ

3, (1.18)

4

On the right side of the equal sign, the first part is about the gravitational effect, and the

second part represent the dark matter and dark energy.

The total energy of the universe is U = ρV , in adiabatic expansion, we have

−PdV = dU = ρdV + V dρ (1.19)

the we get

dρ = −(ρ+ P )dV

V(1.20)

which can be also written like

ρ = −(ρ+ P )V

V. (1.21)

The volume of three-dimensional space V ∝ R3 , so

ρ = −3(ρ+ P )R

R(1.22)

use this equation to fill in the Einstein field equations, we have

R = −4

3πG[3(ρ+ P )− 2ρ]R +

Λ

3R (1.23)

R =4πG

3

ρ

RR2 +

8πG

3ρR +

Λ

3R (1.24)

then times R on both sides,

RR =4πG

3ρR2 +

8πG

3ρRR +

Λ

3RR =

4πG

3

d

dt(ρR2) +

Λ

3RR (1.25)

and then integrate it, we get

1

2R2 =

4πG

3ρR2 +

Λ

6R2 − 1

2k (1.26)

which is

H2 ≡ (R

R)2 =

8πG

3ρ+

Λ

3− k

R2. (1.27)

5

This equation is known as Friedmann equation, it is the basic equation of Standard model.

Denoting their present values, one has

H20 =

8πG

3ρ0 +

Λ

3− k

R20

(1.28)

it can be written like

1 =8πG

3H20

ρ0 +Λ

3H20

− k

H20R

20

(1.29)

or

1 =ρ0ρc

+Λ

3H20

− k

H20R

20

(1.30)

Then we get an important relationship, which is

1 = Ωm + ΩΛ + Ωk. (1.31)

The density parameter is

Ωm ≡ Ω0 =ρ0ρc

=8πGρ03H2

0

, (1.32)

and the cosmological constant parameter is

ΩΛ =Λ

3H20

, (1.33)

last the curvature parameters of the universe is

Ωk = − k

H20R

20

(1.34)

The observed values of the cosmological parameters currently are

Ωm = 0.26± 0.02, (1.35)

ΩΛ = 0.74± 0.03, (1.36)

and

Ωk = 0. (1.37)

for a flat cosmology.

6

1.2 Thermal history in the early universe

We recommend that our universe started to expand from a state of extreme high pressure

and energy density with in quite a tiny volume. If we recall the historical discovery in

physics experiments, we may find that human beings are trying to study clearly about

what the constituents of the matter are. After people discerned that the materials are

made of molecules,it was aware that the molecules are constituted of atoms. Then with

the development of the experimental facilities, we recognized the nuclear force, the quark-

gluon plasma, or perhaps the insight of string theory in the near future. One of the most

important character in this process is the energy region of the matter, the nuclei decomposed

into quarks or even more elementary constituents under conditions resembling those stuff

like colliding particles in LHC. Thus to understand the cosmology in the early radiation era

requires that we must have an exhaustive comprehension during the high-temperature stage.

The general diagram of thermal history is shown in figure 1.1, and later we will discuss the

details period by period.

First of all, we will take a look at the brief review of the thermodynamics and statistical me-

chanics of those matter, which is in thermal equilibrium with negligible chemical potentials.

In the beginning stages, the reaction rates of all the existed particles in chemical equilibri-

um should be much larger than the Hubble expansion rate. Then in standard cosmological

model, we assume the total energy should not change and the pressure in the whole system

is constant. According to the first law of thermodynamics, we have

d[(ρc2 + p)V ] = V dp = 0 (1.38)

since the particles were considered as expanding non-viscous fluid at constant pressure, or

7

Figure 1.1: The thermal history of universe

in a simpler way

dE + pdV = 0 (1.39)

where

E = c2V ρ = c2V (ρm + ρr) (1.40)

the ρmand ρr is the density of matter and radiation.

Also we may rewrite it in term of entropy which is defined as

S =(ρc2 + p)V

kT(1.41)

8

Recall the second law of thermodynamics, the differential form should be

dS =1

kT[d(ρc2V ) + pdV ] =

1

kTV dp = 0 (1.42)

That’s why this process is also called an adiabatic and isentropic expansion.

