radiocarbon dating is a chronological method which uses radioisotope carbon-14 ( c ) to investigate the ages of carbonaceous samples [ 1 , 2 ] .
c atom forms in the nuclear reaction:\documentclass[12pt]{minimal } 
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				 \begin{document}$$ n + { } _ { 7}^{14 } { \text{n}}^{1 + } \to { } _ { 6}^{14 } { \text{c } } + p $ $ \end{document}where p is the proton , n is the atom of nitrogen , n is the neutron resulting from the cosmic ray
. then c spreads throughout the atmosphere and forms carbon dioxide ( co2 ) when it reacts with oxygen .
when sampled , the carbonaceous materials become clocks due to the decaying c in them .
since the half - life of c is moderate , i.e. 5568    30   or 5730    40  years , grosse and libby   at the university of chicago developed the radiocarbon dating method . to achieve high accuracy calendar dates by this method
, scientists had advanced several calibration methods , including tree rings , annually laminated sediments , corals [ 7 , 8 ] , intcal98 , intcal04 , and intcal09 .
the calculations of such calibrations are based on the conventional formula , which was given by stuiver and polach in 1977 . in this formula , the correction of the isotopic fractionation plays a significant role .
the correction is that the difference in c , normalized to  the postulated mean value , i.e. 25  per mil with respect to pdb , of terrestrial wood , simply indicates a double change of that in c within the carbonaceous matters .
however , neither might this quantitative relation always be true for specific processes such as chemical processes in which the factor is 1.9 [ 11 , 12 ] , nor does the conventional formula itself include the effect of temperature , at which the sample formed , on the fractionation correction . the following section tries to relate the equilibrium isotopic fractionation correction , depending on the temperature , to the radiocarbon dating method . and
the derivations in present paper are based on the postulate : samples are well preserved without chemical exchange after they formed .
does not use libby s first assumption because published works shows the variability of c in the atmosphere [ 1117 ] .
this variability is due to the varying intensities of cosmic radiation with time , and the release of c and c into the atmosphere by , for example , burning of fossil ( known as suess effect [ 1921 ] ) and explosion of nuclear bombs , respectively .
as mentioned above , co2 in the atmosphere is the significant medium to preserve c into the carbonaceous matter , xcn , where x stands for the stoichiometry of all atoms except carbon ( c ) in the sample , and n for the number of c in it .
this process involves an isotope reaction [ 23 , 24]:1\documentclass[12pt]{minimal } 
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				 \begin{document}$$ { } ^{14}{\text{co}}_{2 } + \frac{1}{n}x{}^{12}{\text{c}}_{n } \overset k \rightleftharpoons { } ^{12}{\text{co}}_{2 } + \frac{1}{n}x{}^{14}{\text{c}}_{n } $ $ \end{document}where k is the equilibrium constant , depending at most on the temperature t .
incorporation of c into the carbonaceous matter xcn , assuming no further exchange , starts the radiocarbon clock .
the number of c in xcn at t  =  0 , \documentclass[12pt]{minimal } 
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				 \begin{document}$$ n_{{x{\text{c}}_{n } , 0}}^{{{}^{14}{\text{c } } } } $ $ \end{document } , and that at present t  =  t , \documentclass[12pt]{minimal } 
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				 \begin{document}$$ n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } $ $ \end{document } , follow an exponential decay:2\documentclass[12pt]{minimal } 
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				 \begin{document}$$ n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } = n_{{x{\text{c}}_{n } , 0}}^{{{}^{14}{\text{c } } } } \begin{array}{*{20}c } { { \text{e}}^ { - \lambda t } } \\
\end{array } $ $ \end{document}where 1/  =  8033  year ( libby value ) is the mean life of c. now let \documentclass[12pt]{minimal } 
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				 \begin{document}$$ n_{{x{\text{c}}_{n}}}^{{{}^{12}{\text{c } } } } $ $ \end{document } denote the number of c in xcn .
