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<head>1. Introduction<lb/></head>
<p>Longitudinal relaxation time T a is one of the most important parameters in<lb/> modern
NMR spectroscopy. Because of its capability of probing molecular<lb/> mofion in liquids,
it has attracted more and more attention. Nowadays as the<lb/> NMR equipment develops
very quickly, not only the common nuclei, but<lb/> also the less sensitive and even the
extremely rare and weak nuclei such as<lb/> 57Fe <ref type="biblio">[1, 2]</ref> and
2aNe <ref type="biblio">[3]</ref> have also been investigated by relaxation time<lb/>
measurements. In contrast to chemical shifts and coupling constants, which<lb/> can be
precisely and directly read from highly resolved spectra, the relaxa-<lb/>tion times
must be obtained through data fitting. As the Ta experiments are<lb/> rather time
consuming anda sufficient signal-to noise ratio S/N is essential,<lb/> since long much
endeavor has been paid to the development of accurate and<lb/> efficient technique for T
1 measurements <ref type="biblio">[4-6]</ref>. But so lar there is no com-<lb/>mon
criterion for estimating T 1 error. There are too many things which can<lb/> affect the
accuracy of T t. In addition to the vital factor S/N, factors like<lb/> digital
resolution, number of r, techniques in arrangement of r list and the<lb/> method in use
such as inversion-recovery and others, are also important.<lb/> Therefore, many authors
claimed their T a error only by "estimation" without<lb/> saying why. This is
understandable because they really lack a practical crite-<lb/>rion.<lb/></p>
<p>It is the purpose of this paper to propose an empirical criterion for estimat-<lb/>ing T
1 error, which is based on the standard deviafion in data fitting.<lb/></p>
<head>2. Experimental<lb/></head>
<p>170 relaxation times were measured on a Bruker AMX-500 with the sample<lb/> of
"pure" D2O (99.9% in purity). More than 200 expe¡<lb/> were carried<lb/> out
using inversion-recovery technique. The temperature was controlled<lb/> within 25 _+
0.3"C and calibrated by 4% methanoL In order to obtain statisti-<lb/>cal results,
the r fist was ffequently changed, both the interval in between<lb/> and the number of
3, but the shortest r was always kept constant as 0.1 ms.<lb/> The longest r was always
greater than 40 ras. So was the relaxation delay.<lb/> The number of r ranged from 8 to
16. The number of scans in accumula-<lb/>tion was changed from 1 (with S/N = 5) to 640
(with S/N = 120). Some-<lb/>times the data points for acquisition, the size for Fourier
transforrnation, and<lb/> the line broadening factor were also changed, but all
reasonably. All results<lb/> of T a were evaluated using the standard 3-parameter
fitting program in<lb/> Bruker software based on Eq.<ref type="formula"
>(1)</ref>:<lb/></p>
<formula>I = A 1 [1 --A2 exp(-A 3 t)],<lb/>
<label>(1)</label><lb/></formula>
<p>where A 3 = (1/T~ ). After fitting the T~ value was printed out along with the<lb/> value
of the standard deviation.<lb/></p>
<head>3. Results and Discussion<lb/></head>
<p><ref type="figure">Figure 1</ref> shows the statistical correlation between T~ and the
standard devia-<lb/>tion. The standard deviation a can be defined as<lb/></p>
<formula>{1 ~/<lb/> ]1/2<lb/> a= ¡<lb/> [Ii(r0-I(ri)12~ ,<lb/>
<label>(2)</label><lb/> i=1<lb/></formula>
<p>where Ii is the experimental intensity and I the fitted value. Although it is<lb/> well
accepted that the smaller the a is, the smaller the error will be, it is al-<lb/>most
impossible to formulate an explicit relation between a and error. The<lb/> relationship
can, however, be built stafistically by experimental data. The<lb/> error or relafive
error of T 1 can be defined as<lb/></p>
<formula>T 1 -(T1)<lb/> error --<lb/>(T1)<lb/> ,<lb/>
<label>(3)</label><lb/></formula>
<p>where (Ta) denotes the expectation value of T v The expectation (T1) for<lb/> 170 in D20
at 25~ was determined in this work as 5.774 ms. The error for<lb/></p>
Ala Empi¡<lb/> Criterion for Estimating T~ Error<lb/>
<figure>I<lb/> o** .-..A<lb/> ...~<lb/> "7<lb/> ..-''"*<lb/> ...
