Method 2: Using the energy dependence of the ratio of GT to F specific cross sections.

Equation (3) Provides a means of extracting the Fermi strength in a (p,n) spectrum using the energy dependence of the ratio of GT to Fermi specific cross sections.  This procedure requires two spectra measured with two different proton energies.  The two spectra are normalized relative to each other near but not including the Fermi peak as above. Then the Fermi peak appears stronger in the spectrum for the lower proton energy.  The strength ratio is known from equation (3), and iterative subtraction, as described above, can be used to determine the Fermi strength.  The difficulty is that the two spectra have different energy resolutions, so one cannot subract the raw spectra from each other.  It is necessary to do a conventional peak fitting first.  In essence this creates a spectrum in analytical form that can be redisplayed with arbitrary resolution.  Then the two spectra can be overlaid.  The peak fitting must be done with great precision because the analysis usually results in finding a small difference between large numbers, and errors are magnified.

The case of ¹³C(p,n)¹³N provides and interesting example of both the difficulties and the power of the method.  The ground state transition is mixed F and GT.  The strong state at 3.5 MeV in ¹³N is the only state near the ground state that can be used for matching the spectrum near but not including the Fermi peak.
 

Fig. 4 shows a spectrum from ¹³C(p,n)¹³N taken with 120 MeV protons.
 

It is apparent that the peak shape includes a non-gaussian tail.  Less apparent is that the shape of the 3.5 MeV peak is not the same as that of the ground state peak.  In this example one can check the final results because the ft value for the beta-decay transition from the ground state of ¹³N to the ground state of ¹³C is known from beta-decay measurements.  One can also compare the peak fitting with sums, since the peaks are well separated.  With careful fitting, including the tails one can get good results.
 

Fig, 5 shows a fit to the g.s. and 3.5 MeV state in the spectrum of fig. 4.
 

Fig. 6 shows an overlay of  of 120 MeV and 160 MeV spectra reconstructed from peak fits.
 

Application Method 2 to ¹⁶⁰Gd(p,n): The nuclei ¹⁶⁰Gd and ¹⁷⁶Yb have been proposed as neutrino detectors. The relevant GT transition strengths cannot be measured with beta decay.  At IUCF we have measure neutron spectra from ¹⁶⁰Gd(p,n)¹⁶⁰Tb and ¹⁷⁶Yb(p,n)¹⁷⁶Lu with 120 and 160 MeV neutrons.  Fig. 7 shows the spectrum from ¹⁶⁰Gd(p,n) with 120 MeV protons.  The the lower picture shows an expanded view of the low excitation region which contains the levels that would be excited by pp and ⁷Be neutrinos.  The resolution in the (p,n) spectrum is not sufficient to determine the relative strengths of the low-lying energy levels.  An auxiliary measurement was made with the (³He,t) reaction at RCNP in Osaka, Japan [Fujiwara, et al., work in progress] to determine the relative strengths.  The curves shown under the (p,n) data are the relative strengths determined by the (³He,t) experiment displayed at the resolution of the (p,n) reaction to simulate a (p,n) spectrum with those relative strengths.  The good match indicates that both reactions seem to be showing the same structure which we interpret as GT strength.  Since contamination by higher multipole reaction components in the 0-deg. spectra is generally greater in (³He,t) than in (p,n) and varies from level to level, the good match indicates that these spectra are not seriously contaminated.
 

Fig. 6.  The ¹⁶⁰Gd(p,n)¹⁶⁰Tb spectrum with 120 MeV protons.  The lower panel shows and expanded view of the low excitation region with an overlay depicting relative strengths determined from a (³He,t) experiment.

To extract absolute values of the GT strengths we use the energy dependence method described above.  Fig. 8 shows and expanded view of the 120 MeV and 160 Mev spectra overlaid.  The lower traces show the same spectra after subtraction of the Fermi component.  The procedure was to create an analytical representation of the peak shapes for  single levels in both spectra by fitting resolved peaks in ¹³C(p,n).  Then peaks of these shapes were iteratively subtracted point by point from the both spectra using the known energy dependence of the GT/Fermi ratio as a constraint on the relative areas of the subtracted peaks.  Then the sum of the squares of the point-by-point differences of the remaining spectra is minimized.  In this case the underlying GT strength appears as a smooth interpolation from one side to the other of the IAS.  This is not always the case.  Especially in lighter nuclei there may be considerable structure in the remaining GT spectrum.

Fig. 8.  Expanded view of the IAS region for ¹⁶⁰Gd(p,n)¹⁶⁰Tb showing both 120 and 160 MeV spectra before and after subtraction of the Fermi component.