Electron spin resonance spectroscopy

There is disclosed a method of accurately calculating the magnetic field strength at an arbitrary position in an ESR (electron spin resonance) spectrum using Mn.sup.2+ marker. The spectrum contains an ESR signal arising from an unknown sample and an ESR signal arising from a reference sample containing Mn.sup.2+ marker. The resonating magnetic field strengths H.sub.ci and H.sub.cj of two absorption lines of the six absorption lines arising from Mn.sup.2+ are calculated from the applied microwave frequency, the nuclear spin quantum numbers m.sub.Ii and m.sub.Ij producing the two absorption lines, a predetermined g value g.sub.o intrinsic to Mn.sup.2+, and a predetermined isotropic hyperfine coupling constant A. The magnetic field strength H.sub.cx at a position of interest in the ESR spectrum is determined from the distance between the absorption lines, the distance between the position and one of the two absorption lines, and the magnetic field strengths H.sub.ci, H.sub.cj.

FIELD OF THE INVENTION 
The present invention relates to electron spin resonance spectroscopy and, 
more particularly, to electron spin resonance spectroscopy capable of 
determining the magnetic field strength at an arbitrary position in an ESR 
spectrum, using a manganese marker. 
BACKGROUND OF THE INVENTION 
In electron spin resonance spectroscopy, a microwave is applied to a sample 
placed in a static magnetic field that is swept. An ESR (electron spin 
resonance) spectrum is obtained by taking the first-order derivative of a 
microwave absorption curve as a function of the magnetic field. Thus, the 
absorption intensity is plotted on the vertical axis of the spectral chart 
and the magnetic field strength on the horizontal axis. Generally, the 
magnetic field strength (resonating magnetic field) of the resonant 
spectrum is given by 
##EQU1## 
where .nu. is the frequency of microwaves, h is the Planck's constant, 
.beta. is the Bohr magneton, and the value of g is intrinsic to the 
material. The factor g is one of the most important factors, as well as 
the hyperfine coupling constant and the line width, in determining from 
what paramagnetic species the ESR spectrum arises. An unknown sample is 
identified as follows. A known marker is used so that an absorption line 
of the marker may appear in an ESR spectrum together with the absorption 
line of the unknown sample. The magnetic field strength is found from the 
g value of the absorption line of the marker. The g value of the unknown 
sample is calculated from the resonating magnetic field strength H of the 
unknown sample and from the microwave frequency .nu.. 
DPPH (2.0036), TCNQ-Li salt (2.0026), and Cr.sup.3+ (1.98) are used for 
calculating g values. Also, Mn.sup.2+ marker, or manganese marker, has 
been frequently used either in ESR measurement for quantitative analysis 
of a paramagnetic substance contained in an unknown sample or as an 
external reference for measuring the g value. 
Since like substances have like g values, it is required to determine g 
values accurately to identify such substances accurately. Except for 
Mn.sup.2+ marker, however, every marker produces a single absorption line 
and so the magnetic field of a spectrum can be calibrated only at one 
point. Therefore, it is inevitable that g values are calculated at low 
accuracy. 
On the other hand, Mn.sup.2+ marker gives rise to 6 absorption lines 
because of its nuclear spin quantum number I=5/2. If the g values of these 
absorption lines are known, therefore, the horizontal axis of the 
spectrum, or the magnetic field, can be calibrated at plural points. For 
this reason, the magnetic field strength of the absorption line of an 
unknown sample can be accurately calibrated. Hence, the g value can be 
accurately determined. 
However, the g values of the individual absorption lines vary, depending on 
the frequency of the applied microwaves, because the six absorption lines 
of the Mn.sup.2+ marker are produced by its hyperfine structure. The g 
values of the third and the fourth absorption lines change relatively 
little, depending on the applied microwave frequency. In the past, it has 
been assumed that the g values of these two intermediate absorption lines 
are kept constant, irrespective of the applied microwave frequency, and 
the g values of the absorption lines of unknown samples would have been 
calculated from the g values of these two absorption lines. 
