LAMB SHIFT FOR HYDROGEN AND DEUTERIUM*)

E. E. Salpeter

From the translator

This review by Salpeter is devoted to the Lamb shift for hydrogen and deuterium—a touchstone of one of the most interesting problems of modern theoretical and experimental physics: radiative corrections in quantum electrodynamics.**)

Before 1947 no problem in fact existed. The spectrum of the hydrogen atom was then theoretically described by Dirac’s equation for a single electron in an external field. Numerous experimental investigations carried out by optical methods, within the errors of experiment, well confirmed the conclusions of the theory. In particular, the theoretically predicted coincidence of the \(2S_{1/2}\) and \(2P_{1/2}\) levels was confirmed. Experiments by some authors, it is true, indicated the possibility of a small splitting of these levels, but these results were not convincing, since the splitting obtained was of the order of the probable error of measurement.

The exceptionally high accuracy of optical methods (\(\sim 10^{-6}\) relative to the principal term) seemed to set a limit to any substantial new experimental advances in this direction.

However, the use of radio-frequency techniques made it possible to increase the accuracy of measurements by many orders of magnitude. The shift of the \(2P_{1/2}\) and \(2S_{1/2}\) levels, lying at the limit of the possibilities of optical methods, was reliably detected in 1947 by Lamb and Retherford, and in 1952 was measured with an accuracy up to \(10^{-10}\) relative to the principal term of the hydrogen atom. As far as we know, such accuracy (ten significant figures!) had not previously been attained in any physical measurement at all.

The successes of experimental technique stimulated the development of theory. If one proceeds from the exact equations of quantum electrodynamics for the hydrogen atom, it turns out that Dirac’s equation for a particle in a Coulomb field is obtained only as the first nonvanishing approximation of perturbation theory. The next approximations were to give

*) Phys. Rev. 89, 92 (1953), translated by Yu. M. Shirokov.
**) For an additional list of literature in Russian on these questions, see the end of the article.

small corrections, later called radiative. However, because of the internal inconsistency of quantum electrodynamics (which to this day is not a logically closed theory), higher approximations in the calculation yielded not small corrections but infinite, divergent terms.

Until 1947 these divergent expressions were usually simply discarded as meaningless, especially since the first nonvanishing approximation in all cases gave sufficient agreement with experiment. Only one of the radiative corrections, the so-called polarization of the vacuum, was computed in the prewar years. It was not possible, however, to explain the Lamb shift by vacuum polarization, since the corresponding correction was too small in magnitude and, moreover, had the opposite sign.

Since 1947 various authors began to develop methods for extracting small corrections from the infinite expressions for higher approximations in quantum electrodynamics. The main difficulty in this problem consists in the fact that the operation of separating an infinite expression into a finite and an infinite part is, generally speaking, not unique. This difficulty was overcome by casting perturbation theory in a relativistically invariant form. It turned out that the relativistically invariant extraction of finite radiative corrections can be carried out in a definite, unique way in any approximation. When radiative corrections are taken into account in the problem of the hydrogen atom, the magnitude of the Lamb shift proves to be finite and in excellent agreement with the experimental data.

In addition to radiative corrections, a whole series of other factors affect the magnitude of the Lamb shift, and these are discussed in detail in the article cited below. It then turns out that, with all known corrections taken into account, theory agrees with experiment to within one megacycle (9 significant figures relative to the principal atomic term).

It is all the more interesting that, with a further increase in accuracy, a definite discrepancy between theory and experiment appears. This discrepancy proved to be equal to \(0.5\) Mc/sec for an accuracy of \(0.1\) Mc/sec. To explain this discrepancy, apparently, some new physical ideas are needed. Further experimental investigations in this area seem desirable, since the accuracy of measurement already achieved apparently is not the limiting one.

§ 1. INTRODUCTION

In recent years the accuracy of the experimental determination of the Lamb shift has increased considerably. A detailed description of the experimental technique and of the calculations for the precision measurement of the Lamb shift is contained in the series of articles “Fine Structure of the Hydrogen Atom”\(^{1-3}\). The full Lamb shift (denoted by \(S\) and expressed in Mc/sec) is the energy difference between the \(n^2S_{1/2}\) and \(n^2P_{1/2}\) levels of a hydrogen-like atom. In Dirac’s elementary theory for an electron in a Coulomb field these levels coincide. In the present article we shall be concerned mainly with the case \(n=2\) for hydrogen and deuterium. For this case \(S\) has been determined experimentally with an accuracy of one tenth of a Mc/sec. Various authors have calculated the influence of a large number of effects on the Lamb shift. The principal aim of the present article is to collect

together the results of the preceding calculations and to calculate the influence of several more terms. Corrections to the Lamb shift of relative orders \(\alpha^2\), \(\left(\dfrac{\alpha m}{M}\right)\), \(\left(\dfrac{\alpha^{-1}m^2}{M^2}\right)\) and higher will not be considered in the present article.