For V ∝ R3(t), and then what we get from equation 1.40 is the following one

d(R3ρ)

dt+

p

c2d(R3)

dt= 0 (1.43)

It is because that the Big Bang Nucleosynthesis occurred on the radiation domination era,

then let us consider the case of ρr ≫ ρm. At this time, we have

pr =1

3ρrc

2 (1.44)

Then substitute into the first law of thermodynamics shown above, we get

d(ρrR3)

dt+

1

3ρrd(R3)

dt= 0 (1.45)

namely,

ρrR3 + 3ρrR

2R + ρrR2R = 0 =⇒ 1

R

d(ρrR4)

dt= 0 (1.46)

which shows that,

ρr(t) = ρr0[R(t0)

R(t)]4 (1.47)

At the same time, let us recall the Friedmann equation in the first section

R2

R2=

8πGρr3

(1.48)

Combining equation 1.47 and equation 1.48, we can find the relation between the scale factor

R and time t is R ∼ t1/2, and therefore the expansion rate of the universe should be

H =R

R=

1

2t(1.49)

9

On the other hand, if we define the effective degrees of freedom of the mixture as

g∗ =∑

gi(Ti

T)4 +

7

8

∑gj(

Tj

T)4 (1.50)

where gi represented the degrees of freedom for bosons and gj for fermions.The figure below

shows the properties of all the possible particles after big bang.

Figure 1.2: degrees of freedom for the particles in early universe

One more thing we have to mention is that the neutrinos should have 2 spin states in general

cases, but here the right handed neutrino was inert due to the electro weak symmetry break-

10

ing. And as you can see, while the temperature was cooling down, those particles began to

decouple from the thermal equilibrium since the the energy is not sufficient to produce them

any more. Therefore the effective degrees of freedom decrease as the following figure 1.3,

where the temperature is in units of MeV.

Figure 1.3: The evolution of the effective degrees of freedom

Now let us rewrite the energy density in terms of the effective degrees of freedom as following,

ρr =π2

30g∗T

4 (1.51)

Then combined equation 1.48 and equation 1.49 lead the final results of the relations between

time and temperature

t =

√3

16πGg∗aB

1

T 2+ constant (1.52)

11

where the aB is the radiation energy constant, also called Stefan’s constant, defined by∫ ∞

0

hνn(ν)dν = aBT4 (1.53)

or in a more intuitional representation

aB =8π5k4

15h3c3= 7.5657× 10−15erg · cm−3deg−4 (1.54)

And the Big Bang Nucleosynthesis would occur around 3 minutes after Plank time.

1.3 Big Bang Nucleosynthesis

Big Bang Nucleosynthesis is considered as one of the most striking evidences of the Stan-

dard Model of our universe. Our current belief in cosmology is that the nuclei which form

the majority of our lives today originate from nuclei created in a short time at the begin-

ning stages of the universe. During that epoch the temperature decreased from a few MeV

to about 10keV, when leptons, photons and baryons were the main ingredients of the plasma.

Before the synthesis started, the particles are in chemical equilibrium through the three

type of weak interaction shown below:

n+ ν p+ e− (1.55)

n+ e+ p+ ν (1.56)

n p+ e− + ν (1.57)

According to Fermi Golden rule, the weak interaction rate is equal to the number density of

electrons multiply the weak interaction cross section, denoted by

Γ = neσwk =G2

wk(kT )5

~(1.58)

12

And then compare to the expansion rate of our universe

H =1

2t=

√4πGg∗aB

3T 2 + smallconstant (1.59)

You may find that at a high temperature, the weak interaction rate is much larger than

the expansion rate. However, since the space is cooling down, at some point, when the

weak interaction rate just equal to the expansion rate, neutrinos began to decouple from the

chemical equilibrium, and that temperature is called freeze-out temperature.

Tf ≈ 0.7MeV (1.60)

While this time tf = 1.5s+ c At mean time, the neutrons and protons were in a Boltzmann

distribution, and therefore the number ratio of neutron to proton is

nn

np

= (mn

mp

)3/2e−(mn−mp)/Tf (1.61)

Also since there are so many photons in the space with enough high energy to break the

newly created Deuterium, thus the nucleons have to wait until the temperature is about

0.086MeV (about 180s) to make fusion synthesis. During this time, we also have to consider

the neutron decay, then write the ratio as

nn

np

= (mn

mp

)3/2e−(mn−mp)/Tf e−t/τn (1.62)

then substitute the value into this formula, we find the n/p ratio is going to be around 1/7.

Now let us have a brief review of the nuclear properties. The strong interaction force

are so intense that attract the nucleons together inside the nucleus tightly and form a stable

isotope. Therefore in order to break the nucleus into free nucleons, an sufficient amount of

13

energy has to be provided, which is notated by the binding energy B(A,Z), defined as

B(A,Z)

c2= Zmp + (A− Z)mn + Zme −ma(A,Z) (1.63)

where ma is the mass of neutral atom, which is easier to measure in terrestrial experiments.

Also we may rewrite the binding energy in form of mass excess ∆ for universal calculation.