o-''"<lb/> .--9<lb/> 9<lb/> "~<lb/> "i.<lb/> "<lb/>
-<lb/>. ...i---. 2<lb/> :.<lb/> 6<lb/> ""':"<lb/> i._<lb/>
"5<lb/> ~o<lb/> ~176<lb/> .~<lb/> ~<lb/> ~<lb/> ~<lb/> .<lb/> 9<lb/> -',,91176
g,~" .... ~ t<lb/> 9 .<lb/> .<lb/> .<lb/> .<lb/> .<lb/>
-'.'.~~i~~/':: :'~':" "'" "." 9 9
9 " "<lb/> "-~. *.%~<lb/> .~.."<lb/> ".<lb/> .<lb/> 9<lb/>
.<lb/> 9 %<lb/> .... :;-..~..<lb/> .....<lb/> .<lb/> ......<lb/> ~:<lb/> 0131<lb/>
O132<lb/> 0.0,3<lb/> 0.04<lb/> STANDARD DEVIATION<lb/> *02<lb/> 9 at E<lb/> 0
'~<lb/> -DI<lb/> -02<lb/> Fig. 1. Statistical correlation of ZTO T1 data from a
single sample of pure D20 at 25~ with<lb/> the standard deviation. By drawing another
axis of relative error, the figure shows that the<lb/> error limit is approximately
proportional to the statistical standard deviations.<lb/></figure>
<p>each individual T1 can be directly identified by drawing an additional axis of<lb/>
relafive error as shown in <ref type="figure">Fig. 1</ref>. Thus a statistical
correlation between a and<lb/> error limit is achieved.<lb/></p>
<p>We are concemed now in such a question: with a given standard deviation,<lb/> how big
might the error in T~ be expected? From <ref type="figure">Fig. 1</ref> it can be seen
that<lb/></p>
<figure>0136<lb/> ~004<lb/> C3<lb/> .I<lb/> :i<lb/> ~ m<lb/> " "~~ ......
~t-" ..... ,k~ ..............<lb/> |<lb/> | n I n i i |<lb/> i 9 ," I ,<lb/>
40<lb/> BO<lb/> 120<lb/> S/N<lb/> Fig. 2. Plot of the standard deviation against the
signal-to-noise ratio S/N of the spectrum<lb/> with the longest T (3 _ 6 T 1
).<lb/></figure>
<p>the error limit is approximately proportional to o, i.e. the error limit can be<lb/>
represented by two straight fines with slops of _+ 6 as shown in <ref type="figure">Fig.
1</ref>. In<lb/> other words, the error limit can be estimated as big as 6 ~ From
the point of<lb/> view of this paper, if one wants to claim ah error of +_ 2.5% Ÿ his
T1, he<lb/> should reach the standard deviation less than 0.004. This could be
suggested<lb/> as an empi¡<lb/> crite¡<lb/> for estimating the error limit for
experimental T a<lb/> value. As menfioned in the Introduction, there ate numerous
factors which<lb/> can affect the accuracy of the results in T a experiments. People in
different<lb/> laboratories may use different experimental conditions. It would be good
for<lb/> them to have a common criterion which makes it possible to compare the<lb/>
results at the same temperature from different laboratories.<lb/></p>
<p>Among the numerous factors, the signal-to-noise ratio S/N is of central im-<lb/>portance.
In <ref type="figure">Fig. 2</ref> the standard deviation is plotted agalnst S/N for
some<lb/> experiments, where S/Ns were measured from the spectra with longest r<lb/>
(r_-> 6T1). The S/N= 120 was achieved by 640 accumulations, S/N--80<lb/> by 320
accumulations, and so on. <ref type="figure">Fig. 2</ref> indicates that a standard
deviation<lb/> of 0.004 should be ensured by an S/N greater than 40, corresponding
to<lb/> error limits of_+ 2.5%.</p>
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