The displacement of the energy levels of a bound electron in a Coulomb field is expressed as a series in powers of the fine-structure constant \(\alpha\). The results of earlier calculations for all terms of the two lowest orders in \(\alpha\) are collected in § 2. This displacement in the Coulomb field is responsible for the main part of the Lamb shift. Taking into account the finite mass and internal structure of atomic nuclei leads to the appearance of corrections to this displacement. Some corrections, caused by the finite mass of the atomic nucleus, were calculated earlier. The remaining corrections in the lowest order in \(\alpha\), depending on the finite nuclear mass, are discussed in § 3. For deuterium additional corrections were obtained by taking into account the finite dimensions of the nucleus. These corrections are given in § 4. In § 5 the question of the influence of the internal structure of individual nucleons is examined. In § 6 the results of these calculations are compared with the latest experimental data on the Lamb shift. In the appendix a summary table is given of all the corrections to the Lamb shift discussed in the article.

§ 2. THE LAMB SHIFT IN A GIVEN COULOMB FIELD

In this section are collected all the results of calculations, carried out by various authors, of the shift of a level in a given Coulomb field, i.e. for an atom with an infinitely heavy point nucleus.

In these calculations two types of expansions occur. Successive approximations in the virtual radiation field give an expansion in powers of \(\alpha\). Successive approximations in the Coulomb field give an expansion in powers of \(Z\alpha\), where \(Z\) is the atomic number (in our case, unity). For the principal quantum number \(n=2\) and for charge \(Z=1\), the result in the lowest order both in \(\alpha\) and in \(Z\alpha\) has the form:

\[ S_{\infty}^{(1)} = \left(\frac{\alpha^3 \mathrm{Ry}_\infty}{3\pi}\right) \left\{ \left[\ln \frac{mc^2}{k_0(2,0)}-\ln 2+\frac{5}{6}\right] -\frac{1}{5} -\left[\ln \frac{\mathrm{Ry}_\infty}{k_0(2,1)}-\frac{3}{8}\right] \right\}, \tag{1} \]

where \(\mathrm{Ry}_\infty\) is the Rydberg constant for a nucleus of infinite mass, \(m\) is the electron mass, and \(k_0(2,0)\) and \(k_0(2,1)\) are the “mean excitation energies,” defined and calculated in [4]. In (1) the first term in square brackets gives the shift of the \(2S_{1/2}\) level without taking into account the polarization

vacuum. \(-\dfrac{1}{5}\) represents a correction to the shift of the \(2S_{1/2}\) level due to vacuum polarization. The last term in square brackets describes the shift of the \(2P_{1/2}\) level.

In obtaining equation (1), the parts corresponding to virtual photons of “low” and “high” momenta are calculated by different methods. The part corresponding to low momenta is calculated by Bethe’s nonrelativistic method\(^5\), using the exact wave functions of the atom. These calculations give\(^4\)

\[ k_0(2,0)=(16.646\pm0.007)\,\mathrm{Ry}_\infty \]

and

\[ k_0(2,1)=(0.9704\pm0.0002)\,\mathrm{Ry}_\infty . \]

For the part corresponding to high momenta, the electron is treated relativistically, and in the intermediate states the plane-wave approximation is used. These relativistic calculations, leading to equation (1), were carried out by various methods\(^6,\,7\).

The most accurate value of the fine-structure constant can be obtained by measuring the fine\(^8\) and hyperfine structure. In view of the fact that at present the theoretical interpretation of measurements of the fine-structure constant\(^9\) is not strictly definite, we shall write:

\[ \frac{1}{\alpha}=137.0360+\varepsilon_\alpha . \tag{2} \]

The correction \(\varepsilon_\alpha\) does not exceed \(0.002\).

Using the value \(299\,790.9\pm1.0\ \mathrm{km/sec}\) for the speed of light\(^ {10}\), the value of the Lamb constant \(L\), in frequency units, may be written in the form

\[ L=\left(\frac{\alpha^3}{3\pi}\right)\mathrm{Ry}_\infty c =(135.6431-3\varepsilon_\alpha\pm0.0005)\ \mathrm{MHz}. \tag{3} \]

The value of the term in braces in (1) is\(^4\) \(7.7567\pm0.0005\). For the Lamb shift in the lowest order this gives:

\[ S_\infty^{(1)}=(1052.14-22\varepsilon_\alpha\pm0.07)\ \mathrm{MHz}. \tag{4} \]

\(S^{(1)}\) contains a correction \(-27.13\ \mathrm{MHz}\) due to vacuum polarization and \(\dfrac{1}{2}L=+67.82\ \mathrm{MHz}\) due to the anomalous magnetic moment of the electron.