B(A,Z)

c2= −0.000840Z + 0.008665A−∆ (1.64)

We give a table 1.1 of the mass excess of light nuclei we concerned in big bang nucleosynthesis.

From the table 1.1, it is clearly shown that the 42He has the largest binding energy per

nucleon and therefore it is the most stable nucleus during the synthesis process. Since the

initial number ratio of neutron to proton is about 1/7 and most of neutrons would get into

42He through fusion reaction in ideal case. Thus the predicted mass fraction in the end of

Big Bang Nucleosynthesis should be around 25 percents.

After the Deuterium was produced and could exist for a relative long time while the tem-

perature was cooling, they continued to make fusion with the other protons, neutrons or

themselves, and created the 32He, Tritium or other nuclei with gamma radiation in some re-

action channel. The emerging nuclei go forward to participate other nuclear reactions shown

in figure. However, the inverse reaction of those process can also occur, which means the

photons or baryons with enough high energy can also break the newly created nuclei, that

is the reason that only the tightest nuclei can remain in the final stage of the synthesis.

The third parameter of preliminary synthesis is the baryon to photon ratio η, which is

nonlinear proportional to the baryon mass density. The larger baryon density is, the more

acute nuclear reaction rate is. Or vice versa, the lower baryon density, the inert reaction

is. Then people may use different η and fix the other two parameters to testify the relation

with final abundance of each isotope, shown below.

14

Figure 1.4: Diagram of nucleosynthesis

From the figure 1.5, we find the some isotopes are sensitively dependant with the baryon

to photon ratio and that became one of the observational evidence to determine the approx-

imated value of η, which is also a keen interest parameter in other cosmological problems

such as the generation of baryon-antibaryon asymmetry and origin of dark matter, etc. Also

If we choose to use the average value(τn) of neutron lifetime from all the experimental data

around the world in history. Then a numerical approach of the final mass fraction of 4He,

Y4 = 0.228 + 0.012(Nv − 3) + 0.01 ln η10 (1.65)

where η10 ≡ η × 1010 is a dimensionless variable. This means for some isotope like Helium,

the number of neutrino flavors hold a more dominant position in the theoretical model of

synthesis, and this Nv from BBN is mutual controlled with the high energy experimental

result. And some other isotope like Deuterium are seemed more sensitive depend on the

baryon to photon ratio, which can become another effective probe during that era. These

evidence give us powerful constraints on the properties of the universe and the initial con-

ditions of stellar process and therefore also become the supporting evidence of gravitational

theory or equations of states concerned about pulsar or supernovae which stated to evolve

from the same abundance’s situation followed after the Big Bang Nucleosynthesis.

15

Figure 1.5: Relation between η and final abundance

16

Table 1.1: Mass excess of light nuclei

Isotope ∆/MeV

Neutron 8.071

11H 7.289

21H 13.136

31H 14.950

32He 14.931

42He 2.425

63Li 14.086

73Li 14.907

83Li 20.945

74Be 15.769

84Be 4.942

94Be 11.348

140Be 12.607

141Be 20.174

17

Chapter 2

Nuclear reaction rate

During the whole process in Big Bang nucleosynthesis, we have to consider all the possi-

ble nuclear reaction and set up a network started from neutrons and protons. After the

temperature is cold enough to maintain a sufficient abundance of the Deuterium, the Deu-

terium would continued to make nuclear reactions with other protons or neutrons. Then

the newly created nuclei would also participate into another nuclear reaction, see figure 2.1.

At the same time the inverse reaction can also exist but with different Q value, where

Q = (m1 +m2 −m3 −m4)c2 for most cases if we define the particle 1 is incident on a target

nucleus 2 and lead to product of type 3 and 4.

Now let us discuss the details about the reaction rate.

2.1 Nuclear reaction rate

When the nuclei collide each other, they may cause a nuclear reaction, the possibility of

those reaction can be represented by cross section σ(E). For example, suppose the Number

18

Figure 2.1: Network of Big Bang Nucleosynthesis

of incident particles is I particles per and the target has N nuclei per unit area, thus you

may find the created particle number is R,they have the following relations

R(E) = σ(E)IN (2.1)

But unfortunately, the cross section is very complicated if we want to derive it from the

fundamental theory. It may involve the Quantum chromo dynamics and hard to testify the

reality of those mechanism. So the convenient way is to measure the outgoing particles at

different solid angle, then get the differential cross section shown in figure 2.2

d(E, θ, ϕ) =dσ

dΩ(2.2)