division of eq .  2 by \documentclass[12pt]{minimal } 
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				 \begin{document}$$ n_{{x{\text{c}}_{n}}}^{{{}^{12}{\text{c } } } } $ $ \end{document } gives3\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \frac{{n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } } } { { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } = \frac{{n_{{x{\text{c}}_{n } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } { \text{e}}^ { - \lambda t } $ $ \end{document}since \documentclass[12pt]{minimal } 
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				 \begin{document}$$ n_{{x{\text{c}}_{n } } } ^{{^{12 } { \text{c } } } } $ $ \end{document } was not altered after the sample formed . in this equation ,
the left side and the first term of the right side represent the ratio of c / c in the sample at t  =  t and t  =  0 , respectively .
the \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { { n_{{x{\text{c}}_{n } , 0}}^{{{}^{14}{\text{c } } } } } \mathord{\left/ { \vphantom { { n_{{x{\text{c}}_{n } , 0}}^{{{}^{14}{\text{c } } } } } { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } } \right .
\kern-\nulldelimiterspace } { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } $ $ \end{document } ratio in eq . 
3 can be evaluated by studying the equilibrium constant k of reaction  1 at t  =  0 . for reaction  1 , the molality - scale equilibrium constant is 4\documentclass[12pt]{minimal } 
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				 \begin{document}$$ k = \frac{{\left [ { x{}^{14}{\text{c}}_{n } } \right]^{\frac{1}{n } } \left [ { { } ^{12}{\text{co}}_{2 } } \right]}}{{\left [ { x{}^{12}{\text{c}}_{n } } \right]^{\frac{1}{n } } \left [ { { } ^{14}{\text{co}}_{2 } } \right ] } } = \frac{{\left ( { \frac{{\left [ { x{}^{14}{\text{c}}_{n } } \right]}}{{\left [ { x{}^{12}{\text{c}}_{n } } \right ] } } } \right)^{\frac{1}{n } } } } { { \frac{{\left [ { { } ^{14}{\text{co}}_{2 } } \right]}}{{\left [ { { } ^{12}{\text{co}}_{2 } } \right ] } } } } = \frac{{\frac{{n_{{xc_{n } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } } } { { \frac{{n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } } } $ $ \end{document}with the last term expressing the concentration by the number of c and c in those two chemicals : \documentclass[12pt]{minimal } 
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				 \begin{document}$$ n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } $ $ \end{document } and \documentclass[12pt]{minimal } 
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				 \begin{document}$$ n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } $ $ \end{document } are the numbers of c and c in co2 in the atmosphere at t  =  0 , respectively .
this term is the c / c ratio in xcn divided by that in co2 , and it is just the expression of the fractionation factor ,  , defined by experimental chemists , giving k  =  . for theoretical chemists ,  is the ratio of reduced partition function ratios ( rpfrs ) between those two substances [ 23 , 24 ] . since the factor  is unique for a specific process , we have5\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \frac{{\frac{{n_{{x{\text{c}}_{n } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } } } { { \frac{{n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } } } = \alpha = \frac{{{\text{rpfr}}_{{x{}^{14/12}{\text{c}}_{n } } } } } { { { \text{rpfr}}_{{{}^{14/12}{\text{co}}_{2 } } } } } $ $ \end{document}or6\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \frac{{n_{{x{\text{c}}_{n } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } = \frac{{{\text{rpfr}}_{{x{}^{14/12}{\text{c}}_{n } } } } } { { { \text{rpfr}}_{{{}^{14/12}{\text{co}}_{2 } } } } } \frac{{n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } $ $ \end{document}where7\documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text{rpfr } } = \left ( { \frac{{\sigma^{{{}^{12}{\text{c } } } } } } { { \sigma^{{{}^{14}{\text{c } } } } } } \prod\limits_{k}^{3n - 6(5 ) } { \frac{{u_{k}^{{{}^{14}{\text{c } } } } } } { { u_{k}^{{{}^{12}{\text{c } } } } } } } \frac{{\exp \left ( { - u_{k}^{{{}^{14}{\text{c } } } } /2 } \right)}}{{\exp \left ( { - u_{k}^{{{}^{12}{\text{c } } } } /2 } \right)}}\frac{{1 - \exp \left ( { - u_{k}^{{{}^{12}{\text{c } } } } } \right)}}{{1 - \exp \left ( { - u_{k}^{{{}^{14}{\text{c } } } } } \right ) } } } \right)^{{{1 \mathord{\left/ { \vphantom { 1 n } } \right .