In this method, however, high accuracy is not obtained in essence, because 
it neglects the fact that g values vary according to the applied microwave 
frequency. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a method of accurately 
determining the magnetic field strength at an arbitrary position in an ESR 
spectrum by the use of Mn.sup.2+ marker, for accurately calculating g 
values. 
In one feature of the present invention, an ESR spectrometer is 
characterized by the provision of a counter that measures the frequency of 
applied microwaves such that magnetic field strengths and g values can be 
calculated accurately as described above. 
As mentioned previously, Mn.sup.2+ marker gives rise to 6 absorption lines 
in an ESR spectrum. The present invention provides a g value (hereinafter 
referred to as g.sub.o) corresponding to the virtual mean position of the 
6 absorption lines and an isotropic hyperfine coupling constant A. An ESR 
spectrum containing plural absorption lines attributed to Mn.sup.2+ marker 
is obtained. The frequency .nu. of the applied microwaves is measured. The 
magnetic field strength at an arbitrary position in an ESR spectrum is 
found in the manner described below. From equation (1) above, the magnetic 
field strength at the virtual mean position is given by 
EQU H.sub.o =h.nu./g.sub.o .beta. 
where h is the Planck's constant, .nu. is the frequency of the applied 
microwaves, g.sub.o is the aforementioned g value, and .beta. is the Bohr 
magneton. Using the magnetic field strength Ho, the coupling constant A, 
the nuclear spin quantum number I (I=5/2 in the case of Mn.sup.2+), and 
the components m.sub.I of the values of the spin quantum number I 
projected on the static magnetic field, we have 
EQU H.sub.c =H.sub.o -A m.sub.I -A.sup.2 {I(I+1)-m.sub.I.sup.2 }/2H.sub.o 
From this equation, the magnetic field strengths H.sub.ci and H.sub.cj at 
the positions of some of the six absorption lines produced by Mn.sup.2+ 
are calculated. Then, the magnetic field strength at an arbitrary position 
in the ESR spectrum is correctly calculated from the positional relation 
of this arbitrary position to the abovedescribed absorption lines. 
Other objects and features of the invention will appear in the course of 
the description thereof which follows.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Referring to FIG. 1, there is shown an ESR (electron spin resonance) 
spectrometer embodying the concept of the invention. This spectrometer 
includes an electromagnet 1 consisting of the pair of opposite magnetic 
pole pieces 2 and 2', exciting coils 3, and an exciting power supply 4. A 
microwave cavity 5 is disposed at the center of the static magnetic field 
produced between the magnetic pole pieces 2 and 2'. A sample 6 to be 
investigated and a reference sample 7 containing Mn.sup.2+ are placed 
inside the cavity 5. If necessary, a sensor 9 ancillary to a magnetic 
detector 8 is inserted in the static magnetic field. Microwaves produced 
by a microwave generator 10, such as a Gunn oscillator, are supplied into 
the cavity 5 via a circulator 11. When the electron spins of the sample 
resonate, reflecting waves are emitted from the cavity 5 and introduced 
via the circulator 11 into a detector 12, where the waves are detected. 
The resulting ESR signal is converted into digital form, supplied to a 
dataprocessing unit 13, and stored in a memory 14 ancillary to the 
processing unit 13, which is equipped with a display unit 15 and an input 
device 16. 