19

I

polar angle

Figure 2.2: Differential cross section

then integrate it over all the polar angles θ and ϕ by numerical method

σ(E) =

∫d(E, θ, ϕ)dΩ (2.3)

Since the number density is relatively low compare to the photon number density which

means most of those reactions can be considered as 2 bodies nuclear reaction. In the pro-

cess of nucleosynthesis, the relative velocity v between two interacting nuclei followed the

Maxwell-Boltzmann distribution, which can also be represented by temperature. We define

the average value of total cross section multiply the relative velocity as the reaction rates in

astrophysics, which becomes

⟨σv⟩ = (8

πµ)1/2(

1

kT)3/2

∫ ∞

0

σ(E)Eexp(− E

kT)dE (2.4)

where µ is the reduced mass of two colliding nuclei and k is the Boltzmann constant.

This works so well in neutron nuclear interaction, but we have to consider the charged par-

ticles since they were the leading roles in most of the reactions. In classic viewpoint , the

fusion would not occur if the energy E in central mass is lower than the coulomb barrier.

20

However, G.Gamow has given the quantum tunneling solution to this problem in 1928 that

the cross section is proportional to the e−2πη by introducing an parameter called Gamow

energy EG even when the total energy is less than the coulomb barrier.

2πη = (EG

E)1/2 = 31.29Z1Z2(

µ

E)1/2 (2.5)

Pay attention, the η here is not the baryon to photon ratio, but the Sommerfeld parameter,

and in calculation we use numerical value where µ is in unit of amu and E is in keV.

Also according to the wave motion, the wave length of the particle is inverse proportional

to the energy E, with these in mind, we can rewrite the cross section in terms of energy,

Sommerfeld parameter, and S(E) factor which includes all the other possible variables.

σ(E) =1

Eexp(−2πη)S(E) (2.6)

Then if we substitute the S(E) factor into the formula of reaction rate, the reaction rate

can be expressed as

⟨σv⟩ = (8

πµ)1/2(

1

kT)3/2

∫ ∞

0

S(E)exp(− E

kT− E

1/2G

E1/2)dE (2.7)

The advantages of this way is that the S factor doesn’t change a lot with a varying energy,

that simplify our calculation in the discussion of electron screening effect. And since the Big

Bang Nucleosynthesis focus on the light nuclei, the resonant nuclear reaction can be safely

neglected.

Now consider the following nuclear reaction, we may chose the type B of the nuclei as target,

and type A as the projectiles, then followed the reaction A + B −→ C +D,see figure 2.3.

21

Figure 2.3: Nuclear reaction patten

The creation rate of nuclei C should be

dnC

dt= −dnA

dt= −dnB

dt= nAnB⟨σv⟩A,B (2.8)

where nA and nB are the number densities of those two kind of nuclei. But since the universe

is expanding all the time, which means the volume is increasing automatically during the

whole process. For convenience, we change the number density to relative abundance and

get the formula below while Xi is the mass fraction of nuclei i.

ni =ρbXiNA

Ai

= ρbNAYi (2.9)

Then we rewrite the creation rate in forms of abundance

dYC

dt= YAYBρbNA⟨σv⟩A,B (2.10)

which make the calculation much simpler, and from the equation above,it is shown obviously

about how the parameter η determined the baryon density and effect the final abundance of

the network. In real case they may have different reaction channel like C +D −→ M +N

or C −→ i+ e− + νe therefore for each type of nuclei, the differential abundance is expressed

as

dYC

dt= YAYBρbNA⟨σv⟩A,B − YCYDρbNA⟨σv⟩C,D − YCλ (2.11)

22

This is a simple sample of the computation, in real case, one has to consider the permutations

and combinations of possibility of the colliding chances and nuclear channel. Then the

numerical calculation method is to set a tiny time interval, then calculate the abundance

step by step,follow all the possible reaction channel shown in figure 2.4 From the derivation

Figure 2.4: All the possible nuclear reactions

above, It is obvious that the final abundance dependent on the initial neutron to proton

ratio and baryon density in case the reaction rate of each channel is correct. However, we

will discuss the electron screening effect at epoch of Big Bang Nuclesynthesis to assure the

23

reaction network in the next chapter.

2.2 Electron screening effects

In plasma, the kinetic energy is high enough to avoid the formation of bounding states of

electrons and the electric force will repel the ions each other and attract the electrons and

ions together. Thus the electrons and ions would follow the statistical distribution dependent

with the electric potential as follows

ne = ne0exp(−eΦ

Te

) (2.12)

while the Te represented the kinetic energy and Φ is the electric potential. Also in the similar

form, we get the ion number density distribution

ni = ni0exp(−eΦ

Ti

) (2.13)

Because of the charge neutrality, it is obvious that

ne0 = ni0 = n0 (2.14)

Then we substitute them into the Poisson’s equation given by

∇2Φ = 4π(ne − ni)− 4πQ (2.15)

In the case that eΦ ≪ Teand eΦ ≪ Ti, we may expand the exponents and rewrite the Laplace

operator since it is spherical symmetrical.