\kern-\nulldelimiterspace } n } } } $ $ \end{document}in which  is the symmetry number , u  =  hc/kbt , h is the planck s constant , c is the speed of light ,  is the frequency in cm , kb is the boltzmann constant , t is the temperature .
substitution of eq .  6 into eq .  3 gives8\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \frac{{n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } } } { { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } { \text{e}}^{\lambda t } = \frac{{{\text{rpfr}}_{{x{}^{14/12}{\text{c}}_{n } } } } } { { { \text{rpfr}}_{{{}^{14/12}{\text{co}}_{2 } } } } } \frac{{n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } $ $ \end{document}use of the natural logarithm to eq . 
8 leads to9\documentclass[12pt]{minimal } 
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				 \begin{document}$$ t = \frac{1}{\lambda } \left ( { \ln \left ( { { \text{rpfr}}_{{x{}^{14/12}{\text{c}}_{n } } } } \right ) - \ln \left ( { { \text{rpfr}}_{{{}^{14/12}{\text{co}}_{2 } } } } \right ) - \ln \left ( { \frac{{n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } } } { { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } } \right ) + \ln \left ( { \frac{{n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } } \right ) } \right ) $ $ \end{document } this equation shows that the time t the clock recorded is a function of  , t , \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { { n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } } \mathord{\left/ { \vphantom { { n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } } { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } } \right .
\kern-\nulldelimiterspace } { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } $ $ \end{document } , \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { { n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } \mathord{\left/ { \vphantom { { n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } } \right .
\kern-\nulldelimiterspace } { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } $ $ \end{document } and other physical constants .
the first two terms in the right side of above equation is called the isotopic fractionation correction to the radiocarbon dating method . for a specific sample , the accurate value of  and \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { { n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } } \mathord{\left/ { \vphantom { { n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } } { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } } \right .
\kern-\nulldelimiterspace } { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } $ $ \end{document } in eq . 
9 can be directly evaluated by ab initio quantum calculations [ 23 , 24 , 2631 ] and accelerator mass spectrometry ( ams ) technique [ 3242 ] , respectively .
the methods to evaluate t and \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { { n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } \mathord{\left/ { \vphantom { { n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } } \right .
\kern-\nulldelimiterspace } { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } $ $ \end{document } are illustrated below .
9 is the unique temperature at which the sample formed and recorded . according to eq .  5
,  is a function of temperature , and this allow us to find the inverse function of  to evaluate t : that is , if   =  f ( t ) , then we have t  =  f( ) which gives t if  is known .
if  is , however , given by the theoretical calculations , t locates in an interval of temperatures at which xcn can form [ 2629 ] rather than at a unique temperature the specific sample formed .
so we have to experimentally measure the isotope ratios kept in the sample , and then get the unique  and , therefore , the unique t. a good example of such case is to determine t by studying the clumped - isotopes in calcite [ 43 , 44 ] .
the equilibrium constant k3866 ( or 47 , see definitions and details of experiments in refs . [
43 , 44 ] ) of clumped - isotopes in calcite can be given as a function of t from 273.15 to 1273.15  k by theoretical predictions , while a specific sample such as calcite ha4 in ref .  
only formed at 323.15    2(1 )  k with k3866  =  1.00053    0.011(1 ) . therefore , as ref .  
did , one might use the experimental results of clumped - isotope ratios in the calcite rings of the stalagmite to determine k3866 , and then compare this value to the experimental fitted polynomial , t  =  f(k3866 ) , to get t .
since samples ( formed in reaction  1 ) such as stalagmites and trees are in a single phase , ts recorded by them are suggested to be experimentally given by studying the clumped - isotopes [ 4550 ] kept in them .
