Additional analysis under the assumption of a uniformly magnetized sample.

We do not know the exact information about your system and the conditions of its reversal.

But based on Your interests taken from Your RG profile, we can assume that You are interested in the nature of perpendicular magnetic anisotropy of multilayer thin films. Then, most likely, the field Hk is the uniaxial anisotropy field with the easy direction of magnetization along the normal to the layer surface. Further arguments will be treated to the primary measured values, not corrected for Rayleigh (see my previous post).

The above expression can be converted to the form Hs = Hk – 4πMs, where it is clear that the external field must overcome the effect of this anisotropy. And the demagnetizing field of thin plate contributes to the magnetization reversal. In that case, surely Hk>4πMs, and in initial state the magnetization vector is perpendicular to the layer. Then You apply an external field parallel to the film to magnetize it for saturation in the plane. Required field is Hs.

In the case of homogeneous magnetization, it can be mathematically described. In the coordinate system related to the normal to the plane, the angle θ indicates the deviation of the magnetization from easy axis of anisotropy. Then the energy of the system is equal to the sum of the anisotropy energy, demagnetizing energy and the energy in an external field: E = Ksin2θ+2πMs2cos2θ–MsHsinθ.

Minimization leads to the expression:

sinθ = H/(2K/Ms–4πMs),

where 2K/Ms is the anisotropy field, Hk. Ms vector “lies” to the plane when θ = 90 deg., i.e. when H reaches a value of Hs ≡ Hk–4πMs.

If Hk

What are your 4 pies?

Just to be clear - saturation field is an external magnetic field at which a hysteresis loop reaches saturation magnetization. So, first, you can plot your hysteresis and see where it reaches saturation. Socond, you can plot a dM/dH derivative of the magnetization curve and see where it reaches zero. Of course, it's a matter of convention, where these curves "really reach" saturation/zero. 5%? 3%? However, having this estimation of the saturation field you can do any other calculations. 

Cheers!

Your question is not clear. Does saturation field change for in plane and out of plane directions? The shape of hysteresis may change if any anisotropy exists.  

I assume You mean the magnetization in the plane, i.e. the sample is in the form of a plate or film, and you have a curve of magnetization from the externally applied field M(H). According to the formula (my respect for Alan Rawle about 4π_- 4 pies? - It is the budget of pancakes, muffins or cupcakes as a reward for good answers): the Gaussian system of units is used here, where the dimension of Hk and Hs is Oe (Oersted), and magnetization M is measured in Gs (Gauss). If you have magnetization curve, adjusted to the shape of the sample (method of Rayleigh shift), for the saturation field it is need to accept strength of field, in which technical saturation of the specimen is achieved (curve M(H) goes on the horizontal level).

Note that the curve strongly influenced by the shape of the sample due to the emergence of magnetic poles and respective demagnetization field (shape anisotropy).

According to Your formula, it can match the recording to the magnetic induction of the sample at saturation Bs = Hs+4πMs, where Hs is to be understood the true field acting on the sample.

The fact that the sample of limited size is exposed in addition to the external field to the demagnetizing field caused by magnetic poles of a magnetized body, and the latter may reduce the magnitude of the applied field by value equal to Hd=-NM, where N is the demagnetizing factor in the direction of the magnetization of the sample, M is magnetization, reached at this point of the curve. Действующее поле это H-NM. The actual (effective) field is H-NM. Therefore, to obtain the true characteristics, any point of the curve M(H) must be shifted left by this value.

In particular, the true value of the saturation field will be equal to

Hs(corrected) = Hs(original curve) – NMs.

Now you need to determine the shape of Your sample.

If it is a thin film, and it is magnetized in the plane (magnetizing field is applied parallel to the surface), the demagnetizing factor N can be taken as 0, and you will immediately use the true magnetization curve without correction.

If You magnetize the film in perpendicular direction to its plane,

N=4π,

and Hs(corrected) = Hs(original curve) – 4πMs.

In accordance with the comment by Ravi, the sample may also contain the magnetic anisotropy of different nature and this will alter the course of the magnetization curve, but in any case, it can be corrected and the moment of reaching the saturation Hs will be determined with some accuracy .

Additional analysis under the assumption of a uniformly magnetized sample.

We do not know the exact information about your system and the conditions of its reversal.

But based on Your interests taken from Your RG profile, we can assume that You are interested in the nature of perpendicular magnetic anisotropy of multilayer thin films. Then, most likely, the field Hk is the uniaxial anisotropy field with the easy direction of magnetization along the normal to the layer surface. Further arguments will be treated to the primary measured values, not corrected for Rayleigh (see my previous post).

The above expression can be converted to the form Hs = Hk – 4πMs, where it is clear that the external field must overcome the effect of this anisotropy. And the demagnetizing field of thin plate contributes to the magnetization reversal. In that case, surely Hk>4πMs, and in initial state the magnetization vector is perpendicular to the layer. Then You apply an external field parallel to the film to magnetize it for saturation in the plane. Required field is Hs.

In the case of homogeneous magnetization, it can be mathematically described. In the coordinate system related to the normal to the plane, the angle θ indicates the deviation of the magnetization from easy axis of anisotropy. Then the energy of the system is equal to the sum of the anisotropy energy, demagnetizing energy and the energy in an external field: E = Ksin2θ+2πMs2cos2θ–MsHsinθ.

Minimization leads to the expression:

sinθ = H/(2K/Ms–4πMs),

where 2K/Ms is the anisotropy field, Hk. Ms vector “lies” to the plane when θ = 90 deg., i.e. when H reaches a value of Hs ≡ Hk–4πMs.

If Hk

i agree with Mr Alexander T. Morchenko