∇2Φ =1

r21

dr(r2

dΦ

dr) = 4πn0e

2(1

Te

+1

Ti

)Φ (2.16)

If we define the reduced temperature as

T =TeTi

Te + Ti

(2.17)

24

for the case of Te ≫ Ti, the reduced temperature is getting close to Ti.

Figure 2.5: Electron screening effect

Also we may introduce an important physical parameter here, called Debye radius or Debye

length, basically determined the intensity of the electron screening effect.

RD = (T

4πn0e2)12 (2.18)

And if we substitute the Debye Radius into the Poisson’s equation shown above, you may

25

find the solution of electric potential as

Φ =Q

rexp(− r

RD

) (2.19)

Now it is clearly that if an ion is surrounded by the electrons cloud, then the other ions

feel an quickly dropping electric potential outside Debye radius due to the screening effect

and thus cause a larger cross section, see figure 2.5. However since the cross section is quite

tough to be derived from theoretical model, then we may consider this problem in another

way.

We can choose one bare nucleus as the target, and let another nucleus be the projectile,

during the reaction process, it is reasonable to consider that the electrons around the target

are giving the projectile an acceleration. Consequently the projectile will collide the target

in a higher kinetic energy which leading to increase the cross section.

f =σs(E)

σb(E)=

σb(E + Ue)

σb(E)(2.20)

Where f is the enhancement factor and we will discuss whether the electron screening

may result a relative huge final abundance of each isotopes though the reaction network in

the following chapter.

26

Chapter 3

Electron screening effects on Big

Bang Nucleosynthesis

In this chapter, we will describe the effects of electron screening on the preliminary nucle-

osynthesis.

3.1 About the neutron decay

Before the Deuterium were produced, the neutron to proton ratio is given by

nn

np

= e−(mn−mp)/Tf e−t/τn (3.1)

where the Tf is the freeze-out temperature of the neutrinos and it is an inflexible value since

the number of the neutrino families is not changed. Therefore the neutron lifetime became

quite important for determining the initial neutron to proton ratio. Then first of all, we have

to discuss whether the electrons would block the neutron-decay to some extent.

27

At that period, the number of positrons and electrons is much larger than the baryon number.

however after the temperature fall down to lower than 1 MeV, the annihilation of positrons

and electrons are significant. Since the particle number are not conserved any more, each of

those electrons and positrons has a zero chemical potential. Then for the electric neutrality,

we just consider the rest electrons which have the same number density as positive charged

nuclei. At the temperature of T9 = 1.055, the baryon density is about ρb = 2.8E − 05g/cm3

and give out the rest electron number density is 2.4E − 5cm−3. Just substitute it into the

classical fermi energy, we get

EF =~2c2

2mec2(3π2ne)

2/3 = 2.16× 10−8MeV (3.2)

Now let us consider how does it effect the neutron decay. In 1996, Professor Subir Sarkar

has derived a relation between the coupling A and the neutron lifetime τn

1

τn=

A

5

√(m)2 −m2

e[1

6(m)4 − 3

4(m)2m2

e −2

3m4

e] +A

4m4

em cosh−1(m

me

) (3.3)

which leads to

1

τn= 0.0158A(m)5 (3.4)

while m is the mass difference between neutron and proton

m = mn −mp = 939.565MeV − 938.272MeV = 1.293MeV (3.5)

Keep the coupling A as a constant and compare the two values of m and EF , it is easy

to find that the electron blocking effect is so tiny and much smaller than the uncertainty of

global average value of the neutron lifetime measurement. Therefore it is not necessary to

add some modification caused by surrounded electrons on the neutron decay, we may just

use the experimental result of neutron lifetime.

28

3.2 Big Bang NucleoSynthesis with electron screening

During the Big Bang the Universe evolved very rapidly and only the lightest nuclides (e.g.,

D, 3He, 4He, and 7Li) could be synthesized. The abundances of these nuclides are probes of

the conditions of the Universe during the very early stages of its evolution, i.e., the first few

minutes. The conditions during the Big Bang nucleosynthesis (BBN) are believed to be well

described in terms of standard models of cosmology and particle physics which determine

the values of, e.g., temperature, nucleon density, expansion rate, neutrino content, neutrino-

antineutrino asymmetry, etc. Deviations from the BBN test the parameters of these models,

and constrain nonstandard physics or cosmology that may alter the conditions during BBN

[3, 2]. Sensitivity to the several parameters and physics input in the BBN model have been

investigated thoroughly in the past (see, e.g., [4, 5, 6, 7, 8, 9]). In this work we consider if the

screening by electrons would have any impact on the BBN predictions. This work reinforces

the conclusions presented in Ref. [1], namely, that screening is not a relevant ingredient of

the BBN nucleosynthesis.

