Solving a Half-Century-Old Mystery: Why is There Carbon Dating?

Solving a Half-Century-Old Solving a Half-Century-Old Mystery:Mystery:

Why is There Carbon Dating?Why is There Carbon Dating?

R. MachleidtR. Machleidt

University of IdahoUniversity of Idaho

R. MachleidtR. MachleidtCarbon Dating Carbon Dating

UI Colloq. 26-Jan-2009UI Colloq. 26-Jan-2009 22

A plain paper …



and a lot of fuss …



OutlineOutline

• What is C-14 dating?What is C-14 dating?

• What’s the “mystery”?What’s the “mystery”?

• Beta-decay of a nucleusBeta-decay of a nucleus

• Calculating the transition C-14 Calculating the transition C-14 N-14 N-14

• Brown-Rho scaling of meson masses Brown-Rho scaling of meson masses and the resolution of the mysteryand the resolution of the mystery

• ConclusionsConclusions



Some facts about C-14 datingSome facts about C-14 dating• Discovered in 1949 by Willard Libby, Professor at U. Discovered in 1949 by Willard Libby, Professor at U.

Chicago; Chemistry Nobel Prize 1960.Chicago; Chemistry Nobel Prize 1960.• The carbon in the atmosphere (contained in carbon The carbon in the atmosphere (contained in carbon

dioxide) includes a small fraction of C-14 (1 part per dioxide) includes a small fraction of C-14 (1 part per trillion) which is created in the upper atmosphere by the trillion) which is created in the upper atmosphere by the nuclear reactionnuclear reaction

n + N-14 n + N-14 p + C-14 p + C-14 where the incident neutron results from cosmic ray where the incident neutron results from cosmic ray

interactions. interactions. • The half-life of C-14 is 5730 years.The half-life of C-14 is 5730 years.• Organisms, while alive, constantly take up atmospheric Organisms, while alive, constantly take up atmospheric

carbon dioxide through photosynthesis: carbon dioxide through photosynthesis: 6 CO6 CO2 2 + 6 H+ 6 H22O + Energy O + Energy C C66HH1212OO66(Glucose) + 6 O(Glucose) + 6 O22

and, thus, replenish C-14, keeping it at the level of the and, thus, replenish C-14, keeping it at the level of the atmosphere.atmosphere.

• As soon as the organism is dead, the ratio C-14/C-12 drops As soon as the organism is dead, the ratio C-14/C-12 drops because C-14 decays while C-12 is stable. From the ratio, because C-14 decays while C-12 is stable. From the ratio, the age of organic remains can be determined: the age of organic remains can be determined: “Carbon “Carbon dating”dating”

• Because of the long half-life of C-14, the method is good Because of the long half-life of C-14, the method is good for dating over archaeological times (up to about 60,000 for dating over archaeological times (up to about 60,000 years).years).



Caves of Lascaux, Southwestern France; about 15,000 B.C.



“Kennewick Man”, Kennewick, WA; about 9,000 years old



“Shroud of Turin” Is this the shroud in which Jesus Christ was wrapped after his crucifixion?

Radiocarbon dating: 1260-1390 after Christ



What’s the “mystery”?What’s the “mystery”?

10

11

15

1

13

4

16

5730 yea

20 sec.

20 min.

3 sec.

10 min

rs

7 sec.

C

C

C

C

N

N

Half-livesof allowedbeta-decaysof somelight nuclei



14C14N



How to calculate this?How to calculate this?

2

1/ 2

1/ 2

14

Fermi's Golden Rule:

Transition probability per unit time

2 1 ln2| |

with density of final states,

mean life and half-live.

| |

| parent nucleus ( )

| daughter nucleus (

if

if

Ht

t

H f H i

i C

f

14 ) + eN e

1/ 2 2max

max

( , ) | |

with known constant and

( , ) "Fermi integral"

which accounts for Coulomb

distorsion.

if

Kt

f Z E H

K

f Z E



The transition matrix element The transition matrix element

1

| ( ) |A

if V j A j j

j

H f G G i

Fermi Gamow-Teller(GT)

3

Allowed

Fermi transitions

0

0

1

0

J

T

T

3

Allowed

GT transitions

0,1

0,1

1

0

J

T

T

14 14(0 ,1) (1 ,0) 1, 1C N J T GT



14N ← 14C

Hif = GA ⟨f |r j j

+ |i

j =1

A

∑

= GA × MGT

Conclusion –We have to calculate:



⟨.... |r j j

+ |....

j =1

A

∑14N 14C

MGT

1/ 2 2 2max( , ) | |GTA

Kt

f Z E G M



⟨.... |r j j

+ |....

j =1

A

∑14N 14C

All depends on the structure of these two nuclei which, in turn, depends on the forces

between the nucleons (“nuclear forces”).

The bottom line is, The bottom line is, it all depends on nuclear it all depends on nuclear

forces; forces; so, what are they like?so, what are they like?