The next term of the expansion in powers of \(Z\alpha\) (the Coulomb potential acting twice, etc.) was calculated in\(^ {11}\). This

term, taking vacuum polarization into account, is equal to

\[ S(2,Z)=+7.14\ \text{MHz}. \]

The next term of the expansion in powers of \(\alpha\) (two virtual photons) consists of three parts: a) the fourth-order correction to \(S\) due to the anomalous magnetic moment of the electron\(^{12}\), equal to \(-0.94\ \text{MHz}\); b) the fourth-order correction due to vacuum polarization\(^{13}\), equal to \(-0.24\ \text{MHz}\); and c) the other radiative corrections of fourth order\(^{14}\), equal to \((+0.24 \pm 0.10)\ \text{MHz}\). The sum of all these terms of order \(\alpha S^{(1)}\) and \(Z\alpha \cdot S^{(1)}\) is

\[ S^{(2)}=(6.20\pm0.10)\ \text{MHz}. \tag{5} \]

Terms of relative orders \(\alpha^2\), \(Z\alpha^2\), \(Z^2\alpha^3\), and higher have so far not been calculated. Since the term \(S^{(2,Z)}\) of relative order \(Z\alpha\) is numerically much larger than might have been expected, it is not excluded that the term of order \(Z^2\alpha^3\) will have an appreciable magnitude. We shall denote the correction to the Lamb shift due to this term by \(S_Z^{(3)}\); the correction due to the remaining as yet uncalculated terms will be denoted by \(S_R\). The final numerical value of the Lamb shift in a given Coulomb field \((n=2,\ Z=1)\) is

\[ S_\infty=(1058.34-22\varepsilon_\alpha+S_Z^{(3)}+S_R\pm0.12)\text{MHz}. \tag{6} \]

§ 3. CORRECTIONS DUE TO THE FINITE MASS OF THE NUCLEUS

In\(^{15}\) corrections of order \(\alpha\left(\dfrac{m}{M}\right)\) to the fine structure of hydrogen due to the finite mass \(M\) of the proton were calculated. These terms are connected with Feynman diagrams (for example, exchange of two virtual photons between the electron and the proton) that are essentially different from the diagrams pertaining to the Lamb shift proper and to the vacuum-polarization effect. For the levels \(2S_{1/2}\) and \(2P_{1/2}\) these terms are different. Therefore they give a correction to the Lamb shift of order \(\left(\dfrac{m}{M}\right)S\), which is confirmed experimentally.

For hydrogen these terms give a level shift of \(+0.379\ \text{MHz}\) in the \(2S_{1/2}\) state and \(-0.017\ \text{MHz}\) in the \(2P_{1/2}\) state. Thus, the corresponding correction to the Lamb shift is \(+0.396\ \text{MHz}\). If the internal structure of the nucleus is neglected, then for deuterium these terms are twice smaller than for hydrogen. Neglect of the structure of the deuteron gives, for these terms, an error of order \(10\%\).

We must now add corrections due to the reduced mass to expression (6), in order to take into account the finite mass of the atomic

nucleus. We shall introduce these corrections only into the principal term \(S^{(1)}\), defined by equation (4).

As shown in\(^5\), these corrections can in principle be obtained by calculating the Lamb shift as a whole with the aid of a four-dimensional covariant wave equation for a system of two bodies in a bound state. This method is similar to that used in\(^7\) for calculating the Lamb shift, but is more complicated. Both methods contain, for example, an operator defined by equation (12) of paper\(^7\). In the method used in\(^7\), the average of this operator is calculated with the aid of ordinary three-dimensional wave functions, and only transition matrix elements between states satisfying the Dirac equation (to lowest order in \(Z\alpha\)) are used. In the other method, the average of this operator must be calculated with the aid of four-dimensional wave functions of the two-body system, and in so doing matrix elements of transitions between states not satisfying the Dirac equation are already required.

But for the Lamb shift in the lowest order in \(Z\alpha\) the nucleus interacts with the electron only once, so that in the expression for the shift there occur no denominators containing the energy of the nucleus. The mass of the nucleus enters only in a simple form, and, consequently, the effect introduced by the finite mass of the nucleus can be easily obtained by analyzing the ordinary calculations for an electron in a given Coulomb potential.