Modeling the BBN and stellar evolution requires that one includes the information on

nuclear reaction rates ⟨σv⟩ in reaction network calculations, where σ is the nuclear fusion

cross sections and v is the relative velocity between the participant nuclides. Whereas v

is well described by a Maxwell-Boltzmann velocity distribution for a given temperature T ,

the cross section σ is taken from laboratory experiments on earth, some of which are not

as well known as desired [4, 5, 6, 7, 8]. The presence of atomic electrons in laboratory

experiments also influence the measured values of the cross sections in a rather unexpected

way (see, e.g. [10]). In the network calculations for the description of elemental synthesis in

the BBN or in stellar evolution one needs account for the differences of the “bare” nuclear

29

cross sections obtained in laboratory measurements, σb(E), to the corresponding quantities

in stellar interiors, σs(E). One of these corrections is due to stellar electron screening as

light nuclei in the stellar environments are almost completely ionized.

Using the Debye-Huckel model, Salpeter [11] showed that stellar electron screening en-

hances cross sections, reducing the Coulomb barrier that reacting ions must overcome, yield-

ing an enhancement factor

f(E) =σs(E)

σb(E). (3.6)

The Debye-Hueckel model used by Salpeter yields a screened Coulomb potential, valid when

⟨V ⟩ ≪ kT (weak screening),

V (r) =e2Zi

rexp

(− r

RD

), (3.7)

which depends on the ratio of the Coulomb potential at the Debye radius RD to the tem-

perature,

f = exp

(Z1 Z2 e

2

RDkT

)= exp

(0.188Z1 Z2 ζ ρ1/2 T

−3/26

), (3.8)

where

ζRD =

(kT

4πe2n

)1/2

, (3.9)

is the Debye radius, n is the ion number density, ρ is a dimensionless quantity measured in

units of g/cm3,

ζ =

[∑i

XiZ2

i

Ai

+ χ∑i

XiZi

Ai

]1/2

, (3.10)

where Xi is the mass fraction of nuclei of type i, and T6 is the dimensionless temperature in

units of 106 K. The factor χ corrects for the effects of electron degeneracy [11].

Corrections to the Salpeter formula are expected at some level. Nonadiabatic effects

have been suggested as one source, e.g., a high Gamow energy allows reacting nuclei having

30

velocities significantly higher than the typical ion velocity, so that the response of slower

plasma ions might be suppressed. Dynamic corrections were first discussed by [12] and later

studied by [13]. Subsequent work showed that Salpeter’s formula would be valid independent

of the Gamow energy due to the nearly precise thermodynamic equilibrium of the solar

plasma [14, 15, 16]. Later, a number of contradictions were pointed out in investigations

claiming larger corrections, and a field theoretic aproach was shown to lead to the expectation

of only small (∼ 4%) corrections to the standard formula, for solar conditions [17].

Controversies about the magnitude of the screening effect have not entirely died out

and continued in some works [19, 20, 18, 21]. These works are invariably based on molecular

dynamics simulations. Dynamic screening becomes important because the nuclei in a plasma

are much slower than the electrons and are not able to rearrange themselves as quickly around

faster moving ions. Since nuclear reactions require energies several times the average thermal

energy, the ions that are able to engage in nuclear reactions in the stars are such faster moving

ions, which therefore may not be accompanied by their full screening cloud. In fact, dynamic

effects are important when particles react with large relative velocities. It appears that pairs

of ions with greater relative velocities experience less screening than pairs of particles with

lower relative velocities [18]. The correction for dynamical screening can be approximated

by replacing RD in eq. (3.8) by a velocity dependent quantity RD(vp) = RD

√1 + µv2p/kT ,

where µ is the ion pair reduced mass and vp their relative velocity [22]. This effect is relevant

in stellar environments whenever nuclear reactions occur at energies that are greater than

the thermal energy.

Experimentalists have exploited surrogate environments to test our understanding of

plasma screening effects. For example, screening in d(d,p)t has been studied for gaseous

targets and for deuterated metals, insulators, and semiconductors [23, 24]. It is believed

31

that the quasi-free valence electrons in metals create a screening environment quite similar to

that found in stellar plasmas. The experiments in metals seem to have confirmed important

predictions of the Debye model, such as the temperature dependence Ue(T ) ∝ T−1/2 [24].