ESR spectra of nuclei possessing nuclear spins such as Mn.sup.2+ can be 
theoretically explained by considering the interaction among the electron 
spin quantum number S, the nuclear spin quantum number I, and the external 
magnetic field H. Energies of electron spin levels are expressed in terms 
of the following Hamiltonian: 
EQU H=.beta.H g S +I.sub.A S (2) 
where g is a g tensor, and .sub.A is a hyperfine coupling tensor. S, I, and 
H are all vectors. In equation (2), the first term of the right side is 
the electron Zeeman term, and the second term of the right side indicates 
hyperfine coupling. Since the first term is usually much larger than the 
second term, the second term is treated as a perturbation term in 
expanding equation (2). As the contribution of the second term increases 
and can no longer be neglected compared with the contribution of the first 
term, the second term, or the perturbation term, is required to be 
expanded up to a higher order. For ordinary organic free radicals, the 
second term is much smaller than the first term and, therefore, it 
suffices to expand the perturbation term up to the first order. Under this 
condition, the resonating magnetic field is given by 
EQU H.sub.c =H.sub.o -A m.sub.I (3) 
where A is the isotropic hyperfine coupling constant, m.sub.I is the 
component of I projected on H, and H.sub.o is the magnetic field strength 
at the virtual mean position of the spectrum. Using the g value g.sub.o 
corresponding to the virtual mean position, the magnetic field strength 
H.sub.o is given by 
EQU H.sub.o =h.nu./g.sub.o .beta. (5) 
Where the applied microwave frequency lies within the X band of 9000 to 
9500 MHz, the magnetic field strength H.sub.o of Mn.sup.2+ marker in an 
ESR signal is about between 3300 and 3400 G, and the coupling constant A 
is approximately between 80 and 90 G. In this case, the coupling constant 
A is not negligibly small compared with the magnetic field strength 
H.sub.o. As a consequence, the equation is required to be expanded to a 
higher order. By adding the second-order perturbation term to equation 
(3), we have 
##EQU2## 
According to equation (3), hyperfine splittings occur at regular intervals 
of A. According to equation (4), hyperfine splittings do not take place 
regularly because of the third term of the right side. 
The nuclear spin quantum number of Mn.sup.2+ is I=5/2. The nuclear spin 
magnetic quantum number m.sub.I that is the component of the nuclear spin 
quantum number projected on the axis of the static magnetic field can 
assume 6 discrete states of 5/2, 3/2, 1/2, -1/2, -3/2, and -5/2 which 
successively differ by one. Therefore, the ESR signal of Mn.sup.2+ marker 
is split into 6 absorption lines, as shown in FIGS. 2(a) and 2(b). 
FIGS. 2(a) and 2(b) show ESR spectra of contained in the reference sample 
7. The spectra were obtained using the spectrometer shown in FIG. 1. 
During the measurement, the static magnetic field was swept from 3350-250 
G to 3350+250 G. The frequency .nu. of the microwaves was set to 9449.023 
MHz and 9187.39 MHz. When the static magnetic field was swept, the 
strength of the magnetic field was monitored by the magnetic field 
detector 8. Magnetic field strengths H.sub.cl, . . ., H.sub.c6 which are 
put above or below the absorption lines in the spectra were obtained by 
this detector 8. G values g.sub.1, . . . , g.sub.6 that are put below the 
field strengths H.sub.cl, . . ., H.sub.c6 were calculated from the field 
strengths H.sub.cl, . . ., H.sub.c6 and from .nu. in accordance with 
equation (1). 
As can be seen from FIGS. 2(a) and 2(b), changing the microwave frequency 
.nu. varies the positions of the six absorption lines, or the resonating 
magnetic fields, accordingly. 
The g value g.sub.o and the coupling constant A included in equations (4) 
and (5) do not depend on the microwave frequency and assume values 
intrinsic to Mn.sup.2+. Thus, the magnetic field strengths of the six 
absorption lines arising from Mn.sup.2+ can be calculated by previously 
finding g.sub.o and A empirically and substituting the applied microwave 
frequency .nu. into equations (4) and (5). Then, the magnetic field 
strength at an arbitrary position in a spectrum can be calculated from the 
calculated magnetic field strengths of the absorption lines. 
The values of g.sub.o and the coupling constant A can be determined from 
the results of measurements shown in FIGS. 2(a) and 2(b). As an example, 
g.sub.o and A are varied as parameters. The magnetic field strengths of 
the absorption lines which are calculated in accordance with equations (4) 
and (5) are compared with actually measured magnetic field strengths. The 
differences between them are minimized by the self-consistent field 
method. In this way, correct values of g.sub.o and A can be found. 