First we consider the nonrelativistic calculation, carried out in\(^5\), for the part of the Lamb shift corresponding to low momenta. Let \(M\) be the mass of the nucleus, \(m\) the mass of the electron, and \(\mu\) the reduced mass of the electron. We pass to the rest frame of the center of inertia. In this system the momentum of the electron \(\mathbf{p}_e\) and the momentum of the nucleus \(\mathbf{p}_N\) are equal in magnitude and opposite in direction. After the motion of the center of inertia is eliminated, the Schrödinger equation contains the reduced mass \(\mu\) and the “relative momentum” \(\mathbf{p}\). Then \(\mathbf{p}=\mu\mathbf{v}\), where \(\mathbf{v}\) is the relative velocity. \(\mathbf{p}_e\) is equal to the product of \(m\) and the “absolute velocity of the electron,”

\[ \left(\frac{M}{m}+M\right)\mathbf{v}. \]

Thus, \(\mathbf{p}\) is exactly equal to \(\mathbf{p}_e\).

In\(^5\) the Dirac matrix \(\alpha\) is replaced by the velocity of the electron, which is equal to \(\mathbf{p}_e/m\). The expression obtained is proportional (see equation (5) in\(^5\))

\[ \frac{1}{m^{2}}\int dk \sum_{n}' \frac{\left|\langle 0|\mathbf{p}_{e}|n\rangle\right|^{2}(E_{n}-E_{0})}{(E_{n}-E_{0}+k)}, \tag{7} \]

where \(k\) is the energy of the virtual photon, \(E_{0}\) is the eigenvalue

LAMB SHIFT FOR HYDROGEN AND DEUTERIUM

energy of the Schrödinger equation (containing the reduced mass \(\mu\) and relative coordinates and momenta) for the state under consideration; \(E_n\) is the energy of any other state.

Since \(\mathbf{p}_e\) is equal to the relative momentum \(\mathbf{p}\), all quantities in the integral (7), with the exception of the factor \(m^{-1}\), refer to the Schrödinger equation containing the reduced mass \(\mu\). This integral is exactly equal to the equivalent integral for a particle of mass \(\mu\) in a given Coulomb field. The numerical value of this integral contains the factor \(\mu^3\) (this factor is obtained from the square of the atomic wave function). Consequently, we must multiply the principal (nonrelativistic) part \(S_{\infty}^{(1)}\) by \(\mu^3\):

\[ \left(\frac{\mu}{m}\right)^3 \simeq \left(1-\frac{3m}{M}\right). \tag{8} \]

The quantities \(\mathrm{Ry}\), \(k_0(2,0)\), and \(k_0(2,1)\) in the curly brackets in (1) were obtained from a nonrelativistic treatment and are proportional to the first power of the mass. Consequently, these terms must be multiplied by \(\left(\frac{\mu}{m}\right)\), i.e., one must take the numerical value \(\mathrm{Ry}\) for the reduced mass \(\mu\). The remaining terms in the curly brackets (1), including \(mc^2\), are obtained in calculating the part corresponding to high momenta.

In [7], in order to calculate these terms, the 4-potential of the nucleus is expanded in the Fourier integral \(A_\nu(\mathbf{q})\), where \(\mathbf{q}\) is the change in momentum. Then the radiative corrections to the scattering of an electron in this potential are calculated (see equation (24) in [7]).

Taking account of the finite mass of the nucleus leads to the fact that the form of \(A_4(\mathbf{q})\) changes slightly and a small transverse component \(\mathbf{A}(\mathbf{q})\) appears. Equation (24) in [7] contains the true mass of the electron, which does not depend on the form of the potential and therefore also not on the mass of the nucleus. For a given value of \(q\), the ratio of the potential producing the radiative correction to \(A_\nu(\mathbf{q})\) is proportional to \(q^2\) and does not depend on the percentage content of the transverse component for all terms except those due to the anomalous magnetic moment. Then the dependence of the atomic wave function and of \(A_4(\mathbf{q})\) on the nuclear mass leads to the result that the expression for the Lamb shift for an infinite nuclear mass must be multiplied by expression (8). Further, the factor \(mc^2\) under the logarithm in the curly brackets (1) follows from equation (24) in [7] and contains the true mass of the electron, whereas the factor \(k_0(2,0)\) contains the reduced mass \(\mu\), as indicated above. Consequently, we must add to the curly brackets the term

\[ \ln\left(\frac{m}{\mu}\right) \simeq \left(+\frac{m}{M}\right). \]