But there are still controversies on the validity of the experimental analysis and the use of

the Debye screening, or Salpeter formula, to describe the experiments [25, 26].

A good measure of the screening effect is given by the screening parameter given by

Γ = Z1Z2e2/⟨r⟩kT , where ⟨r⟩ = n−1/3. In the core of the sun densities are of the order

of ρ ∼ 150 g/cm3 with temperatures of T ∼ 1.5 × 107 K. For pp reactions in the sun, we

thus get Γ ∼ 1.06 which validates the weak screening approximation, but for p7Be reactions

one gets Γp7Be ∼ 1.5, which is one of the reasons to support modifications of the Salpeter

formula. Also, in the sun the number of ions within a sphere of radius RD (Debye sphere)

is of the order of N ∼ 4. As the Debye-Hueckel approximation is based on the mean field

approximation, i.e., for N = n(4πR3/3) ≫ 1, deviations from the Salpeter approximation

are justifiable.

Based on the discussion above, there is a possibility that the screening enhancement

factor, eq. (3.6), could appreciably differ from the Salpeter formula under several circum-

stances, leading to non-negligible changes in the BBN and stellar evolution predictions. It

is the purpose of this work to verify under what conditions would this statement be true.

In this work, the BBN abundances were calculated with a modified version of the standard

BBN code derived from Wagoner, Fowler, and Hoyle [3] and Kawano [27, 28] (for a public

code, see [29]).

The electron density during the early universe varies strongly with the temperature as

seen in figure 3.1, where T9 is the temperature in units of 109 K (T9). This can compared

with the electron number density at the center of the sun, nsune ∼ 1026/cm3. The figure

32

10 1 0.1

1020

1024

1028

1032

T (109K)

Num

ber d

ensi

ty (

cm-3)

Figure 3.1: Number density of electrons (solid curve) and of baryons (dashed curve) during

the early universe as a function of the temperature in units of billion degrees Kelvin, T9.

shows that, at typical temperatures T9 ∼ 0.1− 1 during the BBN the universe had electron

densities which are much larger that the electron density in the sun. However, in contrast to

the sun, the baryon density in the early universe is much smaller than the electron density.

The large electron density is due to the e+e− production by the abundant photons during

the BBN.

The baryonic density is best seen in figure 3.2. It varies as

ρb ≃ hT 39 , (3.11)

where h is the baryon density parameter [30]. It can be calculated by using Eq. (3.11) of

33

Ref. [30] and the baryon-to-photon ratio η = 6.19× 10−10 at the BBN epoch (from WMAP

data [31]). Around T9 ∼ 2 there is a change of the value of h from h ∼ 2.1 × 10−5 to

h ∼ 5.7× 10−5. Eq. (3.11) with the two values of h are shown as dashed lines in figure 3.2.

10 1 0.110-8

10-6

10-4

10-2

T (109K)

Bary

on d

ensi

ty (

g/cm

3 )

Figure 3.2: Baryon density (solid curve) during the early universe as a function of the

temperature in units of billion degrees Kelvin, T9. The dashed curves are obtained from Eq.

(3.11) with h ∼ 2.1× 10−5 and h ∼ 5.7× 10−5, respectively.

In Ref. [1] a theory was developed to show how the abundant e+e−-pairs during the BBN,

for temperatures below the neutrino decoupling (T9 ∼ 0.7 MeV), result in modifications in

the Salpeter formula. In fact, only a very small fraction of the electrons present in the

medium is needed to neutralize the charge of the protons. The major part of the electrons

are accompanied by the respective positrons created via γγ → e+e− so that the total charge

34

of the universe is zero. At the decoupling temperature the neutron density is only about 1/6

of the total baryon density. Thus, the ion charge density (proton density) at this epoch is

approximately equal to the baryon density. With this assumption the enhancement factor

in Eq. (3.8) becomes independent of the temperature,

fBBN = exp(4.49× 10−8ζZ1Z2

)∼ 1 + 4.49× 10−8ζZ1Z2 for T9 . 1, (3.12)

and

fBBN = exp(2.71× 10−8ζZ1Z2

)∼ 1 + 2.71× 10−8ζZ1Z2 for T9 & 2, (3.13)

which yield small screening corrections for all known nuclear reactions.