In a specific example, parameters g.sub.o and A were found which minimize 
the difference .sigma.(g.sub.o, A) between the actually measured 
resonating magnetic field strength H.sub.R (.nu..sub.i, m.sub.I) and the 
resonating magnetic field strength H.sub.c (.nu..sub.i, m.sub.I) 
calculated in accordance with equation (4). 
##EQU3## 
As a result, a minimum dispersion value .sigma. (g.sub.o, A)=0.0373 was 
obtained at g.sub.o =2.00094 and A=86.77 G. The computer programs used for 
the calculation are shown in FIGS. 3(a) and 3(b). 
Table 1 shows actually measured magnetic field strengths H(measured) of the 
6 absorption lines of Mn.sup.2+ marker at .nu.=9449.02 MHz and 
.nu.=9187.39 MHz, magnetic field strengths H(theoretical) calculated from 
g.sub.o and A found as described above, their differences .DELTA..sub.H, 
and apparent g values g(theoretical) calculated from the above-described 
theoretical values. All the magnetic field strengths are expressed in G. 
TABLE 1 
______________________________________ 
Microwave H 
frequency H (theo- g 
(MH.sub.z) 
mI (measured) 
retical 
.DELTA.H 
(theoretical) 
______________________________________ 
5/2 3154.38 3154.30 
0.08 2.14031 
3/2 3236.60 3236.60 
0.00 2.08588 
9449.02 1/2 3321.06 3321.13 
-0.07 2.03279 
-1/2 3407.90 3407.90 
0.00 1.98104 
-3/2 3496.90 3496.90 
0.00 1.93062 
-5/2 3588.06 3588.12 
-0.06 1.88153 
5/2 3060.67 3060.80 
-0.13 2.14462 
3/2 3142.98 3142.97 
0.01 2.08855 
9187.39 1/2 3227.48 3227.44 
0.04 2.03388 
-1/2 3314.22 3314.21 
0.01 1.98064 
-3/2 3403.32 3403.27 
0.05 1.92880 
-5/2 3494.60 3494.62 
-0.02 1.87838 
______________________________________ 
As can be seen from Table 1, the magnetic field strengths H(measured) of 
the resonance lines of Mn.sup.2+ marker agree very well with the 
corresponding magnetic field strenghts H(theoretical) at either microwave 
frequency. From Table 1, the mean value .DELTA.Hmean of deviations of the 
measured values from the theoretical values of the six absorption lines at 
the two frequencies is given by 
##EQU4## 
Since the order of magnitude of this mean deviation coincides with the 
minimum order of magnitude (of the order of 0.01 G) that can be measured 
by an instrument, it can be regarded as experimental error. Thus, we have 
demonstrated that the aforementioned values g.sub.o =2.00094 and A=86.77 G 
are appropriate. 
One example of a sequence in which the g value of an unknown sample is 
found in accordance with the present invention is next described by 
referring to the flowchart of FIG. 4. 
Step 1 
An ESR spectrum of an unknown sample is obtained, using the instrument 
shown in FIG. 1. This spectrum contains 6 absorption lines S.sub.1 
--S.sub.6 arising from Mn.sup.2+ and an absorption line X arising from the 
unknown sample, as shown in FIG. 5. Data about the spectrum is stored in 
the memory 14 and sent to the display unit 15, where it is displayed. 
Step 2 
The operator enters the applied microwave frequency .nu. into the 
data-processing unit 13 from the input device 16, or the data-processing 
unit can automatically read and store the output from a frequency counter 
during the measurement. 
Step 3 
The operator specifies the positions of arbitrary two (the i-th and j-th) 
of the six absorption lines, using a position-specifying device which 
controls the position of a cursor or the like superimposed on the 
spectrum. At this time, if the displayed spectrum is enlarged in the 
direction of the horizontal axis, then it is easy to specify the center of 
each absorption line. The data-processing unit reads and stores the 
numerals (P.sub.i and P.sub.j) given to the specified positions. 