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				 \begin{document}$$ { { n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } \mathord{\left/ { \vphantom { { n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } } \right .
\kern-\nulldelimiterspace } { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } $ $ \end{document } in eq . 
rewriting eq .  8 , we have10\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \frac{{n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } } { { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } = \frac{{{\text{rpfr}}_{{{}^{14/12}{\text{co}}_{2 } } } } } { { { \text{rpfr}}_{{x{}^{14/12}{\text{c}}_{n } } } } } \frac{{n_{{x{\text{c}}_{n } , t}}^{{{}^{14}{\text{c } } } } } } { { n_{{x{\text{c}}_{n } } } ^{{{}^{12}{\text{c } } } } } } { \text{e}}^{\lambda t } $ $ \end{document } in this equation , the left side is the c / c ratio in the atmosphere and the right side shows information from a sample : the isotope fractionation factor  of c between co2 and the sample , the ams results of c / c ratio in it , and the time t which can be directly given by examining the number of the growth rings of , for example , a tree . with such information on the samples , eq . 
10 gives the ratio of c / c in the atmosphere where the sample formed in the geological time .
if the c / c ratio in the past atmosphere were generated in future literature , we shall obtain a new relationship between this c / c ratio and t. then we expect a reasonable reference value ( which plays the role as absolute international standard activity aabs does in the conventional formula ) of \documentclass[12pt]{minimal } 
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				 \begin{document}$$ \left ( { { { n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } \mathord{\left/ { \vphantom { { n_{{{\text{co}}_{2 } , 0}}^{{{}^{14}{\text{c } } } } } { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } } \right .
\kern-\nulldelimiterspace } { n_{{{\text{co}}_{2 } } } ^{{{}^{12}{\text{c } } } } } } } \right)_{\text{ref } } $ $ \end{document } that makes eq
this is beyond the ability , and thus a limitation , of this theoretical paper : we can not set the last term in eq .  9 down now .
from the above derivations , the conclusions can be drawn as follows : this work introduces us a new dimension , the temperature t , to study radiocarbon dating.the mechanism of isotopic fractionation correction to radiocarbon dating method is clear .
this correction , shown in eq .  9 , is a function of  and t resulting from studying the equilibrium isotopic reaction  1 .
since the fractionation caused by reaction  1 can not happen in the classical mechanism and it can be derived only from quantum mechanism [ 23 , 24 ] , the correction is a quantum - mechanical equilibrium isotopic fractionation correction to radiocarbon dating.one application of this work , as eq . 
9 shows , is to explain why so many calibrations , shown in the introduction , are needed for radiocarbon dating .
the reasons for the need are : the first comes from the difference of carbonaceous matters .
9 when we suppose a constant t and a fixed c / c ratio in the last term .
for example , carbonaceous matters xcn are fiber and calcite in tree rings and corals [ 5 , 51 ] , respectively .
these chemicals have different vibrations  and therefore different rpfrs at the same t;the second is from the temperature . since  between , for example , calcite and carbon dioxide in eq . 
9 is generally not a linear function of t [ 2628 ] , t affects t in a non - linear way if the last two terms in eq
, the temperature of one ring of calcite differs from that of another;the third is due to the variation of c / c ratio in the atmosphere . if the first three terms in eq . 
9 are fixed , then t is a function of the variation of c / c ratio . as published works showed [ 1114 ] , there is no linear function to describe this ratio as a function of t : this ratio affects t in some non - linear way .
briefly , these three parameters separately have a non - linear relationship with t.in the nature , the relationship between t and the three parameters is more complex .
when reaction  1 happened , these three influence factors might change together with the latitude and the altitude or the depth and in seasonable scale or even hundred - year scale [ 1114 , 5254 ] .
consequently , the years given by conventional formula deviate from a straight or linear line and need to be calibrated when high accuracies are required .