On the other hand, the correction \(-\dfrac{1}{2}L\) to \(S^{(1)}\) due to the anomalous magnetic moment of the electron depends in some way on the small transverse component \(A_y\) and is not described by the factor (8). The exact factor for this term has not yet been calculated, and we shall write it in the form\({}^{16}\)

\[ \left[1-(1+\varepsilon_\mu)\frac{m}{M}\right], \tag{9} \]

where \(\varepsilon_\mu\) lies in the interval \(\pm 2\). Then the total correction to \(S^{(1)}\) due to the reduced mass, to the lowest order in \(\dfrac{m}{M}\), is equal to:

\[ \left(\frac{m}{M}\right)\left[-3S^{(1)}+\left(2-\frac{1}{2}\varepsilon_\mu\right)L\right]. \tag{10} \]

The fourth-order corrections to the Lamb shift due to the reduced mass, \(S^{(2)}\), have not yet been calculated; they should not exceed \(0.030\ \mathrm{Mc/sec}\) for hydrogen and \(0.015\ \mathrm{Mc/sec}\) for deuterium. The corrections due to diagrams in which not the electron but the nucleus emits and absorbs a virtual photon are of the order \(\left(\dfrac{m}{M}\right)^2 S\) and, consequently, are negligibly small. Adding the terms calculated in\({}^{15}\) to expression (10), we obtain the total correction to \(S\) due to the finite mass of the nucleus, respectively for hydrogen and deuterium:

\[ \Delta S_{\mathrm{H}}=-[1.175+0.037\varepsilon_\mu \pm 0.030]\ \mathrm{Mc/sec}, \tag{11a} \]

\[ \Delta S_{\mathrm{D}}=-[0.588+0.019\varepsilon_\mu \pm 0.030]\ \mathrm{Mc/sec}. \tag{11b} \]

§ 4. ALLOWANCE FOR THE FINITE RADIUS OF THE DEUTERON

Let us consider the influence of the finite dimensions of the deuteron on the electrostatic nonrelativistic potential energy of the atomic states of deuterium. First we shall calculate this effect, assuming that the atomic electron revolves around the center of inertia of the deuteron, and subsequently we shall give a justification for this assumption. Let \(r\) be the distance from the proton to the center of inertia of the deuteron (\(2r\) is the relative distance), and let \(\gamma^{-1}\) be the deuteron radius, equal to \((4.314\pm 0.004)\times 10^{-13}\ \mathrm{cm}\). Let \(\Phi(r)\) be the deuteron wave function multiplied by \(r\), normalized so that its asymptotic value is \(u(r)=e^{-2\gamma r}\). Let \(\Delta V(r)\) be the difference between the electrostatic potential energy of the electron at a distance \(r\) from the center of the charge distribution in the deuteron and the corresponding energy for a point charge of the nucleus. This correction to

the potential is equal to

\[ \left. \begin{aligned} \Delta V(r)&=+e^2N\int_r^\infty dy\,\Phi^2(y)\{r^{-1}-y^{-1}\};\\ N^{-1}&=\int_0^\infty dr\,\Phi^2(r). \end{aligned} \right\} \tag{12} \]

Since \(\Delta V(r)\) differs from zero only for values of \(r\) much smaller than atomic dimensions, in calculating the mean value \(\langle \Delta V\rangle\) the atomic wave function \(\psi(r)\) may be replaced by its value at the origin \(\psi(0)\):

\[ \left. \begin{aligned} \langle \Delta V\rangle&=|\psi(0)|^2\int d^3r\,\Delta V(r) =\left(2\pi\,\frac{e^2}{3}\right)|\psi(0)|^2\langle r^2\rangle;\\ \langle r^2\rangle&=N\int_0^\infty dr\,r^2\Phi^2(r). \end{aligned} \right\} \tag{13} \]

Equation (13) and an approximate value for \(\langle r^2\rangle\) were first obtained in \(^{11}\). The normalization factor \(N\) is equal to \(4\gamma(1-\gamma r_{0t})^{-1}\), where \(r_{0t}\) is the effective size of the triplet \(n\)-\(p\) potential. \(r_{0t}\) can be obtained \(^{17}\) from scattering experiments at low energies and depends only weakly on the choice of the shape of the potential well. The integral in (13) is equal to

\[ FN\int_0^\infty dr\,r^2u^2(r)=\frac{F}{8\gamma^2}. \]

The factor \(F\) is close to unity, since the maximum of the integrand lies outside the range of nuclear forces, where \(F\) is practically equal to its asymptotic value \(u\). The product \(F(1-\gamma r_{0t})^{-1}\) depends only weakly on the shape of the potential curve and is approximately equal to 1.60 for a rectangular well and 1.62 for the Yukawa potential. We shall take the value \((1.61\pm0.05)\). The error arises from the uncertainty in the value of \(r_{0t}\) and in the shape of the potential curve. Since \(\langle\Delta V\rangle\) in (13) is equal to zero for the \(2P\)-state and is not equal to zero for the \(2S\)-state, there arises from this a correction to the Lamb shift for deuterium, accessible to experimental observation, equal to

\[ \langle\Delta V\rangle=+(0.733\pm0.025)\ \mathrm{MeV}. \tag{14} \]