The operator also enters information indicating what absorption lines are 
the specified absorption lines, i.e., i and j, into the data-processing 
unit from the input device. The data-processing unit finds the spin 
magnetic quantum numbers m.sub.Ii and m.sub.Ij of the i-th and j-th 
absorption lines from a conversion table previously stored in the unit. 
Table 2 shows an example of the conversion table. 
TABLE 2 
______________________________________ 
1 m.sub.I1 +5/2 
2 m.sub.I2 +3/2 
3 m.sub.I3 +1/2 
4 m.sub.I4 -1/2 
5 m.sub.I5 -3/2 
6 m.sub.I6 -5/2 
______________________________________ 
If the operator directly enters the spin magnetic quantum numbers of the 
two absorption lines, such a conversion table is dispensed with. 
Step 4 
The data-processing unit calculates the resonating magnetic field strengths 
H.sub.ci and H.sub.cj of the i-th and j-th absorption lines from g.sub.o 
=2.00094, A=86.77 G, I=5/2 previously stored in the data-processing unit 
and from newly stores .nu., m.sub.Ii, m.sub.Ij in accordance with 
equations (4) and (5). 
Step 5 
The operator specifies the central position of the absorption line X of the 
unknown sample with the position-specifying device. The data-processing 
unit reads and stores the numeral P.sub.x given to the specified position. 
Step 6 
The data-processing unit calculates the resonating magnetic field strength 
H.sub.cx of the absorption line of the unknown sample and the g value 
g.sub.x in accordance with the following formulas. 
EQU H.sub.cx =H.sub.ci +(P.sub.x -P.sub.i)(H.sub.cj -H.sub.ci)/(P.sub.j 
-P.sub.i) (8) 
EQU g.sub.x =h.nu./.beta.H.sub.cx (9) 
In this way, in accordance with the present invention, if the microwave 
frequency .nu. applied during measurement is known, then the g value and 
the resonating magnetic field strength of the absorption line of an 
unknown sample which appears at an arbitrary position in a spectrum can be 
calculated. When the values of g.sub.o and A are found first, a magnetic 
detector is used. This operation can be performed in the manufacturing 
plant of the instrument. consequently, the user can precisely find the 
resonating magnetic field strength of the absorption line of an unknown 
sample and the g value without a magnetic detector. 
In the present invention, the magnetic field strengths of the six 
absorption lines arising from Mn.sup.2+ are precisely found. Utilizing 
these, the distortion of the horizontal axis of the ESR spectrum which is 
caused by nonlinearity of sweep of the magnetic field can be compensated 
for. In particular, the space between the first and second absorption 
lines of the six absorption lines originating from Mn.sup.2+ is A-2A.sup.2 
/H.sub.o. The space between the second and third lines is A-2A.sup.2 
/H.sub.o. The space between the third and fourth lines is A. The space 
between the fourth and fifth lines is A+2A.sup.2 /H.sub.o. The space 
between the fifth and sixth lines is A+2A.sup.2 /H.sub.o. If the magnetic 
field is not swept exactly linearly, the spaces between the successive 
absorption lines deviate from the above values. By comparing the spaces 
between the six successive absorption lines arising from Mn.sup.2+ in a 
spectrum with the above values, the magnetic field strength can be 
calibrated in the regions between the six absorption lines. 
It is to be noted that the present invention is not limited to the above 
example, and that various changes and modifications are possible. In the 
above example, whenever a microwave frequency .nu. is entered, the 
resonating magnetic field strength is calculated. A method utilizing a 
look-up table or the like is also possible. Specifically, calculations are 
performed for various values of .nu. beforehand. A table is prepared 
according to the results of the calculations. When a value of .nu. is 
entered, the results of calculations can be quickly read out by referring 
to the table previously prepared. 
Having thus described my invention with the detail and particularity 
required by the Patent Laws, what is claimed and desired to be protected 
by Letters Patent is set forth in the following claims.