Carbon Dating Carbon Dating UI Colloq. 26-Jan-2009UI Colloq. 26-Jan-2009

Nuclear Forces - Lecture 2 CNS Summer School 2005

Summary:Most important parts of the nuclear force

Tensor force:Spin-orbit force:

Central force

VT (r)⋅S12.......with.......S12 3(

1 ⋅̂r)(

2 ⋅̂r) −

1 ⋅

2

VLS (r)LurgSr

… … and how are those forces made?and how are those forces made?Let’s understand this by analogyLet’s understand this by analogy

to the Coulomb forceto the Coulomb force





The analogy

Atom: Coulomb force Nucleus: Nuclear forces

p eγp n

(138) (600)

(770) ω (782)

γ

+ -

γ

γ

p nmesons


UI Colloq. 26-Jan-2009UI Colloq. 26-Jan-2009 2525Carbon Dating Carbon Dating

UI Colloq. 26-Jan-2009UI Colloq. 26-Jan-2009Nuclear Forces - Lecture 2

CNS Summer School 2005

Summary:Most important parts of the nuclear force

Short Inter-

mediate Long range

ω

ω

Tensor force:Spin-orbit force:

Central force

This was the picture in free space.This was the picture in free space.But what happens when two nucleons interact But what happens when two nucleons interact

inside a nucleus, surrounded by other inside a nucleus, surrounded by other nucleons?nucleons?



γ

p nmesons

γ

p n

γ

p nγ

p nlighter mesonsand nucleons

≅

≅

≅ with lighter masses

“Brown-Rho

Scaling”



Consequences of Brown-Rho Consequences of Brown-Rho scalingscaling

for the nuclear forcefor the nuclear force

• When the mesons change their mass in When the mesons change their mass in the nuclear medium (inside nuclei), then the nuclear medium (inside nuclei), then the force changes.the force changes.

• One important meson is the rho-meson One important meson is the rho-meson which is involved in the tensor force: how which is involved in the tensor force: how does its mass-change affect the tensor does its mass-change affect the tensor force?force?



Tensor forcescreated by rho and pi

The medium effect from BRS:tensor force gets weakerwith increasing density.



How does the weakeningof the tensor force affectthe Gamow-Teller matrixElement?

Back to the C-14 beta decay

1/ 2 2 2max( , ) | |GTA

Kt

f Z E G M



31( )S 3

1( )D

14N14C

Increasing density

Incre

asin

g d

en

sity



14C

14N



SummarySummary

• The C-14 The C-14 N-14 transition depends most N-14 transition depends most sensitively on the structure of the N-14 sensitively on the structure of the N-14 nucleus which, in turn, depends most nucleus which, in turn, depends most sensitively on the strength of the nuclear sensitively on the strength of the nuclear tensor force.tensor force.

• In the nuclear medium, the masses of the In the nuclear medium, the masses of the mesons that “make” nuclear forces change mesons that “make” nuclear forces change as compared to free space.as compared to free space.

• This “Brown-Rho scaling” of, particularly, the This “Brown-Rho scaling” of, particularly, the rho-meson mass weakens the tensor force rho-meson mass weakens the tensor force inside the nucleus.inside the nucleus.

• This change of the tensor force reduces the This change of the tensor force reduces the GT matrix element and, by that, increases the GT matrix element and, by that, increases the predicted half-life of C-14 to the experimental predicted half-life of C-14 to the experimental value.value.

The more general relevance of all The more general relevance of all thisthis

• This is just one example for the fact that, during the This is just one example for the fact that, during the past decade, nuclear theory has made great advances past decade, nuclear theory has made great advances and is now able to produce precise and reliable and is now able to produce precise and reliable predictions for even very complicated nuclear predictions for even very complicated nuclear structure problems.structure problems.

• Thus, in the future, one doesn’t have to do “dirty” and Thus, in the future, one doesn’t have to do “dirty” and expensive experiments, instead one can use the expensive experiments, instead one can use the predictions of a reliable theory.predictions of a reliable theory.

• This will have great spin-off, e.g., for the development This will have great spin-off, e.g., for the development of the “Fourth Generation of Nuclear Reactors”: of the “Fourth Generation of Nuclear Reactors”: Instead of building expensive and potentially Instead of building expensive and potentially dangerous prototypes, one can design the reactor “on dangerous prototypes, one can design the reactor “on the computer” by calculating all the reactions that the computer” by calculating all the reactions that take place inside the reactor. This is now in reach!take place inside the reactor. This is now in reach!





The End



The reasoning underlying Brown-Rho The reasoning underlying Brown-Rho scalingscaling

• At low energy, the chiral symmetry of QCD (in the u/d quark At low energy, the chiral symmetry of QCD (in the u/d quark sector) is spontaneously broken.sector) is spontaneously broken.

• One signature for this is the existence of a quark One signature for this is the existence of a quark condensate, which is density dependent and disappears at condensate, which is density dependent and disappears at sufficiently high density (and temperature).sufficiently high density (and temperature).

• By QCD sum rules, the masses of the low-lying hadrons By QCD sum rules, the masses of the low-lying hadrons (except the pion) are related to the quark condensate.(except the pion) are related to the quark condensate.

• Consequently, hadron masses may depend on the density of Consequently, hadron masses may depend on the density of the nuclear medium and decrease with increasing density.the nuclear medium and decrease with increasing density.

• For vector meson masses, the following simple density For vector meson masses, the following simple density dependence is assumeddependence is assumed

Solving a Half-Century-Old Mystery: Why is There Carbon Dating?

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