We must now justify the assumption made in deriving (14) that the electron is regarded as rotating about the center of inertia of the deuteron, and not about the proton. Consider the problem

in momentum space. Let \(\varphi(\mathbf p)\) be the atomic wave function in momentum space, i.e. the Fourier component of \(\psi(\mathbf r)\), and let \(a_0\) be the Bohr atomic radius. Then, as \(p\) increases for \(p a_0 \gg h\), \(\varphi(\mathbf p)\) falls off rapidly, approximately as \(p^{-4}\). The Fourier component for the Coulomb potential is proportional to \(q^{-2}\), where \(q\) is the change of momentum. The Fourier component \(\Delta V(\mathbf q)\) of \(V(\mathbf r)\) is proportional to a function of order \((q^2 + h^2\gamma^2)^{-1}\), i.e. is practically constant up to momenta of order \(h\gamma\), or about \(50\ \mathrm{Mev}/c\). The approximation in which the electron is regarded as rotating about the center of inertia of the deuteron is sufficiently accurate for momenta \(p\) such that the corresponding kinetic energy of the electron is negligibly small compared with the binding energy of the deuteron, i.e. \(p \ll 2\ \mathrm{Mev}/c\). The expression for \(\langle \Delta V\rangle\) in momentum space is written in the form

\[ \iint d^3p\, d^3q\, \varphi^*(\mathbf p)\,\Delta V(\mathbf q)\,\varphi(\mathbf p+\mathbf q). \tag{15} \]

Since \(p^2\varphi(\mathbf p)\) decreases as \(p^{-2}\) for \(p \gg \dfrac{h}{a_0} \sim 5\ \mathrm{kev}\), the value of the integral (15) depends mainly on \(p, q < \dfrac{h}{a_0} \ll 2\ \mathrm{Mev}/c\). Then the error introduced by our assumption about the electron’s center of rotation will have a relative order of only \(\dfrac{hc}{a_0}\,2\ \mathrm{Mev}\), or less than \(1\%\). In equation (15) \(\Delta V(\mathbf q)\) may be replaced by its value at \(q=0\), which leads exactly to equation (13). Thus, in the present case the result is essentially different from that for allowing for the influence of the finite radius of the deuteron on the hyperfine structure\(^{18}\). In the latter case \(\Delta V(\mathbf q)\) is replaced by an increasing function, and the magnitude of the correction to the corresponding integral depends mainly on large values of \(q\), to which our assumption about the rotation does not extend.

We have calculated only the mean value of the correction potential \(\Delta V(\mathbf r)\) and have neglected the radiative corrections to it. Using equation (24) of \(^{7}\) in momentum space, one can show that these radiative corrections are smaller than the mean value itself by a factor \(\alpha^2\ln\alpha\), and therefore they may be neglected. As shown in \(^{15}\), the influence of the deuteron structure on the corrections due to the finite mass to the fine structure is also small. Neglect of these effects gives an additional error of order \(\pm 0.02\ \mathrm{Mc}\) for the Lamb shift in deuterium quoted in § 3.

§ 5. ALLOWANCE FOR THE INTERNAL STRUCTURE OF THE NUCLEON

Up to now we have assumed that neutrons and protons are point particles obeying Fermi-Dirac statistics and possessing no internal structure whatever. In reality nucleons must behave differently, since

they interact strongly with the virtual meson field. This interaction apparently gives: a) an increase of the anomalous magnetic moment, b) a smearing of the electric charge and magnetic moment over a region of the order of the Compton wavelength of the meson, and c) more complicated relativistic effects. In this paragraph we shall confine ourselves to discussing the order of magnitude of these structural effects without carrying out quantitative calculations.

First we shall discuss effect a), considering the anomalous magnetic moment \(\mu_{\mathrm{an}}\) of the nucleon phenomenologically, i.e. regarding it as proportional to a vector made up of Pauli matrices, without taking into account the origin of this moment and its spatial distribution. This moment, of course, gives a large spin-dependent correction to the hyperfine structure. But, as was shown in \(^{19}\), such a moment also creates a small additional potential, independent of the spin direction. Then for the hydrogen atom the operator of the additional potential energy \(U(\mathbf r)\), due in origin to the anomalous moment of the proton \(\mu_{p,\mathrm{an}}\), is equal to:

\[ U(\mathbf r)=4\pi\mu_{p,\mathrm{an}}\left(\frac{e^2}{4M_p^2}\right)\delta(\mathbf r). \tag{16} \]

The shift of the level \(\langle U\rangle\) due to this potential is zero for all states of nonzero orbital angular momentum and \(+0.025\,Mc^2\) for the \(2S\)-state of hydrogen. Since the anomalous moments of the proton and neutron are almost equal in magnitude and opposite in sign, the corresponding shift of the deuterium level may be neglected.