9 itself contains more information than the conventional one   and , if eq . 
9 were used to determine the years of samples , the deviation would be significant small.another application of present work , as eq . 
10 shows , is to generate the c / c ratio in the atmosphere in the geological history by studying the examples of which the  growth years can be directly examined .
this work introduces us a new dimension , the temperature t , to study radiocarbon dating .
9 , is a function of  and t resulting from studying the equilibrium isotopic reaction  1 . since the fractionation caused by reaction  1 can not happen in the classical mechanism and
it can be derived only from quantum mechanism [ 23 , 24 ] , the correction is a quantum - mechanical equilibrium isotopic fractionation correction to radiocarbon dating .
9 shows , is to explain why so many calibrations , shown in the introduction , are needed for radiocarbon dating .
the reasons for the need are : the first comes from the difference of carbonaceous matters .
9 when we suppose a constant t and a fixed c / c ratio in the last term .
for example , carbonaceous matters xcn are fiber and calcite in tree rings and corals [ 5 , 51 ] , respectively .
these chemicals have different vibrations  and therefore different rpfrs at the same t;the second is from the temperature . since  between , for example , calcite and carbon dioxide in eq . 
9 is generally not a linear function of t [ 2628 ] , t affects t in a non - linear way if the last two terms in eq
, the temperature of one ring of calcite differs from that of another;the third is due to the variation of c / c ratio in the atmosphere . if the first three terms in eq . 
9 are fixed , then t is a function of the variation of c / c ratio .
as published works showed [ 1114 ] , there is no linear function to describe this ratio as a function of t : this ratio affects t in some non - linear way .
briefly , these three parameters separately have a non - linear relationship with t. the first comes from the difference of carbonaceous matters .
9 when we suppose a constant t and a fixed c / c ratio in the last term .
for example , carbonaceous matters xcn are fiber and calcite in tree rings and corals [ 5 , 51 ] , respectively .
these chemicals have different vibrations  and therefore different rpfrs at the same t ; the second is from the temperature . since  between , for example , calcite and carbon dioxide in eq . 
9 is generally not a linear function of t [ 2628 ] , t affects t in a non - linear way if the last two terms in eq . 
, the temperature of one ring of calcite differs from that of another ; the third is due to the variation of c / c ratio in the atmosphere . if the first three terms in eq . 
9 are fixed , then t is a function of the variation of c / c ratio . as published works showed [ 1114 ] , there is no linear function to describe this ratio as a function of t : this ratio affects t in some non - linear way .
briefly , these three parameters separately have a non - linear relationship with t. in the nature , the relationship between t and the three parameters is more complex .
when reaction  1 happened , these three influence factors might change together with the latitude and the altitude or the depth and in seasonable scale or even hundred - year scale [ 1114 , 5254 ] .
consequently , the years given by conventional formula deviate from a straight or linear line and need to be calibrated when high accuracies are required .
9 itself contains more information than the conventional one   and , if eq . 
9 were used to determine the years of samples , the deviation would be significant small .
10 shows , is to generate the c / c ratio in the atmosphere in the geological history by studying the examples of which the  growth years can be directly examined .
 this paper relates the quantum  mechanical equilibrium isotopic fractionation correction to the radiocarbon dating method by eq .  
 9 , and also shows the significant influence of temperature on the method . 
 it is suggested that the correction is a function of the frequencies and temperature of a specific sample and these two variables can be evaluated theoretically by the ab initio quantum calculations and experimentally by analyzing the clumped - isotope ratios in it , respectively . 
 this paper also suggests that the 14c/12c ratio in the atmosphere in geological time can be calculated by eq .  
 10 . 
Introduction
Assumptions and derivations
Conclusions