It is also worthwhile to calculate the Debye radius as a function of the temperature. This

is shown in figure 3.3 where we plot Eq. (3.9) with the ion density equal to the proton

density. The accompanying dashed lines correspond to the approximation of Eq. (3.11),

with h ∼ 2.1 × 10−5 and h ∼ 5.7 × 10−5. This leads to two straight lines in a logarithmic

plot of

RD = R(0)D T−1

9 , (3.14)

with R(0)D ∼ 6.1× 10−5 cm and R

(0)D ∼ 3.7× 10−5 cm, respectively. In figure 3.3 we also show

the inter-ion distance by the lower dashed line. It is clear that the number of ions inside the

Debye sphere is at least of the order 103, which would justify the mean field approximation

for the ions. In contrast to protons, electrons and positrons are mostly relativistic and their

chaotic motion will probably average out the effect of screening around the ions. But because

the number density of electrons is large, an appreciable fraction of them still carry velocities

35

comparable to those of the ions. The effects of such electrons on the modification of the

Debye-Hueckel scenario might be worthwhile to investigate theoretically.

10 1 0.1 0.01

10-7

10-6

10-5

10-4

10-3

T (109K)

r (c

m)

Figure 3.3: Debye radius during the BBN as a function of the temperature in units of billion

degrees Kelvin (solid line). The dotted lines are the approximation given by Eq. (3.14) with

R(0)D ∼ 6.1×10−5 cm and R

(0)D ∼ 3.7×10−5 cm, respectively. The inter-ion distance is shown

by the isolated dashed-line.

Based on the above arguments, the screening by electrons during the BBN is likely a

negligible effect. But one needs to verify arguments that sensitive quantities such as the

Li/H ratio predicted by BBN might be impacted. This ratio is very small but is one of the

major problems for the Big Bang predictions. In fact, there are discrepancies between the

BBN theory and observation for the lithium isotopes, 6Li and 7Li. These discrepancies are

36

substantiated by recent observations of metal-poor halo stars [32] and the high precision

measurement of the baryon-to-photon ratio η of the Universe by WMAP [31, 33, 34]. In

view of the relevance of this topic to BBN and its predictions, it is worthwhile to check the

influence of the electron screening on the elemental abundances.

The BBN is sensitive to certain parameters such as the baryon-to-photon ratio, number

of neutrino families, and the neutron decay lifetime. We use the values η = 6.19 × 10−10,

Nν = 3 and τn = 878.5 s for the baryon/photon ratio, number of neutrino families and

neutron lifetime, respectively. We have included all reactions of the standard BBN model

using the values of the cross sections as published in Ref. [4]. The reaction rates were

modified to include screening factors calculated with Eq. (3.12).

In order to test under which circumstances the screening by electrons would make an

appreciable impact on the BBN predictions, we have artificially modified the Slapeter formula

for fBBN by rescaling it with a fudge factor w, i.e.

f ′ = exp

(wZ1 Z2 e

2

RDkT

), or ln f ′ = w ln f, (3.15)

with w varying between 1 and 104. We quantify the dependence of BBN on the electron

screening by the quantity

∆ =Y ′ − Y

Y, (3.16)

where Y denotes the abundance Y of an element produced during the BBN. The primed

quantities Y ′ denote the abundances modified by the screening effect.

In figure 3.4 we show the Variation (in percent) of the abundances of several light nuclei

as a function of a multiplicative factor w artificially enhancing Salpeter’s formulation of

screening. w is defined in Eq. (3.15), with the values of ∆ (in percent) as a function of

the screening enhancement factor w in Eq. (3.12). For w ∼ 1 any of the abundances are

37

Figure 3.4: Variation of final abundances

modified by less than 10−5 in case that the screening factor is calculated according to Eq.

(3.12). If the w is increased the abundances of 6Li and 7Be increase, while those of D,

T, 3He and 7Li decrease as w increases. This shows that the elemental abundance ratios

have a different sensitivity to the electron screening. These calculations also make it evident

that the relative abundances in the BBN would be somewhat modified only if the electron

screening would be enormously enhanced (by a factor w larger than 104) compared to the

prediction of Salpeter’s model.

In conclusion, using a standard numerical computation of the BBN we have shown that

electron screening cannot be a source of measurable changes in the elemental abundance.

This is verified by artificially increasing the screening obtained by traditional models [11].

38

We back our numerical results with very simple and transparent estimates. This is also

substantiated by the mean-field calculations of screening due to the more abundant free

e+e− pairs published in Ref. [1]. They conclude that screening due to free pairs might yield

a 0.1% change on the BBN abundances. But even if mean field models for electron screening

were not reliable under certain conditions, which we have discussed thoroughly in the text,

it is extremely unlikely that electron screening might have any influence on the predictions

of the standard Big Bang nucleosynthesis.

39

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