For hydrogen a potential similar to (16) is also produced by the Dirac part of the magnetic moment of the proton. This potential is a Darwin term \(^{20,21}\) for the proton and gives a correction of \(0.006\,Mc^2\) to the shift of the \(2S\)-level. The order of this term relative to the fine structure is

\[ \left(\frac{m}{M}\right)^2\alpha. \]

This is only one of several terms of such an order, not yet calculated, which is not of interest to us. We shall omit all these terms.

Next let us consider effect b). The charged meson cloud around the nucleon is apparently smeared over a region of the order of the Compton wavelength of the meson \(\lambda\) (about \(10^{-13}\,\mathrm{cm}\)). This smearing of the charge gives a small electrostatic correction potential (similar to that described in § 4), which is appreciable only at distances of order not greater than \(\lambda\). The Foldy term (16) is a partial approximation to this potential. The Fourier component of this potential for momentum changes \(q\), small in comparison with the meson mass multiplied by \(c\), must be practically independent of \(q\) and proportional to the spatial volume integral of the potential. This is precisely the Fourier component for small values of \(q\), which was measured in experiments on the electron-neutron interaction-

tion. The latest experimental data for the spatial integral of this potential give\({}^{22}\):

\[ \left(\frac{4\pi r_0^3}{3}\right)(4100 \pm 1000)\ \text{ev}. \tag{17} \]

Although the potential (16) is an approximate quantity\({}^{23}\), nevertheless its volume integral agrees exactly with the experimental value (17). Since this potential differs noticeably from zero only in a region much smaller than atomic dimensions, the corresponding level shift depends only on this volume integral, and not on the shape of the potential curve. Thus, the value \(0.025\ \text{Mc}\), calculated earlier, is a sufficiently accurate approximation.

If a reliable meson theory existed, it would be possible to calculate the effect of atomic fine structure of type c), and also b). Such effects arise from processes in which the nucleon emits a virtual meson interacting electromagnetically with the atomic electron and subsequently absorbed again by the nucleon. These effects lead to the occurrence of anomalous magnetic moments of nucleons and, consequently, to the potential (16) and to the smearing of the charge. A relativistic calculation of these effects can in principle give some changes in comparison with the nonrelativistic estimates given above; for example, potentials whose Fourier components are proportional to positive powers of the momentum \(q\) (for \(q\) of the order of \(100\ \text{Mev}/c\)) change substantially. Such potentials cannot be observed in ordinary experiments on neutron–electron interaction, since small values of \(q\) (on the nuclear scale) play a role in them. In the atom, however, atomic electrons have a small but finite probability of possessing very large momenta. Thus it is possible that such effects of relativistic structure will give a noticeable correction to the level shift, but corrections exceeding small fractions of a megacycle are hardly possible.

It appears that all structural effects (except for the Darwin term, which we neglect) will be of opposite sign and approximately equal in magnitude for the neutron and proton, respectively. Therefore we neglect the influence of nuclear structure on the Lamb shift in deuterium. In hydrogen we have obtained the shift

\[ +(0.025+\varepsilon_{st})\ \text{Mc}, \]

where \(\varepsilon_{st}\) denotes the additional effect mentioned in this paragraph.

§ 6. CONCLUSIONS

The final theoretical values for the Lamb shift in hydrogen and deuterium, respectively, are:

\[ S_{\mathrm H}=\left[1057.19-22\varepsilon_a+S_Z^{(3)}+S_R-0.04\varepsilon_\mu+\varepsilon_{st}\pm0.13\right]\ \text{Mc}, \tag{18a} \]

\[ S_{\mathrm D}=\left[1058.49-22\varepsilon_a+S_Z^{(3)}+S_R-0.02\varepsilon_\mu\pm0.13\right]\ \text{Mc}, \tag{18b} \]

whereas the latest experimental results are\(^3\):

\[ S_{\mathrm H}=(1057.77\pm0.10)\ \text{MHz}, \tag{19a} \]

\[ S_{\mathrm D}=(1059.00\pm0.10)\ \text{MHz}. \tag{19b} \]

The theoretical value for the difference between the shifts of the levels in deuterium and hydrogen is

\[ S_{\mathrm D}-S_{\mathrm H}=(1.296+0.019\varepsilon_{\mu}-\varepsilon_{st}\pm0.035)\ \text{MHz}, \tag{20} \]

and the corresponding experimental value is

\[ S_{\mathrm D}-S_{\mathrm H}=(1.23\pm0.15)\ \text{MHz}. \tag{21} \]

Thus, the discrepancy between the theoretical and experimental values for the Lamb shift, both for hydrogen and for deuterium, is approximately one half of a megacycle. The existing uncertainty in \(\varepsilon_\alpha\) and \(\varepsilon_\mu\) only increases the numerical error in (18a) and (18b) to \(\pm 0.16\) MHz. The good agreement between the theoretical (20) and experimental (21) values for \((S_{\mathrm D}-S_{\mathrm H})\) indicates that \(\varepsilon_{st}\) is small and that this discrepancy is apparently not connected with effects depending on the structure and mass of the nuclei.

The discrepancies between the theoretical and experimental quantities for the shifts are three times greater than the probable error and have not yet been explained. If one excludes the possibility of experimental and theoretical errors, then these discrepancies may perhaps be explained by sixth-order corrections. These corrections may have a magnitude of order \(1/2\) MHz in the case where unexpectedly large numerical coefficients stand in front of certain terms. An accurate measurement of the Lamb shift for singly ionized helium may shed some light on the dependence on nuclear charge of some of these terms.

§ 7. ADDENDUM

For the reader’s convenience we give below a brief summary of the various terms discussed in this article, indicating the sections in which they are considered and their contribution to the Lamb shift.

CITED LITERATURE

1. W. E. Lamb and R. C. Retherford, Phys. Rev. 79, 549 (1950), 81, 222 (1951); 86, 1014 (1952).

2. W. E. Lamb, Phys. Rev. 85, 259 (1952).

3. Triebwasser, Dayhoff and Lamb, Phys. Rev. 89, 98 (1953).

4. Bethe, Brown a. Stehn, Phys. Rev. 77, 370 (1950).

5. H. A. Bethe, Phys. Rev. 72, 339 (1947).

6. N. M. Kroll a. W. E. Lamb, Phys. Rev. 75, 388 (1949); J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949); J. Schwinger, Phys. Rev. 76, 790 (1949).

7. R. P. Feynman, Phys. Rev. 76, 769 (1949).

8. Dayhoff, Triebwasser a. Lamb, Phys. Rev. 89, 106 (1953).

9. R. Karplus a. A. Klein, Phys. Rev. 85, 972 (1952); N. M. Kroll and F. Pollock, Phys. Rev. 86, 876 (1952); E. E. Salpeter and W. A. Newcomb, Phys. Rev. 87, 150 (1952).

10. J. W. M. DuMond and E. R. Cohen, Phys. Rev. 82, 555 (1951).

11. M. Baranger, Phys. Rev. 84, 866 (1951); Karplus, Klein, and Schwinger, Phys. Rev. 86, 288 (1952).

12. R. Karplus and N. M. Kroll, Phys. Rev. 77, 536 (1950).

13. Baranger, Dyson a. Salpeter, Phys. Rev. 88, 680 (1952).

14. Bersohn, Weneser a. Kroll, Phys. Rev. 86, 596 (1952).

15. E. E. Salpeter, Phys. Rev. 87, 328 (1952).

16. J. Breit and G. E. Brown, Phys. Rev. 74, 1278 (1949).

17. J. M. Blatt a. J. D. Jackson, Phys. Rev. 76, 18 (1949); H. A. Bethe, Phys. Rev. 76, 38 (1949); E. E. Salpeter, Phys. Rev. 82, 60 (1951).

18. F. Low, Phys. Rev. 77, 361 (1950).

19. L. L. Foldy, Phys. Rev. 83, 688 (1951).

1. C. G. Darwin, Proc. Roy. Soc. (London) A118, 654 (1928).
2. L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950).
3. Hamermesh, Ringo a. Wattenberg, Phys. Rev. 85, 483 (1952).
4. B. D. Fried, Phys. Rev. 86, 434 (1952); S. Borowitz, Phys. Rev. 86, 567 (1952).

Additional list of literature in Russian on questions of the Lamb shift and other radiative corrections

1. V. F. Weisskopf, UFN 41, 165 (1950).
2. Ya. A. Smorodinsky, UFN 39, 325 (1949).
3. Scientific-review collections “Problems of Modern Physics,” issue 6, 1948; issue 1, 1950; issue 11, 1951.
4. Shifts of levels of atomic electrons. Collected volume, IL, Moscow, 1950.
5. A. A. Sokolov and D. D. Ivanenko, Quantum Theory of the Field. Gostekhizdat, Moscow, 1952.
6. U. E. Lamb and R. K. Retherford, Fine Structure of the Hydrogen Atom, UFN 45, 553 (1951).